General introduction to matrix projection models using lefko3
Richard P. Shefferson
This document was built in Markdown in R 4.2.1 and covers package
lefko3 version 5.3.0. This is a long-format vignette,
meaning that we have tried to include a good deal more code and output
than is permissible in the standard CRAN-packaged vignette. Note,
however, that code that results in dramatically long output has been
hashtagged to prevent the inclusion of tens of pages of output that may
confuse the user. In these cases, please remove the hashtag to see the
output.
For reference, please also see the other vignettes included in
package lefko3, as well as our free online e-book called
lefko3: a gentle introduction, available on
the projects page
of the Shefferson lab website.
Datasets and case studies to be used
In this vignette, we will use the cypdata and
cypvert, and lathyrus datasets to illustrate
the estimation of raw MPMs and function-based
MPMs, including IPMs.
Cypripedium candidum
The white lady’s slipper, Cypripedium candidum, is a North American perennial herb in the family Orchidaceae. It is long-lived and of conservation concern. This plant begins life by germinating from a dust seed, and then develops into a protocorm, which is a special subterranean life stage found in orchids and pyroloids. During this stage, the plant is non-photosynthetic and completely parasitic on its mycorrhizal fungi. It spends several years as a protocorm, and previous studies suggest that it typically spends three years before becoming a seedling. As a seedling, it may or may not produce aboveground sprouts, often remaining entirely subterranean and continuing its parasitic lifestyle. It may persist this way for many years before attaining adult size, at which point it may sprout with or without flowers, or may remain underground in a condition referred to as vegetative dormancy. The latter condition may continue for many years, with over a decade of continuous dormancy documented in the literature (Shefferson et al. 2018).
Figure 1. Field work finding and measuring
Cypripedium candidum individuals, yielding the demographic
dataset included with package lefko3. Photo courtesy of R.
Shefferson.
The population from which the dataset originates is located within a wet meadow in a state nature preserve located in northeastern Illinois, USA (Figure 1). The population was monitored annually from 2004 to 2009, with two monitoring sessions per year. Monitoring sessions took roughly two weeks each, and included complete censuses of the population divided into sections referred to as patches. Each monitoring session consisted of searches for previously recorded individuals, which were located according to coordinates relative to fixed stakes at the site, followed by a search for new individuals. Data recorded per individual included: the location, the number of non-flowering sprouts, the number of flowering sprouts, the number of flowers per flowering sprout, and the number of fruit pods per sprout (only in the second monitoring session per year, once fruiting had occurred). Location was used to infer individual identity. More information about this population and its characteristics is given in Shefferson et al. (2001) and Shefferson et al. (2017).
Lathyrus vernus
Lathyrus vernus (family Fabaceae) is a long-lived forest herb, native to Europe and large parts of northern Asia. Individuals increase slowly in size and usually flower only after 10-15 years of vegetative growth. Flowering individuals have an average conditional lifespan of 44.3 years Ehrlén (2002a). L. vernus lacks organs for vegetative spread and individuals are well delimited Ehrlén (2002b). One or several erect shoots of up to 40 cm height emerge from a subterranean rhizome in March and April. Flowering occurs about four weeks after shoot emergence. Shoot growth is determinate, and the number of flowers is determined in the previous year Ehrlén et al. (2001). Individuals do not necessarily produce aboveground structures every year, and instead can remain vegetatively dormant in one or more seasons. L. vernus is self-compatible but requires visits from bumble-bees to produce seeds. Individuals produce few, large seeds and establishment from seeds is relatively frequent Ehrlén and Eriksson (1996). The pre-dispersal seed predator Bruchus atomarius often consumes a large fraction of developing seeds, and roe deer (Capreolus capreolus) sometimes consume the shoots Ehrlén and Munzbergova (2009).
Data for this study were collected from six permanent plots in a population of L. vernus located in a deciduous forest in the Tullgarn area, SE Sweden (58.9496 N, 17.6097 E), from 1988 to 1991 (Ehrlén 1995). The six plots were relatively similar with regard to soil type, elevation, slope, and canopy cover. Within each plot, all individuals were marked with numbered tags that remained over the study period, and their locations were carefully mapped. New individuals were included in the study in each year. Individuals were recorded at least three times every growing season. At the time of shoot emergence, we recorded whether individuals were alive and produced above-ground shoots, and if shoots had been grazed. During flowering, we recorded flower number and the height and diameter of all shoots. At fruit maturation, we counted the number of intact and damaged seeds. To derive a measure of aboveground size for each individual, we calculated the volume of each shoot as \(\pi × (\frac{1}{2} diameter)^2 × height\), and summed the volumes of all shoots. This measure is strongly correlated with the dry mass of aboveground tissues (\(R^2 = 0.924\), \(P < 0.001\), \(n = 50\), log-transformed values; Ehrlén 1995). Size of individuals that had been grazed was estimated based on measures of shoot diameter in grazed shoots, and the relationship between shoot diameter and shoot height in non-grazed individuals. Only individuals with an aboveground volume of more than 230 mm3 flowered and produced fruits during this study. Individuals that lacked aboveground structures in one season but reappeared in the following year were considered dormant. Individuals that lacked aboveground structures in two subsequent seasons were considered dead from the year in which they first lacked aboveground structures. Probabilities of seeds surviving to the next year, and of being present as seedlings or seeds in the soil seed bank, were derived from separate yearly sowing experiments in separate plots adjacent to each subplot (Ehrlén and Eriksson 1996).
Overall goals and initial steps
Our goal in this exercise will be to produce ahistorical and historical raw and function-based matrices for full comparison. We will move through tall of the steps of analysis, and take detours here and there where doing so will add to the educational experience.
Initial set-up
Users should have the latest version of R and R Studio installed. R is available at https://cloud.r-project.org/, while R Studio is available at https://www.rstudio.com/products/rstudio/download/#download.
After installing these two items, please install the R packages markdown, rmarkdown, knitr, and lefko3. You may be prompted if you wish to install dependencies, and in this case, please answer in the affirmative. You may also run the following line to do this.
install.packages(c("markdown", "rmarkdown", "knitr", "lefko3"))After installing these packages, you will need to restart R Studio. Then, check to make sure that you have the following R packages also installed: Rcpp, RcppArmadillo, BH, Matrix, TMB, glmmTMB, lme4, MuMIn, pscl, SparseM, and VGAM. If any are missing, then please install them, or run the following line.
install.packages(c("Rcpp", "RcppArmadillo", "BH", "Matrix", "TMB", "glmmTMB",
"lme4", "MuMIn", "pscl", "SparseM", "VGAM"))Loading lefko3 and the datasets
We will need to load three datasets included in lefko3
for this workshop: cypdata, cypvert, and
lathyrus. Let’s do that now.
rm(list=ls(all=TRUE))
library(lefko3)
data(cypdata)
data(cypvert)
data(lathyrus)It’s worthwhile exploring these datasets a bit. Let’s take a look at
one of the Cypripedium datasets (both cypdata and
cypvert contain the same data, but structured
differently).
dim(cypdata)## [1] 77 29summary(cypdata)## plantid patch X Y censor
## Min. : 164.0 A:23 Min. : 46.5 Min. :-28.00 Min. :1
## 1st Qu.: 265.0 B:35 1st Qu.: 60.1 1st Qu.: 22.50 1st Qu.:1
## Median : 455.0 C:19 Median : 91.4 Median : 75.60 Median :1
## Mean : 669.1 Mean : 92.9 Mean : 55.54 Mean :1
## 3rd Qu.: 829.0 3rd Qu.:142.2 3rd Qu.: 80.10 3rd Qu.:1
## Max. :1560.0 Max. :173.0 Max. :142.40 Max. :1
##
## Inf2.04 Inf.04 Veg.04 Pod.04 Inf2.05
## Min. :0 Min. :0.0000 Min. : 0.000 Min. :0.0 Min. :0.00000
## 1st Qu.:0 1st Qu.:0.0000 1st Qu.: 1.000 1st Qu.:0.0 1st Qu.:0.00000
## Median :0 Median :0.0000 Median : 2.000 Median :0.0 Median :0.00000
## Mean :0 Mean :0.6923 Mean : 2.923 Mean :0.2 Mean :0.04478
## 3rd Qu.:0 3rd Qu.:1.0000 3rd Qu.: 4.000 3rd Qu.:0.0 3rd Qu.:0.00000
## Max. :0 Max. :8.0000 Max. :12.000 Max. :3.0 Max. :1.00000
## NA's :12 NA's :12 NA's :12 NA's :12 NA's :10
## Inf.05 Veg.05 Pod.05 Inf2.06
## Min. : 0.000 Min. :0.000 Min. :0.0000 Min. :0
## 1st Qu.: 0.000 1st Qu.:1.000 1st Qu.:0.0000 1st Qu.:0
## Median : 0.000 Median :1.000 Median :0.0000 Median :0
## Mean : 1.537 Mean :2.134 Mean :0.6567 Mean :0
## 3rd Qu.: 2.000 3rd Qu.:3.000 3rd Qu.:1.0000 3rd Qu.:0
## Max. :18.000 Max. :9.000 Max. :7.0000 Max. :0
## NA's :10 NA's :10 NA's :10 NA's :16
## Inf.06 Veg.06 Pod.06 Inf2.07
## Min. : 0.0000 Min. : 0.000 Min. :0.0000 Mode:logical
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.:0.0000 NA's:77
## Median : 0.0000 Median : 2.000 Median :0.0000
## Mean : 0.9016 Mean : 2.213 Mean :0.3934
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.:0.0000
## Max. :18.0000 Max. :13.000 Max. :4.0000
## NA's :16 NA's :16 NA's :16
## Inf.07 Veg.07 Pod.07 Inf2.08
## Min. :0.0000 Min. : 0.000 Min. :0.0000 Min. :0
## 1st Qu.:0.0000 1st Qu.: 1.000 1st Qu.:0.0000 1st Qu.:0
## Median :0.0000 Median : 2.000 Median :0.0000 Median :0
## Mean :0.6271 Mean : 2.627 Mean :0.0678 Mean :0
## 3rd Qu.:1.0000 3rd Qu.: 4.000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :7.0000 Max. :13.000 Max. :1.0000 Max. :0
## NA's :18 NA's :18 NA's :18 NA's :24
## Inf.08 Veg.08 Pod.08 Inf2.09
## Min. : 0.0000 Min. : 0.00 Min. :0.0000 Min. :0
## 1st Qu.: 0.0000 1st Qu.: 1.00 1st Qu.:0.0000 1st Qu.:0
## Median : 0.0000 Median : 2.00 Median :0.0000 Median :0
## Mean : 0.8868 Mean : 2.83 Mean :0.1509 Mean :0
## 3rd Qu.: 1.0000 3rd Qu.: 4.00 3rd Qu.:0.0000 3rd Qu.:0
## Max. :11.0000 Max. :13.00 Max. :2.0000 Max. :0
## NA's :24 NA's :24 NA's :24 NA's :17
## Inf.09 Veg.09 Pod.09
## Min. : 0.000 Min. : 0.000 Min. :0.000
## 1st Qu.: 0.000 1st Qu.: 1.000 1st Qu.:0.000
## Median : 1.000 Median : 1.000 Median :1.000
## Mean : 1.833 Mean : 2.233 Mean :1.133
## 3rd Qu.: 2.000 3rd Qu.: 3.000 3rd Qu.:1.000
## Max. :11.000 Max. :10.000 Max. :8.000
## NA's :17 NA's :17 NA's :17The dataset cypdata is organized in horizontal format,
meaning that rows correspond to unique individuals and columns
correspond to stage in particular years. You will note a repeating
pattern in the names of the columns. In this dataset, there are 77
individuals, so there are 77 rows with data (not counting the header).
There are 29 columns, which are the variables in the dataset. Note that
the first five columns are variables giving identifying information
about each individual, with each individual’s data entirely restricted
to one row. This is followed by a number of sets of four columns, each
named Inf2.XX, Inf.XX, Veg.XX,
and Pod.XX. The XX in each case corresponds to a specific
year, which are organized consecutively. Thus, columns 4-7 refer to year
04 (short for 2004), columns 8-11 refer to year 05, columns 12-15 refer
to year 06, columns 16-19 refer to year 07, columns 20-23 refer to year
08, and columns 24-27 refer to year 09. Within each set of years, we see
a repeating pattern of our four variables, which makes standardization
easier later (although this repeating pattern is actually not required).
To properly conduct this exercise, we need to know the exact number of
years used, which is six years here (includes all years from 2004 to
2009).
Although this dataset is in horizontal structure, many demographers
prefer to keep their data in a vertical structure. Data
cypvert is essentially the same dataset as
cypdata, but supplied in vertical structure. Let’s look at
a screenshot of the Excel spreadsheet.
dim(cypvert)## [1] 322 14summary(cypvert)## plantid patch X Y censor
## Min. : 164.0 A: 94 Min. : 46.5 Min. :-28.0 Min. :1
## 1st Qu.: 391.0 B:153 1st Qu.: 60.1 1st Qu.: 23.3 1st Qu.:1
## Median : 453.0 C: 75 Median : 91.3 Median : 77.0 Median :1
## Mean : 656.3 Mean : 91.5 Mean : 56.5 Mean :1
## 3rd Qu.: 476.0 3rd Qu.:141.8 3rd Qu.: 80.4 3rd Qu.:1
## Max. :1560.0 Max. :173.0 Max. :142.4 Max. :1
##
## year2 Inf2.2 Inf.2 Veg.2
## Min. :2004 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.:2005 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median :2006 Median :0.000000 Median : 0.000 Median : 2.000
## Mean :2006 Mean :0.009868 Mean : 0.941 Mean : 2.534
## 3rd Qu.:2007 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :2008 Max. :1.000000 Max. :18.000 Max. :13.000
## NA's :18 NA's :17 NA's :17
## Pod.2 Inf2.3 Inf.3 Veg.3
## Min. :0.0000 Min. :0.00000 Min. : 0.000 Min. : 0.000
## 1st Qu.:0.0000 1st Qu.:0.00000 1st Qu.: 0.000 1st Qu.: 1.000
## Median :0.0000 Median :0.00000 Median : 0.000 Median : 2.000
## Mean :0.3049 Mean :0.01034 Mean : 1.192 Mean : 2.436
## 3rd Qu.:0.0000 3rd Qu.:0.00000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :7.0000 Max. :1.00000 Max. :18.000 Max. :13.000
## NA's :17 NA's :32 NA's :31 NA's :31
## Pod.3
## Min. :0.0000
## 1st Qu.:0.0000
## Median :0.0000
## Mean :0.5086
## 3rd Qu.:0.5000
## Max. :8.0000
## NA's :31This dataset is longer and narrower, with more rows and fewer
columns. This is because we constructed the dataset by splitting the
data for each individual across multiple rows. After three columns of
identifying information (plantid, patch, and
censor), a single column designates the time of occasion t,
given as year2. This dataset then includes columns showing
individual state in pairs of consecutive years corresponding to
occasions t and t+1. State in occasion t-1 is not presented because this
is an ahistorical dataset. This dataset includes the
plantid variable, which is an individual identity term that
allows us to associate rows with their individuals and so allows
conversion.
Finally, let’s take a peek at our Lathyrus dataset.
dim(lathyrus)## [1] 1119 38summary(lathyrus)## SUBPLOT GENET Volume88 lnVol88
## Min. :1.000 Min. : 1.0 Min. : 3.4 Min. :1.200
## 1st Qu.:2.000 1st Qu.: 48.0 1st Qu.: 63.0 1st Qu.:4.100
## Median :3.000 Median : 97.0 Median : 732.5 Median :6.600
## Mean :3.223 Mean :110.2 Mean : 749.4 Mean :5.538
## 3rd Qu.:4.000 3rd Qu.:167.5 3rd Qu.:1025.5 3rd Qu.:6.900
## Max. :6.000 Max. :284.0 Max. :7032.0 Max. :8.900
## NA's :404 NA's :404
## FCODE88 Flow88 Intactseed88 Dead1988 Dormant1988
## Min. :0.0000 Min. : 1.00 Min. : 0 Mode:logical Mode:logical
## 1st Qu.:0.0000 1st Qu.: 4.00 1st Qu.: 0 NA's:1119 NA's:1119
## Median :0.0000 Median : 8.00 Median : 0
## Mean :0.3399 Mean :11.86 Mean : 3
## 3rd Qu.:1.0000 3rd Qu.:15.00 3rd Qu.: 4
## Max. :1.0000 Max. :66.00 Max. :34
## NA's :404 NA's :910 NA's :875
## Missing1988 Seedling1988 Volume89 lnVol89
## Mode:logical Min. :1.000 Min. : 1.8 Min. :0.600
## NA's:1119 1st Qu.:2.000 1st Qu.: 15.6 1st Qu.:2.700
## Median :2.000 Median : 118.8 Median :4.800
## Mean :2.144 Mean : 573.3 Mean :4.855
## 3rd Qu.:3.000 3rd Qu.: 968.8 3rd Qu.:6.900
## Max. :3.000 Max. :6539.4 Max. :8.800
## NA's :1022 NA's :294 NA's :294
## FCODE89 Flow89 Intactseed89 Dead1989
## Min. :0.0000 Min. : 1.00 Min. : 0.000 Min. :1
## 1st Qu.:0.0000 1st Qu.: 5.00 1st Qu.: 0.000 1st Qu.:1
## Median :0.0000 Median :11.00 Median : 5.000 Median :1
## Mean :0.2667 Mean :14.88 Mean : 8.273 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:20.00 3rd Qu.:13.000 3rd Qu.:1
## Max. :1.0000 Max. :97.00 Max. :66.000 Max. :1
## NA's :294 NA's :906 NA's :899 NA's :1077
## Dormant1989 Missing1989 Seedling1989 Volume90 lnVol90
## Min. :1 Min. :1 Min. :1.000 Min. : 2.1 Min. :0.700
## 1st Qu.:1 1st Qu.:1 1st Qu.:2.000 1st Qu.: 12.6 1st Qu.:2.500
## Median :1 Median :1 Median :2.000 Median : 61.0 Median :4.100
## Mean :1 Mean :1 Mean :2.136 Mean : 244.1 Mean :4.207
## 3rd Qu.:1 3rd Qu.:1 3rd Qu.:2.000 3rd Qu.: 295.2 3rd Qu.:5.700
## Max. :1 Max. :1 Max. :3.000 Max. :4242.8 Max. :8.400
## NA's :1046 NA's :1112 NA's :1001 NA's :245 NA's :245
## FCODE90 Flow90 Intactseed90 Dead1990
## Min. :0.0000 Min. : 1.000 Min. : 0.000 Min. :1
## 1st Qu.:0.0000 1st Qu.: 3.000 1st Qu.: 0.000 1st Qu.:1
## Median :0.0000 Median : 6.000 Median : 0.000 Median :1
## Mean :0.1581 Mean : 8.104 Mean : 2.514 Mean :1
## 3rd Qu.:0.0000 3rd Qu.:10.750 3rd Qu.: 1.000 3rd Qu.:1
## Max. :1.0000 Max. :54.000 Max. :37.000 Max. :1
## NA's :246 NA's :985 NA's :981 NA's :1007
## Dormant1990 Missing1990 Seedling1990 Volume91 lnVol91
## Min. :1 Min. :1 Min. :1.000 Min. : 4.0 Min. :1.400
## 1st Qu.:1 1st Qu.:1 1st Qu.:2.000 1st Qu.: 12.0 1st Qu.:2.500
## Median :1 Median :1 Median :2.000 Median : 118.5 Median :4.800
## Mean :1 Mean :1 Mean :2.186 Mean : 418.7 Mean :4.642
## 3rd Qu.:1 3rd Qu.:1 3rd Qu.:2.000 3rd Qu.: 689.7 3rd Qu.:6.500
## Max. :1 Max. :1 Max. :3.000 Max. :6645.8 Max. :8.800
## NA's :1054 NA's :1105 NA's :1049 NA's :305 NA's :305
## FCODE91 Flow91 Intactseed91 Dead1991 Dormant1991
## Min. :0.0000 Min. : 1.00 Min. : 0.000 Min. :1 Min. :1
## 1st Qu.:0.0000 1st Qu.: 4.00 1st Qu.: 0.000 1st Qu.:1 1st Qu.:1
## Median :0.0000 Median : 8.00 Median : 3.500 Median :1 Median :1
## Mean :0.2525 Mean :11.12 Mean : 5.805 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:15.00 3rd Qu.:10.000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :48.00 Max. :48.000 Max. :1 Max. :1
## NA's :307 NA's :954 NA's :919 NA's :925 NA's :1034
## Missing1991 Seedling1991
## Min. :1 Min. :1.000
## 1st Qu.:1 1st Qu.:2.000
## Median :1 Median :2.000
## Mean :1 Mean :1.973
## 3rd Qu.:1 3rd Qu.:2.000
## Max. :1 Max. :3.000
## NA's :1095 NA's :1082This dataset is horizontally formatted, and includes information on
1,119 individuals arranged horizontally. So, there are 1,119 rows with
data. There are 38 columns. The first two columns are variables giving
identifying information about each individual. This is followed by four
sets of nine columns, each named VolumeXX,
lnVolXX, FCODEXX, FlowXX,
IntactseedXX, Dead19XX,
DormantXX, Missing19XX, and
SeedlingXX, where XX corresponds to the year
of observation and with years organized consecutively. Thus, columns
3-11 refer to year 1988, columns 12-20 refer to year 1989, etc. This
strictly repeated pattern allows us to manipulate the original dataset
quickly and efficiently via lefko3. We should note the
number of years, which is 4 here (includes all years from and including
1988 to 1991). Ideally, we should also have arranged the columns in the
same order for each year, with years in consecutive order with no extra
columns between them. This order is not required, provided that we are
willing to input all variable names in the correct order when
transforming the dataset later.
Let’s now consider the life history models that we will be using.
Step 1. Life history model development
Our first key decision is how to model the life history of the organism. We generally use the life cycle graph approach, and encourage its use as it simplifies the process of life history model development. The decision of the life history model, however, may be more complicated than simply diagramming the life cycle of the organism. Particularly, we must decide what life stages to include, how these life stages will be defined, and how these life stages are connected to one another via survival and fecundity transitions. These considerations will be strongly influenced by the kind of MPM that we wish to build.
Let’s use the Cypripedium datasets to build raw and function-based versions of ahistorical, historical, and age-by-stage MPMs, as well as Leslie MPMs. Let’s use the Lathyrus dataset to build an IPM.
Let’s explore the range of sizes in our Cypripedium dataset. There is no one variable that encapsulates the entire size of the individual in our dataset, so we will create a series of vectors that sums the numbers of sprouts that are single-flowered flowering, double-flowered flowering, and non-flowering, and use these sums as plant sizes. If we use the horizontal dataset for this purpose, then we get the following exploration.
size.04 <- cypdata$Inf2.04 + cypdata$Inf.04 + cypdata$Veg.04
size.05 <- cypdata$Inf2.05 + cypdata$Inf.05 + cypdata$Veg.05
size.06 <- cypdata$Inf2.06 + cypdata$Inf.06 + cypdata$Veg.06
size.07 <- cypdata$Inf2.07 + cypdata$Inf.07 + cypdata$Veg.07
size.08 <- cypdata$Inf2.08 + cypdata$Inf.08 + cypdata$Veg.08
size.09 <- cypdata$Inf2.09 + cypdata$Inf.09 + cypdata$Veg.09
summary(c(size.04, size.05, size.06, size.07, size.08, size.09))## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 1.000 1.000 2.000 3.644 5.000 24.000 156plot(density(c(size.04, size.05, size.06, size.07, size.08, size.09), na.rm = TRUE))
Figure 2. Density of Cypripedium plant sizes across time
The summary shows that the smallest recorded size is a single sprout, and the largest is 24 sprouts. The 156 NAs are a combination of vegetative dormancy and instances of individuals being dead or not yet born. The density plot shows us that individuals bigger than 10 sprouts are rare.Given this, we might utilize the following life history model for our function-based MPMs.
When deciding on the number and definitions of stages, we need to
find a way to identify natural breaks in the data to separate stages.
There are number of approaches to do this, but one approach that can be
taken in R is to use the Jenks Breaks approach, which uses the Jenks
natural breaks algorithm to identify where to separate stages. To use
this approach, we first need to install the BAMMtools
package.
install.packages("BAMMtools", dependencies = TRUE)Now let’s see where we might fit natural breaks according to specific numbers of assumed breaks.
library(BAMMtools)## Loading required package: apegetJenksBreaks(c(size.04, size.05, size.06, size.07, size.08, size.09), k = 4)## [1] 1 4 12 24getJenksBreaks(c(size.04, size.05, size.06, size.07, size.08, size.09), k = 5)## [1] 1 3 7 14 24getJenksBreaks(c(size.04, size.05, size.06, size.07, size.08, size.09), k = 6)## [1] 1 2 4 7 14 24getJenksBreaks(c(size.04, size.05, size.06, size.07, size.08, size.09), k = 7)## [1] 1 2 4 7 11 16 24The Jenks method gives us the borders of the size classes under different numbers of stages. The first line shows a three stage model, yielding four breaks including the minimum and maximum. This model has the first stage include 1-3 sprouts, the second stage include 4-11 sprouts, and the 3rd stage include 12-24 sprouts. The fourth line is a six stage model with seven breaks shown. Given what we know about the size distribution, we will try to separate the data into five stages (one sprout, 2-4 sprouts, 5-7 sprouts, 8-14 sprouts, and 15-24 sprouts), which will result in 11 total life history stages in our like history (one dormant seed, three protocorm stages of different age, one seedling stage, one vegetatively dormant stage, and five size-classified adult stages). Here is our new life history model). Note that the hashed blue box here refers to the mature stages that are actually observable in our population.
Figure 3. Life history model of Cypripedium candidum for use in raw MPMs
Let’s now build a stageframe using these breaks. We will build vectors describing the stages, particularly the size, reproductive status, observation status, maturity status, immaturity status, propagule status, and associate minimum and maximum ages of each stage. We will use this stageframe to create ahistorical and historical raw MPMs, as well as a raw age-by-stage MPM.
sizevector <- c(0, 0, 0, 0, 0, 0, 1, 3, 6, 11, 19.5)
stagevector <- c("SD", "P1", "P2", "P3", "SL", "D", "XSm", "Sm", "Md", "Lg",
"XLg")
repvector <- c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1)
obsvector <- c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1)
matvector <- c(0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1)
immvector <- c(0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
indataset <- c(0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1)
binvec <- c(0, 0, 0, 0, 0, 0.5, 0.5, 1.5, 1.5, 3.5, 5)
#Age arguments are only needed if an age-by-stage MPM
minage <- c(1, 1, 2, 3, 4, 5, 5, 6, 6, 6, 6)
maxage <- c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA)
comments <- c("Dormant seed", "1st yr protocorm", "2nd yr protocorm",
"3rd yr protocorm", "Seedling", "Dormant adult",
"Extra small adult (1 shoot)", "Small adult (2-4 shoots)",
"Medium adult (5-7 shoots)", "Large adult (8-14 shoots)",
"Extra large adult (>14 shoots)")
cypframe_raw <- sf_create(sizes = sizevector, stagenames = stagevector,
repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
propstatus = propvector, immstatus = immvector, indataset = indataset,
binhalfwidth = binvec, minage = minage, maxage = maxage, comments = comments)
cypframe_raw## stage size size_b size_c min_age max_age repstatus obsstatus propstatus
## 1 SD 0.0 NA NA 1 NA 0 0 1
## 2 P1 0.0 NA NA 1 NA 0 0 0
## 3 P2 0.0 NA NA 2 NA 0 0 0
## 4 P3 0.0 NA NA 3 NA 0 0 0
## 5 SL 0.0 NA NA 4 NA 0 0 0
## 6 D 0.0 NA NA 5 NA 0 0 0
## 7 XSm 1.0 NA NA 5 NA 1 1 0
## 8 Sm 3.0 NA NA 6 NA 1 1 0
## 9 Md 6.0 NA NA 6 NA 1 1 0
## 10 Lg 11.0 NA NA 6 NA 1 1 0
## 11 XLg 19.5 NA NA 6 NA 1 1 0
## immstatus matstatus indataset binhalfwidth_raw sizebin_min sizebin_max
## 1 0 0 0 0.0 0.0 0.0
## 2 1 0 0 0.0 0.0 0.0
## 3 1 0 0 0.0 0.0 0.0
## 4 1 0 0 0.0 0.0 0.0
## 5 1 0 0 0.0 0.0 0.0
## 6 0 1 1 0.5 -0.5 0.5
## 7 0 1 1 0.5 0.5 1.5
## 8 0 1 1 1.5 1.5 4.5
## 9 0 1 1 1.5 4.5 7.5
## 10 0 1 1 3.5 7.5 14.5
## 11 0 1 1 5.0 14.5 24.5
## sizebin_center sizebin_width binhalfwidthb_raw sizebinb_min sizebinb_max
## 1 0.0 0 NA NA NA
## 2 0.0 0 NA NA NA
## 3 0.0 0 NA NA NA
## 4 0.0 0 NA NA NA
## 5 0.0 0 NA NA NA
## 6 0.0 1 NA NA NA
## 7 1.0 1 NA NA NA
## 8 3.0 3 NA NA NA
## 9 6.0 3 NA NA NA
## 10 11.0 7 NA NA NA
## 11 19.5 10 NA NA NA
## sizebinb_center sizebinb_width binhalfwidthc_raw sizebinc_min sizebinc_max
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## sizebinc_center sizebinc_width group comments
## 1 NA NA 0 Dormant seed
## 2 NA NA 0 1st yr protocorm
## 3 NA NA 0 2nd yr protocorm
## 4 NA NA 0 3rd yr protocorm
## 5 NA NA 0 Seedling
## 6 NA NA 0 Dormant adult
## 7 NA NA 0 Extra small adult (1 shoot)
## 8 NA NA 0 Small adult (2-4 shoots)
## 9 NA NA 0 Medium adult (5-7 shoots)
## 10 NA NA 0 Large adult (8-14 shoots)
## 11 NA NA 0 Extra large adult (>14 shoots)In contrast to this, let’s take a look at a life history model that we can use for function-based MPMs.
Figure 4. Life history model of Cypripedium candidum for use in function-based MPMs.
This is a much larger life history model, and we can get away with using such a large model because we know that we will estimate values for every biologically possible matrix element. We can see a variety of transitions within this figure. The juvenile stages have fairly simple transitions. New recruits may enter the population directly from germination of a seed produced the previous year, in which case they start in the protocorm 1 stage, or they may begin as dormant seed. Dormant seed may remain dormant, die, or germinate into the protocorm 1 stage. Protocorms exist for up to three years, yielding the protocorm 1, 2, and 3 stages, without any possibility of staying within each of these stages for more than a single year. Protocorm 3 leads to a seedling stage, in which the plant may persist for many years before becoming mature. Here, maturity does not really refer to reproduction per se, but rather to a morphology indistinguishable from a reproductive plant except for the lack of a flower. The first mature stage is usually either vegetative dormancy (dorm), during which time the plant does not sprout, or a small, non-flowering adult (1V). Once in this portion of the life history, the plant may transition among 49 mature stages, including vegetative dormancy, 1-24 shoots without flowers, or 1-24 shoots with at least one flower.
Let’s now build our stage frame for this model. This will work for our function-based ahistorical, historical, and age-by-stage MPMs.
sizevector <- c(0, 0, 0, 0, 0, seq(from = 0, t = 24), seq(from = 1, to = 24))
stagevector <- c("SD", "P1", "P2", "P3", "SL", "D", "V1", "V2", "V3", "V4", "V5",
"V6", "V7", "V8", "V9", "V10", "V11", "V12", "V13", "V14", "V15", "V16", "V17",
"V18", "V19", "V20", "V21", "V22", "V23", "V24", "F1", "F2", "F3", "F4", "F5",
"F6", "F7", "F8", "F9", "F10", "F11", "F12", "F13", "F14", "F15", "F16", "F17",
"F18", "F19", "F20", "F21", "F22", "F23", "F24")
repvector <- c(0, 0, 0, 0, 0, rep(0, 25), rep(1, 24))
obsvector <- c(0, 0, 0, 0, 0, 0, rep(1, 48))
matvector <- c(0, 0, 0, 0, 0, rep(1, 49))
immvector <- c(0, 1, 1, 1, 1, rep(0, 49))
propvector <- c(1, rep(0, 53))
indataset <- c(0, 0, 0, 0, 0, rep(1, 49))
comments <- c("Dormant seed", "Yr1 protocorm", "Yr2 protocorm", "Yr3 protocorm",
"Seedling", "Veg dorm", "Veg adult 1 stem", "Veg adult 2 stems",
"Veg adult 3 stems", "Veg adult 4 stems", "Veg adult 5 stems",
"Veg adult 6 stems", "Veg adult 7 stems", "Veg adult 8 stems",
"Veg adult 9 stems", "Veg adult 10 stems", "Veg adult 11 stems",
"Veg adult 12 stems", "Veg adult 13 stems", "Veg adult 14 stems",
"Veg adult 15 stems", "Veg adult 16 stems", "Veg adult 17 stems",
"Veg adult 18 stems", "Veg adult 19 stems", "Veg adult 20 stems",
"Veg adult 21 stems", "Veg adult 22 stems", "Veg adult 23 stems",
"Veg adult 24 stems", "Flo adult 1 stem", "Flo adult 2 stems",
"Flo adult 3 stems", "Flo adult 4 stems", "Flo adult 5 stems",
"Flo adult 6 stems", "Flo adult 7 stems", "Flo adult 8 stems",
"Flo adult 9 stems", "Flo adult 10 stems", "Flo adult 11 stems",
"Flo adult 12 stems", "Flo adult 13 stems", "Flo adult 14 stems",
"Flo adult 15 stems", "Flo adult 16 stems", "Flo adult 17 stems",
"Flo adult 18 stems", "Flo adult 19 stems", "Flo adult 20 stems",
"Flo adult 21 stems", "Flo adult 22 stems", "Flo adult 23 stems",
"Flo adult 24 stems")
#The next two arguments are only needed for age-by-stage MPMs
minage <- c(1, 1, 2, 3, 4, 5, 5, rep(6, 47))
maxage <- rep(NA, 54)
cypframe_fb <- sf_create(sizes = sizevector, stagenames = stagevector,
repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
propstatus = propvector, immstatus = immvector, indataset = indataset,
minage = minage, maxage = maxage, comments = comments)
cypframe_fb## stage size size_b size_c min_age max_age repstatus obsstatus propstatus
## 1 SD 0 NA NA 1 NA 0 0 1
## 2 P1 0 NA NA 1 NA 0 0 0
## 3 P2 0 NA NA 2 NA 0 0 0
## 4 P3 0 NA NA 3 NA 0 0 0
## 5 SL 0 NA NA 4 NA 0 0 0
## 6 D 0 NA NA 5 NA 0 0 0
## 7 V1 1 NA NA 5 NA 0 1 0
## 8 V2 2 NA NA 6 NA 0 1 0
## 9 V3 3 NA NA 6 NA 0 1 0
## 10 V4 4 NA NA 6 NA 0 1 0
## 11 V5 5 NA NA 6 NA 0 1 0
## 12 V6 6 NA NA 6 NA 0 1 0
## 13 V7 7 NA NA 6 NA 0 1 0
## 14 V8 8 NA NA 6 NA 0 1 0
## 15 V9 9 NA NA 6 NA 0 1 0
## 16 V10 10 NA NA 6 NA 0 1 0
## 17 V11 11 NA NA 6 NA 0 1 0
## 18 V12 12 NA NA 6 NA 0 1 0
## 19 V13 13 NA NA 6 NA 0 1 0
## 20 V14 14 NA NA 6 NA 0 1 0
## 21 V15 15 NA NA 6 NA 0 1 0
## 22 V16 16 NA NA 6 NA 0 1 0
## 23 V17 17 NA NA 6 NA 0 1 0
## 24 V18 18 NA NA 6 NA 0 1 0
## 25 V19 19 NA NA 6 NA 0 1 0
## 26 V20 20 NA NA 6 NA 0 1 0
## 27 V21 21 NA NA 6 NA 0 1 0
## 28 V22 22 NA NA 6 NA 0 1 0
## 29 V23 23 NA NA 6 NA 0 1 0
## 30 V24 24 NA NA 6 NA 0 1 0
## 31 F1 1 NA NA 6 NA 1 1 0
## 32 F2 2 NA NA 6 NA 1 1 0
## 33 F3 3 NA NA 6 NA 1 1 0
## 34 F4 4 NA NA 6 NA 1 1 0
## 35 F5 5 NA NA 6 NA 1 1 0
## 36 F6 6 NA NA 6 NA 1 1 0
## 37 F7 7 NA NA 6 NA 1 1 0
## 38 F8 8 NA NA 6 NA 1 1 0
## 39 F9 9 NA NA 6 NA 1 1 0
## 40 F10 10 NA NA 6 NA 1 1 0
## 41 F11 11 NA NA 6 NA 1 1 0
## 42 F12 12 NA NA 6 NA 1 1 0
## 43 F13 13 NA NA 6 NA 1 1 0
## 44 F14 14 NA NA 6 NA 1 1 0
## 45 F15 15 NA NA 6 NA 1 1 0
## 46 F16 16 NA NA 6 NA 1 1 0
## 47 F17 17 NA NA 6 NA 1 1 0
## 48 F18 18 NA NA 6 NA 1 1 0
## 49 F19 19 NA NA 6 NA 1 1 0
## 50 F20 20 NA NA 6 NA 1 1 0
## 51 F21 21 NA NA 6 NA 1 1 0
## 52 F22 22 NA NA 6 NA 1 1 0
## 53 F23 23 NA NA 6 NA 1 1 0
## 54 F24 24 NA NA 6 NA 1 1 0
## immstatus matstatus indataset binhalfwidth_raw sizebin_min sizebin_max
## 1 0 0 0 0.5 -0.5 0.5
## 2 1 0 0 0.5 -0.5 0.5
## 3 1 0 0 0.5 -0.5 0.5
## 4 1 0 0 0.5 -0.5 0.5
## 5 1 0 0 0.5 -0.5 0.5
## 6 0 1 1 0.5 -0.5 0.5
## 7 0 1 1 0.5 0.5 1.5
## 8 0 1 1 0.5 1.5 2.5
## 9 0 1 1 0.5 2.5 3.5
## 10 0 1 1 0.5 3.5 4.5
## 11 0 1 1 0.5 4.5 5.5
## 12 0 1 1 0.5 5.5 6.5
## 13 0 1 1 0.5 6.5 7.5
## 14 0 1 1 0.5 7.5 8.5
## 15 0 1 1 0.5 8.5 9.5
## 16 0 1 1 0.5 9.5 10.5
## 17 0 1 1 0.5 10.5 11.5
## 18 0 1 1 0.5 11.5 12.5
## 19 0 1 1 0.5 12.5 13.5
## 20 0 1 1 0.5 13.5 14.5
## 21 0 1 1 0.5 14.5 15.5
## 22 0 1 1 0.5 15.5 16.5
## 23 0 1 1 0.5 16.5 17.5
## 24 0 1 1 0.5 17.5 18.5
## 25 0 1 1 0.5 18.5 19.5
## 26 0 1 1 0.5 19.5 20.5
## 27 0 1 1 0.5 20.5 21.5
## 28 0 1 1 0.5 21.5 22.5
## 29 0 1 1 0.5 22.5 23.5
## 30 0 1 1 0.5 23.5 24.5
## 31 0 1 1 0.5 0.5 1.5
## 32 0 1 1 0.5 1.5 2.5
## 33 0 1 1 0.5 2.5 3.5
## 34 0 1 1 0.5 3.5 4.5
## 35 0 1 1 0.5 4.5 5.5
## 36 0 1 1 0.5 5.5 6.5
## 37 0 1 1 0.5 6.5 7.5
## 38 0 1 1 0.5 7.5 8.5
## 39 0 1 1 0.5 8.5 9.5
## 40 0 1 1 0.5 9.5 10.5
## 41 0 1 1 0.5 10.5 11.5
## 42 0 1 1 0.5 11.5 12.5
## 43 0 1 1 0.5 12.5 13.5
## 44 0 1 1 0.5 13.5 14.5
## 45 0 1 1 0.5 14.5 15.5
## 46 0 1 1 0.5 15.5 16.5
## 47 0 1 1 0.5 16.5 17.5
## 48 0 1 1 0.5 17.5 18.5
## 49 0 1 1 0.5 18.5 19.5
## 50 0 1 1 0.5 19.5 20.5
## 51 0 1 1 0.5 20.5 21.5
## 52 0 1 1 0.5 21.5 22.5
## 53 0 1 1 0.5 22.5 23.5
## 54 0 1 1 0.5 23.5 24.5
## sizebin_center sizebin_width binhalfwidthb_raw sizebinb_min sizebinb_max
## 1 0 1 NA NA NA
## 2 0 1 NA NA NA
## 3 0 1 NA NA NA
## 4 0 1 NA NA NA
## 5 0 1 NA NA NA
## 6 0 1 NA NA NA
## 7 1 1 NA NA NA
## 8 2 1 NA NA NA
## 9 3 1 NA NA NA
## 10 4 1 NA NA NA
## 11 5 1 NA NA NA
## 12 6 1 NA NA NA
## 13 7 1 NA NA NA
## 14 8 1 NA NA NA
## 15 9 1 NA NA NA
## 16 10 1 NA NA NA
## 17 11 1 NA NA NA
## 18 12 1 NA NA NA
## 19 13 1 NA NA NA
## 20 14 1 NA NA NA
## 21 15 1 NA NA NA
## 22 16 1 NA NA NA
## 23 17 1 NA NA NA
## 24 18 1 NA NA NA
## 25 19 1 NA NA NA
## 26 20 1 NA NA NA
## 27 21 1 NA NA NA
## 28 22 1 NA NA NA
## 29 23 1 NA NA NA
## 30 24 1 NA NA NA
## 31 1 1 NA NA NA
## 32 2 1 NA NA NA
## 33 3 1 NA NA NA
## 34 4 1 NA NA NA
## 35 5 1 NA NA NA
## 36 6 1 NA NA NA
## 37 7 1 NA NA NA
## 38 8 1 NA NA NA
## 39 9 1 NA NA NA
## 40 10 1 NA NA NA
## 41 11 1 NA NA NA
## 42 12 1 NA NA NA
## 43 13 1 NA NA NA
## 44 14 1 NA NA NA
## 45 15 1 NA NA NA
## 46 16 1 NA NA NA
## 47 17 1 NA NA NA
## 48 18 1 NA NA NA
## 49 19 1 NA NA NA
## 50 20 1 NA NA NA
## 51 21 1 NA NA NA
## 52 22 1 NA NA NA
## 53 23 1 NA NA NA
## 54 24 1 NA NA NA
## sizebinb_center sizebinb_width binhalfwidthc_raw sizebinc_min sizebinc_max
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## 12 NA NA NA NA NA
## 13 NA NA NA NA NA
## 14 NA NA NA NA NA
## 15 NA NA NA NA NA
## 16 NA NA NA NA NA
## 17 NA NA NA NA NA
## 18 NA NA NA NA NA
## 19 NA NA NA NA NA
## 20 NA NA NA NA NA
## 21 NA NA NA NA NA
## 22 NA NA NA NA NA
## 23 NA NA NA NA NA
## 24 NA NA NA NA NA
## 25 NA NA NA NA NA
## 26 NA NA NA NA NA
## 27 NA NA NA NA NA
## 28 NA NA NA NA NA
## 29 NA NA NA NA NA
## 30 NA NA NA NA NA
## 31 NA NA NA NA NA
## 32 NA NA NA NA NA
## 33 NA NA NA NA NA
## 34 NA NA NA NA NA
## 35 NA NA NA NA NA
## 36 NA NA NA NA NA
## 37 NA NA NA NA NA
## 38 NA NA NA NA NA
## 39 NA NA NA NA NA
## 40 NA NA NA NA NA
## 41 NA NA NA NA NA
## 42 NA NA NA NA NA
## 43 NA NA NA NA NA
## 44 NA NA NA NA NA
## 45 NA NA NA NA NA
## 46 NA NA NA NA NA
## 47 NA NA NA NA NA
## 48 NA NA NA NA NA
## 49 NA NA NA NA NA
## 50 NA NA NA NA NA
## 51 NA NA NA NA NA
## 52 NA NA NA NA NA
## 53 NA NA NA NA NA
## 54 NA NA NA NA NA
## sizebinc_center sizebinc_width group comments
## 1 NA NA 0 Dormant seed
## 2 NA NA 0 Yr1 protocorm
## 3 NA NA 0 Yr2 protocorm
## 4 NA NA 0 Yr3 protocorm
## 5 NA NA 0 Seedling
## 6 NA NA 0 Veg dorm
## 7 NA NA 0 Veg adult 1 stem
## 8 NA NA 0 Veg adult 2 stems
## 9 NA NA 0 Veg adult 3 stems
## 10 NA NA 0 Veg adult 4 stems
## 11 NA NA 0 Veg adult 5 stems
## 12 NA NA 0 Veg adult 6 stems
## 13 NA NA 0 Veg adult 7 stems
## 14 NA NA 0 Veg adult 8 stems
## 15 NA NA 0 Veg adult 9 stems
## 16 NA NA 0 Veg adult 10 stems
## 17 NA NA 0 Veg adult 11 stems
## 18 NA NA 0 Veg adult 12 stems
## 19 NA NA 0 Veg adult 13 stems
## 20 NA NA 0 Veg adult 14 stems
## 21 NA NA 0 Veg adult 15 stems
## 22 NA NA 0 Veg adult 16 stems
## 23 NA NA 0 Veg adult 17 stems
## 24 NA NA 0 Veg adult 18 stems
## 25 NA NA 0 Veg adult 19 stems
## 26 NA NA 0 Veg adult 20 stems
## 27 NA NA 0 Veg adult 21 stems
## 28 NA NA 0 Veg adult 22 stems
## 29 NA NA 0 Veg adult 23 stems
## 30 NA NA 0 Veg adult 24 stems
## 31 NA NA 0 Flo adult 1 stem
## 32 NA NA 0 Flo adult 2 stems
## 33 NA NA 0 Flo adult 3 stems
## 34 NA NA 0 Flo adult 4 stems
## 35 NA NA 0 Flo adult 5 stems
## 36 NA NA 0 Flo adult 6 stems
## 37 NA NA 0 Flo adult 7 stems
## 38 NA NA 0 Flo adult 8 stems
## 39 NA NA 0 Flo adult 9 stems
## 40 NA NA 0 Flo adult 10 stems
## 41 NA NA 0 Flo adult 11 stems
## 42 NA NA 0 Flo adult 12 stems
## 43 NA NA 0 Flo adult 13 stems
## 44 NA NA 0 Flo adult 14 stems
## 45 NA NA 0 Flo adult 15 stems
## 46 NA NA 0 Flo adult 16 stems
## 47 NA NA 0 Flo adult 17 stems
## 48 NA NA 0 Flo adult 18 stems
## 49 NA NA 0 Flo adult 19 stems
## 50 NA NA 0 Flo adult 20 stems
## 51 NA NA 0 Flo adult 21 stems
## 52 NA NA 0 Flo adult 22 stems
## 53 NA NA 0 Flo adult 23 stems
## 54 NA NA 0 Flo adult 24 stemsThe stageframe above is a bit long, and required some long vectors to
create. Fortunately, functionsf_create() includes some
shorthand codes to make large, complex life history models easy to fit
into stageframe format. These shorthand approaches are particularly
useful in building IPMs, which generally require discretized size bins
that act as stages. In the next block, we show a shorter way of building
the above stageframe, using the ipm and
ipmbins options.
sizevector <- c(0, 0, 0, 0, 0, 0, 0.5, 24.5, 0.5, 24.5)
stagevector <- c("SD", "P1", "P2", "P3", "SL", "D", "ipm", "ipm", "ipm", "ipm")
repvector <- c(0, 0, 0, 0, 0, 0, 0, 0, 1, 1)
obsvector <- c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1)
matvector <- c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1)
immvector <- c(0, 1, 1, 1, 1, 0, 0, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
indataset <- c(0, 0, 0, 0, 0, 1, 1, 1, 1, 1)
comments <- c("Dormant seed", "Yr1 protocorm", "Yr2 protocorm", "Yr3 protocorm",
"Seedling", "Veg dorm", "non-reprod", "non-reprod", "reprod", "reprod")
minage <- c(1, 1, 2, 3, 4, 5, 5, 5, 6, 6)
maxage <- rep(NA, 10)
cypframe_ipm <- sf_create(sizes = sizevector, stagenames = stagevector,
repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
propstatus = propvector, immstatus = immvector, indataset = indataset,
comments = comments, minage = minage, maxage = maxage, ipmbins = 24)
cypframe_ipm## stage size size_b size_c min_age max_age repstatus obsstatus
## 1 SD 0 NA NA 1 NA 0 0
## 2 P1 0 NA NA 1 NA 0 0
## 3 P2 0 NA NA 2 NA 0 0
## 4 P3 0 NA NA 3 NA 0 0
## 5 SL 0 NA NA 4 NA 0 0
## 6 D 0 NA NA 5 NA 0 0
## 7 sza_1.0000_0 1 NA NA 5 NA 0 1
## 8 sza_2.0000_0 2 NA NA 5 NA 0 1
## 9 sza_3.0000_0 3 NA NA 5 NA 0 1
## 10 sza_4.0000_0 4 NA NA 5 NA 0 1
## 11 sza_5.0000_0 5 NA NA 5 NA 0 1
## 12 sza_6.0000_0 6 NA NA 5 NA 0 1
## 13 sza_7.0000_0 7 NA NA 5 NA 0 1
## 14 sza_8.0000_0 8 NA NA 5 NA 0 1
## 15 sza_9.0000_0 9 NA NA 5 NA 0 1
## 16 sza_10.000_0 10 NA NA 5 NA 0 1
## 17 sza_11.000_0 11 NA NA 5 NA 0 1
## 18 sza_12.000_0 12 NA NA 5 NA 0 1
## 19 sza_13.000_0 13 NA NA 5 NA 0 1
## 20 sza_14.000_0 14 NA NA 5 NA 0 1
## 21 sza_15.000_0 15 NA NA 5 NA 0 1
## 22 sza_16.000_0 16 NA NA 5 NA 0 1
## 23 sza_17.000_0 17 NA NA 5 NA 0 1
## 24 sza_18.000_0 18 NA NA 5 NA 0 1
## 25 sza_19.000_0 19 NA NA 5 NA 0 1
## 26 sza_20.000_0 20 NA NA 5 NA 0 1
## 27 sza_21.000_0 21 NA NA 5 NA 0 1
## 28 sza_22.000_0 22 NA NA 5 NA 0 1
## 29 sza_23.000_0 23 NA NA 5 NA 0 1
## 30 sza_24.000_0 24 NA NA 5 NA 0 1
## 31 sza_1.0000_0 1 NA NA 6 NA 1 1
## 32 sza_2.0000_0 2 NA NA 6 NA 1 1
## 33 sza_3.0000_0 3 NA NA 6 NA 1 1
## 34 sza_4.0000_0 4 NA NA 6 NA 1 1
## 35 sza_5.0000_0 5 NA NA 6 NA 1 1
## 36 sza_6.0000_0 6 NA NA 6 NA 1 1
## 37 sza_7.0000_0 7 NA NA 6 NA 1 1
## 38 sza_8.0000_0 8 NA NA 6 NA 1 1
## 39 sza_9.0000_0 9 NA NA 6 NA 1 1
## 40 sza_10.000_0 10 NA NA 6 NA 1 1
## 41 sza_11.000_0 11 NA NA 6 NA 1 1
## 42 sza_12.000_0 12 NA NA 6 NA 1 1
## 43 sza_13.000_0 13 NA NA 6 NA 1 1
## 44 sza_14.000_0 14 NA NA 6 NA 1 1
## 45 sza_15.000_0 15 NA NA 6 NA 1 1
## 46 sza_16.000_0 16 NA NA 6 NA 1 1
## 47 sza_17.000_0 17 NA NA 6 NA 1 1
## 48 sza_18.000_0 18 NA NA 6 NA 1 1
## 49 sza_19.000_0 19 NA NA 6 NA 1 1
## 50 sza_20.000_0 20 NA NA 6 NA 1 1
## 51 sza_21.000_0 21 NA NA 6 NA 1 1
## 52 sza_22.000_0 22 NA NA 6 NA 1 1
## 53 sza_23.000_0 23 NA NA 6 NA 1 1
## 54 sza_24.000_0 24 NA NA 6 NA 1 1
## propstatus immstatus matstatus indataset binhalfwidth_raw sizebin_min
## 1 1 0 0 0 0.5 -0.5
## 2 0 1 0 0 0.5 -0.5
## 3 0 1 0 0 0.5 -0.5
## 4 0 1 0 0 0.5 -0.5
## 5 0 1 0 0 0.5 -0.5
## 6 0 0 0 1 0.5 -0.5
## 7 0 0 1 1 0.5 0.5
## 8 0 0 1 1 0.5 1.5
## 9 0 0 1 1 0.5 2.5
## 10 0 0 1 1 0.5 3.5
## 11 0 0 1 1 0.5 4.5
## 12 0 0 1 1 0.5 5.5
## 13 0 0 1 1 0.5 6.5
## 14 0 0 1 1 0.5 7.5
## 15 0 0 1 1 0.5 8.5
## 16 0 0 1 1 0.5 9.5
## 17 0 0 1 1 0.5 10.5
## 18 0 0 1 1 0.5 11.5
## 19 0 0 1 1 0.5 12.5
## 20 0 0 1 1 0.5 13.5
## 21 0 0 1 1 0.5 14.5
## 22 0 0 1 1 0.5 15.5
## 23 0 0 1 1 0.5 16.5
## 24 0 0 1 1 0.5 17.5
## 25 0 0 1 1 0.5 18.5
## 26 0 0 1 1 0.5 19.5
## 27 0 0 1 1 0.5 20.5
## 28 0 0 1 1 0.5 21.5
## 29 0 0 1 1 0.5 22.5
## 30 0 0 1 1 0.5 23.5
## 31 0 0 1 1 0.5 0.5
## 32 0 0 1 1 0.5 1.5
## 33 0 0 1 1 0.5 2.5
## 34 0 0 1 1 0.5 3.5
## 35 0 0 1 1 0.5 4.5
## 36 0 0 1 1 0.5 5.5
## 37 0 0 1 1 0.5 6.5
## 38 0 0 1 1 0.5 7.5
## 39 0 0 1 1 0.5 8.5
## 40 0 0 1 1 0.5 9.5
## 41 0 0 1 1 0.5 10.5
## 42 0 0 1 1 0.5 11.5
## 43 0 0 1 1 0.5 12.5
## 44 0 0 1 1 0.5 13.5
## 45 0 0 1 1 0.5 14.5
## 46 0 0 1 1 0.5 15.5
## 47 0 0 1 1 0.5 16.5
## 48 0 0 1 1 0.5 17.5
## 49 0 0 1 1 0.5 18.5
## 50 0 0 1 1 0.5 19.5
## 51 0 0 1 1 0.5 20.5
## 52 0 0 1 1 0.5 21.5
## 53 0 0 1 1 0.5 22.5
## 54 0 0 1 1 0.5 23.5
## sizebin_max sizebin_center sizebin_width binhalfwidthb_raw sizebinb_min
## 1 0.5 0 1 NA NA
## 2 0.5 0 1 NA NA
## 3 0.5 0 1 NA NA
## 4 0.5 0 1 NA NA
## 5 0.5 0 1 NA NA
## 6 0.5 0 1 NA NA
## 7 1.5 1 1 NA NA
## 8 2.5 2 1 NA NA
## 9 3.5 3 1 NA NA
## 10 4.5 4 1 NA NA
## 11 5.5 5 1 NA NA
## 12 6.5 6 1 NA NA
## 13 7.5 7 1 NA NA
## 14 8.5 8 1 NA NA
## 15 9.5 9 1 NA NA
## 16 10.5 10 1 NA NA
## 17 11.5 11 1 NA NA
## 18 12.5 12 1 NA NA
## 19 13.5 13 1 NA NA
## 20 14.5 14 1 NA NA
## 21 15.5 15 1 NA NA
## 22 16.5 16 1 NA NA
## 23 17.5 17 1 NA NA
## 24 18.5 18 1 NA NA
## 25 19.5 19 1 NA NA
## 26 20.5 20 1 NA NA
## 27 21.5 21 1 NA NA
## 28 22.5 22 1 NA NA
## 29 23.5 23 1 NA NA
## 30 24.5 24 1 NA NA
## 31 1.5 1 1 NA NA
## 32 2.5 2 1 NA NA
## 33 3.5 3 1 NA NA
## 34 4.5 4 1 NA NA
## 35 5.5 5 1 NA NA
## 36 6.5 6 1 NA NA
## 37 7.5 7 1 NA NA
## 38 8.5 8 1 NA NA
## 39 9.5 9 1 NA NA
## 40 10.5 10 1 NA NA
## 41 11.5 11 1 NA NA
## 42 12.5 12 1 NA NA
## 43 13.5 13 1 NA NA
## 44 14.5 14 1 NA NA
## 45 15.5 15 1 NA NA
## 46 16.5 16 1 NA NA
## 47 17.5 17 1 NA NA
## 48 18.5 18 1 NA NA
## 49 19.5 19 1 NA NA
## 50 20.5 20 1 NA NA
## 51 21.5 21 1 NA NA
## 52 22.5 22 1 NA NA
## 53 23.5 23 1 NA NA
## 54 24.5 24 1 NA NA
## sizebinb_max sizebinb_center sizebinb_width binhalfwidthc_raw sizebinc_min
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## 12 NA NA NA NA NA
## 13 NA NA NA NA NA
## 14 NA NA NA NA NA
## 15 NA NA NA NA NA
## 16 NA NA NA NA NA
## 17 NA NA NA NA NA
## 18 NA NA NA NA NA
## 19 NA NA NA NA NA
## 20 NA NA NA NA NA
## 21 NA NA NA NA NA
## 22 NA NA NA NA NA
## 23 NA NA NA NA NA
## 24 NA NA NA NA NA
## 25 NA NA NA NA NA
## 26 NA NA NA NA NA
## 27 NA NA NA NA NA
## 28 NA NA NA NA NA
## 29 NA NA NA NA NA
## 30 NA NA NA NA NA
## 31 NA NA NA NA NA
## 32 NA NA NA NA NA
## 33 NA NA NA NA NA
## 34 NA NA NA NA NA
## 35 NA NA NA NA NA
## 36 NA NA NA NA NA
## 37 NA NA NA NA NA
## 38 NA NA NA NA NA
## 39 NA NA NA NA NA
## 40 NA NA NA NA NA
## 41 NA NA NA NA NA
## 42 NA NA NA NA NA
## 43 NA NA NA NA NA
## 44 NA NA NA NA NA
## 45 NA NA NA NA NA
## 46 NA NA NA NA NA
## 47 NA NA NA NA NA
## 48 NA NA NA NA NA
## 49 NA NA NA NA NA
## 50 NA NA NA NA NA
## 51 NA NA NA NA NA
## 52 NA NA NA NA NA
## 53 NA NA NA NA NA
## 54 NA NA NA NA NA
## sizebinc_max sizebinc_center sizebinc_width group comments
## 1 NA NA NA 0 Dormant seed
## 2 NA NA NA 0 Yr1 protocorm
## 3 NA NA NA 0 Yr2 protocorm
## 4 NA NA NA 0 Yr3 protocorm
## 5 NA NA NA 0 Seedling
## 6 NA NA NA 0 Veg dorm
## 7 NA NA NA 0 non-reprod
## 8 NA NA NA 0 non-reprod
## 9 NA NA NA 0 non-reprod
## 10 NA NA NA 0 non-reprod
## 11 NA NA NA 0 non-reprod
## 12 NA NA NA 0 non-reprod
## 13 NA NA NA 0 non-reprod
## 14 NA NA NA 0 non-reprod
## 15 NA NA NA 0 non-reprod
## 16 NA NA NA 0 non-reprod
## 17 NA NA NA 0 non-reprod
## 18 NA NA NA 0 non-reprod
## 19 NA NA NA 0 non-reprod
## 20 NA NA NA 0 non-reprod
## 21 NA NA NA 0 non-reprod
## 22 NA NA NA 0 non-reprod
## 23 NA NA NA 0 non-reprod
## 24 NA NA NA 0 non-reprod
## 25 NA NA NA 0 non-reprod
## 26 NA NA NA 0 non-reprod
## 27 NA NA NA 0 non-reprod
## 28 NA NA NA 0 non-reprod
## 29 NA NA NA 0 non-reprod
## 30 NA NA NA 0 non-reprod
## 31 NA NA NA 0 reprod
## 32 NA NA NA 0 reprod
## 33 NA NA NA 0 reprod
## 34 NA NA NA 0 reprod
## 35 NA NA NA 0 reprod
## 36 NA NA NA 0 reprod
## 37 NA NA NA 0 reprod
## 38 NA NA NA 0 reprod
## 39 NA NA NA 0 reprod
## 40 NA NA NA 0 reprod
## 41 NA NA NA 0 reprod
## 42 NA NA NA 0 reprod
## 43 NA NA NA 0 reprod
## 44 NA NA NA 0 reprod
## 45 NA NA NA 0 reprod
## 46 NA NA NA 0 reprod
## 47 NA NA NA 0 reprod
## 48 NA NA NA 0 reprod
## 49 NA NA NA 0 reprod
## 50 NA NA NA 0 reprod
## 51 NA NA NA 0 reprod
## 52 NA NA NA 0 reprod
## 53 NA NA NA 0 reprod
## 54 NA NA NA 0 reprodNow let’s build our Lathyrus life history model. Note that IPMs are function-based by default, so we will not create a raw MPM version. Here is our life history model. We will also not create an age-by-stage version, and so will leave age information out.
Figure 5. Life history model of Lathyrus vernus. Not all adult classes are shown. Survival transitions are indicated with solid arrows, while fecundity transitions are indicated with dashed arrows.
In the stageframe code below, we show that we want an IPM by choosing
two stages that serve as the size limits for the IPM’s discretized size
bin classification. These two size classes should have exactly
the same characteristics in the stageframe other than size. The
sizes input into the sizes vector for these two stages
should not be midpoints. Instead, the size for the lower limit should be
the lower limit of the minimum size bin, while the size input for the
upper limit should be the upper limit of the maximum size bin. By
choosing these two size limits, we can skip adding and describing the
many size classes that will fall between these limits - function
sf_create() will create all of these for us. We mark these
limits in the vector that we load into the stagenames
option using the string "ipm". We then input all other
characteristics for these size bins, such as observation status,
maturity status, reproductive status, and these characteristics must be
the same for both the minimum and maximum size bins. Package
lefko3 will then create and name all IPM size classes
according to its own conventions. The default number of size classes is
100 bins, and this can be altered using the ipmbins option.
Note that this is essentially the same procedure described in section
@ref(ipmframe).
sizevector <- c(0, 100, 0, 1, 7100)
stagevector <- c("Sd", "Sdl", "Dorm", "ipm", "ipm")
repvector <- c(0, 0, 0, 1, 1)
obsvector <- c(0, 1, 0, 1, 1)
matvector <- c(0, 0, 1, 1, 1)
immvector <- c(1, 1, 0, 0, 0)
propvector <- c(1, 0, 0, 0, 0)
indataset <- c(0, 1, 1, 1, 1)
binvec <- c(0, 100, 0.5, 1, 1)
comments <- c("Dormant seed", "Seedling", "Dormant", "ipm adult stage",
"ipm adult stage")
lathframe_ipm <- sf_create(sizes = sizevector, stagenames = stagevector,
repstatus = repvector, obsstatus = obsvector, propstatus = propvector,
immstatus = immvector, matstatus = matvector, comments = comments,
indataset = indataset, binhalfwidth = binvec, ipmbins = 100, roundsize = 3)
dim(lathframe_ipm)## [1] 103 29This stageframe has 103 stages - dormant seed, seedling, vegetative dormancy, and 100 size-classified adult stages. Let’s look at this object.
lathframe_ipm## stage size size_b size_c min_age max_age repstatus obsstatus
## 1 Sd 0.000 NA NA NA NA 0 0
## 2 Sdl 100.000 NA NA NA NA 0 1
## 3 Dorm 0.000 NA NA NA NA 0 0
## 4 sza_36.495_0 36.495 NA NA NA NA 1 1
## 5 sza_107.48_0 107.485 NA NA NA NA 1 1
## 6 sza_178.47_0 178.475 NA NA NA NA 1 1
## 7 sza_249.46_0 249.465 NA NA NA NA 1 1
## 8 sza_320.45_0 320.455 NA NA NA NA 1 1
## 9 sza_391.44_0 391.445 NA NA NA NA 1 1
## 10 sza_462.43_0 462.435 NA NA NA NA 1 1
## 11 sza_533.42_0 533.425 NA NA NA NA 1 1
## 12 sza_604.41_0 604.415 NA NA NA NA 1 1
## 13 sza_675.40_0 675.405 NA NA NA NA 1 1
## 14 sza_746.39_0 746.395 NA NA NA NA 1 1
## 15 sza_817.38_0 817.385 NA NA NA NA 1 1
## 16 sza_888.37_0 888.375 NA NA NA NA 1 1
## 17 sza_959.36_0 959.365 NA NA NA NA 1 1
## 18 sza_1030.3_0 1030.355 NA NA NA NA 1 1
## 19 sza_1101.3_0 1101.345 NA NA NA NA 1 1
## 20 sza_1172.3_0 1172.335 NA NA NA NA 1 1
## 21 sza_1243.3_0 1243.325 NA NA NA NA 1 1
## 22 sza_1314.3_0 1314.315 NA NA NA NA 1 1
## 23 sza_1385.3_0 1385.305 NA NA NA NA 1 1
## 24 sza_1456.2_0 1456.295 NA NA NA NA 1 1
## 25 sza_1527.2_0 1527.285 NA NA NA NA 1 1
## 26 sza_1598.2_0 1598.275 NA NA NA NA 1 1
## 27 sza_1669.2_0 1669.265 NA NA NA NA 1 1
## 28 sza_1740.2_0 1740.255 NA NA NA NA 1 1
## 29 sza_1811.2_0 1811.245 NA NA NA NA 1 1
## 30 sza_1882.2_0 1882.235 NA NA NA NA 1 1
## 31 sza_1953.2_0 1953.225 NA NA NA NA 1 1
## 32 sza_2024.2_0 2024.215 NA NA NA NA 1 1
## 33 sza_2095.2_0 2095.205 NA NA NA NA 1 1
## 34 sza_2166.1_0 2166.195 NA NA NA NA 1 1
## 35 sza_2237.1_0 2237.185 NA NA NA NA 1 1
## 36 sza_2308.1_0 2308.175 NA NA NA NA 1 1
## 37 sza_2379.1_0 2379.165 NA NA NA NA 1 1
## 38 sza_2450.1_0 2450.155 NA NA NA NA 1 1
## 39 sza_2521.1_0 2521.145 NA NA NA NA 1 1
## 40 sza_2592.1_0 2592.135 NA NA NA NA 1 1
## 41 sza_2663.1_0 2663.125 NA NA NA NA 1 1
## 42 sza_2734.1_0 2734.115 NA NA NA NA 1 1
## 43 sza_2805.1_0 2805.105 NA NA NA NA 1 1
## 44 sza_2876.0_0 2876.095 NA NA NA NA 1 1
## 45 sza_2947.0_0 2947.085 NA NA NA NA 1 1
## 46 sza_3018.0_0 3018.075 NA NA NA NA 1 1
## 47 sza_3089.0_0 3089.065 NA NA NA NA 1 1
## 48 sza_3160.0_0 3160.055 NA NA NA NA 1 1
## 49 sza_3231.0_0 3231.045 NA NA NA NA 1 1
## 50 sza_3302.0_0 3302.035 NA NA NA NA 1 1
## 51 sza_3373.0_0 3373.025 NA NA NA NA 1 1
## 52 sza_3444.0_0 3444.015 NA NA NA NA 1 1
## 53 sza_3515.0_0 3515.005 NA NA NA NA 1 1
## 54 sza_3585.9_0 3585.995 NA NA NA NA 1 1
## 55 sza_3656.9_0 3656.985 NA NA NA NA 1 1
## 56 sza_3727.9_0 3727.975 NA NA NA NA 1 1
## 57 sza_3798.9_0 3798.965 NA NA NA NA 1 1
## 58 sza_3869.9_0 3869.955 NA NA NA NA 1 1
## 59 sza_3940.9_0 3940.945 NA NA NA NA 1 1
## 60 sza_4011.9_0 4011.935 NA NA NA NA 1 1
## 61 sza_4082.9_0 4082.925 NA NA NA NA 1 1
## 62 sza_4153.9_0 4153.915 NA NA NA NA 1 1
## 63 sza_4224.9_0 4224.905 NA NA NA NA 1 1
## 64 sza_4295.8_0 4295.895 NA NA NA NA 1 1
## 65 sza_4366.8_0 4366.885 NA NA NA NA 1 1
## 66 sza_4437.8_0 4437.875 NA NA NA NA 1 1
## 67 sza_4508.8_0 4508.865 NA NA NA NA 1 1
## 68 sza_4579.8_0 4579.855 NA NA NA NA 1 1
## 69 sza_4650.8_0 4650.845 NA NA NA NA 1 1
## 70 sza_4721.8_0 4721.835 NA NA NA NA 1 1
## 71 sza_4792.8_0 4792.825 NA NA NA NA 1 1
## 72 sza_4863.8_0 4863.815 NA NA NA NA 1 1
## 73 sza_4934.8_0 4934.805 NA NA NA NA 1 1
## 74 sza_5005.7_0 5005.795 NA NA NA NA 1 1
## 75 sza_5076.7_0 5076.785 NA NA NA NA 1 1
## 76 sza_5147.7_0 5147.775 NA NA NA NA 1 1
## 77 sza_5218.7_0 5218.765 NA NA NA NA 1 1
## 78 sza_5289.7_0 5289.755 NA NA NA NA 1 1
## 79 sza_5360.7_0 5360.745 NA NA NA NA 1 1
## 80 sza_5431.7_0 5431.735 NA NA NA NA 1 1
## 81 sza_5502.7_0 5502.725 NA NA NA NA 1 1
## 82 sza_5573.7_0 5573.715 NA NA NA NA 1 1
## 83 sza_5644.7_0 5644.705 NA NA NA NA 1 1
## 84 sza_5715.6_0 5715.695 NA NA NA NA 1 1
## 85 sza_5786.6_0 5786.685 NA NA NA NA 1 1
## 86 sza_5857.6_0 5857.675 NA NA NA NA 1 1
## 87 sza_5928.6_0 5928.665 NA NA NA NA 1 1
## 88 sza_5999.6_0 5999.655 NA NA NA NA 1 1
## 89 sza_6070.6_0 6070.645 NA NA NA NA 1 1
## 90 sza_6141.6_0 6141.635 NA NA NA NA 1 1
## 91 sza_6212.6_0 6212.625 NA NA NA NA 1 1
## 92 sza_6283.6_0 6283.615 NA NA NA NA 1 1
## 93 sza_6354.6_0 6354.605 NA NA NA NA 1 1
## 94 sza_6425.5_0 6425.595 NA NA NA NA 1 1
## 95 sza_6496.5_0 6496.585 NA NA NA NA 1 1
## 96 sza_6567.5_0 6567.575 NA NA NA NA 1 1
## 97 sza_6638.5_0 6638.565 NA NA NA NA 1 1
## 98 sza_6709.5_0 6709.555 NA NA NA NA 1 1
## 99 sza_6780.5_0 6780.545 NA NA NA NA 1 1
## 100 sza_6851.5_0 6851.535 NA NA NA NA 1 1
## 101 sza_6922.5_0 6922.525 NA NA NA NA 1 1
## 102 sza_6993.5_0 6993.515 NA NA NA NA 1 1
## 103 sza_7064.5_0 7064.505 NA NA NA NA 1 1
## propstatus immstatus matstatus indataset binhalfwidth_raw sizebin_min
## 1 1 1 0 0 0.000 0.00
## 2 0 1 0 1 100.000 0.00
## 3 0 0 1 1 0.500 -0.50
## 4 0 0 1 1 35.495 1.00
## 5 0 0 1 1 35.495 71.99
## 6 0 0 1 1 35.495 142.98
## 7 0 0 1 1 35.495 213.97
## 8 0 0 1 1 35.495 284.96
## 9 0 0 1 1 35.495 355.95
## 10 0 0 1 1 35.495 426.94
## 11 0 0 1 1 35.495 497.93
## 12 0 0 1 1 35.495 568.92
## 13 0 0 1 1 35.495 639.91
## 14 0 0 1 1 35.495 710.90
## 15 0 0 1 1 35.495 781.89
## 16 0 0 1 1 35.495 852.88
## 17 0 0 1 1 35.495 923.87
## 18 0 0 1 1 35.495 994.86
## 19 0 0 1 1 35.495 1065.85
## 20 0 0 1 1 35.495 1136.84
## 21 0 0 1 1 35.495 1207.83
## 22 0 0 1 1 35.495 1278.82
## 23 0 0 1 1 35.495 1349.81
## 24 0 0 1 1 35.495 1420.80
## 25 0 0 1 1 35.495 1491.79
## 26 0 0 1 1 35.495 1562.78
## 27 0 0 1 1 35.495 1633.77
## 28 0 0 1 1 35.495 1704.76
## 29 0 0 1 1 35.495 1775.75
## 30 0 0 1 1 35.495 1846.74
## 31 0 0 1 1 35.495 1917.73
## 32 0 0 1 1 35.495 1988.72
## 33 0 0 1 1 35.495 2059.71
## 34 0 0 1 1 35.495 2130.70
## 35 0 0 1 1 35.495 2201.69
## 36 0 0 1 1 35.495 2272.68
## 37 0 0 1 1 35.495 2343.67
## 38 0 0 1 1 35.495 2414.66
## 39 0 0 1 1 35.495 2485.65
## 40 0 0 1 1 35.495 2556.64
## 41 0 0 1 1 35.495 2627.63
## 42 0 0 1 1 35.495 2698.62
## 43 0 0 1 1 35.495 2769.61
## 44 0 0 1 1 35.495 2840.60
## 45 0 0 1 1 35.495 2911.59
## 46 0 0 1 1 35.495 2982.58
## 47 0 0 1 1 35.495 3053.57
## 48 0 0 1 1 35.495 3124.56
## 49 0 0 1 1 35.495 3195.55
## 50 0 0 1 1 35.495 3266.54
## 51 0 0 1 1 35.495 3337.53
## 52 0 0 1 1 35.495 3408.52
## 53 0 0 1 1 35.495 3479.51
## 54 0 0 1 1 35.495 3550.50
## 55 0 0 1 1 35.495 3621.49
## 56 0 0 1 1 35.495 3692.48
## 57 0 0 1 1 35.495 3763.47
## 58 0 0 1 1 35.495 3834.46
## 59 0 0 1 1 35.495 3905.45
## 60 0 0 1 1 35.495 3976.44
## 61 0 0 1 1 35.495 4047.43
## 62 0 0 1 1 35.495 4118.42
## 63 0 0 1 1 35.495 4189.41
## 64 0 0 1 1 35.495 4260.40
## 65 0 0 1 1 35.495 4331.39
## 66 0 0 1 1 35.495 4402.38
## 67 0 0 1 1 35.495 4473.37
## 68 0 0 1 1 35.495 4544.36
## 69 0 0 1 1 35.495 4615.35
## 70 0 0 1 1 35.495 4686.34
## 71 0 0 1 1 35.495 4757.33
## 72 0 0 1 1 35.495 4828.32
## 73 0 0 1 1 35.495 4899.31
## 74 0 0 1 1 35.495 4970.30
## 75 0 0 1 1 35.495 5041.29
## 76 0 0 1 1 35.495 5112.28
## 77 0 0 1 1 35.495 5183.27
## 78 0 0 1 1 35.495 5254.26
## 79 0 0 1 1 35.495 5325.25
## 80 0 0 1 1 35.495 5396.24
## 81 0 0 1 1 35.495 5467.23
## 82 0 0 1 1 35.495 5538.22
## 83 0 0 1 1 35.495 5609.21
## 84 0 0 1 1 35.495 5680.20
## 85 0 0 1 1 35.495 5751.19
## 86 0 0 1 1 35.495 5822.18
## 87 0 0 1 1 35.495 5893.17
## 88 0 0 1 1 35.495 5964.16
## 89 0 0 1 1 35.495 6035.15
## 90 0 0 1 1 35.495 6106.14
## 91 0 0 1 1 35.495 6177.13
## 92 0 0 1 1 35.495 6248.12
## 93 0 0 1 1 35.495 6319.11
## 94 0 0 1 1 35.495 6390.10
## 95 0 0 1 1 35.495 6461.09
## 96 0 0 1 1 35.495 6532.08
## 97 0 0 1 1 35.495 6603.07
## 98 0 0 1 1 35.495 6674.06
## 99 0 0 1 1 35.495 6745.05
## 100 0 0 1 1 35.495 6816.04
## 101 0 0 1 1 35.495 6887.03
## 102 0 0 1 1 35.495 6958.02
## 103 0 0 1 1 35.495 7029.01
## sizebin_max sizebin_center sizebin_width binhalfwidthb_raw sizebinb_min
## 1 0.00 0.000 0.00 NA NA
## 2 200.00 100.000 200.00 NA NA
## 3 0.50 0.000 1.00 NA NA
## 4 71.99 36.495 70.99 NA NA
## 5 142.98 107.485 70.99 NA NA
## 6 213.97 178.475 70.99 NA NA
## 7 284.96 249.465 70.99 NA NA
## 8 355.95 320.455 70.99 NA NA
## 9 426.94 391.445 70.99 NA NA
## 10 497.93 462.435 70.99 NA NA
## 11 568.92 533.425 70.99 NA NA
## 12 639.91 604.415 70.99 NA NA
## 13 710.90 675.405 70.99 NA NA
## 14 781.89 746.395 70.99 NA NA
## 15 852.88 817.385 70.99 NA NA
## 16 923.87 888.375 70.99 NA NA
## 17 994.86 959.365 70.99 NA NA
## 18 1065.85 1030.355 70.99 NA NA
## 19 1136.84 1101.345 70.99 NA NA
## 20 1207.83 1172.335 70.99 NA NA
## 21 1278.82 1243.325 70.99 NA NA
## 22 1349.81 1314.315 70.99 NA NA
## 23 1420.80 1385.305 70.99 NA NA
## 24 1491.79 1456.295 70.99 NA NA
## 25 1562.78 1527.285 70.99 NA NA
## 26 1633.77 1598.275 70.99 NA NA
## 27 1704.76 1669.265 70.99 NA NA
## 28 1775.75 1740.255 70.99 NA NA
## 29 1846.74 1811.245 70.99 NA NA
## 30 1917.73 1882.235 70.99 NA NA
## 31 1988.72 1953.225 70.99 NA NA
## 32 2059.71 2024.215 70.99 NA NA
## 33 2130.70 2095.205 70.99 NA NA
## 34 2201.69 2166.195 70.99 NA NA
## 35 2272.68 2237.185 70.99 NA NA
## 36 2343.67 2308.175 70.99 NA NA
## 37 2414.66 2379.165 70.99 NA NA
## 38 2485.65 2450.155 70.99 NA NA
## 39 2556.64 2521.145 70.99 NA NA
## 40 2627.63 2592.135 70.99 NA NA
## 41 2698.62 2663.125 70.99 NA NA
## 42 2769.61 2734.115 70.99 NA NA
## 43 2840.60 2805.105 70.99 NA NA
## 44 2911.59 2876.095 70.99 NA NA
## 45 2982.58 2947.085 70.99 NA NA
## 46 3053.57 3018.075 70.99 NA NA
## 47 3124.56 3089.065 70.99 NA NA
## 48 3195.55 3160.055 70.99 NA NA
## 49 3266.54 3231.045 70.99 NA NA
## 50 3337.53 3302.035 70.99 NA NA
## 51 3408.52 3373.025 70.99 NA NA
## 52 3479.51 3444.015 70.99 NA NA
## 53 3550.50 3515.005 70.99 NA NA
## 54 3621.49 3585.995 70.99 NA NA
## 55 3692.48 3656.985 70.99 NA NA
## 56 3763.47 3727.975 70.99 NA NA
## 57 3834.46 3798.965 70.99 NA NA
## 58 3905.45 3869.955 70.99 NA NA
## 59 3976.44 3940.945 70.99 NA NA
## 60 4047.43 4011.935 70.99 NA NA
## 61 4118.42 4082.925 70.99 NA NA
## 62 4189.41 4153.915 70.99 NA NA
## 63 4260.40 4224.905 70.99 NA NA
## 64 4331.39 4295.895 70.99 NA NA
## 65 4402.38 4366.885 70.99 NA NA
## 66 4473.37 4437.875 70.99 NA NA
## 67 4544.36 4508.865 70.99 NA NA
## 68 4615.35 4579.855 70.99 NA NA
## 69 4686.34 4650.845 70.99 NA NA
## 70 4757.33 4721.835 70.99 NA NA
## 71 4828.32 4792.825 70.99 NA NA
## 72 4899.31 4863.815 70.99 NA NA
## 73 4970.30 4934.805 70.99 NA NA
## 74 5041.29 5005.795 70.99 NA NA
## 75 5112.28 5076.785 70.99 NA NA
## 76 5183.27 5147.775 70.99 NA NA
## 77 5254.26 5218.765 70.99 NA NA
## 78 5325.25 5289.755 70.99 NA NA
## 79 5396.24 5360.745 70.99 NA NA
## 80 5467.23 5431.735 70.99 NA NA
## 81 5538.22 5502.725 70.99 NA NA
## 82 5609.21 5573.715 70.99 NA NA
## 83 5680.20 5644.705 70.99 NA NA
## 84 5751.19 5715.695 70.99 NA NA
## 85 5822.18 5786.685 70.99 NA NA
## 86 5893.17 5857.675 70.99 NA NA
## 87 5964.16 5928.665 70.99 NA NA
## 88 6035.15 5999.655 70.99 NA NA
## 89 6106.14 6070.645 70.99 NA NA
## 90 6177.13 6141.635 70.99 NA NA
## 91 6248.12 6212.625 70.99 NA NA
## 92 6319.11 6283.615 70.99 NA NA
## 93 6390.10 6354.605 70.99 NA NA
## 94 6461.09 6425.595 70.99 NA NA
## 95 6532.08 6496.585 70.99 NA NA
## 96 6603.07 6567.575 70.99 NA NA
## 97 6674.06 6638.565 70.99 NA NA
## 98 6745.05 6709.555 70.99 NA NA
## 99 6816.04 6780.545 70.99 NA NA
## 100 6887.03 6851.535 70.99 NA NA
## 101 6958.02 6922.525 70.99 NA NA
## 102 7029.01 6993.515 70.99 NA NA
## 103 7100.00 7064.505 70.99 NA NA
## sizebinb_max sizebinb_center sizebinb_width binhalfwidthc_raw sizebinc_min
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## 12 NA NA NA NA NA
## 13 NA NA NA NA NA
## 14 NA NA NA NA NA
## 15 NA NA NA NA NA
## 16 NA NA NA NA NA
## 17 NA NA NA NA NA
## 18 NA NA NA NA NA
## 19 NA NA NA NA NA
## 20 NA NA NA NA NA
## 21 NA NA NA NA NA
## 22 NA NA NA NA NA
## 23 NA NA NA NA NA
## 24 NA NA NA NA NA
## 25 NA NA NA NA NA
## 26 NA NA NA NA NA
## 27 NA NA NA NA NA
## 28 NA NA NA NA NA
## 29 NA NA NA NA NA
## 30 NA NA NA NA NA
## 31 NA NA NA NA NA
## 32 NA NA NA NA NA
## 33 NA NA NA NA NA
## 34 NA NA NA NA NA
## 35 NA NA NA NA NA
## 36 NA NA NA NA NA
## 37 NA NA NA NA NA
## 38 NA NA NA NA NA
## 39 NA NA NA NA NA
## 40 NA NA NA NA NA
## 41 NA NA NA NA NA
## 42 NA NA NA NA NA
## 43 NA NA NA NA NA
## 44 NA NA NA NA NA
## 45 NA NA NA NA NA
## 46 NA NA NA NA NA
## 47 NA NA NA NA NA
## 48 NA NA NA NA NA
## 49 NA NA NA NA NA
## 50 NA NA NA NA NA
## 51 NA NA NA NA NA
## 52 NA NA NA NA NA
## 53 NA NA NA NA NA
## 54 NA NA NA NA NA
## 55 NA NA NA NA NA
## 56 NA NA NA NA NA
## 57 NA NA NA NA NA
## 58 NA NA NA NA NA
## 59 NA NA NA NA NA
## 60 NA NA NA NA NA
## 61 NA NA NA NA NA
## 62 NA NA NA NA NA
## 63 NA NA NA NA NA
## 64 NA NA NA NA NA
## 65 NA NA NA NA NA
## 66 NA NA NA NA NA
## 67 NA NA NA NA NA
## 68 NA NA NA NA NA
## 69 NA NA NA NA NA
## 70 NA NA NA NA NA
## 71 NA NA NA NA NA
## 72 NA NA NA NA NA
## 73 NA NA NA NA NA
## 74 NA NA NA NA NA
## 75 NA NA NA NA NA
## 76 NA NA NA NA NA
## 77 NA NA NA NA NA
## 78 NA NA NA NA NA
## 79 NA NA NA NA NA
## 80 NA NA NA NA NA
## 81 NA NA NA NA NA
## 82 NA NA NA NA NA
## 83 NA NA NA NA NA
## 84 NA NA NA NA NA
## 85 NA NA NA NA NA
## 86 NA NA NA NA NA
## 87 NA NA NA NA NA
## 88 NA NA NA NA NA
## 89 NA NA NA NA NA
## 90 NA NA NA NA NA
## 91 NA NA NA NA NA
## 92 NA NA NA NA NA
## 93 NA NA NA NA NA
## 94 NA NA NA NA NA
## 95 NA NA NA NA NA
## 96 NA NA NA NA NA
## 97 NA NA NA NA NA
## 98 NA NA NA NA NA
## 99 NA NA NA NA NA
## 100 NA NA NA NA NA
## 101 NA NA NA NA NA
## 102 NA NA NA NA NA
## 103 NA NA NA NA NA
## sizebinc_max sizebinc_center sizebinc_width group comments
## 1 NA NA NA 0 Dormant seed
## 2 NA NA NA 0 Seedling
## 3 NA NA NA 0 Dormant
## 4 NA NA NA 0 ipm adult stage
## 5 NA NA NA 0 ipm adult stage
## 6 NA NA NA 0 ipm adult stage
## 7 NA NA NA 0 ipm adult stage
## 8 NA NA NA 0 ipm adult stage
## 9 NA NA NA 0 ipm adult stage
## 10 NA NA NA 0 ipm adult stage
## 11 NA NA NA 0 ipm adult stage
## 12 NA NA NA 0 ipm adult stage
## 13 NA NA NA 0 ipm adult stage
## 14 NA NA NA 0 ipm adult stage
## 15 NA NA NA 0 ipm adult stage
## 16 NA NA NA 0 ipm adult stage
## 17 NA NA NA 0 ipm adult stage
## 18 NA NA NA 0 ipm adult stage
## 19 NA NA NA 0 ipm adult stage
## 20 NA NA NA 0 ipm adult stage
## 21 NA NA NA 0 ipm adult stage
## 22 NA NA NA 0 ipm adult stage
## 23 NA NA NA 0 ipm adult stage
## 24 NA NA NA 0 ipm adult stage
## 25 NA NA NA 0 ipm adult stage
## 26 NA NA NA 0 ipm adult stage
## 27 NA NA NA 0 ipm adult stage
## 28 NA NA NA 0 ipm adult stage
## 29 NA NA NA 0 ipm adult stage
## 30 NA NA NA 0 ipm adult stage
## 31 NA NA NA 0 ipm adult stage
## 32 NA NA NA 0 ipm adult stage
## 33 NA NA NA 0 ipm adult stage
## 34 NA NA NA 0 ipm adult stage
## 35 NA NA NA 0 ipm adult stage
## 36 NA NA NA 0 ipm adult stage
## 37 NA NA NA 0 ipm adult stage
## 38 NA NA NA 0 ipm adult stage
## 39 NA NA NA 0 ipm adult stage
## 40 NA NA NA 0 ipm adult stage
## 41 NA NA NA 0 ipm adult stage
## 42 NA NA NA 0 ipm adult stage
## 43 NA NA NA 0 ipm adult stage
## 44 NA NA NA 0 ipm adult stage
## 45 NA NA NA 0 ipm adult stage
## 46 NA NA NA 0 ipm adult stage
## 47 NA NA NA 0 ipm adult stage
## 48 NA NA NA 0 ipm adult stage
## 49 NA NA NA 0 ipm adult stage
## 50 NA NA NA 0 ipm adult stage
## 51 NA NA NA 0 ipm adult stage
## 52 NA NA NA 0 ipm adult stage
## 53 NA NA NA 0 ipm adult stage
## 54 NA NA NA 0 ipm adult stage
## 55 NA NA NA 0 ipm adult stage
## 56 NA NA NA 0 ipm adult stage
## 57 NA NA NA 0 ipm adult stage
## 58 NA NA NA 0 ipm adult stage
## 59 NA NA NA 0 ipm adult stage
## 60 NA NA NA 0 ipm adult stage
## 61 NA NA NA 0 ipm adult stage
## 62 NA NA NA 0 ipm adult stage
## 63 NA NA NA 0 ipm adult stage
## 64 NA NA NA 0 ipm adult stage
## 65 NA NA NA 0 ipm adult stage
## 66 NA NA NA 0 ipm adult stage
## 67 NA NA NA 0 ipm adult stage
## 68 NA NA NA 0 ipm adult stage
## 69 NA NA NA 0 ipm adult stage
## 70 NA NA NA 0 ipm adult stage
## 71 NA NA NA 0 ipm adult stage
## 72 NA NA NA 0 ipm adult stage
## 73 NA NA NA 0 ipm adult stage
## 74 NA NA NA 0 ipm adult stage
## 75 NA NA NA 0 ipm adult stage
## 76 NA NA NA 0 ipm adult stage
## 77 NA NA NA 0 ipm adult stage
## 78 NA NA NA 0 ipm adult stage
## 79 NA NA NA 0 ipm adult stage
## 80 NA NA NA 0 ipm adult stage
## 81 NA NA NA 0 ipm adult stage
## 82 NA NA NA 0 ipm adult stage
## 83 NA NA NA 0 ipm adult stage
## 84 NA NA NA 0 ipm adult stage
## 85 NA NA NA 0 ipm adult stage
## 86 NA NA NA 0 ipm adult stage
## 87 NA NA NA 0 ipm adult stage
## 88 NA NA NA 0 ipm adult stage
## 89 NA NA NA 0 ipm adult stage
## 90 NA NA NA 0 ipm adult stage
## 91 NA NA NA 0 ipm adult stage
## 92 NA NA NA 0 ipm adult stage
## 93 NA NA NA 0 ipm adult stage
## 94 NA NA NA 0 ipm adult stage
## 95 NA NA NA 0 ipm adult stage
## 96 NA NA NA 0 ipm adult stage
## 97 NA NA NA 0 ipm adult stage
## 98 NA NA NA 0 ipm adult stage
## 99 NA NA NA 0 ipm adult stage
## 100 NA NA NA 0 ipm adult stage
## 101 NA NA NA 0 ipm adult stage
## 102 NA NA NA 0 ipm adult stage
## 103 NA NA NA 0 ipm adult stage
The function sf_create() has created our mesh points and
associated size bins. This is in addition to the discrete stages
covering the dormant seed, seedling, and dormant adult stages. Of
course, we could have made this even more complex. For example, we could
have created two sets of stages to use as the upper and lower bounds of
two sets of continuous size states that differ in some key
characteristic, such as reproductive status. We also could have set up
the IPM using two or three different size metrics and used the
ipm option within each or only some of them. This function
provides a great deal of flexibility and power to create exactly the
life history model that you may want.
Step 2a. Demographic data standardization
Now that we have our life history models, we can create our
standardized datasets. We will standardize these datasets using our
stageframes, and so we need to create two standardized datasets - one
for the raw MPM and another for the function-based MPM. Let’s first
standardize the vertical dataset for the raw MPM. For this purpose, we
will utilize the verticalize3() function.
Note that in these exercises, we will not only standardize the datasets, but also link them to their appropriate life history models. This is eventually necessary if one wishes to create raw (empirical) MPMs, but is not necessary for function-based MPMs and IPMs, and is also not necessary simply to explore a dataset.
Let’s a take a look at the code below to see some of the options
utilized. Because we are lumping reproductive and non-reproductive
individuals into the non-dormant adult classes, we need to set
NRasRep = TRUE. Otherwise, verticalize3() will
attempt to use the reproductive status of individuals in classification,
and will fail due to the presence of non-reproductive adults. We also
need to set NAas0 = TRUE to make sure that NA values in
size are turned into zeros where necessary, and so aid in the assignment
of the vegetative dormancy stage. We will add an age_offset of four
years, since the youngest individual we could possibly see is five years
old and pre-breeding models such as this begin the life history model at
age 1 rather than age 0. Finally, note that we set up three different
size variables here, not as sizea, sizeb, and
sizec, and that we tell R that we want overall size to be
the sum of these (stagesize = "sizeadded").
cypraw_v1 <- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
stageassign = cypframe_raw, stagesize = "sizeadded", age_offset = 4,
NAas0 = TRUE, NRasRep = TRUE)
head(cypraw_v1)## rowid popid patchid individ year2 firstseen lastseen obsage obslifespan
## 1 1 A 164 2004 2004 2004 5 0
## 2 2 A 165 2004 2004 2009 5 5
## 3 3 A 240 2004 2004 2005 5 1
## 4 4 A 242 2004 2004 2009 5 5
## 5 5 A 243 2004 2004 2009 5 5
## 6 6 A 246 2004 2004 2007 5 3
## sizea1 sizeb1 sizec1 size1added repstra1 repstrb1 feca1 juvgiven1 obsstatus1
## 1 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0
## repstatus1 fecstatus1 matstatus1 alive1 stage1 stage1index sizea2 sizeb2
## 1 0 0 0 0 NotAlive 0 0 0
## 2 0 0 0 0 NotAlive 0 0 2
## 3 0 0 0 0 NotAlive 0 0 1
## 4 0 0 0 0 NotAlive 0 0 0
## 5 0 0 0 0 NotAlive 0 0 0
## 6 0 0 0 0 NotAlive 0 0 0
## sizec2 size2added repstra2 repstrb2 feca2 juvgiven2 obsstatus2 repstatus2
## 1 1 1 0 0 0 0 1 0
## 2 1 3 2 0 1 0 1 1
## 3 0 1 1 0 0 0 1 1
## 4 1 1 0 0 0 0 1 0
## 5 5 5 0 0 0 0 1 0
## 6 1 1 0 0 0 0 1 0
## fecstatus2 matstatus2 alive2 stage2 stage2index sizea3 sizeb3 sizec3
## 1 0 1 1 XSm 7 0 0 0
## 2 1 1 1 Sm 8 0 2 0
## 3 0 1 1 XSm 7 0 0 1
## 4 0 1 1 XSm 7 0 0 1
## 5 0 1 1 Md 9 0 0 2
## 6 0 1 1 XSm 7 0 0 1
## size3added repstra3 repstrb3 feca3 juvgiven3 obsstatus3 repstatus3 fecstatus3
## 1 0 0 0 0 0 0 0 0
## 2 2 2 0 0 0 1 1 0
## 3 1 0 0 0 0 1 0 0
## 4 1 0 0 0 0 1 0 0
## 5 2 0 0 0 0 1 0 0
## 6 1 0 0 0 0 1 0 0
## matstatus3 alive3 stage3 stage3index
## 1 1 0 NotAlive 0
## 2 1 1 Sm 8
## 3 1 1 XSm 7
## 4 1 1 XSm 7
## 5 1 1 Sm 8
## 6 1 1 XSm 7In the code above, we used the head() function to take a
look at the first six rows of the standardized data frame. The variable
names and formats work with all functions in lefko3, and of
course this data frame may be used in other packages, as well. Let’s now
take a look at a summary of the full data frame.
summary(cypraw_v1)## rowid popid patchid individ year2
## Min. : 1.00 :320 A: 93 Length:320 Min. :2004
## 1st Qu.:21.00 B:154 Class :character 1st Qu.:2005
## Median :37.50 C: 73 Mode :character Median :2006
## Mean :38.45 Mean :2006
## 3rd Qu.:56.00 3rd Qu.:2007
## Max. :77.00 Max. :2008
## firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 juvgiven1
## Min. : 0.0000 Min. :0.000000 Min. :0.0000 Min. :0
## 1st Qu.: 0.0000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0
## Median : 0.0000 Median :0.000000 Median :0.0000 Median :0
## Mean : 0.7469 Mean :0.009375 Mean :0.2656 Mean :0
## 3rd Qu.: 1.0000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :18.0000 Max. :1.000000 Max. :7.0000 Max. :0
## obsstatus1 repstatus1 fecstatus1 matstatus1
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1.0000
## Median :1.0000 Median :0.0000 Median :0.0000 Median :1.0000
## Mean :0.7469 Mean :0.2875 Mean :0.1344 Mean :0.7688
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## alive1 stage1 stage1index sizea2
## Min. :0.0000 Length:320 Min. : 0.000 Min. :0.000000
## 1st Qu.:1.0000 Class :character 1st Qu.: 6.000 1st Qu.:0.000000
## Median :1.0000 Mode :character Median : 8.000 Median :0.000000
## Mean :0.7688 Mean : 6.144 Mean :0.009375
## 3rd Qu.:1.0000 3rd Qu.: 8.000 3rd Qu.:0.000000
## Max. :1.0000 Max. :11.000 Max. :1.000000
## sizeb2 sizec2 size2added repstra2
## Min. : 0.0000 Min. : 0.000 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.: 0.0000
## Median : 0.0000 Median : 2.000 Median : 2.000 Median : 0.0000
## Mean : 0.8969 Mean : 2.416 Mean : 3.322 Mean : 0.8969
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.: 1.0000
## Max. :18.0000 Max. :13.000 Max. :24.000 Max. :18.0000
## repstrb2 feca2 juvgiven2 obsstatus2
## Min. :0.000000 Min. :0.0000 Min. :0 Min. :0.0000
## 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0 1st Qu.:1.0000
## Median :0.000000 Median :0.0000 Median :0 Median :1.0000
## Mean :0.009375 Mean :0.2906 Mean :0 Mean :0.9531
## 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0 3rd Qu.:1.0000
## Max. :1.000000 Max. :7.0000 Max. :0 Max. :1.0000
## repstatus2 fecstatus2 matstatus2 alive2 stage2
## Min. :0.0000 Min. :0.0000 Min. :1 Min. :1 Length:320
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character
## Median :0.0000 Median :0.0000 Median :1 Median :1 Mode :character
## Mean :0.3688 Mean :0.1562 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :1.0000 Max. :1 Max. :1
## stage2index sizea3 sizeb3 sizec3
## Min. : 6.000 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.000 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median : 8.000 Median :0.000000 Median : 0.000 Median : 1.000
## Mean : 7.919 Mean :0.009375 Mean : 1.069 Mean : 2.209
## 3rd Qu.: 8.000 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :11.000 Max. :1.000000 Max. :18.000 Max. :13.000
## size3added repstra3 repstrb3 feca3
## Min. : 0.000 Min. : 0.000 Min. :0.000000 Min. :0.0000
## 1st Qu.: 1.000 1st Qu.: 0.000 1st Qu.:0.000000 1st Qu.:0.0000
## Median : 2.000 Median : 0.000 Median :0.000000 Median :0.0000
## Mean : 3.288 Mean : 1.069 Mean :0.009375 Mean :0.4562
## 3rd Qu.: 4.000 3rd Qu.: 1.000 3rd Qu.:0.000000 3rd Qu.:0.0000
## Max. :24.000 Max. :18.000 Max. :1.000000 Max. :8.0000
## juvgiven3 obsstatus3 repstatus3 fecstatus3 matstatus3
## Min. :0 Min. :0.0 Min. :0.0 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0 Median :0.0 Median :0.0000 Median :1
## Mean :0 Mean :0.9 Mean :0.4 Mean :0.2219 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0 Max. :1.0 Max. :1.0000 Max. :1
## alive3 stage3 stage3index
## Min. :0.0000 Length:320 Min. : 0.000
## 1st Qu.:1.0000 Class :character 1st Qu.: 7.000
## Median :1.0000 Mode :character Median : 8.000
## Mean :0.9469 Mean : 7.544
## 3rd Qu.:1.0000 3rd Qu.: 8.000
## Max. :1.0000 Max. :11.000Let’s also create our standardized data for the function-based MPM,
as below. Remember that some of the settings need to change here,
primarily because we are now going to separate adults not just by size
but by reproductive status. So, we will NOT set
NRasRep = TRUE here (the default is
NRasRep = FALSE).
cypfb_v1 <- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
stageassign = cypframe_fb, stagesize = "sizeadded", age_offset = 4,
NAas0 = TRUE)
summary_hfv(cypfb_v1)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.
## rowid popid patchid individ year2
## Min. : 1.00 :320 A: 93 Length:320 Min. :2004
## 1st Qu.:21.00 B:154 Class :character 1st Qu.:2005
## Median :37.50 C: 73 Mode :character Median :2006
## Mean :38.45 Mean :2006
## 3rd Qu.:56.00 3rd Qu.:2007
## Max. :77.00 Max. :2008
## firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 juvgiven1
## Min. : 0.0000 Min. :0.000000 Min. :0.0000 Min. :0
## 1st Qu.: 0.0000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0
## Median : 0.0000 Median :0.000000 Median :0.0000 Median :0
## Mean : 0.7469 Mean :0.009375 Mean :0.2656 Mean :0
## 3rd Qu.: 1.0000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :18.0000 Max. :1.000000 Max. :7.0000 Max. :0
## obsstatus1 repstatus1 fecstatus1 matstatus1
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1.0000
## Median :1.0000 Median :0.0000 Median :0.0000 Median :1.0000
## Mean :0.7469 Mean :0.2875 Mean :0.1344 Mean :0.7688
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## alive1 stage1 stage1index sizea2
## Min. :0.0000 Length:320 Min. : 0.00 Min. :0.000000
## 1st Qu.:1.0000 Class :character 1st Qu.: 6.00 1st Qu.:0.000000
## Median :1.0000 Mode :character Median : 8.00 Median :0.000000
## Mean :0.7688 Mean :14.17 Mean :0.009375
## 3rd Qu.:1.0000 3rd Qu.:31.00 3rd Qu.:0.000000
## Max. :1.0000 Max. :51.00 Max. :1.000000
## sizeb2 sizec2 size2added repstra2
## Min. : 0.0000 Min. : 0.000 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.: 0.0000
## Median : 0.0000 Median : 2.000 Median : 2.000 Median : 0.0000
## Mean : 0.8969 Mean : 2.416 Mean : 3.322 Mean : 0.8969
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.: 1.0000
## Max. :18.0000 Max. :13.000 Max. :24.000 Max. :18.0000
## repstrb2 feca2 juvgiven2 obsstatus2
## Min. :0.000000 Min. :0.0000 Min. :0 Min. :0.0000
## 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0 1st Qu.:1.0000
## Median :0.000000 Median :0.0000 Median :0 Median :1.0000
## Mean :0.009375 Mean :0.2906 Mean :0 Mean :0.9531
## 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0 3rd Qu.:1.0000
## Max. :1.000000 Max. :7.0000 Max. :0 Max. :1.0000
## repstatus2 fecstatus2 matstatus2 alive2 stage2
## Min. :0.0000 Min. :0.0000 Min. :1 Min. :1 Length:320
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character
## Median :0.0000 Median :0.0000 Median :1 Median :1 Mode :character
## Mean :0.3688 Mean :0.1562 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :1.0000 Max. :1 Max. :1
## stage2index sizea3 sizeb3 sizec3
## Min. : 6.00 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.00 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median :10.00 Median :0.000000 Median : 0.000 Median : 1.000
## Mean :18.17 Mean :0.009375 Mean : 1.069 Mean : 2.209
## 3rd Qu.:32.00 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :54.00 Max. :1.000000 Max. :18.000 Max. :13.000
## size3added repstra3 repstrb3 feca3
## Min. : 0.000 Min. : 0.000 Min. :0.000000 Min. :0.0000
## 1st Qu.: 1.000 1st Qu.: 0.000 1st Qu.:0.000000 1st Qu.:0.0000
## Median : 2.000 Median : 0.000 Median :0.000000 Median :0.0000
## Mean : 3.288 Mean : 1.069 Mean :0.009375 Mean :0.4562
## 3rd Qu.: 4.000 3rd Qu.: 1.000 3rd Qu.:0.000000 3rd Qu.:0.0000
## Max. :24.000 Max. :18.000 Max. :1.000000 Max. :8.0000
## juvgiven3 obsstatus3 repstatus3 fecstatus3 matstatus3
## Min. :0 Min. :0.0 Min. :0.0 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0 Median :0.0 Median :0.0000 Median :1
## Mean :0 Mean :0.9 Mean :0.4 Mean :0.2219 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0 Max. :1.0 Max. :1.0000 Max. :1
## alive3 stage3 stage3index
## Min. :0.0000 Length:320 Min. : 0.00
## 1st Qu.:1.0000 Class :character 1st Qu.: 7.00
## Median :1.0000 Mode :character Median :10.00
## Mean :0.9469 Mean :18.57
## 3rd Qu.:1.0000 3rd Qu.:33.00
## Max. :1.0000 Max. :54.00Let’s try the same exercise as before, but using the Cypripedium IPM stageframe.
cypipm_v1 <- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
stageassign = cypframe_ipm, stagesize = "sizeadded", age_offset = 4,
NAas0 = TRUE)## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.
## Warning: Some stages occurring in the dataset do not match any characteristics
## in the input stageframe.summary_hfv(cypipm_v1)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.
## rowid popid patchid individ year2
## Min. : 1.00 :320 A: 93 Length:320 Min. :2004
## 1st Qu.:21.00 B:154 Class :character 1st Qu.:2005
## Median :37.50 C: 73 Mode :character Median :2006
## Mean :38.45 Mean :2006
## 3rd Qu.:56.00 3rd Qu.:2007
## Max. :77.00 Max. :2008
## firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 juvgiven1
## Min. : 0.0000 Min. :0.000000 Min. :0.0000 Min. :0
## 1st Qu.: 0.0000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0
## Median : 0.0000 Median :0.000000 Median :0.0000 Median :0
## Mean : 0.7469 Mean :0.009375 Mean :0.2656 Mean :0
## 3rd Qu.: 1.0000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :18.0000 Max. :1.000000 Max. :7.0000 Max. :0
## obsstatus1 repstatus1 fecstatus1 matstatus1
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :1.0000 Median :0.0000 Median :0.0000 Median :1.0000
## Mean :0.7469 Mean :0.2875 Mean :0.1344 Mean :0.7469
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## alive1 stage1 stage1index sizea2
## Min. :0.0000 Length:320 Min. : 0.00 Min. :0.000000
## 1st Qu.:1.0000 Class :character 1st Qu.: 0.00 1st Qu.:0.000000
## Median :1.0000 Mode :character Median : 8.00 Median :0.000000
## Mean :0.7688 Mean :14.04 Mean :0.009375
## 3rd Qu.:1.0000 3rd Qu.:31.00 3rd Qu.:0.000000
## Max. :1.0000 Max. :51.00 Max. :1.000000
## sizeb2 sizec2 size2added repstra2
## Min. : 0.0000 Min. : 0.000 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.: 0.0000
## Median : 0.0000 Median : 2.000 Median : 2.000 Median : 0.0000
## Mean : 0.8969 Mean : 2.416 Mean : 3.322 Mean : 0.8969
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.: 1.0000
## Max. :18.0000 Max. :13.000 Max. :24.000 Max. :18.0000
## repstrb2 feca2 juvgiven2 obsstatus2
## Min. :0.000000 Min. :0.0000 Min. :0 Min. :0.0000
## 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:0 1st Qu.:1.0000
## Median :0.000000 Median :0.0000 Median :0 Median :1.0000
## Mean :0.009375 Mean :0.2906 Mean :0 Mean :0.9531
## 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:0 3rd Qu.:1.0000
## Max. :1.000000 Max. :7.0000 Max. :0 Max. :1.0000
## repstatus2 fecstatus2 matstatus2 alive2 stage2
## Min. :0.0000 Min. :0.0000 Min. :1 Min. :1 Length:320
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character
## Median :0.0000 Median :0.0000 Median :1 Median :1 Mode :character
## Mean :0.3688 Mean :0.1562 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :1.0000 Max. :1 Max. :1
## stage2index sizea3 sizeb3 sizec3
## Min. : 0.00 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.00 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median :10.00 Median :0.000000 Median : 0.000 Median : 1.000
## Mean :17.89 Mean :0.009375 Mean : 1.069 Mean : 2.209
## 3rd Qu.:32.00 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :54.00 Max. :1.000000 Max. :18.000 Max. :13.000
## size3added repstra3 repstrb3 feca3
## Min. : 0.000 Min. : 0.000 Min. :0.000000 Min. :0.0000
## 1st Qu.: 1.000 1st Qu.: 0.000 1st Qu.:0.000000 1st Qu.:0.0000
## Median : 2.000 Median : 0.000 Median :0.000000 Median :0.0000
## Mean : 3.288 Mean : 1.069 Mean :0.009375 Mean :0.4562
## 3rd Qu.: 4.000 3rd Qu.: 1.000 3rd Qu.:0.000000 3rd Qu.:0.0000
## Max. :24.000 Max. :18.000 Max. :1.000000 Max. :8.0000
## juvgiven3 obsstatus3 repstatus3 fecstatus3 matstatus3
## Min. :0 Min. :0.0 Min. :0.0 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0 Median :0.0 Median :0.0000 Median :1
## Mean :0 Mean :0.9 Mean :0.4 Mean :0.2219 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0 Max. :1.0 Max. :1.0000 Max. :1
## alive3 stage3 stage3index
## Min. :0.0000 Length:320 Min. : 0.00
## 1st Qu.:1.0000 Class :character 1st Qu.: 7.00
## Median :1.0000 Mode :character Median :10.00
## Mean :0.9469 Mean :18.29
## 3rd Qu.:1.0000 3rd Qu.:33.00
## Max. :1.0000 Max. :54.00We will do one further standardization to the Cypripedium dataset. This time, to illustrate how we might use individual or environmental covariates in function-based MPM / IPM projection, we will incorporate some climatic covariates into the dataset. Here, we create a new copy of the original vertical dataset, add new variables coding for our climatic variables, and standardize. Note that we will only do this for the function-based case.
cypdata_env <- cypdata
cypdata_env$prec.04 <- 92.2
cypdata_env$prec.05 <- 57.6
cypdata_env$prec.06 <- 96.0
cypdata_env$prec.07 <- 109.8
cypdata_env$prec.08 <- 111.9
cypdata_env$prec.09 <- 106.8
cypfb_env <- verticalize3(data = cypdata_env, noyears = 6, firstyear = 2004,
patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
indcovacol = c("prec.04", "prec.05", "prec.06", "prec.07", "prec.08", "prec.09"),
stageassign = cypframe_fb, stagesize = "sizeadded", NAas0 = TRUE,
age_offset = 4)
summary_hfv(cypfb_env)##
## This hfv dataset contains 320 rows, 57 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.
## rowid popid patchid individ year2
## Min. : 1.00 :320 A: 93 Length:320 Min. :2004
## 1st Qu.:21.00 B:154 Class :character 1st Qu.:2005
## Median :37.50 C: 73 Mode :character Median :2006
## Mean :38.45 Mean :2006
## 3rd Qu.:56.00 3rd Qu.:2007
## Max. :77.00 Max. :2008
## firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 indcova1
## Min. : 0.0000 Min. :0.000000 Min. :0.0000 Min. : 0.00
## 1st Qu.: 0.0000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.: 57.60
## Median : 0.0000 Median :0.000000 Median :0.0000 Median : 92.20
## Mean : 0.7469 Mean :0.009375 Mean :0.2656 Mean : 70.64
## 3rd Qu.: 1.0000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.: 96.00
## Max. :18.0000 Max. :1.000000 Max. :7.0000 Max. :109.80
## juvgiven1 obsstatus1 repstatus1 fecstatus1
## Min. :0 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :0 Median :1.0000 Median :0.0000 Median :0.0000
## Mean :0 Mean :0.7469 Mean :0.2875 Mean :0.1344
## 3rd Qu.:0 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000
## Max. :0 Max. :1.0000 Max. :1.0000 Max. :1.0000
## matstatus1 alive1 stage1 stage1index
## Min. :0.0000 Min. :0.0000 Length:320 Min. : 0.00
## 1st Qu.:1.0000 1st Qu.:1.0000 Class :character 1st Qu.: 6.00
## Median :1.0000 Median :1.0000 Mode :character Median : 8.00
## Mean :0.7688 Mean :0.7688 Mean :14.17
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:31.00
## Max. :1.0000 Max. :1.0000 Max. :51.00
## sizea2 sizeb2 sizec2 size2added
## Min. :0.000000 Min. : 0.0000 Min. : 0.000 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000
## Median :0.000000 Median : 0.0000 Median : 2.000 Median : 2.000
## Mean :0.009375 Mean : 0.8969 Mean : 2.416 Mean : 3.322
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.000 Max. :24.000
## repstra2 repstrb2 feca2 indcova2
## Min. : 0.0000 Min. :0.000000 Min. :0.0000 Min. : 57.60
## 1st Qu.: 0.0000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.: 92.20
## Median : 0.0000 Median :0.000000 Median :0.0000 Median : 96.00
## Mean : 0.8969 Mean :0.009375 Mean :0.2906 Mean : 92.77
## 3rd Qu.: 1.0000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:109.80
## Max. :18.0000 Max. :1.000000 Max. :7.0000 Max. :111.90
## juvgiven2 obsstatus2 repstatus2 fecstatus2 matstatus2
## Min. :0 Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0000 Median :0.0000 Median :0.0000 Median :1
## Mean :0 Mean :0.9531 Mean :0.3688 Mean :0.1562 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1
## alive2 stage2 stage2index sizea3
## Min. :1 Length:320 Min. : 6.00 Min. :0.000000
## 1st Qu.:1 Class :character 1st Qu.: 7.00 1st Qu.:0.000000
## Median :1 Mode :character Median :10.00 Median :0.000000
## Mean :1 Mean :18.17 Mean :0.009375
## 3rd Qu.:1 3rd Qu.:32.00 3rd Qu.:0.000000
## Max. :1 Max. :54.00 Max. :1.000000
## sizeb3 sizec3 size3added repstra3
## Min. : 0.000 Min. : 0.000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 0.000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.: 0.000
## Median : 0.000 Median : 1.000 Median : 2.000 Median : 0.000
## Mean : 1.069 Mean : 2.209 Mean : 3.288 Mean : 1.069
## 3rd Qu.: 1.000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.: 1.000
## Max. :18.000 Max. :13.000 Max. :24.000 Max. :18.000
## repstrb3 feca3 indcova3 juvgiven3 obsstatus3
## Min. :0.000000 Min. :0.0000 Min. : 57.6 Min. :0 Min. :0.0
## 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.: 96.0 1st Qu.:0 1st Qu.:1.0
## Median :0.000000 Median :0.0000 Median :106.8 Median :0 Median :1.0
## Mean :0.009375 Mean :0.4562 Mean : 96.1 Mean :0 Mean :0.9
## 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:109.8 3rd Qu.:0 3rd Qu.:1.0
## Max. :1.000000 Max. :8.0000 Max. :111.9 Max. :0 Max. :1.0
## repstatus3 fecstatus3 matstatus3 alive3
## Min. :0.0 Min. :0.0000 Min. :1 Min. :0.0000
## 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1.0000
## Median :0.0 Median :0.0000 Median :1 Median :1.0000
## Mean :0.4 Mean :0.2219 Mean :1 Mean :0.9469
## 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1.0000
## Max. :1.0 Max. :1.0000 Max. :1 Max. :1.0000
## stage3 stage3index
## Length:320 Min. : 0.00
## Class :character 1st Qu.: 7.00
## Mode :character Median :10.00
## Mean :18.57
## 3rd Qu.:33.00
## Max. :54.00Now we have a new version of the standardized dataset that includes
the variables indcova1, indcova2, ans
indcova3, which correspond to total annual precipitation in
times t-1, t, and t+1, respectively.
Finally, let’s standardize the Lathyrus dataset for use in IPM
development. Prior to doing that, we need to make a small adjustment to
the dataset, by adding an individual identifier variable. Currently, the
GENET variable serves as an individual identifier, but only
within the patch, leading to some individual identifiers being reused
across patches. To make sure that individuals are treated as unique, we
create a new variable that combines the patch name and the individual
name into one unique identifier per plant.
lathyrus$indiv_id <- paste(lathyrus$SUBPLOT, lathyrus$GENET)
lathvert_ipm <- verticalize3(lathyrus, noyears = 4, firstyear = 1988,
individcol = "indiv_id", blocksize = 9, juvcol = "Seedling1988",
sizeacol = "Volume88", repstracol = "FCODE88", fecacol = "Intactseed88",
deadacol = "Dead1988", nonobsacol = "Dormant1988", stageassign = lathframe_ipm,
stagesize = "sizea", censorcol = "Missing1988", censorkeep = NA,
censorRepeat = TRUE, censor = TRUE, NAas0 = TRUE, NRasRep = TRUE)
summary_hfv(lathvert_ipm, full = FALSE)##
## This hfv dataset contains 2527 rows, 42 variables, 1 population,
## 1 patch, 1053 individuals, and 3 time steps.Step 2a.1. Vertical dataset organization
We may also wish to see how to proceed if our original dataset is
already in vertical, but ahistorical, format. This package also includes
dataset cypvert, which is the same dataset as
cypdata but set in ahistorical vertical format. Here, we
will use the historicalize3() function to deal with this
dataset, which uses an individual identity variable in order to put all
of the data for each individual together. Here, the vertical dataset
includes the plantid variable, which is an individual
identity term and must be supplied for conversion. Here is the raw MPM
dataset.
cypraw_v2 <- historicalize3(data = cypvert, patchidcol = "patch",
individcol = "plantid", year2col = "year2", sizea2col = "Inf2.2",
sizea3col = "Inf2.3", sizeb2col = "Inf.2", sizeb3col = "Inf.3",
sizec2col = "Veg.2", sizec3col = "Veg.3", repstra2col = "Inf2.2",
repstra3col = "Inf2.3", repstrb2col = "Inf.2", repstrb3col = "Inf.3",
feca2col = "Pod.2", feca3col = "Pod.3", repstrrel = 2,
stageassign = cypframe_raw, stagesize = "sizeadded", censorcol = "censor",
censorkeep = 1, censor = FALSE, age_offset = 4, NAas0 = TRUE, NRasRep = TRUE,
reduce = TRUE)
summary_hfv(cypraw_v2)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.
## rowid popid patchid individ
## Min. : 0.00 Length:320 Length:320 Length:320
## 1st Qu.: 79.75 Class :character Class :character Class :character
## Median :159.50 Mode :character Mode :character Mode :character
## Mean :159.70
## 3rd Qu.:239.25
## Max. :321.00
## year2 firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2005 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2006 Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2006 Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2007 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 juvgiven1
## Min. :0.000000 Min. : 0.0000 Min. :0.0000 Min. :0
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.:0.0000 1st Qu.:0
## Median :0.000000 Median : 0.0000 Median :0.0000 Median :0
## Mean :0.009375 Mean : 0.7469 Mean :0.2656 Mean :0
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :1.000000 Max. :18.0000 Max. :7.0000 Max. :0
## obsstatus1 repstatus1 fecstatus1 matstatus1
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1.0000
## Median :1.0000 Median :0.0000 Median :0.0000 Median :1.0000
## Mean :0.7469 Mean :0.2875 Mean :0.1344 Mean :0.7688
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## alive1 stage1 stage1index sizea2
## Min. :0.0000 Length:320 Min. : 0.000 Min. :0.000000
## 1st Qu.:1.0000 Class :character 1st Qu.: 6.000 1st Qu.:0.000000
## Median :1.0000 Mode :character Median : 8.000 Median :0.000000
## Mean :0.7688 Mean : 6.144 Mean :0.009375
## 3rd Qu.:1.0000 3rd Qu.: 8.000 3rd Qu.:0.000000
## Max. :1.0000 Max. :11.000 Max. :1.000000
## sizeb2 sizec2 size2added repstra2
## Min. : 0.0000 Min. : 0.000 Min. : 0.000 Min. :0.000000
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.:0.000000
## Median : 0.0000 Median : 2.000 Median : 2.000 Median :0.000000
## Mean : 0.8969 Mean : 2.416 Mean : 3.322 Mean :0.009375
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.:0.000000
## Max. :18.0000 Max. :13.000 Max. :24.000 Max. :1.000000
## repstrb2 feca2 juvgiven2 obsstatus2
## Min. : 0.0000 Min. :0.0000 Min. :0 Min. :0.0000
## 1st Qu.: 0.0000 1st Qu.:0.0000 1st Qu.:0 1st Qu.:1.0000
## Median : 0.0000 Median :0.0000 Median :0 Median :1.0000
## Mean : 0.8969 Mean :0.2906 Mean :0 Mean :0.9531
## 3rd Qu.: 1.0000 3rd Qu.:0.0000 3rd Qu.:0 3rd Qu.:1.0000
## Max. :18.0000 Max. :7.0000 Max. :0 Max. :1.0000
## repstatus2 fecstatus2 matstatus2 alive2 stage2
## Min. :0.0000 Min. :0.0000 Min. :1 Min. :1 Length:320
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character
## Median :0.0000 Median :0.0000 Median :1 Median :1 Mode :character
## Mean :0.3688 Mean :0.1562 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :1.0000 Max. :1 Max. :1
## stage2index sizea3 sizeb3 sizec3
## Min. : 6.000 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.000 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median : 8.000 Median :0.000000 Median : 0.000 Median : 1.000
## Mean : 7.919 Mean :0.009375 Mean : 1.069 Mean : 2.209
## 3rd Qu.: 8.000 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :11.000 Max. :1.000000 Max. :18.000 Max. :13.000
## size3added repstra3 repstrb3 feca3
## Min. : 0.000 Min. :0.000000 Min. : 0.000 Min. :0.0000
## 1st Qu.: 1.000 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.:0.0000
## Median : 2.000 Median :0.000000 Median : 0.000 Median :0.0000
## Mean : 3.288 Mean :0.009375 Mean : 1.069 Mean :0.4562
## 3rd Qu.: 4.000 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.:0.0000
## Max. :24.000 Max. :1.000000 Max. :18.000 Max. :8.0000
## juvgiven3 obsstatus3 repstatus3 fecstatus3 matstatus3
## Min. :0 Min. :0.0 Min. :0.0 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0 Median :0.0 Median :0.0000 Median :1
## Mean :0 Mean :0.9 Mean :0.4 Mean :0.2219 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0 Max. :1.0 Max. :1.0000 Max. :1
## alive3 stage3 stage3index
## Min. :0.0000 Length:320 Min. : 0.000
## 1st Qu.:1.0000 Class :character 1st Qu.: 7.000
## Median :1.0000 Mode :character Median : 8.000
## Mean :0.9469 Mean : 7.544
## 3rd Qu.:1.0000 3rd Qu.: 8.000
## Max. :1.0000 Max. :11.000Let’s also create the function-based MPM version.
cypfb_v2 <- historicalize3(data = cypvert, patchidcol = "patch",
individcol = "plantid", year2col = "year2", sizea2col = "Inf2.2",
sizea3col = "Inf2.3", sizeb2col = "Inf.2", sizeb3col = "Inf.3",
sizec2col = "Veg.2", sizec3col = "Veg.3", repstra2col = "Inf2.2",
repstra3col = "Inf2.3", repstrb2col = "Inf.2", repstrb3col = "Inf.3",
feca2col = "Pod.2", feca3col = "Pod.3", repstrrel = 2,
stageassign = cypframe_fb, stagesize = "sizeadded", censorcol = "censor",
censorkeep = 1, censor = FALSE, age_offset = 4, NAas0 = TRUE, reduce = TRUE)
summary_hfv(cypfb_v2)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.
## rowid popid patchid individ
## Min. : 0.00 Length:320 Length:320 Length:320
## 1st Qu.: 79.75 Class :character Class :character Class :character
## Median :159.50 Mode :character Mode :character Mode :character
## Mean :159.70
## 3rd Qu.:239.25
## Max. :321.00
## year2 firstseen lastseen obsage obslifespan
## Min. :2004 Min. :2004 Min. :2004 Min. :5.000 Min. :0.000
## 1st Qu.:2005 1st Qu.:2004 1st Qu.:2009 1st Qu.:6.000 1st Qu.:5.000
## Median :2006 Median :2004 Median :2009 Median :7.000 Median :5.000
## Mean :2006 Mean :2004 Mean :2009 Mean :6.853 Mean :4.556
## 3rd Qu.:2007 3rd Qu.:2004 3rd Qu.:2009 3rd Qu.:8.000 3rd Qu.:5.000
## Max. :2008 Max. :2008 Max. :2009 Max. :9.000 Max. :5.000
## sizea1 sizeb1 sizec1 size1added
## Min. :0.000000 Min. : 0.0000 Min. : 0.0 Min. : 0.000
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.: 0.0 1st Qu.: 0.000
## Median :0.000000 Median : 0.0000 Median : 1.0 Median : 2.000
## Mean :0.009375 Mean : 0.7469 Mean : 1.9 Mean : 2.656
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.: 3.0 3rd Qu.: 4.000
## Max. :1.000000 Max. :18.0000 Max. :13.0 Max. :21.000
## repstra1 repstrb1 feca1 juvgiven1
## Min. :0.000000 Min. : 0.0000 Min. :0.0000 Min. :0
## 1st Qu.:0.000000 1st Qu.: 0.0000 1st Qu.:0.0000 1st Qu.:0
## Median :0.000000 Median : 0.0000 Median :0.0000 Median :0
## Mean :0.009375 Mean : 0.7469 Mean :0.2656 Mean :0
## 3rd Qu.:0.000000 3rd Qu.: 1.0000 3rd Qu.:0.0000 3rd Qu.:0
## Max. :1.000000 Max. :18.0000 Max. :7.0000 Max. :0
## obsstatus1 repstatus1 fecstatus1 matstatus1
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1.0000
## Median :1.0000 Median :0.0000 Median :0.0000 Median :1.0000
## Mean :0.7469 Mean :0.2875 Mean :0.1344 Mean :0.7688
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## alive1 stage1 stage1index sizea2
## Min. :0.0000 Length:320 Min. : 0.00 Min. :0.000000
## 1st Qu.:1.0000 Class :character 1st Qu.: 6.00 1st Qu.:0.000000
## Median :1.0000 Mode :character Median : 8.00 Median :0.000000
## Mean :0.7688 Mean :14.17 Mean :0.009375
## 3rd Qu.:1.0000 3rd Qu.:31.00 3rd Qu.:0.000000
## Max. :1.0000 Max. :51.00 Max. :1.000000
## sizeb2 sizec2 size2added repstra2
## Min. : 0.0000 Min. : 0.000 Min. : 0.000 Min. :0.000000
## 1st Qu.: 0.0000 1st Qu.: 1.000 1st Qu.: 1.000 1st Qu.:0.000000
## Median : 0.0000 Median : 2.000 Median : 2.000 Median :0.000000
## Mean : 0.8969 Mean : 2.416 Mean : 3.322 Mean :0.009375
## 3rd Qu.: 1.0000 3rd Qu.: 3.000 3rd Qu.: 4.000 3rd Qu.:0.000000
## Max. :18.0000 Max. :13.000 Max. :24.000 Max. :1.000000
## repstrb2 feca2 juvgiven2 obsstatus2
## Min. : 0.0000 Min. :0.0000 Min. :0 Min. :0.0000
## 1st Qu.: 0.0000 1st Qu.:0.0000 1st Qu.:0 1st Qu.:1.0000
## Median : 0.0000 Median :0.0000 Median :0 Median :1.0000
## Mean : 0.8969 Mean :0.2906 Mean :0 Mean :0.9531
## 3rd Qu.: 1.0000 3rd Qu.:0.0000 3rd Qu.:0 3rd Qu.:1.0000
## Max. :18.0000 Max. :7.0000 Max. :0 Max. :1.0000
## repstatus2 fecstatus2 matstatus2 alive2 stage2
## Min. :0.0000 Min. :0.0000 Min. :1 Min. :1 Length:320
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1 1st Qu.:1 Class :character
## Median :0.0000 Median :0.0000 Median :1 Median :1 Mode :character
## Mean :0.3688 Mean :0.1562 Mean :1 Mean :1
## 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:1 3rd Qu.:1
## Max. :1.0000 Max. :1.0000 Max. :1 Max. :1
## stage2index sizea3 sizeb3 sizec3
## Min. : 6.00 Min. :0.000000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.00 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.: 1.000
## Median :10.00 Median :0.000000 Median : 0.000 Median : 1.000
## Mean :18.17 Mean :0.009375 Mean : 1.069 Mean : 2.209
## 3rd Qu.:32.00 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.: 3.000
## Max. :54.00 Max. :1.000000 Max. :18.000 Max. :13.000
## size3added repstra3 repstrb3 feca3
## Min. : 0.000 Min. :0.000000 Min. : 0.000 Min. :0.0000
## 1st Qu.: 1.000 1st Qu.:0.000000 1st Qu.: 0.000 1st Qu.:0.0000
## Median : 2.000 Median :0.000000 Median : 0.000 Median :0.0000
## Mean : 3.288 Mean :0.009375 Mean : 1.069 Mean :0.4562
## 3rd Qu.: 4.000 3rd Qu.:0.000000 3rd Qu.: 1.000 3rd Qu.:0.0000
## Max. :24.000 Max. :1.000000 Max. :18.000 Max. :8.0000
## juvgiven3 obsstatus3 repstatus3 fecstatus3 matstatus3
## Min. :0 Min. :0.0 Min. :0.0 Min. :0.0000 Min. :1
## 1st Qu.:0 1st Qu.:1.0 1st Qu.:0.0 1st Qu.:0.0000 1st Qu.:1
## Median :0 Median :1.0 Median :0.0 Median :0.0000 Median :1
## Mean :0 Mean :0.9 Mean :0.4 Mean :0.2219 Mean :1
## 3rd Qu.:0 3rd Qu.:1.0 3rd Qu.:1.0 3rd Qu.:0.0000 3rd Qu.:1
## Max. :0 Max. :1.0 Max. :1.0 Max. :1.0000 Max. :1
## alive3 stage3 stage3index
## Min. :0.0000 Length:320 Min. : 0.00
## 1st Qu.:1.0000 Class :character 1st Qu.: 7.00
## Median :1.0000 Mode :character Median :10.00
## Mean :0.9469 Mean :18.57
## 3rd Qu.:1.0000 3rd Qu.:33.00
## Max. :1.0000 Max. :54.00We can compare the dimensions of these datasets.
summary_hfv(cypraw_v1, full = FALSE)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.summary_hfv(cypraw_v2, full = FALSE)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.summary_hfv(cypfb_v1, full = FALSE)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.summary_hfv(cypfb_v2, full = FALSE)##
## This hfv dataset contains 320 rows, 54 variables, 1 population,
## 3 patches, 74 individuals, and 5 time steps.The lengths of the datasets are the same in terms of rows and columns, and the variables and data are the same although the order of the columns and rows might not match (see the summaries for comparison).
Let’s now move on to supplying R with the supplemental transitions that we need to properly parameterize our models.
Step 2b. Provide supplemental information for matrix estimation
The next steps involve adding some external data to parameterize the
matrices properly. We will accomplish this with the
supplemental() function. The supplemental()
function provides a means of inputting four kinds of data into MPM
construction:
1. fixed transition values derived from other studies and added as constants to matrices,
2. proxy transition values when data for particular transitions does not exist and other, estimable transitions will be used as proxies,
3. multipliers on proxy transition values, for example to set survival transitions to some fraction of the survival transitions estimated for other, particular transitions, and
4. reproductive multipliers to indicate which stages lead to the production of which stages, and at what level relative to estimated fecundity.
We will start off by creating two supplemental tables taking all of these sorts of data for the raw Cypripedium MPMs. Technically, this first supplement table will be for the ahistorical and age-by-stage raw MPMs.
seeds_per_pod <- 5000
cypsupp2_raw <- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
"XSm", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "rep", "rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "XSm", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "XSm", "XSm", NA, NA),
givenrate = c(0.03, 0.15, 0.1, 0.1, 0.1, 0.05, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, (0.5 * seeds_per_pod),
(0.5 * seeds_per_pod)),
type =c(1, 1, 1, 1, 1, 1, 1, 1, 3, 3),
stageframe = cypframe_raw, historical = FALSE)
cypsupp2_raw## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 SD SD <NA> <NA> <NA> <NA> 0.03 1
## 2 P1 SD <NA> <NA> <NA> <NA> 0.15 1
## 3 P2 P1 <NA> <NA> <NA> <NA> 0.10 1
## 4 P3 P2 <NA> <NA> <NA> <NA> 0.10 1
## 5 SL P3 <NA> <NA> <NA> <NA> 0.10 1
## 6 SL SL <NA> <NA> <NA> <NA> 0.05 1
## 7 D SL <NA> D XSm <NA> NA 1
## 8 XSm SL <NA> XSm XSm <NA> NA 1
## 9 SD rep <NA> <NA> <NA> <NA> NA 2500
## 10 P1 rep <NA> <NA> <NA> <NA> NA 2500
## convtype convtype_t12
## 1 1 1
## 2 1 1
## 3 1 1
## 4 1 1
## 5 1 1
## 6 1 1
## 7 1 1
## 8 1 1
## 9 3 1
## 10 3 1In the above table, we provide information on more than ten general
transitions. The transitions themselves are shown in the
stage2 and stage3 vectors. The first
transition corresponds to the first element in each vector. Since the
first element of stage2 is "SD", and the first
element of stage3 is "SD", the first
transition is the stasis transition for dormant seeds. All elements in
these vectors correspond to stages in the stageframe being used, with
the exceptions of some specific keywords, such as "mat",
"rep", etc. These keywords refer to groups of stages. For
example, "rep" is shorthand for all reproductive stages,
while "mat" is shorthand for all mature stages. Other
keywords exist, and the help file for function
supplemental() defines these. Thus, the transition from
"rep" to "SD" is actually shorthand for all
transitions from reproductive stages to the dormant seed stage.
The first six transitions of the above table are replaced by specific
constants, given as 0.03, 0.15, 0.1, etc. However, the seventh and
eighth transitions listed are to be replaced with proxy values, which
are transitions values estimated for other transitions. The values to
use are provided in the respective places in the eststage2
and eststage3 vectors. Thus, the transitions from
SL to D and from SL to
XSm should be replaced with the values estimated for the
transitions from XSm to D and XSm
to XSm, respectively. The final two transitions, which are
the fecundity steps corresponding to the production of dormant seeds and
first-year protocorms by reproductive individuals, are shown to need to
be multiplied by 0.5 * the number of seeds per pod, via fecundity
multipliers.
Historical MPMs are composed of transitions across three times, and so supplement tables for historical MPMs need to include stage information for three times in all transitions to be worked with. Here, we show a historical supplement table for the raw Cypripedium MPM case.
cypsupp3_raw <- supplemental(stage3 = c("SD", "SD", "P1", "P1", "P2", "P3",
"SL", "SL", "SL", "D", "D", "SD", "P1"),
stage2 = c("SD", "SD", "SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL",
"rep", "rep"),
stage1 = c("SD", "rep", "SD", "rep", "SD", "P1", "P2", "P3", "SL", "P3",
"SL", "mat", "mat"),
eststage3 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "XSm", "D", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "XSm", "XSm", NA, NA),
eststage1 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "XSm", "XSm", NA, NA),
givenrate = c(0.01, 0.05, 0.10, 0.20, 0.1, 0.1, 0.05, 0.05, 0.05, NA, NA,
NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
(0.5 * seeds_per_pod), (0.5 * seeds_per_pod)),
type = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3),
type_t12 = c(1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1),
stageframe = cypframe_raw, historical = TRUE)
cypsupp3_raw## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 SD SD SD <NA> <NA> <NA> 0.01 1
## 2 SD SD rep <NA> <NA> <NA> 0.05 1
## 3 P1 SD SD <NA> <NA> <NA> 0.10 1
## 4 P1 SD rep <NA> <NA> <NA> 0.20 1
## 5 P2 P1 SD <NA> <NA> <NA> 0.10 1
## 6 P3 P2 P1 <NA> <NA> <NA> 0.10 1
## 7 SL P3 P2 <NA> <NA> <NA> 0.05 1
## 8 SL SL P3 <NA> <NA> <NA> 0.05 1
## 9 SL SL SL <NA> <NA> <NA> 0.05 1
## 10 D SL P3 XSm XSm XSm NA 1
## 11 D SL SL D XSm XSm NA 1
## 12 SD rep mat <NA> <NA> <NA> NA 2500
## 13 P1 rep mat <NA> <NA> <NA> NA 2500
## convtype convtype_t12
## 1 1 1
## 2 1 2
## 3 1 1
## 4 1 2
## 5 1 1
## 6 1 1
## 7 1 1
## 8 1 1
## 9 1 1
## 10 1 1
## 11 1 1
## 12 3 1
## 13 3 1Notice that there is a different total number of transitions provided here - there were only 10 transitions detailed in the ahistorical case, but 13 are given here. This is due to the fact that the stage at time t-1 needs to be specified for each transition, and sometimes this requires extra detail to set up properly. For example, a dormant seed in time t that stayed dormant in time t+1 might have also been dormant in time t-1, or might have been produced by a reproductive plant right after the monitoring occasion in time t-1.
Let’s now provide the supplement tables for the function-based ahistorical and historical MPMs. The ahistorical supplement will also be used to create function-based age-by-stage MPMs.
cypsupp2_fb <- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
"V1", "V2", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL", "rep",
"rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "V1", "V2", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "D", "D", "D", NA, NA),
givenrate = c(0.03, 0.15, 0.1, 0.1, 0.1, 0.05, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, (0.5* seeds_per_pod),
(0.5 * seeds_per_pod)),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "R", "R"),
stageframe = cypframe_fb, historical = FALSE)
cypsupp2_fb## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 SD SD <NA> <NA> <NA> <NA> 0.03 1
## 2 P1 SD <NA> <NA> <NA> <NA> 0.15 1
## 3 P2 P1 <NA> <NA> <NA> <NA> 0.10 1
## 4 P3 P2 <NA> <NA> <NA> <NA> 0.10 1
## 5 SL P3 <NA> <NA> <NA> <NA> 0.10 1
## 6 SL SL <NA> <NA> <NA> <NA> 0.05 1
## 7 D SL <NA> D D <NA> NA 1
## 8 V1 SL <NA> V1 D <NA> NA 1
## 9 V2 SL <NA> V2 D <NA> NA 1
## 10 SD rep <NA> <NA> <NA> <NA> NA 2500
## 11 P1 rep <NA> <NA> <NA> <NA> NA 2500
## convtype convtype_t12
## 1 1 1
## 2 1 1
## 3 1 1
## 4 1 1
## 5 1 1
## 6 1 1
## 7 1 1
## 8 1 1
## 9 1 1
## 10 3 1
## 11 3 1cypsupp3_fb <- supplemental(stage3 = c("SD", "SD", "P1", "P1", "P2", "P3", "SL",
"SL", "SL", "D", "V1", "D", "V1", "V2", "SD", "P1"),
stage2 = c("SD", "SD", "SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL",
"SL", "SL", "SL", "rep", "rep"),
stage1 = c("SD", "rep", "SD", "rep", "SD", "P1", "P2", "P3", "SL", "P3", "P3",
"SL", "SL", "SL", "mat", "mat"),
eststage3 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "V1", "D", "V1",
"V2", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D",
"D", NA, NA),
eststage1 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D",
"D", NA, NA),
givenrate = c(0.01, 0.05, 0.10, 0.20, 0.1, 0.1, 0.05, 0.05, 0.05, NA, NA, NA,
NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
(0.5 * seeds_per_pod), (0.5 * seeds_per_pod)),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"R", "R"),
type_t12 = c("S", "F", "S", "F", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"S", "S", "S"), stageframe = cypframe_fb)
cypsupp3_fb## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 SD SD SD <NA> <NA> <NA> 0.01 1
## 2 SD SD rep <NA> <NA> <NA> 0.05 1
## 3 P1 SD SD <NA> <NA> <NA> 0.10 1
## 4 P1 SD rep <NA> <NA> <NA> 0.20 1
## 5 P2 P1 SD <NA> <NA> <NA> 0.10 1
## 6 P3 P2 P1 <NA> <NA> <NA> 0.10 1
## 7 SL P3 P2 <NA> <NA> <NA> 0.05 1
## 8 SL SL P3 <NA> <NA> <NA> 0.05 1
## 9 SL SL SL <NA> <NA> <NA> 0.05 1
## 10 D SL P3 D D D NA 1
## 11 V1 SL P3 V1 D D NA 1
## 12 D SL SL D D D NA 1
## 13 V1 SL SL V1 D D NA 1
## 14 V2 SL SL V2 D D NA 1
## 15 SD rep mat <NA> <NA> <NA> NA 2500
## 16 P1 rep mat <NA> <NA> <NA> NA 2500
## convtype convtype_t12
## 1 1 1
## 2 1 2
## 3 1 1
## 4 1 2
## 5 1 1
## 6 1 1
## 7 1 1
## 8 1 1
## 9 1 1
## 10 1 1
## 11 1 1
## 12 1 1
## 13 1 1
## 14 1 1
## 15 3 1
## 16 3 1Finally, let’s provide the supplemental data for the Lathyrus IPM.
lathsupp2 <- supplemental(stage3 = c("Sd", "Sdl", "Sd", "Sdl"),
stage2 = c("Sd", "Sd", "rep", "rep"),
givenrate = c(0.345, 0.054, NA, NA),
multiplier = c(NA, NA, 0.345, 0.054),
type = c(1, 1, 3, 3), stageframe = lathframe_ipm, historical = FALSE)
lathsupp2## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 Sd Sd <NA> <NA> <NA> <NA> 0.345 1.000
## 2 Sdl Sd <NA> <NA> <NA> <NA> 0.054 1.000
## 3 Sd rep <NA> <NA> <NA> <NA> NA 0.345
## 4 Sdl rep <NA> <NA> <NA> <NA> NA 0.054
## convtype convtype_t12
## 1 1 1
## 2 1 1
## 3 3 1
## 4 3 1lathsupp3 <- supplemental(stage3 = c("Sd", "Sd", "Sdl", "Sdl", "npr", "Sd", "Sdl"),
stage2 = c("Sd", "Sd", "Sd", "Sd", "Sdl", "rep", "rep"),
stage1 = c("Sd", "rep", "Sd", "rep", "Sd", "mat", "mat"),
eststage3 = c(NA, NA, NA, NA, "npr", NA, NA),
eststage2 = c(NA, NA, NA, NA, "Sdl", NA, NA),
eststage1 = c(NA, NA, NA, NA, "Sdl", NA, NA),
givenrate = c(0.345, 0.345, 0.054, 0.054, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, 0.345, 0.054),
type = c(1, 1, 1, 1, 1, 3, 3), type_t12 = c(1, 2, 1, 2, 1, 1, 1),
stageframe = lathframe_ipm, historical = TRUE)
lathsupp3## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 Sd Sd Sd <NA> <NA> <NA> 0.345 1.000
## 2 Sd Sd rep <NA> <NA> <NA> 0.345 1.000
## 3 Sdl Sd Sd <NA> <NA> <NA> 0.054 1.000
## 4 Sdl Sd rep <NA> <NA> <NA> 0.054 1.000
## 5 npr Sdl Sd npr Sdl Sdl NA 1.000
## 6 Sd rep mat <NA> <NA> <NA> NA 0.345
## 7 Sdl rep mat <NA> <NA> <NA> NA 0.054
## convtype convtype_t12
## 1 1 1
## 2 1 2
## 3 1 1
## 4 1 2
## 5 1 1
## 6 3 1
## 7 3 1Step 3. Vital rate modeling (for fbMPMs and IPMs only)
Matrix creation can proceed either as raw (i.e. empirical) matrix creation, as initially outlined in Ehrlén 2000, or via the creation of function-based matrices, in many ways equivalent to complex integral projection models per Ellner and Rees (2006) and as further described in the non-Gaussian case in Shefferson, Warren, and Pulliam (2014). The function-based approach requires the development of vital rate models, and these vital rate models serve not only to parameterize these matrices but also to allow us to test whether history is important to population dynamics in our study system. The importance of other factors, such as size, reproductive history, and age, can also be tested. We will proceed by developing vital rate models that test for history, and use these both to understand the overall demography of the system and to parameterize our function-based MPMs.
Prior to vital rate estimation, a number of key decisions need to be made regarding the assumptions underlying the vital rates, and their relationships with the factors under investigation. These decisions include the general modeling strategy, and the size and fecundity distributions.
Step 3a. General modeling strategy
Most function-based matrices, whether integral projection models or otherwise, are developed using either a generalized linear modeling (GLM) strategy, or a generalized linear mixed modeling (GLMM) strategy. The former is more common and is simpler, but the latter is more theoretically sound because it provides a means of correcting the lack of independence inherent in datasets incorporating repeated sampling of the same individuals. The difference between the two with regards to vital rate modeling is strongly related to assumptions regarding the individual and spatiotemporal variation in vital rates.
In both GLM and GLMM-based MPMs, the underlying dataset utilized is a vertical dataset. Each row of data gives the state of the individual in either two consecutive occasions (the ahistorical case), or three consecutive occasions (the historical case). Under a GLM framework, time in occasion t is a fixed categorical variable, and individual identity is ignored. Treating time as fixed implies that the actual monitoring occasions are the only times for which inference is sought. Thus, if time is treated as fixed, then it would not be correct to infer future population dynamics after 2009 for a dataset collected between 2004 and 2009. Ignoring individual identity treats all transitions as independent, even though data originating from the same sampled individual is clearly not independent of that individual’s previous transitions. This may be interpreted as a form of pseudoreplication because strongly related data is used to create matrices that are assumed to be statistically independent. This might impact demographic modeling by inflating Type 1 error in the linear modeling, yielding more significant terms in the chosen best-fit model and causing the retention of more terms than is warranted.
Under a GLMM (generalized linear mixed model) framework, both time and individual identity can be treated as random categorical terms. This has two major implications. First, both time and individual can be assumed to be random samples from a broader population of times and individuals for which inference is sought. Thus, sampled monitoring occasions represent a greater universe of years for which inference can be made, and so their associated coefficients can be assumed to come from a normal distribution with \(mean = 0\). Second, treating individual as a random categorical term eliminates the pseudoreplication that is inherent in the GLM approach to vital rate estimation when individuals are monitored potentially many times, because each individual is assumed to be randomly drawn and associated with its own response distribution. Subpopulations may also be considered random, in which case they are assumed to have been sampled from all possible spaces that the species might occupy. We encourage researchers to use the GLMM approach in their demographic work, but we have also included easy-to-use GLM functionality, since many will find the GLM approach particularly useful in cases where mixed modeling breaks down.
Step 3b. Size and fecundity distributions
Once a general approach is decided upon, the next step is to choose the underlying distributions. The probabilities of survival, observation, and reproductive status are automatically set to the binomial distribution, and this cannot be altered. However, the probability of size transition and the fecundity rate can be set to the Gaussian, gamma, Poisson, or negative binomial distributions, with zero-inflated and zero-truncated versions of the Poisson and negative binomial also available. If size or fecundity rate is a continuous variable (i.e., not an integer or count variable), then it should typically be set to the Gaussian distribution unless it has no zeros and appears dramatically right-skewed. However, if size or fecundity is a count variable, then it should be set to the Poisson distribution if the mean equals the variance. The negative binomial distribution is provided in cases where the assumption that the mean equals the variance is clearly broken. We do not encourage the use of the negative binomial except in such cases, as the extra parameters estimated for the negative binomial distribution reduce the power of the modeling exercises conducted.
The Poisson and the negative binomial distributions both predict specific numbers of zeros in the response variable. If excess zeros occur within the dataset, then a zero-inflated Poisson or negative binomial distribution may be used. These modeling approaches work by parameterizing a binomial model, typically with a logit link, to predict zero responses. The Poisson or negative binomial is then used to predict non-zero responses. This conditional model ends up really acting as two separate models in which zeros are assumed to be predicted under potentially different processes than the remaining counts. Users should be aware that, because an extra model is built to cover zeros, zero-inflated models are much more complex and can include many more parameters than their non-inflated counterparts. The principle of parsimony suggests that they should only be used when there are significantly more zeros than expected.
Cases may arise in which zeros do not exist in either size or
fecundity. For these situations, we provide zero-truncated
distributions. This may occur in size if all cases of
size = 0 are absorbed by observation status, leaving only
positive integers for the size of observed individuals. For example, if
an unobservable stage such as vegetative dormancy occurs and absorbs all
cases of size = 0, then a zero-truncated Poisson or
negative binomial distribution will be more appropriate than the
equivalent distribution without zero-truncation. It can also occur if
all cases of fecundity = 0 are absorbed by reproductive
status. Such distributions only involve the estimation of single,
conditional models, and so are simpler than zero-inflated models.
Package lefko3 includes a function that can help in
determining which distributions to use: hfv_qc(). Here, we
use it to explore the datasets that we have standardized. Let’s start
with the raw Cypripedium dataset, under ahistorical assumptions.
hfv_qc(data = cypraw_v1, vitalrates = c("surv", "obs", "size", "repst", "fec"),
suite = "full", size = c("size3added", "size2added", "size1added"))## Survival:
##
## Data subset has 55 variables and 320 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 55 variables and 303 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable size3added has 0 missing values.
## Variable size3added appears to be an integer variable.
##
## Variable size3added is fully positive, lacking even 0s.
##
## Overdispersion test:
## Mean size3added is 3.653
## The variance in size3added is 13.41
## The probability of this dispersion level by chance assuming that
## the true mean size3added = variance in size3added,
## and an alternative hypothesis of overdispersion, is 3.721e-138
## Variable size3added is significantly overdispersed.
##
## Zero-inflation and truncation tests:
## Mean lambda in size3added is 0.02592
## The actual number of 0s in size3added is 0
## The expected number of 0s in size3added under the null hypothesis is 7.465
## The probability of this deviation in 0s from expectation by chance is 0.9964
## Variable size3added is not significantly zero-inflated.
##
## Variable size3added does not include 0s, suggesting that a zero-truncated distribution may be warranted.
##
##
## Reproductive status:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 55 variables and 118 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be an integer variable.
##
## Variable feca2 is fully non-negative.
##
## Overdispersion test:
## Mean feca2 is 0.7881
## The variance in feca2 is 1.536
## The probability of this dispersion level by chance assuming that
## the true mean feca2 = variance in feca2,
## and an alternative hypothesis of overdispersion, is 0.1193
## Dispersion level in feca2 matches expectation.
##
## Zero-inflation and truncation tests:
## Mean lambda in feca2 is 0.4547
## The actual number of 0s in feca2 is 68
## The expected number of 0s in feca2 under the null hypothesis is 53.65
## The probability of this deviation in 0s from expectation by chance is 5.904e-06
## Variable feca2 is significantly zero-inflated.We note that size is is composed of positive integers, without any zeroes, and that it is overdispersed. This suggests that we should use the zero-truncated negative binomial distribution. Fecundity is also an integer variable, but it is not overdispersed and it appears to have more zeroes than expected under a Poisson distribution. Thus, we should go with a zero-inflated Poisson distribution.
Let’s compare to the function-based dataset.
hfv_qc(data = cypfb_v1, vitalrates = c("surv", "obs", "size", "repst", "fec"),
suite = "full", size = c("size3added", "size2added", "size1added"))## Survival:
##
## Data subset has 55 variables and 320 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 55 variables and 303 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable size3added has 0 missing values.
## Variable size3added appears to be an integer variable.
##
## Variable size3added is fully positive, lacking even 0s.
##
## Overdispersion test:
## Mean size3added is 3.653
## The variance in size3added is 13.41
## The probability of this dispersion level by chance assuming that
## the true mean size3added = variance in size3added,
## and an alternative hypothesis of overdispersion, is 3.721e-138
## Variable size3added is significantly overdispersed.
##
## Zero-inflation and truncation tests:
## Mean lambda in size3added is 0.02592
## The actual number of 0s in size3added is 0
## The expected number of 0s in size3added under the null hypothesis is 7.465
## The probability of this deviation in 0s from expectation by chance is 0.9964
## Variable size3added is not significantly zero-inflated.
##
## Variable size3added does not include 0s, suggesting that a zero-truncated distribution may be warranted.
##
##
## Reproductive status:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 55 variables and 118 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be an integer variable.
##
## Variable feca2 is fully non-negative.
##
## Overdispersion test:
## Mean feca2 is 0.7881
## The variance in feca2 is 1.536
## The probability of this dispersion level by chance assuming that
## the true mean feca2 = variance in feca2,
## and an alternative hypothesis of overdispersion, is 0.1193
## Dispersion level in feca2 matches expectation.
##
## Zero-inflation and truncation tests:
## Mean lambda in feca2 is 0.4547
## The actual number of 0s in feca2 is 68
## The expected number of 0s in feca2 under the null hypothesis is 53.65
## The probability of this deviation in 0s from expectation by chance is 5.904e-06
## Variable feca2 is significantly zero-inflated.The same results, of course, since we are really using the same size and fecundity data as before (only the stage assignments are different).
Just to reinforce the similarities, let’s compare to the Cypripedium IPM dataset, which will also be the same.
hfv_qc(data = cypipm_v1, vitalrates = c("surv", "obs", "size", "repst", "fec"),
suite = "full", size = c("size3added", "size2added", "size1added"))## Survival:
##
## Data subset has 55 variables and 320 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 55 variables and 303 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable size3added has 0 missing values.
## Variable size3added appears to be an integer variable.
##
## Variable size3added is fully positive, lacking even 0s.
##
## Overdispersion test:
## Mean size3added is 3.653
## The variance in size3added is 13.41
## The probability of this dispersion level by chance assuming that
## the true mean size3added = variance in size3added,
## and an alternative hypothesis of overdispersion, is 3.721e-138
## Variable size3added is significantly overdispersed.
##
## Zero-inflation and truncation tests:
## Mean lambda in size3added is 0.02592
## The actual number of 0s in size3added is 0
## The expected number of 0s in size3added under the null hypothesis is 7.465
## The probability of this deviation in 0s from expectation by chance is 0.9964
## Variable size3added is not significantly zero-inflated.
##
## Variable size3added does not include 0s, suggesting that a zero-truncated distribution may be warranted.
##
##
## Reproductive status:
##
## Data subset has 55 variables and 288 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 55 variables and 118 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be an integer variable.
##
## Variable feca2 is fully non-negative.
##
## Overdispersion test:
## Mean feca2 is 0.7881
## The variance in feca2 is 1.536
## The probability of this dispersion level by chance assuming that
## the true mean feca2 = variance in feca2,
## and an alternative hypothesis of overdispersion, is 0.1193
## Dispersion level in feca2 matches expectation.
##
## Zero-inflation and truncation tests:
## Mean lambda in feca2 is 0.4547
## The actual number of 0s in feca2 is 68
## The expected number of 0s in feca2 under the null hypothesis is 53.65
## The probability of this deviation in 0s from expectation by chance is 5.904e-06
## Variable feca2 is significantly zero-inflated.And finally the function-based Cypripedium dataset with environmental data added.
hfv_qc(data = cypfb_env, vitalrates = c("surv", "obs", "size", "repst", "fec"),
suite = "full", size = c("size3added", "size2added", "size1added"))## Survival:
##
## Data subset has 58 variables and 320 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 58 variables and 303 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 58 variables and 288 transitions.
##
## Variable size3added has 0 missing values.
## Variable size3added appears to be an integer variable.
##
## Variable size3added is fully positive, lacking even 0s.
##
## Overdispersion test:
## Mean size3added is 3.653
## The variance in size3added is 13.41
## The probability of this dispersion level by chance assuming that
## the true mean size3added = variance in size3added,
## and an alternative hypothesis of overdispersion, is 3.721e-138
## Variable size3added is significantly overdispersed.
##
## Zero-inflation and truncation tests:
## Mean lambda in size3added is 0.02592
## The actual number of 0s in size3added is 0
## The expected number of 0s in size3added under the null hypothesis is 7.465
## The probability of this deviation in 0s from expectation by chance is 0.9964
## Variable size3added is not significantly zero-inflated.
##
## Variable size3added does not include 0s, suggesting that a zero-truncated distribution may be warranted.
##
##
## Reproductive status:
##
## Data subset has 58 variables and 288 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 58 variables and 118 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be an integer variable.
##
## Variable feca2 is fully non-negative.
##
## Overdispersion test:
## Mean feca2 is 0.7881
## The variance in feca2 is 1.536
## The probability of this dispersion level by chance assuming that
## the true mean feca2 = variance in feca2,
## and an alternative hypothesis of overdispersion, is 0.1193
## Dispersion level in feca2 matches expectation.
##
## Zero-inflation and truncation tests:
## Mean lambda in feca2 is 0.4547
## The actual number of 0s in feca2 is 68
## The expected number of 0s in feca2 under the null hypothesis is 53.65
## The probability of this deviation in 0s from expectation by chance is 5.904e-06
## Variable feca2 is significantly zero-inflated.Finally, let’s move on to the Lathyrus IPM dataset. Let’s include age for the fun of it, as well.
hfv_qc(data = lathvert_ipm, vitalrates = c("surv", "obs", "size", "fec"),
juvestimate = "Sdl", indiv = "individ", year = "year2", age = "obsage")## Survival:
##
## Data subset has 43 variables and 2246 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 43 variables and 2121 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 43 variables and 1916 transitions.
##
## Variable sizea3 has 0 missing values.
## Variable sizea3 appears to be a floating point variable.
## 1256 elements are not integers.
## The minimum value of sizea3 is 3.4 and the maximum is 6646.
## The mean value of sizea3 is 512.8 and the variance is 507200.
## The value of the Shapiro-Wilk test of normality is 0.7134 with P = 3.014e-49.
## Variable sizea3 differs significantly from a Gaussian distribution.
##
## Variable sizea3 is fully positive, lacking even 0s.
##
##
## Reproductive status:
##
## Data subset has 43 variables and 1916 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 43 variables and 2246 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be a floating point variable.
## 6 elements are not integers.
## The minimum value of feca2 is 0 and the maximum is 66.
## The mean value of feca2 is 1.283 and the variance is 23.22.
## The value of the Shapiro-Wilk test of normality is 0.2968 with P = 1.676e-68.
## Variable feca2 differs significantly from a Gaussian distribution.
##
## Variable feca2 is fully non-negative.
##
##
## Juvenile survival:
##
## Data subset has 43 variables and 281 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Juvenile observation status:
##
## Data subset has 43 variables and 210 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Juvenile primary size:
##
## Data subset has 43 variables and 193 transitions.
##
## Variable sizea3 has 0 missing values.
## Variable sizea3 appears to be a floating point variable.
## 127 elements are not integers.
## The minimum value of sizea3 is 2.1 and the maximum is 61.
## The mean value of sizea3 is 11.23 and the variance is 50.81.
## The value of the Shapiro-Wilk test of normality is 0.5997 with P = 5.72e-21.
## Variable sizea3 differs significantly from a Gaussian distribution.
##
## Variable sizea3 is fully positive, lacking even 0s.
##
##
## Juvenile reproductive status:
##
## Data subset has 43 variables and 193 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Juvenile maturity status:
##
## Data subset has 43 variables and 210 transitions.
##
## Variable matstatus3 has 0 missing values.
## Variable matstatus3 is a binomial variable.Let’s start off by taking a look at a density plot of size, since it appears to deviate from a Gaussian distribution.
plot(density(lathvert_ipm$sizea2))
Indeed, it does not appear to be normally distributed. However, for now, we will ignore this issue and simply assume a Gaussian distribution. Note that in reality, we might wish to try a gamma distribution, or use a logarithmic or other size, in order to make a closer match to the Gaussian or other distribution.
Next, let’s try dealing with fecundity. Our results suggest that fecundity is typically an integer, but that a few values are not integers. This has to do with the fact that although fecundity was typically counted as seeds, there were a few instances in the study in which seed count was estimated from a linear regression. Let’s see this issue below.
length(lathvert_ipm$feca3)## [1] 2527length(which(lathvert_ipm$feca3 != round(lathvert_ipm$feca3)))## [1] 0length(lathvert_ipm$feca2)## [1] 2527length(which(lathvert_ipm$feca2 != round(lathvert_ipm$feca2)))## [1] 6length(lathvert_ipm$feca1)## [1] 2527length(which(lathvert_ipm$feca1 != round(lathvert_ipm$feca1)))## [1] 6As we can see, there are really only six out of 2527 values that are
not integers. The easiest way to deal with this issue is to round them,
as below. Let’s do that, and then re-do the hfv_qc()
run.
lathvert_ipm$feca2 <- round(lathvert_ipm$feca2)
lathvert_ipm$feca1 <- round(lathvert_ipm$feca1)
hfv_qc(data = lathvert_ipm, vitalrates = c("surv", "obs", "size", "fec"),
juvestimate = "Sdl", indiv = "individ", year = "year2", age = "obsage")## Survival:
##
## Data subset has 43 variables and 2246 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Observation status:
##
## Data subset has 43 variables and 2121 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Primary size:
##
## Data subset has 43 variables and 1916 transitions.
##
## Variable sizea3 has 0 missing values.
## Variable sizea3 appears to be a floating point variable.
## 1256 elements are not integers.
## The minimum value of sizea3 is 3.4 and the maximum is 6646.
## The mean value of sizea3 is 512.8 and the variance is 507200.
## The value of the Shapiro-Wilk test of normality is 0.7134 with P = 3.014e-49.
## Variable sizea3 differs significantly from a Gaussian distribution.
##
## Variable sizea3 is fully positive, lacking even 0s.
##
##
## Reproductive status:
##
## Data subset has 43 variables and 1916 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Fecundity:
##
## Data subset has 43 variables and 2246 transitions.
##
## Variable feca2 has 0 missing values.
## Variable feca2 appears to be an integer variable.
##
## Variable feca2 is fully non-negative.
##
## Overdispersion test:
## Mean feca2 is 1.282
## The variance in feca2 is 23.21
## The probability of this dispersion level by chance assuming that
## the true mean feca2 = variance in feca2,
## and an alternative hypothesis of overdispersion, is 0
## Variable feca2 is significantly overdispersed.
##
## Zero-inflation and truncation tests:
## Mean lambda in feca2 is 0.2774
## The actual number of 0s in feca2 is 1980
## The expected number of 0s in feca2 under the null hypothesis is 623
## The probability of this deviation in 0s from expectation by chance is 0
## Variable feca2 is significantly zero-inflated.
##
##
## Juvenile survival:
##
## Data subset has 43 variables and 281 transitions.
##
## Variable alive3 has 0 missing values.
## Variable alive3 is a binomial variable.
##
##
## Juvenile observation status:
##
## Data subset has 43 variables and 210 transitions.
##
## Variable obsstatus3 has 0 missing values.
## Variable obsstatus3 is a binomial variable.
##
##
## Juvenile primary size:
##
## Data subset has 43 variables and 193 transitions.
##
## Variable sizea3 has 0 missing values.
## Variable sizea3 appears to be a floating point variable.
## 127 elements are not integers.
## The minimum value of sizea3 is 2.1 and the maximum is 61.
## The mean value of sizea3 is 11.23 and the variance is 50.81.
## The value of the Shapiro-Wilk test of normality is 0.5997 with P = 5.72e-21.
## Variable sizea3 differs significantly from a Gaussian distribution.
##
## Variable sizea3 is fully positive, lacking even 0s.
##
##
## Juvenile reproductive status:
##
## Data subset has 43 variables and 193 transitions.
##
## Variable repstatus3 has 0 missing values.
## Variable repstatus3 is a binomial variable.
##
##
## Juvenile maturity status:
##
## Data subset has 43 variables and 210 transitions.
##
## Variable matstatus3 has 0 missing values.
## Variable matstatus3 is a binomial variable.According to this, we should treat fecundity as a zero-inflated negative binomial variable. We will assume the Gaussian distribution for size.
Step 3c. Model building and selection
In lefko3, the modelsearch function is the
workhorse that conducts vital rate model estimation. Here, we will
create a full suite of vital rate models for the Cypripedium
candidum dataset. Before proceeding, we need to decide on the
linear model building strategy, the correct vital rates to model, the
proper statistical distributions for estimated vital rates, the proper
parameterizations for each vital rate, and the strategy for
determination of the best-fit models.
First, we must determine the model building strategy. In most cases,
the best procedure will be through mixed linear models in which
monitoring occasion and individual identity are random terms. We will
set monitoring occasion as random because we wish to make inferences for
the population as a whole and do not wish to restrict ourselves to
inference only for the years monitored (i.e. our distribution of
monitoring occasions sampled is itself a sample of the population in
time). We will set individual identity as random because many or most of
the individuals that we have sampled to produce our dataset yield
multiple observation data points across time. Thus, we will set
approach = "mixed". To make sure that time and individual
identity are treated as random, we will set the proper variable names
for indiv and year, corresponding to
individual identity (individ by default), and to occasion t
(year2 by default). The year.as.random option
is set to random by default, and leaving it this way also means that R
will randomly draw coefficient values for years with inestimable
coefficients. Setting year.as.random to FALSE would make
time a fixed categorical variable.
The mixed modeling approach is usually preferable to the GLM
approach. However, a mixed modeling strategy results in lower
statistical power and a greater time used in estimating models (or,
conversely, it yields truer statistical power while the GLM approach
inflates Type I error). Users of package lefko3 wishing to
use a standard generalized linear modeling strategy can set
approach = "glm". In this case, individual identity is not
used, time is a fixed categorical factor, and all observed transitions
are treated as independent.
Next, we must determine which vital rates to model. Function
modelsearch() estimates up to 14 vital rate models:
1. survival probability from occasion t to occasion t+1,
2. observation probability in occasion t+1 assuming survival until that time,
3. primary size in occasion t+1 assuming survival and observation in that time,
4. secondary size in occasion t+1 assuming survival and observation in that time,
5. tertiary size in occasion t+1 assuming survival and observation in that time,
6. reproduction status in occasion t+1 assuming survival and observation until that time,
7. fecundity rate assuming survival until and observation and reproduction in the occasion of production of offspring (occasion t or t+1; mature only),
8. juvenile survival probability from occasion t to occasion t+1,
9. juvenile observation probability in occasion t+1 assuming survival until that time,
10. juvenile primary size in occasion t+1 assuming survival and observation in that time,
11. juvenile secondary size in occasion t+1 assuming survival and observation in that time,
12. juvenile tertiary size in occasion t+1 assuming survival and observation in that time,
13. juvenile transition probability to reproductive status in occasion t+1 assuming survival and observation until that time of a juvenile in occasion t that is becoming mature in occasion t+1, and
14. juvenile transition probability to maturity in occasion t+1 assuming survival until that time of a juvenile in occasion t.
The default settings for modelsearch estimate 1)
survival probability, 3) primary size distribution, and 7) fecundity,
which are the minimum three vital rates required for a full MPM or IPM.
Observation probability (option obs in
vitalrates) should only be included when a life history
stage or size exists that cannot be observed. For example, in the case
of a plant with vegetative dormancy, the observation probability can be
thought of as the sprouting probability, which is a biologically
meaningful vital rate (Shefferson et al. 2001). Further,
reproduction status (option repst in
vitalrates) should only be modeled if size classification
needs to be stratified by the ability to reproduce, as when zero
fecundity occurs within stages that also produce offspring. Juvenile
stages should generally only be used when juveniles exist as a single
stage that should not be size-classified. Otherwise, juvenile stages
should be treated as normal stages (transitions back to juvenile stages
can be prevented later). Since Cypripedium candidum is capable
of long bouts of vegetative dormancy, since we wish to stratify the
population into reproductive and non-reproductive stages of the same
size classes, and since we have no data derived from juvenile
individuals, we will set
vitalrates = c("surv", "obs", "size", "repst", "fec").
Third, we need to set the proper statistical distribution for each parameter. Survival probability, observation probability, and reproductive status are all modeled as binomial variables, and this cannot be changed. In the case of this population of Cypripedium candidum, size was measured as the number of stems and so is a count variable. Likewise, fecundity is actually estimated as the number of fruits produced per plant, and so is also a count variable. We have already performed tests for overdispersion and zero-inflation, and we are also aware that size in observed stages cannot be zero, requiring zero truncation in that parameter. So we will set size to the zero-truncated negative binomial distribution, and fecundity to the zero-inflated Poisson distribution.
Fourth, we need the proper model parameterizations for each vital
rate, using the suite option. The default,
suite = "main", under the mixed model setting
(approach = "mixed") starts with the estimation of global
models that include size and reproductive status in occasions t and t-1
as fixed factors, with individual identity and time in occasion t (year
t) set as random categorical terms. Other terms can be specified,
including individual covariates and age. Setting
suite = "full" will yield global models that also include
all two-way interactions. The global model under
suite = "full" then includes all fixed factors noted
before, plus time in occasion t and all two-way interactions between
fixed factors (“full” is the only setting with interaction terms). If
the population is not stratified by reproductive status, then
suite = "size" will eliminate reproductive status terms and
use all others in the global model. If size is not important, then
suite = "rep" will eliminate size but keep reproductive
status and all other terms. Finally, suite = "cons" will
result in a global model in which neither reproductive status nor size
are considered (although other terms, such as individual covariates and
age, can still be included in the last case).
Finally, we need to determine the proper strategy for the
determination of the best-fit model. Model building proceeds through the
dredge function in package MuMIn Barton
(2014), and each model has an associated AICc value. The default setting
in lefko3 (bestfit = "AICc&k") will
compare all models within 2.0 AICc units of the model with \(\Delta AICc = 0\), and choose the one with
the lowest degrees of freedom. This approach is generally better than
the alternative, which simply uses the model with \(\Delta AICc = 0\)
(bestfit = "AICc"), as all models within 2.0 AICc units of
that model are equally parsimonious and so fewer degrees of freedom
result from fewer parameters estimated Burnham and Anderson (2002).
In the model building exercise below, we will use
suite = "main", which runs all main effects without
interactions among fixed factors. However, we will not actually run this
during our workshop, because this will likely take too long. If you are
willing to wait, then you might try including two-way interactions by
setting suite = "full". In our case, we will load a pre-run
model set called cypmodels3p.
cypmodels3p <- modelsearch(cypfb_env, historical = TRUE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"), patch = "patchid",
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added", "size1added"))
save(cypmodels3p, file = "cypmodels3p.Rdata")load("cypmodels3p.Rdata")
summary(cypmodels3p)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 130.1321 148.9737 -60.0660 120.1321 315
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.199e+00
## year2 (Intercept) 5.161e-05
## patchid (Intercept) 1.113e-05
## Number of obs: 320, groups: individ, 74; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.0356 0.6343
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | patchid) + (1 |
## individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 120.2567 138.8254 -55.1284 110.2567 298
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.0000
## year2 (Intercept) 0.8776
## patchid (Intercept) 0.0000
## Number of obs: 303, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula:
## size3added ~ size1added + size2added + (1 | year2) + (1 | patchid) +
## (1 | individ) + size1added:size2added
## Data: subdata
## AIC BIC logLik df.resid
## 1001.2347 1030.5384 -492.6173 280
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.1747
## patchid (Intercept) 0.1800
## individ (Intercept) 0.7193
##
## Number of obs: 288 / Conditional model: year2, 5; patchid, 3; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 1.4e+07
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size1added size2added
## 0.299228 0.082335 0.079648
## size1added:size2added
## -0.006193
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ repstatus2 + size2added + (1 | year2) + (1 | patchid) +
## (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 333.4037 355.3815 -160.7019 321.4037 282
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.1776
## year2 (Intercept) 0.6636
## patchid (Intercept) 0.3501
## Number of obs: 288, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) repstatus2 size2added
## -1.3836 1.5543 0.1788
##
##
##
## Fecundity model:
## Formula:
## feca2 ~ size1added + size2added + (1 | year2) + (1 | patchid) +
## (1 | individ)
## Zero inflation:
## ~size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 252.3835 282.8610 -115.1917 107
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 6.043e-01
## patchid (Intercept) 2.130e-01
## individ (Intercept) 5.213e-10
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 4.136e-12
## patchid (Intercept) 1.460e-13
## individ (Intercept) 2.951e-04
##
## Number of obs: 118 / Conditional model: year2, 5; patchid, 3; individ, 51 / Zero-inflation model: year2, 5; patchid, 3; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size1added size2added
## -0.51758 -0.03543 0.08305
##
## Zero-inflation model:
## (Intercept) size2added
## 3.857 -1.575
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 113
##
## Number of models in observation table: 113
##
## Number of models in size table: 113
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 113
##
## Number of models in fecundity table: 12169
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 0.947.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is NA.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.74.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.344.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.We can see that historical size is included in the best-fit fecundity
model. This suggests that the historical MPM is the most parsimonious
choice. However, in order to compare MPMs for educational purposes, we
will also create an ahistorical model set. Note that a vital rate model
set that includes historical terms CANNOT be used to make an ahistorical
MPM. To build ahistorical MPMs, we will repeat the above exercise but
with the historical = FALSE setting.
cypmodels2p <- modelsearch(cypfb_env, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"), patch = "patchid",
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added", "size1added"))
save(cypmodels2p, file = "cypmodels2p.Rdata")load("cypmodels2p.Rdata")
summary(cypmodels2p)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 130.1321 148.9737 -60.0660 120.1321 315
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.199e+00
## year2 (Intercept) 5.161e-05
## patchid (Intercept) 1.113e-05
## Number of obs: 320, groups: individ, 74; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.0356 0.6343
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | patchid) + (1 |
## individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 120.2567 138.8254 -55.1284 110.2567 298
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.0000
## year2 (Intercept) 0.8776
## patchid (Intercept) 0.0000
## Number of obs: 303, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula: size3added ~ size2added + (1 | year2) + (1 | patchid) + (1 |
## individ)
## Data: subdata
## AIC BIC logLik df.resid
## 1009.3168 1031.2946 -498.6584 282
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.1215
## patchid (Intercept) 0.2052
## individ (Intercept) 0.9497
##
## Number of obs: 288 / Conditional model: year2, 5; patchid, 3; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 1.23e+09
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size2added
## 0.54038 0.02191
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ repstatus2 + size2added + (1 | year2) + (1 | patchid) +
## (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 333.4037 355.3815 -160.7019 321.4037 282
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.1776
## year2 (Intercept) 0.6636
## patchid (Intercept) 0.3501
## Number of obs: 288, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) repstatus2 size2added
## -1.3836 1.5543 0.1788
##
##
##
## Fecundity model:
## Formula: feca2 ~ (1 | year2) + (1 | patchid) + (1 | individ)
## Zero inflation:
## ~size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 256.6460 281.5822 -119.3230 109
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.5908
## patchid (Intercept) 0.1970
## individ (Intercept) 0.3863
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 2.458e-08
## patchid (Intercept) 3.189e-07
## individ (Intercept) 2.445e-04
##
## Number of obs: 118 / Conditional model: year2, 5; patchid, 3; individ, 51 / Zero-inflation model: year2, 5; patchid, 3; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept)
## -0.2146
##
## Zero-inflation model:
## (Intercept) size2added
## 4.081 -1.597
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 5
##
## Number of models in observation table: 5
##
## Number of models in size table: 5
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 5
##
## Number of models in fecundity table: 21
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 0.947.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is NA.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.74.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.362.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.Let’s also create models that we can use for age and age-by-stage MPM creation.
cypmodels2p_agestage <- modelsearch(cypfb_env, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"), patch = "patchid",
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added", "size1added"),
age = "obsage")
save(cypmodels2p_agestage, file = "cypmodels2p_agestage.Rdata")
cypmodels2p_age <- modelsearch(cypfb_env, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "fec"), patch = "patchid", fecdist = "poisson",
fec.zero = TRUE, suite = "cons", age = "obsage")
save(cypmodels2p_age, file = "cypmodels2p_age.Rdata")load("cypmodels2p_agestage.Rdata")
summary(cypmodels2p_agestage)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 130.1321 148.9737 -60.0660 120.1321 315
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.199e+00
## year2 (Intercept) 5.161e-05
## patchid (Intercept) 1.113e-05
## Number of obs: 320, groups: individ, 74; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.0356 0.6343
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | patchid) + (1 |
## individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 120.2567 138.8254 -55.1284 110.2567 298
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.0000
## year2 (Intercept) 0.8776
## patchid (Intercept) 0.0000
## Number of obs: 303, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula: size3added ~ size2added + (1 | year2) + (1 | patchid) + (1 |
## individ)
## Data: subdata
## AIC BIC logLik df.resid
## 1009.3168 1031.2946 -498.6584 282
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.1215
## patchid (Intercept) 0.2052
## individ (Intercept) 0.9497
##
## Number of obs: 288 / Conditional model: year2, 5; patchid, 3; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 1.23e+09
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size2added
## 0.54038 0.02191
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ repstatus2 + size2added + (1 | year2) + (1 | patchid) +
## (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 333.4037 355.3815 -160.7019 321.4037 282
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.1776
## year2 (Intercept) 0.6636
## patchid (Intercept) 0.3501
## Number of obs: 288, groups: individ, 70; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept) repstatus2 size2added
## -1.3836 1.5543 0.1788
##
##
##
## Fecundity model:
## Formula:
## feca2 ~ obsage + size2added + (1 | year2) + (1 | patchid) + (1 |
## individ) + obsage:size2added
## Zero inflation:
## ~size2added + (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 252.2818 285.5300 -114.1409 106
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.450610
## patchid (Intercept) 0.252398
## individ (Intercept) 0.000849
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 2.528e-04
## patchid (Intercept) 7.348e-03
## individ (Intercept) 7.322e-05
##
## Number of obs: 118 / Conditional model: year2, 5; patchid, 3; individ, 51 / Zero-inflation model: year2, 5; patchid, 3; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) obsage size2added obsage:size2added
## 0.09044 -0.08927 0.19053 -0.01891
##
## Zero-inflation model:
## (Intercept) size2added
## 3.889 -1.589
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 18
##
## Number of models in observation table: 18
##
## Number of models in size table: 18
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 18
##
## Number of models in fecundity table: 270
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 0.947.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is NA.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.74.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.303.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.load("cypmodels2p_age.Rdata")
summary(cypmodels2p_age)## This LefkoMod object includes 2 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ (1 | year2) + (1 | patchid) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 139.0335 154.1068 -65.5167 131.0335 316
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.746
## year2 (Intercept) 0.000
## patchid (Intercept) 0.000
## Number of obs: 320, groups: individ, 74; year2, 5; patchid, 3
## Fixed Effects:
## (Intercept)
## 3.724
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## [1] 1
##
##
##
## Size model:
## [1] 1
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## [1] 1
##
##
##
## Fecundity model:
## Formula: feca2 ~ (1 | year2) + (1 | individ)
## Zero inflation: ~obsage + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 378.7659 405.1441 -182.3829 313
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.6859
## individ (Intercept) 0.4988
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 6.778e-05
## individ (Intercept) 2.737e+00
##
## Number of obs: 320 / Conditional model: year2, 5; individ, 74 / Zero-inflation model: year2, 5; individ, 74
##
## Fixed Effects:
##
## Conditional model:
## (Intercept)
## -0.3712
##
## Zero-inflation model:
## (Intercept) obsage
## 1.95723 0.06176
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 2
##
## Number of models in observation table: 1
##
## Number of models in size table: 1
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 1
##
## Number of models in fecundity table: 2
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 0.947.
## Observation probability not estimated.
## Primary size transition not estimated.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproduction probability not estimated.
## Fecundity estimated with 74 individuals and 320 individual transitions.
## Fecundity pseudo R-squared is 0.416.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.In the preceding vital rate model sets, we created best-fit models
that necessarily included a patch term. This will result necessarily in
the creation of matrices at the patch level. To create matrices for the
population level only, we remove the patch option from the
preceding function calls.
cypmodels3 <- modelsearch(cypfb_env, historical = TRUE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"),
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added", "size1added"),
quiet = TRUE)
save(cypmodels3, file = "cypmodels3.Rdata")
cypmodels2 <- modelsearch(cypfb_env, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"),
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added"), quiet = TRUE)
save(cypmodels2, file = "cypmodels2.Rdata")load("cypmodels3.Rdata")
load("cypmodels2.Rdata")We can also add individual or environmental covariates, by adding
those terms to the call with the indcova,
indcovb, and indcovc options. Let’s try both
of these, for the historical and ahistorical models. However, note that
in the historical set, we set suite = "main" in order to
prevent the model selection protocol from giving an error in fecundity
estimation. The reason for the error is that we have enough fixed terms
that a zero-inflation model, in which two real models (the binomial
model predicting zeros and the conditional model predicting all other
numbers) are actually produced, will yield too many fixed factor
combinations to proceed when interactions are allowed. This would not be
the case if we did not include a zero-inflation model.
cypmodels3_env <- modelsearch(cypfb_env, historical = TRUE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"),
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "main", size = c("size3added", "size2added", "size1added"),
indcova = c("indcova3", "indcova2", "indcova1"), quiet = TRUE)
save(cypmodels3_env, file = "cypmodels3_env.Rdata")
cypmodels2_env <- modelsearch(cypfb_env, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"),
sizedist = "negbin", size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE,
suite = "full", size = c("size3added", "size2added"),
indcova = c("indcova3", "indcova2"), quiet = TRUE)
save(cypmodels2_env, file = "cypmodels2_env.Rdata")load("cypmodels3_env.Rdata")
load("cypmodels2_env.Rdata")
summary(cypmodels3_env)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ indcova1 + indcova2 + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 82.4190 101.2606 -36.2095 72.4190 315
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 242.7
## year2 (Intercept) 0.0
## Number of obs: 320, groups: individ, 74; year2, 5
## Fixed Effects:
## (Intercept) indcova1 indcova2
## 274.3492 -0.9868 -1.3391
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 118.2567 133.1117 -55.1284 110.2567 299
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.078e-05
## year2 (Intercept) 8.776e-01
## Number of obs: 303, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula:
## size3added ~ indcova2 + size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 1001.3778 1023.3555 -494.6889 282
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.06554
## individ (Intercept) 0.96060
##
## Number of obs: 288 / Conditional model: year2, 5; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 1.87e+12
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) indcova2 size2added
## 0.070927 0.004918 0.023979
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ indcova2 + repstatus2 + size2added + (1 | year2) +
## (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 330.0418 352.0196 -159.0209 318.0418 282
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.0002695
## year2 (Intercept) 0.2479120
## Number of obs: 288, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) indcova2 repstatus2 size2added
## -4.23766 0.02954 1.71091 0.17187
##
##
##
## Fecundity model:
## Formula: feca2 ~ indcova2 + size2added + (1 | year2) + (1 | individ)
## Zero inflation: ~size1added + size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 244.2709 271.9777 -112.1355 108
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.2036
## individ (Intercept) 0.2139
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 1.100e-04
## individ (Intercept) 7.855e-05
##
## Number of obs: 118 / Conditional model: year2, 5; individ, 51 / Zero-inflation model: year2, 5; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) indcova2 size2added
## 1.70298 -0.02431 0.06259
##
## Zero-inflation model:
## (Intercept) size1added size2added
## 5.181 -0.710 -1.637
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 55
##
## Number of models in observation table: 64
##
## Number of models in size table: 64
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 64
##
## Number of models in fecundity table: 3582
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 1.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is 0.011.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.719.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.322.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.summary(cypmodels2_env)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ indcova2 + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 87.6075 102.6808 -39.8038 79.6075 316
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 552.9
## year2 (Intercept) 394.6
## Number of obs: 320, groups: individ, 74; year2, 5
## Fixed Effects:
## (Intercept) indcova2
## 476.174 -3.278
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 118.2567 133.1117 -55.1284 110.2567 299
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.078e-05
## year2 (Intercept) 8.776e-01
## Number of obs: 303, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula:
## size3added ~ indcova2 + size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 1001.3778 1023.3555 -494.6889 282
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.06554
## individ (Intercept) 0.96060
##
## Number of obs: 288 / Conditional model: year2, 5; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 1.87e+12
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) indcova2 size2added
## 0.070927 0.004918 0.023979
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ indcova2 + repstatus2 + size2added + (1 | year2) +
## (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 330.0418 352.0196 -159.0209 318.0418 282
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.0002695
## year2 (Intercept) 0.2479120
## Number of obs: 288, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) indcova2 repstatus2 size2added
## -4.23766 0.02954 1.71091 0.17187
##
##
##
## Fecundity model:
## Formula: feca2 ~ size2added + (1 | year2) + (1 | individ)
## Zero inflation: ~size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 248.8609 271.0264 -116.4305 110
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.5760
## individ (Intercept) 0.1639
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 1.642e-06
## individ (Intercept) 3.089e-04
##
## Number of obs: 118 / Conditional model: year2, 5; individ, 51 / Zero-inflation model: year2, 5; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size2added
## -0.54014 0.06174
##
## Zero-inflation model:
## (Intercept) size2added
## 3.865 -1.574
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 17
##
## Number of models in observation table: 18
##
## Number of models in size table: 18
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 18
##
## Number of models in fecundity table: 309
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 1.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is 0.011.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.719.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.32.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.Let’s now take a shot at setting up the Lathyrus models. These models will be used to develop ahistorical and historical IPMs. Although IPMs are typically ahistorical, we can create historical IPMs as well, and need to test historical terms for that purpose.
lathmodels2ipm <- modelsearch(lathvert_ipm, historical = FALSE,
approach = "mixed", suite = "size",
vitalrates = c("surv", "obs", "size", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "negbin",
fec.zero = TRUE, indiv = "individ", year = "year2", year.as.random = TRUE,
juvsize = TRUE)
save(lathmodels2ipm, file = "lathmodels2ipm.Rdata")
lathmodels3ipm <- modelsearch(lathvert_ipm, historical = TRUE,
approach = "mixed", suite = "size",
vitalrates = c("surv", "obs", "size", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "negbin",
fec.zero = TRUE, indiv = "individ", year = "year2", year.as.random = TRUE,
juvsize = TRUE)
save(lathmodels3ipm, file = "lathmodels3ipm.Rdata")load("lathmodels2ipm.Rdata")
load("lathmodels3ipm.Rdata")Let’s now move on to create our first MPMs.
Step 3d. Importing vital rate models for IPMs and fbMPMs
Instead of building vital rate models within lefko3, we
can also import vital rate models that have already been estimated via
other means. The key caveat, in the current version of
lefko3, is that the vital rate models need to be some
variant of linear, generalized linear, or generalized linear mixed
model. For this purpose, we will work with the Lathyrus dataset.
Let’s start off by building a skeleton vrm_input object.
Note that we input only the years capable of being transitioned from in
the dataset, which means the three years 1988, 1989, and 1990. We will
make this object a bit bigger than before, because we will allow
fecundity to be zero-inflated (setting zi = TRUE will add
seven new columns governing the zero-inflation binomial model for all
parameters capable of being zero-inflated). We will also set
use.juv = TRUE to change some defaults to allow the use of
juvenile transitions. Here is the code to produce our skeleton
object.
lath_vrm <- vrm_import(years = c(1988:1990), zi = TRUE, dist.fec = "negbin",
use.juv = TRUE)
lath_vrm## $vrm_frame
## main_effect_1 main_1_defined surv obs sizea sizeb sizec
## 1 intercept y-intercept 0 0 0 0 0
## 2 size2 sizea in time t 0 0 0 0 0
## 3 size1 sizea in time t-1 0 0 0 0 0
## 4 sizeb2 sizeb in time t 0 0 0 0 0
## 5 sizeb1 sizeb in time t-1 0 0 0 0 0
## 6 sizec2 sizec in time t 0 0 0 0 0
## 7 sizec1 sizec in time t-1 0 0 0 0 0
## 8 repst2 reproductive status in time t 0 0 0 0 0
## 9 repst1 reproductive status in time t-1 0 0 0 0 0
## 10 age age in time t 0 0 0 0 0
## 11 density density in time t 0 0 0 0 0
## 12 indcova2 individual covariate a in time t 0 0 0 0 0
## 13 indcova1 individual covariate a in time t-1 0 0 0 0 0
## 14 indcovb2 individual covariate b in time t 0 0 0 0 0
## 15 indcovb1 individual covariate b in time t-1 0 0 0 0 0
## 16 indcovc2 individual covariate c in time t 0 0 0 0 0
## 17 indcovc1 individual covariate c in time t-1 0 0 0 0 0
## repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi sizeb_zi
## 1 0 0 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0 0 0 0 0
## 9 0 0 0 0 0 0 0 0 0 0 0
## 10 0 0 0 0 0 0 0 0 0 0 0
## 11 0 0 0 0 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0 0 0 0 0
## 14 0 0 0 0 0 0 0 0 0 0 0
## 15 0 0 0 0 0 0 0 0 0 0 0
## 16 0 0 0 0 0 0 0 0 0 0 0
## 17 0 0 0 0 0 0 0 0 0 0 0
## sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0
## 2 0 0 0 0 0
## 3 0 0 0 0 0
## 4 0 0 0 0 0
## 5 0 0 0 0 0
## 6 0 0 0 0 0
## 7 0 0 0 0 0
## 8 0 0 0 0 0
## 9 0 0 0 0 0
## 10 0 0 0 0 0
## 11 0 0 0 0 0
## 12 0 0 0 0 0
## 13 0 0 0 0 0
## 14 0 0 0 0 0
## 15 0 0 0 0 0
## 16 0 0 0 0 0
## 17 0 0 0 0 0
##
## $year_frame
## years surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 1988 0 0 0 0 0 0 0 0 0 0 0 0
## 2 1989 0 0 0 0 0 0 0 0 0 0 0 0
## 3 1990 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0
##
## $patch_frame
## patches surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group2_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group1_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $dist_frame
## response dist
## 1 surv binom
## 2 obs constant
## 3 sizea gaussian
## 4 sizeb constant
## 5 sizec constant
## 6 repst constant
## 7 fec negbin
## 8 jsurv binom
## 9 jobs constant
## 10 jsizea gaussian
## 11 jsizeb constant
## 12 jsizec constant
## 13 jrepst constant
## 14 jmatst constant
##
## $st_frame
## surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb
## 1 1 1 1 1 1 1 1 1 1 1
## jsizec jrepst jmatst
## 1 1 1
##
## attr(,"class")
## [1] "vrm_input"Now that we have our skeleton vrm_input object, let’s
take a look at the vital rate models. In a typical publication, the
authors might present equations showing the linear relationships among
terms. Alternatively, they may present some of the output from modeling,
giving us the slope coefficients. Below, we see how the survival and
sprouting models might be presented, using the real estimated terms.
\(logit(s(x_i, t)) = 2.32571 + 0.00109 size(t) + year(t) + indiv(i)\)
\(logit(r(x_j, t+1)) = 2.230 + year(t) + indiv(i)\)
In these models, we see that both are binomial models using the logit
link. Both have y-intercepts (2.32571 in the case of survival, and 2.230
in the case of sprouting). The survival model involves a relationship
with size in time t, and that relationship is linear with a slope of
0.00109. Both equations include categorical values for year in time t
and individual, because both survival and sprouting probabilities were
estimated as mixed models with year in time t and individual as random
terms. Let’s input all of the main terms into our skeleton
vrm_input object, particularly getting the y-intercepts and
slope coefficients into the appropriate parts of the
vrm_frame. Let’s also change the distribution of the
sprouting model from constant (the current setting) to
binom.
int.elem <- which(lath_vrm$vrm_frame$main_effect_1 == "intercept")
size2.elem <- which(lath_vrm$vrm_frame$main_effect_1 == "size2")
lath_vrm$vrm_frame$surv[int.elem] <- 2.32571
lath_vrm$vrm_frame$surv[size2.elem] <- 0.00109
lath_vrm$vrm_frame$obs[int.elem] <- 2.230
lath_vrm$dist_frame$dist[2] <- "binom"
lath_vrm$vrm_frame## main_effect_1 main_1_defined surv obs sizea sizeb
## 1 intercept y-intercept 2.32571 2.23 0 0
## 2 size2 sizea in time t 0.00109 0.00 0 0
## 3 size1 sizea in time t-1 0.00000 0.00 0 0
## 4 sizeb2 sizeb in time t 0.00000 0.00 0 0
## 5 sizeb1 sizeb in time t-1 0.00000 0.00 0 0
## 6 sizec2 sizec in time t 0.00000 0.00 0 0
## 7 sizec1 sizec in time t-1 0.00000 0.00 0 0
## 8 repst2 reproductive status in time t 0.00000 0.00 0 0
## 9 repst1 reproductive status in time t-1 0.00000 0.00 0 0
## 10 age age in time t 0.00000 0.00 0 0
## 11 density density in time t 0.00000 0.00 0 0
## 12 indcova2 individual covariate a in time t 0.00000 0.00 0 0
## 13 indcova1 individual covariate a in time t-1 0.00000 0.00 0 0
## 14 indcovb2 individual covariate b in time t 0.00000 0.00 0 0
## 15 indcovb1 individual covariate b in time t-1 0.00000 0.00 0 0
## 16 indcovc2 individual covariate c in time t 0.00000 0.00 0 0
## 17 indcovc1 individual covariate c in time t-1 0.00000 0.00 0 0
## sizec repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi
## 1 0 0 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0 0 0 0 0
## 9 0 0 0 0 0 0 0 0 0 0 0
## 10 0 0 0 0 0 0 0 0 0 0 0
## 11 0 0 0 0 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0 0 0 0 0
## 14 0 0 0 0 0 0 0 0 0 0 0
## 15 0 0 0 0 0 0 0 0 0 0 0
## 16 0 0 0 0 0 0 0 0 0 0 0
## 17 0 0 0 0 0 0 0 0 0 0 0
## sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0
## 2 0 0 0 0 0 0
## 3 0 0 0 0 0 0
## 4 0 0 0 0 0 0
## 5 0 0 0 0 0 0
## 6 0 0 0 0 0 0
## 7 0 0 0 0 0 0
## 8 0 0 0 0 0 0
## 9 0 0 0 0 0 0
## 10 0 0 0 0 0 0
## 11 0 0 0 0 0 0
## 12 0 0 0 0 0 0
## 13 0 0 0 0 0 0
## 14 0 0 0 0 0 0
## 15 0 0 0 0 0 0
## 16 0 0 0 0 0 0
## 17 0 0 0 0 0 0lath_vrm$dist_frame## response dist
## 1 surv binom
## 2 obs binom
## 3 sizea gaussian
## 4 sizeb constant
## 5 sizec constant
## 6 repst constant
## 7 fec negbin
## 8 jsurv binom
## 9 jobs constant
## 10 jsizea gaussian
## 11 jsizeb constant
## 12 jsizec constant
## 13 jrepst constant
## 14 jmatst constantWe have added the fixed main effects to our survival and sprouting models. Next, we will add the appropriate year terms. Year is a random term in both cases, meaning that the average effect of year has actually already been absorbed by the y-intercept and the mean of the year terms should be approximately zero. So, if we cannot find these terms in the paper, then we can simply assume it is 0, or we can produce random numbers if we have information on the variance of the year term in the model. In our case, we see in the output for the survival and sprouting models in the IPM chapter that the standard deviation of the year term is 0, meaning that these coefficients were inestimable under the mixed structure used. So, we will skip adding these terms here. Because we are not interested in predicting individual survival probabilities, we will also not incorporate any individual terms.
Let’s move on to size. Our size model has a Gaussian response and so uses the identity link. Thus, our predicted size in time t+1 is given by the equation below.
\(E(size(x_j, t+1)) = 164.0695 + 0.6211 size(z_i, t) + year(t) + indiv(i)\)
The probability of becoming size j in time t+1 assuming a Gaussian distribution is the following.
\(g(x_j, x_i) = \frac{1}{\sqrt{2 \pi} \sigma(x_j)} e^{-E(size(x_j, t+1))}\)
Our size model includes year terms, which are 96.3244, -240.8036, and 144.4792 for years 1988, 1989, and 1990. We also see that \(\sigma = 503.6167\), which is shown as the standard deviation of the residual component in the random effects section of the model summary output. We will add these terms below. Note that primary size is set to the Gaussian distribution by default.
lath_vrm$vrm_frame$sizea[int.elem] <- 164.0695
lath_vrm$vrm_frame$sizea[size2.elem] <- 0.6211
lath_vrm$year_frame$sizea <- c(96.3244, -240.8036, 144.4792)
lath_vrm$st_frame[3] <- 503.6167
lath_vrm$vrm_frame## main_effect_1 main_1_defined surv obs sizea sizeb
## 1 intercept y-intercept 2.32571 2.23 164.0695 0
## 2 size2 sizea in time t 0.00109 0.00 0.6211 0
## 3 size1 sizea in time t-1 0.00000 0.00 0.0000 0
## 4 sizeb2 sizeb in time t 0.00000 0.00 0.0000 0
## 5 sizeb1 sizeb in time t-1 0.00000 0.00 0.0000 0
## 6 sizec2 sizec in time t 0.00000 0.00 0.0000 0
## 7 sizec1 sizec in time t-1 0.00000 0.00 0.0000 0
## 8 repst2 reproductive status in time t 0.00000 0.00 0.0000 0
## 9 repst1 reproductive status in time t-1 0.00000 0.00 0.0000 0
## 10 age age in time t 0.00000 0.00 0.0000 0
## 11 density density in time t 0.00000 0.00 0.0000 0
## 12 indcova2 individual covariate a in time t 0.00000 0.00 0.0000 0
## 13 indcova1 individual covariate a in time t-1 0.00000 0.00 0.0000 0
## 14 indcovb2 individual covariate b in time t 0.00000 0.00 0.0000 0
## 15 indcovb1 individual covariate b in time t-1 0.00000 0.00 0.0000 0
## 16 indcovc2 individual covariate c in time t 0.00000 0.00 0.0000 0
## 17 indcovc1 individual covariate c in time t-1 0.00000 0.00 0.0000 0
## sizec repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi
## 1 0 0 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0 0 0 0 0
## 9 0 0 0 0 0 0 0 0 0 0 0
## 10 0 0 0 0 0 0 0 0 0 0 0
## 11 0 0 0 0 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0 0 0 0 0
## 14 0 0 0 0 0 0 0 0 0 0 0
## 15 0 0 0 0 0 0 0 0 0 0 0
## 16 0 0 0 0 0 0 0 0 0 0 0
## 17 0 0 0 0 0 0 0 0 0 0 0
## sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0
## 2 0 0 0 0 0 0
## 3 0 0 0 0 0 0
## 4 0 0 0 0 0 0
## 5 0 0 0 0 0 0
## 6 0 0 0 0 0 0
## 7 0 0 0 0 0 0
## 8 0 0 0 0 0 0
## 9 0 0 0 0 0 0
## 10 0 0 0 0 0 0
## 11 0 0 0 0 0 0
## 12 0 0 0 0 0 0
## 13 0 0 0 0 0 0
## 14 0 0 0 0 0 0
## 15 0 0 0 0 0 0
## 16 0 0 0 0 0 0
## 17 0 0 0 0 0 0lath_vrm$year_frame## years surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb
## 1 1988 0 0 96.3244 0 0 0 0 0 0 0 0
## 2 1989 0 0 -240.8036 0 0 0 0 0 0 0 0
## 3 1990 0 0 144.4792 0 0 0 0 0 0 0 0
## jsizec jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi
## 1 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0
## jsizec_zi
## 1 0
## 2 0
## 3 0lath_vrm$st_frame## surv obs sizea sizeb sizec repst fec jsurv
## 1.0000 1.0000 503.6167 1.0000 1.0000 1.0000 1.0000 1.0000
## jobs jsizea jsizeb jsizec jrepst jmatst
## 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000The next model to add is the fecundity model. This will be a zero-inflated negative binomial mixed model. Zero-inflation models are actually composed of two linear models - a binomial model governing the occurrence of zeros, and a second model with the target distribution covering all non-zeros (in this case, a negative binomial with a log link). So, we will need to parameterize both.
\(logit(f(x_i) = 0) = 6.252765 - 0.007313 size(t) + year(t) + indiv(i)\)
\(log(f(x_i) > 0) = 1.517 + year(t) + indiv(i)\)
The dispersion parameter for the negative binomial is \(\theta\), and we see in the IPM chapter that this is given as 0.2342114. The year terms for the zero-inflation binomial model are \(3.741475 \times 10^{-7}\), \(-7.804715 \times 10^{-8}\), and \(-2.533755 \times 10^{-7}\) for 1988, 1989, and 1990, respectively. The year terms for the conditional model (governing non-zero responses) are -0.41749627, 0.51421684, and -0.07964038, respectively. Let’s incorporate all of these values.
lath_vrm$vrm_frame$fec[int.elem] <- 1.517
lath_vrm$vrm_frame$fec_zi[int.elem] <- 6.252765
lath_vrm$vrm_frame$fec_zi[size2.elem] <- -0.007313
lath_vrm$year_frame$fec <- c(-0.41749627, 0.51421684, -0.07964038)
lath_vrm$year_frame$fec_zi <- c(3.741475e-07, -7.804715e-08, -2.533755e-07)
lath_vrm$st_frame[7] <- 0.2342114
lath_vrm## $vrm_frame
## main_effect_1 main_1_defined surv obs sizea sizeb
## 1 intercept y-intercept 2.32571 2.23 164.0695 0
## 2 size2 sizea in time t 0.00109 0.00 0.6211 0
## 3 size1 sizea in time t-1 0.00000 0.00 0.0000 0
## 4 sizeb2 sizeb in time t 0.00000 0.00 0.0000 0
## 5 sizeb1 sizeb in time t-1 0.00000 0.00 0.0000 0
## 6 sizec2 sizec in time t 0.00000 0.00 0.0000 0
## 7 sizec1 sizec in time t-1 0.00000 0.00 0.0000 0
## 8 repst2 reproductive status in time t 0.00000 0.00 0.0000 0
## 9 repst1 reproductive status in time t-1 0.00000 0.00 0.0000 0
## 10 age age in time t 0.00000 0.00 0.0000 0
## 11 density density in time t 0.00000 0.00 0.0000 0
## 12 indcova2 individual covariate a in time t 0.00000 0.00 0.0000 0
## 13 indcova1 individual covariate a in time t-1 0.00000 0.00 0.0000 0
## 14 indcovb2 individual covariate b in time t 0.00000 0.00 0.0000 0
## 15 indcovb1 individual covariate b in time t-1 0.00000 0.00 0.0000 0
## 16 indcovc2 individual covariate c in time t 0.00000 0.00 0.0000 0
## 17 indcovc1 individual covariate c in time t-1 0.00000 0.00 0.0000 0
## sizec repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi
## 1 0 0 1.517 0 0 0 0 0 0 0 0
## 2 0 0 0.000 0 0 0 0 0 0 0 0
## 3 0 0 0.000 0 0 0 0 0 0 0 0
## 4 0 0 0.000 0 0 0 0 0 0 0 0
## 5 0 0 0.000 0 0 0 0 0 0 0 0
## 6 0 0 0.000 0 0 0 0 0 0 0 0
## 7 0 0 0.000 0 0 0 0 0 0 0 0
## 8 0 0 0.000 0 0 0 0 0 0 0 0
## 9 0 0 0.000 0 0 0 0 0 0 0 0
## 10 0 0 0.000 0 0 0 0 0 0 0 0
## 11 0 0 0.000 0 0 0 0 0 0 0 0
## 12 0 0 0.000 0 0 0 0 0 0 0 0
## 13 0 0 0.000 0 0 0 0 0 0 0 0
## 14 0 0 0.000 0 0 0 0 0 0 0 0
## 15 0 0 0.000 0 0 0 0 0 0 0 0
## 16 0 0 0.000 0 0 0 0 0 0 0 0
## 17 0 0 0.000 0 0 0 0 0 0 0 0
## sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 6.252765 0 0 0
## 2 0 0 -0.007313 0 0 0
## 3 0 0 0.000000 0 0 0
## 4 0 0 0.000000 0 0 0
## 5 0 0 0.000000 0 0 0
## 6 0 0 0.000000 0 0 0
## 7 0 0 0.000000 0 0 0
## 8 0 0 0.000000 0 0 0
## 9 0 0 0.000000 0 0 0
## 10 0 0 0.000000 0 0 0
## 11 0 0 0.000000 0 0 0
## 12 0 0 0.000000 0 0 0
## 13 0 0 0.000000 0 0 0
## 14 0 0 0.000000 0 0 0
## 15 0 0 0.000000 0 0 0
## 16 0 0 0.000000 0 0 0
## 17 0 0 0.000000 0 0 0
##
## $year_frame
## years surv obs sizea sizeb sizec repst fec jsurv jobs jsizea
## 1 1988 0 0 96.3244 0 0 0 -0.41749627 0 0 0
## 2 1989 0 0 -240.8036 0 0 0 0.51421684 0 0 0
## 3 1990 0 0 144.4792 0 0 0 -0.07964038 0 0 0
## jsizeb jsizec jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi
## 1 0 0 0 0 0 0 0 3.741475e-07
## 2 0 0 0 0 0 0 0 -7.804715e-08
## 3 0 0 0 0 0 0 0 -2.533755e-07
## jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0
## 2 0 0 0
## 3 0 0 0
##
## $patch_frame
## patches surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group2_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group1_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $dist_frame
## response dist
## 1 surv binom
## 2 obs binom
## 3 sizea gaussian
## 4 sizeb constant
## 5 sizec constant
## 6 repst constant
## 7 fec negbin
## 8 jsurv binom
## 9 jobs constant
## 10 jsizea gaussian
## 11 jsizeb constant
## 12 jsizec constant
## 13 jrepst constant
## 14 jmatst constant
##
## $st_frame
## surv obs sizea sizeb sizec repst
## 1.0000000 1.0000000 503.6167000 1.0000000 1.0000000 1.0000000
## fec jsurv jobs jsizea jsizeb jsizec
## 0.2342114 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
## jrepst jmatst
## 1.0000000 1.0000000
##
## attr(,"class")
## [1] "vrm_input"Next we will need to input the associated parameters for any other model included in the IPM. Particularly, we know that vital rate models were also estimated for seedlings, and that these models were incorporated as juvenile vital rate models. The juvenile vital rates include survival, sprouting, and size transition. The equations for these models are as follows.
\(logit(s_{juv}(x_i, t)) = 1.03 + year(t) + indiv(i)\)
\(logit(r_{juv}(x_j, t+1)) = 10.390 + year(t) + indiv(i)\)
\(E_{juv}(size(x_j, t+1)) = 3.0559 + 0.8482 size(t) + year(t) + indiv(i)\)
Together with the residual \(\sigma\) for the size model and year terms
for sprouting and size, let’s add all of these terms to our
vrm_input object.
lath_vrm$vrm_frame$jsurv[int.elem] <- 1.03
lath_vrm$vrm_frame$jobs[int.elem] <- 10.390
lath_vrm$vrm_frame$jsizea[int.elem] <- 3.0559
lath_vrm$vrm_frame$jsizea[size2.elem] <- 0.8482
lath_vrm$st_frame[10] <- 5.831
lath_vrm$year_frame$jobs <- c(-0.7459843, 0.6118826, -0.9468618)
lath_vrm$year_frame$jsizea <- c(0.5937962, 1.4551236, -2.0489198)
lath_vrm$dist_frame$dist[9] <- "binom"
lath_vrm## $vrm_frame
## main_effect_1 main_1_defined surv obs sizea sizeb
## 1 intercept y-intercept 2.32571 2.23 164.0695 0
## 2 size2 sizea in time t 0.00109 0.00 0.6211 0
## 3 size1 sizea in time t-1 0.00000 0.00 0.0000 0
## 4 sizeb2 sizeb in time t 0.00000 0.00 0.0000 0
## 5 sizeb1 sizeb in time t-1 0.00000 0.00 0.0000 0
## 6 sizec2 sizec in time t 0.00000 0.00 0.0000 0
## 7 sizec1 sizec in time t-1 0.00000 0.00 0.0000 0
## 8 repst2 reproductive status in time t 0.00000 0.00 0.0000 0
## 9 repst1 reproductive status in time t-1 0.00000 0.00 0.0000 0
## 10 age age in time t 0.00000 0.00 0.0000 0
## 11 density density in time t 0.00000 0.00 0.0000 0
## 12 indcova2 individual covariate a in time t 0.00000 0.00 0.0000 0
## 13 indcova1 individual covariate a in time t-1 0.00000 0.00 0.0000 0
## 14 indcovb2 individual covariate b in time t 0.00000 0.00 0.0000 0
## 15 indcovb1 individual covariate b in time t-1 0.00000 0.00 0.0000 0
## 16 indcovc2 individual covariate c in time t 0.00000 0.00 0.0000 0
## 17 indcovc1 individual covariate c in time t-1 0.00000 0.00 0.0000 0
## sizec repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi
## 1 0 0 1.517 1.03 10.39 3.0559 0 0 0 0 0
## 2 0 0 0.000 0.00 0.00 0.8482 0 0 0 0 0
## 3 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 4 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 5 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 6 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 7 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 8 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 9 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 10 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 11 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 12 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 13 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 14 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 15 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 16 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 17 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 6.252765 0 0 0
## 2 0 0 -0.007313 0 0 0
## 3 0 0 0.000000 0 0 0
## 4 0 0 0.000000 0 0 0
## 5 0 0 0.000000 0 0 0
## 6 0 0 0.000000 0 0 0
## 7 0 0 0.000000 0 0 0
## 8 0 0 0.000000 0 0 0
## 9 0 0 0.000000 0 0 0
## 10 0 0 0.000000 0 0 0
## 11 0 0 0.000000 0 0 0
## 12 0 0 0.000000 0 0 0
## 13 0 0 0.000000 0 0 0
## 14 0 0 0.000000 0 0 0
## 15 0 0 0.000000 0 0 0
## 16 0 0 0.000000 0 0 0
## 17 0 0 0.000000 0 0 0
##
## $year_frame
## years surv obs sizea sizeb sizec repst fec jsurv jobs
## 1 1988 0 0 96.3244 0 0 0 -0.41749627 0 -0.7459843
## 2 1989 0 0 -240.8036 0 0 0 0.51421684 0 0.6118826
## 3 1990 0 0 144.4792 0 0 0 -0.07964038 0 -0.9468618
## jsizea jsizeb jsizec jrepst jmatst sizea_zi sizeb_zi sizec_zi
## 1 0.5937962 0 0 0 0 0 0 0
## 2 1.4551236 0 0 0 0 0 0 0
## 3 -2.0489198 0 0 0 0 0 0 0
## fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 3.741475e-07 0 0 0
## 2 -7.804715e-08 0 0 0
## 3 -2.533755e-07 0 0 0
##
## $patch_frame
## patches surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group2_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group1_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $dist_frame
## response dist
## 1 surv binom
## 2 obs binom
## 3 sizea gaussian
## 4 sizeb constant
## 5 sizec constant
## 6 repst constant
## 7 fec negbin
## 8 jsurv binom
## 9 jobs binom
## 10 jsizea gaussian
## 11 jsizeb constant
## 12 jsizec constant
## 13 jrepst constant
## 14 jmatst constant
##
## $st_frame
## surv obs sizea sizeb sizec repst
## 1.0000000 1.0000000 503.6167000 1.0000000 1.0000000 1.0000000
## fec jsurv jobs jsizea jsizeb jsizec
## 0.2342114 1.0000000 1.0000000 5.8310000 1.0000000 1.0000000
## jrepst jmatst
## 1.0000000 1.0000000
##
## attr(,"class")
## [1] "vrm_input"Voilà! Here, we have built a simple vrm_input object for
use later in MPM building. We could also have created a more complex
one, with interaction terms, individual covariates, etc.
Step 4. MPM estimation
Now we will work on developing the MPMs themselves. Note that these will be objects that include sets of matrices characterizing the years and subpopulations within the dataset. Our exploration will include the development of both raw and function-based versions of ahistorical, historical, age-by-stage, and Leslie MPMs. We will also make IPMs.
Step 4a. Modeling the main MPMs
We will begin with the creation of a set of ahistorical matrices for
the Cypripedium candidum dataset. The rlefko2
function was created to deal with the construction of ahistorical MPMs
using raw data. Matrices may strongly differ, particularly if the
demographic dataset is somewhat sparse. This happens because there may
not be enough individuals per year to encounter all possible
transitions, leading to seemingly random shifts in the location of
non-zero elements within matrices across time. We strongly advise
readers to build life history models that reflect the sample size that
they are working with to prevent this issue from causing odd results in
MPM analysis.
cypmatrix2rp <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
year = "all", patch = "all", stages = c("stage3", "stage2"),
size = c("size3added", "size2added"), supplement = cypsupp2_raw,
yearcol = "year2", patchcol = "patchid", indivcol = "individ")
cypmatrix2rp## $A
## $A$`1`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 500.0 1666.6666667 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 500.0 1666.6666667 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0 0.0000000 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0
## [7,] 0.00 0.0 0.0 0.0 0.6363636 0 0.6363636 0.2 0.0000000 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.2727273 0.6 0.6666667 1
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.2 0.3333333 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $A$`2`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.000 0 312.500 1111.1111111 1250 0 0
## [2,] 0.15 0.0 0.0 0.0 0.000 0 312.500 1111.1111111 1250 0 0
## [3,] 0.00 0.1 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.000 0 0.000 0.0000000 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.050 0 0.000 0.0000000 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.125 0 0.125 0.0000000 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.750 0 0.750 0.3333333 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.4444444 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.2222222 1 0 0
## [10,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [11,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
##
## $A$`3`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0 0.00 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0 0.00 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 1 0.3333333 1 0.25 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.75 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
##
## $A$`4`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.0000000 0
## [6,] 0.00 0.0 0.0 0.0 0.1666667 0 0.1666667 0.2222222 0.0000000 0
## [7,] 0.00 0.0 0.0 0.0 0.5000000 0 0.5000000 0.4444444 0.3333333 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.3333333 0.3333333 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.3333333 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $A$`5`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 500.0 5000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 500.0 5000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0.0000000 0.0000000 0.0 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.4444444 0.3333333 0.4444444 0.0 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0.6666667 0.4444444 0.6 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 1 0 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
##
## $A$`6`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 166.6666667 625.00 1875.00
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 166.6666667 625.00 1875.00
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.00 0.00
## [6,] 0.00 0.0 0.0 0.0 0.1111111 0 0.1111111 0.0000000 0.00 0.00
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0.2666667 0.00 0.00
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.6000000 0.50 0.00
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.1333333 0.25 0.25
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.0000000 0.25 0.50
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.25
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $A$`7`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0 0.00000000 1.666667e+03 4375.00
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0 0.00000000 1.666667e+03 4375.00
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0 0.00000000 0.000000e+00 0.00
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0 0.00000000 0.000000e+00 0.00
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0 0.00000000 0.000000e+00 0.00
## [6,] 0.00 0.0 0.0 0.0 0.09090909 0 0.09090909 8.333333e-02 0.00
## [7,] 0.00 0.0 0.0 0.0 0.45454545 0 0.45454545 2.500000e-01 0.25
## [8,] 0.00 0.0 0.0 0.0 0.00000000 1 0.36363636 5.833333e-01 0.50
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.000000e+00 0.25
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.000000e+00 0.00
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.000000e+00 0.00
## [,10] [,11]
## [1,] 5000.00 17500
## [2,] 5000.00 17500
## [3,] 0.00 0
## [4,] 0.00 0
## [5,] 0.00 0
## [6,] 0.00 0
## [7,] 0.00 0
## [8,] 0.50 0
## [9,] 0.25 0
## [10,] 0.25 0
## [11,] 0.00 1
##
## $A$`8`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 277.7777778 781.2500 6250.0 5000
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 277.7777778 781.2500 6250.0 5000
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000 0.0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000 0.0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.1875 0.0 0
## [7,] 0.00 0.0 0.0 0.0 0.3333333 1 0.3333333 0.3125 0.0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.3750 0.5 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.0625 0.0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0625 0.5 1
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0
## [,11]
## [1,] 10000
## [2,] 10000
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 1
## [11,] 0
##
## $A$`9`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 250.0 0.0 1250.00
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 250.0 0.0 1250.00
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0.0000000 0.0000000 0.0 0.0 0.00
## [6,] 0.00 0.0 0.0 0.0 0.1818182 0.3333333 0.1818182 0.0 0.0 0.00
## [7,] 0.00 0.0 0.0 0.0 0.2727273 0.3333333 0.2727273 0.2 0.0 0.00
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.4545455 0.2 0.5 0.00
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.4 0.5 0.50
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0.3333333 0.0000000 0.0 0.0 0.25
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.25
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $A$`10`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 714.2857143 1250
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 714.2857143 1250
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.000 0.0000000 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0
## [7,] 0.00 0.0 0.0 0.0 0.5714286 1 0.5714286 0.125 0.0000000 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.2857143 0.750 0.0000000 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.125 0.7142857 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.2857143 1
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0
## [,11]
## [1,] 2500
## [2,] 2500
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 1
##
## $A$`11`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 625.00 833.3333333 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 625.00 833.3333333 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.00 0.0000000 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0.25 0.0000000 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.50 0.3333333 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.25 0.6666667 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 1
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0
## [,11]
## [1,] 7500
## [2,] 7500
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 1
##
## $A$`12`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0 0.00 0.0000000 4166.6666667 1250 5000
## [2,] 0.15 0.0 0.0 0.0 0.00 0 0.00 0.0000000 4166.6666667 1250 5000
## [3,] 0.00 0.1 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0 0.00 0.0000000 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.75 0 0.75 0.1666667 0.0000000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.8333333 0.6666667 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.3333333 1 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 1
## [11,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
##
## $A$`13`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0 0.00 1071.4285714 2500.0000000 2500 0
## [2,] 0.15 0.0 0.0 0.0 0.00 0 0.00 1071.4285714 2500.0000000 2500 0
## [3,] 0.00 0.1 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0 0.00 0.0000000 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.50 0 0.50 0.0000000 0.3333333 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0 0.25 0.5714286 0.3333333 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.2857143 0.3333333 0 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.1428571 0.0000000 1 0
## [11,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
##
## $A$`14`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 416.6666667 0.0000000 0.0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 416.6666667 0.0000000 0.0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.0000000 0.0
## [6,] 0.00 0.0 0.0 0.0 0.3333333 0 0.3333333 0.1666667 0.0000000 0.0
## [7,] 0.00 0.0 0.0 0.0 0.3333333 0 0.3333333 0.1666667 0.3333333 0.0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.3333333 0.3333333 0.5
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.3333333 0.3333333 0.0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.5
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $A$`15`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0.0 0.00 0.00 625 0 0
## [2,] 0.15 0.0 0.0 0.0 0.00 0.0 0.00 0.00 625 0 0
## [3,] 0.00 0.1 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0.0 0.00 0.00 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0.0 0.00 0.00 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.25 0.5 0.25 0.00 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0.5 0.75 0.50 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.25 1 0 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.25 0 0 0
## [11,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 1
##
##
## $U
## $U$`1`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.6363636 0 0.6363636 0.2 0.0000000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.2727273 0.6 0.6666667 1 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.2 0.3333333 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0 0.0000000 0 0
##
## $U$`2`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [2,] 0.15 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [3,] 0.00 0.1 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.000 0 0.000 0.0000000 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.050 0 0.000 0.0000000 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.125 0 0.125 0.0000000 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.750 0 0.750 0.3333333 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.4444444 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.2222222 1 0 0
## [10,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
## [11,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0000000 0 0 0
##
## $U$`3`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0 0.00 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0 0.00 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 1 0.3333333 1 0.25 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.75 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0 0.00 0 0
##
## $U$`4`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.0000000 0
## [6,] 0.00 0.0 0.0 0.0 0.1666667 0 0.1666667 0.2222222 0.0000000 0
## [7,] 0.00 0.0 0.0 0.0 0.5000000 0 0.5000000 0.4444444 0.3333333 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.3333333 0.3333333 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.3333333 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $U$`5`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0.0000000 0.0000000 0.0 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.4444444 0.3333333 0.4444444 0.0 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0.6666667 0.4444444 0.6 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 1 0 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0 0 0
##
## $U$`6`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.00 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.00 0.00 0
## [6,] 0.00 0.0 0.0 0.0 0.1111111 0 0.1111111 0.0000000 0.00 0.00 0
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0.2666667 0.00 0.00 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.6000000 0.50 0.00 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.1333333 0.25 0.25 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.0000000 0.25 0.50 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.00 0.25 0
##
## $U$`7`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.00
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.00
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.00
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.00
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0 0.00000000 0.00000000 0.00 0.00
## [6,] 0.00 0.0 0.0 0.0 0.09090909 0 0.09090909 0.08333333 0.00 0.00
## [7,] 0.00 0.0 0.0 0.0 0.45454545 0 0.45454545 0.25000000 0.25 0.00
## [8,] 0.00 0.0 0.0 0.0 0.00000000 1 0.36363636 0.58333333 0.50 0.50
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.25 0.25
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.25
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0 0.00000000 0.00000000 0.00 0.00
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 1
##
## $U$`8`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000 0.0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000 0.0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.1875 0.0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.3333333 1 0.3333333 0.3125 0.0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.3750 0.5 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.1111111 0.0625 0.0 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0625 0.5 1 1
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000 0.0 0 0
##
## $U$`9`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.00 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0.0000000 0.0000000 0.0 0.0 0.00 0
## [6,] 0.00 0.0 0.0 0.0 0.1818182 0.3333333 0.1818182 0.0 0.0 0.00 0
## [7,] 0.00 0.0 0.0 0.0 0.2727273 0.3333333 0.2727273 0.2 0.0 0.00 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.4545455 0.2 0.5 0.00 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.4 0.5 0.50 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0.3333333 0.0000000 0.0 0.0 0.25 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0 0.0 0.25 0
##
## $U$`10`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.000 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.5714286 1 0.5714286 0.125 0.0000000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.2857143 0.750 0.0000000 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.125 0.7142857 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.2857143 1 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.000 0.0000000 0 1
##
## $U$`11`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.00 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.6666667 0 0.6666667 0.25 0.0000000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.50 0.3333333 0 0
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.25 0.6666667 0 0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 1 0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.00 0.0000000 0 1
##
## $U$`12`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0 0.00 0.0000000 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.75 0 0.75 0.1666667 0.0000000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.8333333 0.6666667 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.3333333 1 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 1
## [11,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
##
## $U$`13`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [2,] 0.15 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [3,] 0.00 0.1 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0 0.00 0.0000000 0.0000000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.50 0 0.50 0.0000000 0.3333333 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0 0.25 0.5714286 0.3333333 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.2857143 0.3333333 0 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.1428571 0.0000000 1 0
## [11,] 0.00 0.0 0.0 0.0 0.00 0 0.00 0.0000000 0.0000000 0 0
##
## $U$`14`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [2,] 0.15 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [3,] 0.00 0.1 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [4,] 0.00 0.0 0.1 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [5,] 0.00 0.0 0.0 0.1 0.0500000 0 0.0000000 0.0000000 0.0000000 0.0
## [6,] 0.00 0.0 0.0 0.0 0.3333333 0 0.3333333 0.1666667 0.0000000 0.0
## [7,] 0.00 0.0 0.0 0.0 0.3333333 0 0.3333333 0.1666667 0.3333333 0.0
## [8,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.3333333 0.3333333 0.5
## [9,] 0.00 0.0 0.0 0.0 0.0000000 0 0.3333333 0.3333333 0.3333333 0.0
## [10,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.0
## [11,] 0.00 0.0 0.0 0.0 0.0000000 0 0.0000000 0.0000000 0.0000000 0.5
## [,11]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
##
## $U$`15`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [2,] 0.15 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [3,] 0.00 0.1 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [4,] 0.00 0.0 0.1 0.0 0.00 0.0 0.00 0.00 0 0 0
## [5,] 0.00 0.0 0.0 0.1 0.05 0.0 0.00 0.00 0 0 0
## [6,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 0
## [7,] 0.00 0.0 0.0 0.0 0.25 0.5 0.25 0.00 0 0 0
## [8,] 0.00 0.0 0.0 0.0 0.00 0.5 0.75 0.50 0 0 0
## [9,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.25 1 0 0
## [10,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.25 0 0 0
## [11,] 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.00 0 0 1
##
##
## $F
## $F$`1`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 500 1666.667 0 0
## [2,] 0 0 0 0 0 0 0 500 1666.667 0 0
## [3,] 0 0 0 0 0 0 0 0 0.000 0 0
## [4,] 0 0 0 0 0 0 0 0 0.000 0 0
## [5,] 0 0 0 0 0 0 0 0 0.000 0 0
## [6,] 0 0 0 0 0 0 0 0 0.000 0 0
## [7,] 0 0 0 0 0 0 0 0 0.000 0 0
## [8,] 0 0 0 0 0 0 0 0 0.000 0 0
## [9,] 0 0 0 0 0 0 0 0 0.000 0 0
## [10,] 0 0 0 0 0 0 0 0 0.000 0 0
## [11,] 0 0 0 0 0 0 0 0 0.000 0 0
##
## $F$`2`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 312.5 1111.111 1250 0 0
## [2,] 0 0 0 0 0 0 312.5 1111.111 1250 0 0
## [3,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [4,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [5,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [6,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [7,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [8,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [9,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [10,] 0 0 0 0 0 0 0.0 0.000 0 0 0
## [11,] 0 0 0 0 0 0 0.0 0.000 0 0 0
##
## $F$`3`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 0 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0
##
## $F$`4`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 0 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0
##
## $F$`5`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 500 5000 0 0
## [2,] 0 0 0 0 0 0 0 500 5000 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0
##
## $F$`6`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 166.6667 625 1875 0
## [2,] 0 0 0 0 0 0 0 166.6667 625 1875 0
## [3,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [4,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [5,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [6,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [7,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [8,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [9,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [10,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [11,] 0 0 0 0 0 0 0 0.0000 0 0 0
##
## $F$`7`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 1666.667 4375 5000 17500
## [2,] 0 0 0 0 0 0 0 1666.667 4375 5000 17500
## [3,] 0 0 0 0 0 0 0 0.000 0 0 0
## [4,] 0 0 0 0 0 0 0 0.000 0 0 0
## [5,] 0 0 0 0 0 0 0 0.000 0 0 0
## [6,] 0 0 0 0 0 0 0 0.000 0 0 0
## [7,] 0 0 0 0 0 0 0 0.000 0 0 0
## [8,] 0 0 0 0 0 0 0 0.000 0 0 0
## [9,] 0 0 0 0 0 0 0 0.000 0 0 0
## [10,] 0 0 0 0 0 0 0 0.000 0 0 0
## [11,] 0 0 0 0 0 0 0 0.000 0 0 0
##
## $F$`8`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 277.7778 781.25 6250 5000 10000
## [2,] 0 0 0 0 0 0 277.7778 781.25 6250 5000 10000
## [3,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [4,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [5,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [6,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [7,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [8,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [9,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [10,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
## [11,] 0 0 0 0 0 0 0.0000 0.00 0 0 0
##
## $F$`9`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 250 0 1250 0
## [2,] 0 0 0 0 0 0 0 250 0 1250 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0
##
## $F$`10`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 0 714.2857 1250 2500
## [2,] 0 0 0 0 0 0 0 0 714.2857 1250 2500
## [3,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [4,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [5,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [6,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [7,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [8,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [9,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [10,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [11,] 0 0 0 0 0 0 0 0 0.0000 0 0
##
## $F$`11`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 625 833.3333 0 7500
## [2,] 0 0 0 0 0 0 0 625 833.3333 0 7500
## [3,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [4,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [5,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [6,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [7,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [8,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [9,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [10,] 0 0 0 0 0 0 0 0 0.0000 0 0
## [11,] 0 0 0 0 0 0 0 0 0.0000 0 0
##
## $F$`12`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 0 4166.667 1250 5000
## [2,] 0 0 0 0 0 0 0 0 4166.667 1250 5000
## [3,] 0 0 0 0 0 0 0 0 0.000 0 0
## [4,] 0 0 0 0 0 0 0 0 0.000 0 0
## [5,] 0 0 0 0 0 0 0 0 0.000 0 0
## [6,] 0 0 0 0 0 0 0 0 0.000 0 0
## [7,] 0 0 0 0 0 0 0 0 0.000 0 0
## [8,] 0 0 0 0 0 0 0 0 0.000 0 0
## [9,] 0 0 0 0 0 0 0 0 0.000 0 0
## [10,] 0 0 0 0 0 0 0 0 0.000 0 0
## [11,] 0 0 0 0 0 0 0 0 0.000 0 0
##
## $F$`13`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 1071.429 2500 2500 0
## [2,] 0 0 0 0 0 0 0 1071.429 2500 2500 0
## [3,] 0 0 0 0 0 0 0 0.000 0 0 0
## [4,] 0 0 0 0 0 0 0 0.000 0 0 0
## [5,] 0 0 0 0 0 0 0 0.000 0 0 0
## [6,] 0 0 0 0 0 0 0 0.000 0 0 0
## [7,] 0 0 0 0 0 0 0 0.000 0 0 0
## [8,] 0 0 0 0 0 0 0 0.000 0 0 0
## [9,] 0 0 0 0 0 0 0 0.000 0 0 0
## [10,] 0 0 0 0 0 0 0 0.000 0 0 0
## [11,] 0 0 0 0 0 0 0 0.000 0 0 0
##
## $F$`14`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 416.6667 0 0 0
## [2,] 0 0 0 0 0 0 0 416.6667 0 0 0
## [3,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [4,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [5,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [6,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [7,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [8,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [9,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [10,] 0 0 0 0 0 0 0 0.0000 0 0 0
## [11,] 0 0 0 0 0 0 0 0.0000 0 0 0
##
## $F$`15`
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 0 625 0 0
## [2,] 0 0 0 0 0 0 0 0 625 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0
##
##
## $hstages
## [1] NA
##
## $agestages
## [1] NA
##
## $ahstages
## stage_id stage original_size original_size_b original_size_c min_age max_age
## 1 1 SD 0.0 NA NA 1 NA
## 2 2 P1 0.0 NA NA 1 NA
## 3 3 P2 0.0 NA NA 2 NA
## 4 4 P3 0.0 NA NA 3 NA
## 5 5 SL 0.0 NA NA 4 NA
## 6 6 D 0.0 NA NA 5 NA
## 7 7 XSm 1.0 NA NA 5 NA
## 8 8 Sm 3.0 NA NA 6 NA
## 9 9 Md 6.0 NA NA 6 NA
## 10 10 Lg 11.0 NA NA 6 NA
## 11 11 XLg 19.5 NA NA 6 NA
## repstatus obsstatus propstatus immstatus matstatus entrystage indataset
## 1 0 0 1 0 0 1 0
## 2 0 0 0 1 0 1 0
## 3 0 0 0 1 0 0 0
## 4 0 0 0 1 0 0 0
## 5 0 0 0 1 0 0 0
## 6 0 0 0 0 1 0 1
## 7 1 1 0 0 1 0 1
## 8 1 1 0 0 1 0 1
## 9 1 1 0 0 1 0 1
## 10 1 1 0 0 1 0 1
## 11 1 1 0 0 1 0 1
## binhalfwidth_raw sizebin_min sizebin_max sizebin_center sizebin_width
## 1 0.0 0.0 0.0 0.0 0
## 2 0.0 0.0 0.0 0.0 0
## 3 0.0 0.0 0.0 0.0 0
## 4 0.0 0.0 0.0 0.0 0
## 5 0.0 0.0 0.0 0.0 0
## 6 0.5 -0.5 0.5 0.0 1
## 7 0.5 0.5 1.5 1.0 1
## 8 1.5 1.5 4.5 3.0 3
## 9 1.5 4.5 7.5 6.0 3
## 10 3.5 7.5 14.5 11.0 7
## 11 5.0 14.5 24.5 19.5 10
## binhalfwidthb_raw sizebinb_min sizebinb_max sizebinb_center sizebinb_width
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## binhalfwidthc_raw sizebinc_min sizebinc_max sizebinc_center sizebinc_width
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## group comments alive almostborn
## 1 0 Dormant seed 1 0
## 2 0 1st yr protocorm 1 0
## 3 0 2nd yr protocorm 1 0
## 4 0 3rd yr protocorm 1 0
## 5 0 Seedling 1 0
## 6 0 Dormant adult 1 0
## 7 0 Extra small adult (1 shoot) 1 0
## 8 0 Small adult (2-4 shoots) 1 0
## 9 0 Medium adult (5-7 shoots) 1 0
## 10 0 Large adult (8-14 shoots) 1 0
## 11 0 Extra large adult (>14 shoots) 1 0
##
## $labels
## pop patch year2
## 1 1 A 2004
## 2 1 A 2005
## 3 1 A 2006
## 4 1 A 2007
## 5 1 A 2008
## 6 1 B 2004
## 7 1 B 2005
## 8 1 B 2006
## 9 1 B 2007
## 10 1 B 2008
## 11 1 C 2004
## 12 1 C 2005
## 13 1 C 2006
## 14 1 C 2007
## 15 1 C 2008
##
## $matrixqc
## [1] 255 70 15
##
## $dataqc
## [1] 74 320
##
## attr(,"class")
## [1] "lefkoMat"The output from this analysis is a lefkoMat object,
which is a list object with the following elements:
A: a list of full population projection
matrices, in order of population, patch, and year
U: a list of matrices showing only
survival-transition elements, in the same order as A
F: a list of matrices showing only
fecundity elements, in the same order as A
hstages: a data frame showing the order
of paired stages (given if matrices are historical, otherwise
NA)
agestages: this is a data frame showing
the order of age-stages (if an age-by-stage MPM has been created,
otherwise NA)
ahstages: this is the stageframe used
in analysis, with stages reordered and edited as they occur in the
matrix
labels: a table showing the order of
matrices, according to population, patch, and year
matrixqc: a short vector used in
summary() statements to describe the overall quality of
each matrix
dataqc: a short vector used in
summary() statements to describe key sampling aspects of
the dataset (in raw MPMs)
modelqc: a short vector used in
summary() statements to describe the vital rate models (in
function-based MPMs)
Calling particular values and elements within lefkoMat
objects is not complicated, once you know the structure. The figure
below illustrates how to call a particular element from one of the A
matrices, for example.
Figure 6. Organization of a lefkoMat
object, and how to call a specific element.
Objects of class lefkoMat have their own
summary() statements, which we can use to understand more
about them.
summary(cypmatrix2rp)##
## This ahistorical lefkoMat object contains 15 matrices.
##
## Each matrix is square with 11 rows and columns, and a total of 121 elements.
## A total of 255 survival transitions were estimated, with 17 per matrix.
## A total of 70 fecundity transitions were estimated, with 4.667 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## Min. 0.000 0.000 0.000 0.000 0.000 0.000 0.100 0.100 0.000 0.100 0.000 0.000
## 1st Qu. 0.100 0.050 0.100 0.050 0.100 0.100 0.140 0.140 0.100 0.140 0.100 0.100
## Median 0.180 0.100 0.180 0.100 0.180 0.180 0.909 0.778 0.505 0.857 0.717 0.750
## Mean 0.461 0.389 0.472 0.351 0.406 0.483 0.627 0.604 0.518 0.633 0.563 0.548
## 3rd Qu. 0.955 0.900 1.000 0.692 0.744 1.000 1.000 1.000 0.955 1.000 1.000 1.000
## Max. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## [,13] [,14] [,15]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.100 0.100 0.100
## Median 0.180 0.180 0.300
## Mean 0.435 0.472 0.525
## 3rd Qu. 0.875 1.000 1.000
## Max. 1.000 1.000 1.000We start off learning that 15 matrices have been estimated. This is
followed by the matrix dimensions. Of note here is the output telling us
how many elements were actually estimated as non-zero, both overall and
per matrix, and the number of individuals and transitions the matrices
are based on. It is typical for population ecologists to consider the
total number of transitions in a dataset as a measure of the statistical
power of a matrix, but the number of individuals used is just as
important because each transition that an individual experiences is
dependent on the other transitions that it also experiences. Then we see
the number of populations, subpopulations / patches, and time steps
covered by the MPM. The final bit of the summary shows us the range of
survival probabilities of stages in the matrices, where the survival
probabilities are calculated as column sums of each U
matrix. Since there are 15 matrices, there are 15 column sum summaries.
It is important to check to see that no stage survives outside the realm
of possibility (i.e. no probability should be greater than 1.0 or lower
than 0.0). Unusual stage survival probabilities will result in a warning
as a part of the summary() output.
The input for the rlefko2() function includes
patch = "all" and year = "all", but can be set
to focus on any set of patches / subpopulations or years included within
the data. Package lefko3 includes a great deal of
flexibility here, and can estimate many matrices covering all of the
populations, patches, and years occurring in a specific dataset. For
example, if we had wished to skip the patch divisions within the
population and instead estimate only annual matrices at the population
level, then we could have eliminated the patch option
altogether from the input, as below.
cypmatrix2r <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
year = "all", stages = c("stage3", "stage2"),
size = c("size3added", "size2added"), supplement = cypsupp2_raw,
yearcol = "year2", patchcol = "patchid", indivcol = "individ")
summary(cypmatrix2r)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 11 rows and columns, and a total of 121 elements.
## A total of 115 survival transitions were estimated, with 23 per matrix.
## A total of 40 fecundity transitions were estimated, with 8 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.100 0.100 0.000 0.100
## 1st Qu. 0.100 0.140 0.140 0.100 0.140
## Median 0.746 0.870 0.864 0.600 0.882
## Mean 0.562 0.642 0.627 0.532 0.615
## 3rd Qu. 1.000 1.000 1.000 0.960 1.000
## Max. 1.000 1.000 1.000 1.000 1.000The first call to rlefko2() yielded 15 matrices, because
there are three patches in our dataset, and there are a total of six
years of data, yielding five transitions between years (also referred to
as time steps or periods). So, there are \(3
\times 5 = 15\) matrices. But in the second call, we no longer
recognize patches and so have only estimated one set of five matrices
covering the whole population. We can also focus in on specific patches
and specific sets of years, setting the options appropriately.
Now let’s estimate a raw, historical MPM, using the
rlefko3() function.
cypmatrix3rp <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
year = "all", patch = "all", stages = c("stage3", "stage2", "stage1"),
size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw,
yearcol = "year2", patchcol = "patchid", indivcol = "individ")
summary(cypmatrix3rp)##
## This historical lefkoMat object contains 12 matrices.
##
## Each matrix is square with 121 rows and columns, and a total of 14641 elements.
## A total of 386 survival transitions were estimated, with 32.167 per matrix.
## A total of 70 fecundity transitions were estimated, with 5.833 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 4 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## Min. 0.0000 0.0000 0.0000 0.0000 0.000 0.00 0.000 0.000 0.0000 0.000 0.0000
## 1st Qu. 0.0000 0.0000 0.0000 0.0000 0.000 0.00 0.000 0.000 0.0000 0.000 0.0000
## Median 0.0000 0.0000 0.0000 0.0000 0.000 0.00 0.000 0.000 0.0000 0.000 0.0000
## Mean 0.0862 0.0706 0.0665 0.0754 0.124 0.13 0.123 0.135 0.0968 0.072 0.0968
## 3rd Qu. 0.0000 0.0000 0.0000 0.0000 0.000 0.00 0.000 0.000 0.0000 0.000 0.0000
## Max. 1.0000 1.0000 1.0000 1.0000 1.000 1.00 1.000 1.000 1.0500 1.000 1.0000
## [,12]
## Min. 0.000
## 1st Qu. 0.000
## Median 0.000
## Mean 0.113
## 3rd Qu. 0.000
## Max. 1.000## Warning: Some matrices include stages with survival probability greater than
## 1.0.Quickly scanning this output shows a number of important differences. First, there are three fewer matrices here than in the ahistorical case. There are three patches that we are estimating matrices for, and six years of data for each patch, leading to five possible ahistorical time steps and 15 possible ahistorical matrices. Since historical matrices require three consecutive years of transition data to estimate a single matrix element, only four historical transitions are possible per patch, leading to 12 total historical matrices. Second, the dimensionality of the matrices is the square of the dimensions of the ahistorical matrices. This leads to vastly more matrix elements within each matrix, although it turns out that most of these matrix elements are structural zeros because they reflect impossible transitions. Indeed, in this case, although there are 14,641 elements in each matrix, on average only 38 are actually estimated to values greater than zero. Finally, we see that one of our matrices has a survival probability greater than 1.0. This is a problem for us, and normally we would need to correct the situation via the supplemental table. However, for the time being we will proceed without any correction (no other MPMs here will have this problem).
Let’s look at the first historical matrix, corresponding to the transition from 2004 and 2005 to 2006 in the first patch. Because this is a huge matrix, we will only look at the top corner, followed by a middle section. The full matrix is not shown here, but we can focus on portions of it if we wish. These matrices may also be exported to Excel or another spreadsheet program to look over in detail. Particularly note the sparseness - most elements are zeros, because most transitions are actually impossible.
cypmatrix3rp$A[[1]][1:20,1:10]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.01 0.0 0 0 0 0 0 0 0 0
## [2,] 0.10 0.0 0 0 0 0 0 0 0 0
## [3,] 0.00 0.0 0 0 0 0 0 0 0 0
## [4,] 0.00 0.0 0 0 0 0 0 0 0 0
## [5,] 0.00 0.0 0 0 0 0 0 0 0 0
## [6,] 0.00 0.0 0 0 0 0 0 0 0 0
## [7,] 0.00 0.0 0 0 0 0 0 0 0 0
## [8,] 0.00 0.0 0 0 0 0 0 0 0 0
## [9,] 0.00 0.0 0 0 0 0 0 0 0 0
## [10,] 0.00 0.0 0 0 0 0 0 0 0 0
## [11,] 0.00 0.0 0 0 0 0 0 0 0 0
## [12,] 0.00 0.0 0 0 0 0 0 0 0 0
## [13,] 0.00 0.0 0 0 0 0 0 0 0 0
## [14,] 0.00 0.1 0 0 0 0 0 0 0 0
## [15,] 0.00 0.0 0 0 0 0 0 0 0 0
## [16,] 0.00 0.0 0 0 0 0 0 0 0 0
## [17,] 0.00 0.0 0 0 0 0 0 0 0 0
## [18,] 0.00 0.0 0 0 0 0 0 0 0 0
## [19,] 0.00 0.0 0 0 0 0 0 0 0 0
## [20,] 0.00 0.0 0 0 0 0 0 0 0 0print(cypmatrix3rp$A[[1]][66:85,73:81], digits = 3)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.000 0.000 0 0 0 0 0 0 0
## [2,] 357.143 0.000 0 0 0 0 0 0 0
## [3,] 357.143 0.000 0 0 0 0 0 0 0
## [4,] 0.000 0.000 0 0 0 0 0 0 0
## [5,] 0.000 0.000 0 0 0 0 0 0 0
## [6,] 0.000 0.000 0 0 0 0 0 0 0
## [7,] 0.143 0.000 0 0 0 0 0 0 0
## [8,] 0.714 0.000 0 0 0 0 0 0 0
## [9,] 0.000 0.000 0 0 0 0 0 0 0
## [10,] 0.000 0.000 0 0 0 0 0 0 0
## [11,] 0.000 0.000 0 0 0 0 0 0 0
## [12,] 0.000 0.000 0 0 0 0 0 0 0
## [13,] 0.000 1666.667 0 0 0 0 0 0 0
## [14,] 0.000 1666.667 0 0 0 0 0 0 0
## [15,] 0.000 0.000 0 0 0 0 0 0 0
## [16,] 0.000 0.000 0 0 0 0 0 0 0
## [17,] 0.000 0.000 0 0 0 0 0 0 0
## [18,] 0.000 0.000 0 0 0 0 0 0 0
## [19,] 0.000 0.667 0 0 0 0 0 0 0
## [20,] 0.000 0.333 0 0 0 0 0 0 0Continuing with the styles of MPM, we might next wish to create a Leslie MPM. Age-based MPMs are inherently ahistorical, and do not require stageframes since only the ages to be modeled and whether life can continue past the final age need to be specified.
cypleslie_r <- rleslie(cypraw_v1, fecage_min = 5, yearcol = "year2",
indivcol = "individ")
summary(cypleslie_r)##
## This ahistorical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 10 rows and columns, and a total of 100 elements.
## A total of 11 survival transitions were estimated, with 2.75 per matrix.
## A total of 7 fecundity transitions were estimated, with 1.75 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 4 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.000 0.000 0.000 0.000
## 1st Qu. 0.000 0.000 0.000 0.000
## Median 0.000 0.000 0.000 0.000
## Mean 0.170 0.163 0.292 0.346
## 3rd Qu. 0.000 0.000 0.686 0.847
## Max. 0.953 0.967 1.000 1.000Let’s take a peek at the first matrix.
cypleslie_r$A[[1]]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0 0 0 0 0.00 0.687500 0 0 0 0
## [2,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [3,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [4,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [5,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [6,] 0 0 0 0 0.75 0.000000 0 0 0 0
## [7,] 0 0 0 0 0.00 0.953125 0 0 0 0
## [8,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [9,] 0 0 0 0 0.00 0.000000 0 0 0 0
## [10,] 0 0 0 0 0.00 0.000000 0 0 0 0Here we have a problem because we have not incorporated supplemental
information into the matrix construction. So, we do not have the first
few years of survival, and we do not have any fecundity multipliers
included. There is currently no way to incorporate supplemental info in
the main matrix construction calls for Leslie MPMs in
lefko3, so we will do so manually. This turns out to be
relatively simple, and can be done using for loops.
for (i in c(1:length(cypleslie_r$A))) {
cypleslie_r$U[[i]][2, 1] <- 0.1
cypleslie_r$U[[i]][3, 2] <- 0.1
cypleslie_r$U[[i]][4, 3] <- 0.1
cypleslie_r$U[[i]][5, 4] <- cypleslie_r$U[[i]][6, 5]
cypleslie_r$F[[i]][1, 6] <- cypleslie_r$F[[i]][1, 6] * seeds_per_pod * 0.15
cypleslie_r$F[[i]][1, 7] <- cypleslie_r$F[[i]][1, 7] * seeds_per_pod * 0.15
cypleslie_r$F[[i]][1, 8] <- cypleslie_r$F[[i]][1, 8] * seeds_per_pod * 0.15
cypleslie_r$F[[i]][1, 9] <- cypleslie_r$F[[i]][1, 9] * seeds_per_pod * 0.15
cypleslie_r$F[[i]][1, 10] <- cypleslie_r$F[[i]][1, 10] * seeds_per_pod * 0.15
cypleslie_r$A[[i]] <- cypleslie_r$U[[i]] + cypleslie_r$F[[i]]
}
cypleslie_r$A[[1]]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.0 0.0 0.0 0.00 0.00 515.625000 0 0 0 0
## [2,] 0.1 0.0 0.0 0.00 0.00 0.000000 0 0 0 0
## [3,] 0.0 0.1 0.0 0.00 0.00 0.000000 0 0 0 0
## [4,] 0.0 0.0 0.1 0.00 0.00 0.000000 0 0 0 0
## [5,] 0.0 0.0 0.0 0.75 0.00 0.000000 0 0 0 0
## [6,] 0.0 0.0 0.0 0.00 0.75 0.000000 0 0 0 0
## [7,] 0.0 0.0 0.0 0.00 0.00 0.953125 0 0 0 0
## [8,] 0.0 0.0 0.0 0.00 0.00 0.000000 0 0 0 0
## [9,] 0.0 0.0 0.0 0.00 0.00 0.000000 0 0 0 0
## [10,] 0.0 0.0 0.0 0.00 0.00 0.000000 0 0 0 0Now let’s take a look at a summary.
summary(cypleslie_r)##
## This ahistorical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 10 rows and columns, and a total of 100 elements.
## A total of 11 survival transitions were estimated, with 2.75 per matrix.
## A total of 7 fecundity transitions were estimated, with 1.75 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 4 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.000 0.000 0.000 0.000
## 1st Qu. 0.000 0.000 0.025 0.100
## Median 0.100 0.050 0.100 0.300
## Mean 0.275 0.193 0.422 0.426
## 3rd Qu. 0.588 0.100 0.979 0.847
## Max. 0.953 0.967 1.000 1.000Finally, before we head to function-based MPMs and IPMs, we might
wish to construct an age-by-stage MPM. These are large matrices, and so
require more data to parameterize properly, but we will give it a shot
anyway. Note that we will use the stages option, using the
stageframe in this context, where non-reproductive individuals are still
treated as reproductive, may yield errors otherwise.
cypagestage_r <- arlefko2(cypraw_v1, cypframe_raw, indivcol = "individ",
yearcol = "year2", size = c("size3added", "size2added", "size1added"),
stages = c("stage3", "stage2", "stage1"), supplement = cypsupp2_raw)
summary(cypagestage_r)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 99 rows and columns, and a total of 9801 elements.
## A total of 313 survival transitions were estimated, with 62.6 per matrix.
## A total of 38 fecundity transitions were estimated, with 7.6 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.0000 0.00 0.000 0.000 0.0000
## 1st Qu. 0.0000 0.00 0.000 0.000 0.0000
## Median 0.0000 0.00 0.000 0.000 0.0000
## Mean 0.0603 0.12 0.119 0.127 0.0908
## 3rd Qu. 0.1000 0.10 0.100 0.100 0.1000
## Max. 0.9565 1.00 1.000 1.050 1.0500## Warning: Some matrices include stages with survival probability greater than
## 1.0.As before, we have an issue with some survival probabilities greater than 1.0. This is caused by our supplemental info, and ideally we should fix this prior to using this matrix in an analysis. For now, however, we will push ahead to other MPMs.
Now let’s estimate the function-based matrices. Let’s start off with
the ahistorical function-based matrix. Note that we have far fewer terms
required to build a basic function-based MPM than to build a raw MPM,
because so much of the parameterization is coded within the
lefkoMod object holding the vital rate models.
cypmatrix2fp <- flefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
modelsuite = cypmodels2p, data = cypfb_env)
summary(cypmatrix2fp)##
## This ahistorical lefkoMat object contains 15 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 36150 survival transitions were estimated, with 2410 per matrix.
## A total of 720 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## Min. 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100
## 1st Qu. 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991
## Median 0.999 1.000 1.000 1.000 0.999 0.998 1.000 0.999 0.999 0.999 0.997 0.999
## Mean 0.918 0.920 0.920 0.920 0.916 0.917 0.920 0.919 0.919 0.916 0.915 0.919
## 3rd Qu. 1.000 1.000 1.000 1.000 1.000 0.999 1.000 1.000 1.000 0.999 0.999 1.000
## Max. 1.000 1.000 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 0.999 1.000
## [,13] [,14] [,15]
## Min. 0.100 0.100 0.100
## 1st Qu. 0.991 0.991 0.991
## Median 0.999 0.998 0.997
## Mean 0.918 0.918 0.914
## 3rd Qu. 0.999 0.999 0.999
## Max. 1.000 0.999 0.999Here we have the same number of patches as in the raw ahistorical MPM, and patch-level matrices were estimated without us needing to specify anything because patch was a factor in the vital rate models. The same number of matrices were created as in the raw case, but the raw MPM process yielded an average of 21.667 estimated transitions and only 11 rows and 11 columns per matrix. In contrast, the function-based MPM process led to 2,458 elements and 54 rows and 54 columns per matrix. This happens because we are using a different life history model with many more stages and because we are using our vital rate models to propagate every matrix element that is biologically possible. To see the impact, let’s compare the first raw matrix to the first function-based matrix, as below.
writeLines("First matrix in raw ahMPM:")## First matrix in raw ahMPM:print(cypmatrix2rp$A[[1]], digits = 3)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0.03 0.0 0.0 0.0 0.000 0 0.000 500.0 1666.667 0 0
## [2,] 0.15 0.0 0.0 0.0 0.000 0 0.000 500.0 1666.667 0 0
## [3,] 0.00 0.1 0.0 0.0 0.000 0 0.000 0.0 0.000 0 0
## [4,] 0.00 0.0 0.1 0.0 0.000 0 0.000 0.0 0.000 0 0
## [5,] 0.00 0.0 0.0 0.1 0.050 0 0.000 0.0 0.000 0 0
## [6,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0 0.000 0 0
## [7,] 0.00 0.0 0.0 0.0 0.636 0 0.636 0.2 0.000 0 0
## [8,] 0.00 0.0 0.0 0.0 0.000 0 0.273 0.6 0.667 1 0
## [9,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.2 0.333 0 0
## [10,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0 0.000 0 0
## [11,] 0.00 0.0 0.0 0.0 0.000 0 0.000 0.0 0.000 0 0writeLines("\nFirst matrix in function-based ahMPM:")##
## First matrix in function-based ahMPM:print(cypmatrix2fp$A[[1]], digits = 3)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [2,] 0.15 0.0 0.0 0.0 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [3,] 0.00 0.1 0.0 0.0 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00 0.0 0.1 0.0 0.000 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00 0.0 0.0 0.1 0.050 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 0.00 0.0 0.0 0.0 0.047 4.70e-02 3.69e-02 2.81e-02 2.10e-02 1.56e-02
## [7,] 0.00 0.0 0.0 0.0 0.237 2.37e-01 2.40e-01 2.34e-01 2.23e-01 2.09e-01
## [8,] 0.00 0.0 0.0 0.0 0.149 1.49e-01 1.52e-01 1.50e-01 1.44e-01 1.36e-01
## [9,] 0.00 0.0 0.0 0.0 0.000 9.37e-02 9.63e-02 9.55e-02 9.25e-02 8.80e-02
## [10,] 0.00 0.0 0.0 0.0 0.000 5.89e-02 6.11e-02 6.10e-02 5.96e-02 5.71e-02
## [11,] 0.00 0.0 0.0 0.0 0.000 3.71e-02 3.87e-02 3.90e-02 3.84e-02 3.71e-02
## [12,] 0.00 0.0 0.0 0.0 0.000 2.33e-02 2.45e-02 2.49e-02 2.47e-02 2.41e-02
## [13,] 0.00 0.0 0.0 0.0 0.000 1.47e-02 1.56e-02 1.59e-02 1.59e-02 1.56e-02
## [14,] 0.00 0.0 0.0 0.0 0.000 9.22e-03 9.86e-03 1.02e-02 1.02e-02 1.01e-02
## [15,] 0.00 0.0 0.0 0.0 0.000 5.80e-03 6.25e-03 6.50e-03 6.60e-03 6.58e-03
## [16,] 0.00 0.0 0.0 0.0 0.000 3.64e-03 3.96e-03 4.16e-03 4.25e-03 4.27e-03
## [17,] 0.00 0.0 0.0 0.0 0.000 2.29e-03 2.51e-03 2.66e-03 2.74e-03 2.77e-03
## [18,] 0.00 0.0 0.0 0.0 0.000 1.44e-03 1.59e-03 1.70e-03 1.76e-03 1.80e-03
## [19,] 0.00 0.0 0.0 0.0 0.000 9.06e-04 1.01e-03 1.08e-03 1.14e-03 1.17e-03
## [20,] 0.00 0.0 0.0 0.0 0.000 5.70e-04 6.40e-04 6.93e-04 7.31e-04 7.58e-04
## [21,] 0.00 0.0 0.0 0.0 0.000 3.58e-04 4.06e-04 4.43e-04 4.71e-04 4.92e-04
## [22,] 0.00 0.0 0.0 0.0 0.000 2.25e-04 2.57e-04 2.83e-04 3.03e-04 3.19e-04
## [23,] 0.00 0.0 0.0 0.0 0.000 1.42e-04 1.63e-04 1.81e-04 1.95e-04 2.07e-04
## [24,] 0.00 0.0 0.0 0.0 0.000 8.91e-05 1.03e-04 1.15e-04 1.26e-04 1.34e-04
## [25,] 0.00 0.0 0.0 0.0 0.000 5.60e-05 6.55e-05 7.38e-05 8.10e-05 8.73e-05
## [26,] 0.00 0.0 0.0 0.0 0.000 3.52e-05 4.15e-05 4.72e-05 5.22e-05 5.66e-05
## [27,] 0.00 0.0 0.0 0.0 0.000 2.21e-05 2.63e-05 3.01e-05 3.36e-05 3.68e-05
## [28,] 0.00 0.0 0.0 0.0 0.000 1.39e-05 1.67e-05 1.93e-05 2.16e-05 2.39e-05
## [29,] 0.00 0.0 0.0 0.0 0.000 8.76e-06 1.06e-05 1.23e-05 1.39e-05 1.55e-05
## [30,] 0.00 0.0 0.0 0.0 0.000 5.51e-06 6.70e-06 7.86e-06 8.98e-06 1.00e-05
## [31,] 0.00 0.0 0.0 0.0 0.000 7.38e-02 8.92e-02 1.04e-01 1.19e-01 1.33e-01
## [32,] 0.00 0.0 0.0 0.0 0.000 4.64e-02 5.65e-02 6.65e-02 7.64e-02 8.63e-02
## [33,] 0.00 0.0 0.0 0.0 0.000 2.92e-02 3.58e-02 4.25e-02 4.92e-02 5.60e-02
## [34,] 0.00 0.0 0.0 0.0 0.000 1.83e-02 2.27e-02 2.72e-02 3.17e-02 3.63e-02
## [35,] 0.00 0.0 0.0 0.0 0.000 1.15e-02 1.44e-02 1.74e-02 2.04e-02 2.36e-02
## [36,] 0.00 0.0 0.0 0.0 0.000 7.25e-03 9.13e-03 1.11e-02 1.31e-02 1.53e-02
## [37,] 0.00 0.0 0.0 0.0 0.000 4.56e-03 5.79e-03 7.09e-03 8.47e-03 9.94e-03
## [38,] 0.00 0.0 0.0 0.0 0.000 2.87e-03 3.67e-03 4.53e-03 5.45e-03 6.45e-03
## [39,] 0.00 0.0 0.0 0.0 0.000 1.80e-03 2.33e-03 2.89e-03 3.51e-03 4.18e-03
## [40,] 0.00 0.0 0.0 0.0 0.000 1.13e-03 1.47e-03 1.85e-03 2.26e-03 2.72e-03
## [41,] 0.00 0.0 0.0 0.0 0.000 7.13e-04 9.34e-04 1.18e-03 1.46e-03 1.76e-03
## [42,] 0.00 0.0 0.0 0.0 0.000 4.48e-04 5.92e-04 7.55e-04 9.38e-04 1.14e-03
## [43,] 0.00 0.0 0.0 0.0 0.000 2.82e-04 3.75e-04 4.82e-04 6.04e-04 7.43e-04
## [44,] 0.00 0.0 0.0 0.0 0.000 1.77e-04 2.38e-04 3.08e-04 3.89e-04 4.82e-04
## [45,] 0.00 0.0 0.0 0.0 0.000 1.11e-04 1.51e-04 1.97e-04 2.51e-04 3.13e-04
## [46,] 0.00 0.0 0.0 0.0 0.000 7.01e-05 9.56e-05 1.26e-04 1.61e-04 2.03e-04
## [47,] 0.00 0.0 0.0 0.0 0.000 4.41e-05 6.06e-05 8.04e-05 1.04e-04 1.32e-04
## [48,] 0.00 0.0 0.0 0.0 0.000 2.77e-05 3.84e-05 5.14e-05 6.69e-05 8.55e-05
## [49,] 0.00 0.0 0.0 0.0 0.000 1.74e-05 2.44e-05 3.28e-05 4.31e-05 5.55e-05
## [50,] 0.00 0.0 0.0 0.0 0.000 1.10e-05 1.54e-05 2.10e-05 2.78e-05 3.60e-05
## [51,] 0.00 0.0 0.0 0.0 0.000 6.89e-06 9.79e-06 1.34e-05 1.79e-05 2.34e-05
## [52,] 0.00 0.0 0.0 0.0 0.000 4.33e-06 6.21e-06 8.56e-06 1.15e-05 1.52e-05
## [53,] 0.00 0.0 0.0 0.0 0.000 2.72e-06 3.93e-06 5.47e-06 7.41e-06 9.85e-06
## [54,] 0.00 0.0 0.0 0.0 0.000 1.71e-06 2.49e-06 3.50e-06 4.77e-06 6.39e-06
## [,11] [,12] [,13] [,14] [,15] [,16] [,17] [,18]
## [1,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [2,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 1.15e-02 8.47e-03 6.21e-03 4.55e-03 3.33e-03 2.44e-03 1.78e-03 1.30e-03
## [7,] 1.93e-01 1.77e-01 1.60e-01 1.43e-01 1.27e-01 1.12e-01 9.81e-02 8.52e-02
## [8,] 1.26e-01 1.16e-01 1.06e-01 9.58e-02 8.57e-02 7.61e-02 6.70e-02 5.86e-02
## [9,] 8.26e-02 7.67e-02 7.04e-02 6.41e-02 5.78e-02 5.16e-02 4.58e-02 4.03e-02
## [10,] 5.40e-02 5.05e-02 4.67e-02 4.28e-02 3.89e-02 3.50e-02 3.13e-02 2.77e-02
## [11,] 3.53e-02 3.33e-02 3.10e-02 2.86e-02 2.62e-02 2.38e-02 2.14e-02 1.91e-02
## [12,] 2.31e-02 2.19e-02 2.06e-02 1.92e-02 1.76e-02 1.61e-02 1.46e-02 1.31e-02
## [13,] 1.51e-02 1.45e-02 1.37e-02 1.28e-02 1.19e-02 1.09e-02 9.97e-03 9.03e-03
## [14,] 9.89e-03 9.53e-03 9.08e-03 8.57e-03 8.01e-03 7.42e-03 6.81e-03 6.21e-03
## [15,] 6.47e-03 6.28e-03 6.03e-03 5.73e-03 5.39e-03 5.03e-03 4.65e-03 4.27e-03
## [16,] 4.23e-03 4.14e-03 4.00e-03 3.83e-03 3.63e-03 3.41e-03 3.18e-03 2.94e-03
## [17,] 2.77e-03 2.73e-03 2.66e-03 2.56e-03 2.45e-03 2.32e-03 2.17e-03 2.02e-03
## [18,] 1.81e-03 1.80e-03 1.76e-03 1.71e-03 1.65e-03 1.57e-03 1.48e-03 1.39e-03
## [19,] 1.18e-03 1.18e-03 1.17e-03 1.15e-03 1.11e-03 1.07e-03 1.01e-03 9.56e-04
## [20,] 7.74e-04 7.80e-04 7.77e-04 7.66e-04 7.48e-04 7.23e-04 6.92e-04 6.57e-04
## [21,] 5.06e-04 5.14e-04 5.16e-04 5.12e-04 5.04e-04 4.90e-04 4.73e-04 4.52e-04
## [22,] 3.31e-04 3.39e-04 3.42e-04 3.43e-04 3.39e-04 3.33e-04 3.23e-04 3.11e-04
## [23,] 2.16e-04 2.23e-04 2.27e-04 2.29e-04 2.28e-04 2.26e-04 2.21e-04 2.14e-04
## [24,] 1.42e-04 1.47e-04 1.51e-04 1.53e-04 1.54e-04 1.53e-04 1.51e-04 1.47e-04
## [25,] 9.26e-05 9.69e-05 1.00e-04 1.02e-04 1.04e-04 1.04e-04 1.03e-04 1.01e-04
## [26,] 6.05e-05 6.38e-05 6.65e-05 6.85e-05 6.98e-05 7.04e-05 7.03e-05 6.96e-05
## [27,] 3.96e-05 4.21e-05 4.42e-05 4.58e-05 4.70e-05 4.78e-05 4.81e-05 4.79e-05
## [28,] 2.59e-05 2.77e-05 2.93e-05 3.06e-05 3.17e-05 3.24e-05 3.28e-05 3.29e-05
## [29,] 1.69e-05 1.83e-05 1.95e-05 2.05e-05 2.13e-05 2.20e-05 2.24e-05 2.27e-05
## [30,] 1.11e-05 1.20e-05 1.29e-05 1.37e-05 1.44e-05 1.49e-05 1.53e-05 1.56e-05
## [31,] 1.47e-01 1.61e-01 1.74e-01 1.86e-01 1.98e-01 2.09e-01 2.18e-01 2.27e-01
## [32,] 9.61e-02 1.06e-01 1.15e-01 1.25e-01 1.33e-01 1.41e-01 1.49e-01 1.56e-01
## [33,] 6.28e-02 6.97e-02 7.66e-02 8.33e-02 8.98e-02 9.60e-02 1.02e-01 1.07e-01
## [34,] 4.11e-02 4.59e-02 5.08e-02 5.57e-02 6.05e-02 6.51e-02 6.95e-02 7.37e-02
## [35,] 2.69e-02 3.03e-02 3.37e-02 3.72e-02 4.07e-02 4.42e-02 4.75e-02 5.07e-02
## [36,] 1.76e-02 2.00e-02 2.24e-02 2.49e-02 2.74e-02 3.00e-02 3.25e-02 3.49e-02
## [37,] 1.15e-02 1.31e-02 1.49e-02 1.67e-02 1.85e-02 2.03e-02 2.22e-02 2.40e-02
## [38,] 7.52e-03 8.66e-03 9.87e-03 1.11e-02 1.24e-02 1.38e-02 1.51e-02 1.65e-02
## [39,] 4.92e-03 5.71e-03 6.55e-03 7.45e-03 8.38e-03 9.35e-03 1.03e-02 1.13e-02
## [40,] 3.22e-03 3.76e-03 4.35e-03 4.98e-03 5.65e-03 6.34e-03 7.07e-03 7.81e-03
## [41,] 2.10e-03 2.48e-03 2.89e-03 3.33e-03 3.80e-03 4.30e-03 4.83e-03 5.37e-03
## [42,] 1.38e-03 1.63e-03 1.92e-03 2.23e-03 2.56e-03 2.92e-03 3.30e-03 3.69e-03
## [43,] 9.00e-04 1.08e-03 1.27e-03 1.49e-03 1.73e-03 1.98e-03 2.25e-03 2.54e-03
## [44,] 5.88e-04 7.09e-04 8.45e-04 9.96e-04 1.16e-03 1.34e-03 1.54e-03 1.75e-03
## [45,] 3.85e-04 4.67e-04 5.61e-04 6.66e-04 7.83e-04 9.11e-04 1.05e-03 1.20e-03
## [46,] 2.52e-04 3.08e-04 3.72e-04 4.45e-04 5.27e-04 6.18e-04 7.18e-04 8.27e-04
## [47,] 1.65e-04 2.03e-04 2.47e-04 2.98e-04 3.55e-04 4.19e-04 4.91e-04 5.69e-04
## [48,] 1.08e-04 1.34e-04 1.64e-04 1.99e-04 2.39e-04 2.84e-04 3.35e-04 3.91e-04
## [49,] 7.04e-05 8.81e-05 1.09e-04 1.33e-04 1.61e-04 1.93e-04 2.29e-04 2.69e-04
## [50,] 4.60e-05 5.81e-05 7.23e-05 8.91e-05 1.09e-04 1.31e-04 1.56e-04 1.85e-04
## [51,] 3.01e-05 3.83e-05 4.80e-05 5.96e-05 7.31e-05 8.88e-05 1.07e-04 1.27e-04
## [52,] 1.97e-05 2.52e-05 3.19e-05 3.98e-05 4.92e-05 6.02e-05 7.30e-05 8.76e-05
## [53,] 1.29e-05 1.66e-05 2.12e-05 2.66e-05 3.32e-05 4.09e-05 4.98e-05 6.02e-05
## [54,] 8.42e-06 1.09e-05 1.40e-05 1.78e-05 2.23e-05 2.77e-05 3.41e-05 4.14e-05
## [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
## [1,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [2,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 9.54e-04 6.97e-04 5.10e-04 3.73e-04 2.72e-04 1.99e-04 1.46e-04 1.06e-04
## [7,] 7.35e-02 6.30e-02 5.38e-02 4.56e-02 3.86e-02 3.25e-02 2.72e-02 2.28e-02
## [8,] 5.09e-02 4.39e-02 3.77e-02 3.22e-02 2.74e-02 2.32e-02 1.96e-02 1.65e-02
## [9,] 3.53e-02 3.06e-02 2.65e-02 2.28e-02 1.95e-02 1.66e-02 1.41e-02 1.19e-02
## [10,] 2.44e-02 2.14e-02 1.86e-02 1.61e-02 1.38e-02 1.19e-02 1.02e-02 8.65e-03
## [11,] 1.69e-02 1.49e-02 1.30e-02 1.14e-02 9.85e-03 8.50e-03 7.31e-03 6.26e-03
## [12,] 1.17e-02 1.04e-02 9.15e-03 8.02e-03 7.00e-03 6.08e-03 5.26e-03 4.54e-03
## [13,] 8.11e-03 7.24e-03 6.42e-03 5.67e-03 4.98e-03 4.35e-03 3.79e-03 3.29e-03
## [14,] 5.62e-03 5.05e-03 4.51e-03 4.00e-03 3.54e-03 3.11e-03 2.73e-03 2.38e-03
## [15,] 3.89e-03 3.52e-03 3.16e-03 2.83e-03 2.51e-03 2.23e-03 1.96e-03 1.72e-03
## [16,] 2.69e-03 2.45e-03 2.22e-03 2.00e-03 1.79e-03 1.59e-03 1.41e-03 1.25e-03
## [17,] 1.87e-03 1.71e-03 1.56e-03 1.41e-03 1.27e-03 1.14e-03 1.02e-03 9.04e-04
## [18,] 1.29e-03 1.19e-03 1.09e-03 9.96e-04 9.03e-04 8.15e-04 7.32e-04 6.54e-04
## [19,] 8.95e-04 8.31e-04 7.67e-04 7.04e-04 6.42e-04 5.83e-04 5.27e-04 4.74e-04
## [20,] 6.19e-04 5.79e-04 5.38e-04 4.97e-04 4.56e-04 4.17e-04 3.79e-04 3.43e-04
## [21,] 4.29e-04 4.04e-04 3.78e-04 3.51e-04 3.24e-04 2.98e-04 2.73e-04 2.49e-04
## [22,] 2.97e-04 2.82e-04 2.65e-04 2.48e-04 2.31e-04 2.13e-04 1.96e-04 1.80e-04
## [23,] 2.06e-04 1.96e-04 1.86e-04 1.75e-04 1.64e-04 1.53e-04 1.41e-04 1.30e-04
## [24,] 1.42e-04 1.37e-04 1.31e-04 1.24e-04 1.17e-04 1.09e-04 1.02e-04 9.44e-05
## [25,] 9.87e-05 9.54e-05 9.16e-05 8.74e-05 8.28e-05 7.81e-05 7.32e-05 6.84e-05
## [26,] 6.83e-05 6.65e-05 6.43e-05 6.17e-05 5.89e-05 5.59e-05 5.27e-05 4.95e-05
## [27,] 4.73e-05 4.64e-05 4.51e-05 4.36e-05 4.19e-05 4.00e-05 3.79e-05 3.58e-05
## [28,] 3.28e-05 3.23e-05 3.17e-05 3.08e-05 2.98e-05 2.86e-05 2.73e-05 2.60e-05
## [29,] 2.27e-05 2.25e-05 2.22e-05 2.18e-05 2.12e-05 2.04e-05 1.97e-05 1.88e-05
## [30,] 1.57e-05 1.57e-05 1.56e-05 1.54e-05 1.50e-05 1.46e-05 1.41e-05 1.36e-05
## [31,] 2.34e-01 2.40e-01 2.44e-01 2.48e-01 2.51e-01 2.52e-01 2.53e-01 2.53e-01
## [32,] 1.62e-01 1.67e-01 1.71e-01 1.75e-01 1.78e-01 1.80e-01 1.82e-01 1.83e-01
## [33,] 1.12e-01 1.16e-01 1.20e-01 1.24e-01 1.27e-01 1.29e-01 1.31e-01 1.33e-01
## [34,] 7.76e-02 8.12e-02 8.44e-02 8.74e-02 9.00e-02 9.23e-02 9.43e-02 9.61e-02
## [35,] 5.37e-02 5.66e-02 5.93e-02 6.17e-02 6.40e-02 6.60e-02 6.79e-02 6.96e-02
## [36,] 3.72e-02 3.95e-02 4.16e-02 4.36e-02 4.55e-02 4.72e-02 4.89e-02 5.04e-02
## [37,] 2.58e-02 2.75e-02 2.92e-02 3.08e-02 3.23e-02 3.38e-02 3.52e-02 3.65e-02
## [38,] 1.78e-02 1.92e-02 2.05e-02 2.17e-02 2.30e-02 2.42e-02 2.53e-02 2.64e-02
## [39,] 1.24e-02 1.34e-02 1.44e-02 1.54e-02 1.63e-02 1.73e-02 1.82e-02 1.91e-02
## [40,] 8.56e-03 9.32e-03 1.01e-02 1.09e-02 1.16e-02 1.24e-02 1.31e-02 1.39e-02
## [41,] 5.93e-03 6.50e-03 7.08e-03 7.66e-03 8.25e-03 8.85e-03 9.44e-03 1.00e-02
## [42,] 4.11e-03 4.53e-03 4.97e-03 5.41e-03 5.87e-03 6.33e-03 6.80e-03 7.27e-03
## [43,] 2.84e-03 3.16e-03 3.49e-03 3.82e-03 4.17e-03 4.53e-03 4.89e-03 5.26e-03
## [44,] 1.97e-03 2.20e-03 2.45e-03 2.70e-03 2.97e-03 3.24e-03 3.52e-03 3.81e-03
## [45,] 1.36e-03 1.54e-03 1.72e-03 1.91e-03 2.11e-03 2.32e-03 2.53e-03 2.76e-03
## [46,] 9.44e-04 1.07e-03 1.20e-03 1.35e-03 1.50e-03 1.66e-03 1.82e-03 2.00e-03
## [47,] 6.54e-04 7.46e-04 8.45e-04 9.52e-04 1.07e-03 1.19e-03 1.31e-03 1.45e-03
## [48,] 4.53e-04 5.20e-04 5.93e-04 6.72e-04 7.57e-04 8.48e-04 9.45e-04 1.05e-03
## [49,] 3.14e-04 3.63e-04 4.16e-04 4.75e-04 5.38e-04 6.07e-04 6.80e-04 7.59e-04
## [50,] 2.17e-04 2.53e-04 2.92e-04 3.35e-04 3.83e-04 4.34e-04 4.90e-04 5.50e-04
## [51,] 1.50e-04 1.76e-04 2.05e-04 2.37e-04 2.72e-04 3.10e-04 3.52e-04 3.98e-04
## [52,] 1.04e-04 1.23e-04 1.44e-04 1.67e-04 1.93e-04 2.22e-04 2.54e-04 2.88e-04
## [53,] 7.21e-05 8.57e-05 1.01e-04 1.18e-04 1.37e-04 1.59e-04 1.83e-04 2.09e-04
## [54,] 5.00e-05 5.97e-05 7.09e-05 8.35e-05 9.77e-05 1.14e-04 1.31e-04 1.51e-04
## [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34]
## [1,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 2.31e+03 1.77e+03 8.24e+02 2.26e+02
## [2,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 2.31e+03 1.77e+03 8.24e+02 2.26e+02
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 7.78e-05 5.69e-05 4.16e-05 3.04e-05 3.69e-02 2.81e-02 2.10e-02 1.56e-02
## [7,] 1.90e-02 1.58e-02 1.31e-02 1.09e-02 1.19e-01 1.09e-01 9.72e-02 8.53e-02
## [8,] 1.38e-02 1.16e-02 9.69e-03 8.09e-03 7.55e-02 6.96e-02 6.26e-02 5.53e-02
## [9,] 1.01e-02 8.50e-03 7.14e-03 6.00e-03 4.79e-02 4.45e-02 4.03e-02 3.59e-02
## [10,] 7.35e-03 6.23e-03 5.27e-03 4.44e-03 3.04e-02 2.84e-02 2.60e-02 2.33e-02
## [11,] 5.35e-03 4.56e-03 3.88e-03 3.30e-03 1.92e-02 1.82e-02 1.67e-02 1.51e-02
## [12,] 3.90e-03 3.34e-03 2.86e-03 2.44e-03 1.22e-02 1.16e-02 1.08e-02 9.82e-03
## [13,] 2.84e-03 2.45e-03 2.11e-03 1.81e-03 7.73e-03 7.41e-03 6.93e-03 6.37e-03
## [14,] 2.07e-03 1.80e-03 1.55e-03 1.34e-03 4.90e-03 4.74e-03 4.46e-03 4.14e-03
## [15,] 1.51e-03 1.32e-03 1.15e-03 9.95e-04 3.11e-03 3.03e-03 2.88e-03 2.68e-03
## [16,] 1.10e-03 9.65e-04 8.45e-04 7.38e-04 1.97e-03 1.93e-03 1.85e-03 1.74e-03
## [17,] 8.01e-04 7.07e-04 6.23e-04 5.47e-04 1.25e-03 1.24e-03 1.19e-03 1.13e-03
## [18,] 5.83e-04 5.18e-04 4.59e-04 4.06e-04 7.91e-04 7.90e-04 7.68e-04 7.34e-04
## [19,] 4.25e-04 3.80e-04 3.38e-04 3.01e-04 5.02e-04 5.04e-04 4.95e-04 4.76e-04
## [20,] 3.10e-04 2.78e-04 2.49e-04 2.23e-04 3.18e-04 3.22e-04 3.19e-04 3.09e-04
## [21,] 2.26e-04 2.04e-04 1.84e-04 1.65e-04 2.02e-04 2.06e-04 2.05e-04 2.01e-04
## [22,] 1.64e-04 1.49e-04 1.36e-04 1.22e-04 1.28e-04 1.32e-04 1.32e-04 1.30e-04
## [23,] 1.20e-04 1.10e-04 9.99e-05 9.08e-05 8.10e-05 8.41e-05 8.51e-05 8.45e-05
## [24,] 8.72e-05 8.03e-05 7.36e-05 6.73e-05 5.14e-05 5.37e-05 5.48e-05 5.49e-05
## [25,] 6.35e-05 5.88e-05 5.43e-05 4.99e-05 3.26e-05 3.43e-05 3.53e-05 3.56e-05
## [26,] 4.63e-05 4.31e-05 4.00e-05 3.70e-05 2.06e-05 2.19e-05 2.27e-05 2.31e-05
## [27,] 3.37e-05 3.16e-05 2.95e-05 2.74e-05 1.31e-05 1.40e-05 1.46e-05 1.50e-05
## [28,] 2.46e-05 2.32e-05 2.17e-05 2.03e-05 8.29e-06 8.96e-06 9.43e-06 9.74e-06
## [29,] 1.79e-05 1.70e-05 1.60e-05 1.51e-05 5.26e-06 5.72e-06 6.07e-06 6.32e-06
## [30,] 1.30e-05 1.24e-05 1.18e-05 1.12e-05 3.33e-06 3.66e-06 3.91e-06 4.10e-06
## [31,] 2.52e-01 2.51e-01 2.50e-01 2.48e-01 2.10e-01 2.29e-01 2.45e-01 2.57e-01
## [32,] 1.84e-01 1.84e-01 1.84e-01 1.84e-01 1.33e-01 1.46e-01 1.57e-01 1.67e-01
## [33,] 1.34e-01 1.35e-01 1.36e-01 1.36e-01 8.43e-02 9.36e-02 1.01e-01 1.08e-01
## [34,] 9.76e-02 9.89e-02 1.00e-01 1.01e-01 5.34e-02 5.98e-02 6.53e-02 7.02e-02
## [35,] 7.11e-02 7.25e-02 7.37e-02 7.48e-02 3.39e-02 3.82e-02 4.21e-02 4.55e-02
## [36,] 5.18e-02 5.31e-02 5.43e-02 5.55e-02 2.15e-02 2.44e-02 2.71e-02 2.96e-02
## [37,] 3.77e-02 3.89e-02 4.01e-02 4.11e-02 1.36e-02 1.56e-02 1.74e-02 1.92e-02
## [38,] 2.75e-02 2.85e-02 2.95e-02 3.05e-02 8.63e-03 9.97e-03 1.12e-02 1.24e-02
## [39,] 2.00e-02 2.09e-02 2.18e-02 2.26e-02 5.47e-03 6.37e-03 7.24e-03 8.08e-03
## [40,] 1.46e-02 1.53e-02 1.60e-02 1.68e-02 3.47e-03 4.07e-03 4.66e-03 5.24e-03
## [41,] 1.06e-02 1.12e-02 1.18e-02 1.24e-02 2.20e-03 2.60e-03 3.00e-03 3.40e-03
## [42,] 7.75e-03 8.23e-03 8.72e-03 9.21e-03 1.39e-03 1.66e-03 1.93e-03 2.21e-03
## [43,] 5.64e-03 6.03e-03 6.43e-03 6.83e-03 8.83e-04 1.06e-03 1.24e-03 1.43e-03
## [44,] 4.11e-03 4.42e-03 4.74e-03 5.06e-03 5.60e-04 6.79e-04 8.02e-04 9.31e-04
## [45,] 3.00e-03 3.24e-03 3.49e-03 3.75e-03 3.55e-04 4.34e-04 5.16e-04 6.04e-04
## [46,] 2.18e-03 2.37e-03 2.57e-03 2.78e-03 2.25e-04 2.77e-04 3.33e-04 3.92e-04
## [47,] 1.59e-03 1.74e-03 1.90e-03 2.06e-03 1.43e-04 1.77e-04 2.14e-04 2.54e-04
## [48,] 1.16e-03 1.27e-03 1.40e-03 1.53e-03 9.04e-05 1.13e-04 1.38e-04 1.65e-04
## [49,] 8.44e-04 9.34e-04 1.03e-03 1.13e-03 5.73e-05 7.23e-05 8.88e-05 1.07e-04
## [50,] 6.15e-04 6.85e-04 7.60e-04 8.40e-04 3.63e-05 4.62e-05 5.72e-05 6.96e-05
## [51,] 4.48e-04 5.02e-04 5.60e-04 6.23e-04 2.30e-05 2.95e-05 3.68e-05 4.51e-05
## [52,] 3.26e-04 3.68e-04 4.13e-04 4.62e-04 1.46e-05 1.89e-05 2.37e-05 2.93e-05
## [53,] 2.38e-04 2.69e-04 3.04e-04 3.42e-04 9.25e-06 1.20e-05 1.53e-05 1.90e-05
## [54,] 1.73e-04 1.97e-04 2.24e-04 2.54e-04 5.87e-06 7.70e-06 9.84e-06 1.23e-05
## [,35] [,36] [,37] [,38] [,39] [,40] [,41] [,42]
## [1,] 4.94e+01 1.02e+01 2.06e+00 4.18e-01 8.47e-02 1.71e-02 3.47e-03 7.03e-04
## [2,] 4.94e+01 1.02e+01 2.06e+00 4.18e-01 8.47e-02 1.71e-02 3.47e-03 7.03e-04
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 1.15e-02 8.47e-03 6.21e-03 4.55e-03 3.33e-03 2.44e-03 1.78e-03 1.30e-03
## [7,] 7.40e-02 6.36e-02 5.43e-02 4.61e-02 3.89e-02 3.27e-02 2.75e-02 2.30e-02
## [8,] 4.84e-02 4.19e-02 3.60e-02 3.08e-02 2.62e-02 2.22e-02 1.88e-02 1.58e-02
## [9,] 3.16e-02 2.76e-02 2.39e-02 2.06e-02 1.77e-02 1.51e-02 1.28e-02 1.09e-02
## [10,] 2.07e-02 1.82e-02 1.59e-02 1.38e-02 1.19e-02 1.02e-02 8.75e-03 7.47e-03
## [11,] 1.35e-02 1.20e-02 1.05e-02 9.21e-03 8.01e-03 6.93e-03 5.98e-03 5.14e-03
## [12,] 8.85e-03 7.90e-03 7.00e-03 6.16e-03 5.40e-03 4.70e-03 4.08e-03 3.54e-03
## [13,] 5.79e-03 5.21e-03 4.65e-03 4.12e-03 3.63e-03 3.19e-03 2.79e-03 2.43e-03
## [14,] 3.79e-03 3.43e-03 3.08e-03 2.76e-03 2.45e-03 2.16e-03 1.91e-03 1.67e-03
## [15,] 2.48e-03 2.26e-03 2.05e-03 1.84e-03 1.65e-03 1.47e-03 1.30e-03 1.15e-03
## [16,] 1.62e-03 1.49e-03 1.36e-03 1.23e-03 1.11e-03 9.96e-04 8.90e-04 7.91e-04
## [17,] 1.06e-03 9.81e-04 9.02e-04 8.24e-04 7.48e-04 6.76e-04 6.08e-04 5.44e-04
## [18,] 6.92e-04 6.47e-04 5.99e-04 5.51e-04 5.04e-04 4.58e-04 4.15e-04 3.74e-04
## [19,] 4.53e-04 4.26e-04 3.98e-04 3.68e-04 3.39e-04 3.11e-04 2.84e-04 2.58e-04
## [20,] 2.96e-04 2.81e-04 2.64e-04 2.46e-04 2.29e-04 2.11e-04 1.94e-04 1.77e-04
## [21,] 1.94e-04 1.85e-04 1.75e-04 1.65e-04 1.54e-04 1.43e-04 1.32e-04 1.22e-04
## [22,] 1.27e-04 1.22e-04 1.16e-04 1.10e-04 1.04e-04 9.71e-05 9.04e-05 8.38e-05
## [23,] 8.28e-05 8.03e-05 7.72e-05 7.37e-05 6.98e-05 6.58e-05 6.17e-05 5.76e-05
## [24,] 5.42e-05 5.29e-05 5.13e-05 4.93e-05 4.70e-05 4.47e-05 4.22e-05 3.97e-05
## [25,] 3.54e-05 3.49e-05 3.40e-05 3.29e-05 3.17e-05 3.03e-05 2.88e-05 2.73e-05
## [26,] 2.32e-05 2.30e-05 2.26e-05 2.20e-05 2.13e-05 2.06e-05 1.97e-05 1.88e-05
## [27,] 1.52e-05 1.51e-05 1.50e-05 1.47e-05 1.44e-05 1.39e-05 1.34e-05 1.29e-05
## [28,] 9.91e-06 9.98e-06 9.96e-06 9.85e-06 9.68e-06 9.46e-06 9.19e-06 8.88e-06
## [29,] 6.48e-06 6.58e-06 6.61e-06 6.59e-06 6.52e-06 6.42e-06 6.27e-06 6.11e-06
## [30,] 4.24e-06 4.33e-06 4.39e-06 4.41e-06 4.39e-06 4.35e-06 4.29e-06 4.20e-06
## [31,] 2.66e-01 2.74e-01 2.79e-01 2.83e-01 2.86e-01 2.88e-01 2.89e-01 2.89e-01
## [32,] 1.74e-01 1.80e-01 1.85e-01 1.90e-01 1.93e-01 1.95e-01 1.97e-01 1.99e-01
## [33,] 1.14e-01 1.19e-01 1.23e-01 1.27e-01 1.30e-01 1.33e-01 1.35e-01 1.37e-01
## [34,] 7.45e-02 7.83e-02 8.17e-02 8.48e-02 8.75e-02 8.99e-02 9.21e-02 9.40e-02
## [35,] 4.87e-02 5.16e-02 5.42e-02 5.67e-02 5.89e-02 6.10e-02 6.29e-02 6.46e-02
## [36,] 3.18e-02 3.40e-02 3.60e-02 3.79e-02 3.97e-02 4.14e-02 4.30e-02 4.45e-02
## [37,] 2.08e-02 2.24e-02 2.39e-02 2.53e-02 2.67e-02 2.81e-02 2.93e-02 3.06e-02
## [38,] 1.36e-02 1.48e-02 1.59e-02 1.69e-02 1.80e-02 1.90e-02 2.00e-02 2.10e-02
## [39,] 8.91e-03 9.73e-03 1.05e-02 1.13e-02 1.21e-02 1.29e-02 1.37e-02 1.45e-02
## [40,] 5.83e-03 6.41e-03 6.99e-03 7.58e-03 8.17e-03 8.76e-03 9.36e-03 9.95e-03
## [41,] 3.81e-03 4.22e-03 4.64e-03 5.07e-03 5.50e-03 5.94e-03 6.39e-03 6.85e-03
## [42,] 2.49e-03 2.78e-03 3.08e-03 3.39e-03 3.71e-03 4.03e-03 4.37e-03 4.71e-03
## [43,] 1.63e-03 1.83e-03 2.05e-03 2.27e-03 2.50e-03 2.73e-03 2.98e-03 3.24e-03
## [44,] 1.07e-03 1.21e-03 1.36e-03 1.52e-03 1.68e-03 1.86e-03 2.04e-03 2.23e-03
## [45,] 6.97e-04 7.96e-04 9.02e-04 1.01e-03 1.13e-03 1.26e-03 1.39e-03 1.53e-03
## [46,] 4.56e-04 5.25e-04 5.98e-04 6.78e-04 7.63e-04 8.54e-04 9.51e-04 1.05e-03
## [47,] 2.98e-04 3.46e-04 3.97e-04 4.53e-04 5.14e-04 5.79e-04 6.49e-04 7.25e-04
## [48,] 1.95e-04 2.28e-04 2.64e-04 3.03e-04 3.46e-04 3.93e-04 4.44e-04 4.99e-04
## [49,] 1.28e-04 1.50e-04 1.75e-04 2.03e-04 2.33e-04 2.66e-04 3.03e-04 3.43e-04
## [50,] 8.34e-05 9.89e-05 1.16e-04 1.36e-04 1.57e-04 1.81e-04 2.07e-04 2.36e-04
## [51,] 5.45e-05 6.52e-05 7.72e-05 9.06e-05 1.06e-04 1.23e-04 1.41e-04 1.62e-04
## [52,] 3.57e-05 4.29e-05 5.12e-05 6.06e-05 7.12e-05 8.32e-05 9.66e-05 1.12e-04
## [53,] 2.33e-05 2.83e-05 3.40e-05 4.05e-05 4.80e-05 5.64e-05 6.60e-05 7.68e-05
## [54,] 1.53e-05 1.86e-05 2.26e-05 2.71e-05 3.23e-05 3.83e-05 4.51e-05 5.28e-05
## [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
## [1,] 1.42e-04 2.88e-05 5.84e-06 1.18e-06 2.39e-07 4.85e-08 9.81e-09 1.99e-09
## [2,] 1.42e-04 2.88e-05 5.84e-06 1.18e-06 2.39e-07 4.85e-08 9.81e-09 1.99e-09
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 9.54e-04 6.97e-04 5.10e-04 3.73e-04 2.72e-04 1.99e-04 1.46e-04 1.06e-04
## [7,] 1.92e-02 1.59e-02 1.32e-02 1.10e-02 9.11e-03 7.54e-03 6.23e-03 5.15e-03
## [8,] 1.33e-02 1.11e-02 9.30e-03 7.76e-03 6.47e-03 5.39e-03 4.49e-03 3.73e-03
## [9,] 9.19e-03 7.75e-03 6.52e-03 5.48e-03 4.60e-03 3.86e-03 3.23e-03 2.70e-03
## [10,] 6.36e-03 5.40e-03 4.58e-03 3.87e-03 3.27e-03 2.76e-03 2.32e-03 1.96e-03
## [11,] 4.41e-03 3.77e-03 3.21e-03 2.74e-03 2.33e-03 1.97e-03 1.67e-03 1.42e-03
## [12,] 3.05e-03 2.63e-03 2.25e-03 1.93e-03 1.65e-03 1.41e-03 1.20e-03 1.03e-03
## [13,] 2.11e-03 1.83e-03 1.58e-03 1.36e-03 1.18e-03 1.01e-03 8.67e-04 7.43e-04
## [14,] 1.46e-03 1.28e-03 1.11e-03 9.64e-04 8.35e-04 7.22e-04 6.24e-04 5.38e-04
## [15,] 1.01e-03 8.90e-04 7.79e-04 6.81e-04 5.94e-04 5.17e-04 4.49e-04 3.90e-04
## [16,] 7.02e-04 6.20e-04 5.47e-04 4.81e-04 4.22e-04 3.70e-04 3.23e-04 2.82e-04
## [17,] 4.86e-04 4.32e-04 3.84e-04 3.40e-04 3.00e-04 2.64e-04 2.33e-04 2.04e-04
## [18,] 3.37e-04 3.01e-04 2.69e-04 2.40e-04 2.13e-04 1.89e-04 1.67e-04 1.48e-04
## [19,] 2.33e-04 2.10e-04 1.89e-04 1.70e-04 1.52e-04 1.35e-04 1.21e-04 1.07e-04
## [20,] 1.61e-04 1.47e-04 1.33e-04 1.20e-04 1.08e-04 9.68e-05 8.68e-05 7.76e-05
## [21,] 1.12e-04 1.02e-04 9.31e-05 8.46e-05 7.66e-05 6.93e-05 6.24e-05 5.62e-05
## [22,] 7.74e-05 7.12e-05 6.53e-05 5.97e-05 5.45e-05 4.95e-05 4.49e-05 4.07e-05
## [23,] 5.36e-05 4.97e-05 4.58e-05 4.22e-05 3.87e-05 3.54e-05 3.24e-05 2.95e-05
## [24,] 3.71e-05 3.46e-05 3.22e-05 2.98e-05 2.75e-05 2.53e-05 2.33e-05 2.13e-05
## [25,] 2.57e-05 2.41e-05 2.26e-05 2.11e-05 1.96e-05 1.81e-05 1.68e-05 1.55e-05
## [26,] 1.78e-05 1.68e-05 1.58e-05 1.49e-05 1.39e-05 1.30e-05 1.21e-05 1.12e-05
## [27,] 1.23e-05 1.17e-05 1.11e-05 1.05e-05 9.89e-06 9.28e-06 8.68e-06 8.10e-06
## [28,] 8.54e-06 8.18e-06 7.80e-06 7.42e-06 7.03e-06 6.64e-06 6.25e-06 5.87e-06
## [29,] 5.91e-06 5.70e-06 5.48e-06 5.24e-06 5.00e-06 4.75e-06 4.50e-06 4.25e-06
## [30,] 4.09e-06 3.98e-06 3.84e-06 3.70e-06 3.55e-06 3.40e-06 3.24e-06 3.08e-06
## [31,] 2.88e-01 2.87e-01 2.85e-01 2.83e-01 2.80e-01 2.77e-01 2.74e-01 2.71e-01
## [32,] 1.99e-01 2.00e-01 2.00e-01 2.00e-01 1.99e-01 1.98e-01 1.97e-01 1.96e-01
## [33,] 1.38e-01 1.39e-01 1.40e-01 1.41e-01 1.41e-01 1.42e-01 1.42e-01 1.42e-01
## [34,] 9.57e-02 9.71e-02 9.84e-02 9.96e-02 1.01e-01 1.01e-01 1.02e-01 1.03e-01
## [35,] 6.62e-02 6.77e-02 6.91e-02 7.03e-02 7.15e-02 7.26e-02 7.35e-02 7.44e-02
## [36,] 4.59e-02 4.72e-02 4.85e-02 4.97e-02 5.08e-02 5.19e-02 5.29e-02 5.39e-02
## [37,] 3.18e-02 3.29e-02 3.40e-02 3.51e-02 3.61e-02 3.71e-02 3.81e-02 3.90e-02
## [38,] 2.20e-02 2.29e-02 2.39e-02 2.48e-02 2.57e-02 2.66e-02 2.74e-02 2.83e-02
## [39,] 1.52e-02 1.60e-02 1.68e-02 1.75e-02 1.83e-02 1.90e-02 1.97e-02 2.05e-02
## [40,] 1.06e-02 1.12e-02 1.18e-02 1.24e-02 1.30e-02 1.36e-02 1.42e-02 1.48e-02
## [41,] 7.31e-03 7.78e-03 8.25e-03 8.73e-03 9.22e-03 9.72e-03 1.02e-02 1.07e-02
## [42,] 5.06e-03 5.42e-03 5.79e-03 6.17e-03 6.56e-03 6.95e-03 7.36e-03 7.78e-03
## [43,] 3.50e-03 3.78e-03 4.06e-03 4.36e-03 4.66e-03 4.97e-03 5.30e-03 5.63e-03
## [44,] 2.43e-03 2.64e-03 2.85e-03 3.08e-03 3.31e-03 3.56e-03 3.81e-03 4.08e-03
## [45,] 1.68e-03 1.84e-03 2.00e-03 2.17e-03 2.36e-03 2.55e-03 2.74e-03 2.95e-03
## [46,] 1.16e-03 1.28e-03 1.40e-03 1.54e-03 1.67e-03 1.82e-03 1.98e-03 2.14e-03
## [47,] 8.06e-04 8.93e-04 9.86e-04 1.08e-03 1.19e-03 1.30e-03 1.42e-03 1.55e-03
## [48,] 5.58e-04 6.23e-04 6.92e-04 7.66e-04 8.46e-04 9.32e-04 1.02e-03 1.12e-03
## [49,] 3.87e-04 4.34e-04 4.85e-04 5.41e-04 6.02e-04 6.67e-04 7.37e-04 8.12e-04
## [50,] 2.68e-04 3.03e-04 3.41e-04 3.82e-04 4.28e-04 4.77e-04 5.30e-04 5.88e-04
## [51,] 1.85e-04 2.11e-04 2.39e-04 2.70e-04 3.04e-04 3.41e-04 3.82e-04 4.26e-04
## [52,] 1.28e-04 1.47e-04 1.68e-04 1.91e-04 2.16e-04 2.44e-04 2.75e-04 3.08e-04
## [53,] 8.89e-05 1.03e-04 1.18e-04 1.35e-04 1.54e-04 1.75e-04 1.98e-04 2.23e-04
## [54,] 6.16e-05 7.15e-05 8.26e-05 9.52e-05 1.09e-04 1.25e-04 1.42e-04 1.62e-04
## [,51] [,52] [,53] [,54]
## [1,] 4.02e-10 8.15e-11 1.65e-11 3.34e-12
## [2,] 4.02e-10 8.15e-11 1.65e-11 3.34e-12
## [3,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [4,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [5,] 0.00e+00 0.00e+00 0.00e+00 0.00e+00
## [6,] 7.78e-05 5.69e-05 4.16e-05 3.04e-05
## [7,] 4.25e-03 3.51e-03 2.89e-03 2.39e-03
## [8,] 3.10e-03 2.57e-03 2.13e-03 1.77e-03
## [9,] 2.26e-03 1.88e-03 1.57e-03 1.31e-03
## [10,] 1.64e-03 1.38e-03 1.16e-03 9.72e-04
## [11,] 1.20e-03 1.01e-03 8.54e-04 7.20e-04
## [12,] 8.73e-04 7.42e-04 6.30e-04 5.34e-04
## [13,] 6.36e-04 5.43e-04 4.64e-04 3.96e-04
## [14,] 4.63e-04 3.98e-04 3.42e-04 2.94e-04
## [15,] 3.37e-04 2.92e-04 2.52e-04 2.18e-04
## [16,] 2.46e-04 2.14e-04 1.86e-04 1.61e-04
## [17,] 1.79e-04 1.57e-04 1.37e-04 1.20e-04
## [18,] 1.30e-04 1.15e-04 1.01e-04 8.87e-05
## [19,] 9.51e-05 8.42e-05 7.44e-05 6.57e-05
## [20,] 6.93e-05 6.17e-05 5.49e-05 4.87e-05
## [21,] 5.05e-05 4.52e-05 4.04e-05 3.61e-05
## [22,] 3.68e-05 3.31e-05 2.98e-05 2.68e-05
## [23,] 2.68e-05 2.43e-05 2.20e-05 1.99e-05
## [24,] 1.95e-05 1.78e-05 1.62e-05 1.47e-05
## [25,] 1.42e-05 1.30e-05 1.19e-05 1.09e-05
## [26,] 1.04e-05 9.56e-06 8.80e-06 8.09e-06
## [27,] 7.54e-06 7.00e-06 6.49e-06 6.00e-06
## [28,] 5.50e-06 5.13e-06 4.78e-06 4.45e-06
## [29,] 4.00e-06 3.76e-06 3.53e-06 3.30e-06
## [30,] 2.92e-06 2.76e-06 2.60e-06 2.44e-06
## [31,] 2.67e-01 2.64e-01 2.60e-01 2.56e-01
## [32,] 1.95e-01 1.93e-01 1.92e-01 1.90e-01
## [33,] 1.42e-01 1.42e-01 1.41e-01 1.41e-01
## [34,] 1.03e-01 1.04e-01 1.04e-01 1.04e-01
## [35,] 7.53e-02 7.60e-02 7.68e-02 7.74e-02
## [36,] 5.48e-02 5.57e-02 5.66e-02 5.74e-02
## [37,] 4.00e-02 4.08e-02 4.17e-02 4.25e-02
## [38,] 2.91e-02 2.99e-02 3.07e-02 3.15e-02
## [39,] 2.12e-02 2.19e-02 2.27e-02 2.34e-02
## [40,] 1.54e-02 1.61e-02 1.67e-02 1.73e-02
## [41,] 1.13e-02 1.18e-02 1.23e-02 1.28e-02
## [42,] 8.20e-03 8.63e-03 9.07e-03 9.53e-03
## [43,] 5.97e-03 6.33e-03 6.69e-03 7.06e-03
## [44,] 4.35e-03 4.64e-03 4.93e-03 5.24e-03
## [45,] 3.17e-03 3.40e-03 3.63e-03 3.88e-03
## [46,] 2.31e-03 2.49e-03 2.68e-03 2.88e-03
## [47,] 1.68e-03 1.82e-03 1.97e-03 2.13e-03
## [48,] 1.23e-03 1.34e-03 1.46e-03 1.58e-03
## [49,] 8.93e-04 9.80e-04 1.07e-03 1.17e-03
## [50,] 6.51e-04 7.18e-04 7.91e-04 8.69e-04
## [51,] 4.74e-04 5.26e-04 5.83e-04 6.44e-04
## [52,] 3.45e-04 3.86e-04 4.30e-04 4.78e-04
## [53,] 2.52e-04 2.83e-04 3.17e-04 3.54e-04
## [54,] 1.83e-04 2.07e-04 2.33e-04 2.62e-04Next, let’s estimate the function-based historical MPM.
cypmatrix3fp <- flefko3(stageframe = cypframe_fb, supplement = cypsupp3_fb,
modelsuite = cypmodels3p, data = cypfb_env)
summary(cypmatrix3fp)##
## This historical lefkoMat object contains 15 matrices.
##
## Each matrix is square with 2916 rows and columns, and a total of 8503056 elements.
## A total of 1765635 survival transitions were estimated, with 117709 per matrix.
## A total of 35280 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## Min. 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.936 0.964 0.963 0.959 0.951 0.925 0.962 0.952 0.945 0.935 0.903 0.953
## Median 0.988 0.997 0.996 0.994 0.992 0.979 0.995 0.992 0.990 0.986 0.969 0.993
## Mean 0.809 0.818 0.816 0.815 0.813 0.801 0.816 0.813 0.811 0.807 0.794 0.813
## 3rd Qu. 0.994 0.999 0.999 0.998 0.997 0.988 0.998 0.997 0.995 0.993 0.982 0.997
## Max. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 1.000
## [,13] [,14] [,15]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.938 0.935 0.925
## Median 0.988 0.985 0.979
## Mean 0.810 0.807 0.801
## 3rd Qu. 0.994 0.992 0.989
## Max. 1.000 1.000 1.000Let’s note the contrasts with the other MPMs in the output above. First, we see 15 matrices produced again. However, we created 3 matrices fewer in the raw historical MPM, because we need three monitoring occasions of data to parameterize each raw transition (six monitoring occasions total means four sets of three monitoring occasions, or four total time steps, in the historical case). Since this is a function-based MPM, and since it resulted from a mixed modeling strategy where year was treated as random, we can actually use our functions to estimate transitions in the first year even without a full set of three years of data. The result is five time steps for which we can estimate transitions.
Second, these matrices are utterly huge. They have \(54^2 = 2916\) rows and columns, yielding \(2916^2 = 8503056\) total elements per matrix. Our raw hMPM had \(11^2 = 121\) rows and columns, yielding 14,641 elements per matrix. Finally, although the raw hMPMs only had an average of 37.83 elements estimated per matrix, here we have 120,061 elements estimated per matrix. This is vastly more than in the raw case and in the ahistorical function-based case. However, it is also just a small fraction of the total number of elements in the matrix. In fact, in an unreduced hMPM estimated in Ehrlén format, the total number of elements that can be estimated is equal to \(\frac{r \times c}{m}\), where \(r\) and \(c\) are the numbers of rows and columns, respectively, and \(m\) is the number of stages in the stageframe. For example, if there are ten stages in the stageframe, then only 1000 of the total 10,000 elements in the hMPM are estimable (10% of the elements).
Now that we have created our MPMs, we might wish to create
element-wise arithmetic mean matrices to aid inference and further
analysis. For example, we might be interested in developing patch-level
means and an overall population mean, but one in which the element means
treat each patch and each year as equal in proportional effect. For this
purpose, we can use the lmean() function. Let’s take a look
at the mean raw ahMPM first.
cyp2rp_mean <- lmean(cypmatrix2rp)
cyp2rp_mean## $A
## $A[[1]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 62.50000000 422.22222222
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 62.50000000 422.22222222
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.00000000 0.00000000
## [6,] 0.00 0.0 0.0 0.0 0.05833333 0.00000000 0.05833333 0.04444444
## [7,] 0.00 0.0 0.0 0.0 0.59949495 0.06666667 0.59949495 0.19555556
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.33333333 0.21010101 0.59555556
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.08444444
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [,9] [,10] [,11]
## [1,] 1.583333e+03 0.0 0
## [2,] 1.583333e+03 0.0 0
## [3,] 0.000000e+00 0.0 0
## [4,] 0.000000e+00 0.0 0
## [5,] 0.000000e+00 0.0 0
## [6,] 0.000000e+00 0.0 0
## [7,] 6.666667e-02 0.0 0
## [8,] 2.500000e-01 0.2 0
## [9,] 4.833333e-01 0.0 0
## [10,] 2.000000e-01 0.0 0
## [11,] 0.000000e+00 0.0 0
##
## $A[[2]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 55.55555556 572.91666667
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 55.55555556 572.91666667
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.00000000 0.00000000
## [6,] 0.00 0.0 0.0 0.0 0.07676768 0.06666667 0.07676768 0.05416667
## [7,] 0.00 0.0 0.0 0.0 0.45974026 0.46666667 0.45974026 0.23083333
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.20000000 0.30966811 0.50166667
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.02222222 0.14416667
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.06666667 0.02222222 0.01250000
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [,9] [,10] [,11]
## [1,] 2392.8571429 2875.0 6e+03
## [2,] 2392.8571429 2875.0 6e+03
## [3,] 0.0000000 0.0 0e+00
## [4,] 0.0000000 0.0 0e+00
## [5,] 0.0000000 0.0 0e+00
## [6,] 0.0000000 0.0 0e+00
## [7,] 0.0500000 0.0 0e+00
## [8,] 0.4000000 0.1 0e+00
## [9,] 0.3428571 0.2 0e+00
## [10,] 0.2071429 0.6 2e-01
## [11,] 0.0000000 0.1 4e-01
##
## $A[[3]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.0 0.00000000 422.61904762 1625.0000000
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.0 0.00000000 422.61904762 1625.0000000
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.0 0.00000000 0.00000000 0.0000000
## [6,] 0.00 0.0 0.0 0.0 0.06666667 0.0 0.06666667 0.03333333 0.0000000
## [7,] 0.00 0.0 0.0 0.0 0.50000000 0.1 0.50000000 0.11666667 0.1333333
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.1 0.26666667 0.54761905 0.3333333
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.06666667 0.22380952 0.5333333
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.07857143 0.0000000
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000
## [,10] [,11]
## [1,] 750.0 2500.0
## [2,] 750.0 2500.0
## [3,] 0.0 0.0
## [4,] 0.0 0.0
## [5,] 0.0 0.0
## [6,] 0.0 0.0
## [7,] 0.0 0.0
## [8,] 0.1 0.0
## [9,] 0.2 0.0
## [10,] 0.4 0.2
## [11,] 0.1 0.4
##
## $A[[4]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 39.351851852 472.58597884
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 39.351851852 472.58597884
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.000000000 0.00000000
## [6,] 0.00 0.0 0.0 0.0 0.06725589 0.02222222 0.067255892 0.04398148
## [7,] 0.00 0.0 0.0 0.0 0.51974507 0.21111111 0.519745070 0.18101852
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.21111111 0.262145262 0.54828042
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.029629630 0.15080688
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.02222222 0.007407407 0.03035714
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [,9] [,10] [,11]
## [1,] 1.867063e+03 1.208333e+03 2833.3333333
## [2,] 1.867063e+03 1.208333e+03 2833.3333333
## [3,] 0.000000e+00 0.000000e+00 0.0000000
## [4,] 0.000000e+00 0.000000e+00 0.0000000
## [5,] 0.000000e+00 0.000000e+00 0.0000000
## [6,] 0.000000e+00 0.000000e+00 0.0000000
## [7,] 8.333333e-02 0.000000e+00 0.0000000
## [8,] 3.277778e-01 1.333333e-01 0.0000000
## [9,] 4.531746e-01 1.333333e-01 0.0000000
## [10,] 1.357143e-01 3.333333e-01 0.1333333
## [11,] 0.000000e+00 6.666667e-02 0.2666667
##
##
## $U
## $U[[1]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.00000000 0.00000000
## [6,] 0.00 0.0 0.0 0.0 0.05833333 0.00000000 0.05833333 0.04444444
## [7,] 0.00 0.0 0.0 0.0 0.59949495 0.06666667 0.59949495 0.19555556
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.33333333 0.21010101 0.59555556
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.08444444
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000
## [,9] [,10] [,11]
## [1,] 0.00000000 0.0 0
## [2,] 0.00000000 0.0 0
## [3,] 0.00000000 0.0 0
## [4,] 0.00000000 0.0 0
## [5,] 0.00000000 0.0 0
## [6,] 0.00000000 0.0 0
## [7,] 0.06666667 0.0 0
## [8,] 0.25000000 0.2 0
## [9,] 0.48333333 0.0 0
## [10,] 0.20000000 0.0 0
## [11,] 0.00000000 0.0 0
##
## $U[[2]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.00000000 0.00000000 0.0000000
## [6,] 0.00 0.0 0.0 0.0 0.07676768 0.06666667 0.07676768 0.05416667 0.0000000
## [7,] 0.00 0.0 0.0 0.0 0.45974026 0.46666667 0.45974026 0.23083333 0.0500000
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.20000000 0.30966811 0.50166667 0.4000000
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.02222222 0.14416667 0.3428571
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.06666667 0.02222222 0.01250000 0.2071429
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000
## [,10] [,11]
## [1,] 0.0 0.0
## [2,] 0.0 0.0
## [3,] 0.0 0.0
## [4,] 0.0 0.0
## [5,] 0.0 0.0
## [6,] 0.0 0.0
## [7,] 0.0 0.0
## [8,] 0.1 0.0
## [9,] 0.2 0.0
## [10,] 0.6 0.2
## [11,] 0.1 0.4
##
## $U[[3]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000 0.0
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000 0.0
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000 0.0
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000 0.0
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.0 0.00000000 0.00000000 0.0000000 0.0
## [6,] 0.00 0.0 0.0 0.0 0.06666667 0.0 0.06666667 0.03333333 0.0000000 0.0
## [7,] 0.00 0.0 0.0 0.0 0.50000000 0.1 0.50000000 0.11666667 0.1333333 0.0
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.1 0.26666667 0.54761905 0.3333333 0.1
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.06666667 0.22380952 0.5333333 0.2
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.07857143 0.0000000 0.4
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.0 0.00000000 0.00000000 0.0000000 0.1
## [,11]
## [1,] 0.0
## [2,] 0.0
## [3,] 0.0
## [4,] 0.0
## [5,] 0.0
## [6,] 0.0
## [7,] 0.0
## [8,] 0.0
## [9,] 0.0
## [10,] 0.2
## [11,] 0.4
##
## $U[[4]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.03 0.0 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [2,] 0.15 0.0 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [3,] 0.00 0.1 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [4,] 0.00 0.0 0.1 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [5,] 0.00 0.0 0.0 0.1 0.05000000 0.00000000 0.000000000 0.00000000
## [6,] 0.00 0.0 0.0 0.0 0.06725589 0.02222222 0.067255892 0.04398148
## [7,] 0.00 0.0 0.0 0.0 0.51974507 0.21111111 0.519745070 0.18101852
## [8,] 0.00 0.0 0.0 0.0 0.00000000 0.21111111 0.262145262 0.54828042
## [9,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.029629630 0.15080688
## [10,] 0.00 0.0 0.0 0.0 0.00000000 0.02222222 0.007407407 0.03035714
## [11,] 0.00 0.0 0.0 0.0 0.00000000 0.00000000 0.000000000 0.00000000
## [,9] [,10] [,11]
## [1,] 0.00000000 0.00000000 0.0000000
## [2,] 0.00000000 0.00000000 0.0000000
## [3,] 0.00000000 0.00000000 0.0000000
## [4,] 0.00000000 0.00000000 0.0000000
## [5,] 0.00000000 0.00000000 0.0000000
## [6,] 0.00000000 0.00000000 0.0000000
## [7,] 0.08333333 0.00000000 0.0000000
## [8,] 0.32777778 0.13333333 0.0000000
## [9,] 0.45317460 0.13333333 0.0000000
## [10,] 0.13571429 0.33333333 0.1333333
## [11,] 0.00000000 0.06666667 0.2666667
##
##
## $F
## $F[[1]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 62.5 422.2222 1583.333 0 0
## [2,] 0 0 0 0 0 0 62.5 422.2222 1583.333 0 0
## [3,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [4,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [5,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [6,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [7,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [8,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [9,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [10,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
## [11,] 0 0 0 0 0 0 0.0 0.0000 0.000 0 0
##
## $F[[2]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 55.55556 572.9167 2392.857 2875 6000
## [2,] 0 0 0 0 0 0 55.55556 572.9167 2392.857 2875 6000
## [3,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [4,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [5,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [6,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [7,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [8,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [9,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [10,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
## [11,] 0 0 0 0 0 0 0.00000 0.0000 0.000 0 0
##
## $F[[3]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 0 422.619 1625 750 2500
## [2,] 0 0 0 0 0 0 0 422.619 1625 750 2500
## [3,] 0 0 0 0 0 0 0 0.000 0 0 0
## [4,] 0 0 0 0 0 0 0 0.000 0 0 0
## [5,] 0 0 0 0 0 0 0 0.000 0 0 0
## [6,] 0 0 0 0 0 0 0 0.000 0 0 0
## [7,] 0 0 0 0 0 0 0 0.000 0 0 0
## [8,] 0 0 0 0 0 0 0 0.000 0 0 0
## [9,] 0 0 0 0 0 0 0 0.000 0 0 0
## [10,] 0 0 0 0 0 0 0 0.000 0 0 0
## [11,] 0 0 0 0 0 0 0 0.000 0 0 0
##
## $F[[4]]
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 0 0 0 0 0 0 39.35185 472.586 1867.063 1208.333 2833.333
## [2,] 0 0 0 0 0 0 39.35185 472.586 1867.063 1208.333 2833.333
## [3,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [4,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [5,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [6,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [7,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [8,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [9,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [10,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
## [11,] 0 0 0 0 0 0 0.00000 0.000 0.000 0.000 0.000
##
##
## $hstages
## [1] NA
##
## $agestages
## NA
## 1 NA
##
## $ahstages
## stage_id stage original_size original_size_b original_size_c min_age max_age
## 1 1 SD 0.0 NA NA 1 NA
## 2 2 P1 0.0 NA NA 1 NA
## 3 3 P2 0.0 NA NA 2 NA
## 4 4 P3 0.0 NA NA 3 NA
## 5 5 SL 0.0 NA NA 4 NA
## 6 6 D 0.0 NA NA 5 NA
## 7 7 XSm 1.0 NA NA 5 NA
## 8 8 Sm 3.0 NA NA 6 NA
## 9 9 Md 6.0 NA NA 6 NA
## 10 10 Lg 11.0 NA NA 6 NA
## 11 11 XLg 19.5 NA NA 6 NA
## repstatus obsstatus propstatus immstatus matstatus entrystage indataset
## 1 0 0 1 0 0 1 0
## 2 0 0 0 1 0 1 0
## 3 0 0 0 1 0 0 0
## 4 0 0 0 1 0 0 0
## 5 0 0 0 1 0 0 0
## 6 0 0 0 0 1 0 1
## 7 1 1 0 0 1 0 1
## 8 1 1 0 0 1 0 1
## 9 1 1 0 0 1 0 1
## 10 1 1 0 0 1 0 1
## 11 1 1 0 0 1 0 1
## binhalfwidth_raw sizebin_min sizebin_max sizebin_center sizebin_width
## 1 0.0 0.0 0.0 0.0 0
## 2 0.0 0.0 0.0 0.0 0
## 3 0.0 0.0 0.0 0.0 0
## 4 0.0 0.0 0.0 0.0 0
## 5 0.0 0.0 0.0 0.0 0
## 6 0.5 -0.5 0.5 0.0 1
## 7 0.5 0.5 1.5 1.0 1
## 8 1.5 1.5 4.5 3.0 3
## 9 1.5 4.5 7.5 6.0 3
## 10 3.5 7.5 14.5 11.0 7
## 11 5.0 14.5 24.5 19.5 10
## binhalfwidthb_raw sizebinb_min sizebinb_max sizebinb_center sizebinb_width
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## binhalfwidthc_raw sizebinc_min sizebinc_max sizebinc_center sizebinc_width
## 1 NA NA NA NA NA
## 2 NA NA NA NA NA
## 3 NA NA NA NA NA
## 4 NA NA NA NA NA
## 5 NA NA NA NA NA
## 6 NA NA NA NA NA
## 7 NA NA NA NA NA
## 8 NA NA NA NA NA
## 9 NA NA NA NA NA
## 10 NA NA NA NA NA
## 11 NA NA NA NA NA
## group comments alive almostborn
## 1 0 Dormant seed 1 0
## 2 0 1st yr protocorm 1 0
## 3 0 2nd yr protocorm 1 0
## 4 0 3rd yr protocorm 1 0
## 5 0 Seedling 1 0
## 6 0 Dormant adult 1 0
## 7 0 Extra small adult (1 shoot) 1 0
## 8 0 Small adult (2-4 shoots) 1 0
## 9 0 Medium adult (5-7 shoots) 1 0
## 10 0 Large adult (8-14 shoots) 1 0
## 11 0 Extra large adult (>14 shoots) 1 0
##
## $labels
## pop patch
## 1 1 A
## 2 1 B
## 3 1 C
## 4 1 0
##
## $matrixqc
## [1] 114 34 4
##
## $dataqc
## [1] 74 320
##
## attr(,"class")
## [1] "lefkoMat"A quick scan through our output shows that we have four matrices. The
labels element shows us the order of these. There is no
time term in the labels element, because all matrices are
temporal means. Instead, we see that the first three matrices are the
patch-level means for patches A, B, and
C. This is followed by the overall population mean matrix,
listed as patch 0. It also pays to look at the summary.
summary(cyp2rp_mean)##
## This ahistorical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 11 rows and columns, and a total of 121 elements.
## A total of 114 survival transitions were estimated, with 28.5 per matrix.
## A total of 34 fecundity transitions were estimated, with 8.5 per matrix.
## This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.000 0.100 0.100 0.100
## 1st Qu. 0.100 0.140 0.140 0.140
## Median 0.200 0.600 0.600 0.467
## Mean 0.416 0.573 0.509 0.499
## 3rd Qu. 0.788 0.917 0.850 0.776
## Max. 1.000 1.000 1.000 1.000And so we see that we have four matrices, and we see that these matrices have slightly more elements estimated, on average, than in the raw ahistorical MPM. This happened because some of the zeros in the original raw MPM were zeros only because of a sampling issue - no individuals actually transitioned through a particular transition in a particular year, but may have transitioned in other years, yielding higher numbers of non-zero elements in the arithmetic mean matrices than in the original raw matrices. Unfortunately these zeros will nonetheless drag these mean element values down artificially, potentially impacting our analyses. A comparison with the function-based means should be interesting in this regard.
cyp2fp_mean <- lmean(cypmatrix2fp)
summary(cyp2fp_mean)##
## This ahistorical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 9640 survival transitions were estimated, with 2410 per matrix.
## A total of 192 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.100 0.100 0.100 0.100
## 1st Qu. 0.991 0.991 0.991 0.991
## Median 1.000 0.999 0.998 0.999
## Mean 0.919 0.918 0.916 0.918
## 3rd Qu. 1.000 0.999 0.999 0.999
## Max. 1.000 1.000 0.999 1.000The function-based mean matrices have the same number of estimated transitions as their constituent matrices, because all elements that are estimable are actually estimated using the vital rate models supplied.
Now let’s create the final sets of arithmetic mean matrices for the raw and function-based hMPMs.
cyp3rp_mean <- lmean(cypmatrix3rp)
cyp3fp_mean <- lmean(cypmatrix3fp)
summary(cyp3rp_mean)##
## This historical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 121 rows and columns, and a total of 14641 elements.
## A total of 242 survival transitions were estimated, with 60.5 per matrix.
## A total of 74 fecundity transitions were estimated, with 18.5 per matrix.
## This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.0000 0.000 0.0000 0.0000
## 1st Qu. 0.0000 0.000 0.0000 0.0000
## Median 0.0000 0.000 0.0000 0.0000
## Mean 0.0747 0.128 0.0947 0.0991
## 3rd Qu. 0.0000 0.250 0.0000 0.1000
## Max. 1.0000 1.000 1.0000 0.9907summary(cyp3fp_mean)##
## This historical lefkoMat object contains 4 matrices.
##
## Each matrix is square with 2916 rows and columns, and a total of 8503056 elements.
## A total of 470836 survival transitions were estimated, with 117709 per matrix.
## A total of 9408 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 4 patches, and 0 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4]
## Min. 0.000 0.000 0.000 0.000
## 1st Qu. 0.957 0.941 0.933 0.941
## Median 0.993 0.988 0.982 0.988
## Mean 0.814 0.810 0.805 0.810
## 3rd Qu. 0.997 0.994 0.991 0.994
## Max. 1.000 1.000 1.000 1.000Now let’s create the remaining population-level MPMs that have not been created. These will be MPMs that do not separate matrices by patch. Note that these matrices will be different than the mean matrices produced earlier, because the latter were created by taking unweighted arithmetic means of matrix elements across patches. The MPMs we will create now will not separate individuals by patch, and so larger patches may have a greater influence on the resulting matrices.
cypmatrix3r <- rlefko3(data = cypraw_v1, stageframe = cypframe_raw,
year = "all", stages = c("stage3", "stage2", "stage1"),
size = c("size3added", "size2added", "size1added"), supplement = cypsupp3_raw,
yearcol = "year2", patchcol = "patchid", indivcol = "individ")
cypmatrix3f <- flefko3(stageframe = cypframe_fb, supplement = cypsupp3_fb,
modelsuite = cypmodels3, data = cypfb_env)
cypmatrix2f <- flefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
modelsuite = cypmodels2, data = cypfb_env)
cyp3r_mean <- lmean(cypmatrix3r)
cyp3f_mean <- lmean(cypmatrix3f)
cyp2r_mean <- lmean(cypmatrix2r)
cyp2f_mean <- lmean(cypmatrix2f)Before moving further, let’s show how we may use our individual
covariate-related vital rate models to create MPMs contingent on some
values of those individual covariates. Here, we create one more
function-based MPMs in which we use the lefkoMod object
created to handle precipitation, along with a specific value of total
annual precipitation.
cypmatrix2f_env <- flefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
modelsuite = cypmodels2_env, data = cypfb_env, inda = 90)
summary(cypmatrix2f_env)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 12050 survival transitions were estimated, with 2410 per matrix.
## A total of 240 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.100 3.92e-08 0.100 0.100 0.100
## 1st Qu. 0.999 3.92e-08 1.000 1.000 0.999
## Median 1.000 3.92e-08 1.000 1.000 1.000
## Mean 0.926 9.81e-03 0.928 0.929 0.926
## 3rd Qu. 1.000 3.93e-08 1.000 1.000 1.000
## Max. 1.000 1.80e-01 1.000 1.000 1.000Here we see some differences in the column sum distributions of the matrices. If we used different values of annual precipitation, we would find more dramatic differences. For example, here we will create one more MPM that uses the actual precipitation values originally loaded.
precip_vector <- c(92.2, 57.6, 96.0, 109.8, 111.9)
cypmatrix2f_env1 <- flefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
modelsuite = cypmodels2_env, data = cypfb_env, inda = precip_vector)
summary(cypmatrix2f_env1)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 12050 survival transitions were estimated, with 2410 per matrix.
## A total of 240 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.100 0.100 0.100 0.100 0.100
## 1st Qu. 0.999 1.000 1.000 0.999 0.999
## Median 1.000 1.000 1.000 1.000 1.000
## Mean 0.926 0.929 0.927 0.927 0.924
## 3rd Qu. 1.000 1.000 1.000 1.000 1.000
## Max. 1.000 1.000 1.000 1.000 1.000Now let’s move on to creating function-based Leslie MPMs. Here, we will use the Cypripedium data to do that, and then take a look at the first survival-transition and fecundity matrices.
cypleslie_f <- fleslie(data = cypfb_v1, modelsuite = cypmodels2p_age, start_age = 0,
fecage_min = 5)
cypleslie_f$U[[1]]## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [2,] 0.9764238 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [3,] 0.0000000 0.9764238 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.9764238 0.0000000 0.0000000 0.0000000 0.0000000
## [5,] 0.0000000 0.0000000 0.0000000 0.9764238 0.0000000 0.0000000 0.0000000
## [6,] 0.0000000 0.0000000 0.0000000 0.0000000 0.9764238 0.0000000 0.0000000
## [7,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9764238 0.0000000
## [8,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9764238
## [9,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [10,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [,8] [,9] [,10] [,11]
## [1,] 0.0000000 0.0000000 0.0000000 0.0000000
## [2,] 0.0000000 0.0000000 0.0000000 0.0000000
## [3,] 0.0000000 0.0000000 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000 0.0000000
## [5,] 0.0000000 0.0000000 0.0000000 0.0000000
## [6,] 0.0000000 0.0000000 0.0000000 0.0000000
## [7,] 0.0000000 0.0000000 0.0000000 0.0000000
## [8,] 0.0000000 0.0000000 0.0000000 0.0000000
## [9,] 0.9764238 0.0000000 0.0000000 0.0000000
## [10,] 0.0000000 0.9764238 0.0000000 0.0000000
## [11,] 0.0000000 0.0000000 0.9764238 0.9764238cypleslie_f$F[[1]]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0 0 0 0 0 0.9060239 0.9111521 0.9160264 0.9206566
## [2,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [3,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [4,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [5,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [6,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [7,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [8,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [9,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [10,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0 0 0 0 0 0.0000000 0.0000000 0.0000000 0.0000000
## [,10] [,11]
## [1,] 0.9250524 0.9292234
## [2,] 0.0000000 0.0000000
## [3,] 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000
## [5,] 0.0000000 0.0000000
## [6,] 0.0000000 0.0000000
## [7,] 0.0000000 0.0000000
## [8,] 0.0000000 0.0000000
## [9,] 0.0000000 0.0000000
## [10,] 0.0000000 0.0000000
## [11,] 0.0000000 0.0000000As before, we do not have a means of incorporating supplemental information into the Leslie MPM estimation process yet, so we do so manually. One easy approach is with a simple loop. Let’s do that, and take a look at the first matrix.
for (i in c(1:length(cypleslie_f$A))) {
cypleslie_f$U[[i]][2, 1] <- 0.1
cypleslie_f$U[[i]][3, 2] <- 0.1
cypleslie_f$U[[i]][4, 3] <- 0.1
cypleslie_f$F[[i]][1, 6] <- cypleslie_f$F[[i]][1, 6] * seeds_per_pod * 0.15
cypleslie_f$F[[i]][1, 7] <- cypleslie_f$F[[i]][1, 7] * seeds_per_pod * 0.15
cypleslie_f$F[[i]][1, 8] <- cypleslie_f$F[[i]][1, 8] * seeds_per_pod * 0.15
cypleslie_f$F[[i]][1, 9] <- cypleslie_f$F[[i]][1, 9] * seeds_per_pod * 0.15
cypleslie_f$F[[i]][1, 10] <- cypleslie_f$F[[i]][1, 10] * seeds_per_pod * 0.15
cypleslie_f$A[[i]] <- cypleslie_f$U[[i]] + cypleslie_f$F[[i]]
}
cypleslie_f$A[[1]]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,] 0.0 0.0 0.0 0.0000000 0.0000000 679.5178919 683.3640702 687.0198170
## [2,] 0.1 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [3,] 0.0 0.1 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [4,] 0.0 0.0 0.1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [5,] 0.0 0.0 0.0 0.9764238 0.0000000 0.0000000 0.0000000 0.0000000
## [6,] 0.0 0.0 0.0 0.0000000 0.9764238 0.0000000 0.0000000 0.0000000
## [7,] 0.0 0.0 0.0 0.0000000 0.0000000 0.9764238 0.0000000 0.0000000
## [8,] 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.9764238 0.0000000
## [9,] 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.9764238
## [10,] 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0.0 0.0 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [,9] [,10] [,11]
## [1,] 690.4924689 693.7893093 0.9292234
## [2,] 0.0000000 0.0000000 0.0000000
## [3,] 0.0000000 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000
## [5,] 0.0000000 0.0000000 0.0000000
## [6,] 0.0000000 0.0000000 0.0000000
## [7,] 0.0000000 0.0000000 0.0000000
## [8,] 0.0000000 0.0000000 0.0000000
## [9,] 0.0000000 0.0000000 0.0000000
## [10,] 0.9764238 0.0000000 0.0000000
## [11,] 0.0000000 0.9764238 0.9764238We can also take a peek at a summary.
summary(cypleslie_f)##
## This ahistorical lefkoMat object contains 15 matrices.
##
## Each matrix is square with 11 rows and columns, and a total of 121 elements.
## A total of 165 survival transitions were estimated, with 11 per matrix.
## A total of 90 fecundity transitions were estimated, with 6 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation probability not estimated.
## Primary size transition not estimated.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproduction probability not estimated.
## Fecundity estimated with 74 individuals and 320 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## Min. 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100
## 1st Qu. 0.538 0.538 0.538 0.538 0.538 0.538 0.538 0.538 0.538 0.538 0.538 0.538
## Median 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976
## Mean 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737
## 3rd Qu. 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976
## Max. 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976
## [,13] [,14] [,15]
## Min. 0.100 0.100 0.100
## 1st Qu. 0.538 0.538 0.538
## Median 0.976 0.976 0.976
## Mean 0.737 0.737 0.737
## 3rd Qu. 0.976 0.976 0.976
## Max. 0.976 0.976 0.976Now let’s try a function-based age-by-stage MPM.
cypagestage_f <- aflefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
data = cypfb_v1, modelsuite = cypmodels2p_agestage)
summary(cypagestage_f)##
## This ahistorical lefkoMat object contains 15 matrices.
##
## Each matrix is square with 486 rows and columns, and a total of 236196 elements.
## A total of 146475 survival transitions were estimated, with 9765 per matrix.
## A total of 2880 fecundity transitions were estimated, with 192 per matrix.
## This lefkoMat object covers 1 population, 3 patches, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## Min. 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Median 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Mean 0.417 0.418 0.418 0.418 0.416 0.417 0.418 0.418 0.418 0.416 0.416 0.418
## 3rd Qu. 0.999 1.000 1.000 1.000 0.999 0.998 0.999 0.999 0.999 0.998 0.997 0.999
## Max. 1.000 1.000 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 0.999 1.000
## [,13] [,14] [,15]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.000 0.000 0.000
## Median 0.000 0.000 0.000
## Mean 0.417 0.417 0.415
## 3rd Qu. 0.998 0.998 0.997
## Max. 1.000 0.999 0.999These are very large matrices, and are quite sparse. However, all estimable elements have been estimated.
Now let’s finally turn our attention to IPMs. Let’s build a standard, ahistorical IPM for Lathyrus, as below.
lathmat2_ipm <- flefko2(stageframe = lathframe_ipm, modelsuite = lathmodels2ipm,
data = lathvert_ipm, supplement = lathsupp2)
summary(lathmat2_ipm)##
## This ahistorical lefkoMat object contains 3 matrices.
##
## Each matrix is square with 103 rows and columns, and a total of 10609 elements.
## A total of 26947 survival transitions were estimated, with 8982.333 per matrix.
## A total of 600 fecundity transitions were estimated, with 200 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 3 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 931 individuals and 2246 individual transitions.
## Observation estimated with 858 individuals and 2121 individual transitions.
## Primary size estimated with 845 individuals and 1916 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproduction probability not estimated.
## Fecundity estimated with 931 individuals and 2246 individual transitions.
## Juvenile survival estimated with 281 individuals and 281 individual transitions.
## Juvenile observation estimated with 210 individuals and 210 individual transitions.
## Juvenile primary size estimated with 193 individuals and 193 individual transitions.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.994 0.970 0.996
## Median 1.000 1.000 1.000
## Mean 0.961 0.929 0.965
## 3rd Qu. 1.000 1.000 1.000
## Max. 1.000 1.000 1.000This discretized IPM is composed of rather large, very dense matrices. Indeed, we see the matrices are composed of 10,609 elements each, and on average 8982.3 (84.7%) are estimated as non-zero. If interested, we can build historical IPMs as well, which will be far larger and very sparse, as below.
lathmat3_ipm <- flefko3(stageframe = lathframe_ipm, modelsuite = lathmodels3ipm,
data = lathvert_ipm, supplement = lathsupp3)
summary(lathmat3_ipm)##
## This historical lefkoMat object contains 3 matrices.
##
## Each matrix is square with 10609 rows and columns, and a total of 112550881 elements.
## A total of 2684709 survival transitions were estimated, with 894903 per matrix.
## A total of 60600 fecundity transitions were estimated, with 20200 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 3 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 931 individuals and 2246 individual transitions.
## Observation estimated with 858 individuals and 2121 individual transitions.
## Primary size estimated with 845 individuals and 1916 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproduction probability not estimated.
## Fecundity estimated with 931 individuals and 2246 individual transitions.
## Juvenile survival estimated with 281 individuals and 281 individual transitions.
## Juvenile observation estimated with 210 individuals and 210 individual transitions.
## Juvenile primary size estimated with 193 individuals and 193 individual transitions.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.993 0.986 0.992
## Median 0.998 0.998 0.998
## Mean 0.958 0.947 0.957
## 3rd Qu. 0.999 0.999 0.999
## Max. 1.000 1.000 1.000The resulting set of matrices is utterly huge - 10,609 rows and 10,609 columns, yielding 112,550,881 elements per matrix for three matrices. Although many more elements are estimated than in the ahistorical case, the increase in elements is overwhelmingly in zero elements, with only 0.8% of elements estimated as non-zero. This fact makes these matrices especially sparse, with greater and greater sparseness as the number of stages increases.
Our newest historical IPM in particular takes up a lot of memory. To prevent exhausting the available RAM for further analysis, we will remove this object now.
rm("lathmat3_ipm")Let’s now go on to look at a few issues in modeling MPMs and IPMs.
Step 4b. Issues in modeling MPMs.
In this section, we wish to look at three issues that have arisen in MPM building so far - the importance of proper parameterization of size and fecundity, matrix dimensionality reduction, and Ehrlén format vs. deVries format. We will also consider a fourth issue - vital rate model accuracy.
The first issue is the proper parameterization of size and fecundity
in the function-based approach. In this example, we used the
sf_distrib() function to assess the distributions to use,
since we knew that both size and fecundity are count variables. That
assessment led to the choice of the zero-truncated negative binomial
distribution to model size, and the zero-inflated Poisson distribution
to model fecundity. The impact of the choices we made for these
distributions can be quite profound, and can strongly impact the results
of analysis. To allow us to make comparisons later, let’s create an
extra set of vital rate models and one more function-based ahMPM
utilizing the Poisson distribution for both size and fecundity. Note
that this choice may have a large effect because the Poisson
distribution assumes that the mean and variance are equal (the negative
binomial only assumes that they are related), and actually predicts
specific numbers of zeros (zero truncation and zero inflation make this
far more flexible).
cypmodels2_wrong <- modelsearch(cypfb_v1, historical = FALSE, approach = "mixed",
vitalrates = c("surv", "obs", "size", "repst", "fec"),
sizedist = "poisson", fecdist = "poisson", suite = "full",
size = c("size3added", "size2added", "size1added"), quiet = TRUE)
cypmatrix2f_wrong <- flefko2(stageframe = cypframe_fb, supplement = cypsupp2_fb,
modelsuite = cypmodels2_wrong, data = cypfb_env)Now let’s compare the new MPM to the correct MPM.
summary(cypmatrix2f)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 12050 survival transitions were estimated, with 2410 per matrix.
## A total of 240 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.100 0.100 0.100 0.100 0.100
## 1st Qu. 0.991 0.991 0.991 0.991 0.991
## Median 1.000 1.000 1.000 1.000 1.000
## Mean 0.917 0.920 0.919 0.919 0.916
## 3rd Qu. 1.000 1.000 1.000 1.000 1.000
## Max. 1.000 1.000 1.000 1.000 1.000summary(cypmatrix2f_wrong)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 54 rows and columns, and a total of 2916 elements.
## A total of 12048 survival transitions were estimated, with 2409.6 per matrix.
## A total of 240 fecundity transitions were estimated, with 48 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.100 0.100 0.100 0.100 0.100
## 1st Qu. 0.944 0.910 0.923 0.938 0.942
## Median 0.982 0.963 0.971 0.978 0.981
## Mean 0.896 0.882 0.888 0.895 0.894
## 3rd Qu. 0.995 0.988 0.991 0.994 0.995
## Max. 0.999 0.997 0.998 0.999 0.999We see a number of issues arising from the new choice in distribution. First, the distribution of transition probabilities is lower, leading to lower mean and maximum survival column sums. Second, because of the former issue, a few more elements are estimated as zero. We will come back to this set later in our analyses.
The second issue is matrix dimensionality reduction. Occasionally,
historical matrices end up being produced in which stage-pairs exist
that are not associated with any transitions. This is particularly the
case with raw MPMs. The result is that the rows and columns associated
with such a stage pair are completely full of zeros. In this situation,
the matrix can be reduced by eliminating the stage-pair altogether. The
result is a set of smaller matrices, which is useful considering that
historical MPMs are generally large and can take up lots of memory. We
can tell R to develop reduced MPMs by using the
reduce = TRUE option in the matrix generating function that
we are using. Note that this option is available in ALL matrix
generating functions, even for those creating ahistorical MPMs, and that
rows and columns will be reduced only if ALL matrices within the
lefkoMat object have empty rows and columns associated with
a particular stage or stage-pair. Here, we create a reduced
function-based hMPM, and then compare it to the unreduced function-based
hMPM.
cypmatrix3f_red <- flefko3(stageframe = cypframe_fb, supplement = cypsupp3_fb,
modelsuite = cypmodels3, data = cypfb_env, reduce = TRUE)
summary(cypmatrix3f_red)##
## This historical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 2458 rows and columns, and a total of 6041764 elements.
## A total of 588545 survival transitions were estimated, with 117709 per matrix.
## A total of 11760 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.994 0.997 0.997 0.996 0.994
## Median 0.998 1.000 0.999 0.999 0.998
## Mean 0.971 0.972 0.971 0.971 0.971
## 3rd Qu. 0.999 1.000 1.000 0.999 0.999
## Max. 1.000 1.000 1.000 1.000 1.000summary(cypmatrix3f)##
## This historical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 2916 rows and columns, and a total of 8503056 elements.
## A total of 588545 survival transitions were estimated, with 117709 per matrix.
## A total of 11760 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.965 0.965 0.965 0.965 0.965
## Median 0.998 0.999 0.999 0.998 0.998
## Mean 0.818 0.819 0.819 0.819 0.818
## 3rd Qu. 0.999 1.000 1.000 0.999 0.999
## Max. 1.000 1.000 1.000 1.000 1.000Our new hMPM has matrices with 2458 rows and columns, while the original unreduced hMPM has matrices with 2916 rows and columns. This is a reduction of 458 rows and columns, with an overall reduction in the size of the matrix by \(8503056-6041764 = 2461292\) elements. In memory, the unreduced hMPM takes up 1GB while the reduced hMPM takes up 725MB. So, clearly this reduced set is preferable to the unreduced set.
Third, we should consider the issue of the format of the hMPM. By
default, we use Ehrlén format, which we find more intuitive and simpler
to use. However, some prefer the format outlined in de Vries and Caswell
(2018). There is only one real difference between these formats, and
that is the treatment of an individual that newly recruits in time t. de
Vries and Caswell (2018) argued that it is not logical to consider
maternal state as the stage in time t-1 for an individual transitioning
from newborn in time t to whatever its next stage is in time t+1. They
therefore suggested that a prior “not yet born” stage should be added to
the matrix to deal with issue. Thus, any individuals that is newly-born
in time t is treated as having been in this extra stage in time t-1
(note that this stage is not considered as a part of the fecundity
transition, but is instead counted ONLY in survival transitions for
individuals new to the population in time t). This extra stage yields
more stage-pairs in the hMPM, and causes some transitions to be split
that would otherwise not be split in Ehrlén format. In any case,
analyses using deVries format are not expected to yield any differences
analytically from Ehrlén format, but the different matrix dimensions and
small differences in treatment of some transitions will likely yield
small differences overall here and there. Functions
rlefko3() and flefko3() both produce Ehrlén
format matrices by default, but users can also produce deVries format
matrices by setting format = "deVries", as below.
cypmatrix3f_dev <- flefko3(stageframe = cypframe_fb, supplement = cypsupp3_fb,
modelsuite = cypmodels3, data = cypfb_env, format = "deVries")
summary(cypmatrix3f_dev)##
## This historical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 2970 rows and columns, and a total of 8820900 elements.
## A total of 588305 survival transitions were estimated, with 117661 per matrix.
## A total of 11760 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.964 0.965 0.965 0.965 0.964
## Median 0.997 0.999 0.999 0.998 0.998
## Mean 0.801 0.802 0.802 0.802 0.801
## 3rd Qu. 0.999 1.000 1.000 0.999 0.999
## Max. 1.000 1.000 1.000 1.000 1.000summary(cypmatrix3f)##
## This historical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 2916 rows and columns, and a total of 8503056 elements.
## A total of 588545 survival transitions were estimated, with 117709 per matrix.
## A total of 11760 fecundity transitions were estimated, with 2352 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## Vital rate modeling quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Observation estimated with 70 individuals and 303 individual transitions.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity transition probability not estimated.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.000 0.000 0.000 0.000
## 1st Qu. 0.965 0.965 0.965 0.965 0.965
## Median 0.998 0.999 0.999 0.998 0.998
## Mean 0.818 0.819 0.819 0.819 0.818
## 3rd Qu. 0.999 1.000 1.000 0.999 0.999
## Max. 1.000 1.000 1.000 1.000 1.000Our new matrices have 54 more rows and columns, and over 300,000 more
elements. To see what is different, compare the hstages
element in each on your own, and you will see the inclusion of some new
stage pairs with the AlmostBorn stage in the prior
position. Some transitions were split, as well, yielding different
numbers of transitions that were estimated as non-zero.
Finally, we will address the issue of vital rate model accuracy. This
issue is generally ignored in most of the function-based MPM / IPM
literature, but the truth of the matter is that function-based MPMs are
only as good as the vital rate models that they are parameterized with.
If the best-fit models determining vital rates are themselves not
particularly good descriptions of the demographic data, then they will
yield erroneous predictions and analytical results. Ideally, users
interested in parameterizing function-based MPMs will assess the overall
quality of each of their vital rate models.The quality of vital rate
models utilizing a binomial response can be assessed using
accuracy, which refers to the ability of a binomial
response model to predict the original dataset. Other models can be
assessed with pseudo-R2. To see the quality of a set of
models, look at the summary of a lefkoMod object.
summary(cypmodels3)## This LefkoMod object includes 5 linear models.
## Best-fit model criterion used: aicc&k
##
##
##
## Survival model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: alive3 ~ size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 128.1324 143.2057 -60.0662 120.1324 316
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.198361
## year2 (Intercept) 0.008826
## Number of obs: 320, groups: individ, 74; year2, 5
## Fixed Effects:
## (Intercept) size2added
## 2.0352 0.6344
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Observation model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: obsstatus3 ~ size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 118.2567 133.1117 -55.1284 110.2567 299
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 1.078e-05
## year2 (Intercept) 8.776e-01
## Number of obs: 303, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) size2added
## 2.4904 0.3134
## optimizer (Nelder_Mead) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings
##
##
##
## Size model:
## Formula:
## size3added ~ repstatus1 + repstatus2 + size1added + size2added +
## (1 | year2) + (1 | individ) + repstatus2:size1added
## Data: subdata
## AIC BIC logLik df.resid
## -15046.078 -15013.112 7532.039 279
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.1247
## individ (Intercept) 0.9368
##
## Number of obs: 288 / Conditional model: year2, 5; individ, 70
##
## Dispersion parameter for truncated_nbinom2 family (): 4.81e+15
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) repstatus1 repstatus2
## 0.46241 0.05328 0.13092
## size1added size2added repstatus2:size1added
## 0.01237 0.02458 -0.02016
##
##
##
## Secondary size model:
## [1] 1
##
##
##
## Tertiary size model:
## [1] 1
##
##
##
## Reproductive status model:
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: repstatus3 ~ repstatus2 + size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik deviance df.resid
## 333.6176 351.9324 -161.8088 323.6176 283
## Random effects:
## Groups Name Std.Dev.
## individ (Intercept) 0.1829
## year2 (Intercept) 0.6250
## Number of obs: 288, groups: individ, 70; year2, 5
## Fixed Effects:
## (Intercept) repstatus2 size2added
## -1.4630 1.6457 0.1715
##
##
##
## Fecundity model:
## Formula: feca2 ~ size2added + (1 | year2) + (1 | individ)
## Zero inflation: ~size2added + (1 | year2) + (1 | individ)
## Data: subdata
## AIC BIC logLik df.resid
## 248.8609 271.0264 -116.4305 110
## Random-effects (co)variances:
##
## Conditional model:
## Groups Name Std.Dev.
## year2 (Intercept) 0.5760
## individ (Intercept) 0.1639
##
## Zero-inflation model:
## Groups Name Std.Dev.
## year2 (Intercept) 1.642e-06
## individ (Intercept) 3.089e-04
##
## Number of obs: 118 / Conditional model: year2, 5; individ, 51 / Zero-inflation model: year2, 5; individ, 51
##
## Fixed Effects:
##
## Conditional model:
## (Intercept) size2added
## -0.54014 0.06174
##
## Zero-inflation model:
## (Intercept) size2added
## 3.865 -1.574
##
##
## Juvenile survival model:
## [1] 1
##
##
##
## Juvenile observation model:
## [1] 1
##
##
##
## Juvenile size model:
## [1] 1
##
##
##
## Juvenile secondary size model:
## [1] 1
##
##
##
## Juvenile tertiary size model:
## [1] 1
##
##
##
## Juvenile reproduction model:
## [1] 1
##
##
##
## Juvenile maturity model:
## [1] 1
##
##
##
##
##
## Number of models in survival table: 113
##
## Number of models in observation table: 113
##
## Number of models in size table: 113
##
## Number of models in secondary size table: 1
##
## Number of models in tertiary size table: 1
##
## Number of models in reproduction status table: 113
##
## Number of models in fecundity table: 12437
##
## Number of models in juvenile survival table: 1
##
## Number of models in juvenile observation table: 1
##
## Number of models in juvenile size table: 1
##
## Number of models in juvenile secondary size table: 1
##
## Number of models in juvenile tertiary size table: 1
##
## Number of models in juvenile reproduction table: 1
##
## Number of models in juvenile maturity table: 1
##
##
##
##
##
## General model parameter names (column 1), and
## specific names used in these models (column 2):
## parameter_names mainparams
## 1 time t year2
## 2 individual individ
## 3 patch patch
## 4 alive in time t+1 surv3
## 5 observed in time t+1 obs3
## 6 sizea in time t+1 size3
## 7 sizeb in time t+1 sizeb3
## 8 sizec in time t+1 sizec3
## 9 reproductive status in time t+1 repst3
## 10 fecundity in time t+1 fec3
## 11 fecundity in time t fec2
## 12 sizea in time t size2
## 13 sizea in time t-1 size1
## 14 sizeb in time t sizeb2
## 15 sizeb in time t-1 sizeb1
## 16 sizec in time t sizec2
## 17 sizec in time t-1 sizec1
## 18 reproductive status in time t repst2
## 19 reproductive status in time t-1 repst1
## 20 maturity status in time t+1 matst3
## 21 maturity status in time t matst2
## 22 age in time t age
## 23 density in time t density
## 24 individual covariate a in time t indcova2
## 25 individual covariate a in time t-1 indcova1
## 26 individual covariate b in time t indcovb2
## 27 individual covariate b in time t-1 indcovb1
## 28 individual covariate c in time t indcovc2
## 29 individual covariate c in time t-1 indcovc1
## 30 stage group in time t group2
## 31 stage group in time t-1 group1
##
##
##
##
##
## Quality control:
##
## Survival estimated with 74 individuals and 320 individual transitions.
## Survival accuracy is 0.947.
## Observation estimated with 70 individuals and 303 individual transitions.
## Observation accuracy is 0.95.
## Primary size estimated with 70 individuals and 288 individual transitions.
## Primary size pseudo R-squared is NA.
## Secondary size transition not estimated.
## Tertiary size transition not estimated.
## Reproductive status estimated with 70 individuals and 288 individual transitions.
## Reproductive status accuracy is 0.715.
## Fecundity estimated with 51 individuals and 118 individual transitions.
## Fecundity pseudo R-squared is 0.32.
## Juvenile survival not estimated.
## Juvenile observation probability not estimated.
## Juvenile primary size transition not estimated.
## Juvenile secondary size transition not estimated.
## Juvenile tertiary size transition not estimated.
## Juvenile reproduction probability not estimated.
## Juvenile maturity probability not estimated.Our accuracy results are at the bottom of the summary. They suggest that the survival and observation models are pretty good, but the reproduction model is wrong almost 1/3 of the time. Our primary size model has a very low pseudo-R2, as does fecundity. This may all affect the performance of this model down the line.
Step 4c. Importing IPMs and fbMPMs
In step 3c, we imported vital rate models for use in creating a
Lathyrus MPM. At this point, we have created all of our main IPM vital
rate model inputs. However, there are still a few missing pieces that we
need to fit in order to create the IPM properly. Our next step is to
include instructions that tell lefko3 how the vital rate
models fit together. Let’s see how this works.
By default, elements in survival-transition matrices created by
functions flefko2(), flefko3(),
aflefko2(), and fleslie() are estimated via
the following two kernels, the first for adults and the second for
juveniles.
\(e_{j, i} = surv_{kernel} obs_{kernel} sizea_{kernel} sizeb_{kernel} sizec_{kernel} repr_{kernel}\)
\(e_{j, i} = jsurv_{kernel} jobs_{kernel} jsizea_{kernel} jsizeb_{kernel} jsizec_{kernel} jrepr_{kernel} jmatst_{kernel}\)
Here, \(e_{j, i}\) refers to the element at the jth row and the ith column. The terms to the right of the equal sign refer to the values developed by kernels representing different vital rates. So, \(surv_{kernel}\) is the kernel developing the probability of survival, \(obs_{kernel}\) is the kernel developing the probability of observation (or its complement), \(sizea_{kernel}\) is the probability of size transition in the primary size variable, \(sizeb_{kernel}\) is the probability of size transition in the secondary size variable, \(sizec_{kernel}\) is the probability of size transition in the tertiary size variable, and \(repst_{kernel}\) is the probability of becoming reproductive or its complement. The juvenile kernel is composed of similar vital rate kernels, but also includes \(jmatst_{kernel}\), which is the probability of becoming mature or its complement.
Package lefko3 uses the vital rate kernels above to
produce values in essentially all cases, and so vital rate kernel values
can shift to a constant value of \(1\)
if a vital rate is not used. There are also circumstances in which vital
rates may be fixed to 0 or even other constants, though these are
relatively rare (this may occur is a juvenile size class is not
utilized, because then juvenile vital rates should generally equal zero
during matrix creation). The key is to fix unused vital rates to
constant values of \(1\). We do this by
changing the y-intercept of unused vital rate models to exactly
1, and changing the distribution listed for the vital rate
in dist_frame as constant. Note that we do not
need to bother doing this with zero-inflation components of unused vital
rates. In our case, all unused vital rate models are already set to
constant, so we will just change the intercepts to
1.
lath_vrm$vrm_frame$sizeb[1] <- 1
lath_vrm$vrm_frame$sizec[1] <- 1
lath_vrm$vrm_frame$repst[1] <- 1
lath_vrm$vrm_frame$jsizeb[1] <- 1
lath_vrm$vrm_frame$jsizec[1] <- 1
lath_vrm$vrm_frame$jrepst[1] <- 1
lath_vrm$vrm_frame$jmatst[1] <- 1
lath_vrm## $vrm_frame
## main_effect_1 main_1_defined surv obs sizea sizeb
## 1 intercept y-intercept 2.32571 2.23 164.0695 1
## 2 size2 sizea in time t 0.00109 0.00 0.6211 0
## 3 size1 sizea in time t-1 0.00000 0.00 0.0000 0
## 4 sizeb2 sizeb in time t 0.00000 0.00 0.0000 0
## 5 sizeb1 sizeb in time t-1 0.00000 0.00 0.0000 0
## 6 sizec2 sizec in time t 0.00000 0.00 0.0000 0
## 7 sizec1 sizec in time t-1 0.00000 0.00 0.0000 0
## 8 repst2 reproductive status in time t 0.00000 0.00 0.0000 0
## 9 repst1 reproductive status in time t-1 0.00000 0.00 0.0000 0
## 10 age age in time t 0.00000 0.00 0.0000 0
## 11 density density in time t 0.00000 0.00 0.0000 0
## 12 indcova2 individual covariate a in time t 0.00000 0.00 0.0000 0
## 13 indcova1 individual covariate a in time t-1 0.00000 0.00 0.0000 0
## 14 indcovb2 individual covariate b in time t 0.00000 0.00 0.0000 0
## 15 indcovb1 individual covariate b in time t-1 0.00000 0.00 0.0000 0
## 16 indcovc2 individual covariate c in time t 0.00000 0.00 0.0000 0
## 17 indcovc1 individual covariate c in time t-1 0.00000 0.00 0.0000 0
## sizec repst fec jsurv jobs jsizea jsizeb jsizec jrepst jmatst sizea_zi
## 1 1 1 1.517 1.03 10.39 3.0559 1 1 1 1 0
## 2 0 0 0.000 0.00 0.00 0.8482 0 0 0 0 0
## 3 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 4 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 5 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 6 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 7 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 8 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 9 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 10 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 11 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 12 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 13 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 14 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 15 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 16 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## 17 0 0 0.000 0.00 0.00 0.0000 0 0 0 0 0
## sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 6.252765 0 0 0
## 2 0 0 -0.007313 0 0 0
## 3 0 0 0.000000 0 0 0
## 4 0 0 0.000000 0 0 0
## 5 0 0 0.000000 0 0 0
## 6 0 0 0.000000 0 0 0
## 7 0 0 0.000000 0 0 0
## 8 0 0 0.000000 0 0 0
## 9 0 0 0.000000 0 0 0
## 10 0 0 0.000000 0 0 0
## 11 0 0 0.000000 0 0 0
## 12 0 0 0.000000 0 0 0
## 13 0 0 0.000000 0 0 0
## 14 0 0 0.000000 0 0 0
## 15 0 0 0.000000 0 0 0
## 16 0 0 0.000000 0 0 0
## 17 0 0 0.000000 0 0 0
##
## $year_frame
## years surv obs sizea sizeb sizec repst fec jsurv jobs
## 1 1988 0 0 96.3244 0 0 0 -0.41749627 0 -0.7459843
## 2 1989 0 0 -240.8036 0 0 0 0.51421684 0 0.6118826
## 3 1990 0 0 144.4792 0 0 0 -0.07964038 0 -0.9468618
## jsizea jsizeb jsizec jrepst jmatst sizea_zi sizeb_zi sizec_zi
## 1 0.5937962 0 0 0 0 0 0 0
## 2 1.4551236 0 0 0 0 0 0 0
## 3 -2.0489198 0 0 0 0 0 0 0
## fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 3.741475e-07 0 0 0
## 2 -7.804715e-08 0 0 0
## 3 -2.533755e-07 0 0 0
##
## $patch_frame
## patches surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group2_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $group1_frame
## groups surv obs sizea sizeb sizec repst fec jsurv jobs jsizea jsizeb jsizec
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## jrepst jmatst sizea_zi sizeb_zi sizec_zi fec_zi jsizea_zi jsizeb_zi jsizec_zi
## 1 0 0 0 0 0 0 0 0 0
##
## $dist_frame
## response dist
## 1 surv binom
## 2 obs binom
## 3 sizea gaussian
## 4 sizeb constant
## 5 sizec constant
## 6 repst constant
## 7 fec negbin
## 8 jsurv binom
## 9 jobs binom
## 10 jsizea gaussian
## 11 jsizeb constant
## 12 jsizec constant
## 13 jrepst constant
## 14 jmatst constant
##
## $st_frame
## surv obs sizea sizeb sizec repst
## 1.0000000 1.0000000 503.6167000 1.0000000 1.0000000 1.0000000
## fec jsurv jobs jsizea jsizeb jsizec
## 0.2342114 1.0000000 1.0000000 5.8310000 1.0000000 1.0000000
## jrepst jmatst
## 1.0000000 1.0000000
##
## attr(,"class")
## [1] "vrm_input"Now let’s build our IPM using the flefko2()
function.
lathmat2_importipm <- flefko2(stageframe = lathframe_ipm, modelsuite = lath_vrm,
supplement = lathsupp2, reduce = FALSE)
summary(lathmat2_importipm)##
## This ahistorical lefkoMat object contains 3 matrices.
##
## Each matrix is square with 103 rows and columns, and a total of 10609 elements.
## A total of 26926 survival transitions were estimated, with 8975.333 per matrix.
## A total of 600 fecundity transitions were estimated, with 200 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 3 time steps.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3]
## Min. 0.000 0.000 0.000
## 1st Qu. 0.979 0.956 0.980
## Median 0.998 0.998 0.998
## Mean 0.951 0.922 0.954
## 3rd Qu. 1.000 1.000 1.000
## Max. 1.000 1.000 1.000Users exploring the output and comparing against the original published IPM will notice that the element values are almost the same, but differ by tiny amounts (often on the order of 10-5 or smaller). These small differences occur because of the rounding that happens when authors publish their models, and are not a cause of concern nor will they cause dramatic differences in inference.
Step 5. MPM analysis
We will now explore analyses that we might conduct with MPMs and IPMs. The code for the analyses is essentially the same regardless of the kind of MPM that we utilize, so we will only focus on the Cypripedium MPMs. We will start off by looking at typical, or “classical”, projection analyses, and then we will introduce complex, custom analyses.
Step 5a. “Classical” analyses
Let’s estimate the asymptotic deterministic population growth rate for each of the four sets of patch-level MPMs. We will first focus on the raw patch-level MPMs.
cyp2rp_lam <- lambda3(cypmatrix2rp)
cyp3rp_lam <- lambda3(cypmatrix3rp)
plot(lambda ~ year2, data = subset(cyp2rp_lam, patch == "A"),
ylim = c(0, 1.25), type = "l", lwd = 2, bty = "n")
lines(lambda ~ year2, data = subset(cyp2rp_lam, patch == "B"), type = "l",
lwd = 2, lty = 2)
lines(lambda ~ year2, data = subset(cyp2rp_lam, patch == "C"), type = "l",
lwd = 2, lty = 3)
lines(lambda ~ year2, data = subset(cyp3rp_lam, patch == "A"), type = "l",
lwd = 2, lty = 1, col = "red")
lines(lambda ~ year2, data = subset(cyp3rp_lam, patch == "B"), type = "l",
lwd = 2, lty = 2, col = "red")
lines(lambda ~ year2, data = subset(cyp3rp_lam, patch == "C"), type = "l",
lwd = 2, lty = 3, col = "red")
legend("bottomleft", c("A ahistorical", "B ahistorical", "C ahistorical",
"A historical", "B historical", "C historical"), lty = c(1, 2, 3, 1, 2, 3),
col = c("black", "black", "black", "red", "red", "red"), lwd = 2, cex = 0.7,
bty = "n")
Here we see that \(\lambda\) is different between the ahistorical and historical cases in the raw MPMs. Let’s compare the results above to the asymptotic deterministic population growth rate in the function-based MPMs.
cyp2fp_lam <- lambda3(cypmatrix2fp)
cyp3fp_lam <- lambda3(cypmatrix3fp)
plot(lambda ~ year2, data = subset(cyp2fp_lam, patch == "A"),
ylim = c(0.75, 1.20), type = "l", lwd = 2, bty = "n")
lines(lambda ~ year2, data = subset(cyp2fp_lam, patch == "B"), type = "l",
lwd = 2, lty = 2)
lines(lambda ~ year2, data = subset(cyp2fp_lam, patch == "C"), type = "l",
lwd = 2, lty = 3)
lines(lambda ~ year2, data = subset(cyp3fp_lam, patch == "A"), type = "l",
lwd = 2, lty = 1, col = "red")
lines(lambda ~ year2, data = subset(cyp3fp_lam, patch == "B"), type = "l",
lwd = 2, lty = 2, col = "red")
lines(lambda ~ year2, data = subset(cyp3fp_lam, patch == "C"), type = "l",
lwd = 2, lty = 3, col = "red")
legend("bottomleft", c("A ahistorical", "B ahistorical", "C ahistorical",
"A historical", "B historical", "C historical"), lty = c(1, 2, 3, 1, 2, 3),
col = c("black", "black", "black", "red", "red", "red"), lwd = 2, cex = 0.7,
bty = "n")
Note that we see very different predictions from the function-based MPMs than from the raw MPMs here. There are several reasons for this. First, there are strong impacts of the quality and size of the dataset, and of the vital rate models. Raw MPMs are parameterized by the actual transitions that were documented during the study, which can include “artificial” zeroes resulting from low sample size, while the function-based MPMs are parameterized with the vital rate models, which represent process variance and exclude error that cannot be attributed to the modeled factors. In a sufficiently small dataset, many transitions will be lacking in the raw MPMs because there are too few individual to propagate all transition estimates, and in turn these estimates get treated as zeros that artificially drag down some matrix element means and can decrease \(\lambda\). The function-based approach, in contrast, is parameterized by vital rate models that use the dataset as a whole to identify trends, rather than being susceptible to the seemingly random nature in which individuals contribute to transitions. However, smaller datasets will also affect function-based MPMs by yielding vital rate models that are less variable and less influenced by tested factors, as a result of lost statistical power. They are also only as good as their vital rate models, and so there may be an influence of some of the loss of accuracy that we witnessed in the vital rate model for reproductive status, and imperfect prediction of size and fecundity. Remember that poor accuracy in linear modeling affects the predictions of models by making them more “horizontal”.
A second issue that is less obvious is that our MPMs generally involve case-specific assumptions that can have great effects on predicted population dynamics. The biggest assumption that we make in these MPMs is the nature of fecundity - our model assumes that individuals become seedlings at least three years after seed production, and that the numbers of seed produced, the germination rate, the survival probabilities of juvenile stages, and the entry rates into the adult population (i.e. the recruitment rate) are all constant. In truth, the literature suggests that these are all problematic assumptions - the range in seeds produced per pod is likely to range from 500 to over 50,000, and adequate estimates of survival probability of any juvenile stage are essentially lacking in wild orchid populations. The most profound break likely comes from the assumptions about the germination rate, because what little literature exists on this topic suggests that germination rate is strongly variable across years, potentially reflecting the strong influence of variability in weather from year to year, and that germination is actually strongly density dependent. These assumptions should be addressed through more intensive research and exploration.
The literature has steadily suggested a preference to function-based models like IPMs over the years, but which approach is more appropriate is actually likely to be a strong function of how large a dataset is, and whether the factors utilized truly explain variation in vital rates well.
Now let’s compare the overall \(\lambda\) of the patch-level arithmetic
mean matrices, and the stochastic log growth rate, \(a = \text{log} \lambda _{S}\). To make sure
that our stochastic growth rate estimates equal yours, we will use the
set.seed() function prior to each stochastic run. We will
also use the defaults, which include estimation via 10,000 time steps
and equivalent time weights.
writeLines("Raw ahistorical lambda:")## Raw ahistorical lambda:lambda3(cyp2rp_mean)## pop patch lambda
## 1 1 A 1.035653
## 2 1 B 1.151105
## 3 1 C 1.112109
## 4 1 0 1.092164writeLines("\nRaw historical lambda:")##
## Raw historical lambda:lambda3(cyp3rp_mean)## pop patch lambda
## 1 1 A 0.8220534
## 2 1 B 0.8157691
## 3 1 C 0.9050548
## 4 1 0 0.7819680writeLines("\nFunction-based ahistorical lambda:")##
## Function-based ahistorical lambda:lambda3(cyp2fp_mean)## pop patch lambda
## 1 1 A 1.124856
## 2 1 B 1.099774
## 3 1 C 1.116857
## 4 1 0 1.114978writeLines("\nFunction-based historical lambda:")##
## Function-based historical lambda:lambda3(cyp3fp_mean)## pop patch lambda
## 1 1 A 0.9601966
## 2 1 B 0.9631450
## 3 1 C 0.9645564
## 4 1 0 0.9627838writeLines("Raw ahistorical stochastic log lambda:")## Raw ahistorical stochastic log lambda:set.seed(42)
slambda3(cypmatrix2rp)## pop patch a var sd se
## 1 1 A 0.01755652 4.4183458 2.1019862 0.021019862
## 2 1 B 0.13829938 1.3681927 1.1696977 0.011696977
## 3 1 C 0.10429814 1.2296795 1.1089092 0.011089092
## 4 1 0 0.08601101 0.7931451 0.8905869 0.008905869writeLines("\nRaw historical stochastic log lambda:")##
## Raw historical stochastic log lambda:set.seed(42)
slambda3(cypmatrix3rp)## pop patch a var sd se
## 1 1 A -0.2238227 6.3215036 2.5142600 0.025142600
## 2 1 B -0.2352251 2.3058704 1.5185093 0.015185093
## 3 1 C -0.1237223 3.8913169 1.9726421 0.019726421
## 4 1 0 -0.2582071 0.8835078 0.9399509 0.009399509writeLines("\nFunction-based ahistorical stochastic log lambda:")##
## Function-based ahistorical stochastic log lambda:set.seed(42)
slambda3(cypmatrix2fp)## pop patch a var sd se
## 1 1 A 0.11774998 0.07497810 0.2738213 0.002738213
## 2 1 B 0.09489865 0.09217605 0.3036051 0.003036051
## 3 1 C 0.11049388 0.04553841 0.2133973 0.002133973
## 4 1 0 0.10905382 0.06876260 0.2622262 0.002622262writeLines("\nFunction-based historical stochastic log lambda:")##
## Function-based historical stochastic log lambda:set.seed(42)
slambda3(cypmatrix3fp)## pop patch a var sd se
## 1 1 A -0.04054553 0.04812858 0.2193823 0.002193823
## 2 1 B -0.03752284 0.06369572 0.2523801 0.002523801
## 3 1 C -0.03602980 0.02500557 0.1581315 0.001581315
## 4 1 0 -0.03781516 0.04353458 0.2086494 0.002086494Users will notice at least two trends in the output above. First of all, the most obvious differences are between ahistorical and historical \(\lambda\) and \(\text{log} \lambda _{S}\) values. Generally, ahistorical values of \(\lambda\) are higher than those for historical MPMs. Part of the reason is that our historical MPMs have lost virtually all sensitivity to fecundity rates, and are now basically driven by survival terms (try changing the numbers of seeds per pod at the start of the stage frame input to see the impact of this). However, it is also likely that the difference is driven by long-term trade-offs captured within the historical models, but not within the ahistorical models. Of particular note is the growth trade-off - we have found large individuals that grew to large size in time t from small size in time t-1 have much lower survival probability to time t+1 (Shefferson, Warren, and Pulliam 2014).
In addition to the difference between historical and ahistorical growth rate estimates, we see that function-based estimates of \(\lambda\) and \(\text{log} \lambda _{S}\) are higher than those of the raw MPMs. This is likely to be an effect of the extra variance in matrix elements caused by the occasional lack of individuals making some transitions in the raw MPM case, and of the elimination of some vital rate variance from vital rate models.
Under the circumstances, users may wonder which approach is better. While we always advocate using the historical approach over the ahistorical when modeling suggests that history is important, the question of raw vs. function-based is more challenging. All else being equal, larger datasets with more years and more individuals will make both approaches more accurate, but should have a greater influence on inference through the function-based approach than the raw approach. In general, astute, parsimonious life history model construction can make the raw MPM approach very valuable in small datasets, where the raw approach will at least preserve temporal trends in population dynamics better. In contrast, large datasets can give users the ability to explain trends in population dynamics more fully via the parameterization of very good vital rate models that can inform patterns in \(\lambda\) and other metrics.
As a comparison, we might wish to see what our growth rate estimates are when we assume the wrong distributions for size and fecundity.
wrong_mean_f <- lmean(cypmatrix2f_wrong)
lambda3(wrong_mean_f)## pop patch lambda
## 1 1 1 1.049226lambda3(cyp2f_mean)## pop patch lambda
## 1 1 1 1.110766set.seed(42)
slambda3(cypmatrix2f_wrong)## pop patch a var sd se
## 1 1 1 0.04742637 0.614831 0.7841116 0.007841116slambda3(cypmatrix2f)## pop patch a var sd se
## 1 1 1 0.104924 0.06845388 0.2616369 0.002616369Notice that our estimates are a little lower than here than for the other function-based MPMs. The fact that they are lower is not necessarily expected - we only expect that they may be different.
Now let’s take a look at the stable stage distributions at the population level. We will look at deterministic and stochastic versions of the raw ahistorical and historical MPMs first.
tm2ss_r <- stablestage3(cyp2r_mean)
tm3ss_r <- stablestage3(cyp3r_mean)
tm2ss_rs <- stablestage3(cypmatrix2r, stochastic = TRUE, seed = 42)
tm3ss_rs <- stablestage3(cypmatrix3r, stochastic = TRUE, seed = 42)
ss_put_together <- cbind.data.frame(tm2ss_r$ss_prop, tm3ss_r$ahist$ss_prop,
tm2ss_rs$ss_prop, tm3ss_rs$ahist$ss_prop)
names(ss_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(ss_put_together) <- tm2ss_r$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.60), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, cex = 0.9, bty = "n")
Overall, these are generally similar patterns but with some key differences. Whether ahistorical or historical, deterministic or stochastic, our analyses suggest that dormant seeds and 1st year protocorms take up the greatest share of the stable stage structure, with 2nd year protocorms coming next. However, the proportion of second-year protocorms drops dramatically in historical analyses relative to ahistorical analyses, and stochastic analyses show a drop in dormant seeds. Adults contribute little to the stable stage structure.
Let’s now compare to the results from the function-based analyses.
tm2ss_f <- stablestage3(cyp2f_mean)
tm3ss_f <- stablestage3(cyp3f_mean)
tm2ss_fs <- stablestage3(cypmatrix2f, stochastic = TRUE, seed = 42)
tm3ss_fs <- stablestage3(cypmatrix3f, stochastic = TRUE, seed = 42)
ss_put_together <- cbind.data.frame(tm2ss_f$ss_prop, tm3ss_f$ahist$ss_prop,
tm2ss_fs$ss_prop, tm3ss_fs$ahist$ss_prop)
names(ss_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(ss_put_together) <- tm2ss_f$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.55), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
If we look over the stageframe used for these MPMs (see the
ahstages element of any of the function-based MPMs), we can
see that at equilibrium, the population should be composed of the same
stages shown to dominate in the raw case - 1st year protocorms and
dormant seeds. Second-year protocorms also drop in proportion in
historical analyses, as before. So, no dramatic differences from the raw
analyses here.
Finally, let’s look over the predicted stable stage distribution if we use the wrong size and fecundity distributions.
tm2w_f <- stablestage3(lmean(cypmatrix2f_wrong))
tm2w_fs <- stablestage3(cypmatrix2f_wrong, stochastic = TRUE, seed = 42)
ss_put_together <- cbind.data.frame(tm2w_f$ss_prop, tm2w_fs$ss_prop)
names(ss_put_together) <- c("det ahist", "sto ahist")
rownames(ss_put_together) <- tm2w_f$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.55), col = c("black", "grey"), bty = "n")
legend("topright", c("det ahist", "sto ahist"), col = c("black", "grey"),
pch = 15, bty = "n")
A comparison with the previous graphs shows little difference for now.
Next let’s look at the reproductive values associated with both ahistorical and historical approaches, starting with raw MPMs. We will standardize against the maximum value in each case to make these comparable (note that this is NOT standard practice - we only do it here because of the strong difference in scale across the analyses).
tm2rv_r <- repvalue3(cyp2r_mean)
tm3rv_r <- repvalue3(cyp3r_mean)
tm2rv_rs <- repvalue3(cypmatrix2r, stochastic = TRUE, seed = 42)
tm3rv_rs <- repvalue3(cypmatrix3r, stochastic = TRUE, seed = 42)
rv_put_together <- cbind.data.frame((tm2rv_r$rep_value / max(tm2rv_r$rep_value)),
(tm3rv_r$ahist$rep_value / max(tm3rv_r$ahist$rep_value)),
(tm2rv_rs$rep_value / max(tm2rv_rs$rep_value)),
(tm3rv_rs$ahist$rep_value / max(tm3rv_rs$ahist$rep_value)))
names(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- tm2rv_r$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
ylim = c(0, 1.1), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
There are some big differences here, particularly between ahistorical and historical analyses. Indeed, in the ahistorical case, reproductive value increases with size, reaching its peak in the largest adults. In contrast, in the historical case, the greatest reproductive values by far are associated with small and medium adults. So, history appears to have large effects here. Interesting results in need of further study!
Now let’s compare with the function-based case.
tm2rv_f <- repvalue3(cyp2f_mean)
tm3rv_f <- repvalue3(cyp3f_mean)
tm2rv_fs <- repvalue3(cypmatrix2f, stochastic = TRUE, seed = 42)
tm3rv_fs <- repvalue3(cypmatrix3f, stochastic = TRUE, seed = 42)
rv_put_together <- cbind.data.frame((tm2rv_f$rep_value / max(tm2rv_f$rep_value)),
(tm3rv_f$ahist$rep_value / max(tm3rv_f$ahist$rep_value)),
(tm2rv_fs$rep_value / max(tm2rv_fs$rep_value)),
(tm3rv_fs$ahist$rep_value / max(tm3rv_fs$ahist$rep_value)))
names(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- tm2rv_f$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
ylim = c(0, 1.5), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, cex = 0.8, bty = "n")
We see the ahistorical MPMs showing increasing reproductive value with size again, except that the very highest values are associated with one- and two-sprouted flowering adults. In contrast, historical analyses show most adult stages with high reproductive value, and lower values for the absolute smallest stages.
Let’s now do a sensitivity analysis, again using both deterministic and stochastic approaches. First we will look at the raw ahistorical MPM.
tm2sens_r <- sensitivity3(cyp2r_mean)
set.seed(42)
tm2sens_rs <- sensitivity3(cypmatrix2r, stochastic = TRUE)
writeLines("\nThe highest deterministic sensitivity value: ")##
## The highest deterministic sensitivity value:max(tm2sens_r$ah_sensmats[[1]][which(cyp2r_mean$A[[1]] > 0)])## [1] 0.9109618writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm2sens_r$ah_sensmats[[1]] == max(tm2sens_r$ah_sensmats[[1]][which(cyp2r_mean$A[[1]] > 0)])), which(cyp2r_mean$A[[1]] > 0))## [1] 38writeLines("\nThe highest stochastic sensitivity value: ")##
## The highest stochastic sensitivity value:max(tm2sens_rs$ah_sensmats[[1]][which(cyp2r_mean$A[[1]] > 0)])## [1] 0.8025366writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm2sens_rs$ah_sensmats[[1]] == max(tm2sens_rs$ah_sensmats[[1]][which(cyp2r_mean$A[[1]] > 0)])), which(cyp2r_mean$A[[1]] > 0))## [1] 14The highest sensitivity value appears to be associated with element
38 in the deterministic case, and element 14 in the stochastic case.
Since there are 11 stages in the stageframe, this means that \(\lambda\) and \(log \lambda\) are most sensitive to the
transition from the 4th stage (3rd year protocorm) to the 5th stage
(seedling) in the deterministic case, and from the 2nd stage (1st year
protocorm) to the 3rd stage (2nd year protocorm) in the stochastic case.
You can check this by typing, for example, ceiling(14/11)
to get the correct column number and 14 %% 11 to get the
correct row number for element 14.
Let’s compare this to the historical case.
tm3sens_r <- sensitivity3(cyp3r_mean)
set.seed(42)
tm3sens_rs <- sensitivity3(cypmatrix3r, stochastic = TRUE)
writeLines("\nThe highest deterministic sensitivity value: ")##
## The highest deterministic sensitivity value:max(tm3sens_r$h_sensmats[[1]][which(cyp3r_mean$A[[1]] > 0)])## [1] 0.3033102writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm3sens_r$h_sensmats[[1]] == max(tm3sens_r$h_sensmats[[1]][which(cyp3r_mean$A[[1]] > 0)])), which(cyp3r_mean$A[[1]] > 0))## [1] 10249writeLines("\nThe highest stochastic sensitivity value: ")##
## The highest stochastic sensitivity value:max(tm3sens_rs$h_sensmats[[1]][which(cyp3r_mean$A[[1]] > 0)])## [1] 0.3411136writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm3sens_rs$h_sensmats[[1]] == max(tm3sens_rs$h_sensmats[[1]][which(cyp3r_mean$A[[1]] > 0)])), which(cyp3r_mean$A[[1]] > 0))## [1] 10249Here we find that \(\lambda\) and
\(log \lambda\) are both most sensitive
to element 10249. Looking over the hstages portion, which
outlines the 121 stage-pairs corresponding to the rows and column in
these matrices, this corresponds to the the transition from the 85th
stage pair (Small adult in times t-1 and t), to the 85th stage pair
(Small adult in times t and t+1). So, we find here that stasis as a
Small adult is most important. This sort of result suggests the
importance of history - incorporating a single extra time step in
transitions shows sensitivities shifting from juveniles to adults in
this case study.
Here is some simple code to help determine the exact stages for a specific element.
# For element 14 in the ahistorical case
writeLines(paste0("Element 14 in cyp2r_mean is associated with column ",
ceiling(14 / dim(cyp2r_mean$ahstages)[1]), " and row ",
(14 %% dim(cyp2r_mean$ahstages)[1])))## Element 14 in cyp2r_mean is associated with column 2 and row 3# For element 10249 in the historical case
writeLines(paste0("Element 10249 in cyp3r_mean is associated with column ",
ceiling(10249 / dim(cyp3r_mean$hstages)[1]), " and row ",
(10249 %% dim(cyp3r_mean$hstages)[1])))## Element 10249 in cyp3r_mean is associated with column 85 and row 85Let’s now compare to the function-based case. We will run both ahistorical and historical portions at the same time.
tm2sens_f <- sensitivity3(cyp2f_mean)
set.seed(42)
tm2sens_fs <- sensitivity3(cypmatrix2f, stochastic = TRUE)
tm3sens_f <- sensitivity3(cyp3f_mean)
set.seed(42)
tm3sens_fs <- sensitivity3(cypmatrix3f, stochastic = TRUE, steps = 500)
writeLines("\nThe highest deterministic ahistorical sensitivity value: ")##
## The highest deterministic ahistorical sensitivity value:max(tm2sens_f$ah_sensmats[[1]][which(cyp2f_mean$A[[1]] > 0)])## [1] 0.8446548writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm2sens_f$ah_sensmats[[1]] == max(tm2sens_f$ah_sensmats[[1]][which(cyp2f_mean$A[[1]] > 0)])), which(cyp2f_mean$A[[1]] > 0))## [1] 167writeLines("\nThe highest stochastic ahistorical sensitivity value: ")##
## The highest stochastic ahistorical sensitivity value:max(tm2sens_fs$ah_sensmats[[1]][which(cyp2f_mean$A[[1]] > 0)])## [1] 0.7599012writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm2sens_fs$ah_sensmats[[1]] == max(tm2sens_fs$ah_sensmats[[1]][which(cyp2f_mean$A[[1]] > 0)])), which(cyp2f_mean$A[[1]] > 0))## [1] 57writeLines("\nThe highest deterministic historical sensitivity value: ")##
## The highest deterministic historical sensitivity value:max(tm3sens_f$h_sensmats[[1]][which(cyp3f_mean$A[[1]] > 0)])## [1] 0.05071351writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm3sens_f$h_sensmats[[1]] == max(tm3sens_f$h_sensmats[[1]][which(cyp3f_mean$A[[1]] > 0)])), which(cyp3f_mean$A[[1]] > 0))## [1] 962658writeLines("\nThe highest stochastic historical sensitivity value: ")##
## The highest stochastic historical sensitivity value:max(tm3sens_fs$h_sensmats[[1]][which(cyp3f_mean$A[[1]] > 0)])## [1] 0.05189656writeLines("\nThis value is associated with element: ")##
## This value is associated with element:intersect(which(tm3sens_fs$h_sensmats[[1]] == max(tm3sens_fs$h_sensmats[[1]][which(cyp3f_mean$A[[1]] > 0)])), which(cyp3f_mean$A[[1]] > 0))## [1] 962658The highest sensitivity values in deterministic and stochastic
ahistorical analysis appear to be the transitions between different
protocorm stages (ceiling(167 / 54) = 4 and
167 %% 54 = 5, and ceiling(57 / 54) = 2 and
57 %% 54 = 3). Inspecting each sensitivity matrix also
shows that transitions near that element in both matrices are also
associated with rather high sensitivities. Both deterministic and
stochastic historical analyses suggest the strongest sensitivity in
response to element 962658. Element 962658 is the transition from
1-sprouted vegetative adult in times t-1 and t to 24-sprouted flowering
adult in time t+1 (column 331 and row 378). So, we continue to see that
ahistorical analyses are suggesting a strong influence of the earliest
stages on life, while historical analyses are suggesting a stronger
influence of later adult stages.
Let’s now assess the elasticity of \(\lambda\) to matrix elements, comparing the ahistorical to the historically-corrected case in both deterministic and stochastic analyses.
tm2elas_r <- elasticity3(cyp2r_mean)
tm3elas_r <- elasticity3(cyp3r_mean)
set.seed(42)
tm2elas_rs <- elasticity3(cypmatrix2r, stochastic = TRUE)
set.seed(42)
tm3elas_rs <- elasticity3(cypmatrix3r, stochastic = TRUE)
writeLines("\nThe largest ahistorical deterministic elasticity is associated with element: ")##
## The largest ahistorical deterministic elasticity is associated with element:which(tm2elas_r$ah_elasmats[[1]] == max(tm2elas_r$ah_elasmats[[1]]))## [1] 85writeLines("\nThe largest historically-corrected deterministic elasticity is associated with element: ")##
## The largest historically-corrected deterministic elasticity is associated with element:which(tm3elas_r$ah_elasmats[[1]] == max(tm3elas_r$ah_elasmats[[1]]))## [1] 85writeLines("\nThe largest historical deterministic elasticity is associated with element: ")##
## The largest historical deterministic elasticity is associated with element:which(tm3elas_r$h_elasmats[[1]] == max(tm3elas_r$h_elasmats[[1]]))## [1] 10249writeLines("\nThe largest ahistorical stochastic elasticity is associated with element: ")##
## The largest ahistorical stochastic elasticity is associated with element:which(tm2elas_rs$ah_elasmats[[1]] == max(tm2elas_rs$ah_elasmats[[1]]))## [1] 85writeLines("\nThe largest historically-corrected stochastic elasticity is associated with element: ")##
## The largest historically-corrected stochastic elasticity is associated with element:which(tm3elas_rs$ah_elasmats[[1]] == max(tm3elas_rs$ah_elasmats[[1]]))## [1] 85writeLines("\nThe largest historical stochastic elasticity is associated with element: ")##
## The largest historical stochastic elasticity is associated with element:which(tm3elas_rs$h_elasmats[[1]] == max(tm3elas_rs$h_elasmats[[1]]))## [1] 10249Here we see something interesting - the ahistorical and historical analyses are generally agreeing about which transitions \(\lambda\) is most elastic in response to, as are the deterministic and stochastic. Here we see that \(\lambda\) and \(\text{log} \lambda _{S}\) are most elastic across the board to the ahistorical stasis transition from Small adult in time t to Small adult in time t+1, and the historical stasis transition as Small adult in times t-1, t, and t+1. This is a very different result from sensitivity analysis, and likely relates to differences the nature of the perturbation assessed in each case.
Let’s now look at the function-based case. Note that we have set the number of stochastic steps in the historical case to only 500, to make sure that this does not take too long.
tm2elas_f <- elasticity3(cyp2f_mean)
tm3elas_f <- elasticity3(cyp3f_mean)
set.seed(42)
tm2elas_fs <- elasticity3(cypmatrix2f, stochastic = TRUE)
set.seed(42)
tm3elas_fs <- elasticity3(cypmatrix3f, stochastic = TRUE, steps = 500)
writeLines("\nThe largest ahistorical deterministic elasticity is associated with element: ")##
## The largest ahistorical deterministic elasticity is associated with element:which(tm2elas_f$ah_elasmats[[1]] == max(tm2elas_f$ah_elasmats[[1]]))## [1] 167writeLines("\nThe largest historically-corrected deterministic elasticity is associated with element: ")##
## The largest historically-corrected deterministic elasticity is associated with element:which(tm3elas_f$ah_elasmats[[1]] == max(tm3elas_f$ah_elasmats[[1]]))## [1] 331writeLines("\nThe largest historical deterministic elasticity is associated with element: ")##
## The largest historical deterministic elasticity is associated with element:which(tm3elas_f$h_elasmats[[1]] == max(tm3elas_f$h_elasmats[[1]]))## [1] 962611writeLines("\nThe largest ahistorical stochastic elasticity is associated with element: ")##
## The largest ahistorical stochastic elasticity is associated with element:which(tm2elas_fs$ah_elasmats[[1]] == max(tm2elas_fs$ah_elasmats[[1]]))## [1] 57writeLines("\nThe largest historically-corrected stochastic elasticity is associated with element: ")##
## The largest historically-corrected stochastic elasticity is associated with element:which(tm3elas_fs$ah_elasmats[[1]] == max(tm3elas_fs$ah_elasmats[[1]]))## [1] 331writeLines("\nThe largest historical stochastic elasticity is associated with element: ")##
## The largest historical stochastic elasticity is associated with element:which(tm3elas_fs$h_elasmats[[1]] == max(tm3elas_fs$h_elasmats[[1]]))## [1] 962611Element 57 is the transition from 1st year to 2nd year protocorm, and element 167 is the transition from 2nd year protocorm to 3rd year protocorm. In contrast, element 331 corresponds to the ahistorical stasis transition in one-sprouted vegetative adults, and element 962611 corresponds to the historical stasis transition in this stage. So, here we see that historical analyses generally favor the adult stages, while ahistorical analyses emphasize the early stages.
Now let’s compare the elasticity of population growth rate in relation to the core life history stages, via a barplot comparison.
elas_put_together <- cbind.data.frame(colSums(tm2elas_f$ah_elasmats[[1]]),
colSums(tm3elas_f$ah_elasmats[[1]]), colSums(tm2elas_fs$ah_elasmats[[1]]),
colSums(tm3elas_fs$ah_elasmats[[1]]))
names(elas_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_put_together) <- tm2elas_f$ah_stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered", "grey", "darkred"), ylim = c(0, 0.20), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Elasticity analyses in this plot look quite different. All analyses show that \(\lambda\) and \(\text{log} \lambda _{S}\) are most elastic to small adult stages. However, the ahistorical analyses show that \(\lambda\) and \(\text{log} \lambda _{S}\) should be quite elastic in response to shifts in transitions from the juvenile stages. In contrast, these stages have little elasticity in historical analyses. We tend to side with the historical interpretation, but leave it up to the user to make their own determination.
Finally, let’s take a look at how the importance of different kinds of transitions changes, by looking at elasticity sums.
tm2elas_f_sums <- summary(tm2elas_f)
tm3elas_f_sums <- summary(tm3elas_f)
tm2elas_fs_sums <- summary(tm2elas_fs)
tm3elas_fs_sums <- summary(tm3elas_fs)
elas_sums_together <- cbind.data.frame(tm2elas_f_sums$ahist[,2],
tm3elas_f_sums$ahist[,2], tm2elas_fs_sums$ahist[,2], tm3elas_fs_sums$ahist[,2])
names(elas_sums_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_sums_together) <- tm2elas_f_sums$ahist$category
barplot(t(elas_sums_together), beside=T, ylab = "Elasticity", xlab = "Transition",
col = c("black", "orangered", "grey", "darkred"), ylim = c(0, 0.50), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
Fecundity makes least difference in all cases, although it does influence ahistorical analysis a reasonable amount. Growth, in contrast, appears to be among the most important transition types across the board. Stasis is more important in ahistorical than in historical analyses, and shrinkage shows the reverse pattern.
Next, we will see which historical transitions are most important.
elas_hist2plot <- cbind.data.frame(tm3elas_f_sums$hist[,2],
tm3elas_fs_sums$hist[,2])
names(elas_hist2plot) <- c("det hist", "sto hist")
rownames(elas_hist2plot) <- tm3elas_f_sums$hist$category
par(mar = c(7, 4, 2, 2) + 0.2)
barplot(t(elas_hist2plot), beside = T, ylab = "Elasticity", xlab = "", xaxt = "n",
col = c("orangered", "darkred"), bty = "n")
text(cex=0.75, x=seq(from = 0.2, to = 2.9*length(tm3elas_f_sums$hist$category), by = 2.95),
y=-0.06, tm3elas_f_sums$hist$category, xpd=TRUE, srt=45)
legend("topright", c("det hist", "sto hist"),
col = c("orangered", "darkred"), pch = 15, bty = "n")
We can see that growth from occasion t-1 to t followed by shrinkage to occasion t+1 is associated with the greatest summed elasticities, while the inverse, shrinkage from occasion t-1 to t followed by growth to occasion t+1 is the next most important. Transitions associated with fecundity are associated with the lowest summed elasticities.
Step 5b. Advanced analysis - projecting existing MPMs
The most recent versions of lefko3 incorporate two
projection functions, projection3() and
f_projection3(). We will start with
projection3(), which is used to project an MPM or IPM in
which the matrices have already been created. In other words, it uses an
existing lefkoMat object and projects it forward, under
user-defined conditions. Originally, this function was developed to
create deterministic and stochastic projections of varying lengths and
complexity, with output including growth rate, stage structure, and
reproductive value per time step, with a number of options designed to
streamline analysis. The function now has replication, allows different
forms of density dependence, and includes the ability to define the
order of matrices explicitly.
We will demonstrate this function by conducting a simple
quasi-extinction analysis. Let’s use the function-based ahistorical MPM
to illustrate this, using this function to create 100 stochastic
replicates of a population projection. In the code below, we set the
length of time to project using times (defaults to 10,000
time steps), set the number of replicates with nreps, set
that we wish to use a stochastic projection with
stochastic = TRUE, and force all reproduction to yield
integers only (i.e. all decimal individual values rounded down).
set.seed(42)
cypproj_2r <- projection3(cypmatrix2r, nreps = 100, stochastic = TRUE, integeronly = TRUE)Now let’s take a look at the resulting object, which is a list of
class lefkoProj. We will not output the whole object
because it is quite long. Instead, let’s first look at the elements that
constitute it.
names(cypproj_2r)## [1] "projection" "stage_dist" "rep_value" "pop_size" "labels"
## [6] "ahstages" "hstages" "agestages" "control"We see a list of nine elements, with five seemingly familiar -
ahstages is the stage frame from the input MPM,
hstages is a data frame showing the order of historical
stage pairs (only provided if an hMPM is used as input),
agestages is a data frame showing the order of age-stages
(only provided if an age-by-stage MPM is used as input),
labels is the order of matrices in the input MPM, and
control is a basic vector summarizing the number of time
steps and replicates in the projection.
The most important element is projection, which is a
list with elements equal to the number of patches / populations in the
MPM. Let’s take a look at its length, and compare to the
labels object.
length(cypproj_2r$projection)## [1] 1cypproj_2r$labels## pop patch
## 1 1 1So we see that there is only one element in this list. This element is also a list with the number of elements equal to the number of replicates. Here, there are 100 replicates, as we can see below.
length(cypproj_2r$projection[[1]])## [1] 100The next level of this object is composed of matrices - each replicate produces a single matrix. Let’s take a look at its dimensions.
dim(cypproj_2r$projection[[1]][[1]])## [1] 11 10001We see that there are 11 rows and 10,001 columns. The columns correspond to the discrete times that we have projected (equals the number of time steps plus one, the latter denoting the start time). The rows correspond to the 11 stages in the stageframe.
Lets’s take a look at the first few time steps of replicate 1.
cypproj_2r$projection[[1]][[1]][,1:15]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 1 3688 2757 16193 485 14 114 3 0 200 206 119 3
## [2,] 1 3688 3200 16524 2428 72 115 17 0 200 230 144 17
## [3,] 1 0 368 320 1652 242 7 11 1 0 20 23 14
## [4,] 1 0 0 36 32 165 24 0 1 0 0 2 2
## [5,] 1 0 0 0 3 3 16 3 0 0 0 0 0
## [6,] 1 0 0 0 0 0 0 3 0 1 0 0 0
## [7,] 1 1 0 0 0 1 2 6 5 2 1 0 0
## [8,] 1 1 1 0 0 0 0 0 1 1 0 0 0
## [9,] 1 0 0 0 0 0 0 0 0 0 0 0 0
## [10,] 1 1 1 0 0 0 0 0 0 0 0 0 0
## [11,] 1 1 1 0 0 0 0 0 0 0 0 0 0
## [,14] [,15]
## [1,] 0 0
## [2,] 0 0
## [3,] 1 0
## [4,] 1 0
## [5,] 0 0
## [6,] 0 0
## [7,] 0 0
## [8,] 0 0
## [9,] 0 0
## [10,] 0 0
## [11,] 0 0We see an interesting pattern in which we start with a single individual of each stage, and then see a growth of juvenile stages, leading to a decline overall in the near-term.
Let’s use the summary.lefkoProj() function to get a
better handle on this.
summary(cypproj_2r)##
## The input lefkoProj object covers 1 population-patches.
## It includes 10000 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 2501 0
## 5001 0
## 7501 0
## 10001 0
##
## $extinction_times
## [1] NAThis summary gives us some basic information about our projection,
and also shows us the number of replicates with more than one individual
projected alive at a series of projected times (times 1, 2501, 5001,
7501, and 10001). We can change these milepost times using either
proportions or explicit times, as below. We can also get an estimate of
the time to extinction by using the ext_time option.
summary(cypproj_2r, milepost = c(1, 3, 5, 8, 9, 10, 15, 25, 50, 100),
ext_time = TRUE)##
## The input lefkoProj object covers 1 population-patches.
## It includes 10000 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 3 100
## 5 100
## 8 75
## 9 75
## 10 73
## 15 52
## 25 2
## 50 0
## 100 0
##
## $extinction_times
## ext_reps ext_time
## 1 100 14.64The output above gives us a sense that the population is declining
quite quickly - indeed, only 2 replicates still has at least one
individual alive in time 25. The average time to extinction across our
100 replicates is 14.64 years. We can view the actual population sizes
via the pop_size element of the lefkoProj
object. This element is also list, but with a slightly simpler layering
that projection. Here, pop_size is a list with
the number of elements equal to the number of patches / populations.
Each element in this list is a matrix, with the rows coresponding to
replicates and the columns corresponding to the time. Let’s view the
first 10 times in the projection.
cypproj_2r$pop_size[[1]][,c(1:10)]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 11 7380 6328 33073 4600 497 278 43 8 404
## [2,] 11 33656 8443 1033 106 17 9 9897 3854 10158
## [3,] 11 7380 30242 19201 7555 15975 6881 2642 972 932
## [4,] 11 19770 24475 10837 1408 153 25 637 5558 1955
## [5,] 11 39665 17437 3973 479 55 15 409 1935 1069
## [6,] 11 39665 13187 3892 486 57 14 634 681 1696
## [7,] 11 2068 2727 371 38 2 0 0 0 0
## [8,] 11 19770 24475 5838 3209 2926 2067 576 2575 5714
## [9,] 11 2068 687 85 7 0 0 0 0 0
## [10,] 11 33656 10006 18475 4918 611 1659 4562 6602 1494
## [11,] 11 2068 2727 371 38 2 0 0 0 0
## [12,] 11 39665 8803 2764 347 45 2513 6620 2095 2766
## [13,] 11 2068 913 117 11 0 0 0 0 0
## [14,] 11 2068 913 117 11 0 0 0 0 0
## [15,] 11 19770 27706 3789 407 43 11 302 9804 2255
## [16,] 11 39665 13187 3892 486 56 1830 1060 1391 814
## [17,] 11 19770 27706 3789 407 44 1100 161 6206 6401
## [18,] 11 19770 18391 24986 3414 370 43 14 1210 4048
## [19,] 11 19770 5562 22171 10578 8885 2867 767 1188 1049
## [20,] 11 2068 584 2293 317 31 2 0 0 0
## [21,] 11 39665 8803 2764 347 45 2623 6993 9161 9007
## [22,] 11 2068 1997 268 27 2 0 0 0 0
## [23,] 11 33656 8443 1033 106 19 11 6407 1700 2867
## [24,] 11 39665 6353 1338 159 26 2402 1592 1415 2863
## [25,] 11 39665 8803 2764 347 42 12 1184 1638 9330
## [26,] 11 2068 584 365 47 3 0 0 0 0
## [27,] 11 7380 18800 17598 4880 23105 3435 386 444 868
## [28,] 11 39665 13187 2070 230 29 1317 7707 2252 9605
## [29,] 11 19770 5562 3484 15463 4652 3086 3206 837 103
## [30,] 11 7380 30242 19200 9775 1285 142 20 2626 8560
## [31,] 11 33656 9623 3964 511 62 15 808 3989 1144
## [32,] 11 33656 9623 3964 511 59 11 301 7581 6158
## [33,] 11 33656 9623 10919 1485 165 1613 4555 4731 1891
## [34,] 11 39665 8803 1455 163 26 2737 1587 3462 1067
## [35,] 11 19770 3166 354 34 7 5 3646 804 2318
## [36,] 11 33656 10006 17735 17440 6658 10635 17158 3246 2151
## [37,] 11 33656 8443 1033 106 17 1826 6007 6149 1127
## [38,] 11 33656 22151 5329 647 73 1835 1514 2085 8040
## [39,] 11 19770 5562 16315 4758 23096 4353 3605 786 9863
## [40,] 11 7380 18800 25097 3428 368 40 691 502 5166
## [41,] 11 39665 13187 2295 262 31 1750 650 675 5219
## [42,] 11 19770 5562 25411 3531 386 1127 1070 775 5856
## [43,] 11 39665 6353 1008 406 58 2056 6040 1426 4845
## [44,] 11 33656 9623 10179 8881 1206 138 6422 1495 8957
## [45,] 11 39665 17437 2665 296 34 11 8195 1949 635
## [46,] 11 33656 22151 3351 371 44 14 1257 977 2067
## [47,] 11 2068 1997 268 27 2 0 0 0 0
## [48,] 11 2068 1997 268 27 2 0 0 0 0
## [49,] 11 39665 17437 2889 328 38 11 1256 976 1840
## [50,] 11 2068 584 2293 317 31 2 0 0 0
## [51,] 11 19770 5562 25411 3531 385 40 2887 809 5417
## [52,] 11 2068 913 117 11 0 0 0 0 0
## [53,] 11 33656 22151 12396 2036 239 34 1614 10487 4726
## [54,] 11 7380 6328 18620 2576 278 245 489 1206 461
## [55,] 11 39665 8224 1601 188 26 10 7294 6149 5707
## [56,] 11 19770 5562 16315 24757 3396 1506 2590 3209 1241
## [57,] 11 2068 913 117 11 0 0 0 0 0
## [58,] 11 19770 5562 3484 20463 10352 2497 706 2751 684
## [59,] 11 33656 10006 18475 2941 334 1562 3969 6955 8930
## [60,] 11 7380 3097 399 41 3 0 0 0 0
## [61,] 11 7380 18800 2597 279 28 5 3 1 226
## [62,] 11 33656 9623 3964 511 60 11 2850 989 7010
## [63,] 11 19770 24475 5003 592 64 921 760 695 6703
## [64,] 11 33656 8443 1033 106 16 8 7757 2337 586
## [65,] 11 7380 33472 6874 814 84 233 39 7297 8342
## [66,] 11 19770 3166 354 34 7 5 3093 831 101
## [67,] 11 19770 18391 17487 22366 5201 635 4399 1210 10784
## [68,] 11 39665 17437 2560 681 94 20 3135 7961 10862
## [69,] 11 2068 2727 371 38 2 0 0 0 0
## [70,] 11 33656 8443 1033 106 17 1826 6007 6149 1232
## [71,] 11 2068 584 696 93 8 0 0 0 0
## [72,] 11 7380 30242 9201 6155 5109 2360 311 330 5401
## [73,] 11 19770 18391 2486 265 29 8 4 296 664
## [74,] 11 39665 8224 1269 141 24 12 1260 1380 3439
## [75,] 11 2068 2727 371 38 2 0 0 0 0
## [76,] 11 2068 1997 268 27 2 0 0 0 0
## [77,] 11 19770 24475 5838 2374 311 1175 794 4663 6003
## [78,] 11 19770 3166 354 34 6 871 122 11 0
## [79,] 11 2068 1997 268 27 2 0 0 0 0
## [80,] 11 33656 10006 4985 652 74 13 1882 5847 5938
## [81,] 11 33656 9623 10179 8881 1206 2094 1094 1397 780
## [82,] 11 2068 687 85 7 0 0 0 0 0
## [83,] 11 2068 913 117 11 0 0 0 0 0
## [84,] 11 7380 18800 22598 10580 3049 601 77 602 10064
## [85,] 11 7380 3097 399 41 3 217 29 2 0
## [86,] 11 2068 1997 268 27 2 0 0 0 0
## [87,] 11 39665 6353 2422 310 42 14 2010 10082 2287
## [88,] 11 7380 33472 5277 590 59 231 2311 2203 1500
## [89,] 11 19770 3166 354 34 6 684 391 2489 344
## [90,] 11 39665 8803 2764 347 42 10 3748 1152 2307
## [91,] 11 7380 30242 8486 3196 410 52 12 7295 2899
## [92,] 11 33656 21421 7850 16003 6656 2610 4327 1780 10718
## [93,] 11 39665 17437 2665 296 38 15 1882 1466 786
## [94,] 11 33656 22151 5329 647 73 19 307 14903 12064
## [95,] 11 39665 17437 4704 582 68 1322 1370 4568 13625
## [96,] 11 2068 1997 268 27 2 0 0 0 0
## [97,] 11 39665 8224 1269 141 24 10 4105 5930 4229
## [98,] 11 2068 687 85 7 0 0 0 0 0
## [99,] 11 2068 584 365 47 3 0 0 0 0
## [100,] 11 33656 8443 1033 106 16 8 630 5403 1047We may also plot these trends using the plot() function,
as below.
plot(cypproj_2r, xlim = c(0, 30), lwd = 2, bty = "n")
In the above plot, we show the results of each population projection for the first 30 projected occasions. All 100 replicates go to extinction, of course. If our MPM had higher population growth rates, then we might have had only some go to extinction. In that case, we could see the number of replicates with surviving individuals at particular points in time, and estimate the time to extinction. For example, if 95% of replicates have died out by 100 occasions, then this analysis would suggest extinction within that time. Quasi-extinction analysis would use this number against another series of replicates set up under a different set of assumptions to assess extinction. For example, we could run another analysis of data from the same population but a different patch that has been subjected to a different management regime, and compare the time to extinction in each. If the management regime is associated with increased population lifespan, then the probability of extinction is considered lower under the management scenario.
As a comparison point, let’s also look at a projection of the function-based ahistorical MPM and compare it. Then, we will plot our results as before.
set.seed(42)
cypproj_2f <- projection3(cypmatrix2f, nreps = 100, stochastic = TRUE,
integeronly = TRUE)
plot(cypproj_2f, xlim = c(0, 30), lwd = 2, bty = "n")
The function-based ahistorical analysis also suggests inevitable extinction, and quite quickly. Let’s see a summary.
summary(cypproj_2f, ext_time = TRUE)##
## The input lefkoProj object covers 1 population-patches.
## It includes 10000 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 2501 0
## 5001 0
## 7501 0
## 10001 0
##
## $extinction_times
## ext_reps ext_time
## 1 100 17.96This leaves us to wonder how to boost the growth rate.
As a final analysis, we might try a comparison of our population using the raw MPM, under the current conditions against a scenario where we somehow double germination to around 30%. To do this analysis, we need to create a new set of MPMs under this new assumption. First, we will set up a new supplement table using this assumption. Then we will take a look at it relative to the original supplement table.
cypsupp2_raw1 <- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
"XSm", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "rep", "rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "XSm", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "XSm", "XSm", NA, NA),
givenrate = c(0.03, 0.30, 0.1, 0.1, 0.1, 0.05, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, (0.5 * seeds_per_pod),
(0.5 * seeds_per_pod)),
type =c(1, 1, 1, 1, 1, 1, 1, 1, 3, 3),
stageframe = cypframe_raw, historical = FALSE)
cypsupp2_raw1## stage3 stage2 stage1 eststage3 eststage2 eststage1 givenrate multiplier
## 1 SD SD <NA> <NA> <NA> <NA> 0.03 1
## 2 P1 SD <NA> <NA> <NA> <NA> 0.30 1
## 3 P2 P1 <NA> <NA> <NA> <NA> 0.10 1
## 4 P3 P2 <NA> <NA> <NA> <NA> 0.10 1
## 5 SL P3 <NA> <NA> <NA> <NA> 0.10 1
## 6 SL SL <NA> <NA> <NA> <NA> 0.05 1
## 7 D SL <NA> D XSm <NA> NA 1
## 8 XSm SL <NA> XSm XSm <NA> NA 1
## 9 SD rep <NA> <NA> <NA> <NA> NA 2500
## 10 P1 rep <NA> <NA> <NA> <NA> NA 2500
## convtype convtype_t12
## 1 1 1
## 2 1 1
## 3 1 1
## 4 1 1
## 5 1 1
## 6 1 1
## 7 1 1
## 8 1 1
## 9 3 1
## 10 3 1Now let’s create the new ahMPMs.
cypmatrix2r_1 <- rlefko2(data = cypraw_v1, stageframe = cypframe_raw,
year = "all", stages = c("stage3", "stage2", "stage1"),
size = c("size3added", "size2added", "size1added"),
supplement = cypsupp2_raw1, yearcol = "year2", patchcol = "patchid",
indivcol = "individ")
summary(cypmatrix2r_1)##
## This ahistorical lefkoMat object contains 5 matrices.
##
## Each matrix is square with 11 rows and columns, and a total of 121 elements.
## A total of 115 survival transitions were estimated, with 23 per matrix.
## A total of 40 fecundity transitions were estimated, with 8 per matrix.
## This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
##
## The dataset contains a total of 74 unique individuals and 320 unique transitions.
##
## Survival probability sum check (each matrix represented by column in order):
## [,1] [,2] [,3] [,4] [,5]
## Min. 0.000 0.100 0.100 0.000 0.100
## 1st Qu. 0.100 0.215 0.215 0.100 0.215
## Median 0.746 0.870 0.864 0.600 0.882
## Mean 0.576 0.655 0.640 0.545 0.628
## 3rd Qu. 1.000 1.000 1.000 0.960 1.000
## Max. 1.000 1.000 1.000 1.000 1.000Let’s take a quick look at our \(\lambda\) estimates to see if the population growth rare has increased.
lambda3(cypmatrix2r_1)## pop patch year2 lambda
## 1 1 1 2004 1.1798304
## 2 1 1 2005 1.1761380
## 3 1 1 2006 1.2104526
## 4 1 1 2007 0.9908389
## 5 1 1 2008 1.1282830lambda3(cypmatrix2r)## pop patch year2 lambda
## 1 1 1 2004 1.1714145
## 2 1 1 2005 1.1639008
## 3 1 1 2006 1.2002440
## 4 1 1 2007 0.9834515
## 5 1 1 2008 1.1208121We see small increases, but then again this is an assessment of deterministic growth rate while our projection will be stochastic. In any case, let’s conduct our new projections, and then plot them against the original hMPM projections.
set.seed(42)
cypproj_2r_1 <- projection3(cypmatrix2r_1, nreps = 100, stochastic = TRUE,
integeronly = TRUE)
par(mfrow = c(1, 2))
plot(cypproj_2r, xlim = c(0, 50), bty = "n")
plot(cypproj_2r_1, xlim = c(0, 50), bty = "n")
We see only small improvement in the lifespan of the population with boosted fecundity (shown on the right). To get a better sense of this, let’s look at the mileposts again.
summary(cypproj_2r, milepost = c(1, 3, 5, 8, 9, 10, 12, 15, 18, 20, 30, 50, 100),
ext_time = TRUE)##
## The input lefkoProj object covers 1 population-patches.
## It includes 10000 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 3 100
## 5 100
## 8 75
## 9 75
## 10 73
## 12 72
## 15 52
## 18 25
## 20 9
## 30 0
## 50 0
## 100 0
##
## $extinction_times
## ext_reps ext_time
## 1 100 14.64summary(cypproj_2r_1, milepost = c(1, 3, 5, 8, 9, 10, 12, 15, 18, 20, 30, 50, 100),
ext_time = TRUE)##
## The input lefkoProj object covers 1 population-patches.
## It includes 10000 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 3 100
## 5 100
## 8 76
## 9 76
## 10 75
## 12 73
## 15 66
## 18 45
## 20 27
## 30 2
## 50 0
## 100 0
##
## $extinction_times
## ext_reps ext_time
## 1 100 16.18Users will note a small difference here, with the inference that boosting germination has little real impact on the population dynamics. It is at this point that we consider other strategies, such as boosting adult survival through other means.
Step 5c. Advanced analysis - projecting function-based MPMs and IPMs
Let’s say that we are interested in projecting an MPM or IPM forward,
but changing some of the inputs to values not originally observed during
monitoring. The function f_projection3() allows us to do
this. Essentially, this function builds a new function-based matrix at
each time, assuming whatever conditions are given. This can be quite
useful if we wish to project an IPM forward under changing conditions,
or if we have density dependence in the vital rates themselves rather
than in the matrix elements.
Let’s try one such projection. Let’s assume that we are interested in
projecting an MPM forward based on changing values of climate. Perhaps
we have predicted climate data that we wish to use in our projection. In
the Cypripedium candidum case, for example, what would happen
to the population dynamics if precipitation declined by 0.2cm per year
for a century, assuming a starting point of 100cm per year? Let’s set up
this vector, and then run a stochastic projection. Note that the
construction of a new projection matrix at each time step makes this
projection take a bit more time that function projection3()
takes.
Let’s start by creating the precipitation input table.
pred_precip <- seq(from = 100, by = -0.2, length.out = 100)
ind_frame <- cbind.data.frame(inda = pred_precip, indb = 0, indc = 0)
ind_frame## inda indb indc
## 1 100.0 0 0
## 2 99.8 0 0
## 3 99.6 0 0
## 4 99.4 0 0
## 5 99.2 0 0
## 6 99.0 0 0
## 7 98.8 0 0
## 8 98.6 0 0
## 9 98.4 0 0
## 10 98.2 0 0
## 11 98.0 0 0
## 12 97.8 0 0
## 13 97.6 0 0
## 14 97.4 0 0
## 15 97.2 0 0
## 16 97.0 0 0
## 17 96.8 0 0
## 18 96.6 0 0
## 19 96.4 0 0
## 20 96.2 0 0
## 21 96.0 0 0
## 22 95.8 0 0
## 23 95.6 0 0
## 24 95.4 0 0
## 25 95.2 0 0
## 26 95.0 0 0
## 27 94.8 0 0
## 28 94.6 0 0
## 29 94.4 0 0
## 30 94.2 0 0
## 31 94.0 0 0
## 32 93.8 0 0
## 33 93.6 0 0
## 34 93.4 0 0
## 35 93.2 0 0
## 36 93.0 0 0
## 37 92.8 0 0
## 38 92.6 0 0
## 39 92.4 0 0
## 40 92.2 0 0
## 41 92.0 0 0
## 42 91.8 0 0
## 43 91.6 0 0
## 44 91.4 0 0
## 45 91.2 0 0
## 46 91.0 0 0
## 47 90.8 0 0
## 48 90.6 0 0
## 49 90.4 0 0
## 50 90.2 0 0
## 51 90.0 0 0
## 52 89.8 0 0
## 53 89.6 0 0
## 54 89.4 0 0
## 55 89.2 0 0
## 56 89.0 0 0
## 57 88.8 0 0
## 58 88.6 0 0
## 59 88.4 0 0
## 60 88.2 0 0
## 61 88.0 0 0
## 62 87.8 0 0
## 63 87.6 0 0
## 64 87.4 0 0
## 65 87.2 0 0
## 66 87.0 0 0
## 67 86.8 0 0
## 68 86.6 0 0
## 69 86.4 0 0
## 70 86.2 0 0
## 71 86.0 0 0
## 72 85.8 0 0
## 73 85.6 0 0
## 74 85.4 0 0
## 75 85.2 0 0
## 76 85.0 0 0
## 77 84.8 0 0
## 78 84.6 0 0
## 79 84.4 0 0
## 80 84.2 0 0
## 81 84.0 0 0
## 82 83.8 0 0
## 83 83.6 0 0
## 84 83.4 0 0
## 85 83.2 0 0
## 86 83.0 0 0
## 87 82.8 0 0
## 88 82.6 0 0
## 89 82.4 0 0
## 90 82.2 0 0
## 91 82.0 0 0
## 92 81.8 0 0
## 93 81.6 0 0
## 94 81.4 0 0
## 95 81.2 0 0
## 96 81.0 0 0
## 97 80.8 0 0
## 98 80.6 0 0
## 99 80.4 0 0
## 100 80.2 0 0Now let’s run the projection. We will make it a stochastic projection in which the year terms in the linear models are essentially shuffled, and assess 100 replicates over 100 years each.
set.seed(42)
cypfproj_2f_env <- f_projection3(data = cypfb_env, format = 3, stageframe = cypframe_fb,
supplement = cypsupp2_fb, modelsuite = cypmodels2_env, ind_terms = ind_frame,
stochastic = TRUE, integeronly = TRUE, nreps = 100, times = 100)## Warning: Option patch not set, so will set to first patch/population.summary(cypfproj_2f_env, ext_time = TRUE)##
## The input lefkoProj object covers 1 population-patches.
## It includes 100 projected steps per replicate and 100 replicates.
## The number of replicates with population size above the threshold size of 1 is as in
## the following matrix, with pop-patches given by column and milepost times given by row:## $milepost_sums
## 1 1
## 1 100
## 26 1
## 51 0
## 76 0
## 101 0
##
## $extinction_times
## ext_reps ext_time
## 1 100 13.52Clearly, things did not go very well for our population under the new climate scenario. Let’s take a peek at a plot of the replicates.
plot(cypfproj_2f_env)
This look rather similar to the preceding scenario, also leading to extinction very quickly.
Function f_projection3() includes many further options,
including the ability to start with different numbers of individuals in
different stages, density dependence, cyclical projection, ordered
projection, substochasticity checking, differential matrix weighting in
stochastic simulation, and more. Please see Chapter 10 of
lefko3: a gentle introduction for further details.