Take Home Exercise 1
Po Theodora
5/7/2020
Link to Rpubs: https://rpubs.com/theodorapo/take-home_ex-01
Objective
1. Calibrating a simple linear regression to reveal the relation between public bus commuters’ flows (i.e. tap-in or tap-out) data and residential population at the planning sub-zone level.
2. Performing spatial autocorrelation analysis on the residual of the regression model to test if the model conforms to the randomization assumption.
3. Performing localized geospatial statistics analysis by using commuters’ tap-in and tap-out data to identify geographical clustering.
The data used are:
2. Planning sub-zone level data (URA)
3. Bus stop Location data (LTA)
4. 2019 Residential population data by subzone (Singstat)
GEOSPATIAL DATA WRANGLING
DATA CLEANING - To derive appropriate variables to lead to the analysis of the relation between public bus commuters’ flows and residential population at the planning subzone level
Install packages
packages <- c('sf', 'tidyverse')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}## Loading required package: sf## Linking to GEOS 3.8.0, GDAL 3.0.4, PROJ 6.3.1## Loading required package: tidyverse## -- Attaching packages ------------------------------------------------------------------------------- tidyverse 1.3.0 --## v ggplot2 3.3.0 v purrr 0.3.4
## v tibble 3.0.1 v dplyr 0.8.5
## v tidyr 1.0.3 v stringr 1.4.0
## v readr 1.3.1 v forcats 0.5.0## -- Conflicts ---------------------------------------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()SUBZONE DATA
Import data
subzone <- st_read("data/geospatial/MP14_SUBZONE_NO_SEA_PL.shp")## Reading layer `MP14_SUBZONE_NO_SEA_PL' from data source `C:\Users\amoss\OneDrive\Documents\data\geospatial\MP14_SUBZONE_NO_SEA_PL.shp' using driver `ESRI Shapefile'
## Simple feature collection with 323 features and 15 fields
## geometry type: MULTIPOLYGON
## dimension: XY
## bbox: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
## projected CRS: SVY21Check CRS
st_crs(subzone)## Coordinate Reference System:
## User input: SVY21
## wkt:
## PROJCRS["SVY21",
## BASEGEOGCRS["SVY21[WGS84]",
## DATUM["World Geodetic System 1984",
## ELLIPSOID["WGS 84",6378137,298.257223563,
## LENGTHUNIT["metre",1]],
## ID["EPSG",6326]],
## PRIMEM["Greenwich",0,
## ANGLEUNIT["Degree",0.0174532925199433]]],
## CONVERSION["unnamed",
## METHOD["Transverse Mercator",
## ID["EPSG",9807]],
## PARAMETER["Latitude of natural origin",1.36666666666667,
## ANGLEUNIT["Degree",0.0174532925199433],
## ID["EPSG",8801]],
## PARAMETER["Longitude of natural origin",103.833333333333,
## ANGLEUNIT["Degree",0.0174532925199433],
## ID["EPSG",8802]],
## PARAMETER["Scale factor at natural origin",1,
## SCALEUNIT["unity",1],
## ID["EPSG",8805]],
## PARAMETER["False easting",28001.642,
## LENGTHUNIT["metre",1],
## ID["EPSG",8806]],
## PARAMETER["False northing",38744.572,
## LENGTHUNIT["metre",1],
## ID["EPSG",8807]]],
## CS[Cartesian,2],
## AXIS["(E)",east,
## ORDER[1],
## LENGTHUNIT["metre",1,
## ID["EPSG",9001]]],
## AXIS["(N)",north,
## ORDER[2],
## LENGTHUNIT["metre",1,
## ID["EPSG",9001]]]]Convert to latlong
subzone <- st_transform(subzone, 4326)Check CRS again
st_crs(subzone)## Coordinate Reference System:
## User input: EPSG:4326
## wkt:
## GEOGCRS["WGS 84",
## DATUM["World Geodetic System 1984",
## ELLIPSOID["WGS 84",6378137,298.257223563,
## LENGTHUNIT["metre",1]]],
## PRIMEM["Greenwich",0,
## ANGLEUNIT["degree",0.0174532925199433]],
## CS[ellipsoidal,2],
## AXIS["geodetic latitude (Lat)",north,
## ORDER[1],
## ANGLEUNIT["degree",0.0174532925199433]],
## AXIS["geodetic longitude (Lon)",east,
## ORDER[2],
## ANGLEUNIT["degree",0.0174532925199433]],
## USAGE[
## SCOPE["unknown"],
## AREA["World"],
## BBOX[-90,-180,90,180]],
## ID["EPSG",4326]]Plot subzone
plot(subzone, max.plot=1)
BUS STOP DATA
Import data
busstop <- st_read("data/geospatial/BusStop.shp")## Reading layer `BusStop' from data source `C:\Users\amoss\OneDrive\Documents\data\geospatial\BusStop.shp' using driver `ESRI Shapefile'
## Simple feature collection with 5040 features and 3 fields
## geometry type: POINT
## dimension: XY
## bbox: xmin: 4427.938 ymin: 26482.1 xmax: 48282.5 ymax: 52983.82
## projected CRS: SVY21busstop$BUS_STOP_N = as.numeric(as.character(busstop$BUS_STOP_N))Check CRS
st_crs(busstop)## Coordinate Reference System:
## User input: SVY21
## wkt:
## PROJCRS["SVY21",
## BASEGEOGCRS["WGS 84",
## DATUM["World Geodetic System 1984",
## ELLIPSOID["WGS 84",6378137,298.257223563,
## LENGTHUNIT["metre",1]],
## ID["EPSG",6326]],
## PRIMEM["Greenwich",0,
## ANGLEUNIT["Degree",0.0174532925199433]]],
## CONVERSION["unnamed",
## METHOD["Transverse Mercator",
## ID["EPSG",9807]],
## PARAMETER["Latitude of natural origin",1.36666666666667,
## ANGLEUNIT["Degree",0.0174532925199433],
## ID["EPSG",8801]],
## PARAMETER["Longitude of natural origin",103.833333333333,
## ANGLEUNIT["Degree",0.0174532925199433],
## ID["EPSG",8802]],
## PARAMETER["Scale factor at natural origin",1,
## SCALEUNIT["unity",1],
## ID["EPSG",8805]],
## PARAMETER["False easting",28001.642,
## LENGTHUNIT["metre",1],
## ID["EPSG",8806]],
## PARAMETER["False northing",38744.572,
## LENGTHUNIT["metre",1],
## ID["EPSG",8807]]],
## CS[Cartesian,2],
## AXIS["(E)",east,
## ORDER[1],
## LENGTHUNIT["metre",1,
## ID["EPSG",9001]]],
## AXIS["(N)",north,
## ORDER[2],
## LENGTHUNIT["metre",1,
## ID["EPSG",9001]]]]Convert to latlong
busstop <- st_transform(busstop, 4326)Check CRS again
st_crs(busstop)## Coordinate Reference System:
## User input: EPSG:4326
## wkt:
## GEOGCRS["WGS 84",
## DATUM["World Geodetic System 1984",
## ELLIPSOID["WGS 84",6378137,298.257223563,
## LENGTHUNIT["metre",1]]],
## PRIMEM["Greenwich",0,
## ANGLEUNIT["degree",0.0174532925199433]],
## CS[ellipsoidal,2],
## AXIS["geodetic latitude (Lat)",north,
## ORDER[1],
## ANGLEUNIT["degree",0.0174532925199433]],
## AXIS["geodetic longitude (Lon)",east,
## ORDER[2],
## ANGLEUNIT["degree",0.0174532925199433]],
## USAGE[
## SCOPE["unknown"],
## AREA["World"],
## BBOX[-90,-180,90,180]],
## ID["EPSG",4326]]View(busstop)
PASSENGER DATA
Import data
passenger <- read_csv("data/aspatial/passenger_volume_by_busstop.csv")## Parsed with column specification:
## cols(
## PT_CODE = col_double(),
## TOTAL_TAP_IN_VOLUME = col_double(),
## TOTAL_TAP_OUT_VOLUME = col_double()
## )# CHANGE VARIABLE NAME
colnames(passenger)[1] = "BUS_STOP_N"Join passenger and bus stop data
passbus <- inner_join(passenger,busstop,by="BUS_STOP_N")glimpse(passbus)## Rows: 194,869
## Columns: 6
## $ BUS_STOP_N <dbl> 67551, 67551, 66541, 66541, 54209, 54209, 6104...
## $ TOTAL_TAP_IN_VOLUME <dbl> 224, 3922, 648, 127, 736, 1388, 92, 140, 442, ...
## $ TOTAL_TAP_OUT_VOLUME <dbl> 22, 122, 364, 109, 325, 920, 72, 143, 117, 284...
## $ BUS_ROOF_N <chr> "B01", "B01", "B01", "B01", "B06", "B06", "B10...
## $ LOC_DESC <chr> "BLK 471A", "BLK 471A", "BLK 980C", "BLK 980C"...
## $ geometry <POINT [°]> POINT (103.8805 1.395861), POINT (103.88...Convert geometry to latlong
passbus <- passbus %>% extract(geometry, c('lat', 'lon'), '\\((.*), (.*)\\)', convert = TRUE)glimpse(passbus)## Rows: 194,869
## Columns: 7
## $ BUS_STOP_N <dbl> 67551, 67551, 66541, 66541, 54209, 54209, 6104...
## $ TOTAL_TAP_IN_VOLUME <dbl> 224, 3922, 648, 127, 736, 1388, 92, 140, 442, ...
## $ TOTAL_TAP_OUT_VOLUME <dbl> 22, 122, 364, 109, 325, 920, 72, 143, 117, 284...
## $ BUS_ROOF_N <chr> "B01", "B01", "B01", "B01", "B06", "B06", "B10...
## $ LOC_DESC <chr> "BLK 471A", "BLK 471A", "BLK 980C", "BLK 980C"...
## $ lat <dbl> 103.8805, 103.8805, 103.8814, 103.8814, 103.83...
## $ lon <dbl> 1.395861, 1.395861, 1.379980, 1.379980, 1.3763...Convert into simple feature
map2(passbus$lat, passbus$lon, ~st_point(c(.x, .y))) %>%
st_sfc(crs = 4326) %>%
st_sf(passbus[,-(5:6)], .) -> passbus_sfpassbus_subzone <- bind_cols(
passbus,
subzone[as.numeric(st_within(passbus_sf, subzone)),]
) %>%
dplyr::select(BUS_STOP_N,TOTAL_TAP_IN_VOLUME,TOTAL_TAP_OUT_VOLUME,LOC_DESC,lat, lon, subzone_name=SUBZONE_N) %>%
mutate(subzone_name = str_to_title(subzone_name))## although coordinates are longitude/latitude, st_within assumes that they are planarRemove NA variables
passbus_subzone <- na.omit(passbus_subzone)View(passbus_subzone)
Add population column
population <- read_csv("data/aspatial/residentialpopulation_2019.csv")## Parsed with column specification:
## cols(
## SZ = col_character(),
## Pop = col_double()
## )subzonepop <- population %>%
group_by(SZ) %>%
summarise(Pop = sum(Pop))
colnames(subzonepop)[1] = "subzone_name"
passbussubpop <- inner_join(passbus_subzone,subzonepop,by="subzone_name")View(passbussubpop)
Group by subzone
Sum total tapin, total tapout
library(dplyr)
tapin_ <- passbussubpop %>%
group_by(subzone_name) %>%
summarise(TOTAL_TAP_IN_VOLUME = sum(TOTAL_TAP_IN_VOLUME))
tapout_ <- passbussubpop %>%
group_by(subzone_name) %>%
summarise(TOTAL_TAP_OUT_VOLUME = sum(TOTAL_TAP_OUT_VOLUME)) Join tapin and subzonepop
tapin <- inner_join(tapin_, subzonepop, by="subzone_name")Join tapout and subzonepop
tapout <- inner_join(tapout_, subzonepop, by="subzone_name")View(tapin) View(tapout)
Linear model
TAP IN
Since the adjusted R-squared = 0.5647, we can conclude that there is a weak linear relationship between population and tap in volume.
datatest <- tapin
tap_in <- datatest$TOTAL_TAP_IN_VOLUME
pop <- datatest$Pop
regfit <- lm(tap_in~pop)
summary(regfit)##
## Call:
## lm(formula = tap_in ~ pop)
##
## Residuals:
## Min 1Q Median 3Q Max
## -841313 -122165 -63062 46001 1952899
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.222e+05 2.130e+04 5.738 2.32e-08 ***
## pop 1.884e+01 9.472e-01 19.886 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 301000 on 303 degrees of freedom
## Multiple R-squared: 0.5662, Adjusted R-squared: 0.5647
## F-statistic: 395.4 on 1 and 303 DF, p-value: < 2.2e-16model <- plot(pop, tap_in, main="Linear relationship between population and tap in", xlab="population", ylab="tapin")
abline(regfit, lwd=3, col="red")
TAP OUT
Since the adjusted R-squared = 0.5784, we can conclude that there is a weak linear relationship between population and tap out volume.
datatest2 <- tapout
tap_out <- datatest2$TOTAL_TAP_OUT_VOLUME
popout <- datatest2$Pop
regfit2 <- lm(tap_out~popout)
summary(regfit2)##
## Call:
## lm(formula = tap_out ~ popout)
##
## Residuals:
## Min 1Q Median 3Q Max
## -802936 -120462 -58450 36240 1647268
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.220e+05 2.062e+04 5.915 8.94e-09 ***
## popout 1.875e+01 9.172e-01 20.446 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 291500 on 303 degrees of freedom
## Multiple R-squared: 0.5798, Adjusted R-squared: 0.5784
## F-statistic: 418.1 on 1 and 303 DF, p-value: < 2.2e-16plot(popout, tap_out, main="linear relationship between population and tap out", xlab="population", ylab="tapout")
abline(regfit2, lwd=3, col="red")
Residual analysis
To test for 4 assumptions:
1. Normality
2. Independence
3. Linearity
4. Homogeneity
TAP IN
# plot residuals against population, to get the residuals plot
plot(pop, residuals(regfit), main="Relationship between population and residuals",xlab="population", ylab="residuals")
library(fitdistrplus)## Loading required package: MASS##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select## Loading required package: survival## Loading required package: npsurv## Loading required package: lseifnorm <- fitdist(residuals(regfit), "norm")
plot(fnorm)
result <- gofstat(fnorm, discrete=FALSE)
result## Goodness-of-fit statistics
## 1-mle-norm
## Kolmogorov-Smirnov statistic 0.2055747
## Cramer-von Mises statistic 4.2289931
## Anderson-Darling statistic 22.2859309
##
## Goodness-of-fit criteria
## 1-mle-norm
## Akaike's Information Criterion 8562.645
## Bayesian Information Criterion 8570.086# Test KScrit value to see if null hypothesis should be rejected or not
KScritvalue <- 1.36/sqrt(length(tap_in))
KScritvalue## [1] 0.07787337## KS stat > KS crit, reject null hypothesis. sufficient evidence that it is not normally distributed
TOTAL_TAP_IN_VOLUME.Pop.lm <- lm(TOTAL_TAP_IN_VOLUME ~ Pop, data = tapin)
summary(TOTAL_TAP_IN_VOLUME.Pop.lm)##
## Call:
## lm(formula = TOTAL_TAP_IN_VOLUME ~ Pop, data = tapin)
##
## Residuals:
## Min 1Q Median 3Q Max
## -841313 -122165 -63062 46001 1952899
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.222e+05 2.130e+04 5.738 2.32e-08 ***
## Pop 1.884e+01 9.472e-01 19.886 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 301000 on 303 degrees of freedom
## Multiple R-squared: 0.5662, Adjusted R-squared: 0.5647
## F-statistic: 395.4 on 1 and 303 DF, p-value: < 2.2e-16plot(TOTAL_TAP_IN_VOLUME.Pop.lm)
From the residual analysis, it shows a distinctive clustering at the point where population = 0, this shows that the two variables population and residuals are not statistically independent. P-P plot and Q-Q plot shows slight deviation from normal assumption.
Testing KScrit value, since KS stat value of 0.1864818 > KS critical value of 0.07787337, reject null hypothesis. There is sufficient evidence that the data is not normally distributed.
Combining the two evaluation, we can conclude that the data is weakly normally distributed.
The residuals vs fitted plot shows a distinct clustering, hence affirming the conclusion that there is a weak linear relationship between population and tap in volume.
Q-Q plot shows some points lying on the line, with several points away from it too. This affirms the previous conclusion that the data is weakly normally distributed.
Homogeneity of variance assumption is violated, as it can be seen that the variances of the residual points increases with the value of the fitted outcome variable. Hence, this suggests non-constant variances in the residuals errors, which is also heteroscedasticity.
As previously concluded, the variables are not statistically independent
Overall, the assumptions of residuals are violated.
TAP OUT
plot(popout, residuals(regfit2), main="Relationship between population and residuals",xlab="population", ylab="residuals")
library(fitdistrplus)
fnorm <- fitdist(residuals(regfit2), "norm")
plot(fnorm)
result <- gofstat(fnorm, discrete=FALSE)
result## Goodness-of-fit statistics
## 1-mle-norm
## Kolmogorov-Smirnov statistic 0.2100444
## Cramer-von Mises statistic 3.9543646
## Anderson-Darling statistic 20.5222382
##
## Goodness-of-fit criteria
## 1-mle-norm
## Akaike's Information Criterion 8542.983
## Bayesian Information Criterion 8550.423KScritvalue <- 1.36/sqrt(length(tap_out))
KScritvalue## [1] 0.07787337## KS stat > KS crit, reject null hypothesis. sufficient evidence that it is not normally distributed
TOTAL_TAP_OUT_VOLUME.Pop.lm <- lm(TOTAL_TAP_OUT_VOLUME ~ Pop, data = tapout)
summary(TOTAL_TAP_OUT_VOLUME.Pop.lm)##
## Call:
## lm(formula = TOTAL_TAP_OUT_VOLUME ~ Pop, data = tapout)
##
## Residuals:
## Min 1Q Median 3Q Max
## -802936 -120462 -58450 36240 1647268
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.220e+05 2.062e+04 5.915 8.94e-09 ***
## Pop 1.875e+01 9.172e-01 20.446 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 291500 on 303 degrees of freedom
## Multiple R-squared: 0.5798, Adjusted R-squared: 0.5784
## F-statistic: 418.1 on 1 and 303 DF, p-value: < 2.2e-16plot(TOTAL_TAP_OUT_VOLUME.Pop.lm)
From the residual analysis, it shows a distinctive clustering at the point where population = 0, this shows that the two variables population and residuals are not statistically independent. P-P plot and Q-Q plot shows slight deviation from normal assumption.
Testing KScrit value, since KS stat value of 0.1864818 > KS critical value of 0.07787337, reject null hypothesis. There is sufficient evidence that the data is not normally distributed.
Combining the two evaluation, we can conclude that the data is weakly normally distributed.
The residuals vs fitted plot shows a distinct clustering, hence affirming the conclusion that there is a weak linear relationship between population and tap out volume.
Q-Q plot shows some points lying on the line, with several points away from it too. This affirms the previous conclusion that the data is weakly normally distributed.
Homogeneity of variance assumption is violated, as it can be seen that the variances of the residual points increases with the value of the fitted outcome variable. Hence, this suggests non-constant variances in the residuals errors, which is also heteroscedasticity.
As previously concluded, the variables are not statistically independent
Overall, the assumptions of residuals are violated.
Evaluation of the relation between public bus commuters’ flows (tap-in or tap-out) data and residential population at the planning subzone level:
The output between tap in and residential population is similar to the output between tap out and residential population, this could be since the data of tap in and tap out are similar (a person who taps in has to tap out)
So, we can just take the two main variables of public commuters’ flows and residential population
There is a weak linear relationship between the two variables, and the residuals assumption are violated. Hence, we conclude that the two variables are not statistically independent.
GEOSPATIAL ANALYSIS - to analyse geospatial data provided
FOR TAP IN
View(subzone) View(tapin)
Join both subzone and tapin
# Rename subzone_n
colnames(tapin)[1] = "SUBZONE_N"
# uppercase all observations in SUBZONE_N
tapin$SUBZONE_N = toupper(tapin$SUBZONE_N)
# change character to attribute
subzone$SUBZONE_N = as.character(subzone$SUBZONE_N)
# join both data set
subzone_tapin <- left_join(tapin,subzone)## Joining, by = "SUBZONE_N"# omit NA values
subzone_tapin <- na.omit(subzone_tapin)
# select variables
subzone_tapin <- dplyr::select(subzone_tapin,SUBZONE_N,Pop, TOTAL_TAP_IN_VOLUME,X_ADDR,Y_ADDR,geometry)View(subzone_tapin)
Install packages again
packages = c('rgdal', 'sf', 'spdep', 'tmap', 'tidyverse')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}## Loading required package: rgdal## Loading required package: sp## rgdal: version: 1.4-8, (SVN revision 845)
## Geospatial Data Abstraction Library extensions to R successfully loaded
## Loaded GDAL runtime: GDAL 2.2.3, released 2017/11/20
## Path to GDAL shared files: C:/R/R-4.0.0/library/rgdal/gdal
## GDAL binary built with GEOS: TRUE
## Loaded PROJ.4 runtime: Rel. 4.9.3, 15 August 2016, [PJ_VERSION: 493]
## Path to PROJ.4 shared files: C:/R/R-4.0.0/library/rgdal/proj
## Linking to sp version: 1.4-1## Loading required package: spdep## Loading required package: spData## To access larger datasets in this package, install the spDataLarge
## package with: `install.packages('spDataLarge',
## repos='https://nowosad.github.io/drat/', type='source')`## Loading required package: tmapChange to sf
subzone_tapin = st_as_sf(subzone_tapin)Plot qtm graph to analysis different regions
The tap in volume graph shows a high tap in volume of 3min to 4min at the east part of the map, which is supported by the population graph, which shows the highest population at the same part of the map
st_make_valid(subzone_tapin)## Simple feature collection with 305 features and 5 fields
## geometry type: GEOMETRY
## dimension: XY
## bbox: xmin: 103.6057 ymin: 1.210655 xmax: 104.0336 ymax: 1.470775
## geographic CRS: WGS 84
## # A tibble: 305 x 6
## SUBZONE_N Pop TOTAL_TAP_IN_VOL~ X_ADDR Y_ADDR geometry
## <chr> <dbl> <dbl> <dbl> <dbl> <MULTIPOLYGON [°]>
## 1 ADMIRALTY 14110 178509 27091. 48334. (((103.8285 1.458775, 103.8~
## 2 AIRPORT R~ 0 22692 35133. 37075. (((103.9014 1.356171, 103.9~
## 3 ALEXANDRA~ 13780 497549 25359. 29991. (((103.8144 1.285474, 103.8~
## 4 ALEXANDRA~ 2120 59359 26548. 30519. (((103.8174 1.294306, 103.8~
## 5 ALJUNIED 40190 1613882 33593. 32971. (((103.8913 1.321318, 103.8~
## 6 ANAK BUKIT 22250 799579 21127. 35562. (((103.771 1.347908, 103.77~
## 7 ANCHORVALE 46610 472053 34219. 41815. (((103.896 1.399924, 103.89~
## 8 ANG MO KI~ 4890 1100348 29502. 39419. (((103.8485 1.368786, 103.8~
## 9 ANSON 0 73701 29145. 28467. (((103.8441 1.274875, 103.8~
## 10 BALESTIER 32760 1061622 29975. 34194. (((103.8623 1.329565, 103.8~
## # ... with 295 more rowspar(mfrow=c(1,2))
qtm(subzone_tapin, "TOTAL_TAP_IN_VOLUME")## Warning: The shape subzone_tapin is invalid. See sf::st_is_valid
qtm(subzone_tapin, "Pop")## Warning: The shape subzone_tapin is invalid. See sf::st_is_valid
Creating (QUEEN) contiguity based neighbours
The summary report above shows that there are 305 area units in subzone tap in.
The most connected area unit has 17 neighbours.
There are no area units with only on neighbour.
wm_q_in <- poly2nb(subzone_tapin, queen=TRUE)
summary(wm_q_in)## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1848
## Percentage nonzero weights: 1.986563
## Average number of links: 6.059016
## Link number distribution:
##
## 2 3 4 5 6 7 8 9 10 11 12 14 17
## 6 11 27 79 74 51 36 14 2 2 1 1 1
## 6 least connected regions:
## 2 41 132 170 228 275 with 2 links
## 1 most connected region:
## 39 with 17 linksCreating (ROOK) contiguity based neighbours
The summary report above shows that there are 305 area units in subzone tap in.
The most connected area unit has 14 neighbours.
There is 1 area unit with only one neighbour.
wm_r_in <- poly2nb(subzone_tapin, queen=FALSE)
summary(wm_r_in)## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1596
## Percentage nonzero weights: 1.715668
## Average number of links: 5.232787
## Link number distribution:
##
## 1 2 3 4 5 6 7 8 9 10 13 14
## 1 5 24 70 91 62 29 15 4 2 1 1
## 1 least connected region:
## 170 with 1 link
## 1 most connected region:
## 39 with 14 linksPlotting Queen and Rook based on neighbour map
centroids <- sf::st_centroid(subzone_tapin$geometry)## Warning in st_centroid.sfc(subzone_tapin$geometry): st_centroid does not give
## correct centroids for longitude/latitude dataplot(subzone_tapin$geometry, border="lightgrey")
plot(wm_q_in, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red", main="Queen Contiguity")
plot(subzone_tapin$geometry, border="lightgrey")
plot(wm_r_in, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red", main="Rook Contiguity")
Queen contiguity is preferred since it has more edges than Rook’s, as shown by more red lines connecting to the points. Queen will hence give a more accurate reading.
Computing distance based neighbours
The summary shows that the largest first nearest neighbour distance is 5.4040km, so using this as an upper threshold gives certainty that all units will have at least one neighbour
coords <-st_coordinates(centroids)
k1 <- knn2nb(knearneigh(coords))
k1dists <- unlist(nbdists(k1, coords, longlat=TRUE))
summary(k1dists)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1825 0.6167 0.8917 0.9414 1.1699 5.4040Computing fixed distance weight matrix
wm_d_in <- dnearneigh(coords,0,6,longlat=TRUE)
wm_d_in## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 19302
## Percentage nonzero weights: 20.74926
## Average number of links: 63.28525To display weight matrix
str(wm_d_in)## List of 305
## $ : int [1:29] 85 123 139 145 146 147 158 166 168 169 ...
## $ : int [1:70] 5 7 10 14 15 18 20 21 23 25 ...
## $ : int [1:96] 4 9 12 17 22 26 28 30 35 36 ...
## $ : int [1:103] 3 9 10 12 17 18 22 25 26 28 ...
## $ : int [1:92] 2 10 12 15 17 18 20 21 22 23 ...
## $ : int [1:61] 11 29 31 32 33 34 39 48 54 55 ...
## $ : int [1:46] 2 8 44 50 60 64 75 96 97 98 ...
## $ : int [1:59] 7 10 18 20 21 25 27 39 44 50 ...
## $ : int [1:85] 3 4 10 12 17 18 22 23 26 28 ...
## $ : int [1:115] 2 4 5 8 9 12 17 18 20 21 ...
## $ : int [1:37] 6 29 31 32 33 34 39 48 49 63 ...
## $ : int [1:94] 3 4 5 9 10 17 18 22 23 25 ...
## $ : int [1:18] 14 15 16 69 76 78 81 109 118 121 ...
## $ : int [1:33] 2 13 15 16 64 69 76 78 81 109 ...
## $ : int [1:46] 2 5 13 14 16 20 64 69 76 78 ...
## $ : int [1:22] 13 14 15 69 76 78 81 109 116 118 ...
## $ : int [1:110] 3 4 5 9 10 12 18 20 21 22 ...
## $ : int [1:113] 2 4 5 8 9 10 12 17 20 21 ...
## $ : int [1:25] 24 45 87 88 94 103 106 107 108 124 ...
## $ : int [1:92] 2 5 8 10 15 17 18 21 23 25 ...
## $ : int [1:88] 2 5 8 10 17 18 20 23 25 26 ...
## $ : int [1:99] 3 4 5 9 10 12 17 18 23 25 ...
## $ : int [1:102] 2 5 9 10 12 17 18 20 21 22 ...
## $ : int [1:42] 19 29 31 32 33 34 45 48 71 83 ...
## $ : int [1:106] 2 4 5 8 10 12 17 18 20 21 ...
## $ : int [1:115] 3 4 5 9 10 12 17 18 21 22 ...
## $ : int [1:97] 2 5 8 10 17 18 20 21 22 23 ...
## $ : int [1:107] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:53] 6 11 24 31 32 33 34 45 48 49 ...
## $ : int [1:108] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:57] 6 11 24 29 32 33 34 45 48 49 ...
## $ : int [1:58] 6 11 24 29 31 33 34 39 45 48 ...
## $ : int [1:60] 6 11 24 29 31 32 34 45 48 49 ...
## $ : int [1:56] 6 11 24 29 31 32 33 45 48 49 ...
## $ : int [1:101] 3 4 9 10 12 17 18 22 23 25 ...
## $ : int [1:94] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:87] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:83] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:39] 6 8 11 32 44 61 63 68 72 79 ...
## $ : int [1:9] 41 42 76 140 141 180 236 248 293
## $ : int [1:12] 40 42 76 140 141 179 180 181 182 183 ...
## $ : int [1:16] 40 41 76 140 141 179 180 181 182 183 ...
## $ : int [1:109] 3 4 9 10 12 17 18 22 23 25 ...
## $ : int [1:65] 2 7 8 10 18 20 21 25 27 39 ...
## $ : int [1:41] 19 24 29 31 32 33 34 71 86 87 ...
## $ : int [1:97] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:93] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:40] 6 11 24 29 31 32 33 34 49 63 ...
## $ : int [1:35] 11 29 31 32 33 34 48 63 72 79 ...
## $ : int [1:67] 2 7 8 10 18 20 21 23 25 27 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 20 22 ...
## $ : int [1:77] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:54] 3 6 29 31 32 33 34 55 56 57 ...
## $ : int [1:57] 3 6 29 31 32 33 34 54 56 57 ...
## $ : int [1:52] 3 6 31 32 33 34 54 55 57 59 ...
## $ : int [1:54] 3 4 6 31 32 33 34 54 55 56 ...
## $ : int [1:95] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:89] 3 4 6 9 17 22 26 28 35 36 ...
## $ : int [1:53] 2 7 8 20 21 44 50 64 75 96 ...
## $ : int [1:83] 3 4 6 10 25 26 27 32 33 35 ...
## $ : int [1:103] 4 5 9 10 12 17 18 20 21 22 ...
## $ : int [1:43] 6 11 29 31 32 33 34 39 48 49 ...
## $ : int [1:71] 2 5 7 8 10 14 15 18 20 21 ...
## $ : int [1:92] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:110] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:63] 3 4 6 26 31 32 33 35 36 43 ...
## $ : int [1:113] 3 4 6 8 10 17 18 21 22 23 ...
## $ : int [1:38] 2 5 12 13 14 15 16 18 20 23 ...
## $ : int [1:90] 3 4 9 10 12 17 18 22 23 26 ...
## $ : int [1:56] 6 24 29 31 32 33 34 45 54 55 ...
## $ : int [1:36] 6 11 29 31 32 33 34 39 48 49 ...
## $ : int [1:97] 3 4 6 10 17 22 25 26 27 28 ...
## $ : int [1:111] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:54] 2 7 8 20 21 44 50 60 64 96 ...
## $ : int [1:21] 13 14 15 16 40 41 42 140 141 179 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:35] 2 5 13 14 15 16 20 23 64 69 ...
## $ : int [1:36] 11 29 31 32 39 48 49 63 72 83 ...
## $ : int [1:105] 2 5 10 12 17 18 20 21 22 23 ...
## $ : int [1:66] 2 5 10 12 13 14 15 16 17 18 ...
## $ : int [1:73] 3 4 6 26 31 32 33 35 36 43 ...
## $ : int [1:44] 6 11 24 29 31 32 33 34 39 48 ...
## $ : int [1:119] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:29] 1 79 123 127 145 146 147 158 166 168 ...
## $ : int [1:54] 6 11 24 29 31 32 33 34 39 45 ...
## $ : int [1:19] 19 45 88 103 106 108 124 134 194 208 ...
## $ : int [1:20] 19 24 45 87 103 106 108 124 134 194 ...
## $ : int [1:99] 3 4 9 10 12 17 18 22 26 28 ...
## $ : int [1:83] 3 4 6 10 17 21 25 26 27 32 ...
## $ : int [1:54] 6 11 24 29 31 32 33 34 39 48 ...
## $ : int [1:79] 3 4 6 26 33 35 36 43 53 54 ...
## $ : int [1:68] 3 4 6 11 26 29 31 32 33 34 ...
## $ : int [1:48] 6 19 24 29 31 32 33 34 45 48 ...
## $ : int [1:48] 6 11 24 29 31 32 33 34 45 48 ...
## $ : int [1:68] 2 7 8 15 20 21 25 27 44 50 ...
## $ : int [1:57] 2 7 8 15 20 21 44 50 60 64 ...
## $ : int [1:65] 2 7 8 15 20 21 25 27 44 50 ...
## $ : int [1:107] 3 4 5 9 10 12 17 18 22 23 ...
## [list output truncated]
## - attr(*, "class")= chr "nb"
## - attr(*, "nbtype")= chr "distance"
## - attr(*, "distances")= num [1:2] 0 6
## - attr(*, "region.id")= chr [1:305] "1" "2" "3" "4" ...
## - attr(*, "call")= language dnearneigh(x = coords, d1 = 0, d2 = 6, longlat = TRUE)
## - attr(*, "dnn")= num [1:2] 0 6
## - attr(*, "bounds")= chr [1:2] "GT" "LE"
## - attr(*, "sym")= logi TRUEPlotting fixed distance weight matrix
The red lines show the links of 1st nearest neighbours and the black lines shows the links of neighbours within cut-off distance of 6km.
plot(subzone_tapin$geometry, border="lightgrey")
plot(wm_d_in, coords, add=TRUE)
plot(k1, coords, add=TRUE, col="red", length=0.08)
Computing adaptive distance weight matrix
wm_knn_in <- knn2nb(knearneigh(coords, k=6))
wm_knn_in## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1830
## Percentage nonzero weights: 1.967213
## Average number of links: 6
## Non-symmetric neighbours listPlotting distance based neighbours
plot(subzone_tapin$geometry, border="lightgrey")
plot(wm_knn_in, coords, pch=19, cex=0.6, add=TRUE, col="red")
Comparing between fixed and adaptive
par(mfrow=c(1,2))
plot(subzone_tapin$geometry, border="lightgrey")
plot(wm_d_in, coords, add=TRUE)
plot(k1, coords, add=TRUE, col="red", length=0.08)
plot(subzone_tapin$geometry, border="lightgrey")
plot(wm_knn_in, coords, pch=19, cex=0.6, add=TRUE, col="red")
From the two plots, we can conclude that adaptive weight matrix should be used instead of fixed weight matrix, as there are several missing values from the fixed weight matrix.
Adaptive weight matrix will give a more accurate result.
Distance based neighbours of adaptive weight matrix shows a wider spread.
Weights based on IDW
To derive spatial weight matrix based on Inversed Distance
Compute the distances between areas
dist <- nbdists(wm_q_in, coords, longlat=TRUE)
ids <- lapply(dist, function(x)1/(x))
ids## [[1]]
## [1] 0.8844741 1.0480106 0.8884195 0.7273930 0.7194021 0.8145234
##
## [[2]]
## [1] 5.4804873 0.5914406
##
## [[3]]
## [1] 0.7684313 0.8535230 1.6794906 0.5228568 0.9209379 1.0335332 0.8362314
## [8] 1.1858760
##
## [[4]]
## [1] 0.7684313 0.8561584 1.5861382 1.5851576 0.6885791 1.6141956
##
## [[5]]
## [1] 0.7967051 0.5944818 0.5997141 0.6629291 0.6203724 0.4515680 0.6945245
## [8] 0.6050715 0.5702452
##
## [[6]]
## [1] 0.6512450 0.4671241 0.5523379 0.5381036 0.7563190 0.4293571
##
## [[7]]
## [1] 0.9454551 0.7069300 0.7923350 1.0038999
##
## [[8]]
## [1] 1.2338839 0.7348002 1.2417769 0.9286924 1.1898496 0.7053242 0.7185641
## [8] 0.8303788
##
## [[9]]
## [1] 0.9959641 1.4465246 0.8333802 1.2148883 2.8731441
##
## [[10]]
## [1] 0.7966416 0.7644062 0.9104840 0.5199711 0.7906893 0.5484124 1.0309079
##
## [[11]]
## [1] 0.3340457 1.0083355 1.0893203 1.1046888
##
## [[12]]
## [1] 0.9203617 2.0296518 1.0517067 1.1788886 0.7483162
##
## [[13]]
## [1] 1.0456883 0.3182148 0.3952152 0.3518170
##
## [[14]]
## [1] 0.6605051 0.7274535 0.4081463 0.3335142 0.4247973 0.4347253 0.4996058
## [8] 0.2714662
##
## [[15]]
## [1] 0.6605051 0.5181626 0.4845160 0.5729661 0.3245971 0.7023854
##
## [[16]]
## [1] 1.0456883 0.7274535 0.3591743 0.4215225 0.3491247
##
## [[17]]
## [1] 1.494798 1.310005 1.254374 1.634446 5.480304 1.374106 3.113369
##
## [[18]]
## [1] 0.7966416 1.0681731 1.0154484 0.6284209 0.8855030 0.7727805 0.7862932
## [8] 0.5582583
##
## [[19]]
## [1] 0.4808461 0.5021124 0.8552760 0.4146835 0.4872360 0.8520672 0.3744038
##
## [[20]]
## [1] 1.0407714 0.6501177 1.1361333 1.5437228 1.0316895
##
## [[21]]
## [1] 0.9133713 0.6159350 0.9185460 0.8728618 0.9670262 0.4796687 0.6021557
## [8] 0.5586217 0.5457509
##
## [[22]]
## [1] 2.786039 1.267237 2.117575 1.005309 4.246211 1.961564
##
## [[23]]
## [1] 0.7967051 1.2804026 1.3084566 0.7795739 1.2994920
##
## [[24]]
## [1] 0.8162941 0.8398473 0.8406951 0.4617172 0.5947887 0.8492181 0.5505385
##
## [[25]]
## [1] 1.435667 1.373967 1.301129 1.553840 1.196840
##
## [[26]]
## [1] 1.0772240 1.1926237 0.7632259 1.3811674 1.7409411 2.1360097 1.1732691
## [8] 1.3218132
##
## [[27]]
## [1] 0.9133713 1.4356674 1.1960461 1.0402323 1.1480015
##
## [[28]]
## [1] 1.494798 1.688457 1.806317 2.215308 1.827245
##
## [[29]]
## [1] 0.8033619 1.1060481 0.7749737 0.9886477 0.5712238
##
## [[30]]
## [1] 1.238054 1.127686 1.875910 1.768634 2.907615
##
## [[31]]
## [1] 0.8033619 1.2456534 0.8677066 1.0995651 1.1008159
##
## [[32]]
## [1] 1.2456534 1.1139321 1.2296401 0.9886387
##
## [[33]]
## [1] 0.6512450 0.8677066 1.1139321 0.5251549 0.5749132 0.4975997 0.7172742
## [8] 0.9562401 0.4655908
##
## [[34]]
## [1] 1.1060481 1.0995651 0.5251549 0.4251808 0.8437940 1.1504476 0.8727465
##
## [[35]]
## [1] 0.8561584 0.8849391 1.0580419 0.9130916 1.3190915 1.5405780 1.3468824
##
## [[36]]
## [1] 0.8535230 0.8953934 1.4431053 0.9473712 1.5587781 1.8066475 1.0600227
##
## [[37]]
## [1] 1.945150 1.681763 2.867192 1.639472 2.216894
##
## [[38]]
## [1] 0.9959641 0.9203617 1.9451504 0.4801978 0.8909323 0.6389542 1.1068127
## [8] 1.1632581 1.3729115
##
## [[39]]
## [1] 0.3340457 0.2968846 0.2888334 0.2008093 0.2052084 0.1755475 0.2010533
## [8] 0.1768133 0.2737723 0.2683509 0.3031680 0.2675447 0.2437008 0.3102618
## [15] 0.1645856 0.1596139 0.2470689
##
## [[40]]
## [1] 0.1928125 0.2883690 0.2455807
##
## [[41]]
## [1] 0.1928125 0.4024330
##
## [[42]]
## [1] 0.2883690 0.4024330 0.6129275 1.9940343 0.8012110 0.2990791 0.2256625
##
## [[43]]
## [1] 1.5861382 0.8849391 0.8398056 0.5903987 1.4993784 0.9754874 0.6645181
## [8] 0.6068086 1.1098669
##
## [[44]]
## [1] 1.2338839 1.0523191 0.5523030 0.5613916 0.8274346 0.9741141 0.8390431
##
## [[45]]
## [1] 0.8162941 0.5354624 0.7349362 0.5731777 1.1214091 0.8920002
##
## [[46]]
## [1] 2.786039 1.655131 1.585099 1.188247 4.539164 2.151226
##
## [[47]]
## [1] 1.4465246 1.6817633 1.6551312 1.0095825 4.0676512 1.2752529 2.4095598
## [8] 1.0939357 0.9929413 1.9481332
##
## [[48]]
## [1] 1.2058561 0.9429210 0.8453777 1.1247852 0.1549636
##
## [[49]]
## [1] 1.2058561 0.4999200 0.5894127 0.6608199 0.1555487 1.0815070
##
## [[50]]
## [1] 0.7348002 0.6159350 1.0523191 0.5157744 0.6161005 0.8084862 0.6482094
## [8] 1.2375038
##
## [[51]]
## [1] 1.2672374 1.6884572 1.2380540 0.9940303 1.2837980 0.6653816 1.2277930
## [8] 0.7652348 1.3663294 1.0067668 1.2508523
##
## [[52]]
## [1] 0.8333802 0.4801978 0.9365332 0.5600298 0.3833439 0.5271297 0.3948815
## [8] 0.6004264
##
## [[53]]
## [1] 2.1175745 1.5850988 0.9940303 1.9383979 1.3815174 1.1669983
##
## [[54]]
## [1] 1.2679553 0.9472898 0.9463619 0.7722828 0.9961068
##
## [[55]]
## [1] 1.2679553 0.6445698 0.8307125 0.7361825 1.0745144 0.5206418
##
## [[56]]
## [1] 0.9472898 1.6527189 0.8805075 0.3870804 0.3140803 0.5674396
##
## [[57]]
## [1] 0.9463619 0.6445698 1.6527189 0.7989980 0.7772218 0.4282428 0.7856067
##
## [[58]]
## [1] 2.0296518 0.8909323 1.2837980 1.1211354 2.1805833
##
## [[59]]
## [1] 1.4706887 0.7308158 0.7947949 0.7562514 1.8549252
##
## [[60]]
## [1] 0.9454551 0.5320455 0.7959180 0.7495172 0.5882395 0.8816489 0.9588370
##
## [[61]]
## [1] 0.9569594 0.8914390 0.5038010 0.9232634 0.5109726 0.5691983
##
## [[62]]
## [1] 1.1276855 0.6653816 0.9056074 2.0375888 0.8137809 0.9199540 1.2646240
## [8] 0.7443998
##
## [[63]]
## [1] 1.0083355 0.2968846 0.6160743 0.5959538 0.9512227 0.5635195
##
## [[64]]
## [1] 5.4804873 0.4252456 0.4092100 0.4999176 0.8560016 0.4577477 0.3484185
## [8] 0.5391147 0.7922589
##
## [[65]]
## [1] 1.6794906 0.8953934 0.5532178 0.7800484 1.6215842 0.6991945
##
## [[66]]
## [1] 1.310005 1.806317 1.848831 1.382043 1.810943 1.620788 1.357020
##
## [[67]]
## [1] 0.7989980 0.8487075 1.0756715 0.7005317 1.1515548
##
## [[68]]
## [1] 0.6551446 0.6323499 0.7688724 0.6923502 0.6630384
##
## [[69]]
## [1] 0.3576763 2.7288438 0.5130455 0.4573773
##
## [[70]]
## [1] 1.2148883 1.0095825 0.9365332 0.9376335 1.7858516 0.7647919
##
## [[71]]
## [1] 0.7722828 0.8307125 0.9182911 0.9217925 1.5196015
##
## [[72]]
## [1] 1.0893203 0.2888334 0.5223506 0.8925147 1.7640783
##
## [[73]]
## [1] 0.9569594 1.0927183 0.6092194 1.1052748
##
## [[74]]
## [1] 1.254374 1.847188 1.246549 2.947320 1.905523 1.215226
##
## [[75]]
## [1] 0.7069300 0.5320455 0.4114400 0.6135504 0.4537765 0.5824593 0.6725423
## [8] 0.6052865
##
## [[76]]
## [1] 0.6129275 0.8043294 0.4952155 0.7646293 0.5671159
##
## [[77]]
## [1] 2.2153080 1.2277930 1.9383979 1.8488310 1.0542144 1.4284612 0.9593715
##
## [[78]]
## [1] 0.3182148 0.4081463 0.3591743 0.4050295 0.4590652 0.8120932 0.5132071
## [8] 0.8547808
##
## [[79]]
## [1] 0.2008093 0.4999200 0.5223506 0.3195949 0.4659314 0.6380523 0.6422838
## [8] 0.4497935 0.4809803 0.5992606
##
## [[80]]
## [1] 0.5944818 1.0681731 1.2804026 1.0097213 1.2503065 0.7338037 0.7472477
##
## [[81]]
## [1] 0.5997141 0.4050295 0.4150285 0.7066867 0.8635013 0.4740998 0.5368732
## [8] 0.5367962
##
## [[82]]
## [1] 0.8487075 1.1616463 0.6850966 0.8707461 1.0969734
##
## [[83]]
## [1] 0.6160743 0.6319974 0.5990246 0.7803611 0.8192146 0.7113396 0.5287903
## [8] 0.7192815 0.3073443
##
## [[84]]
## [1] 1.0772240 0.9472528 0.9530090 0.9036667 1.8975126 1.6983369
##
## [[85]]
## [1] 0.9052538 1.6882874 0.7239262 0.4968966 1.0396107 0.6690701
##
## [[86]]
## [1] 0.7749737 1.1008159 1.2296401 0.6319974 0.8888807 0.9712211
##
## [[87]]
## [1] 0.4808461 0.6798021 0.6710674 0.5866048
##
## [[88]]
## [1] 0.5021124 0.6798021 0.6095744 0.5423227 0.3218916 0.3581644 0.4299867
##
## [[89]]
## [1] 1.5851576 1.0580419 1.4431053 0.9068886 1.7085096 1.6868741
##
## [[90]]
## [1] 0.2052084 0.8914390 0.6551446 0.4331565 0.5114785 0.4718265
##
## [[91]]
## [1] 0.4671241 0.9886387 0.5749132 0.5959538 0.5990246 0.8888807 0.6391110
##
## [[92]]
## [1] 1.4706887 1.1616463 1.0185734 0.8388392 1.2334553 0.5743472
##
## [[93]]
## [1] 0.5523379 0.5038010 0.5434165 0.5945844 0.5624522 0.8207609
##
## [[94]]
## [1] 0.8398473 0.5354624 0.4679993 0.5878285 0.5670518 0.5945566 0.7970978
##
## [[95]]
## [1] 0.9886477 0.7803611 0.9712211 0.7024756 0.4660007
##
## [[96]]
## [1] 1.0135684 1.3229506 1.3302127 0.7080008 0.6583602
##
## [[97]]
## [1] 0.7959180 1.0135684 0.7605919 1.3386343 0.6622192
##
## [[98]]
## [1] 0.7495172 1.3229506 0.7605919 0.7633608 0.4080910 0.7176809 1.0333706
##
## [[99]]
## [1] 1.054214 1.728954 2.081489 3.270822
##
## [[100]]
## [1] 0.9182911 1.0416720 0.5005635 0.7230575 0.7987768 0.9121866 0.7359705
##
## [[101]]
## [1] 1.3820435 0.8777783 2.1551137 2.3549864 1.3134107
##
## [[102]]
## [1] 1.1046888 0.9512227 0.8925147 0.8192146 0.8793560 0.7729020 0.7086318
##
## [[103]]
## [1] 0.8552760 0.6095744 0.4655936 0.3033305 0.9296686 0.2975364 0.3175127
## [8] 0.3022505 0.1573922
##
## [[104]]
## [1] 1.0407714 0.7262922 0.5670105 0.9349039 0.9646251 1.0121334 0.8005541
##
## [[105]]
## [1] 1.0416720 0.8205733 1.6226101 1.0939244
##
## [[106]]
## [1] 0.5742682 0.5943254 0.4113321 0.5452498
##
## [[107]]
## [1] 0.5742682 0.6140380 0.4362903 0.4658673 0.5344569 0.5967864 0.4382571
##
## [[108]]
## [1] 0.8406951 0.7349362 0.9415510 0.5528436 1.0636402
##
## [[109]]
## [1] 0.3335142 0.5181626 0.4252456 0.4150285 0.6292237 0.7308923 0.3675036
## [8] 0.4111101
##
## [[110]]
## [1] 1.015448 1.308457 1.009721 1.162562 1.263443
##
## [[111]]
## [1] 0.6629291 0.7795739 1.2503065 0.7262922 0.9737761 1.1552076
##
## [[112]]
## [1] 0.6203724 1.2994920 0.9056074 1.1625619 0.7706998 0.9697024
##
## [[113]]
## [1] 1.8759095 0.7652348 2.0375888 3.2176501
##
## [[114]]
## [1] 0.7644062 0.6284209 1.8471885 0.8777783 0.9677324 1.1733551 0.8659227
## [8] 1.0628311
##
## [[115]]
## [1] 0.9473712 0.5600298 0.9376335 0.9068886 1.2115791 1.7619035 1.2938641
## [8] 1.3185284 1.5874208
##
## [[116]]
## [1] 0.4515680 0.4092100 0.7066867 0.5670105 0.6292237 0.4577219 0.7756175
## [8] 0.5388115
##
## [[117]]
## [1] 0.4999176 1.3302127 1.3386343 0.4791165 0.5220114 1.5436506 0.4699383
##
## [[118]]
## [1] 0.4590652 0.8635013 1.3106704 0.7257642
##
## [[119]]
## [1] 0.9429210 0.7113396 0.7024756 1.0310969 0.8670278 0.4247892
##
## [[120]]
## [1] 1.2417769 1.0061644 1.6064429 0.7500858 0.6527241 1.1553013
##
## [[121]]
## [1] 0.4247973 0.4845160 0.8120932 0.4740998 0.7308923 0.4577219
##
## [[122]]
## [1] 0.5228568 0.5532178 0.4659881 0.6848407 0.7154689 1.1606192 1.2516079
## [8] 0.4898260
##
## [[123]]
## [1] 0.4560956 0.5015524 0.9737212 0.5147668 0.5251365 0.7195327 0.9792542
## [8] 0.6773042
##
## [[124]]
## [1] 0.4617172 0.5731777 0.4655936 0.9415510 0.5971054 0.7045412 0.6752555
##
## [[125]]
## [1] 0.9104840 0.8855030 1.3739673 1.8447553 1.2660919 0.7090568
##
## [[126]]
## [1] 0.7080008 0.7633608 0.4791165 1.0600947 0.6155955 0.6409793 0.6648319
##
## [[127]]
## [1] 0.3347517 0.3706974 0.7000040 0.5612803 0.4524644 0.5409671
##
## [[128]]
## [1] 0.4679993 0.5005635 0.8205733 0.6140380 0.8493902 0.4077919 0.9326988
## [8] 0.6865580
##
## [[129]]
## [1] 0.7727805 0.8137809 1.2465487 1.2634434 0.7706998 0.9677324 1.3857451
## [8] 1.2956422 1.5903328
##
## [[130]]
## [1] 0.7308158 0.9232634 1.0927183 1.0185734 0.5434165 0.3973167 0.4535570
## [8] 0.5543265 0.5751480
##
## [[131]]
## [1] 1.1926237 0.8398056 1.7289543 1.1439498 1.3789246 2.1873170 1.2376825
## [8] 1.5469512
##
## [[132]]
## [1] 0.3268232 0.1589236
##
## [[133]]
## [1] 1.634446 2.947320 1.385745 1.410838 3.151098 1.256131
##
## [[134]]
## [1] 0.4146835 0.3033305 0.5943254 0.4362903 0.7655235 0.4130353 0.4338324
## [8] 1.2109539
##
## [[135]]
## [1] 0.9185460 1.3011294 1.1960461 0.9251273 2.0740006 0.7783528 0.9728857
##
## [[136]]
## [1] 0.8560016 0.6583602 0.5220114 1.0600947 0.5421828 0.6904218 0.6428618
##
## [[137]]
## [1] 0.6501177 0.8728618 0.5157744 0.9251273 1.0775249 0.6159859 0.6438742
## [8] 1.0063526
##
## [[138]]
## [1] 0.4577477 1.5436506 1.1808305 0.6359548 0.5019490
##
## [[139]]
## [1] 0.4560956 0.4381241 0.4159762 0.3833159 0.3158822 0.5882482 0.4798430
## [8] 0.7144880
##
## [[140]]
## [1] 1.9940343 0.8043294 0.9596644
##
## [[141]]
## [1] 0.8012110 0.4952155 0.9596644 0.6448882 0.5569022
##
## [[142]]
## [1] 1.905523 2.155114 1.173355 2.117224
##
## [[143]]
## [1] 0.6945245 0.5368732 0.9349039 0.9737761 0.7756175 0.6925473 0.5453684
##
## [[144]]
## [1] 0.5199711 0.6323499 0.9472528 0.8631758 0.7656371 0.6757964 0.5871453
##
## [[145]]
## [1] 0.1755475 0.5015524 0.7442450 0.3007840 0.4677231 0.4484203 0.4853337
## [8] 0.3315351 0.3264268 0.4951431 0.4649258 0.4851777
##
## [[146]]
## [1] 0.9737212 0.7442450 1.2016692
##
## [[147]]
## [1] 0.2010533 0.3195949 0.3007840 0.2794967 0.4319578 0.7285834 0.6137348
##
## [[148]]
## [1] 0.9209379 0.6885791 0.5903987 0.7947949 0.3973167 1.4487496 0.6863953
## [8] 1.2854792 0.7389841
##
## [[149]]
## [1] 1.051707 1.366329 1.121135 0.919954 0.707254
##
## [[150]]
## [1] 0.3576763 0.3565229 0.7828311 0.7434382
##
## [[151]]
## [1] 1.1788886 0.6389542 0.7072540 0.7820824
##
## [[152]]
## [1] 2.7288438 0.5132071 1.3106704 0.3565229 0.5498483 0.4522282
##
## [[153]]
## [1] 0.3833439 0.4290775 0.2315013 0.3627578 0.9756103 0.7358156 0.8145211
##
## [[154]]
## [1] 0.9670262 0.6161005 0.5989028 0.6123236 0.7815656 0.6897698
##
## [[155]]
## [1] 0.7923350 0.4114400 0.5517190 1.1535795 0.5293861 0.5392787 0.8134473
## [8] 0.5051229
##
## [[156]]
## [1] 2.867192 4.067651 2.561208
##
## [[157]]
## [1] 1.0335332 0.7562514 1.4487496 0.6895619 1.3707566 0.8877237
##
## [[158]]
## [1] 0.9052538 0.7286530 0.5374064 0.8571719 1.2635250 0.8494379
##
## [[159]]
## [1] 2.3549864 0.8659227 0.9323119 1.9315370 1.2191397
##
## [[160]]
## [1] 0.7906893 0.9530090 1.0628311 0.8631758 0.9323119 1.2216780
##
## [[161]]
## [1] 0.1768133 0.7688724 0.4331565 0.7656371 0.5266164 0.6332063
##
## [[162]]
## [1] 0.6050715 0.5130455 0.5367962 0.7257642 0.7828311 0.5498483 0.5835158
##
## [[163]]
## [1] 0.7632259 0.6923502 0.9036667 0.6757964 1.2700510 0.5433986 0.9875742
## [8] 0.7858094
##
## [[164]]
## [1] 0.7772218 1.0756715 0.4659881 0.6138716 0.9035689
##
## [[165]]
## [1] 0.5381036 0.2737723 0.5635195 0.6391110 0.4653930
##
## [[166]]
## [1] 0.2683509 0.5147668 0.4677231 0.2794967 0.8021035 0.3635262
##
## [[167]]
## [1] 1.897513 1.931537 1.221678
##
## [[168]]
## [1] 1.6882874 0.3347517 0.3440207 0.5632434 0.6394194
##
## [[169]]
## [1] 0.7863828 0.6765798 0.7936921 0.6766562 0.9144950
##
## [[170]]
## [1] 1.0237727 0.5317908
##
## [[171]]
## [1] 0.7005317 0.6850966 0.8388392 0.6848407 0.6895619 0.6138716 0.7246413
## [8] 1.4065933
##
## [[172]]
## [1] 1.381167 1.499378 1.143950 2.013966 2.670432
##
## [[173]]
## [1] 1.740941 1.698337 1.270051
##
## [[174]]
## [1] 1.810943 1.428461 2.081489 1.378925 2.105451
##
## [[175]]
## [1] 0.8805075 0.6080403 0.2383592 0.6806124 1.0994479
##
## [[176]]
## [1] 0.4659314 0.3706974 0.4714593 0.5463106 0.1363204 0.7572748
##
## [[177]]
## [1] 0.3870804 0.4282428 0.7154689 0.9035689 0.5264702 0.8238056
##
## [[178]]
## [1] 1.1606192 0.4290775 0.5264702 0.4718700 0.6285524
##
## [[179]]
## [1] 0.7549454 0.8548739 0.8288461 0.5041562 0.8457130
##
## [[180]]
## [1] 0.7646293 0.6448882 0.7549454 0.7522203 0.5443706 0.4372582
##
## [[181]]
## [1] 0.5569022 0.8548739 0.7522203 0.3250162 0.5951573
##
## [[182]]
## [1] 0.3250162 0.6141856 0.5673076 0.3761027 0.4890482 0.3790200 0.3634999
## [8] 0.4832962
##
## [[183]]
## [1] 0.8288461 0.5951573 0.6141856 0.6921918
##
## [[184]]
## [1] 2.1360097 0.9754874 2.1873170 2.0139658 1.2233765
##
## [[185]]
## [1] 0.5729661 0.3675036 0.5117486 0.7198171 0.4840912 0.6385329
##
## [[186]]
## [1] 0.5673076 0.5117486 0.6083635 0.4240189 0.6027918
##
## [[187]]
## [1] 0.3484185 1.1808305 0.3761027 0.6083635 0.5976273
##
## [[188]]
## [1] 1.0053088 0.9130916 1.1882469 1.2752529 1.3815174 2.5428180 1.4708078
## [8] 1.1895831 1.0944313
##
## [[189]]
## [1] 1.553840 1.844755 2.074001 1.130316 1.077190
##
## [[190]]
## [1] 0.8453777 0.5894127 1.0310969 0.4911626 0.1834646
##
## [[191]]
## [1] 0.4113321 0.4658673 0.6080403 0.7957456
##
## [[192]]
## [1] 2.409560 2.542818 1.346062
##
## [[193]]
## [1] 4.246211 4.539164 3.190791
##
## [[194]]
## [1] 0.6710674 0.5423227 0.3598882
##
## [[195]]
## [1] 0.5914406 0.5391147 0.4111101 0.6359548 0.7198171 0.4240189 0.5976273
##
## [[196]]
## [1] 0.3140803 0.2315013 0.2383592 0.8238056 0.4718700
##
## [[197]]
## [1] 0.5484124 0.7862932 0.7338037 1.2660919 0.7783528 1.1303162 1.1249031
## [8] 1.2898274
##
## [[198]]
## [1] 0.5517190 0.4890482 0.5237262 0.9521497 0.6454933 1.2202489
##
## [[199]]
## [1] 1.1535795 1.0237727 0.5237262 0.4482643 0.5900350
##
## [[200]]
## [1] 0.8362314 0.7800484 1.2516079 1.3707566 0.7246413
##
## [[201]]
## [1] 1.961564 1.639472 1.106813 2.151226 1.093936 1.006767 2.180583 3.190791
##
## [[202]]
## [1] 1.1858760 1.6141956 1.5587781 1.7085096 0.6863953
##
## [[203]]
## [1] 0.7000040 0.3268232 0.4714593 0.4248008 0.1192066
##
## [[204]]
## [1] 0.6645181 0.6092194 0.4535570 1.2854792 0.5433986 0.6242597 0.8038132
##
## [[205]]
## [1] 0.5882395 0.6622192 0.4699383 0.5019490 0.5293861 0.3790200 0.9521497
## [8] 1.0807459
##
## [[206]]
## [1] 1.3190915 0.6068086 1.1669983 0.9593715 3.2708222 1.2376825 1.4708078
##
## [[207]]
## [1] 1.768634 1.264624 3.217650 1.295642 4.857047 1.431108
##
## [[208]]
## [1] 0.9296686 0.5971054 0.1884797 0.5864134
##
## [[209]]
## [1] 0.4872360 0.5452498 0.7655235 0.6898072
##
## [[210]]
## [1] 1.7640783 0.6380523 0.8793560 1.5652699
##
## [[211]]
## [1] 5.480304 1.620788 1.215226 1.410838 2.117224
##
## [[212]]
## [1] 0.6135504 0.5392787 0.4482643 0.5809053 0.3945352
##
## [[213]]
## [1] 0.4537765 0.4381241 0.5809053 0.5272022
##
## [[214]]
## [1] 0.5523030 0.5824593 0.7244121 0.7850972 1.0231996 0.5897564 0.6931957
## [8] 0.4085719
##
## [[215]]
## [1] 0.8844741 0.4484203 0.8353524 0.9941691 0.7224300 0.5402783
##
## [[216]]
## [1] 1.0480106 0.7863828 0.9848955 0.7364404 0.5525513
##
## [[217]]
## [1] 0.3031680 1.0061644 0.8851455 0.7331908 0.5283589 0.7911798
##
## [[218]]
## [1] 0.8884195 0.8353524 0.6392443 1.0130489 0.8182763
##
## [[219]]
## [1] 0.7273930 0.4853337 0.6765798 0.9941691 0.9848955 0.7850920
##
## [[220]]
## [1] 0.7194021 0.7364404 0.6477426
##
## [[221]]
## [1] 1.0038999 0.8816489 0.8134473 0.6454933 1.0807459
##
## [[222]]
## [1] 0.6725423 0.4159762 0.3945352 0.5272022 0.7244121 0.5478539
##
## [[223]]
## [1] 0.6422838 0.5287903 0.7729020 1.5652699 1.2445951
##
## [[224]]
## [1] 0.5582583 1.1361333 0.7472477 0.9646251 1.1552076 0.6925473 1.1249031
## [8] 0.9030380
##
## [[225]]
## [1] 0.3440207 0.6392443 0.8143723 0.6328791 0.4385275
##
## [[226]]
## [1] 0.3315351 0.7224300 1.0130489 0.8143723 0.8089478 0.6705902
##
## [[227]]
## [1] 0.7239262 0.7286530 0.5632434 0.6328791 0.8089478 0.8042739
##
## [[228]]
## [1] 0.5271297 0.3627578
##
## [[229]]
## [1] 0.6155955 0.5421828 1.0775249 0.8035498 1.0567880
##
## [[230]]
## [1] 0.4796687 0.5613916 0.8084862 0.4080910 0.6409793 0.6159859 0.8035498
## [8] 0.6716319 0.7414336
##
## [[231]]
## [1] 0.7176809 0.6648319 0.7850972 0.6716319 1.0406271 0.8628155
##
## [[232]]
## [1] 0.8274346 0.6482094 1.0231996 0.7414336 1.0406271 0.6297705
##
## [[233]]
## [1] 0.9286924 1.6064429 0.5989028 0.8851455 0.8631963 0.7535825
##
## [[234]]
## [1] 0.8520672 0.5866048 0.4130353 0.6898072
##
## [[235]]
## [1] 0.3952152 0.4573773 0.8547808 0.4522282
##
## [[236]]
## [1] 0.4347253 0.3245971 0.4215225 0.6441040 0.4568251 0.4758056
##
## [[237]]
## [1] 0.9929413 1.7858516 1.2115791 1.1895831 1.3460620 1.7106658
##
## [[238]]
## [1] 0.7361825 0.7856067 1.1515548 0.8707461 0.5253496 1.3972628
##
## [[239]]
## [1] 1.173269 1.357020 1.313411 1.546951 1.219140 2.105451 1.223376
##
## [[240]]
## [1] 0.2675447 0.5251365 0.3833159 0.3264268 0.8021035 0.5940072 0.4117553
##
## [[241]]
## [1] 0.7483162 1.1632581 0.3948815 0.7820824
##
## [[242]]
## [1] 1.374106 1.590333 3.151098 4.857047 1.274913
##
## [[243]]
## [1] 0.7563190 0.4975997 1.0745144 0.5945844 0.5253496 0.6668643 0.5599429
##
## [[244]]
## [1] 0.4293571 0.2437008 0.5109726 0.5114785 0.5624522 0.4653930
##
## [[245]]
## [1] 0.3102618 0.3158822 0.3635262 0.7331908 0.5940072 0.4993618 0.4552457
## [8] 0.6941495
##
## [[246]]
## [1] 0.7922589 1.0121334 0.5388115 0.6904218 0.5453684 0.8220740
##
## [[247]]
## [1] 0.5947887 1.1214091 0.5878285 0.5344569 0.8493902 0.4338324 0.6055280
##
## [[248]]
## [1] 0.2990791 0.5671159 0.5041562 0.5443706 0.6441040 0.5306867 0.5488513
## [8] 0.2922441
##
## [[249]]
## [1] 0.8457130 0.4372582 0.3634999 0.6921918 0.4840912 0.6027918 0.5306867
## [8] 0.5028630
##
## [[250]]
## [1] 0.4996058 0.7023854 0.6385329 0.4568251 0.5488513 0.5028630
##
## [[251]]
## [1] 1.3218132 1.1098669 0.9875742 2.6704319 0.6242597
##
## [[252]]
## [1] 1.8549252 1.2334553 0.7389841 0.8877237 1.4065933
##
## [[253]]
## [1] 2.873144 2.216894 1.372911 1.948133 2.561208
##
## [[254]]
## [1] 0.5702452 0.7443998 0.9697024 0.7434382 0.5835158
##
## [[255]]
## [1] 0.7230575 0.5967864 0.4077919 0.6806124 0.7957456 0.7478303
##
## [[256]]
## [1] 1.1247852 0.6608199 0.4497935 0.7192815 0.7086318 0.8670278 1.2445951
##
## [[257]]
## [1] 1.6215842 0.4898260 0.9756103 0.6285524 0.7735387
##
## [[258]]
## [1] 0.6004264 0.7647919 1.7619035 0.7358156 1.6947872
##
## [[259]]
## [1] 1.8066475 0.6991945 1.2938641 0.8145211 0.7735387 1.6947872
##
## [[260]]
## [1] 0.5712238 0.4251808 0.3073443 0.5670518 0.4660007 0.4247892 0.4911626
## [8] 0.3887759 0.2087434 0.3542026 0.5275972
##
## [[261]]
## [1] 0.3218916 0.2975364 0.8291476 0.5049576 0.1542575
##
## [[262]]
## [1] 0.8145234 0.8182763 0.6477426 0.4385275
##
## [[263]]
## [1] 1.540578 1.318528 1.094431 1.710666 1.966668
##
## [[264]]
## [1] 1.346882 1.060023 1.686874 1.587421 1.966668
##
## [[265]]
## [1] 1.0309079 1.1968400 1.0402323 0.7090568 0.5871453 0.5266164 1.0746367
##
## [[266]]
## [1] 0.6021557 1.1480015 0.1645856 0.6123236 0.6332063 1.0746367 0.4485643
##
## [[267]]
## [1] 0.7172742 0.8437940 0.7987768 1.6226101 0.7671860 1.1476841
##
## [[268]]
## [1] 0.9562401 0.5206418 0.9217925 0.9121866 0.6668643 0.7671860
##
## [[269]]
## [1] 1.1898496 0.5586217 0.9741141 1.2375038 0.7500858 0.7815656 0.8631963
##
## [[270]]
## [1] 0.9588370 0.6052865 1.0333706 0.5897564 0.8628155
##
## [[271]]
## [1] 0.3581644 0.3598882 0.8084843 0.5520072 0.4484357
##
## [[272]]
## [1] 0.4299867 0.3175127 0.8291476 0.8084843 0.5240704
##
## [[273]]
## [1] 0.5049576 0.5520072 0.5240704 0.2929905 0.1134900 0.1213000
##
## [[274]]
## [1] 0.4484357 0.2929905 0.1850478
##
## [[275]]
## [1] 0.1134900 0.1850478
##
## [[276]]
## [1] 0.3744038 0.8920002 0.3022505 0.4382571 0.7045412 1.2109539 0.6055280
##
## [[277]]
## [1] 0.1596139 0.4809803 0.5612803 0.4319578 0.5463106 0.4248008 0.4915513
## [8] 0.4054007
##
## [[278]]
## [1] 0.5691983 0.6630384 1.1052748 0.4718265 0.5543265 0.7858094 0.8038132
##
## [[279]]
## [1] 1.0969734 0.5743472 0.8207609 0.5751480 1.3972628 0.5599429
##
## [[280]]
## [1] 1.5437228 0.8005541 0.6428618 0.6438742 1.0567880 0.8220740 0.7157434
##
## [[281]]
## [1] 0.2470689 0.6897698 0.5283589 0.7535825 0.4485643
##
## [[282]]
## [1] 3.113369 1.827245 2.907615 1.250852 1.256131 1.431108 1.274913
##
## [[283]]
## [1] 0.5051229 0.5317908 0.4832962 1.2202489 0.5900350
##
## [[284]]
## [1] 0.8492181 0.5945566 0.5528436 0.3887759 0.2854144 0.5447089
##
## [[285]]
## [1] 0.9961068 0.5674396 1.5196015 0.7359705 1.0994479 0.7478303
##
## [[286]]
## [1] 0.1549636 0.1555487 0.1573922 0.1589236 0.1363204 0.1834646 0.1192066
## [8] 0.1884797 0.2087434 0.1542575 0.1213000 0.2854144 0.1471734 0.2384254
##
## [[287]]
## [1] 0.4524644 0.7285834 0.5374064 0.4915513 0.8148601 0.5866934 1.0408171
##
## [[288]]
## [1] 0.4968966 0.4951431 0.8571719 0.5402783 0.6705902 0.8042739 0.6335440
##
## [[289]]
## [1] 1.0396107 1.2635250 0.8148601 0.6916850 1.0034162
##
## [[290]]
## [1] 0.4649258 0.6137348 0.8494379 0.5866934 0.6335440 0.6916850
##
## [[291]]
## [1] 0.6690701 0.5409671 0.6394194 0.4054007 1.0408171 1.0034162
##
## [[292]]
## [1] 1.0316895 0.5457509 0.9728857 1.0063526 1.0771904 1.2898274 0.9030380
## [8] 0.7157434
##
## [[293]]
## [1] 0.3518170 0.2714662 0.3491247 0.2455807 0.2256625 0.4758056 0.2922441
##
## [[294]]
## [1] 1.0815070 0.5992606 0.7572748 0.1471734
##
## [[295]]
## [1] 0.7053242 0.6527241 0.4993618 1.2130854 1.1116498 1.1896508
##
## [[296]]
## [1] 0.7185641 0.8390431 0.6931957 0.6297705 1.2130854 0.7434640
##
## [[297]]
## [1] 0.5882482 0.4085719 0.5478539 0.4552457 1.1116498 0.7434640
##
## [[298]]
## [1] 0.8303788 1.1553013 0.7911798 0.6941495 1.1896508
##
## [[299]]
## [1] 0.7195327 0.7936921 0.9976961 1.0967103 1.1267590
##
## [[300]]
## [1] 0.4798430 0.6766562 0.9976961 0.6996838
##
## [[301]]
## [1] 0.9792542 0.7144880 0.4117553 1.0967103 0.6996838 0.6276027
##
## [[302]]
## [1] 0.6773042 0.4851777 1.2016692 0.9144950 0.5525513 0.7850920 1.1267590
## [8] 0.6276027
##
## [[303]]
## [1] 0.4655908 1.1504476 1.0939244 0.9326988 0.3542026 1.1476841 0.9467308
##
## [[304]]
## [1] 0.8727465 0.7970978 0.6865580 0.5275972 0.9467308
##
## [[305]]
## [1] 0.5505385 1.0636402 0.6752555 0.5864134 0.5447089 0.2384254Row-standardised weights matrix of queens
rswm_q <- nb2listw(wm_q_in, style="W", zero.policy = TRUE)
rswm_q## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1848
## Percentage nonzero weights: 1.986563
## Average number of links: 6.059016
##
## Weights style: W
## Weights constants summary:
## n nn S0 S1 S2
## W 305 93025 305 105.9392 1253.617Measure of Global Spatial Autocorrelation
Computing Moran’s I
Since pvalue = 0.6067 > 0.05 (assumed level of significance), do not reject null hypothesis.
Hence, we conclude that the spatial distribution is randomly distributed
lm.morantest(regfit, listw=rswm_q, zero.policy = TRUE)##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = tap_in ~ pop)
## weights: rswm_q
##
## Moran I statistic standard deviate = -0.27068, p-value = 0.6067
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## -0.013381618 -0.004364448 0.001109728GEOSPATIAL ANALYSIS - FOR TAP OUT
As mentioned above, the data set of tap in and tap out are similar, but the analysis will be run again to confirm the conclusion.
View(subzone) View(tapout)
Join both subzone and tapout
# Rename subzone_n
colnames(tapout)[1] = "SUBZONE_N"
# uppercase all observations in SUBZONE_N
tapout$SUBZONE_N = toupper(tapout$SUBZONE_N)
# change character to attribute
subzone$SUBZONE_N = as.character(subzone$SUBZONE_N)
# join both data set
subzone_tapout <- left_join(tapout,subzone)## Joining, by = "SUBZONE_N"# omit NA values
subzone_tapout <- na.omit(subzone_tapout)
# select variables
subzone_tapout <- dplyr::select(subzone_tapout,SUBZONE_N,Pop, TOTAL_TAP_OUT_VOLUME,X_ADDR,Y_ADDR,geometry)View(subzone_tapout)
Change to sf
subzone_tapout = st_as_sf(subzone_tapout)Plot qtm graph to analysis different regions
The tap in volume graph shows a high tap in volume of 3min to 4min at the east part of the map, which is supported by the population graph, which shows the highest population at the same part of the map
st_make_valid(subzone_tapout)## Simple feature collection with 305 features and 5 fields
## geometry type: GEOMETRY
## dimension: XY
## bbox: xmin: 103.6057 ymin: 1.210655 xmax: 104.0336 ymax: 1.470775
## geographic CRS: WGS 84
## # A tibble: 305 x 6
## SUBZONE_N Pop TOTAL_TAP_OUT_VO~ X_ADDR Y_ADDR geometry
## <chr> <dbl> <dbl> <dbl> <dbl> <MULTIPOLYGON [°]>
## 1 ADMIRALTY 14110 228889 27091. 48334. (((103.8285 1.458775, 103.8~
## 2 AIRPORT R~ 0 24524 35133. 37075. (((103.9014 1.356171, 103.9~
## 3 ALEXANDRA~ 13780 603813 25359. 29991. (((103.8144 1.285474, 103.8~
## 4 ALEXANDRA~ 2120 41292 26548. 30519. (((103.8174 1.294306, 103.8~
## 5 ALJUNIED 40190 1704099 33593. 32971. (((103.8913 1.321318, 103.8~
## 6 ANAK BUKIT 22250 745657 21127. 35562. (((103.771 1.347908, 103.77~
## 7 ANCHORVALE 46610 474260 34219. 41815. (((103.896 1.399924, 103.89~
## 8 ANG MO KI~ 4890 1052820 29502. 39419. (((103.8485 1.368786, 103.8~
## 9 ANSON 0 59522 29145. 28467. (((103.8441 1.274875, 103.8~
## 10 BALESTIER 32760 1065839 29975. 34194. (((103.8623 1.329565, 103.8~
## # ... with 295 more rowspar(mfrow=c(1,2))
qtm(subzone_tapout, "TOTAL_TAP_OUT_VOLUME")## Warning: The shape subzone_tapout is invalid. See sf::st_is_valid
qtm(subzone_tapout, "Pop")## Warning: The shape subzone_tapout is invalid. See sf::st_is_valid
Creating (QUEEN) contiguity based neighbours
The summary report above shows that there are 305 area units in subzone tap out.
The most connected area unit has 17 neighbours.
There are no area units with only one neighbour.
wm_q_out <- poly2nb(subzone_tapout, queen=TRUE)
summary(wm_q_out)## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1848
## Percentage nonzero weights: 1.986563
## Average number of links: 6.059016
## Link number distribution:
##
## 2 3 4 5 6 7 8 9 10 11 12 14 17
## 6 11 27 79 74 51 36 14 2 2 1 1 1
## 6 least connected regions:
## 2 41 132 170 228 275 with 2 links
## 1 most connected region:
## 39 with 17 linksCreating (ROOK) contiguity based neighbours
The summary report above shows that there are 305 area units in subzone tap out.
The most connected area unit has 14 neighbours.
There is 1 area unit with only one neighbour.
wm_r_out <- poly2nb(subzone_tapout, queen=FALSE)
summary(wm_r_out)## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1596
## Percentage nonzero weights: 1.715668
## Average number of links: 5.232787
## Link number distribution:
##
## 1 2 3 4 5 6 7 8 9 10 13 14
## 1 5 24 70 91 62 29 15 4 2 1 1
## 1 least connected region:
## 170 with 1 link
## 1 most connected region:
## 39 with 14 linksPlotting both Queen and Rook based on neighbour map
Queen contiguity is preferred since it has more edges than Rook’s, as shown by more red lines connecting to the points. Queen will hence give a more accurate reading.
centroids <- sf::st_centroid(subzone_tapout$geometry)## Warning in st_centroid.sfc(subzone_tapout$geometry): st_centroid does not give
## correct centroids for longitude/latitude dataplot(subzone_tapout$geometry, border="lightgrey")
plot(wm_q_out, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red", main="Queen Contiguity")
plot(subzone_tapout$geometry, border="lightgrey")
plot(wm_r_out, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red", main="Rook Contiguity")
Computing distance based neighbours
The summary shows that the largest first nearest neighbour distance is 5.4040km, so using this as an upper threshold gives certainty that all units will have at least one neighbour
coords_out <-st_coordinates(centroids)
k1_out <- knn2nb(knearneigh(coords_out))
k1dists_out <- unlist(nbdists(k1_out, coords_out, longlat=TRUE))
summary(k1dists_out)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1825 0.6167 0.8917 0.9414 1.1699 5.4040Computing fixed distance weight matrix
wm_d_out <- dnearneigh(coords_out,0,6,longlat=TRUE)
wm_d_out## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 19302
## Percentage nonzero weights: 20.74926
## Average number of links: 63.28525Row-standardised weights matrix of queens
rswm_q_out <- nb2listw(wm_q_out, style="W", zero.policy = TRUE)
rswm_q_out## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1848
## Percentage nonzero weights: 1.986563
## Average number of links: 6.059016
##
## Weights style: W
## Weights constants summary:
## n nn S0 S1 S2
## W 305 93025 305 105.9392 1253.617To display weight matrix
str(wm_d_in)## List of 305
## $ : int [1:29] 85 123 139 145 146 147 158 166 168 169 ...
## $ : int [1:70] 5 7 10 14 15 18 20 21 23 25 ...
## $ : int [1:96] 4 9 12 17 22 26 28 30 35 36 ...
## $ : int [1:103] 3 9 10 12 17 18 22 25 26 28 ...
## $ : int [1:92] 2 10 12 15 17 18 20 21 22 23 ...
## $ : int [1:61] 11 29 31 32 33 34 39 48 54 55 ...
## $ : int [1:46] 2 8 44 50 60 64 75 96 97 98 ...
## $ : int [1:59] 7 10 18 20 21 25 27 39 44 50 ...
## $ : int [1:85] 3 4 10 12 17 18 22 23 26 28 ...
## $ : int [1:115] 2 4 5 8 9 12 17 18 20 21 ...
## $ : int [1:37] 6 29 31 32 33 34 39 48 49 63 ...
## $ : int [1:94] 3 4 5 9 10 17 18 22 23 25 ...
## $ : int [1:18] 14 15 16 69 76 78 81 109 118 121 ...
## $ : int [1:33] 2 13 15 16 64 69 76 78 81 109 ...
## $ : int [1:46] 2 5 13 14 16 20 64 69 76 78 ...
## $ : int [1:22] 13 14 15 69 76 78 81 109 116 118 ...
## $ : int [1:110] 3 4 5 9 10 12 18 20 21 22 ...
## $ : int [1:113] 2 4 5 8 9 10 12 17 20 21 ...
## $ : int [1:25] 24 45 87 88 94 103 106 107 108 124 ...
## $ : int [1:92] 2 5 8 10 15 17 18 21 23 25 ...
## $ : int [1:88] 2 5 8 10 17 18 20 23 25 26 ...
## $ : int [1:99] 3 4 5 9 10 12 17 18 23 25 ...
## $ : int [1:102] 2 5 9 10 12 17 18 20 21 22 ...
## $ : int [1:42] 19 29 31 32 33 34 45 48 71 83 ...
## $ : int [1:106] 2 4 5 8 10 12 17 18 20 21 ...
## $ : int [1:115] 3 4 5 9 10 12 17 18 21 22 ...
## $ : int [1:97] 2 5 8 10 17 18 20 21 22 23 ...
## $ : int [1:107] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:53] 6 11 24 31 32 33 34 45 48 49 ...
## $ : int [1:108] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:57] 6 11 24 29 32 33 34 45 48 49 ...
## $ : int [1:58] 6 11 24 29 31 33 34 39 45 48 ...
## $ : int [1:60] 6 11 24 29 31 32 34 45 48 49 ...
## $ : int [1:56] 6 11 24 29 31 32 33 45 48 49 ...
## $ : int [1:101] 3 4 9 10 12 17 18 22 23 25 ...
## $ : int [1:94] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:87] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:83] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:39] 6 8 11 32 44 61 63 68 72 79 ...
## $ : int [1:9] 41 42 76 140 141 180 236 248 293
## $ : int [1:12] 40 42 76 140 141 179 180 181 182 183 ...
## $ : int [1:16] 40 41 76 140 141 179 180 181 182 183 ...
## $ : int [1:109] 3 4 9 10 12 17 18 22 23 25 ...
## $ : int [1:65] 2 7 8 10 18 20 21 25 27 39 ...
## $ : int [1:41] 19 24 29 31 32 33 34 71 86 87 ...
## $ : int [1:97] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:93] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:40] 6 11 24 29 31 32 33 34 49 63 ...
## $ : int [1:35] 11 29 31 32 33 34 48 63 72 79 ...
## $ : int [1:67] 2 7 8 10 18 20 21 23 25 27 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 20 22 ...
## $ : int [1:77] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:54] 3 6 29 31 32 33 34 55 56 57 ...
## $ : int [1:57] 3 6 29 31 32 33 34 54 56 57 ...
## $ : int [1:52] 3 6 31 32 33 34 54 55 57 59 ...
## $ : int [1:54] 3 4 6 31 32 33 34 54 55 56 ...
## $ : int [1:95] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:89] 3 4 6 9 17 22 26 28 35 36 ...
## $ : int [1:53] 2 7 8 20 21 44 50 64 75 96 ...
## $ : int [1:83] 3 4 6 10 25 26 27 32 33 35 ...
## $ : int [1:103] 4 5 9 10 12 17 18 20 21 22 ...
## $ : int [1:43] 6 11 29 31 32 33 34 39 48 49 ...
## $ : int [1:71] 2 5 7 8 10 14 15 18 20 21 ...
## $ : int [1:92] 3 4 9 12 17 22 26 28 30 35 ...
## $ : int [1:110] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:63] 3 4 6 26 31 32 33 35 36 43 ...
## $ : int [1:113] 3 4 6 8 10 17 18 21 22 23 ...
## $ : int [1:38] 2 5 12 13 14 15 16 18 20 23 ...
## $ : int [1:90] 3 4 9 10 12 17 18 22 23 26 ...
## $ : int [1:56] 6 24 29 31 32 33 34 45 54 55 ...
## $ : int [1:36] 6 11 29 31 32 33 34 39 48 49 ...
## $ : int [1:97] 3 4 6 10 17 22 25 26 27 28 ...
## $ : int [1:111] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:54] 2 7 8 20 21 44 50 60 64 96 ...
## $ : int [1:21] 13 14 15 16 40 41 42 140 141 179 ...
## $ : int [1:103] 3 4 5 9 10 12 17 18 22 23 ...
## $ : int [1:35] 2 5 13 14 15 16 20 23 64 69 ...
## $ : int [1:36] 11 29 31 32 39 48 49 63 72 83 ...
## $ : int [1:105] 2 5 10 12 17 18 20 21 22 23 ...
## $ : int [1:66] 2 5 10 12 13 14 15 16 17 18 ...
## $ : int [1:73] 3 4 6 26 31 32 33 35 36 43 ...
## $ : int [1:44] 6 11 24 29 31 32 33 34 39 48 ...
## $ : int [1:119] 3 4 5 9 10 12 17 18 20 21 ...
## $ : int [1:29] 1 79 123 127 145 146 147 158 166 168 ...
## $ : int [1:54] 6 11 24 29 31 32 33 34 39 45 ...
## $ : int [1:19] 19 45 88 103 106 108 124 134 194 208 ...
## $ : int [1:20] 19 24 45 87 103 106 108 124 134 194 ...
## $ : int [1:99] 3 4 9 10 12 17 18 22 26 28 ...
## $ : int [1:83] 3 4 6 10 17 21 25 26 27 32 ...
## $ : int [1:54] 6 11 24 29 31 32 33 34 39 48 ...
## $ : int [1:79] 3 4 6 26 33 35 36 43 53 54 ...
## $ : int [1:68] 3 4 6 11 26 29 31 32 33 34 ...
## $ : int [1:48] 6 19 24 29 31 32 33 34 45 48 ...
## $ : int [1:48] 6 11 24 29 31 32 33 34 45 48 ...
## $ : int [1:68] 2 7 8 15 20 21 25 27 44 50 ...
## $ : int [1:57] 2 7 8 15 20 21 44 50 60 64 ...
## $ : int [1:65] 2 7 8 15 20 21 25 27 44 50 ...
## $ : int [1:107] 3 4 5 9 10 12 17 18 22 23 ...
## [list output truncated]
## - attr(*, "class")= chr "nb"
## - attr(*, "nbtype")= chr "distance"
## - attr(*, "distances")= num [1:2] 0 6
## - attr(*, "region.id")= chr [1:305] "1" "2" "3" "4" ...
## - attr(*, "call")= language dnearneigh(x = coords, d1 = 0, d2 = 6, longlat = TRUE)
## - attr(*, "dnn")= num [1:2] 0 6
## - attr(*, "bounds")= chr [1:2] "GT" "LE"
## - attr(*, "sym")= logi TRUEPlotting fixed distance weight matrix
The red lines show the links of 1st nearest neighbours and the black lines shows the links of neighbours within cut-off distance of 6km.
plot(subzone_tapout$geometry, border="lightgrey")
plot(wm_d_out, coords_out, add=TRUE)
plot(k1_out, coords_out, add=TRUE, col="red", length=0.08)
Computing adaptive distance weight matrix
wm_knn_out <- knn2nb(knearneigh(coords_out, k=6))
wm_knn_out## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1830
## Percentage nonzero weights: 1.967213
## Average number of links: 6
## Non-symmetric neighbours listPlotting distance based neighbours
plot(subzone_tapout$geometry, border="lightgrey")
plot(wm_knn_out, coords_out, pch=19, cex=0.6, add=TRUE, col="red")
Combining fixed and adaptive
par(mfrow=c(1,2))
plot(subzone_tapout$geometry, border="lightgrey")
plot(wm_d_out, coords_out, add=TRUE)
plot(k1_out, coords_out, add=TRUE, col="red", length=0.08)
plot(subzone_tapout$geometry, border="lightgrey")
plot(wm_knn_out, coords_out, pch=19, cex=0.6, add=TRUE, col="red")
From the two plots, we can conclude that adaptive weight matrix should be used instead of fixed weight matrix, as there are several missing values from the fixed weight matrix.
Adaptive weight matrix will give a more accurate result.
Distance based neighbours of adaptive weight matrix shows a wider spread.
Weights based on IDW
To derive spatial weight matrix based on Inversed Distance
Compute the distances between areas
dist_out <- nbdists(wm_q_out, coords_out, longlat=TRUE)
ids_out <- lapply(dist_out, function(x)1/(x))
ids_out## [[1]]
## [1] 0.8844741 1.0480106 0.8884195 0.7273930 0.7194021 0.8145234
##
## [[2]]
## [1] 5.4804873 0.5914406
##
## [[3]]
## [1] 0.7684313 0.8535230 1.6794906 0.5228568 0.9209379 1.0335332 0.8362314
## [8] 1.1858760
##
## [[4]]
## [1] 0.7684313 0.8561584 1.5861382 1.5851576 0.6885791 1.6141956
##
## [[5]]
## [1] 0.7967051 0.5944818 0.5997141 0.6629291 0.6203724 0.4515680 0.6945245
## [8] 0.6050715 0.5702452
##
## [[6]]
## [1] 0.6512450 0.4671241 0.5523379 0.5381036 0.7563190 0.4293571
##
## [[7]]
## [1] 0.9454551 0.7069300 0.7923350 1.0038999
##
## [[8]]
## [1] 1.2338839 0.7348002 1.2417769 0.9286924 1.1898496 0.7053242 0.7185641
## [8] 0.8303788
##
## [[9]]
## [1] 0.9959641 1.4465246 0.8333802 1.2148883 2.8731441
##
## [[10]]
## [1] 0.7966416 0.7644062 0.9104840 0.5199711 0.7906893 0.5484124 1.0309079
##
## [[11]]
## [1] 0.3340457 1.0083355 1.0893203 1.1046888
##
## [[12]]
## [1] 0.9203617 2.0296518 1.0517067 1.1788886 0.7483162
##
## [[13]]
## [1] 1.0456883 0.3182148 0.3952152 0.3518170
##
## [[14]]
## [1] 0.6605051 0.7274535 0.4081463 0.3335142 0.4247973 0.4347253 0.4996058
## [8] 0.2714662
##
## [[15]]
## [1] 0.6605051 0.5181626 0.4845160 0.5729661 0.3245971 0.7023854
##
## [[16]]
## [1] 1.0456883 0.7274535 0.3591743 0.4215225 0.3491247
##
## [[17]]
## [1] 1.494798 1.310005 1.254374 1.634446 5.480304 1.374106 3.113369
##
## [[18]]
## [1] 0.7966416 1.0681731 1.0154484 0.6284209 0.8855030 0.7727805 0.7862932
## [8] 0.5582583
##
## [[19]]
## [1] 0.4808461 0.5021124 0.8552760 0.4146835 0.4872360 0.8520672 0.3744038
##
## [[20]]
## [1] 1.0407714 0.6501177 1.1361333 1.5437228 1.0316895
##
## [[21]]
## [1] 0.9133713 0.6159350 0.9185460 0.8728618 0.9670262 0.4796687 0.6021557
## [8] 0.5586217 0.5457509
##
## [[22]]
## [1] 2.786039 1.267237 2.117575 1.005309 4.246211 1.961564
##
## [[23]]
## [1] 0.7967051 1.2804026 1.3084566 0.7795739 1.2994920
##
## [[24]]
## [1] 0.8162941 0.8398473 0.8406951 0.4617172 0.5947887 0.8492181 0.5505385
##
## [[25]]
## [1] 1.435667 1.373967 1.301129 1.553840 1.196840
##
## [[26]]
## [1] 1.0772240 1.1926237 0.7632259 1.3811674 1.7409411 2.1360097 1.1732691
## [8] 1.3218132
##
## [[27]]
## [1] 0.9133713 1.4356674 1.1960461 1.0402323 1.1480015
##
## [[28]]
## [1] 1.494798 1.688457 1.806317 2.215308 1.827245
##
## [[29]]
## [1] 0.8033619 1.1060481 0.7749737 0.9886477 0.5712238
##
## [[30]]
## [1] 1.238054 1.127686 1.875910 1.768634 2.907615
##
## [[31]]
## [1] 0.8033619 1.2456534 0.8677066 1.0995651 1.1008159
##
## [[32]]
## [1] 1.2456534 1.1139321 1.2296401 0.9886387
##
## [[33]]
## [1] 0.6512450 0.8677066 1.1139321 0.5251549 0.5749132 0.4975997 0.7172742
## [8] 0.9562401 0.4655908
##
## [[34]]
## [1] 1.1060481 1.0995651 0.5251549 0.4251808 0.8437940 1.1504476 0.8727465
##
## [[35]]
## [1] 0.8561584 0.8849391 1.0580419 0.9130916 1.3190915 1.5405780 1.3468824
##
## [[36]]
## [1] 0.8535230 0.8953934 1.4431053 0.9473712 1.5587781 1.8066475 1.0600227
##
## [[37]]
## [1] 1.945150 1.681763 2.867192 1.639472 2.216894
##
## [[38]]
## [1] 0.9959641 0.9203617 1.9451504 0.4801978 0.8909323 0.6389542 1.1068127
## [8] 1.1632581 1.3729115
##
## [[39]]
## [1] 0.3340457 0.2968846 0.2888334 0.2008093 0.2052084 0.1755475 0.2010533
## [8] 0.1768133 0.2737723 0.2683509 0.3031680 0.2675447 0.2437008 0.3102618
## [15] 0.1645856 0.1596139 0.2470689
##
## [[40]]
## [1] 0.1928125 0.2883690 0.2455807
##
## [[41]]
## [1] 0.1928125 0.4024330
##
## [[42]]
## [1] 0.2883690 0.4024330 0.6129275 1.9940343 0.8012110 0.2990791 0.2256625
##
## [[43]]
## [1] 1.5861382 0.8849391 0.8398056 0.5903987 1.4993784 0.9754874 0.6645181
## [8] 0.6068086 1.1098669
##
## [[44]]
## [1] 1.2338839 1.0523191 0.5523030 0.5613916 0.8274346 0.9741141 0.8390431
##
## [[45]]
## [1] 0.8162941 0.5354624 0.7349362 0.5731777 1.1214091 0.8920002
##
## [[46]]
## [1] 2.786039 1.655131 1.585099 1.188247 4.539164 2.151226
##
## [[47]]
## [1] 1.4465246 1.6817633 1.6551312 1.0095825 4.0676512 1.2752529 2.4095598
## [8] 1.0939357 0.9929413 1.9481332
##
## [[48]]
## [1] 1.2058561 0.9429210 0.8453777 1.1247852 0.1549636
##
## [[49]]
## [1] 1.2058561 0.4999200 0.5894127 0.6608199 0.1555487 1.0815070
##
## [[50]]
## [1] 0.7348002 0.6159350 1.0523191 0.5157744 0.6161005 0.8084862 0.6482094
## [8] 1.2375038
##
## [[51]]
## [1] 1.2672374 1.6884572 1.2380540 0.9940303 1.2837980 0.6653816 1.2277930
## [8] 0.7652348 1.3663294 1.0067668 1.2508523
##
## [[52]]
## [1] 0.8333802 0.4801978 0.9365332 0.5600298 0.3833439 0.5271297 0.3948815
## [8] 0.6004264
##
## [[53]]
## [1] 2.1175745 1.5850988 0.9940303 1.9383979 1.3815174 1.1669983
##
## [[54]]
## [1] 1.2679553 0.9472898 0.9463619 0.7722828 0.9961068
##
## [[55]]
## [1] 1.2679553 0.6445698 0.8307125 0.7361825 1.0745144 0.5206418
##
## [[56]]
## [1] 0.9472898 1.6527189 0.8805075 0.3870804 0.3140803 0.5674396
##
## [[57]]
## [1] 0.9463619 0.6445698 1.6527189 0.7989980 0.7772218 0.4282428 0.7856067
##
## [[58]]
## [1] 2.0296518 0.8909323 1.2837980 1.1211354 2.1805833
##
## [[59]]
## [1] 1.4706887 0.7308158 0.7947949 0.7562514 1.8549252
##
## [[60]]
## [1] 0.9454551 0.5320455 0.7959180 0.7495172 0.5882395 0.8816489 0.9588370
##
## [[61]]
## [1] 0.9569594 0.8914390 0.5038010 0.9232634 0.5109726 0.5691983
##
## [[62]]
## [1] 1.1276855 0.6653816 0.9056074 2.0375888 0.8137809 0.9199540 1.2646240
## [8] 0.7443998
##
## [[63]]
## [1] 1.0083355 0.2968846 0.6160743 0.5959538 0.9512227 0.5635195
##
## [[64]]
## [1] 5.4804873 0.4252456 0.4092100 0.4999176 0.8560016 0.4577477 0.3484185
## [8] 0.5391147 0.7922589
##
## [[65]]
## [1] 1.6794906 0.8953934 0.5532178 0.7800484 1.6215842 0.6991945
##
## [[66]]
## [1] 1.310005 1.806317 1.848831 1.382043 1.810943 1.620788 1.357020
##
## [[67]]
## [1] 0.7989980 0.8487075 1.0756715 0.7005317 1.1515548
##
## [[68]]
## [1] 0.6551446 0.6323499 0.7688724 0.6923502 0.6630384
##
## [[69]]
## [1] 0.3576763 2.7288438 0.5130455 0.4573773
##
## [[70]]
## [1] 1.2148883 1.0095825 0.9365332 0.9376335 1.7858516 0.7647919
##
## [[71]]
## [1] 0.7722828 0.8307125 0.9182911 0.9217925 1.5196015
##
## [[72]]
## [1] 1.0893203 0.2888334 0.5223506 0.8925147 1.7640783
##
## [[73]]
## [1] 0.9569594 1.0927183 0.6092194 1.1052748
##
## [[74]]
## [1] 1.254374 1.847188 1.246549 2.947320 1.905523 1.215226
##
## [[75]]
## [1] 0.7069300 0.5320455 0.4114400 0.6135504 0.4537765 0.5824593 0.6725423
## [8] 0.6052865
##
## [[76]]
## [1] 0.6129275 0.8043294 0.4952155 0.7646293 0.5671159
##
## [[77]]
## [1] 2.2153080 1.2277930 1.9383979 1.8488310 1.0542144 1.4284612 0.9593715
##
## [[78]]
## [1] 0.3182148 0.4081463 0.3591743 0.4050295 0.4590652 0.8120932 0.5132071
## [8] 0.8547808
##
## [[79]]
## [1] 0.2008093 0.4999200 0.5223506 0.3195949 0.4659314 0.6380523 0.6422838
## [8] 0.4497935 0.4809803 0.5992606
##
## [[80]]
## [1] 0.5944818 1.0681731 1.2804026 1.0097213 1.2503065 0.7338037 0.7472477
##
## [[81]]
## [1] 0.5997141 0.4050295 0.4150285 0.7066867 0.8635013 0.4740998 0.5368732
## [8] 0.5367962
##
## [[82]]
## [1] 0.8487075 1.1616463 0.6850966 0.8707461 1.0969734
##
## [[83]]
## [1] 0.6160743 0.6319974 0.5990246 0.7803611 0.8192146 0.7113396 0.5287903
## [8] 0.7192815 0.3073443
##
## [[84]]
## [1] 1.0772240 0.9472528 0.9530090 0.9036667 1.8975126 1.6983369
##
## [[85]]
## [1] 0.9052538 1.6882874 0.7239262 0.4968966 1.0396107 0.6690701
##
## [[86]]
## [1] 0.7749737 1.1008159 1.2296401 0.6319974 0.8888807 0.9712211
##
## [[87]]
## [1] 0.4808461 0.6798021 0.6710674 0.5866048
##
## [[88]]
## [1] 0.5021124 0.6798021 0.6095744 0.5423227 0.3218916 0.3581644 0.4299867
##
## [[89]]
## [1] 1.5851576 1.0580419 1.4431053 0.9068886 1.7085096 1.6868741
##
## [[90]]
## [1] 0.2052084 0.8914390 0.6551446 0.4331565 0.5114785 0.4718265
##
## [[91]]
## [1] 0.4671241 0.9886387 0.5749132 0.5959538 0.5990246 0.8888807 0.6391110
##
## [[92]]
## [1] 1.4706887 1.1616463 1.0185734 0.8388392 1.2334553 0.5743472
##
## [[93]]
## [1] 0.5523379 0.5038010 0.5434165 0.5945844 0.5624522 0.8207609
##
## [[94]]
## [1] 0.8398473 0.5354624 0.4679993 0.5878285 0.5670518 0.5945566 0.7970978
##
## [[95]]
## [1] 0.9886477 0.7803611 0.9712211 0.7024756 0.4660007
##
## [[96]]
## [1] 1.0135684 1.3229506 1.3302127 0.7080008 0.6583602
##
## [[97]]
## [1] 0.7959180 1.0135684 0.7605919 1.3386343 0.6622192
##
## [[98]]
## [1] 0.7495172 1.3229506 0.7605919 0.7633608 0.4080910 0.7176809 1.0333706
##
## [[99]]
## [1] 1.054214 1.728954 2.081489 3.270822
##
## [[100]]
## [1] 0.9182911 1.0416720 0.5005635 0.7230575 0.7987768 0.9121866 0.7359705
##
## [[101]]
## [1] 1.3820435 0.8777783 2.1551137 2.3549864 1.3134107
##
## [[102]]
## [1] 1.1046888 0.9512227 0.8925147 0.8192146 0.8793560 0.7729020 0.7086318
##
## [[103]]
## [1] 0.8552760 0.6095744 0.4655936 0.3033305 0.9296686 0.2975364 0.3175127
## [8] 0.3022505 0.1573922
##
## [[104]]
## [1] 1.0407714 0.7262922 0.5670105 0.9349039 0.9646251 1.0121334 0.8005541
##
## [[105]]
## [1] 1.0416720 0.8205733 1.6226101 1.0939244
##
## [[106]]
## [1] 0.5742682 0.5943254 0.4113321 0.5452498
##
## [[107]]
## [1] 0.5742682 0.6140380 0.4362903 0.4658673 0.5344569 0.5967864 0.4382571
##
## [[108]]
## [1] 0.8406951 0.7349362 0.9415510 0.5528436 1.0636402
##
## [[109]]
## [1] 0.3335142 0.5181626 0.4252456 0.4150285 0.6292237 0.7308923 0.3675036
## [8] 0.4111101
##
## [[110]]
## [1] 1.015448 1.308457 1.009721 1.162562 1.263443
##
## [[111]]
## [1] 0.6629291 0.7795739 1.2503065 0.7262922 0.9737761 1.1552076
##
## [[112]]
## [1] 0.6203724 1.2994920 0.9056074 1.1625619 0.7706998 0.9697024
##
## [[113]]
## [1] 1.8759095 0.7652348 2.0375888 3.2176501
##
## [[114]]
## [1] 0.7644062 0.6284209 1.8471885 0.8777783 0.9677324 1.1733551 0.8659227
## [8] 1.0628311
##
## [[115]]
## [1] 0.9473712 0.5600298 0.9376335 0.9068886 1.2115791 1.7619035 1.2938641
## [8] 1.3185284 1.5874208
##
## [[116]]
## [1] 0.4515680 0.4092100 0.7066867 0.5670105 0.6292237 0.4577219 0.7756175
## [8] 0.5388115
##
## [[117]]
## [1] 0.4999176 1.3302127 1.3386343 0.4791165 0.5220114 1.5436506 0.4699383
##
## [[118]]
## [1] 0.4590652 0.8635013 1.3106704 0.7257642
##
## [[119]]
## [1] 0.9429210 0.7113396 0.7024756 1.0310969 0.8670278 0.4247892
##
## [[120]]
## [1] 1.2417769 1.0061644 1.6064429 0.7500858 0.6527241 1.1553013
##
## [[121]]
## [1] 0.4247973 0.4845160 0.8120932 0.4740998 0.7308923 0.4577219
##
## [[122]]
## [1] 0.5228568 0.5532178 0.4659881 0.6848407 0.7154689 1.1606192 1.2516079
## [8] 0.4898260
##
## [[123]]
## [1] 0.4560956 0.5015524 0.9737212 0.5147668 0.5251365 0.7195327 0.9792542
## [8] 0.6773042
##
## [[124]]
## [1] 0.4617172 0.5731777 0.4655936 0.9415510 0.5971054 0.7045412 0.6752555
##
## [[125]]
## [1] 0.9104840 0.8855030 1.3739673 1.8447553 1.2660919 0.7090568
##
## [[126]]
## [1] 0.7080008 0.7633608 0.4791165 1.0600947 0.6155955 0.6409793 0.6648319
##
## [[127]]
## [1] 0.3347517 0.3706974 0.7000040 0.5612803 0.4524644 0.5409671
##
## [[128]]
## [1] 0.4679993 0.5005635 0.8205733 0.6140380 0.8493902 0.4077919 0.9326988
## [8] 0.6865580
##
## [[129]]
## [1] 0.7727805 0.8137809 1.2465487 1.2634434 0.7706998 0.9677324 1.3857451
## [8] 1.2956422 1.5903328
##
## [[130]]
## [1] 0.7308158 0.9232634 1.0927183 1.0185734 0.5434165 0.3973167 0.4535570
## [8] 0.5543265 0.5751480
##
## [[131]]
## [1] 1.1926237 0.8398056 1.7289543 1.1439498 1.3789246 2.1873170 1.2376825
## [8] 1.5469512
##
## [[132]]
## [1] 0.3268232 0.1589236
##
## [[133]]
## [1] 1.634446 2.947320 1.385745 1.410838 3.151098 1.256131
##
## [[134]]
## [1] 0.4146835 0.3033305 0.5943254 0.4362903 0.7655235 0.4130353 0.4338324
## [8] 1.2109539
##
## [[135]]
## [1] 0.9185460 1.3011294 1.1960461 0.9251273 2.0740006 0.7783528 0.9728857
##
## [[136]]
## [1] 0.8560016 0.6583602 0.5220114 1.0600947 0.5421828 0.6904218 0.6428618
##
## [[137]]
## [1] 0.6501177 0.8728618 0.5157744 0.9251273 1.0775249 0.6159859 0.6438742
## [8] 1.0063526
##
## [[138]]
## [1] 0.4577477 1.5436506 1.1808305 0.6359548 0.5019490
##
## [[139]]
## [1] 0.4560956 0.4381241 0.4159762 0.3833159 0.3158822 0.5882482 0.4798430
## [8] 0.7144880
##
## [[140]]
## [1] 1.9940343 0.8043294 0.9596644
##
## [[141]]
## [1] 0.8012110 0.4952155 0.9596644 0.6448882 0.5569022
##
## [[142]]
## [1] 1.905523 2.155114 1.173355 2.117224
##
## [[143]]
## [1] 0.6945245 0.5368732 0.9349039 0.9737761 0.7756175 0.6925473 0.5453684
##
## [[144]]
## [1] 0.5199711 0.6323499 0.9472528 0.8631758 0.7656371 0.6757964 0.5871453
##
## [[145]]
## [1] 0.1755475 0.5015524 0.7442450 0.3007840 0.4677231 0.4484203 0.4853337
## [8] 0.3315351 0.3264268 0.4951431 0.4649258 0.4851777
##
## [[146]]
## [1] 0.9737212 0.7442450 1.2016692
##
## [[147]]
## [1] 0.2010533 0.3195949 0.3007840 0.2794967 0.4319578 0.7285834 0.6137348
##
## [[148]]
## [1] 0.9209379 0.6885791 0.5903987 0.7947949 0.3973167 1.4487496 0.6863953
## [8] 1.2854792 0.7389841
##
## [[149]]
## [1] 1.051707 1.366329 1.121135 0.919954 0.707254
##
## [[150]]
## [1] 0.3576763 0.3565229 0.7828311 0.7434382
##
## [[151]]
## [1] 1.1788886 0.6389542 0.7072540 0.7820824
##
## [[152]]
## [1] 2.7288438 0.5132071 1.3106704 0.3565229 0.5498483 0.4522282
##
## [[153]]
## [1] 0.3833439 0.4290775 0.2315013 0.3627578 0.9756103 0.7358156 0.8145211
##
## [[154]]
## [1] 0.9670262 0.6161005 0.5989028 0.6123236 0.7815656 0.6897698
##
## [[155]]
## [1] 0.7923350 0.4114400 0.5517190 1.1535795 0.5293861 0.5392787 0.8134473
## [8] 0.5051229
##
## [[156]]
## [1] 2.867192 4.067651 2.561208
##
## [[157]]
## [1] 1.0335332 0.7562514 1.4487496 0.6895619 1.3707566 0.8877237
##
## [[158]]
## [1] 0.9052538 0.7286530 0.5374064 0.8571719 1.2635250 0.8494379
##
## [[159]]
## [1] 2.3549864 0.8659227 0.9323119 1.9315370 1.2191397
##
## [[160]]
## [1] 0.7906893 0.9530090 1.0628311 0.8631758 0.9323119 1.2216780
##
## [[161]]
## [1] 0.1768133 0.7688724 0.4331565 0.7656371 0.5266164 0.6332063
##
## [[162]]
## [1] 0.6050715 0.5130455 0.5367962 0.7257642 0.7828311 0.5498483 0.5835158
##
## [[163]]
## [1] 0.7632259 0.6923502 0.9036667 0.6757964 1.2700510 0.5433986 0.9875742
## [8] 0.7858094
##
## [[164]]
## [1] 0.7772218 1.0756715 0.4659881 0.6138716 0.9035689
##
## [[165]]
## [1] 0.5381036 0.2737723 0.5635195 0.6391110 0.4653930
##
## [[166]]
## [1] 0.2683509 0.5147668 0.4677231 0.2794967 0.8021035 0.3635262
##
## [[167]]
## [1] 1.897513 1.931537 1.221678
##
## [[168]]
## [1] 1.6882874 0.3347517 0.3440207 0.5632434 0.6394194
##
## [[169]]
## [1] 0.7863828 0.6765798 0.7936921 0.6766562 0.9144950
##
## [[170]]
## [1] 1.0237727 0.5317908
##
## [[171]]
## [1] 0.7005317 0.6850966 0.8388392 0.6848407 0.6895619 0.6138716 0.7246413
## [8] 1.4065933
##
## [[172]]
## [1] 1.381167 1.499378 1.143950 2.013966 2.670432
##
## [[173]]
## [1] 1.740941 1.698337 1.270051
##
## [[174]]
## [1] 1.810943 1.428461 2.081489 1.378925 2.105451
##
## [[175]]
## [1] 0.8805075 0.6080403 0.2383592 0.6806124 1.0994479
##
## [[176]]
## [1] 0.4659314 0.3706974 0.4714593 0.5463106 0.1363204 0.7572748
##
## [[177]]
## [1] 0.3870804 0.4282428 0.7154689 0.9035689 0.5264702 0.8238056
##
## [[178]]
## [1] 1.1606192 0.4290775 0.5264702 0.4718700 0.6285524
##
## [[179]]
## [1] 0.7549454 0.8548739 0.8288461 0.5041562 0.8457130
##
## [[180]]
## [1] 0.7646293 0.6448882 0.7549454 0.7522203 0.5443706 0.4372582
##
## [[181]]
## [1] 0.5569022 0.8548739 0.7522203 0.3250162 0.5951573
##
## [[182]]
## [1] 0.3250162 0.6141856 0.5673076 0.3761027 0.4890482 0.3790200 0.3634999
## [8] 0.4832962
##
## [[183]]
## [1] 0.8288461 0.5951573 0.6141856 0.6921918
##
## [[184]]
## [1] 2.1360097 0.9754874 2.1873170 2.0139658 1.2233765
##
## [[185]]
## [1] 0.5729661 0.3675036 0.5117486 0.7198171 0.4840912 0.6385329
##
## [[186]]
## [1] 0.5673076 0.5117486 0.6083635 0.4240189 0.6027918
##
## [[187]]
## [1] 0.3484185 1.1808305 0.3761027 0.6083635 0.5976273
##
## [[188]]
## [1] 1.0053088 0.9130916 1.1882469 1.2752529 1.3815174 2.5428180 1.4708078
## [8] 1.1895831 1.0944313
##
## [[189]]
## [1] 1.553840 1.844755 2.074001 1.130316 1.077190
##
## [[190]]
## [1] 0.8453777 0.5894127 1.0310969 0.4911626 0.1834646
##
## [[191]]
## [1] 0.4113321 0.4658673 0.6080403 0.7957456
##
## [[192]]
## [1] 2.409560 2.542818 1.346062
##
## [[193]]
## [1] 4.246211 4.539164 3.190791
##
## [[194]]
## [1] 0.6710674 0.5423227 0.3598882
##
## [[195]]
## [1] 0.5914406 0.5391147 0.4111101 0.6359548 0.7198171 0.4240189 0.5976273
##
## [[196]]
## [1] 0.3140803 0.2315013 0.2383592 0.8238056 0.4718700
##
## [[197]]
## [1] 0.5484124 0.7862932 0.7338037 1.2660919 0.7783528 1.1303162 1.1249031
## [8] 1.2898274
##
## [[198]]
## [1] 0.5517190 0.4890482 0.5237262 0.9521497 0.6454933 1.2202489
##
## [[199]]
## [1] 1.1535795 1.0237727 0.5237262 0.4482643 0.5900350
##
## [[200]]
## [1] 0.8362314 0.7800484 1.2516079 1.3707566 0.7246413
##
## [[201]]
## [1] 1.961564 1.639472 1.106813 2.151226 1.093936 1.006767 2.180583 3.190791
##
## [[202]]
## [1] 1.1858760 1.6141956 1.5587781 1.7085096 0.6863953
##
## [[203]]
## [1] 0.7000040 0.3268232 0.4714593 0.4248008 0.1192066
##
## [[204]]
## [1] 0.6645181 0.6092194 0.4535570 1.2854792 0.5433986 0.6242597 0.8038132
##
## [[205]]
## [1] 0.5882395 0.6622192 0.4699383 0.5019490 0.5293861 0.3790200 0.9521497
## [8] 1.0807459
##
## [[206]]
## [1] 1.3190915 0.6068086 1.1669983 0.9593715 3.2708222 1.2376825 1.4708078
##
## [[207]]
## [1] 1.768634 1.264624 3.217650 1.295642 4.857047 1.431108
##
## [[208]]
## [1] 0.9296686 0.5971054 0.1884797 0.5864134
##
## [[209]]
## [1] 0.4872360 0.5452498 0.7655235 0.6898072
##
## [[210]]
## [1] 1.7640783 0.6380523 0.8793560 1.5652699
##
## [[211]]
## [1] 5.480304 1.620788 1.215226 1.410838 2.117224
##
## [[212]]
## [1] 0.6135504 0.5392787 0.4482643 0.5809053 0.3945352
##
## [[213]]
## [1] 0.4537765 0.4381241 0.5809053 0.5272022
##
## [[214]]
## [1] 0.5523030 0.5824593 0.7244121 0.7850972 1.0231996 0.5897564 0.6931957
## [8] 0.4085719
##
## [[215]]
## [1] 0.8844741 0.4484203 0.8353524 0.9941691 0.7224300 0.5402783
##
## [[216]]
## [1] 1.0480106 0.7863828 0.9848955 0.7364404 0.5525513
##
## [[217]]
## [1] 0.3031680 1.0061644 0.8851455 0.7331908 0.5283589 0.7911798
##
## [[218]]
## [1] 0.8884195 0.8353524 0.6392443 1.0130489 0.8182763
##
## [[219]]
## [1] 0.7273930 0.4853337 0.6765798 0.9941691 0.9848955 0.7850920
##
## [[220]]
## [1] 0.7194021 0.7364404 0.6477426
##
## [[221]]
## [1] 1.0038999 0.8816489 0.8134473 0.6454933 1.0807459
##
## [[222]]
## [1] 0.6725423 0.4159762 0.3945352 0.5272022 0.7244121 0.5478539
##
## [[223]]
## [1] 0.6422838 0.5287903 0.7729020 1.5652699 1.2445951
##
## [[224]]
## [1] 0.5582583 1.1361333 0.7472477 0.9646251 1.1552076 0.6925473 1.1249031
## [8] 0.9030380
##
## [[225]]
## [1] 0.3440207 0.6392443 0.8143723 0.6328791 0.4385275
##
## [[226]]
## [1] 0.3315351 0.7224300 1.0130489 0.8143723 0.8089478 0.6705902
##
## [[227]]
## [1] 0.7239262 0.7286530 0.5632434 0.6328791 0.8089478 0.8042739
##
## [[228]]
## [1] 0.5271297 0.3627578
##
## [[229]]
## [1] 0.6155955 0.5421828 1.0775249 0.8035498 1.0567880
##
## [[230]]
## [1] 0.4796687 0.5613916 0.8084862 0.4080910 0.6409793 0.6159859 0.8035498
## [8] 0.6716319 0.7414336
##
## [[231]]
## [1] 0.7176809 0.6648319 0.7850972 0.6716319 1.0406271 0.8628155
##
## [[232]]
## [1] 0.8274346 0.6482094 1.0231996 0.7414336 1.0406271 0.6297705
##
## [[233]]
## [1] 0.9286924 1.6064429 0.5989028 0.8851455 0.8631963 0.7535825
##
## [[234]]
## [1] 0.8520672 0.5866048 0.4130353 0.6898072
##
## [[235]]
## [1] 0.3952152 0.4573773 0.8547808 0.4522282
##
## [[236]]
## [1] 0.4347253 0.3245971 0.4215225 0.6441040 0.4568251 0.4758056
##
## [[237]]
## [1] 0.9929413 1.7858516 1.2115791 1.1895831 1.3460620 1.7106658
##
## [[238]]
## [1] 0.7361825 0.7856067 1.1515548 0.8707461 0.5253496 1.3972628
##
## [[239]]
## [1] 1.173269 1.357020 1.313411 1.546951 1.219140 2.105451 1.223376
##
## [[240]]
## [1] 0.2675447 0.5251365 0.3833159 0.3264268 0.8021035 0.5940072 0.4117553
##
## [[241]]
## [1] 0.7483162 1.1632581 0.3948815 0.7820824
##
## [[242]]
## [1] 1.374106 1.590333 3.151098 4.857047 1.274913
##
## [[243]]
## [1] 0.7563190 0.4975997 1.0745144 0.5945844 0.5253496 0.6668643 0.5599429
##
## [[244]]
## [1] 0.4293571 0.2437008 0.5109726 0.5114785 0.5624522 0.4653930
##
## [[245]]
## [1] 0.3102618 0.3158822 0.3635262 0.7331908 0.5940072 0.4993618 0.4552457
## [8] 0.6941495
##
## [[246]]
## [1] 0.7922589 1.0121334 0.5388115 0.6904218 0.5453684 0.8220740
##
## [[247]]
## [1] 0.5947887 1.1214091 0.5878285 0.5344569 0.8493902 0.4338324 0.6055280
##
## [[248]]
## [1] 0.2990791 0.5671159 0.5041562 0.5443706 0.6441040 0.5306867 0.5488513
## [8] 0.2922441
##
## [[249]]
## [1] 0.8457130 0.4372582 0.3634999 0.6921918 0.4840912 0.6027918 0.5306867
## [8] 0.5028630
##
## [[250]]
## [1] 0.4996058 0.7023854 0.6385329 0.4568251 0.5488513 0.5028630
##
## [[251]]
## [1] 1.3218132 1.1098669 0.9875742 2.6704319 0.6242597
##
## [[252]]
## [1] 1.8549252 1.2334553 0.7389841 0.8877237 1.4065933
##
## [[253]]
## [1] 2.873144 2.216894 1.372911 1.948133 2.561208
##
## [[254]]
## [1] 0.5702452 0.7443998 0.9697024 0.7434382 0.5835158
##
## [[255]]
## [1] 0.7230575 0.5967864 0.4077919 0.6806124 0.7957456 0.7478303
##
## [[256]]
## [1] 1.1247852 0.6608199 0.4497935 0.7192815 0.7086318 0.8670278 1.2445951
##
## [[257]]
## [1] 1.6215842 0.4898260 0.9756103 0.6285524 0.7735387
##
## [[258]]
## [1] 0.6004264 0.7647919 1.7619035 0.7358156 1.6947872
##
## [[259]]
## [1] 1.8066475 0.6991945 1.2938641 0.8145211 0.7735387 1.6947872
##
## [[260]]
## [1] 0.5712238 0.4251808 0.3073443 0.5670518 0.4660007 0.4247892 0.4911626
## [8] 0.3887759 0.2087434 0.3542026 0.5275972
##
## [[261]]
## [1] 0.3218916 0.2975364 0.8291476 0.5049576 0.1542575
##
## [[262]]
## [1] 0.8145234 0.8182763 0.6477426 0.4385275
##
## [[263]]
## [1] 1.540578 1.318528 1.094431 1.710666 1.966668
##
## [[264]]
## [1] 1.346882 1.060023 1.686874 1.587421 1.966668
##
## [[265]]
## [1] 1.0309079 1.1968400 1.0402323 0.7090568 0.5871453 0.5266164 1.0746367
##
## [[266]]
## [1] 0.6021557 1.1480015 0.1645856 0.6123236 0.6332063 1.0746367 0.4485643
##
## [[267]]
## [1] 0.7172742 0.8437940 0.7987768 1.6226101 0.7671860 1.1476841
##
## [[268]]
## [1] 0.9562401 0.5206418 0.9217925 0.9121866 0.6668643 0.7671860
##
## [[269]]
## [1] 1.1898496 0.5586217 0.9741141 1.2375038 0.7500858 0.7815656 0.8631963
##
## [[270]]
## [1] 0.9588370 0.6052865 1.0333706 0.5897564 0.8628155
##
## [[271]]
## [1] 0.3581644 0.3598882 0.8084843 0.5520072 0.4484357
##
## [[272]]
## [1] 0.4299867 0.3175127 0.8291476 0.8084843 0.5240704
##
## [[273]]
## [1] 0.5049576 0.5520072 0.5240704 0.2929905 0.1134900 0.1213000
##
## [[274]]
## [1] 0.4484357 0.2929905 0.1850478
##
## [[275]]
## [1] 0.1134900 0.1850478
##
## [[276]]
## [1] 0.3744038 0.8920002 0.3022505 0.4382571 0.7045412 1.2109539 0.6055280
##
## [[277]]
## [1] 0.1596139 0.4809803 0.5612803 0.4319578 0.5463106 0.4248008 0.4915513
## [8] 0.4054007
##
## [[278]]
## [1] 0.5691983 0.6630384 1.1052748 0.4718265 0.5543265 0.7858094 0.8038132
##
## [[279]]
## [1] 1.0969734 0.5743472 0.8207609 0.5751480 1.3972628 0.5599429
##
## [[280]]
## [1] 1.5437228 0.8005541 0.6428618 0.6438742 1.0567880 0.8220740 0.7157434
##
## [[281]]
## [1] 0.2470689 0.6897698 0.5283589 0.7535825 0.4485643
##
## [[282]]
## [1] 3.113369 1.827245 2.907615 1.250852 1.256131 1.431108 1.274913
##
## [[283]]
## [1] 0.5051229 0.5317908 0.4832962 1.2202489 0.5900350
##
## [[284]]
## [1] 0.8492181 0.5945566 0.5528436 0.3887759 0.2854144 0.5447089
##
## [[285]]
## [1] 0.9961068 0.5674396 1.5196015 0.7359705 1.0994479 0.7478303
##
## [[286]]
## [1] 0.1549636 0.1555487 0.1573922 0.1589236 0.1363204 0.1834646 0.1192066
## [8] 0.1884797 0.2087434 0.1542575 0.1213000 0.2854144 0.1471734 0.2384254
##
## [[287]]
## [1] 0.4524644 0.7285834 0.5374064 0.4915513 0.8148601 0.5866934 1.0408171
##
## [[288]]
## [1] 0.4968966 0.4951431 0.8571719 0.5402783 0.6705902 0.8042739 0.6335440
##
## [[289]]
## [1] 1.0396107 1.2635250 0.8148601 0.6916850 1.0034162
##
## [[290]]
## [1] 0.4649258 0.6137348 0.8494379 0.5866934 0.6335440 0.6916850
##
## [[291]]
## [1] 0.6690701 0.5409671 0.6394194 0.4054007 1.0408171 1.0034162
##
## [[292]]
## [1] 1.0316895 0.5457509 0.9728857 1.0063526 1.0771904 1.2898274 0.9030380
## [8] 0.7157434
##
## [[293]]
## [1] 0.3518170 0.2714662 0.3491247 0.2455807 0.2256625 0.4758056 0.2922441
##
## [[294]]
## [1] 1.0815070 0.5992606 0.7572748 0.1471734
##
## [[295]]
## [1] 0.7053242 0.6527241 0.4993618 1.2130854 1.1116498 1.1896508
##
## [[296]]
## [1] 0.7185641 0.8390431 0.6931957 0.6297705 1.2130854 0.7434640
##
## [[297]]
## [1] 0.5882482 0.4085719 0.5478539 0.4552457 1.1116498 0.7434640
##
## [[298]]
## [1] 0.8303788 1.1553013 0.7911798 0.6941495 1.1896508
##
## [[299]]
## [1] 0.7195327 0.7936921 0.9976961 1.0967103 1.1267590
##
## [[300]]
## [1] 0.4798430 0.6766562 0.9976961 0.6996838
##
## [[301]]
## [1] 0.9792542 0.7144880 0.4117553 1.0967103 0.6996838 0.6276027
##
## [[302]]
## [1] 0.6773042 0.4851777 1.2016692 0.9144950 0.5525513 0.7850920 1.1267590
## [8] 0.6276027
##
## [[303]]
## [1] 0.4655908 1.1504476 1.0939244 0.9326988 0.3542026 1.1476841 0.9467308
##
## [[304]]
## [1] 0.8727465 0.7970978 0.6865580 0.5275972 0.9467308
##
## [[305]]
## [1] 0.5505385 1.0636402 0.6752555 0.5864134 0.5447089 0.2384254rswm_q_out <- nb2listw(wm_q_out, style="W", zero.policy = TRUE)
rswm_q_out## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 1848
## Percentage nonzero weights: 1.986563
## Average number of links: 6.059016
##
## Weights style: W
## Weights constants summary:
## n nn S0 S1 S2
## W 305 93025 305 105.9392 1253.617Measure of Global Spatial Autocorrelation
Computing Moran’s I
Since pvalue = 0.3971 > 0.05 (assumed level of significance), do not reject null hypothesis.
Hence, we conclude that spatial distribution is randomly distributed.
lm.morantest(regfit2, rswm_q_out, alternative="greater", spChk=NULL, zero.policy = TRUE)##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = tap_out ~ popout)
## weights: rswm_q_out
##
## Moran I statistic standard deviate = 0.26085, p-value = 0.3971
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## 0.004325173 -0.004364448 0.001109728GEOVISUALISATION - FOR TAP IN
To communicate the complex spatial statistics results in business friendly visual representations
Perform localise geospatial statistics analysis to identify geographical clustering
Install packages again
packages = c('rgdal', 'spdep', 'tmap', 'tidyverse')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}Computing local Moran’s I
fips <- order(subzone_tapin$Pop)
localMI <- localmoran(subzone_tapin$TOTAL_TAP_IN_VOLUME, rswm_q)
head(localMI)## Ii E.Ii Var.Ii Z.Ii Pr(z > 0)
## 1 0.10398996 -0.003289474 0.1552387 0.2722805 0.39270318
## 2 0.46528109 -0.003289474 0.4716059 0.6823157 0.24751967
## 3 -0.10380168 -0.003289474 0.1156928 -0.2955056 0.61619616
## 4 0.07510718 -0.003289474 0.1552387 0.1989746 0.42114132
## 5 0.45915802 -0.003289474 0.1025108 1.4443673 0.07431785
## 6 -0.27572788 -0.003289474 0.1552387 -0.6914621 0.75536240Mapping the local Moran’s I
subzone_tapin.localMI <- cbind(subzone_tapin, localMI)Mapping local Moran’s I values
tm_shape(subzone_tapin.localMI)+
tm_fill(col="Ii",
style="pretty",
title="local moran tap in statistics")+
tm_borders(alpha=0.5)## Warning: The shape subzone_tapin.localMI is invalid. See sf::st_is_valid## Variable(s) "Ii" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
Mapping local Moran’s I p-values
The choropleth shows there is evidence for both positive and negative values but consider the pvalues too
tm_shape(subzone_tapin.localMI) +
tm_fill(col="Pr.z...0.",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title="local Moran's I tap in p-values") +
tm_borders(alpha=0.5)## Warning: The shape subzone_tapin.localMI is invalid. See sf::st_is_valid
Mapping both local Moran’s I values and p-values for effective interpretation
localMI.map <- tm_shape(subzone_tapin.localMI) +
tm_fill(col="Ii",
style="pretty",
title="local moran tap in statistics")+
tm_borders(alpha=0.5)
pvalue.map <- tm_shape(subzone_tapin.localMI)+
tm_fill(col="Pr.z...0.",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title="local Moran's I tap in p-values")+
tm_borders(alpha=0.5)
tmap_arrange(localMI.map, pvalue.map, asp=1, ncol=2)## Warning: The shape subzone_tapin.localMI is invalid. See sf::st_is_valid## Variable(s) "Ii" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.## Warning: The shape subzone_tapin.localMI is invalid. See sf::st_is_valid
From the two graphs, we can tell that the regions with a higher local Moran tap in statistics and smaller local Moran’s I tap in p-values are the same, which coincides with the same area of high tap in volume and high population, in the qtm graph that have been plotted in the previous part.
The small p values indicate that the cluster identified is statistically significant.
Creating a LISA Cluster Map
nci <- moran.plot(subzone_tapin$TOTAL_TAP_IN_VOLUME, rswm_q, labels=as.character(subzone_tapin$Pop), xlab="TOTAL TAP IN VOLUME", ylab="Spatially Lag Tap in Volume")
Plotting Moran scatterplot with standardised variable
subzone_tapin$z.TOTAL_TAP_IN_VOLUME <- scale(subzone_tapin$TOTAL_TAP_IN_VOLUME) %>% as.vectornci2 <- moran.plot(subzone_tapin$z.TOTAL_TAP_IN_VOLUME, rswm_q, labels=as.character(subzone_tapin$Pop, xlab="z-Total tap in volume", ylab="Spatially lag tap in volume"))
Code chunks to prepare a LISA cluster map
quadrant <- vector(mode="numeric", length=nrow(localMI))Centers the variable of interest around its mean
DV <- subzone_tapin$TOTAL_TAP_IN_VOLUME - mean(subzone_tapin$TOTAL_TAP_IN_VOLUME)Centering local Moran’s around the mean
C_mI <- localMI[,1] - mean(localMI[,1])Set statistical significance level for the local Moran
signif <- 0.05Define high-high, low-low, low-high, high-low
quadrant[DV >0 & C_mI>0] <- 4
quadrant[DV <0 & C_mI<0] <- 1
quadrant[DV <0 & C_mI>0] <- 2
quadrant[DV >0 & C_mI<0] <- 3Non significant Moran in category 0
quadrant[localMI[,5]>signif] <- 0Build LISA map
subzone_tapin.localMI$quadrant <- quadrant
colors <- c("#ffffff", "#2c7bb6", "#abd9e9", "#fdae61", "#d7191c")
clusters <- c("insignificant", "low-low", "low-high", "high-low", "high-high")
tm_shape(subzone_tapin.localMI) +
tm_fill(col = "quadrant", style = "cat", palette = colors[c(sort(unique(quadrant)))+1], labels = clusters[c(sort(unique(quadrant)))+1], popup.vars = c("Postal.Code")) +
tm_view(set.zoom.limits = c(11,17)) +
tm_borders(alpha=0.5)## Warning: The shape subzone_tapin.localMI is invalid. See sf::st_is_valid
There are areas of high-high, but no areas of low-high, majority of the areas are insignificant
We can identify that the main area of high-high is at the east of the map.
Hot Spot and Cold Spot Area Analysis
Computing GI statistics
GI statistics using fixed distance
centroids <- sf::st_centroid(subzone_tapin$geometry)## Warning in st_centroid.sfc(subzone_tapin$geometry): st_centroid does not give
## correct centroids for longitude/latitude datadnb <- dnearneigh(st_coordinates(centroids), 0, 26, longlat=TRUE)
dnb## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 90590
## Percentage nonzero weights: 97.38242
## Average number of links: 297.0164Display the data and neighbours
plot(subzone_tapin$geometry, border='lightgrey')
plot(dnb,st_coordinates(centroids), add=TRUE, col='red')
dnb_lw <- nb2listw(dnb, style='B')
summary(dnb_lw)## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 90590
## Percentage nonzero weights: 97.38242
## Average number of links: 297.0164
## Link number distribution:
##
## 169 211 227 230 240 247 248 251 262 263 266 268 269 270 272 273 276 277 279 280
## 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 3 4 1
## 281 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
## 1 1 1 4 2 1 3 2 4 5 1 1 5 8 4 12 6 6 7 16
## 302 303 304
## 35 53 100
## 1 least connected region:
## 275 with 169 links
## 100 most connected regions:
## 3 4 6 9 11 12 17 22 26 28 30 35 36 37 38 39 43 46 47 51 52 53 58 59 61 63 65 66 67 68 70 72 73 74 77 82 84 89 90 92 93 99 101 114 115 122 130 131 133 142 144 147 148 153 156 157 159 160 161 163 164 165 167 171 172 173 174 177 178 184 188 192 193 200 201 202 204 206 211 228 237 238 239 241 243 244 251 252 253 257 258 259 263 264 265 266 278 279 281 282 with 304 links
##
## Weights style: B
## Weights constants summary:
## n nn S0 S1 S2
## B 305 93025 90590 181180 107893080Creating adaptive proximity matrix
coords <- st_coordinates(centroids)
knb2 <- knn2nb(knearneigh(coords, k=8, longlat = TRUE), row.names=row.names(subzone_tapin$TOTAL_TAP_IN_VOLUME))
knb_lw2 <- nb2listw(knb2, style = 'B')
summary(knb_lw2)## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 2440
## Percentage nonzero weights: 2.622951
## Average number of links: 8
## Non-symmetric neighbours list
## Link number distribution:
##
## 8
## 305
## 305 least connected regions:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 with 8 links
## 305 most connected regions:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 with 8 links
##
## Weights style: B
## Weights constants summary:
## n nn S0 S1 S2
## B 305 93025 2440 4412 79922Plot centroids
plot(subzone_tapin$geometry, border="lightgrey")
plot(knb2, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red")
Combining fixed and adaptive
par(mfrow=c(1,2))
plot(subzone_tapin$geometry, border = 'lightgrey')
plot(dnb, st_coordinates(centroids), add=TRUE, col='red')
plot(subzone_tapin$geometry, border="lightgrey")
plot(knb2, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red")
There are several missing values (NaN) in the Gi using fixed distance, hence adaptive proximity matrix is preferred.
Computing Gi statistics
Gi statistics using fixed distance
fips <- order(subzone_tapin$Pop)
gi.fixed <- localG(subzone_tapin$TOTAL_TAP_IN_VOLUME, dnb_lw)
gi.fixed## [1] 0.986446710 1.762346256 NaN NaN 1.241930239
## [6] -Inf 1.865853392 0.724835938 NaN 0.724797759
## [11] NaN Inf 1.186524111 0.846301775 2.240845641
## [16] 1.042956282 -Inf 0.725536075 -2.980556693 1.108830638
## [21] 0.724780247 NaN 1.104765289 -2.404336686 0.728067579
## [26] NaN 0.727836691 NaN 0.013145881 NaN
## [31] -0.222686286 -0.851940559 -0.852167866 0.013691599 NaN
## [36] NaN NaN NaN NaN -0.560557589
## [41] 0.685282445 0.788827900 NaN 0.724802606 -2.130828955
## [46] NaN NaN -0.225331235 -0.215763798 0.725362951
## [51] NaN NaN NaN -0.865926921 -0.855698534
## [56] -0.215501401 -0.852530356 NaN NaN 1.755073667
## [61] NaN 0.727828229 NaN 1.758941040 NaN
## [66] NaN NaN NaN 1.977920210 NaN
## [71] -0.213581887 NaN NaN NaN 1.561170053
## [76] 0.822370715 NaN 2.108372441 -0.851453874 1.106026234
## [81] 1.749550064 NaN -0.851790232 NaN -0.085777422
## [86] -0.851633929 -3.009274389 -3.051075626 NaN NaN
## [91] -0.851941918 NaN NaN 0.508521884 -0.216860987
## [96] 1.748079616 2.028327523 1.747515521 NaN -0.212990045
## [101] NaN -0.861891380 -2.657815924 1.248183473 0.003233697
## [106] -2.366790864 0.531080964 -4.320417302 2.020977166 0.727487650
## [111] 1.106618685 1.108137950 0.729636439 NaN NaN
## [116] 1.749263206 1.867986467 1.869689506 -0.216115156 0.726622830
## [121] 1.881493966 NaN 0.726921806 -3.800510057 0.728778727
## [126] 1.243839809 -0.288065251 0.853544952 0.725675446 NaN
## [131] NaN -3.548653746 NaN -2.277950026 0.727916206
## [136] 1.586594575 0.987960021 2.038478422 0.986183050 0.909087356
## [141] 0.851232710 NaN 1.589532179 Inf 0.729297424
## [146] 0.730155622 NaN NaN 0.728346810 1.600974606
## [151] 0.885839928 1.861481733 NaN 0.725322009 2.021220255
## [156] NaN NaN 0.726513034 -Inf NaN
## [161] NaN 1.596675345 NaN NaN NaN
## [166] 0.730002758 NaN -0.098611424 0.982741331 2.303257692
## [171] NaN NaN NaN NaN -0.213259489
## [176] 0.238560917 NaN NaN 1.287551030 0.667976764
## [181] 0.697670949 1.977020787 1.061064786 NaN 2.257809866
## [186] 2.254033046 1.898492457 NaN 0.728315834 0.508080973
## [191] 0.614031038 NaN -Inf -3.330740402 1.897548984
## [196] -0.213532397 0.985780680 2.291858197 2.035505819 NaN
## [201] NaN NaN 0.516084721 NaN 1.883519492
## [206] NaN 0.729739633 -2.980480024 -3.799319727 -0.851446842
## [211] NaN 1.759123425 1.250588905 1.108206301 0.981101877
## [216] 0.987653224 0.729498542 0.984424551 0.986411317 1.448684016
## [221] 2.024149863 1.110215353 -0.851938193 1.105353154 0.987109644
## [226] 0.728528815 0.728917673 NaN 1.100888155 1.101680736
## [231] 1.244608491 1.108180091 0.727542653 -2.618271605 1.869243861
## [236] 1.294055822 NaN NaN NaN 0.730068311
## [241] NaN 0.729515121 Inf NaN 0.729031677
## [246] 1.593458596 0.512227603 1.304141200 0.861629597 0.696118506
## [251] NaN NaN NaN 1.107117156 0.607935663
## [256] -0.853269770 -Inf NaN NaN 0.856744999
## [261] -3.414901958 0.988107470 NaN NaN NaN
## [266] -Inf 0.012247102 -0.851464363 0.726530491 1.747905541
## [271] -3.335909415 -3.397654692 -2.838280440 -3.118145358 -3.676544439
## [276] -2.440483580 -0.851685474 NaN Inf 1.246791805
## [281] NaN NaN 2.295908133 -2.443011296 -0.212835652
## [286] -3.166206058 -0.853346547 0.726237680 -0.108702229 0.726193728
## [291] -0.092251900 0.988091930 1.074896267 -0.217833265 0.726051121
## [296] 0.987135660 0.729524150 0.727026526 0.982334852 0.981055226
## [301] 0.981202094 0.981943093 0.606825972 0.846300534 -3.645476751
## attr(,"gstari")
## [1] FALSE
## attr(,"call")
## localG(x = subzone_tapin$TOTAL_TAP_IN_VOLUME, listw = dnb_lw)
## attr(,"class")
## [1] "localG"The Gi statistics is represented as a Z-score. Greater values represent a greater intensity of clust
Visualising local Gi
Gi statistics using fixed distance
subzone.gi <- cbind(subzone_tapin, as.matrix(gi.fixed))
names(subzone.gi)[7] <- "gstat"tm_shape(subzone.gi)+
tm_fill(col="gstat",
style="pretty",
palette="-RdBu",
title="local Gi")+
tm_borders(alpha=0.5)## Warning: The shape subzone.gi is invalid. See sf::st_is_valid## Variable(s) "gstat" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
The Gi statistics using fixed value has many missing areas, hence we try the adaptive distance
Gi statistics using adaptive distance
gi.adaptive <- localG(subzone_tapin$TOTAL_TAP_IN_VOLUME, knb_lw2)
subzone.gi_adaptive <- cbind(subzone_tapin, as.matrix(gi.adaptive))
names(subzone.gi_adaptive)[7] <- "gstat_adaptive"tm_shape(subzone.gi_adaptive) +
tm_fill(col= "gstat_adaptive",
style = "pretty",
palette="-RdBu",
title = "local Gi") +
tm_borders(alpha = 0.5)## Warning: The shape subzone.gi_adaptive is invalid. See sf::st_is_valid## Variable(s) "gstat_adaptive" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
The Gi plot allows us to identify the various clusters, according to the colour scheme.
The darkness of the colours represents the intensity of the Gi values.
Areas shaded in red are hot spot area, and areas shaded in blue are cold spot area.
We can identify that there are more cold spot than hot spot area.
GEOVISUALISATION - FOR TAP IN
Install packages
packages = c('rgdal', 'spdep', 'tmap', 'tidyverse')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}Computing local Moran’s I
fips_out <- order(subzone_tapout$Pop)
localMI_out <- localmoran(subzone_tapout$TOTAL_TAP_OUT_VOLUME, rswm_q_out)
head(localMI_out)## Ii E.Ii Var.Ii Z.Ii Pr(z > 0)
## 1 0.07984235 -0.003289474 0.1554506 0.2108488 0.4165026
## 2 0.44108414 -0.003289474 0.4722590 0.6466334 0.2589346
## 3 -0.19563860 -0.003289474 0.1158495 -0.5651231 0.7140050
## 4 0.08695043 -0.003289474 0.1554506 0.2288772 0.4094822
## 5 0.40540291 -0.003289474 0.1026492 1.2756126 0.1010462
## 6 -0.21606701 -0.003289474 0.1554506 -0.5396716 0.7052883Mapping the local Moran’s I
subzone_tapout.localMI <- cbind(subzone_tapout, localMI_out)Mapping local Moran’s I values
tm_shape(subzone_tapout.localMI)+
tm_fill(col="Ii",
style="pretty",
title="local moran tap out statistics")+
tm_borders(alpha=0.5)## Warning: The shape subzone_tapout.localMI is invalid. See sf::st_is_valid## Variable(s) "Ii" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
Mapping local Moran’s I p-values
The choropleth shows there is evidence for both positive and negative values but consider the pvalues too
tm_shape(subzone_tapout.localMI) +
tm_fill(col="Pr.z...0.",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title="local Moran's I tap out p-values") +
tm_borders(alpha=0.5)## Warning: The shape subzone_tapout.localMI is invalid. See sf::st_is_valid
Mapping both local Moran’s I values and p-values for effective interpretation
localMI.map_out <- tm_shape(subzone_tapout.localMI) +
tm_fill(col="Ii",
style="pretty",
title="local moran tap out statistics")+
tm_borders(alpha=0.5)
pvalue.map_out <- tm_shape(subzone_tapout.localMI)+
tm_fill(col="Pr.z...0.",
breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
palette="-Blues",
title="local Moran's I tap outp-values")+
tm_borders(alpha=0.5)
tmap_arrange(localMI.map_out, pvalue.map_out, asp=1, ncol=2)## Warning: The shape subzone_tapout.localMI is invalid. See sf::st_is_valid## Variable(s) "Ii" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.## Warning: The shape subzone_tapout.localMI is invalid. See sf::st_is_valid
From the two graphs, we can tell that the regions with a higher local Moran tap in statistics and higher local Moran’s I tap in p-values are the same, which coincides with the same area of high tap in volume and high population, in the qtm graph that have been plotted in the previous part.
Creating a LISA Cluster Map
nci_out <- moran.plot(subzone_tapout$TOTAL_TAP_OUT_VOLUME, rswm_q_out, labels=as.character(subzone_tapout$Pop), xlab="TOTAL TAP OUT VOLUME", ylab="Spatially Lag Tap Out Volume")
Plotting Moran scatterplot with standardised variable
subzone_tapout$z.TOTAL_TAP_OUT_VOLUME <- scale(subzone_tapout$TOTAL_TAP_OUT_VOLUME) %>% as.vector nci2_out <- moran.plot(subzone_tapout$z.TOTAL_TAP_OUT_VOLUME, rswm_q_out, labels=as.character(subzone_tapout$Pop, xlab="z-Total tap out volume", ylab="Spatially lag tap out volume"))
Code chunks to prepare a LISA cluster map
quadrant_out <- vector(mode="numeric", length=nrow(localMI_out))Centers the variable of interest around its mean
DV_out <- subzone_tapout$TOTAL_TAP_OUT_VOLUME - mean(subzone_tapout$TOTAL_TAP_OUT_VOLUME)Centering local Moran’s around the mean
C_mI_out <- localMI_out[,1] - mean(localMI_out[,1])Set statistical significance level for the local Moran
signif <- 0.05Define high-high, low-low, low-high, high-low
quadrant_out[DV_out >0 & C_mI_out>0] <- 4
quadrant_out[DV_out <0 & C_mI_out<0] <- 1
quadrant_out[DV_out <0 & C_mI_out>0] <- 2
quadrant_out[DV_out >0 & C_mI_out<0] <- 3Non significant Moran in category 0
quadrant_out[localMI_out[,5]>signif] <- 0Build LISA map
subzone_tapout.localMI$quadrant_out <- quadrant_out
colors <- c("#ffffff", "#2c7bb6", "#abd9e9", "#fdae61", "#d7191c")
clusters <- c("insignificant", "low-low", "low-high", "high-low", "high-high")
tm_shape(subzone_tapout.localMI) +
tm_fill(col = "quadrant_out", style = "cat", palette = colors[c(sort(unique(quadrant_out)))+1], labels = clusters[c(sort(unique(quadrant_out)))+1], popup.vars = c("Postal.Code")) +
tm_view(set.zoom.limits = c(11,17)) +
tm_borders(alpha=0.5)## Warning: The shape subzone_tapout.localMI is invalid. See sf::st_is_valid
There are areas of low-low,low-high, high-low, and high-high
Hot Spot and Cold Spot Area Analysis
Computing GI statistics
GI statistics using fixed distance
centroids <- sf::st_centroid(subzone_tapout$geometry)## Warning in st_centroid.sfc(subzone_tapout$geometry): st_centroid does not give
## correct centroids for longitude/latitude datadnb_out <- dnearneigh(st_coordinates(centroids), 0, 26, longlat=TRUE)
dnb_out## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 90590
## Percentage nonzero weights: 97.38242
## Average number of links: 297.0164Display the data and neighbours
plot(subzone_tapout$geometry, border='lightgrey')
plot(dnb,st_coordinates(centroids), add=TRUE, col='red')
dnb_lw_out <- nb2listw(dnb_out, style='B')
summary(dnb_lw_out)## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 90590
## Percentage nonzero weights: 97.38242
## Average number of links: 297.0164
## Link number distribution:
##
## 169 211 227 230 240 247 248 251 262 263 266 268 269 270 272 273 276 277 279 280
## 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 3 4 1
## 281 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
## 1 1 1 4 2 1 3 2 4 5 1 1 5 8 4 12 6 6 7 16
## 302 303 304
## 35 53 100
## 1 least connected region:
## 275 with 169 links
## 100 most connected regions:
## 3 4 6 9 11 12 17 22 26 28 30 35 36 37 38 39 43 46 47 51 52 53 58 59 61 63 65 66 67 68 70 72 73 74 77 82 84 89 90 92 93 99 101 114 115 122 130 131 133 142 144 147 148 153 156 157 159 160 161 163 164 165 167 171 172 173 174 177 178 184 188 192 193 200 201 202 204 206 211 228 237 238 239 241 243 244 251 252 253 257 258 259 263 264 265 266 278 279 281 282 with 304 links
##
## Weights style: B
## Weights constants summary:
## n nn S0 S1 S2
## B 305 93025 90590 181180 107893080Creating adaptive proximity matrix
coords <- st_coordinates(centroids)
knb2_out <- knn2nb(knearneigh(coords, k=8, longlat = TRUE), row.names=row.names(subzone_tapout$TOTAL_TAP_OUT_VOLUME))
knb_lw2_out <- nb2listw(knb2_out, style = 'B')
summary(knb_lw2_out)## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 305
## Number of nonzero links: 2440
## Percentage nonzero weights: 2.622951
## Average number of links: 8
## Non-symmetric neighbours list
## Link number distribution:
##
## 8
## 305
## 305 least connected regions:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 with 8 links
## 305 most connected regions:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 with 8 links
##
## Weights style: B
## Weights constants summary:
## n nn S0 S1 S2
## B 305 93025 2440 4412 79922Plot centroids
plot(subzone_tapout$geometry, border="lightgrey")
plot(knb2_out, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red")
Combining fixed and adaptive
par(mfrow=c(1,2))
plot(subzone_tapout$geometry, border = 'lightgrey')
plot(dnb_out, st_coordinates(centroids), add=TRUE, col='red')
plot(subzone_tapout$geometry, border="lightgrey")
plot(knb2_out, st_coordinates(centroids), pch=19, cex=0.6, add=TRUE, col="red")
adaptive is preferred
Computing Gi statistics
Gi statistics using fixed distance
fips <- order(subzone_tapout$Pop)
gi.fixed_out <- localG(subzone_tapout$TOTAL_TAP_OUT_VOLUME, dnb_lw_out)
gi.fixed_out## [1] 0.88419980 1.76586457 NaN NaN 1.20569304 NaN
## [7] 1.88856562 0.71589342 NaN 0.71590375 NaN NaN
## [13] 1.21007452 0.88253282 2.29895744 1.07380166 NaN 0.71651538
## [19] -3.09028720 1.08318928 0.71597955 NaN 1.07853468 -2.42521653
## [25] 0.71863018 NaN 0.71904073 Inf 0.04861064 -Inf
## [31] -0.29649697 -0.94212161 -0.94271955 0.04892443 NaN NaN
## [37] NaN NaN NaN -0.55780734 0.62621178 0.72564732
## [43] NaN 0.71617190 -2.13924132 NaN NaN -0.29876179
## [49] -0.28941745 0.71598799 NaN NaN NaN -0.95423736
## [55] -0.94624717 -0.29007742 -0.94293808 NaN NaN 1.75844627
## [61] NaN 0.71876493 NaN 1.76086700 NaN NaN
## [67] NaN NaN 1.97636096 NaN -0.28757733 NaN
## [73] NaN NaN 1.56085536 0.77598896 NaN 2.15570653
## [79] -0.94199490 1.07953933 1.75157453 NaN -0.94174311 NaN
## [85] -0.15625916 -0.94263508 -3.11863301 -3.12938599 NaN NaN
## [91] -0.94253728 NaN NaN 0.48772492 -0.29020778 1.75120494
## [97] 2.02597107 1.75115820 NaN -0.28704005 NaN -0.95282539
## [103] -2.73678533 1.21071062 0.03943779 -2.41785857 0.51093489 -4.37739260
## [109] 2.01793001 0.71806456 1.08076022 1.08139032 0.72083020 NaN
## [115] NaN 1.75213380 1.89113738 1.89214489 -0.29141061 0.71834590
## [121] 1.91642802 -Inf 0.71624829 -3.87020969 0.72079978 1.20727031
## [127] -0.38793068 0.83124504 0.71693571 NaN Inf -3.57359487
## [133] NaN -2.24918076 0.71876381 1.56919147 0.88620548 2.03572386
## [139] 0.88494130 0.84955249 0.76214716 -Inf 1.57232620 -Inf
## [145] 0.72092298 0.72144736 NaN NaN 0.71938244 1.58385665
## [151] 0.94895131 1.88490774 NaN 0.71688949 2.01803528 NaN
## [157] NaN 0.71774547 NaN NaN NaN 1.57937576
## [163] NaN NaN NaN 0.72115840 NaN -0.16798473
## [169] 0.88122972 2.36189149 NaN NaN Inf NaN
## [175] -0.28720161 0.24429834 NaN NaN 1.29767168 0.64623242
## [181] 0.67339635 1.98935234 1.09008825 NaN 2.31514330 2.31121861
## [187] 1.93328186 NaN 0.71907499 0.54011168 0.59896748 NaN
## [193] NaN -3.37355984 1.93341576 -0.28752933 0.88365043 2.35109684
## [199] 2.08248837 NaN NaN NaN 0.54731697 NaN
## [205] 1.91813309 NaN 0.72088078 -3.08954597 -3.86858981 -0.94194378
## [211] NaN 1.76259039 1.21351107 1.08084099 0.87958962 0.88539770
## [217] 0.72021858 0.88216101 0.88493915 1.43857190 2.02055021 1.08384784
## [223] -0.94304602 1.07955955 0.88508813 0.71994622 0.72010200 NaN
## [229] 1.07365592 1.07532319 1.20685744 1.08176853 0.71803806 -2.61438362
## [235] 1.93517575 1.30535872 NaN NaN NaN 0.72109124
## [241] NaN 0.72061139 NaN NaN 0.72010326 1.57762385
## [247] 0.49053884 1.31549688 0.89540198 0.72172879 NaN NaN
## [253] NaN 1.08081386 0.59171024 -0.94348374 NaN NaN
## [259] NaN 0.83487298 -3.45604018 0.88715287 NaN NaN
## [265] NaN NaN 0.04778962 -0.94217748 0.71841932 1.75155125
## [271] -3.44867371 -3.43098347 -2.87224114 -3.13797093 -3.81761633 -2.44353424
## [277] -0.94232041 NaN NaN 1.20811915 NaN NaN
## [283] 2.35315992 -2.44565117 -0.28735270 -3.27691412 -0.94281484 0.71838050
## [289] -0.17651944 0.71743297 -0.16683263 0.88571759 1.04761860 -0.29241754
## [295] 0.71717464 0.88503678 0.72075058 0.71834419 0.88048392 0.87942421
## [301] 0.88059065 0.87915744 0.59145231 0.82545672 -3.68404413
## attr(,"gstari")
## [1] FALSE
## attr(,"call")
## localG(x = subzone_tapout$TOTAL_TAP_OUT_VOLUME, listw = dnb_lw_out)
## attr(,"class")
## [1] "localG"The Gi statistics is represented as a Z-score. Greater values represent a greater intensity of clust
Visualising local Gi
Gi statistics using fixed distance
subzone.gi_out <- cbind(subzone_tapout, as.matrix(gi.fixed_out))
names(subzone.gi_out)[6] <- "gstat"tm_shape(subzone.gi_out)+
tm_fill(col="gstat",
stype="pretty",
palette="-RdBu",
title="local Gi")+
tm_borders(alpha=0.5)## Warning: The shape subzone.gi_out is invalid. See sf::st_is_valid## Variable(s) "gstat" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
The Gi statistics using fixed value has many missing areas, hence we try the adaptive distance
Gi statistics using adaptive distance
gi.adaptive_out <- localG(subzone_tapout$TOTAL_TAP_OUT_VOLUME, knb_lw2_out)
subzone.gi_adaptive_out <- cbind(subzone_tapout, as.matrix(gi.adaptive_out))
names(subzone.gi_adaptive_out)[6] <- "gstat_adaptive"tm_shape(subzone.gi_adaptive_out) +
tm_fill(col= "gstat_adaptive",
style = "pretty",
palette="-RdBu",
title = "local Gi") +
tm_borders(alpha = 0.5)## Warning: The shape subzone.gi_adaptive_out is invalid. See sf::st_is_valid## Variable(s) "gstat_adaptive" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
The Gi plot allows us to identify the various clusters, according to the colour scheme.
The darkness of the colours represents the intensity of the Gi values.
Areas shaded in red are hot spot area, and areas shaded in blue are cold spot area.
We can identify that there are more cold spot than hot spot area.
BONUS - Gain insights from the data
From the data and plots derived above, we can conclude that it is mainly the East area, followed by the West, with a higher public bus commuter flow. This could likely be due to the high population of local residents in those area. Areas such as Tampines East, Woodlands East, or Bedok North are shown to be spots with the highest public bus commuter flow. This could also largely be due to the large population in these areas.
Working only on tap in since tap in and tap out data are similar,
Create an interactive map to show tap in volume per residential area, to see where the main clusters are
Plotted based on the longitude and lattitude, with clustering options to group by neighbourhood
Install packages required
packages <- c('tmap', 'sf', 'rgdal', 'lwgeom', 'leaflet')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}## Loading required package: lwgeom## Linking to liblwgeom 3.0.0beta1 r16016, GEOS 3.6.1, PROJ 4.9.3## Warning in fun(libname, pkgname): GEOS versions differ: lwgeom has 3.6.1 sf has
## 3.8.0## Warning in fun(libname, pkgname): PROJ versions differ: lwgeom has 4.9.3 sf has
## 6.3.1## Loading required package: leaflettmap_mode("view")## tmap mode set to interactive viewinglisting_distribution <- leaflet(passbus_subzone) %>%
addTiles() %>%
addMarkers(~lat, ~lon, labelOptions = labelOptions(noHide = F),
clusterOptions = markerClusterOptions(),
popup = paste0("<b> subzone_name: </b>", passbus_subzone$subzone_name,
"<br/><b> bus stop number: </b>", passbus_subzone$BUS_STOP_N,
"<br> <b> Location Description: </b>", passbus_subzone$LOC_DESC,
"<br/> <b> Total tap in volume: </b>",subzone_tapin$TOTAL_TAP_IN_VOLUME)) %>%
addProviderTiles("CartoDB.Positron")
# To mute the background, and highlight the cluster details.
print(listing_distribution)To view the specific clusters in the interactive map, simply click to zoom.
The interactive map shows us that the largest cluster of tap in volume is in central east, likely because of the high residential population, resulting in a high public bus commuter flow.
This interactive map is useful in us visualising which areas have the highest tap in volume, which is also easy to use, as the areas can be identified simply by clicking on the location spots.
tmap_mode("plot")## tmap mode set to plotting