Applied Spatial Statistics: Problem Set # 2

Holly Widen

date()

## [1] "Wed Mar 20 14:01:09 2013"

Due Date: March 20, 2013

Total Points: 40

Using the ants data set

1. Plot the nest locations.

require(spatstat)

## Loading required package: spatstat

## Warning: package 'spatstat' was built under R version 2.15.2

## Loading required package: mgcv

## This is mgcv 1.7-18. For overview type 'help("mgcv-package")'.

## Loading required package: deldir

## Warning: package 'deldir' was built under R version 2.15.2

## deldir 0.0-21

## spatstat 1.31-0 Type 'help(spatstat)' for an overview of spatstat
## 'latest.news()' for news on latest version 'licence.polygons()' for
## licence information on polygon calculations

data(ants)
summary(ants)

## Marked planar point pattern: 97 points
## Average intensity 0.000226 points per square unit (one unit = 0.5 feet) 
## 
## Coordinates are integers 
## i.e. rounded to the nearest 1 unit (one unit = 0.5 feet) 
## 
## Multitype:
##             frequency proportion intensity
## Cataglyphis        29      0.299  6.76e-05
## Messor             68      0.701  1.59e-04
## 
## Window: polygonal boundary
## single connected closed polygon with 11 vertices
## enclosing rectangle: [-25, 803] x [-49, 717] units 
## Window area =  428922 square units 
## Unit of length: 0.5 feet

plot(ants)

## Cataglyphis      Messor 
##           1           2

plot(split(ants))

1. Create separate ppp objects for the two types of ants.
   

cat = ants[ants$marks == "Cataglyphis"]
mes = ants[ants$marks == "Messor"]

1. Plot the G function for the Cataglyphis and Messor ants separately. What do you find?
   

G1 = Gest(cat)
plot(G1)

##      lty col  key           label                           meaning
## km     1   1   km   hat(G)[km](r)     Kaplan-Meier estimate of G(r)
## rs     2   2   rs hat(G)[bord](r) border corrected estimate of G(r)
## han    3   3  han  hat(G)[han](r)          Hanisch estimate of G(r)
## theo   4   4 theo      G[pois](r)          theoretical Poisson G(r)

G2 = Gest(mes)
plot(G2)

##      lty col  key           label                           meaning
## km     1   1   km   hat(G)[km](r)     Kaplan-Meier estimate of G(r)
## rs     2   2   rs hat(G)[bord](r) border corrected estimate of G(r)
## han    3   3  han  hat(G)[han](r)          Hanisch estimate of G(r)
## theo   4   4 theo      G[pois](r)          theoretical Poisson G(r)

For the Cataglyphis ants: At first, the observed value of G is higher than the theoretical value of G (from about 0 to 10 feet). This indicates that there are more nests in the vicinity of other nests than is expected by chance, suggesting clustering nearby the nests. However, as the radius increases to about 15 feet, the observed value of G becomes less than the theoretical value of G. This indicates that there are fewer nests in the vicinity of other nests than is expected by chance which suggests spatial regularity.

For the Messor ants: The observed value of G is less than the theoretical value of G for the most part. This indicates that there are fewer nests in the vicinity of other nexts than is expected, suggesting spatial regularity. At a distance of about 27 feet, the observed and theoretical values of G flip and demonstrate clustering of nests.

1. Plot the Kcross function and describe the evidence for inter-species clustering.
   

plot(Kcross(ants, "Cataglyphis", "Messor"))

##        lty col    key
## iso      1   1    iso
## trans    2   2  trans
## border   3   3 border
## theo     4   4   theo
##                                                           label
## iso              hat(K[list(Cataglyphis, Messor)]^{    iso})(r)
## trans          hat(K[list(Cataglyphis, Messor)]^{    trans})(r)
## border          hat(K[list(Cataglyphis, Messor)]^{    bord})(r)
## theo   {    K[list(Cataglyphis, Messor)]^{        pois    }}(r)
##                                                                           meaning
## iso    Ripley isotropic correction estimate of Kcross["Cataglyphis", "Messor"](r)
## trans        translation-corrected estimate of Kcross["Cataglyphis", "Messor"](r)
## border            border-corrected estimate of Kcross["Cataglyphis", "Messor"](r)
## theo                       theoretical Poisson Kcross["Cataglyphis", "Messor"](r)

Since the theoretical value of K is fairly equal to the observed value of K, we can say that the events are independent of each other. There is no evidence of inter-species clustering. Also, with distance, we tend to find more Messor nests than expected.

1. Create an umarked ppp object and model the nests using a Strauss process with interaction distance of 100 and border correction distance of 100. Interpret the first order term and interaction parameter.

ANTS = unmark(ants)
plot(Gest(ANTS))

##      lty col  key           label                           meaning
## km     1   1   km   hat(G)[km](r)     Kaplan-Meier estimate of G(r)
## rs     2   2   rs hat(G)[bord](r) border corrected estimate of G(r)
## han    3   3  han  hat(G)[han](r)          Hanisch estimate of G(r)
## theo   4   4 theo      G[pois](r)          theoretical Poisson G(r)

ANTS.model = ppm(ANTS, trend = ~1, interaction = Strauss(r = 100), rbord = 100)
ANTS.model

## Stationary Strauss process
## 
## First order term:
##      beta 
## 0.0009962 
## 
## Interaction: Strauss process 
## interaction distance:    100
## Fitted interaction parameter gamma:  0.8189
## 
## Relevant coefficients:
## Interaction 
##     -0.1998 
## 
## For standard errors, type coef(summary(x))

The first-order term is about 0.000996 which represents the intensity of the proposal events. It is larger than the actual intensity (0.000226 points per square unit) so it suggests an inhibition process (because of the “thinning”). The interaction parameter is 0.8189 and as it is less than one, it also indicates an inhibition process. Thus, the presence of a nest at a given location makes it less likely that another nest will be nearby.