Applied Spatial Statistics: Problem Set # 2

Loury Migliorelli

date()

## [1] "Wed Mar 20 12:53:00 2013"

Due Date: March 20, 2013

Total Points: 40

Using the ants data set

1. Plot the nest locations.

suppressMessages(require(spatstat))
SP = ants
plot(SP)

## Cataglyphis      Messor 
##           1           2

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

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

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

plot(Gest(Cataglyphis))

##      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)

plot(Gest(Messor))

##      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)

It is observed that the Cataglyphis nests exhibit G values greater than the theoretical values at distances less than 15 feet. This indicates a randomized distribution. At distances greater than 15 feet, G values are less than the theoretical values. This indicates spatial regularity. The Messor nests exhibit G values less than the theoretical values indicating spacial regularity.

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

plot(Kcross(SP, "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)

It is evident that the observed value is greater than the theoretical value. Therefore, there is no evidence for inter-species clustering. Nest locations for each species are independent of eachother.

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)
SP.model = ppm(ANTS, trend = ~1, interaction = Strauss(r = 100), rbord = 100)
SP.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 beta is 0.0009961705, which is greater than the actual intensity of 0.000226. This indicates an inhibition process. The interaction parameter (gamma) is 0.8189, therefore it indicates no interaction since it is less than 1. The presence of a nest at a specific location makes it unlikely that another nest will be in the same location.