Applied Spatial Statistics: Problem Set # 2

Brendan Mulholand

date()

## [1] "Wed Mar 20 15:26:48 2013"

Due Date: March 20, 2013

Total Points: 40

Using the ants data set

1. Plot the nest locations.

suppressMessages(require(spatstat))

## Warning: package 'spatstat' was built under R version 2.15.3

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

Nests = ants
plot(Nests)

## Cataglyphis      Messor 
##           1           2

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

Obj = split(Nests)
Cat = Obj$Cataglyphis
Mes = Obj$Messor

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

plot(Gest(Cat))

##      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(Mes))

##      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)

At small distances, there is indication of randomized distribution. But as the distance increase; for the Catagylphis ants the observed G value is less than theoretical G.This is indicatory of spatial regularity. Meaning there are fewer nests n the vicinity of other nests than is expected by chance. The Messor ants has a observed G value that is for the most part less than theoretical G, indicating spatial regularity.

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

plot(Kcross(Nests, "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)

Estimates are close to the theoretical poisson functions but are higher than the theoretical curce. At smaller radii the two types appear to be independent. But as distance is increasing, more Cataglyphis ants are close to Messor ants than was 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.

Nests.u = unmark(Nests)
Nests.model = ppm(Nests.u, trend = ~1, interaction = Strauss(r = 100), rbord = 100)
Nests.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))

summary(Nests)

## 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

Beta, the first order term for the model is lower than the first order term for the data. Because the model value is greater than the data, an inhibition proces may be occuring. This process can be confirmed by the interaction parameter, which is found to be lower than one, indicating no interaction. As a result is unlikely that multiple nests will be in the same area.