Analysing spatial point patterns in R - CSIRO

28.2 Residuals for Gibbs processes 173
28.2 Residuals for Gibbs processes
28.2.1 Definition
Residuals for a general Gibbs model were defined only recently [12, 6]. The total residual in a
region B ⊂ R 2 is defined as
∫
R(B) = n(x ∩ B) − ̂λ(u,x)du (47)
B
where again n(x ∩ B) is the observed number of points in the region B, and ̂λ(u,x) is the
conditional intensity of the fitted model, evaluated for the data point pattern x. If the fitted
model is correct, the residuals have mean zero.
This definition is similar to the definition of residuals for Poisson processes (Section 16.2)
except that the intensity ̂λ(u) of the fitted Poisson process has been replaced by the conditional
intensity ̂λ(u,x) of the fitted Gibbs process evaluated for the data point pattern x.
28.2.2 Residual plots
Residuals for Gibbs processes can be plotted using the same techniques as in Section 16.2. Here
is the four-panel plot:
> diagnose.ppm(fit, type = "Pearson")
cumulative sum of Pearson residuals
0.5 0 −0.5 −1 −1.5
cumulative sum of Pearson residuals
−2 −1.5 −1 −0.5 0 0.5
0 0.2 0.4 0.6 0.8 1
y coordinate
5
−4
−2
3
4
1
2
0
−2
−3
−1
−5
−4
−3
−3
−5
−4
−2
−1
0
4
6
5
1
2
7
3
0
−1
0 0.2 0.4 0.6 0.8 1
x coordinate
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