Analysing spatial point patterns in R - CSIRO

16.2 Validation using residuals 109
This is an image plot of the ‘smoothed residual field’
where ̂λ(u) is the nonparametric, kernel estimate of the intensity,
s(u) = ̂λ(u) − λ † (u) (12)
n(x)
∑
̂λ(u) = e(u) κ(u − x i )
i=1
while λ † (u) is a correspondingly-smoothed version of the parametric estimate of the intensity
according to the fitted model,
∫
λ † (u) = e(u) κ(u − v)λˆθ(v)dv.
W
Here κ is the smoothing kernel and e(u) is the edge correction (2) on page 79. The difference
(12) should be approximately zero if the model is true.
In this example the smoothed residual image contains a visible trend, suggesting that the
model is inappropriate.
16.2.4 Lurking variable plot
If there is a spatial covariate Z(u) that plays an important role in the analysis, it may be useful
to display a lurking variable plot of the residuals against Z. This is a plot of C(z) = R(B(z))
against z, where
B(z) = {u ∈ W : Z(u) ≤ z}
is the region of space where the covariate value is less than or equal to z.
> grad lurking(fitx, grad, type = "raw")
cumulative raw residuals
−600 −400 −200 0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
covariate
Note that the lurking variable plot typically starts and ends at the horizontal axis, since (for
any model with an intercept term) the total residual for the entire window W must equal zero.
This is analogous to the fact that the residuals in linear regression sum to zero.
The plot also shows approximate 5% significance bands for the cumulative residual C(x) or
C(y), obtained from the asymptotic variance under the model.
CopyrightCSIRO 2010