residuals.lppm: Residuals for Fitted Point Process Model on a Network

In spatstat.linnet: Linear Networks Functionality of the 'spatstat' Family

Residuals for Fitted Point Process Model on a Network

Description

Given a point process model fitted to a point pattern on a linear network, compute residuals of the fitted model.

Usage

  ## S3 method for class 'lppm'
residuals(object, type="raw", ...)

Arguments

object	The fitted point process model (an object of class "ppm") for which residuals should be calculated.
type	String indicating the type of residuals to be calculated. Current options are "raw", "inverse", "pearson" and "score". A partial match is adequate.
...	Other arguments are currently ignored.

Details

This function computes several kinds of residuals for the fit of a point process model to a spatial point pattern on a linear network. It is an extension of the method of Baddeley et al (2005) to point process models on a network. Use plot.msr to plot the residuals directly.

The argument object must be a fitted point process model on a network (object of class "lppm"). Such objects are produced by the model-fitting algorithm lppm. This fitted model object contains complete information about the original data pattern.

Residuals are attached both to the data points and to some other points in the window of observation (namely, to the dummy points of the quadrature scheme used to fit the model). If the fitted model is correct, then the sum of the residuals over all (data and dummy) points in a spatial region B has mean zero. For further explanation, see Baddeley et al (2005).

The type of residual is chosen by the argument type. Current options are

"raw":

the raw residuals

r_j = z_j - w_j \lambda_j

at the quadrature points u_j, where z_j is the indicator equal to 1 if u_j is a data point and 0 if u_j is a dummy point; w_j is the quadrature weight attached to u_j; and

\lambda_j = \hat\lambda(u_j,x)

is the conditional intensity of the fitted model at u_j. These are the spatial analogue of the martingale residuals of a one-dimensional counting process.

"inverse":

the ‘inverse-lambda’ residuals (Baddeley et al, 2005)

r^{(I)}_j = \frac{r_j}{\lambda_j} = \frac{z_j}{\lambda_j} - w_j

obtained by dividing the raw residuals by the fitted conditional intensity. These are a counterpart of the exponential energy marks (see eem).

"pearson":

the Pearson residuals (Baddeley et al, 2005)

r^{(P)}_j = \frac{r_j}{\sqrt{\lambda_j}} = \frac{z_j}{\sqrt{\lambda_j}} - w_j \sqrt{\lambda_j}

obtained by dividing the raw residuals by the square root of the fitted conditional intensity. The Pearson residuals are standardised, in the sense that if the model (true and fitted) is Poisson, then the sum of the Pearson residuals in a spatial region B has variance equal to the area of B.

"score":

the score residuals (Baddeley et al, 2005)

r_j = (z_j - w_j \lambda_j) x_j

obtained by multiplying the raw residuals r_j by the covariates x_j for quadrature point j. The score residuals always sum to zero.

The result of residuals.ppm is a measure (object of class "msr"). Use plot.msr to plot the residuals directly. Use integral.msr to compute the total residual.

Value

An object of class "msr" representing a signed measure or vector-valued measure (see msr). This object can be plotted.

Author(s)

\spatstatAuthors

.

References

Baddeley, A., Turner, R., \Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.

See Also

lppm, msr, residuals.ppm.

Examples

   fit <- lppm(unmark(chicago) ~ x + y)

   # raw residuals
   rr <- residuals(fit)
   rr

   # Pearson residuals
   rp <- residuals(fit, type="pe")
   rp
   plot(rp, main="Pearson residuals")

   ## multitype data
   fitm <- lppm(chicago ~ (x+y) * marks, eps=100)
   rpm <- residuals(fitm, type="pe")
   ## plot(rpm) would display 7 panels, one for each crime type
   ## Select residuals for crime type = Assault
   plot(split(rpm)[["assault"]], markscale=2)

spatstat.linnet documentation built on Jan. 31, 2026, 5:06 p.m.