residuals.ppm: Residuals for Fitted Point Process Model

In spatstat.model: Parametric Statistical Modelling and Inference for the 'spatstat' Family

Residuals for Fitted Point Process Model

Description

Given a point process model fitted to a point pattern, compute residuals.

Usage

  ## S3 method for class 'ppm'
residuals(object, type="raw", ...,
                check=TRUE, drop=FALSE,
                fittedvalues=NULL, 
                new.coef=NULL, dropcoef=FALSE,
                quad=NULL)

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.
...	Ignored.
check	Logical value indicating whether to check the internal format of object. If there is any possibility that this object has been restored from a dump file, or has otherwise lost track of the environment where it was originally computed, set check=TRUE.
drop	Logical value determining whether to delete quadrature points that were not used to fit the model. See quad.ppm for explanation.
fittedvalues	Vector of fitted values for the conditional intensity at the quadrature points, from which the residuals will be computed. For expert use only.
new.coef	Optional. Numeric vector of coefficients for the model, replacing coef(object). See the section on Modified Residuals below.
dropcoef	Internal use only.
quad	Optional. Data specifying how to re-fit the model. A list of arguments passed to quadscheme. See the section on Modified Residuals below.

Details

This function computes several kinds of residuals for the fit of a point process model to a spatial point pattern dataset (Baddeley et al, 2005). Use plot.msr to plot the residuals directly, or diagnose.ppm to produce diagnostic plots based on these residuals.

The argument object must be a fitted point process model (object of class "ppm"). Such objects are produced by the maximum pseudolikelihood fitting algorithm ppm. 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, or diagnose.ppm to produce diagnostic plots based on these residuals. Use integral.msr to compute the total residual.

By default, the window of the measure is the same as the original window of the data. If drop=TRUE then the window is the domain of integration of the pseudolikelihood or composite likelihood. This only matters when the model object was fitted using the border correction: in that case, if drop=TRUE the window of the residuals is the erosion of the original data window by the border correction distance rbord.

Value

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

Modified Residuals

Sometimes we want to modify the calculation of residuals by using different values for the model parameters. This capability is provided by the arguments new.coef and quad.

If new.coef is given, then the residuals will be computed by taking the model parameters to be new.coef. This should be a numeric vector of the same length as the vector of fitted model parameters coef(object).

If new.coef is missing and quad is given, then the model parameters will be determined by re-fitting the model using a new quadrature scheme specified by quad. Residuals will be computed for the original model object using these new parameter values.

The argument quad should normally be a list of arguments in name=value format that will be passed to quadscheme (together with the original data points) to determine the new quadrature scheme. It may also be a quadrature scheme (object of class "quad") to which the model should be fitted, or a point pattern (object of class "ppp") specifying the dummy points in a new quadrature scheme.

Author(s)

\adrian

and \rolf

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.

Baddeley, A., \Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.

See Also

msr, diagnose.ppm, ppm.object, ppm

Examples

   fit <- ppm(cells, ~x, Strauss(r=0.15))

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

   # simulated data
   X <- rStrauss(100,0.7,0.05)
   # fit Strauss model 
   fit <- ppm(X, ~1, Strauss(0.05))
   res.fit <- residuals(fit)

   # check that total residual is 0 
   integral.msr(residuals(fit, drop=TRUE))

   # true model parameters
   truecoef <- c(log(100), log(0.7))
   res.true <- residuals(fit, new.coef=truecoef)  

spatstat.model documentation built on Sept. 30, 2024, 9:26 a.m.