cdf.test.mppm {spatstat.model}

Spatial Distribution Test for Multiple Point Process Model

Description

Performs a spatial distribution test of a point process model fitted to multiple spatial point patterns. The test compares the observed and predicted distributions of the values of a spatial covariate, using either the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling test of goodness-of-fit.

Usage

## S3 method for class 'mppm'
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ...,
            nsim=19, verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)

Arguments

model

An object of class "mppm" representing a point process model fitted to multiple spatial point patterns.

covariate

The spatial covariate on which the test will be based. A function, a pixel image, a list of functions, a list of pixel images, a hyperframe, a character string containing the name of one of the covariates in model, or one of the strings "x" or "y".

test

Character string identifying the test to be performed: "ks" for Kolmogorov-Smirnov test, "cvm" for Cramer-von Mises test or "ad" for Anderson-Darling test.

...

Arguments passed to cdf.test to control the test.

nsim

Number of simulated realisations which should be generated, if a Monte Carlo test is required.

verbose

Logical flag indicating whether to print progress reports.

interpolate

Logical flag indicating whether to interpolate between pixel values when covariate is a pixel image. See Details.

fast

Logical flag. If TRUE, values of the covariate are only sampled at the original quadrature points used to fit the model. If FALSE, values of the covariate are sampled at all pixels, which can be slower by three orders of magnitude.

jitter

Logical flag. If TRUE, observed values of the covariate are perturbed by adding small random values, to avoid tied observations.

Details

This function is a method for the generic function cdf.test for the class mppm.

This function performs a goodness-of-fit test of a point process model that has been fitted to multiple point patterns. The observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same values under the model, are compared using the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling test of goodness-of-fit. These are exact tests if the model is Poisson; otherwise, for a Gibbs model, a Monte Carlo p-value is computed by generating simulated realisations of the model and applying the selected goodness-of-fit test to each simulation.

The argument model should be a fitted point process model fitted to multiple point patterns (object of class "mppm").

The argument covariate contains the values of a spatial function. It can be

• a function(x,y)

• a pixel image (object of class "im"

• a list of function(x,y), one for each point pattern

• a list of pixel images, one for each point pattern

• a hyperframe (see hyperframe) of which the first column will be taken as containing the covariate

• a character string giving the name of one of the covariates in model

• one of the character strings "x" or "y", indicating the spatial coordinates.

If covariate is an image, it should have numeric values, and its domain should cover the observation window of the model. If covariate is a function, it should expect two arguments x and y which are vectors of coordinates, and it should return a numeric vector of the same length as x and y.

First the original data point pattern is extracted from model. The values of the covariate at these data points are collected.

The predicted distribution of the values of the covariate under the fitted model is computed as follows. The values of the covariate at all locations in the observation window are evaluated, weighted according to the point process intensity of the fitted model, and compiled into a cumulative distribution function F using ewcdf.

The probability integral transformation is then applied: the values of the covariate at the original data points are transformed by the predicted cumulative distribution function F into numbers between 0 and 1. If the model is correct, these numbers are i.i.d. uniform random numbers. A goodness-of-fit test of the uniform distribution is applied to these numbers using ks.test, cvm.test or ad.test.

The argument interpolate determines how pixel values will be handled when covariate is a pixel image. The value of the covariate at a data point is obtained by looking up the value of the nearest pixel if interpolate=FALSE, or by linearly interpolating between the values of the four nearest pixels if interpolate=TRUE. Linear interpolation is slower, but is sometimes necessary to avoid tied values of the covariate arising when the pixel grid is coarse.

If model is a Poisson point process, then the Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling tests are theoretically exact. This test was apparently first described (in the context of spatial data, and for Kolmogorov-Smirnov) by Berman (1986). See also Baddeley et al (2005).

If model is not a Poisson point process, then the Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling tests are biased. Instead they are used as the basis of a Monte Carlo test. First nsim simulated realisations of the model will be generated. Each simulated realisation consists of a list of simulated point patterns, one for each of the original data patterns. This can take a very long time. The model is then re-fitted to each simulation, and the refitted model is subjected to the goodness-of-fit test described above. A Monte Carlo p-value is then computed by comparing the p-value of the original test with the p-values obtained from the simulations.

Value

An object of class "cdftest" and "htest" containing the results of the test. See cdf.test for details.

Author(s)

Adrian Baddeley [e-mail], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

References

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

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.

Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54–62.

See Also

cdf.test, quadrat.test, mppm

Examples

   # three i.i.d. realisations of nonuniform Poisson process
   lambda <- as.im(function(x,y) { 200 * exp(x) }, square(1))
   dat <- hyperframe(X=list(rpoispp(lambda), rpoispp(lambda), rpoispp(lambda)))

   # fit uniform Poisson process
   fit0 <- mppm(X~1, dat)
   # fit correct nonuniform Poisson process
   fit1 <- mppm(X~x, dat)

   # test wrong model
   cdf.test(fit0, "x")
   # test right model
   cdf.test(fit1, "x")

   # Gibbs model
   fitGibbs <- update(fit0, interaction=Strauss(0.05))
   ns <- if(interactive()) 19 else 2
   cdf.test(fitGibbs, "x", nsim=ns)