berman.test.ppm {spatstat.model}

Berman's Tests for Point Process Model

Description

Tests the goodness-of-fit of a Poisson point process model using methods of Berman (1986).

Usage

## S3 method for class 'ppm'
berman.test(model, covariate,
                         which = c("Z1", "Z2"),
               alternative = c("two.sided", "less", "greater"), ...)

Arguments

model

A fitted point process model (object of class "ppm" or "lppm").

covariate

The spatial covariate on which the test will be based. An image (object of class "im") or a function.

which

Character string specifying the choice of test.

alternative

Character string specifying the alternative hypothesis.

...

Additional arguments controlling the pixel resolution (arguments dimyx, eps and rule.eps passed to as.mask) or other undocumented features.

Details

These functions perform a goodness-of-fit test of a Poisson point process model fitted to point pattern data. 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 either of two test statistics Z_1 and Z_2 proposed by Berman (1986). The Z_1 test is also known as the Lawson-Waller test.

The function berman.test is generic, with methods for point patterns ("ppp" or "lpp") and point process models ("ppm" or "lppm").

• If X is a point pattern dataset (object of class "ppp" or "lpp"), then berman.test(X, ...) performs a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness, CSR) for this dataset.

• If model is a fitted point process model (object of class "ppm" or "lppm") then berman.test(model, ...) performs a test of goodness-of-fit for this fitted model. In this case, model should be a Poisson point process.

The test is performed by comparing the observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same covariate under the model. Thus, you must nominate a spatial covariate for this test.

The argument covariate should be either a function(x,y) or a pixel image (object of class "im" containing the values of a spatial function. 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.

Next the values of the covariate at all locations in the observation window are evaluated. The point process intensity of the fitted model is also evaluated at all locations in the window.

• If which="Z1", the test statistic Z_1 is computed as follows. The sum S of the covariate values at all data points is evaluated. The predicted mean \mu and variance \sigma^2 of S are computed from the values of the covariate at all locations in the window. Then we compute Z_1 = (S-\mu)/\sigma. Closely-related tests were proposed independently by Waller et al (1993) and Lawson (1993) so this test is often termed the Lawson-Waller test in epidemiological literature.

• If which="Z2", the test statistic Z_2 is computed as follows. The values of the covariate at all locations in the observation window, weighted by the point process intensity, are compiled into a cumulative distribution function F. 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. The standardised sample mean of these numbers is the statistic Z_2.

In both cases the null distribution of the test statistic is the standard normal distribution, approximately.

The return value is an object of class "htest" containing the results of the hypothesis test. The print method for this class gives an informative summary of the test outcome.

Value

An object of class "htest" (hypothesis test) and also of class "bermantest", containing the results of the test. The return value can be plotted (by plot.bermantest) or printed to give an informative summary of the test.

Warning

The meaning of a one-sided test must be carefully scrutinised: see the printed output.

Author(s)

Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

References

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

Lawson, A.B. (1993) On the analysis of mortality events around a prespecified fixed point. Journal of the Royal Statistical Society, Series A 156 (3) 363–377.

Waller, L., Turnbull, B., Clark, L.C. and Nasca, P. (1992) Chronic Disease Surveillance and testing of clustering of disease and exposure: Application to leukaemia incidence and TCE-contaminated dumpsites in upstate New York. Environmetrics 3, 281–300.

See Also

cdf.test, quadrat.test, ppm

Examples

   # Berman's data
   X <- copper$SouthPoints
   L <- copper$SouthLines
   D <- distmap(L, eps=1)
   # test of fitted model
   fit <- ppm(X ~ x+y)
   berman.test(fit, D)