spatstat (version 1.31-3)

bermantest: 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

bermantest(...)
## S3 method for class 'ppp':
bermantest(X, covariate,
                         which = c("Z1", "Z2"),
        alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'ppm':
bermantest(model, covariate,
                         which = c("Z1", "Z2"),
               alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'lpp':
bermantest(X, covariate,
                         which = c("Z1", "Z2"),
        alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'lppm':
bermantest(model, covariate,
                         which = c("Z1", "Z2"),
               alternative = c("two.sided", "less", "greater"), ...)

Arguments

X

A point pattern (object of class "ppp" or "lpp").

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.

...

Ignored.

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.

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

• IfXis a point pattern dataset (object of class"ppp"or"lpp"), thenbermantest(X, ...)performs a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness, CSR) for this dataset.
• Ifmodelis a fitted point process model (object of class"ppm"or"lppm") thenbermantest(model, ...)performs a test of goodness-of-fit for this fitted model. In this case,modelshould 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.

• Ifwhich="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$Z_1 = (S-\mu)/\sigma$.
• Ifwhich="Z2", the test statistic$Z_2$is computed as follows. The values of thecovariateat 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 thecovariateat 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.

References

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

See Also

kstest, quadrat.test, ppm

Examples

# Berman's data
   data(copper)
   X <- copper$SouthPoints
   L <- copper$SouthLines
   D <- distmap(L, eps=1)
   # test of CSR
   bermantest(X, D)
   bermantest(X, D, "Z2")