R: Berman's Tests for Point Process Model

berman.test {spatstat.core} | R Documentation

## 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

    berman.test(...)

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

    ## S3 method for class 'ppm'
    berman.test(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.
`...` | Additional arguments controlling the pixel resolution (arguments `dimyx` and `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 μ and variance σ^2 of S are computed from the values of the covariate at all locations in the window. Then we compute Z[1]=(S-μ)/σ. 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 Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### 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
       data(copper)
       X <- copper$SouthPoints
       L <- copper$SouthLines
       D <- distmap(L, eps=1)
       # test of CSR
       berman.test(X, D)
       berman.test(X, D, "Z2")

[Package spatstat.core version 2.3-1 Index]