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##### Dispersion Test for Spatial Point Pattern Based on Quadrat Counts

Performs a test of Complete Spatial Randomness for a given point pattern, based on quadrat counts. Alternatively performs a goodness-of-fit test of a fitted inhomogeneous Poisson model. By default performs chi-squared tests; can also perform Monte Carlo based tests.

Keywords
htest, spatial
##### Usage
quadrat.test(X, ...)## S3 method for class 'ppp':
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)## S3 method for class 'ppm':
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)## S3 method for class 'quadratcount':
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
nsim=1999)
##### Arguments
X
A point pattern (object of class "ppp") to be subjected to the goodness-of-fit test. Alternatively a fitted point process model (object of class "ppm") to be tested. Alternatively X can be the result
nx,ny
Numbers of quadrats in the $x$ and $y$ directions. Incompatible with xbreaks and ybreaks.
alternative
Character string (partially matched) specifying the alternative hypothesis.
method
Character string (partially matched) specifying the test to use: either method="Chisq" for the chi-squared test (the default), or method="MonteCarlo" for a Monte Carlo test.
conditional
Logical. Should the Monte Carlo test be conducted conditionally upon the observed number of points of the pattern? Ignored if method="Chisq".
...
Ignored.
xbreaks
Optional. Numeric vector giving the $x$ coordinates of the boundaries of the quadrats. Incompatible with nx.
ybreaks
Optional. Numeric vector giving the $y$ coordinates of the boundaries of the quadrats. Incompatible with ny.
tess
Tessellation (object of class "tess") determining the quadrats. Incompatible with nx, ny, xbreaks, ybreaks.
nsim
The number of simulated samples to generate when method="MonteCarlo".
##### Details

These functions perform $\chi^2$ tests or Monte Carlo tests of goodness-of-fit for a point process model, based on quadrat counts.

The function quadrat.test is generic, with methods for point patterns (class "ppp"), split point patterns (class "splitppp"), point process models (class "ppm") and quadrat count tables (class "quadratcount").

• ifXis a point pattern, we test the null hypothesis that the data pattern is a realisation of Complete Spatial Randomness (the uniform Poisson point process). Marks in the point pattern are ignored.
• ifXis a split point pattern, then for each of the component point patterns (taken separately) we test the null hypotheses of Complete Spatial Randomness. Seequadrat.test.splitpppfor documentation.
• IfXis a fitted point process model, then it should be a Poisson point process model. The data to which this model was fitted are extracted from the model object, and are treated as the data point pattern for the test. We test the null hypothesis that the data pattern is a realisation of the (inhomogeneous) Poisson point process specified byX.

In all cases, the window of observation is divided into tiles, and the number of data points in each tile is counted, as described in quadratcount. The quadrats are rectangular by default, or may be regions of arbitrary shape specified by the argument tess. The expected number of points in each quadrat is also calculated, as determined by CSR (in the first case) or by the fitted model (in the second case). Then we perform either the $\chi^2$ test of goodness-of-fit to the quadrat counts (if method="Chisq") or a Monte Carlo test (if method="MonteCarlo").

If method="Chisq" then the $\chi^2$ test of goodness-of-fit is performed. The Pearson $X^2$ statistic $$X^2 = sum((observed - expected)^2/expected)$$ is computed, and compared to the $\chi^2$ distribution with $m-k$ degrees of freedom, where m is the number of quadrats and $k$ is the number of fitted parameters (equal to 1 for quadrat.test.ppp). The default is to compute the two-sided $p$-value, so that the test will be declared significant if $X^2$ is either very large or very small. One-sided $p$-values can be obtained by specifying the alternative. An important requirement of the $\chi^2$ test is that the expected counts in each quadrat be greater than 5.

If method="MonteCarlo" then a Monte Carlo test is performed, obviating the need for all expected counts to be at least 5. In the Monte Carlo test, nsim random point patterns are generated from the null hypothesis (either CSR or the fitted point process model). The Pearson $X^2$ statistic is computed as above. The $p$-value is determined by comparing the $X^2$ statistic for the observed point pattern, with the values obtained from the simulations. Again the default is to compute the two-sided $p$-value.

If conditional is TRUE then the simulated samples are generated from the multinomial distribution with the number of trials equal to the number of observed points and the vector of probabilities equal to the expected counts divided by the sum of the expected counts. Otherwise the simulated samples are independent Poisson counts, with means equal to the expected counts.

The return value is an object of class "htest". Printing the object gives comprehensible output about the outcome of the test.

The return value also belongs to the special class "quadrat.test". Plotting the object will display the quadrats, annotated by their observed and expected counts and the Pearson residuals. See the examples.

##### Value

• An object of class "htest". See chisq.test for explanation.

The return value is also an object of the special class "quadrattest", and there is a plot method for this class. See the examples.

quadrat.test.splitppp, quadratcount, quadrats, quadratresample, chisq.test, kstest.

To test a Poisson point process model against a specific alternative, use anova.ppm.

##### Examples
data(simdat)

# Using Monte Carlo p-values
quadrat.test(swedishpines) # Get warning, small expected values.

# fitted model: inhomogeneous Poisson
fitx <- ppm(simdat, ~x, Poisson())

residuals(te)  # Pearson residuals

plot(te)

plot(simdat, pch="+", cols="green", lwd=2)
plot(te, add=TRUE, col="red", cex=1.4, lty=2, lwd=3)

sublab <- eval(substitute(expression(p[chi^2]==z),
list(z=signif(te$p.value,3)))) title(sub=sublab, cex.sub=3) # quadrats of irregular shape B <- dirichlet(runifpoint(6, simdat$window))
plot(qB, add=TRUE, col="red", lwd=2, cex=1.2)