tau.est {NTSS}

(Partial) correlation coefficient between a point process and a covariate

Description

Correlation coefficient between a point process and a random field (covariate of interest), taking into account the possible effect of nuisance covariates, see Dvořák and Mrkvička (2022). The random shift test based on this test statistics is given in the function tau.test.

Usage

tau.est(
  X,
  covariate.interest,
  covariates.nuisance,
  bws,
  n.test.points = 1000,
  nonparametric = TRUE,
  verbose = FALSE,
  bw.factor.rhonhat = 1
)

Arguments

X

point pattern dataset (object of class ppp)

covariate.interest

random field (object of class im)

covariates.nuisance

list of covariates (objects of class im) determining the nuisance covariates

bws

vector of positive real values from which the bandwidth is adaptively chosen if at least one nuisance covariate is present; if no nuisance covariates are present, the first value is used

n.test.points

the number of independent test points used in computing the test statistic value

nonparametric

logical value indicating whether nonparametric residuals should be used when computing the test statistic

verbose

logical value indicating whether auxiliary information should be printed and auxiliary figures plotted during the computation

bw.factor.rhonhat

multiplicative factor used when determining the bandwidth in the nonparametric estimation of the intensity function depending on the nuisance covariates (defaults to 1)

Details

This function computes the Kendall's correlation coefficient between the covariate of interest and the smoothed residual field, sampled at a given number of test points scattered independently in the observation window, see the paper Dvořák and Mrkvička (2022). If no nuisance covariates are given, a constant intensity function of the point process is assumed when constructing the residuals. If one or more nuisance covariates are provided, an intensity function depending on the nuisance covariates (but not on the covariate of interest) is assumed and the residuals are constructed using this intensity function.

For constructing the smoothed residual field an adaptive bandwidth selection method can be used, see Dvořák and Mrkvička (2022). A vector of candidate bandwidth values can be provided using the argument bws.

The residuals can be constructed in a nonparametric way (see Baddeley et al. (2012)) or in a parametric way (using the ppm function from the spatstat package, see Baddeley et al. (2015)). This choice is given by the argument nonparametric. The raw residuals are considered here.

The observed point pattern should be supplied using the argument X. The realization of the covariate of interest should be supplied using the argument covariate.interest. The set of nuisance covariates should be supplied as a list using the argument covariates.nuisance. This list can be empty if no nuisance covariates are considered.

Value

Value of the (partial) Kendall's correlation coefficient.

References

J. Dvořák, T. Mrkvička (2022): Nonparametric testing of the covariate significance for spatial point patterns under the presence of nuisance covariates. https://arxiv.org/abs/2210.05424

T. Mrkvička, J. Dvořák, J.A. González, J. Mateu (2021): Revisiting the random shift approach for testing in spatial statistics. Spatial Statistics 42, 100430.

A. Baddeley, E. Rubak, R. Turner (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman & Hall Interdisciplinary Statistics Series. CRC Press, Boca Raton, Florida.

A. Baddeley, Y.-M. Chang, Y. Song, R. Turner (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5(2), 221?236.

Examples

library(spatstat)
library(ks)

# the point pattern
X <- bei
plot(X)

# two covariates are available
elevation <- bei.extra$elev
slope <- bei.extra$grad
plot(elevation)
plot(slope)

# candidate values for adaptive bandwidth selection
bws <- seq(from=12.5, to=100, by=12.5)

# no nuisance covariates
out1 <- tau.est(X, covariate.interest=elevation, covariates.nuisance=NULL,
                bws=bws, verbose=TRUE)
out1

# one nuisance covariate
out2 <- tau.est(X, covariate.interest=elevation, covariates.nuisance=list(slope=slope),
                bws=bws, verbose=TRUE)
out2