GeoTestIsotropy: Parametric bootstrap test for isotropy

In GeoModels: Procedures for Gaussian and Non Gaussian Geostatistical (Large) Data Analysis

Parametric bootstrap test for isotropy

Description

Performs an isotropy test based on parametric bootstrap by comparing an isotropic model (H0) with an anisotropic model (H1). A user-chosen spatial correlation model is fitted to spatial data, and the test statistic is a likelihood ratio between H1 (with anisotropy estimated) and H0 (anisotropy fixed at ratio = 1 and angle = 0). The p-value is computed via parametric bootstrap.

Usage

GeoTestIsotropy(data, coordx,
                start, fixed,
                optimizer = "bobyqa",
                model = "Gaussian",
                corrmodel = "Matern",
                lower = NULL, upper = NULL,
                B = 1000,
                likelihood = NULL,
                type = NULL,
                copula = NULL,
                neighb = 5,
                parallel = TRUE,
                ncores = NULL,
                progress = TRUE)

Arguments

data	Numeric vector of observations (length n).
coordx	n x d matrix with the spatial coordinates of the sites (at least 2 columns for anisotropy analysis).
start	Named list with the parameters to be estimated; names must match those expected by CorrParam(corrmodel) and NuisParam(model).
fixed	Named list with the parameters kept fixed during estimation; together with start it must exactly cover the required set of parameters.
optimizer	Optimization algorithm to be used in GeoFit (e.g., "bobyqa").
model	Marginal model for the data (e.g., "Gaussian").
corrmodel	Spatial correlation model (e.g., "Matern").
lower, upper	Named lists with lower/upper bounds for parameters in start only. If NULL, no extra bounds beyond defaults.
B	Number of parametric bootstrap replications (default 1000).
likelihood	Type of (composite) likelihood to pass to GeoFit (e.g., "Marginal").
type	Composite-likelihood type (e.g., "Pairwise").
copula	Copula object for simulation via GeoSimCopula; if NULL Gaussian simulation is used.
neighb	Number of neighbors for composite-likelihood estimation, where applicable.
parallel	If TRUE, bootstrap fits are distributed in parallel through future.
ncores	Number of cores to use in parallel; if NULL it is chosen automatically up to B or parallel::detectCores()-1.
progress	If TRUE, show progress bars using progressr.

Details

GeoTestIsotropy implements a likelihood ratio test (LRT) between:

• H0: isotropic model with anisotropy parameters fixed at angle = 0, ratio = 1;

• H1: anisotropic model with angle and ratio estimated.

The observed statistic is \mathrm{LRT}_{\text{obs}} = 2(\ell_1 - \ell_0), truncated at zero if negative. The null distribution is approximated by simulating B datasets under H0 with parameters equal to the H0 fit, and recomputing the LRT. The p-value is computed as (1 + \#\{\mathrm{LRT}_b \ge \mathrm{LRT}_{\text{obs}}\}) / (1 + B_{\text{valid}}). The function includes robust checks on parameter names, bounds, simulation method choice (Cholesky vs TB), and parallelization.

Value

An (invisible) list with components:

statistic	Observed LRT statistic.
pvalue	Bootstrap p-value.
ratio_hat	Estimated anisotropy ratio (> 1 indicates stretching along the principal axis).
angle_hat	Estimated anisotropy angle (radians/degrees according to GeoFit convention).
corrmodel	Correlation model used.
model	Marginal model used.
fit_H0	GeoFit object fitted under H0.
fit_H1	GeoFit object fitted under H1.
B_rep	Vector of LRT values obtained on the valid bootstrap replications.

Author(s)

Moreno Bevilacqua [e-mail] https://sites.google.com/view/moreno-bevilacqua/home
Víctor Morales Oñate [e-mail] https://sites.google.com/site/moralesonatevictor/
Christian Caamaño-Carrillo [e-mail] https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit, GeoSim, GeoSimapprox, GeoVarestbootstrap

GeoModels documentation built on Nov. 25, 2025, 1:06 a.m.