GeoTestIsotropy: Parametric Bootstrap Tests for Spatial Isotropy

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

Parametric Bootstrap Tests for Spatial Isotropy

Description

Performs two parametric-bootstrap tests of spatial isotropy by comparing an isotropic model under the null hypothesis with a geometrically anisotropic model under the alternative. The first test (MWCLT) is based on the estimated anisotropy ratio. The second test (WCLRT) is based on a likelihood-ratio or composite-likelihood-ratio statistic. Their null distributions are estimated by simulation from the fitted isotropic model.

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,
 method = c("cholesky", "TB"),
 L = 10000,
 n = 1,
 angle_starts = c(pi/4, 3*pi/4),
 batch_size = NULL,
 max_attempts = NULL,
 parallel = FALSE,
 ncores = 6,
 progress = TRUE,
 seed = NULL)

Arguments

data	Numeric vector of observations.
coordx	Numeric matrix with exactly two columns containing the spatial coordinates. The number of rows must equal length(data).
start	Named list containing the covariance and marginal parameters to be estimated. Parameters angle and ratio must not be included, because they are handled internally by the isotropy test.
fixed	Named list containing the covariance and marginal parameters kept fixed. Together, start and fixed must contain exactly the parameters required by CorrParam(corrmodel) and NuisParam(model).
optimizer	Optimization algorithm passed to GeoFit.
model	Marginal random-field model.
corrmodel	Spatial correlation model.
lower, upper	Named lists of lower and upper bounds for the parameters in start. Missing bounds are replaced by -Inf and Inf, respectively.
B	Number of valid parametric-bootstrap replications used to calibrate each test. Failed fitting attempts are replaced by newly simulated samples.
likelihood	Likelihood type passed to GeoFit. If both likelihood and type are NULL, full likelihood is used for at most 2000 observations and marginal pairwise composite likelihood is used otherwise.
type	Likelihood type passed to GeoFit; typically "Standard" for full likelihood or "Pairwise" for composite likelihood. It must be supplied together with likelihood.
copula	Optional copula specification. With a supported purely spatial correlation model, both method="cholesky" and method="TB" are available.
neighb	Number of nearest neighbours used when type = "Pairwise". Ignored otherwise.
method	Simulation method under the null hypothesis. Use "cholesky" for exact simulation or "TB" for approximate turning-bands simulation. When copula is supplied, the corresponding latent engine is selected internally by GeoSimCopula.
L	Number of spectral components used when method = "TB".
n	Common number of trials/successes used by models that require an integer n, such as Binomial-type models. It can be a scalar or, where supported, a vector with one value per observation. For "Bernoulli" and "Binary", n is internally fixed to 1.
angle_starts	Finite numeric vector of starting angles used for the anisotropic fit. All supplied angles are tried (modulo \pi) and the fit with the largest objective value is retained. The default uses two starts, \pi/4 and 3\pi/4, for both the observed fit and every bootstrap refit.
batch_size	Number of bootstrap datasets simulated and processed at one time. If NULL, a modest batch size is selected automatically. Batching limits memory use for large spatial datasets.
max_attempts	Maximum number of simulated bootstrap datasets allowed in order to collect exactly B valid replications. If NULL, the default is max(B + 20, ceiling(1.5 * B)). The function stops if this limit is reached before B valid replications are obtained.
parallel	Logical; if TRUE, bootstrap refits are distributed across multiple R sessions. With method="TB", bootstrap simulation may also use multiple workers when supported by the simulation routine.
ncores	Positive integer or NULL. With parallel=TRUE, an explicit integer requests that many workers, capped by detected cores and the current number of jobs. Set ncores=NULL for automatic selection.
progress	Logical; if TRUE, display the same progressr text progress bar used by other parallel GeoModels functions. A single bar is shown for the whole bootstrap and counts valid bootstrap replications in both sequential and parallel execution.
seed	Optional finite numeric seed. The previous random-number-generator state is restored when the function exits.

Details

The null and alternative hypotheses are

H_0: \mathrm{ratio}=1,

and

H_1: \mathrm{ratio}>1,

with the anisotropy angle estimated under the alternative. Under H_0, the angle is not identifiable and the ratio is on the boundary of its parameter space. Consequently, a standard chi-squared reference distribution is not assumed; both tests are calibrated by parametric bootstrap.

The MWCLT statistic is the unconstrained anisotropy-ratio estimate \widehat\tau. If \widehat\tau_b^* denotes the corresponding estimate in bootstrap replication b, the upper-tail bootstrap p-value is

\widehat p_\tau = \frac{1 + \sum_{b=1}^B I(\widehat\tau_b^* \geq \widehat\tau)}{B+1}.

The WCLRT statistic is

T_{\mathrm{obs}}=\max\{0,2(\ell_1-\ell_0)\},

where \ell_0 and \ell_1 are the maximized likelihoods or composite likelihoods under the isotropic and anisotropic models. Its bootstrap p-value is

\widehat p_T = \frac{1 + \sum_{b=1}^B I(T_b \geq T_{\mathrm{obs}})}{B+1}.

The returned components statistic and pvalue remain aliases for this WCLRT statistic and p-value for backward compatibility.

For every observed or bootstrap anisotropic fit, all angles in angle_starts are used as starting values and the valid fit with the largest objective value is retained. During bootstrap refitting, the fitted observed anisotropy angle is also included as a starting angle. If the best numerical anisotropic fit has an objective value below the isotropic fit by more than numerical tolerance, the boundary solution is selected, giving ratio 1 and WCLRT statistic 0.

For pairwise composite likelihood, the isotropic and anisotropic fits must use the same retained pairs. The function checks this condition before computing the WCLRT statistic.

Bootstrap datasets are generated from the fitted isotropic model in batches. Failed bootstrap fits do not enter the empirical null distribution: new samples are generated until exactly B valid replications have been collected, or until max_attempts is reached. This makes the denominator of both bootstrap p-values exactly B+1.

The argument n is propagated consistently to GeoFit, GeoSim, GeoSimapprox, and GeoSimCopula. Direct latent Gaussian simulation of some non-Gaussian models imposes additional structural constraints. In particular, direct "Gamma" simulation requires the fitted shape to be a positive integer because it is the number of squared latent Gaussian fields. For the direct Gamma model used in the latent construction, it is therefore recommended to keep shape fixed at a positive integer during the isotropy test. This integer restriction does not apply merely because the marginal distribution is Gamma when a supported copula construction is used.

Value

An invisible list containing, among others:

statistic, pvalue	Backward-compatible aliases for the observed WCLRT statistic and its bootstrap p-value.
statistic_lrt, pvalue_lrt	Observed WCLRT statistic and p-value.
statistic_ratio, pvalue_ratio	Observed MWCLT anisotropy-ratio statistic and p-value.
pvalues	Named vector containing the MWCLT and WCLRT p-values.
ratio_hat, angle_hat	Estimated anisotropy ratio and angle. The angle is NA when the boundary solution is selected.
parallel, ncores	Whether parallel bootstrap refitting was used and the number of workers selected by the function.
fit_H0, fit_H1	Fitted isotropic and best anisotropic GeoFit objects for the observed data.
B_rep, B_rep_lrt	The B valid bootstrap WCLRT statistics.
B_rep_ratio	The B valid bootstrap anisotropy-ratio estimates.
B_rep_angle	Bootstrap angle estimates; boundary solutions are NA.
bootstrap_B	Requested and obtained number of valid bootstrap replications.
bootstrap_total_attempts	Total number of simulated datasets needed to obtain B valid replications.
bootstrap_failed_attempts	Number of failed attempts that were replaced.
bootstrap_failures	Data frame describing failed attempts.
bootstrap_boundary_solutions	Number of valid bootstrap replications for which the boundary solution was selected.
observed_multistart_angles	Starting angles used for the observed H1 fit.
batch_size, max_attempts	Bootstrap batching and retry settings used by the function.

Author(s)

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

See Also

GeoFit, GeoSim, GeoSimapprox, GeoVariogramDir

Examples

## Not run: 
## Isotropy tests for a Gaussian random field simulated under H0
set.seed(123)
nsite <- 500
coords <- cbind(runif(nsite), runif(nsite))

param <- list(
  mean = 0,
  nugget = 0,
  sill = 1,
  scale = 0.20,
  smooth = 0.5
)

z <- GeoSim(
  coordx = coords,
  corrmodel = "Matern",
  model = "Gaussian",
  param = param
)$data

start <- list(
  mean = 0,
  sill = 1,
  scale = 0.20,
  smooth = 0.5
)
fixed <- list(nugget = 0)

lower <- list(mean = -5, sill = 0.01, scale = 0.01, smooth = 0.05)
upper <- list(mean =  5, sill = 5,    scale = 1,    smooth = 2.5)

iso_test <- GeoTestIsotropy(
  data = z,
  coordx = coords,
  start = start,
  fixed = fixed,
  model = "Gaussian",
  corrmodel = "Matern",
  lower = lower,
  upper = upper,
  B = 99,
  likelihood = "Full",
  type = "Standard",
  method = "cholesky",
  seed = 321
)

iso_test$pvalue_ratio  # MWCLT
iso_test$pvalue_lrt    # WCLRT

## End(Not run)

GeoModels documentation built on Sept. 23, 2026, 5:07 p.m.