clt_test {covsep}

Test for separability of covariance operators for Gaussian process.

Description

This function performs the asymptotic test for the separability of the covariance operator for a random surface generated from a Gaussian process (described in the paper http://arxiv.org/abs/1505.02023).

Usage

clt_test(Data, L1, L2)

Arguments

Data

a (non-empty) N x d1 x d2 array of data values. The first direction indices the N observations, each consisting of a d1 x d2 discretization of the surface, so that Data[i,,] corresponds to the i-th observed surface.

L1

an integer or vector of integers in 1:p indicating the eigenfunctions in the first direction to be used for the test.

L2

an integer or vector of integers in 1:q indicating the eigenfunctions in the second direction to be used for the test.

Value

The p-value of the test for each pair (l1,l2) = (L1[k], L2[k]), for k = 1:length(L1).

Details

If L1 and L2 are vectors, they need to be of the same length.

The function tests for separability using the projection of the covariance operator in the separable eigenfunctions u_i tensor v_j : i = 1, ..., l1; j = 1, ..., l2, for each pair (l1,l2) = (L1[k], L2[k]), for k = 1:length(L1).

The test works by using asymptotics, and is only valid if the data is assumed to be Gaussian.

The surface data needs to be measured or resampled on a common regular grid or on common basis functions.

References

Aston, John A. D.; Pigoli, Davide; Tavakoli, Shahin. Tests for separability in nonparametric covariance operators of random surfaces. Ann. Statist. 45 (2017), no. 4, 1431–1461. doi:10.1214/16-AOS1495. https://projecteuclid.org/euclid.aos/1498636862

See Also

empirical_bootstrap_test, gaussian_bootstrap_test

Examples

data(SurfacesData)
clt_test(SurfacesData, L1=c(1,2), L2=c(1,4))