empirical_bootstrap_test {covsep}

Projection-based empirical bootstrap test for separability of covariance structure

Description

This function performs the test for the separability of covariance structure of a random surface based on the empirical bootstrap procedure described in the paper http://arxiv.org/abs/1505.02023.

Usage

empirical_bootstrap_test(Data, L1 = 1, L2 = 1, studentize = "full",
  B = 1000, verbose = TRUE)

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.

studentize

parameter to specify which type of studentization is performed. Possible options are 'no', 'diag' or 'full' (see details section).

B

number of bootstrap replicates to be used.

verbose

logical parameter for printing progress

Value

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

Details

This function performs the test of separability of the covariance structure for a random surface (introduced in the paper http://arxiv.org/abs/1505.02023), when generated from a Gaussian process. The sample surfaces need to be measured on a common regular grid. The test consider a subspace formed by the tensor product of eigenfunctions of the separable covariances. It is possible to specify the number of eigenfunctions to be considered in each direction.

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 x v_j : i = 1, ..., l1; j = 1, ..., l2, for each pair (l1,l2) = (L1[k], L2[k]), for k = 1:length(L1).

studentize can take the values

'full'

default & recommended method. Yhe projection coordinates are renormalized by an estimate of their joint covariance

'no'

NOT RECOMMENDED. No studentization is performed

'diag'

NOT RECOMMENDED. Each projection coordinate is renormalized by an estimate of its standard deviation

B the number of bootstrap replicates (1000 by default).

verbose to print the progress of the computations (TRUE by default)

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

gaussian_bootstrap_test, clt_test

Examples

data(SurfacesData)
empirical_bootstrap_test(SurfacesData)
empirical_bootstrap_test(SurfacesData, B=100)
empirical_bootstrap_test(SurfacesData,L1=2,L2=2, B=1000, studentize='full')