kroncov: The covariance estimation of tensor normal distribution

In TRES: Tensor Regression with Envelope Structure

Description

This function provides the MLE of the covariance matrix of tensor normal distribution, where the covariance has a separable Kronecker structure, i.e. Σ=Σ_{m}\otimes … \otimesΣ_{1}. The algorithm is a generalization of the MLE algorithm in Manceur, A. M., & Dutilleul, P. (2013).

Usage

kroncov(Tn, tol = 1e-06, maxiter = 10)

Arguments

Tn	A p_1\times\cdots p_m\times n matrix, array or tensor, where n is the sample size.
tol	The convergence tolerance with default value 1e-6. The iteration terminates when ||Σ_i^{(t+1)} - Σ_i^{(t)}||_F < tol for some covariance matrix Σ_i.
maxiter	The maximal number of iterations. The default value is 10.

Details

The individual component covariance matrices Σ_i, i=1,…, m are not identifiable. To overcome the identifiability issue, each matrix Σ_i is normalized at the end of the iteration such that ||Σ_i||_F = 1. And an overall normalizing constant λ is extracted so that the overall covariance matrix Σ is defined as

Σ = λ Σ_m \otimes \cdots \otimes Σ_1.

If Tn is a p \times n design matrix for a multivariate random variable, then lambda = 1 and S is a length-one list containing the sample covariance matrix.

Value

lambda	The normalizing constant.
S	A matrix list, consisting of each normalized covariance matrix Σ_1,…,Σ_m.

References

Manceur, A.M. and Dutilleul, P., 2013. Maximum likelihood estimation for the tensor normal distribution: Algorithm, minimum sample size, and empirical bias and dispersion. Journal of Computational and Applied Mathematics, 239, pp.37-49.

TRES documentation built on Oct. 20, 2021, 9:06 a.m.