In this section
Most of the members of this group are from the Statistics Section and Biomaths research group of the Department of Mathematics. Below you can find a list of research areas that members of this group are currently working on and/or would like to work on by applying their developed mathematical and statistical methods.
@article{Battey:2017,
author = {Battey, HS},
journal = {Bernoulli},
pages = {3166--3177},
title = {Eigen structure of a new class of structured covariance and inverse covariance matrices},
url = {http://www.bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers},
volume = {23},
year = {2017}
}
TY - JOUR
AB - There is a one to one mapping between a p dimensional strictly positive definite covariancematrix Σ and its matrix logarithm L. We exploit this relationship to study thestructure induced on Σ through a sparsity constraint on L. Consider L as a randommatrix generated through a basis expansion, with the support of the basis coefficientstaken as a simple random sample of size s = s∗from the index set [p(p + 1)/2] ={1, . . . , p(p + 1)/2}. We find that the expected number of non-unit eigenvalues of Σ, denotedE[|A|], is approximated with near perfect accuracy by the solution of the equation4p + p(p − 1)2(p + 1)hlog pp − d −d2p(p − d)i− s∗ = 0.Furthermore, the corresponding eigenvectors are shown to possess only p − |Ac| nonzeroentries. We use this result to elucidate the precise structure induced on Σ and Σ−1.We demonstrate that a positive definite symmetric matrix whose matrix logarithm issparse is significantly less sparse in the original domain. This finding has importantimplications in high dimensional statistics where it is important to exploit structure inorder to construct consistent estimators in non-trivial norms. An estimator exploitingthe structure of the proposed class is presented.
AU - Battey,HS
EP - 3177
PY - 2017///
SP - 3166
TI - Eigen structure of a new class of structured covariance and inverse covariance matrices
T2 - Bernoulli
UR - http://www.bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers
VL - 23
ER -
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