Sparse Cholesky covariance parametrization for recovering latent structure in ordered data

TL;DR

This paper introduces a novel sparse covariance Cholesky factor estimation method for ordered data, interpreting it as a hidden variable model, and evaluates its performance on simulated and real spatial and temporal datasets.

Contribution

It proposes a new matrix loss penalization approach for estimating sparse covariance Cholesky factors, extending the model to unordered data scenarios.

Findings

The new estimator outperforms regression-based methods in simulations.

Effective in spatial and temporal data with natural ordering.

Guidelines provided for method selection based on empirical results.

Abstract

The sparse Cholesky parametrization of the inverse covariance matrix can be interpreted as a Gaussian Bayesian network; however its counterpart, the covariance Cholesky factor, has received, with few notable exceptions, little attention so far, despite having a natural interpretation as a hidden variable model for ordered signal data. To fill this gap, in this paper we focus on arbitrary zero patterns in the Cholesky factor of a covariance matrix. We discuss how these models can also be extended, in analogy with Gaussian Bayesian networks, to data where no apparent order is available. For the ordered scenario, we propose a novel estimation method that is based on matrix loss penalization, as opposed to the existing regression-based approaches. The performance of this sparse model for the Cholesky factor, together with our novel estimator, is assessed in a simulation setting, as well as over spatial and temporal real data where a natural ordering arises among the variables. We give guidelines, based on the empirical results, about which of the methods analysed is more appropriate for each setting.