# Stationary subspace analysis of nonstationary covariance processes: eigenstructure description and testing

## TL;DR

This paper develops eigenstructure-based methods to identify and test stationary subspaces within nonstationary covariance processes, providing a computationally efficient alternative to optimization techniques.

## Contribution

It characterizes stationary subspaces via eigenvalues and eigenvectors, and introduces statistical tests and estimation methods for their dimensions.

## Key findings

- Eigenstructure characterization of stationary subspaces.
- Proposed eigenstructure-based estimation methods.
- Validated methods on simulated and real data.

## Abstract

Stationary subspace analysis (SSA) searches for linear combinations of the components of nonstationary vector time series that are stationary. These linear combinations and their number defne an associated stationary subspace and its dimension. SSA is studied here for zero mean nonstationary covariance processes. We characterize stationary subspaces and their dimensions in terms of eigenvalues and eigenvectors of certain symmetric matrices. This characterization is then used to derive formal statistical tests for estimating dimensions of stationary subspaces. Eigenstructure-based techniques are also proposed to estimate stationary subspaces, without relying on previously used computationally intensive optimization-based methods. Finally, the introduced methodologies are examined on simulated and real data.