Finding Stationary Subspaces in Multivariate Time Series

Abstract

Identifying temporally invariant components in complex multivariate time series is key to understanding the underlying dynamical system and predict its future behavior. In this Letter, we propose a novel technique, stationary subspace analysis (SSA), that decomposes a multivariate time series into its stationary and nonstationary part. The method is based on two assumptions: (a) the observed signals are linear superpositions of stationary and nonstationary sources; and (b) the nonstationarity is measurable in the first two moments. We characterize theoretical and practical properties of SSA and study it in simulations and cortical signals measured by electroencephalography. Here, SSA succeeds in finding stationary components that lead to a significantly improved prediction accuracy and meaningful topographic maps which contribute to a better understanding of the underlying nonstationary brain processes.

Feature Extraction for Change-Point Detection Using Stationary Subspace Analysis

“…Stationary Subspace Analysis [35] factorizes a multivariate time series x(t) ∈ R D into stationary and non-stationary sources according to the linear mixing model,…”

“…In the SSA algorithms [35,11], a time series X t is considered stationary if its mean and covariance is constant over time, i.e…”

“…This is a variant of weak stationarity [28] where we do not take time structure into account. Following this concept of stationarity, the SSA algorithm [35] finds the s-projectionB s that minimizes the difference between the first two moments of the estimated s-sourcesŝ s (t) across epochs of the time series, since we cannot estimate the mean and covariance at a single time point. Thus we divide the samples from x(t) into n non-overlapping epochs defined by the index sets T 1 , .…”

“…In this paper we show how to extract useful features for change point detection by finding the most non-stationary directions using Stationary Subspace Analysis [35]. Even though there exists a wide range of feature extraction methods for classification and regression [10], to date no specialized procedure for feature extraction or for general signal processing [12] has been proposed for change point detection.…”

Learning a common substructure of multiple graphical Gaussian models

“…The scope of the present paper is in the latter context where the relationship among datasets is the objective we want to analyze. For the purpose, we focus on invariance of the data against the underlying changes which provides partial yet important aspects of the data behaviors (von Bünau et al, 2009;Hara et al, 2012). We provide a technique for finding one of such invariance, specifically constant interactions or dependencies among variables across several different conditions.…”