Finding Stationary Subspaces in Multivariate Time Series

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.

Figure 1

Nonstationary and stationary source with time-variable covariance illustrated by amplitude scatter plots for four epochs.

Figure 2

Median error of SSA measured as the angle between the true and estimated $\mathfrak{n}$-space (vertical axis) for varying numbers of epochs (horizontal axis) on synthetic 20-dimensional data. The red and blue curves show the performance for 14 and 6 nonstationary sources (mixed with a random matrix), respectively. We consider two scenarios: only the covariance matrix of the $\mathfrak{n}$-sources is varied between epochs (solid curves); and both the mean and the covariance matrix are chosen at random (dashed curves). The impact of small sample statistics is excluded by using the exact mean and covariance matrix directly in the SSA procedure. The error bars stretch from the 25% to the 75% quantile estimated over 1000 random realizations of the data.

Figure 3

Median error of SSA measured as the angle between the true and estimated $\mathfrak{n}$-space (vertical axis) for varying numbers of epochs (horizontal axis) that a fixed number of samples is split into. The data consists of four $\mathfrak{s}$-sources embedded in eight dimensions using a random mixing matrix. The total size of the sample is $K=1000$; it is composed of seven segments with random length, each with a random covariance matrix for the $\mathfrak{n}$-sources. The error bars stretch from the 25% to the 75% quantile computed over 1000 random realizations of the data.

Figure 4

The left panel shows the classification error on the original EEG BCI data set with and without preprocessing by SSA, the middle panel the evolution of the error for different amounts of alpha oscillations that have been added. In the right panel, the relative differences in signal power between calibration—and test set is plotted as a function of the electrode position. The field pattern shows the head from above, each cross is an electrode. The number of $\mathfrak{s}$-sources is $d=44$; the number of epochs formed as consecutive trials is $N=30$. The result is robust with respect to reasonable parameter variations.