Stationary subspace analysis of nonstationary processes

Abstract

Stationary subspace analysis (SSA) is a recent technique for finding linear transformations of nonstationary processes that are stationary in the limited sense that the first two moments or means and lag‐0 covariances are time‐invariant. It finds a matrix that projects the nonstationary data onto a stationary subspace by minimizing a Kullback–Leibler divergence between Gaussian distributions measuring the nonconstancy of the means and covariances across several segments. We propose an SSA procedure for general multivariate, second‐order nonstationary processes. It relies on the asymptotic uncorrelatedness of the discrete Fourier transform of a stationary time series to define a measure of departure from stationarity, which is then minimized to find the stationary subspace. The dimension of the subspace is estimated using a sequential testing procedure, and its asymptotic properties are discussed. We illustrate the broader applicability and better performance of our method in comparison to existing SSA methods through simulations and discuss an application in analyzing electroencephalogram (EEG) data from brain–computer interface (BCI) experiments.

Reducing Brain Signal Noise in the Prediction of Economic Choices: A Case Study in Neuroeconomics

“…where for a matrix A ∈ ℝ d × d , denotes its Frobenius norm. A solution is obtained by minimizing D Y ( B ) subject to the orthonormality assumption , see Sundararajan and Pourahmadi ( 2017 ) for more details. In section 2.2.5 of the previous work a sequential technique for estimating the unknown dimension d of the stationary subspace is described.…”

“…Since the actual dimension d is unknown, we present the results for d = 4, 5, 6, 7, 8. We also applied the sequential technique in Sundararajan and Pourahmadi ( 2017 ) to detect d for each subject and each food choice task. Here we obtained a mode of d = 8 as an estimate of the dimension of the stationary subspace.…”

“…In this paper, we apply dependent SSA (DSSA, for short) method proposed in Sundararajan and Pourahmadi ( 2017 ) for the general class of multi-dimensional nonstationary time series to data from a neuroeconomics case study. DSSA relies on the asymptotic uncorrelatedness of the discrete Fourier transform (DFT) of a second-order stationary time series at different Fourier frequencies and unlike the well-known cointegration theory that is restricted to parametric models such as VAR, DSSA avoids parametric model assumptions.…”

“…The ability of DSSA and ISSA in detecting the true dimension of the stationary subspace process is illustrated for different sample sizes and dimensions and it was observed that in most cases DSSA performs better than ISSA. See section 3 of Sundararajan and Pourahmadi ( 2017 ) for more details.…”

Beyond Integration: A Symmetry-Based Approach to Reaching Stationarity in Economic Time Series

“…How can we find such coefficients? To find the corresponding coefficients, we can use well-developed co-integration techniques (see, e.g., [3]) or, better yet, the newly developed techniques of stationary subspace analysis (see, e.g., [4] and references therein). These techniques find stationary lineae combinations of non-stationary processes.…”