Stationary subspace analysis of nonstationary covariance processes: Eigenstructure description and testing

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 define 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.

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

“…The figures show that VC method performs better than DSSA even "locally" for most values of u under measure D 2 whereas DSSA performs better under measure D 1 . Analogous figures for Models 2-4 can be found in Sundararajan et al (2019). The results are similar to Figure 6, especially for larger sample sizes.…”

“…It is noted that DSSA always provides a lower estimate of d than the VC method. Similar plots for all 9 subjects can be found Sundararajan et al (2019) Table 3: Left: Out-of-sample classification accuracy (in %) for the 3 subjects S3, S5 and S8 for the two indicated methods with p = 5. Right: Out-of-sample classification accuracy (in %) for 3 subjects S3, S5 and S8 corresponding to d= 7, 9, 11, and 13 for the VC method with p = 22.…”

“…But for simplicity and better readability, we shall assume that µ 4 is known. Construction of consistent estimators µ 4 is discussed in the supplementary technical appendix (Sundararajan et al (2019)), and would be sufficient for the results below to hold assuming that µ 4 is estimated through µ 4 . Furthermore, if one is willing to assume Gaussianity of Y t , note that µ 4 = 2 can also be used.…”

“…The estimation results for the two methods and the four considered models are presented through violin plots in Figure 3. Violin plots are intended as a somewhat qualitative visualization of the results -perhaps slightly more informative histograms for the results can be found in Sundararajan et al (2019). The plots show that estimation improves with increasing sample size T , and that the proposed VC method performs better than the competing DSSA method in detecting the true dimension d.…”

Modeling Spectral Properties in Stationary Processes of Varying Dimensions with Applications to Brain Local Field Potential Signals

“…The next goal in the data analysis is to estimate the epoch-evolving dimension and the latent stationary time series where . In Figure 4 , we apply SSA and plot the estimates of the stationary subspace dimension across epochs using the method in Sundararajan et al [ 22 ].…”