Stationary subspace analysis

In many settings, the measured time series contains contributions from various underlying sources that cannot be measured directly. For instance, in EEG analysis, the electrodes on the scalp record the activity of a large number of sources located inside the brain.Niedermeyer E, da Silva F L. Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. Lippincott Williams & Wilkins, 2004. {{ISBN|0-7817-5126-8}} These sources can be stationary or non-stationary, but they are not discernible in the electrode signals, which are a mixture of these sources. SSA allows the separation of the stationary from the non-stationary sources in an observed time series.

According to the SSA model, the observed multivariate time series $x(t)$ is assumed to be generated as a linear superposition of stationary sources $s^\backslash mathfrak\{s\}(t)$ and non-stationary sources $s^\backslash mathfrak\{n\}(t)$,

:$$

x(t) = A s(t) = \begin{bmatrix} A^\mathfrak{s} & A^\mathfrak{n} \end{bmatrix} \begin{bmatrix} s^\mathfrak{s}(t) \\ s^\mathfrak{n}(t) \\ \end{bmatrix},

where $A$ is an unknown but time-constant mixing matrix; $A^\backslash mathfrak\{s\}$ and $A^\backslash mathfrak\{n\}$ are the basis of the stationary and non-stationary subspace respectively.

Given samples from the time series $x(t)$, the aim of Stationary Subspace Analysis is to estimate the inverse mixing matrix $A^\{-1\}$ separating the stationary from non-stationary sources in the mixture $x(t)$.

The true stationary sources $s^\backslash mathfrak\{s\}(t)$ are identifiable (up to a linear transformation) and the true non-stationary subspace $A^\backslash mathfrak\{n\}$ is identifiable. The true non-stationary sources $s^\backslash mathfrak\{n\}(t)$ and the true stationary subspace $A^\backslash mathfrak\{s\}$ cannot be identified, because arbitrary contributions from the stationary sources do not change the non-stationary nature of a non-stationary source.

Stationary subspace analysis has been successfully applied to Brain-computer interfacing,von Bünau P, Meinecke F C, Scholler S, Müller K-R. [https://www.ncbi.nlm.nih.gov/pubmed/21096218 Finding Stationary Brain Sources in EEG Data], IEEE EMBC 2010, Buenos Aires computer visionMeinecke F, von Bünau P, Kawanabe M, Müller K-R.

[https://dx.doi.org/10.1109/ICCVW.2009.5457715 "Learning Invariances with Stationary Subspace Analysis"], Proc. Subspace Workshop of the ICCV 2009, Kyoto and temporal segmentation. There are variants of the SSA problem that can be solved analytically in closed form, without numerical optimization.Hara S, Kawahara Y, Washio T, von Bünau P. [https://dx.doi.org/10.1007/978-3-642-17537-4_52 "Stationary Subspace Analysis as a Generalized Eigenvalue Problem"] Lecture Notes in Computer Science, 2010, Volume 6443/2010, 422-429

• Blind signal separation (BSS)
• Factor analysis
• Independent component analysis
• Cointegration

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Category:Multivariate time series