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{{multiple issues|context=November 2010|onesource=November 2010}}
'''Stationary Subspace Analysis (SSA)'''<ref name="ssaprl">
von Bünau P, Meinecke F C, Király F J, Müller K-R (2009). [http://dx.doi.org/10.1103/PhysRevLett.103.214101 Finding Stationary Subspaces in Multivariate Time Series] ''Phys. Rev. Letter'' 103, 214101.
</ref> is a [[blind source separation]] [[algorithm]] which factorizes a [[multivariate]] [[time series]] into [[Stationary process|stationary]] and [[non-stationary]] components.
 
== Introduction ==
 
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.<ref>Niedermeyer E, da Silva F L. Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. Lippincott Williams & Wilkins, 2004. ISBN 0-7817-5126-8</ref> 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,<ref name="ssaprl"/> the observed multivariate time series <math>x(t)</math> is assumed to be generated as a linear superposition of stationary sources <math>s^\mathfrak{s}(t)</math> and non-stationary sources <math>s^\mathfrak{n}(t)</math>,
:<math>
  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},
</math>
where <math>A</math> is an unknown but time-constant mixing matrix; <math>A^\mathfrak{s}</math> and <math>A^\mathfrak{n}</math> are the basis of the stationary and non-stationary subspace respectively.  
 
Given samples from the time series <math>x(t)</math>, the aim of Stationary Subspace Analysis is to estimate the inverse mixing matrix <math>A^{-1}</math> separating the stationary from non-stationary sources in the mixture <math>x(t)</math>.
 
== Identifiability of the solution ==
 
The true stationary sources <math>s^\mathfrak{s}(t)</math> are identifiable (up to a linear transformation) and the true non-stationary subspace <math>A^\mathfrak{n}</math> is identifiable. The true non-stationary sources <math>s^\mathfrak{n}(t)</math> and the true stationary subspace <math>A^\mathfrak{s}</math> cannot be identified, because arbitrary contributions from the stationary sources do not change the non-stationary nature of a non-stationary source<ref name="ssaprl"/>
 
== Applications and extensions ==
 
Stationary subspace analysis has been successfully applied to [[Brain-computer interface|Brain-computer interfacing]],<ref>von Bünau P, Meinecke F C, Scholler S, Müller K-R. [http://www.ncbi.nlm.nih.gov/pubmed/21096218 Finding Stationary Brain Sources in EEG Data], IEEE EMBC 2010, Buenos Aires</ref> [[computer vision]]<ref>Meinecke F, von Bünau P, Kawanabe M, Müller K-R.
[http://dx.doi.org/10.1109/ICCVW.2009.5457715  "Learning Invariances with Stationary Subspace Analysis"], Proc. Subspace Workshop of the ICCV 2009, Kyoto</ref> and temporal segmentation. There are variants of the SSA problem that can be solved analytically in closed form, without numerical optimization.<ref>Hara S, Kawahara Y, Washio T, von Bünau P. [http://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</ref>
 
== See also ==
 
* [[Blind signal separation|Blind signal separation (BSS)]]
* [[Factor analysis]]
* [[Independent component analysis]]
* [[Cointegration]]
 
== References ==
 
{{Reflist}}
 
{{DEFAULTSORT:Stationary Subspace Analysis}}
[[Category:Multivariate time series analysis]]
[[Category:Signal processing]]
[[Category:Data analysis]]
[[Category:Statistical models]]
[[Category:Articles created via the Article Wizard]]

Revision as of 18:54, 26 May 2013

Template:Multiple issues Stationary Subspace Analysis (SSA)[1] is a blind source separation algorithm which factorizes a multivariate time series into stationary and non-stationary components.

Introduction

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.[2] 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,[1] the observed multivariate time series x(t) is assumed to be generated as a linear superposition of stationary sources s𝔰(t) and non-stationary sources s𝔫(t),

x(t)=As(t)=[A𝔰A𝔫][s𝔰(t)s𝔫(t)],

where A is an unknown but time-constant mixing matrix; A𝔰 and A𝔫 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).

Identifiability of the solution

The true stationary sources s𝔰(t) are identifiable (up to a linear transformation) and the true non-stationary subspace A𝔫 is identifiable. The true non-stationary sources s𝔫(t) and the true stationary subspace A𝔰 cannot be identified, because arbitrary contributions from the stationary sources do not change the non-stationary nature of a non-stationary source[1]

Applications and extensions

Stationary subspace analysis has been successfully applied to Brain-computer interfacing,[3] computer vision[4] and temporal segmentation. There are variants of the SSA problem that can be solved analytically in closed form, without numerical optimization.[5]

See also

References

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  1. ↑ 1.0 1.1 1.2 von Bünau P, Meinecke F C, Király F J, Müller K-R (2009). Finding Stationary Subspaces in Multivariate Time Series Phys. Rev. Letter 103, 214101.
  2. ↑ Niedermeyer E, da Silva F L. Electroencephalography: Basic Principles, Clinical Applications, and Related Fields. Lippincott Williams & Wilkins, 2004. ISBN 0-7817-5126-8
  3. ↑ von Bünau P, Meinecke F C, Scholler S, Müller K-R. Finding Stationary Brain Sources in EEG Data, IEEE EMBC 2010, Buenos Aires
  4. ↑ Meinecke F, von Bünau P, Kawanabe M, Müller K-R. "Learning Invariances with Stationary Subspace Analysis", Proc. Subspace Workshop of the ICCV 2009, Kyoto
  5. ↑ Hara S, Kawahara Y, Washio T, von Bünau P. "Stationary Subspace Analysis as a Generalized Eigenvalue Problem" Lecture Notes in Computer Science, 2010, Volume 6443/2010, 422-429
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