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| '''Stationary Subspace Analysis (SSA)'''<ref name="ssaprl">
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| 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.
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| </ref> is a [[blind source separation]] [[algorithm]] which factorizes a [[multivariate]] [[time series]] into [[Stationary process|stationary]] and [[non-stationary]] components.
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| == Introduction ==
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| 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.
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| 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>,
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| :<math>
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| 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},
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| </math>
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| 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.
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| 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>.
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| == Identifiability of the solution ==
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| 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"/>
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| == Applications and extensions ==
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| 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.
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| [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> | |
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| == See also ==
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| * [[Blind signal separation|Blind signal separation (BSS)]]
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| * [[Factor analysis]]
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| * [[Independent component analysis]]
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| * [[Cointegration]]
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Stationary Subspace Analysis}}
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| [[Category:Multivariate time series analysis]]
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| [[Category:Signal processing]]
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| [[Category:Data analysis]]
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| [[Category:Statistical models]]
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| [[Category:Articles created via the Article Wizard]]
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