 | This article may be too technical for most readers to understand.(December 2021) |
Stationary Subspace Analysis (SSA) [1] in statistics is a blind source separation algorithm which factorizes a multivariate time series into stationary and non-stationary components.
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 is assumed to be generated as a linear superposition of stationary sources and non-stationary sources ,
where is an unknown but time-constant mixing matrix; and are the basis of the stationary and non-stationary subspace respectively.
Given samples from the time series , the aim of Stationary Subspace Analysis is to estimate the inverse mixing matrix separating the stationary from non-stationary sources in the mixture .
The true stationary sources are identifiable (up to a linear transformation) and the true non-stationary subspace is identifiable. The true non-stationary sources and the true stationary subspace cannot be identified, because arbitrary contributions from the stationary sources do not change the non-stationary nature of a non-stationary source. [1]
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]
In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.
Source separation, blind signal separation (BSS) or blind source separation, is the separation of a set of source signals from a set of mixed signals, without the aid of information about the source signals or the mixing process. It is most commonly applied in digital signal processing and involves the analysis of mixtures of signals; the objective is to recover the original component signals from a mixture signal. The classical example of a source separation problem is the cocktail party problem, where a number of people are talking simultaneously in a room, and a listener is trying to follow one of the discussions. The human brain can handle this sort of auditory source separation problem, but it is a difficult problem in digital signal processing.
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In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.
In mathematics, a unipotent elementr of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n.
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe/model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.
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In mathematics, a Lie algebra is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra is the subalgebra of , denoted
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In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. Its roots lie in the classical Karhunen (1946)–Loève spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem. SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition.
Common spatial pattern (CSP) is a mathematical procedure used in signal processing for separating a multivariate signal into additive subcomponents which have maximum differences in variance between two windows.
Brain connectivity estimators represent patterns of links in the brain. Connectivity can be considered at different levels of the brain's organisation: from neurons, to neural assemblies and brain structures. Brain connectivity involves different concepts such as: neuroanatomical or structural connectivity, functional connectivity and effective connectivity.
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