Independent component analysis

In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that the subcomponents are non-Gaussian signals and that they are statistically independent from each other. ICA is a special case of blind source separation. A common example application is the "cocktail party problem" of listening in on one person's speech in a noisy room.

Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results. It is also used for signals that are not supposed to be generated by a mixing for analysis purposes.

A simple application of ICA is the "cocktail party problem", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes.

An important note to consider is that if N sources are present, at least N observations (e.g. microphones) are needed to recover the original signals. This constitutes the square case (J = D, where D is the input dimension of the data and J is the dimension of the model). Other cases of underdetermined (J > D) and overdetermined (J < D) have been investigated.

That the ICA separation of mixed signals gives very good results is based on two assumptions and three effects of mixing source signals. Two assumptions:

Three effects of mixing source signals:

Those principles contribute to the basic establishment of ICA. If the signals we happen to extract from a set of mixtures are independent like sources signals, or have non-Gaussian histograms like source signals, or have low complexity like source signals, then they must be source signals.

ICA finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components. We may choose one of many ways to define a proxy for independence, and this choice governs the form of the ICA algorithm. The two broadest definitions of independence for ICA are

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