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Independent component analysis P18

Methods using Time Structure The model of independent component analysis (ICA) that we have considered so far consists of mixing independent random variables, usually linearly. In many applications, however, what is mixed is not random variables but time signals, or time series. This is in contrast to the basic ICA model in which the samples of have no particular order: We could shuffle them in any way we like, and this would have no effect on the validity of the model, nor on the estimation methods we have discussed. If the independent components (ICs) are time signals, the situation is quite. | Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic 18 Methods using Time Structure The model of independent component analysis ICA that we have considered so far consists of mixing independent random variables usually linearly. In many applications however what is mixed is not random variables but time signals or time series. This is in contrast to the basic ICA model in which the samples of x have no particular order We could shuffle them in any way we like and this would have no effect on the validity of the model nor on the estimation methods we have discussed. If the independent components ICs are time signals the situation is quite different. In fact if the ICs are time signals they may contain much more structure than simple random variables. For example the autocovariances covariances over different time lags of the ICs are then well-defined statistics. One can then use such additional statistics to improve the estimation of the model. This additional information can actually make the estimation of the model possible in cases where the basic ICA methods cannot estimate it for example if the ICs are gaussian but correlated over time. In this chapter we consider the estimation of the ICA model when the ICs are time signals Si t t 1 T where is the time index. In the previous chapters we denoted by t the sample index but here has a more precise meaning since it defines an order between the ICs. The model is then expressed by x i As i 18.1 where A is assumed to be square as usual and the ICs are of course independent. In contrast the ICs need not be nongaussian. In the following we shall make some assumptions on the time structure of the ICs that allow for the estimation of the model. These assumptions are alternatives to the 341 342 METHODS USING TIME STRUCTURE assumption of nongaussianity made in other chapters of this book. First we shall assume that .