Optimal feature selection and decision rules in classification problems with time series

Abstract
The problem to be considered is that of classifying a given time seriesZ_{N} = (y(1), cdots ,y(N))into one ofrclassesC_{i}, i= 1, cdots ,r. The stochastic processy(cdot)is assumed to obey an autoregressive structure involving a parameter vectortheta, whose probability densityp(theta|C_{i})depends on the class to whichZory(cdot)belongs. Assuming appropriate expressions forp(theta|C_{i}), it is shown that the probability density ofZ_{N}characterizing each class, namelyp(Z_{N}|C_{i}), possesses a vectorbar{theta}of sufficient statistics, i.e., all the information aboutZ_{N}needed for the discrimination between the various classes is contained in the vectorbar{theta}=(bar{theta}_{1}(Z_{N}), cdots , bar{theta}_{m+1}(Z_{N}))^{T}, where the functionsbar{theta}_{i}(Z_{N}), i=1, cdots ,m+1have the same structure for allN. Thus the best possible feature set for the problem isbar{theta}From this is deduced the optimal decision rule to minimize the average probability of error. The optimal feature set and the corresponding optimal decision rule are compared with other feature sets and decision rules mentioned in the literature on speech recognition.

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