A class of feature extraction criteria and its relation to the Bayes risk estimate

Feature extraction criteria of the form are considered where and are the conditional means and covariances. The function is assumed to be invariant under nonsingular linear transformations and coordinate shifts. For the case , is shown to depend only upon the distance between classes. The class case, , is shown to depend upon the between-class distances . This criterion is also shown to be equivalent to the mean-square-error of the general Bayes risk estimate. The most general is reduced to a function of sets of between-class distances with metrics induced by the conditional covariances. When this dependence is reduced to two parameters which may be regarded as different between-class distance measures. Finally, the linear mapping is reduced to a one-parameter problem as in the work of Peterson and Mattson.