To provide a reference, two eight-dimensional normal distributions (not time-sampled gaussian pulses) were generated, both with mean vector equal to the zero vector and covariance matrix equal to the identity matrix. Refer to Experiments 6-2 and 6-3 for additional information about these data sets. normal results, a distributional difference is evident.Įxperiment 6: In the second experiment the time sampled-gaussian pulse was compared with the time-sampled double exponential pulse. The mean value and standard deviation results of the eigenvalue calculations for four trials are presented in Table 10-1. As before, 100 samples per class were used, k was chosen as three, and α was chosen as zero. A normal distribution was compared with a uniform distribution, both two-dimensional with mean vector equal to the zero vector and covariance matrix equal to the identity matrix. To complete the experiment a second comparison was performed. The test of this section requires no distributional assumptions, and produces an eigenvalue and eigenvector decomposition that is ranked by distributional similarity.

However, this method fails to indicate the subspace in which the differences are most prominent, or what type of differences exists. The Bayes error could be estimated by nonparametric techniques such as the Parzen and kNN approaches as discussed in Chapter 7. Although these tests are simple and adequate for most applications, more sophisticated tests are needed for distributions severly distorted from normal.Ī second alternative is to estimate the degree of overlap of the two distributions. Also, the Bhattacharyya distance is used to measure the closeness of two distributions. Two covariance matrices can be more easily compared by simultaneously diagonalizing them and checking whether or not the diagonal terms of the second matrix are close to one. One method of testing the similarity of two distributions in a high-dimensional space is to compare the mean vectors and covariance matrices of the distributions. However, multiple measurements present a problem. When only a single measurement is taken, the problem can be solved in a fairly straightforward manner. One can gather data sets at different times or under different conditions and distributional testing can then be used to determine whether they are similar or not. The ability to compare two distributions has numerous applications.

Keinosuke Fukunaga, in Introduction to Statistical Pattern Recognition (Second Edition), 1990 Distributional TestingĪnother application of the nonparametric scatter matrices is in the testing of structural similarity of two distributions.

Scatter matrix