Joint Distributions for Two Useful Classes of Statistics, With Applications to Classification and Hypothesis Testing

摘要

In this paper, we analyze the statistics of two general classes of statistics. The first class is 'M quadratic and linear forms of correlated Gaussian random variables'. Examples include both cyclic and non-cyclic autocorrelation function (ACF) estimates of a correlated Gaussian process or the magnitude-squared of the output samples of a filtered Gaussian process. The second class consists of a subset of order statistics together with a remainder term. An example is the largest M - 1 bins of a discrete Fourier transform (DFT) or discrete wavelet transform (DWT), together with the sum of the remaining energies, forming an M-dimensional statistic. Both classes of statistics are useful in classification and detection of signals. In this paper, we solve for the joint probability density functions (PDFs) of both classes. Using the PDF projection method, these results can be used to transform the feature PDFs into the corresponding high-dimensional PDFs of the raw input data.