## Features

Let be a set of independent random variables (RVs) distributed under hypothesis according to the common PDF . The joint probability density function (PDF) of is

Now let be ordered in decreasing order into the set where . We now choose a set of indexes , with to form a selected collection of order statistics . To this set, we add the residual energy",

 (7.1)

where the set are the integers from to not including the values , and is a function which is needed to insure that has units of energy and controls the energy statistic. We then form the complete feature vector of length ():

By appending the residual energy to the feature vector, we insure that contains the energy statistic. We consider two important cases:
1. if is positive intensity or spectral data and has approximate chi-square statistics (resulting from sums of squares of Gaussian RV), then is sufficient. The resulting energy statistic and reference hypotheses are the Exponential" in Table 3.1. For this case, we consider to be a set of magnitude-squared DFT bin outputs, which are exponentially distributed.
2. if are raw measurements and have approximate Gaussian statistics, use or . The resulting energy statistic and reference hypotheses are the Gaussian" in Table 3.1. For this case, we let be a set of absolute values of zero-mean Gaussian RVs. Then, .

Baggenstoss 2017-05-19