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:
 if is positive intensity
or spectral data and has approximate chisquare 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 magnitudesquared DFT bin outputs,
which are exponentially distributed.
 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
zeromean Gaussian RVs. Then, .
Baggenstoss
20170519