Example 2
Let be a set of samples of positivevalued
data, such as intensity or power spectrum measurements.
Let
, where
matrix is .
This is a simple linear operation producing the
dimensional feature .
To compute the Jfunction for this
feature transformation, we need to select
a reference hypothesis
for which we can compute
the feature PDF
.
We are tempted to use the Gaussian PDF
since the Gaussian PDF of would be trivial to derive.
Although (2.2) would result in a valid projected PDF,
would have support for negative
values of , which would violate our prior knowledge
that was positive. Other reasons why this is a bad choice
of are discussed on chapter 3.
A much better choice is the exponential PDF
(1/2 times the standard random variable
: chisquare distribution with 2 degrees of freedom, scaled by 0.5)
Unfortunately, the PDF
is not known in closed
form. This problem was adressed in
2000 by Steven Kay who suggested the use of the saddle point approximation
(SPA) [16].
Although
is not known in closed
form, the moment generating function (MGF) is known.
Is is just a matter of numerical inversion of the MGF.
We address this problem in detail in Section 16.6.
The function software/pdf_A_chisq_m.m implements
the algorithm that computes
.
To test the approach, we generated data
as independent samples of from the distribution,
for , then used the matrix
We compared the histogram of the values of with the
SPA for each histogram bin. Figure 2.2 shows the result.
Figure 2.2:
Comparing histogram with PDF approximated using SPA.
