General solution using CLT (module_A_chisq_clt.m)
The SPA is the preferred solution to solve
for
, but for completeness,
and to serve as a comparison of the SPA approach, we cover the CLT solution.
Because we intend to use the CLT to approximate
, we need to use a floating reference hypothesis,
(See Sections 2.3.3 and 2.3.4)
and corresponding Jfunction
We seek a
such that the mean of under
is
equal to or close to the given feature value, which we
designate .
There are two possible methods. For arbitrary
matrices , this can be done by projecting the input vector
upon the column space of ,
or
This will satisfy the constraint
,
however may be invalid if it has negative values.
The problem of negative values can be solved
by finding a suitable positivevalued approximation,
using quadratic programming to
find the positivevalued (nonzero)
that minimizes
under the constraint that
.
For example MATLAB optimization toolkit has quadprog.m which
solves the above problem with the syntax
x = quadprog(eye(N),zeros(N,1),[],[],A',z,meps*ones(N,1));
where meps is a small positive number (minimum allowed value of ).
A good
also results if we minimize the spectral entropy

(5.5) 
under the same constraints.
Another particularly elegant solution to find
can be found in timeseries analysis
where we use the autoregressive spectrum estimate for
.
We will discuss this in section 5.2.5.
Regardless of which method we used to obtain
,
let
be the hypothesis that has mean
,
with
satisfying (or nearly satisfying)
the constraint
.
Therefore, under
, the mean of is near to itself.
In section 2.3.3, we discussed the concept of the region of sufficiency (ROS).
When applying the central limit theorem, we need only consider the mean
and variance of . For the
chisquared RV, the mean is and the variance
is
.
The mean of is therefore
The covariance of under
is
where is the diagonal matrix with diagonal elements
. We can immediately write down the Jfunction denominator,
If
, then the last term disappears.
Example 9
We now reexamine example 8, however this time we
apply the CLT method.
Once again, projected PDF values are plotted against the true values of
in Figure 5.3.
The acid test passes.
The function software/module_A_chisq implements reature trasformation and Jfunction.
The function software/module_A_chisq_test.m runs the acid test
(with variable CLT set to 1) using syntax:
module_A_chisq_test('acid',100,2,2,'',1);
It is interesting to compare
the Jout values from the SPA and CLT methods.
In Figure 5.4, we see there is close agreement.
Mathematically, we have
where the right hand side is the Jfunction from the CLT method using
the floating reference hypothesis. Clearly then,
Note that this provides an alternative to the SPA
for computing the PDF of
under a fixed reference hypothesis.
Figure 5.3:
Acid test results for module_A_chisq_clt.m.

Figure 5.4:
Jout comparison for SPA and CLT.

Baggenstoss
20170519