The SPA is the preferred solution to solve
, 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 J-function
General solution using CLT (module_A_chisq_clt.m)
We seek a
such that the mean of under
equal to or close to the given feature value, which we
There are two possible methods. For arbitrary
matrices , this can be done by projecting the input vector
upon the column space of ,
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 positive-valued approximation,
using quadratic programming to
find the positive-valued (non-zero)
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 ).
also results if we minimize the spectral entropy
under the same constraints.
Another particularly elegant solution to find
can be found in time-series 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
be the hypothesis that has mean
satisfying (or nearly satisfying)
, 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
chi-squared RV, the mean is and the variance
The mean of is therefore
The covariance of under
where is the diagonal matrix with diagonal elements
. We can immediately write down the J-function denominator,
, then the last term disappears.
We now re-examine 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 J-function.
The function software/module_A_chisq_test.m runs the acid test
(with variable CLT set to 1) using syntax:
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 J-function 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.
Acid test results for module_A_chisq_clt.m.
Jout comparison for SPA and CLT.