## 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 J-function

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 positive-valued approximation, using quadratic programming to find the positive-valued (non-zero) 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 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 , 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 chi-squared 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 J-function denominator,

If , then the last term disappears.

Example 9   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:
       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 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.

Baggenstoss 2017-05-19