### Module

The function software/module_A_chisq.m implements a module for the feature calculation (5.1). The module uses a floating reference hypothesis for normalization (Section 2.3.4). The function software/module_A_chisq_synth.m implements UMS and function software/module_A_chisq_test.m tests both functions.

Example 8   We now provide an example of the application of the SPA to a linear combination of exponentials. We performed an acid test (see Section 2.3.8) by generating 1000 samples of a 100-by-1 vector of independent exponentially distributed RVs. The elements of were scaled such that the expected value of the -th element was , . Let the PDF of under these conditions be denoted by . Although the elements of have different means, they are independent, so is easily obtained from the joint PDF from product of chi-square distributions with 2 degrees of freedom (Section 16.1.2).

Next, we applied the linear transformation , where

Notice that the columns of form a linear subspace which contains both the special scaling function applied to under as well as constant scaling under . We can assume, therefore, that will be approximately sufficient for vs. . We then estimated the PDF using a Gaussian mixture model (Section 13.2.1).

Using the module software/module_A_chisq.m, we obtained the projected PDF:

where

Projected PDF values are plotted against the true values of in Figure 5.1. The agreement is very close. The script software/module_A_chisq_test.m runs the example with the following syntax:
               module_A_chisq_test('acid',100,2,2);


We then changed matrix to include only the first column (a constant). This makes a scalar and no longer an approximate sufficient statistic for vs. . The result is shown in Figure 5.2. Note the worsening of the error. The script software/module_A_chisq_test.m runs this test with the following syntax:

               module_A_chisq_test('acid',100,1,2);


Baggenstoss 2017-05-19