a module for the feature calculation (5.1).
The module uses a floating reference hypothesis
for normalization (Section 2.3.4).
software/module_A_chisq_synth.m implements UMS and function
software/module_A_chisq_test.m tests both functions.
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
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
Gaussian mixture model (Section 13.2.1).
Using the module software/module_A_chisq.m, we obtained the
Projected PDF values are plotted against the true values of
5.1. The agreement is very close.
The script software/module_A_chisq_test.m runs the example with the following
Acid test results for module_A_chisq.m.
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.
software/module_A_chisq_test.m runs this test with the following
Acid test results for module_A_chisq.m with insufficient features.