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 100by1 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 chisquare 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);
Figure 5.1:
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.
The script
software/module_A_chisq_test.m runs this test with the following
syntax:
module_A_chisq_test('acid',100,1,2);
Figure 5.2:
Acid test results for module_A_chisq.m with insufficient features.
