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 ${\bf x}$ of independent exponentially distributed RVs. The elements of ${\bf x}$ were scaled such that the expected value of the $i$-th element was $100+i$, $i=1,2 \ldots N$. Let the PDF of ${\bf x}$ under these conditions be denoted by $p({\bf x}\vert H_a)$. Although the elements of ${\bf x}$ have different means, they are independent, so $p({\bf x}\vert H_a)$ is easily obtained from the joint PDF from product of chi-square distributions with 2 degrees of freedom (Section 17.1.2).

Next, we applied the linear transformation ${\bf z}= {\bf A} {\bf x}$, where

$${\bf A} = \left[ \begin{array}{rr} 1 & .01 1 & .02 1 & .03 \\ \vdots & \vdots 1 & 1 \end{array}\right].$$

Notice that the columns of ${\bf A}$ form a linear subspace which contains both the special scaling function applied to ${\bf x}$ under $H_a$ as well as constant scaling under $H_0$. We can assume, therefore, that ${\bf z}$ will be approximately sufficient for $H_a$ vs. $H_0$. We then estimated the PDF $p({\bf z}\vert H_a)$ using a Gaussian mixture model (Section 13.2.1).

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

$$\log p_p({\bf x}\vert H_a) = \log J({\bf x}) + \log p({\bf z}\vert H_a),$$

where

$$\log J({\bf x}) = \log p({\bf x}\vert H_0$$$({\bf z})$$$) - \log p({\bf z}\vert H_0$$$({\bf z})$$$).$$

Projected PDF values are plotted against the true values of $p({\bf x}\vert H_a)$ 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 ${\bf A}$ to include only the first column (a constant). This makes ${\bf z}$ a scalar and no longer an approximate sufficient statistic for $H_a$ vs. $H_0$. 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.