Squaring of Gaussian data

Consider the transformation

$$z_i=x_i^2,\;\;\; 1\leq i \leq N.$$

Although dimension-preserving, the sign of the input data is lost. This problem clearly belongs to the “Gaussian" row in Table 3.1 since the associated energy statistic can indeed be computed from ${\bf z}$. The PDF $p({\bf x}\vert H_0)$ is found in the “Gaussian" row in Table 3.1, from which we have

$$\log p({\bf x}\vert H_0) = -\sum_{i=1}^N \left\{ \frac{ \log(2\pi) +x_i^2}{2}\right\}.$$

Under $H_0$, ${\bf z}$ is a set of independent Chi-square RV with one degree of freedom (See Section 17.1.2), therefore,

$$\log p({\bf z}\vert H_0) = -\sum_{i=1}^N \left\{ \frac{ \log(2\pi z_i) - z_i}{2}\right\}.$$

Performing the calcellations in

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

we get

$$\log J = \sum_{i=1}^N \left\{\frac{ \log(z_i)}{2}\right\}.$$

For more information, refer to the software module software/module_square.m.

Re-synthesis of ${\bf x}$ using UMS (Section 3.3) from ${\bf z}$ is accomplished by assigning

$$x_i = z_i \cdot s_i,\;\;\; 1\leq i \leq N,$$

where sign $s_i$ takes a value of $1$ or $-1$, each with probability 1/2. For more information, refer to the software module software/module_square_synth.m.