Squaring of real data

Consider the transformation

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

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

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

Under $ H_0$, $ {\bf z}$ is a set of independent Chi-square RV with one degree of freedom (See Section 16.1.2), therefore,

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

Performing the calcellations in

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

we get

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

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

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

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.

Baggenstoss 2017-05-19