Feature transformation and J-function

The feature transformation is the same as (5.1) in Section 5.1:

$${\bf z}= {\bf A}^\prime {\bf x},$$ (7.1)

but the similarity ends here. Because ${\bf x}$ is limited to a compact set, the unit hypercube ${\cal U}^N$, an energy statistic is theoretically not needed for MEPP (See Theorem 2 in Section 3.2.1) ^7.1Whereas the exponential distribution played a central role in the analysis of the singly-bounded case, the mutivariate truncated exponential distributon (TED) plays an important role in the unit hypercube ${\cal U}^N$ (all elements of ${\bf x}$ in $[0,1]$). The uniform reference distribution, $p({\bf x}\vert H_0)=1$ results in maximum entropy (See Theorem 2 in Section 3.2.1), and is a special case of the mutivariate TED. When $p({\bf x}\vert H_0)=1$, the J-function simplifies to

$$J({\bf x}, T) = \frac{1}{p({\bf z}\vert H_0)}.$$

So, computing the J-function is primarily a task of computing $p({\bf z}\vert H_0)$, which can be done using two methods as described in Section 17.7. The J-function is implemented by software/module_A_dbound.m, and tested by software/module_A_dbound_test.m. The re-synthesis approach using UMS is described in Secion 7.2 and implemented by software/module_A_dbound_synth.m.