J-Function

The PDF $p({\bf z}\vert H_0)$ can be derived if we first orthogonalize the features. Let $\rho = {\bf x}^\prime \left\{ {\bf I}-{\bf A} \left({\bf A}^\prime{\bf A} \right)^{-1} {\bf A}^\prime\right\} {\bf x},$ which can also be written $\rho = t_2({\bf x}) - {\bf x}^\prime {\bf A} \left({\bf A}^\prime{\bf A} \right)^{-1} {\bf A}^\prime {\bf x}.$ This is the energy in ${\bf x}$ orthogonal to the columns of ${\bf A}$. Accordingly, $\rho$ is statistically independent of ${\bf z}_A$ under $H_0$. Thus, $p(\rho, {\bf z}_A\vert H_0) = p(\rho\vert H_0) \; p({\bf z}_A \vert H_0).$ Also, $p(\rho\vert H_0)$ is chi-squared with $N-D$ degrees of freedom,

$$p(\rho\vert H_0)=\frac{\rho^{(\kappa/2-1)} \; e^{-\rho/2}}{2^{\kappa/2} \; \Gamma(\kappa/2)},$$

where $\kappa=N-D$ and $p({\bf z}_A \vert H_0)$ is the Gaussian distribution with mean ${\bf0}$ and co-variance ${\bf A}^\prime{\bf A}$:

$$p({\bf z}_A\vert H_0) = \left(2\pi\right)^{-\frac{D}{2}} \; \vert... ...frac{1}{2}{\bf z}_A^\prime \left({\bf A}^\prime{\bf A} \right)^{-1} {\bf z}_A}$$

And, since $(\rho,{\bf z}_A)$ can be obtained from ${\bf z}$ using a linear transformation with Jacobian of determinant 1, we can write $p({\bf z}\vert H_0) = p(\rho\vert H_0) \; p({\bf z}_A \vert H_0).$ The J-function is just $p({\bf x}\vert H_0)/p({\bf z}\vert H_0).$ For more information see the function software/module_lin_gauss.m.