Feature Transformation and Reference hypothesis

Let ${\bf z}_A = {\bf A}^\prime {\bf x},$ where ${\bf A}$ is any non-singular $N\times D$ matrix. This feature does not contain an energy statistic, since no linear function of ${\bf x}$ can lead to a norm on ${\cal R}^N$. To add an ES, we augment ${\bf z}_A$. The feature ${\bf z}$ is the union of ${\bf z}_A$ with $t_2({\bf x}) = \sum_{i=1}^N x_i^2,$ so ${\bf z}= [ t_2({\bf x}),\; {\bf z}_A].$ The reference PDF $p({\bf x}\vert H_0)$ is shown in Table 3.1 (“Gaussian"), and $t_2({\bf x})$ is the ES.