Feature transformation and Energy Statistic (ES approach)

Let

$${\bf z}= {\bf A}^\prime {\bf x},$$ (5.1)

where ${\bf A}$ is a non-singular $N\times D$ matrix. Because the data is positive, a positive-weighted sum of the input data meets the conditions of a norm on ${\cal P}^N$, which is the positive quadrant of ${\cal R}^N$. We apply row “Exponential" in Table 3.1. The ES is the sum of the input samples, which is also a linear transformation, and so can be integrated into matrix ${\bf A}$. In order that ${\bf z}$ contain the ES, we need that

$${\bf 1} = {\bf A} \left({\bf A}^\prime{\bf A} \right)^{-1} {\bf A}^\prime {\bf 1},$$ (5.2)

where ${\bf 1}=[1,1,1,\ldots 1]^\prime$.

An example of this is MEL frequency band analysis, covered in Section 5.2.6, where the columns of ${{\bf A}}$ are the MEL band functions. The sum of the standard MEL band functions is a constant, if the zero and Nyquist frequency bands are present, meeting requirement (5.2). Another example is in auto-regressive spectral analysis [31] where matrix ${\bf A}$ computes the auto-correlation function (ACF), covered in Section 5.2.2, where the zero-th lag ACF coefficient is $t_2({\bf x})$ defined in Table 3.1.