Feature transformation and Energy Statistic

Let

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

$\displaystyle {\bf 1} = {\bf A} \left({\bf A}^\prime{\bf A} \right)^{-1} {\bf A}^\prime {\bf 1},$ (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 [25] 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.

Baggenstoss 2017-05-19