Hanning-3 Segmentation

Block segmentation was necessary for strict application of PDF projection to segmented data. In 2012, a means was discovered to circumvent this restriction, and still obtain “exact" projected likelihood comparison between differently-segmented branches, but it is only possible for 2/3 overlap, 3/4 overlap, etc. It is not possible for the commonly-used 50% overlap. Since 3/4 overlap and more is rarely used, we use 2/3 overlap and call the method Hanning-3. Consider overlapped segments with segment size $K$ and window time shift $S$, overlapping by $O = K-S$ samples. If we circularly-index the data such that $x_{N+i}=x_i$, we will obtain exactly $T = N/S$ segments. Let ${\bf x}_i=[x_{(1+Si)} w_1, x_{(2+Si)} w_2, \ldots x_{(K+Si)} w_K],$ be the $i$-th segment where $[w_1,w_2\ldots w_K]$ are the Hanning weights. The Hanning weights must be periodic with a period of $K$, not $K-1$ as is typically used to avoid any zero weights.

$$w_t=\frac{1-\cos(2\pi(t-1)/K)}{c}, \;\; 1\leq t \leq K,$$ (12.3)

where $c=\frac{3}{\sqrt{2}}.$ The special scaling factor $c$ is non-standard and results in the desired property that follows. Let the complete hanning-3 segmentation be denoted by ${\bf X}^{K,h3}=[{\bf x}_1 , {\bf x}_2 \ldots {\bf x}_T].$ Note that ${\bf X}^{K,h3}$ has a total dimension of $K\times T = (3S)\times (N/S)$, so has a total dimension of $3N$, three times as large as the dimension of ${\bf x}$.

To use various hanning-3 segmentations together in a class-specific classifier, we need to apply the concept of virtual input data. Consider two hanning-3 segmentations with different segment sizes $K_l$ and $K_m$, denoted by ${\bf X}^{K_l,h3}$ and ${\bf X}^{K_m,h3}$. It has been shown [57] that with weights $w_t$ as defined in (12.3), that ${\bf X}^{K_l,h3}$ and ${\bf X}^{K_m,h3}$ are related by an orthogonal linear transformation. Specifically, there exists an ortho-normal matrix ${\bf U}$ such that ${\bf X}^{K_l,h3} = {\bf U} \; {\bf X}^{K_m,h3}.$ In Figure 12.2, the output of each segmentation operation is considered as the “virtual input data" of each branch. Each branch has a different virtual input data, but they are considered “equivalent". Therefore, the projected likelihood function for ${\bf X}^{K_l,h3}$ may be compared to the projected the likelihood function for ${\bf X}^{K_m,h3}$.

Each block “feature calculation" in the figure normally consists of more than one stage, organized as a chain (See Section 2.2.4). The starting reference hypothesis ($H_{0x}$ in equation 2.9) is typically canonical reference hypothesis, exponential or Gaussian, in which all the elements of ${\bf x}$ are independent. Thus, all the elements of ${\bf X}^{K,h3}$ are assumed independent under $H_0$. Because each data space can be converted to another using an orthonormal rotation, the projected PDFs of all the branches can be considered PDFs in some common data space (a virtual data space).

Figure 12.2: The concept of virtual input data illustrated for three hanning-3 segmenation sizes.

The mathematical postulations of Hanning-3 can be tested using the function software/hanning3_wts.m, with syntax [w,W,A]=hanning3_wts(K,N);. The outputs include w, which is the weight vector ${\bf w}$, W which is the $N\times K$ matrix of window functions, and A is the $3N\times N$ linear expansion matrix that creates ${\bf X}^{K_l,h3}$ from ${\bf x}$ in one column. In Figure 12.3, we plotted W as an image for $K=192$, $N=1536,$ for which there are $T=24$ segments, and for $K=768$, $N=1536,$ for which there are $T=6$ segments. In the figure, you can see the circular indexing.

Let ${\bf A}_K$ be the matrix A produced for the given value of $K$. It is easy to verify in either case that

1. the product ${\bf A}_K{\bf x}$ produces the concatenated segmentation ${\bf X}^{K,h3}$,
2. that ${\bf A}_K$ is orthonormal, so that ${\bf A}_K^\prime {\bf A}_K={\bf I},$
3. that to transform between the segmentations $K$ and $\tilde{K}$, we can use
   $${\bf A}_{\tilde{K}}{\bf x}= {\bf A}_{\tilde{K}} \left({\bf A}_{K... ...\bf x}= \left({\bf A}_{\tilde{K}} {\bf A}_{K}^\prime\right) {\bf A}_{K}{\bf x},$$
   where ${\bf A}_{\tilde{K}} {\bf A}_{K}^\prime$ is an orthonormal transformation since it is the product of two orthonormal transformations.

Figure: Window functions for Hanning-3 segmentation for $K=192$, $N=1536$ (left) and $K=768$, $N=1536$ (right).