MR-HMM Definition

The MR-HMM is related to a segmental HMM [75,70]. While both MR-HMM and segmental HMMs generate segments of random length, the MR-HMM operates in the time-series domain, generating the time-series of a segment. In contrast, the segmental HMM operates in the feature space, generating a segment as a sequence of features. These features exist in a feature space created by extracting a fixed feature type from uniformly-segmented frames. The MRHMM can be seen as a graphical model [76] and can be defined by its data generation process. We assume that the MR-HMM produces a sequence of variable-length time-series segments, ${\bf x}_s$, indexed by segment counter $s$, which, when concatenated, produce the complete input time-series ${\bf x}$. We assume that the length of these segments can selected from a set $n$ possible segment sizes, each a multiple of the base segment size $K$. For example, for $K=12$ and $n=10$, a possible choice of segment sizes is $\{24, 36, 48, 72, 96, 144, 192, 288, 384, 768\}$, which spans a wide range of segment sizes, approximately geometrically spaced. Assume that there are $L$ sub-classes, defining distinct spectral or temporal character, approximately analogous to the discrete states of a HMM.

The following is the generation process for the MR-HMM.

1. Initialize segment counter $s$ to 1.
2. Select an discrete sub-class index for segment $s$, denoted by $m_s$. For the first segment, use the initial probability distribution $\pi_m, \; 1\leq m \leq L$. For subsequent segments, use the state transition matrix (STM) $A_{m_{s-1},m}, \; 1\leq m \leq L$.
3. Select a random segment size for segment $s$ by choosing from $n$ available segment sizes according to the discrete probability $\rho_{m_s,i}$, where $\sum_{i=1}^n \; \rho_{m_s,i} = 1$. Let this segment size be $k_s$ base segments, or $K k_s$ samples.
4. Generate a segment time-series ${\bf x}_s$ of $K k_s$ samples according to the PDF $p({\bf x}_s \vert m_s,k_s)$, which takes the form of equation (2.2). For each combination of state and segment size ($m,k$), we assume a pre-defined feature transformation type $f$, so ${\bf z}_f=T_f({\bf x}_s)$. To generate ${\bf x}_s$, draw a feature sample ${\bf z}_f^*$ from the feature PDF $p({\bf z}_f; m_s, k_s)$, then draw sample ${\bf x}_s$ from the uniform distribution on the manifold ${{\bf x}: T_f({\bf x})={\bf z}_f^*}.$ This technique is covered in detail in Chapter 3.
5. Increment segment counter $s$ and go to step 2.

The MR-HMM parameters $\Lambda$ include $\{\pi_m, \; 1\leq m \leq L\}$, $\{\rho_{m,i}, \; 1\leq m \leq L, \; 1\leq i \leq n\}$, ${\bf A}= {\bf A}_{m,i}, \; 1\leq m \leq L, \; 1\leq i \leq L$, and the feature PDFs by sub-class and segment size, $\{ p({\bf z}_f; m, k), \; 1\leq m \leq L, \; 1\leq k \leq n\}$, where feature type $f$ is determined uniquely from $\{m,k\}$.