MR-HMM Definition

The MR-HMM is related to a segmental HMM [67,62]. 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 [68] 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\}$.

Baggenstoss 2017-05-19