## 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, , indexed by segment counter , which, when concatenated, produce the complete input time-series . We assume that the length of these segments can selected from a set possible segment sizes, each a multiple of the base segment size . For example, for and , a possible choice of segment sizes is , which spans a wide range of segment sizes, approximately geometrically spaced. Assume that there are 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 to 1.
2. Select an discrete sub-class index for segment , denoted by . For the first segment, use the initial probability distribution . For subsequent segments, use the state transition matrix (STM) .
3. Select a random segment size for segment by choosing from available segment sizes according to the discrete probability , where . Let this segment size be base segments, or samples.
4. Generate a segment time-series of samples according to the PDF , which takes the form of equation (2.2). For each combination of state and segment size (), we assume a pre-defined feature transformation type , so . To generate , draw a feature sample from the feature PDF , then draw sample from the uniform distribution on the manifold This technique is covered in detail in Chapter 3.
5. Increment segment counter and go to step 2.

The MR-HMM parameters include , , , and the feature PDFs by sub-class and segment size, , where feature type is determined uniquely from .

Baggenstoss 2017-05-19