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 
and can be defined by its data generation process.
We assume that the MR-HMM produces a sequence of
segments, , indexed by segment counter ,
which, when concatenated, produce the complete input
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.
- Initialize segment counter to 1.
- Select an discrete sub-class index for segment ,
denoted by . For the first segment, use the initial
For subsequent segments, use the
state transition matrix (STM)
- Select a random segment size for segment by choosing
from available segment sizes according to the discrete
Let this segment size be base segments, or samples.
- 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
This technique is covered in detail in Chapter 3.
- Increment segment counter and go to step 2.
The MR-HMM parameters
and the feature PDFs by sub-class and segment size,
where feature type is determined uniquely from .