MR-HMM Terminology

The following definitions make the connection between MATLAB variables, the terms upon which they are based, and the concepts we have presented so far.

• Base Segment Size or variable $K$. These all mean the same thing. It is the finest sub-division of data measured in time samples. All segment sizes are measured in units of the base segment size. It is also called base shift because all processing windows are shifted in increments of the base segment size.

• Signal Class. A physical signal condition distinguished by a spectrum, amplitude, statistical distribution, or any other combination of observable characteristics. There are assumed to be $L$ signal classes.

• State. One of the discrete states that may correspond to a signal class in a Markov process. When signal classes do not always occur in the same sequence, we may wish to assign separate occurrences of the same signal class to a different state. Yet, because they have the same signal characteristics, we want to assign them to the same signal class. The number of states is always as least as large as the number of signal classes. For example, a signal consisting of two clicks would require two signal classes, one for noise or absence of click, and another for click. But it would require four states: (1) noise (at rest), (2) first click, (3) inter-click gap, and (4) second click.

• Partition. A partition is one of several processing window sizes assigned to a state. It is called a partition for reasons that will become clear when discussing wait states below.

• PDF. A probability density function (PDF) estimator assigned to a particular combination of signal class and segment size.

• Segment Size or Segment Length. The segment size is the processing window size, equal to the base segment size $K$ times the number of base segments for the segment. In the MR-HMM, an input data time record is segmented into a set of processing windows whose segment sizes add up to the total number of samples.

• Wait State. Each combination of state and segment size corresponds to a partition of states in the expanded STM. The expanded STM is made up of many wait-states whose purpose is just to index (advance) the Markov state through the $k$ base segments that make up the segment size.

Just to re-iterate (in case you're confused). Whereas in the HMM, there was just one “state" concept, now we have five separate concepts. In order from most general to most specific:

1. signal class. The definition of some statistical phenomena (characterized by a distinct spectral or temporal character). Speech example: a phoneme.
2. state. Can be the same as a signal class. But, if signal classes can occur in more than one typical sequence (speech example: if the same phoneme can occur in more than one word), then a different state must be assigned to the same signal class when ocurring in a different sequence (word).
3. partition. One distinct combination of state and segment size.
4. PDF. One distinct combination of signal class and segment size - therefore associated with a given PDF estimate. Can be the same as “partition" if the states and signal classes coincide.
5. wait state. Forced-transition states used to count to the end of a partition.