The proxy HMM is a hypothetical simple HMM that
provides structure to the MR-HMM. It
assumes that the input data is broken into
small non-overlapping segments (called base segments)
of samples. Let
be the base segments,
where is the total number of base segments.
We don't actually carve up the data into base segments -
they are only used to understand the proxy HMM.
Refer to Figure 14.1 , which illustrates
the proxy HMM structure for sub-classes
with designations ``background" and ``noise burst".
On the top is the time-series of length base segments
wherein we see two bursts.
Let the allowable segment sizes be
Then, two possible segmentations
for the time-series
out of thousands of possible ones are , in format , are:
Illustration of the relationship between MR-HMM
segmentation and proxy HMM trellis path.
These two segmentations are represented as dotted boxes drawn on top
of the time-series and differ only in the way that the
gap between the two bursts is divided, either as
We stress that these are just two of the many possible segmentations,
all of which are considewred by the MR-HMM.
In Figure 14.1 in the part
labeled ``Available Segments", we see
all the allowable segment sizes and time shifts.
The state trellis (``proxy state index" in figure 14.1)
is divided into ``partitions", each representing
a choice of sub-class and segment size with vertical extent
equal to the segment length.
The paths corresponding to segmentations
and are shown as dotted lines
and filled-in circles, which are the proxy HMM states
visited by the two candidate segmentations.
The diagonal patterns are caused because
once the system transitions into the first
state of a partition, it is forced to complete the segment,
counting out the states called wait-states.
Note that all possible segmentations map to a unique path through the
proxy HMM trellis.
The proxy HMM parameters are created from the MR-HMM parameters
The proxy state transition matrix (STM) is highly structured.
The proxy has states consisting of all wait-states
( in Figure 14.1).
The states are sub-divided into sub-classes and partitions,
as explained above. The STM is sparse, mostly consisting of
zeros and ones, which force the state to increment
through the partition.
Due to the forced wait-state counting,
the proxy HMM has a very structured state transition matrix (STM).
The proxy STM corresponding to Figure 14.1
is shown in Figure 14.2.
Note the subdivision of proxy states into sub-class
(solid lines) and partitions (dotted lines).
The black circles indicate that once the system transitions
to the start of a partition, it must increment to the end of the partition.
The shaded circles indicate that the last wait-state of each partition may
transition to the first wait-state of any partition.
Since it involves a transition to a particular
sub-class and segment size,
the probability value is equal to the product of the sub-class transition
and the segment size probability
The proxy initial state probabilities are conformal with any
column of the STM, with values determined by the
product of the of the sub-class initial
probability and the segment size probability
The proxy STM corresponding to Figure 14.1.
Empty circles are zero, black circles are 1.0 and shaded circles
take a value between 0 and 1.