Proxy HMM

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 $ K$ samples. Let $ {\bf x}^b_i, \;\;\; 1\leq i \leq T,$ be the base segments, where $ T$ 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 $ L=2$ sub-classes with designations ``background" and ``noise burst".
Figure 14.1: Illustration of the relationship between MR-HMM segmentation and proxy HMM trellis path.
\includegraphics[width=4.5in]{trellis.eps}
On the top is the time-series of length $ T=34$ base segments wherein we see two bursts. Let the allowable segment sizes be $ k\in\{1, \;\; 2, \;\; 3, \;\; 4, \;\; 5\}.$ Then, two possible segmentations $ {\bf q}_1, {\bf q}_2$ for the time-series out of thousands of possible ones are , in format $ (m,k)$, are:

\begin{displaymath}
{\small
\begin{array}{l}
$ \mbox{\small ${\bf q}_1=$} \{(1,...
...5), (1,3), (1,4), (2,3), (1,4), (1,4), (1,4)\}.$
\end{array}}
\end{displaymath}

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 $ (1,4), (1,3)$, or $ (1,3), (1,4)$. 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 $ {\bf q}_1$ and $ {\bf q}_2$ 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 $ \Lambda$. The proxy state transition matrix (STM) is highly structured. The proxy has $ N_p$ states consisting of all wait-states $ N_p = \sum_{m=1}^L \; \sum_{i=1}^{n} \; k_i,$ ($ N_p=25$ in Figure 14.1). The $ N_p$ 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 probability $ {\bf A}_{i,j}$ and the segment size probability $ \rho_{m,i}$. 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 $ \pi_{i}$ and the segment size probability $ \rho_{m,i}$.
Figure 14.2: 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.
\includegraphics[width=4.0in]{mrhmm_stm.eps}

Baggenstoss 2017-05-19