CMEX function

The main processing engine of the MR-HMM is the CMEX function alphbetacompress.c. It can be operated in stand-alone manner separate from the Class-Specific Toolkit. The calling syntax is:

   function [alphas,alognorm,betas,blognorm,gammaN,gamma] =
       alphabetacompress(Lin,S,hparm);

Definition of inputs:

• Lin: the input log-likelihoods (partial PDF values). Each value in Lin is the assumed log-likelihood for a segment of data for a given signal class, a given segment length, at a specified starting base segment. The log-likelihood value must be scaled by $1/k$ where $k$ is the number of base segments in the data segment. Matrix Lin is dimensioned nseg by Npdf, where nseg is the total number of base segments in the input timeseries, and Npdf is the number of PDFs (there is one PDF for each combination of signal class and segment length). Each PDF is evaluated at various time shifts. It does not need to be evaluated at all time shifts if not necessary. These positions in the matrix that are not evaluated must be then set to negative infinity. Since Lin contains the log-PDF of the segment starting at the specified time, there are many “impossible" values, which are set to negative infinity, for example for a long segment starting near the end of the time-series. An example of Lin is provided in Figure 14.4 which illustrates the case when there are 16 base segments and there are two possible segment sizes: 8 and 14 base segments. It shows where the segment log-likelihood values (scaled by $1/k$) are placed in matrix Lin. On the left is shown a time grid with 16 base segments. The letters indicate all the possible segments that can exist with lengths 8 and 14 base segments.

  The column location of the likelihood value in Lin indicates the index of the PDF. You can put the columns in any order as long as input variable S is correctly set (see next item). The row location of the likelihood value in Lin indicates the time shift. The rule is that the first position (first row) of Lin corresponds to zero shift - such that the processing window starts with the first base segment. Shift values for which the processing window extends beyond nseg are invalid. Unavailable or invalid values are indicated by -inf value. If the input time-series is smaller than the processing window length, then there will be no valid likelihood values in that column.

  IMPORTANT: The likelihood values in Lin are normalized (divided) by the number of base segments in the processing window (i.e. they are partial PDF values).

• S: this is a matrix of indexes that tells alphabetacompress in which column of Lin to look for given combination of state and partition. Thus, S(istate,ipartition)=index where index is the column number (starting with 1, not with 0). The column width of S is the maximum number of partitions across all states. Since some states may have more partititons, some of the values of the matrix S are invalid and should be set to 0.

• hparm: this is the MR-HMM parameter structure with the following fields:
  • N. The number of states.
  • Pi. The vector of N initial state probabilities.
  • A. The state transition matrix. A(i,j) is the probabity of transitioning from state i to state j.
  • state_to_class_index. This is the length-N vector of signal class indexes. This indicates the signal class corresponding to each state. Often, each state corresponds to a class, so state_to_class_index = [1:N];.
  • pdf_to_class_index. This vector is of length Npdf and indicates the index of the class to which each PDF belongs.
  • pdf_entry. This is the vector of entry flags of length Npdf equal to 1 or 0. A one means the given partition can be accessed from another state (i.e. the system can start in the given signal class with that segment length). When in doubt, set all to 1.
  • ksegment: This vector is of length Npdf and indicates the number of base segments for the corresponding pdf.
  • beta_end: dimension nstate by 1, normally all ones. If any elelemt is zero, prevents MR-HMM from ending in that state.
  • PartitionDistrib. A matrix of dimension maxnpartition by N where maxnpartition is the largest number of partitions assigned to any class, and N is the number of states. PartitionDistrib specifies the probability that a given partition will be chosen conditioned on the specified state being chosen (see variable $\rho_{m_s,i}$ above).

Figure 14.4: Illustration of Likelihood Input matrix Lin for a simple example with just two partitions. Time divisions are one base segment. Entries with “*" are set to negative infinity. The example is trivial because there is only one valid partition of the data : using segments A and I.

Definition of outputs:

• alphas: a nseg by Ntot matrix of expanded forward probabilities where Ntot is the total number of wait states. The columns of alphas are arranged in wait state counting order, which is the order counted out if we loop over states, then loop over the number of partitions in the signal class corresponding to the state, then loop over the wait states in the partition. For example, the elements of alphas for time step t are set to zero in wait state counting order in the following loop:

     icount = 1;
     for i=1:N,
       for j=1:P(i),
            for iwait = 1:K(i,j),
               alphas(t,icount)=0;
               icount = icount+1;
            end;
       end;
     end;
  

  where P(i) is the number of partititons in the signal class assigned to state i, and K(i,j) is the segment length in base segments for the partition.

• alognorm: a nseg by 1 vector of log-normalization values for the forward probabilities. It is the accumulated log-correction factors after completion of the forward procedure. The likelihood value output for the forward procedure is computed by:

        log_p_pout = log(sum(alphas(end,:)))+alognorm(end);
  

• betas: a nseg by Ntot matrix of expanded backward probabilities in wait state counting order.

• blognorm: a nseg by 1 vector of log-normalization values for the backward probabilities.

• gammaN: a nseg by N matrix. gammaN(t,i) is the posteriori probability that the system is in state i at time step t.

• gamma: a nseg by Ntot matrix of expanded posteriori probabilities. gamma(t,i) is the posteriori probability that the system is in wait state i at time step t. It is the normalized product of alphas and betas. Columns are in wait state counting order.