Viterbi

A UGen performing a generalized Viterbi algorithm. The Viterbi algorithm tries to find the best path among sequences of states, by evaluating transition probabilities. It runs over a predefined number of frames, accumulating data of different states. It maximizes the likelihood of the terminal state, and then backtracks to reconstruct the likely path (sequence of states). The output is the sequence of state indices (from zero inclusive to numStates exclusive).

'''Note:''' This UGen must run until numFrames or the inputs are exhausted, before it can begin outputting values.

This implementation is generalized in the sense that instead of the canonical matrices "sequences of observations", "initial probabilities", "transition matrix", and "emission matrix", it takes two large matrices mul and add that contain the equivalent information. These two matrices allow the UGen to operate in two different modes:

• multiplicative (as in the Wikipedia article) by damping the probabilities over time
• accumulative (as used in Praat for pitch tracking) by adding up the weights (if you have "costs", feed in their negative values).

Basically the internal delta matrix is created by the update function delta = (delta_prev * mul) + add (with the corresponding matrix indices).

The initial delta state is zero. Therefore, in order to provide the initial state, you obtain an initial vector v by providing an add matrix of numStates x numStates cells, which is zero except for the first column filled by v (alternatively, each row filled with the next value of v, or a diagonal matrix of v; if v can take negative value, make sure to fill the initial numStates x numStates completely by duplicating v).

For the classical data, set add just to the initial matrix as explained above (multiplying emitted first observations with initial probabilities), and then use exclusively mul by passing in emitted observations multiplied by their transition probabilities:

   mul[t][i][j] = transitionProb[i][j] * emissionProb[observation[t]][i]

See https://en.wikipedia.org/wiki/Viterbi_algorithm