Description and implementation of the Viterbi algorithm for hidden Markov models using Scala.

**Overview**

A significant number of algorithms used in machine learning relies on dynamic programming and hidden Markov model (HMM) is no exception. Scala tail elimination represents an excellent alternative to the traditional iterative implementation of the 3 canonical forms of the HMM.

**Introduction to HMM**

Markov processes, and more specifically HMM, are commonly used in speech recognition, language translation, and text classification, document tagging, data compression and decoding.

A HMM algorithm uses 3 key components

A HMM algorithm uses 3 key components

- A set of observations
- A sequence of hidden states
- A model that maximizes the joint probability of the observations and hidden states, known as the Lambda model

There are 3 use cases (or canonical forms) of the HMM.

**Evaluation**: Evaluate the probability of a given sequence of observations, given a model**Training**: Identify (or learn) a model given a set of observations**Decoding**Estimate the state sequence with the highest probability to generate a given as set of observations and a model

**Viterbi recursive implementation**

Given a sequence of states {qt} and sequence of observations {oj}, the probability δt(i) for any sequence to have the highest probability path for the first T observations is defined for the state Si.

**Delta**: sequence to have the highest probability path for the first i observations is defined for a specific test δt(i)**psi**Matrix that contains the indices used in the maximization of the probabilities of pre-defined states**Qstar**: the optimum sequence q* of states Q0:T-1

**HMMState**class1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | final class HMMState( val lambda: HMMLambda, val maxIters: Int) { // Matrix of elements (t, i) that defines the highest // probability of a single path of t // observations reaching state S(i) val delta = Matrix[Double](lambda.getT, lambda.getN) // Auxiliary matrix of indices that maximize the probability //of a given sequence of states val psi = Matrix[Int](lambda.getT, lambda.getN) // Singleton to compute the sequence Q* of states with // the highest probability given a sequence // of observations. object QStar { private val qStar = Array.fill(lambda.getT)(0) // Update Q* the optimum sequence of state using backtracking.. def update(t: Int, index: Int): Unit ... } } |

The class

The algorithm is conveniently illustrated by the following diagram.

**HMMLambda**contains the three components of the Markov processes- State transition matrix A
- Observation emission matrix B
- Initial state probabilities, pi

The algorithm is conveniently illustrated by the following diagram.

First, Let's create the key member and method for the Lambda model for HMM. The model is defined as a tuple of the transition probability matrix A, emission probability matrix B and the initial probability π

Then let's define the basic components for implementing the Viterbi algorithm. The class

The Viterbi algorithm is fully defined with

**ViterbiPath**encapsulates the member variables critical to the recursion.The Viterbi algorithm is fully defined with

**lambda**: Lambda model as described in the previous section (line 2)**state**: State of the computation (line 3)**obsIndx**: Index of observed states (line 4)

1 2 3 4 5 6 7 8 | class ViterbiPath( lambda: HMMLambda, state: HMMState, obsIndx: Array[Int]) { val maxDelta = recurse(lambda.config._T, 0) // ... } |

The recursive method,

**recurse**that implements the formula or steps defined earlier. The method relies on the tail recursion. Tail recursion or tail elimination algorithm avoid the creation of a new stack frame for each invocation, improving the performance of the entire recursion.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | @scala.annotation.tailrec private def recurse(t: Int): Double = { // Initialization of the delta value, // return -1.0 in case of error if( t == 0) initial // then for the subsequent observations ... else { // Update the maximum delta value and its state index // for the observation t Range(0, lambda.getN).foreach( updateMaxDelta(t, _) ) // If we reached the last observation... //exit by backtracing the computation of the if( t == obs.size-1) { val idxMaxDelta = Range(0, lambda.getN).map(i => (i, state.delta(t, i))).maxBy(_._2) // Update the Q* value with the index that maximize //the delta.A state.QStar.update(t+1, idxMaxDelta._1) idxMaxDelta._2 } else recurse(t+1) } } |

Once initialized (line 7), the maximum value of delta,

**maxDelta**, is computed through the method**updateMaxDelta**after the first iteration (line 13). Once the step**t**reaches the maximum number of observation labels, last index in the sequence of observations**obs.size-1**) (line 17), the optimum sequence of state**q* / state.QStar**is updated (line 23). The index of the column of the transition matrix A,**idxMaxDelta**corresponding to the maximum of delta is computed (lines 18, 19). The last step is to update the matrix**QStar**(line 23).**References**

Scala for machine learning Packt Publishing 2014

*Pattern Recognition and Machine Learning; Chap 13 Sequential Data/Hidden Markov Models*C. Bishop Springer 2009
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