Viterbi algorithm

The Viterbi algorithm is a dynamic programming algorithm for finding the most likely sequence of hidden states – called the Viterbi path – that results in a sequence of observed events, especially in the context of Markov information sources and hidden Markov models.

The terms "Viterbi path" and "Viterbi algorithm" are also applied to related dynamic programming algorithms that discover the single most likely explanation for an observation. For example, in statistical parsing a dynamic programming algorithm can be used to discover the single most likely context-free derivation (parse) of a string, which is sometimes called the "Viterbi parse".

The Viterbi algorithm was proposed by Andrew Viterbi in 1967 as a decoding algorithm for convolutional codes over noisy digital communication links.^[1] The algorithm has found universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.11 wireless LANs. It is now also commonly used in speech recognition, keyword spotting, computational linguistics, and bioinformatics. For example, in speech-to-text (speech recognition), the acoustic signal is treated as the observed sequence of events, and a string of text is considered to be the "hidden cause" of the acoustic signal. The Viterbi algorithm finds the most likely string of text given the acoustic signal.

Algorithm

Suppose we are given a Hidden Markov Model (HMM) with state space $S$ , initial probabilities $\pi _{i}$ of being in state $i$ and transition probabilities $a_{i,j}$ of transitioning from state $i$ to state $j$ . Say we observe outputs $y_{1},\dots ,y_{T}$ . The most likely state sequence $x_{1},\dots ,x_{T}$ that produces the observations is given by the recurrence relations:^[2]

{\begin{array}{rcl}V_{1,k}&=&\mathrm {P} {\big (}y_{1}\ |\ k{\big )}\cdot \pi _{k}\\V_{t,k}&=&\mathrm {P} {\big (}y_{t}\ |\ k{\big )}\cdot \max _{x\in S}\left(a_{x,k}\cdot V_{t-1,x}\right)\end{array}}

Here $V_{t,k}$ is the probability of the most probable state sequence responsible for the first $t$ observations that has $k$ as its final state. The Viterbi path can be retrieved by saving back pointers that remember which state $x$ was used in the second equation. Let $\mathrm {Ptr} (k,t)$ be the function that returns the value of $x$ used to compute $V_{t,k}$ if $t>1$ , or $k$ if $t=1$ . Then:

{\begin{array}{rcl}x_{T}&=&\arg \max _{x\in S}(V_{T,x})\\x_{t-1}&=&\mathrm {Ptr} (x_{t},t)\end{array}}

Here we're using the standard definition of arg max.
The complexity of this algorithm is $O(T\times \left|{S}\right|^{2})$ .

Pseudocode

Here is some necessary set up for the problem. Given the observation space $O=\{o_{1},o_{2},\dots ,o_{N}\}$ , the state space $S=\{s_{1},s_{2},\dots ,s_{K}\}$ , a sequence of observations $Y=\{y_{1},y_{2},\ldots ,y_{T}\}$ , transition matrix $A$ of size $K\times K$ such that $A_{ij}$ stores the transition probability of transiting from state $s_{i}$ to state $s_{j}$ , emission matrix $B$ of size $K\times N$ such that $B_{ij}$ stores the probability of observing $o_{j}$ from state $s_{i}$ , an array of initial probabilities $\pi$ of size $K$ such that $\pi _{i}$ stores the probability that $x_{1}==s_{i}$ .We say a path $X=\{x_{1},x_{2},\ldots ,x_{T}\}$ is a sequence of states that generate the observations $Y=\{y_{1},y_{2},\ldots ,y_{T}\}$ .

In this dynamic programming problem, we construct two 2-dimensional tables $T_{1},T_{2}$ of size $K\times T$ . Each element of $T_{1}$ , $T_{1}[i,j]$ , stores the probability of the most likely path so far ${\hat {X}}=\{{\hat {x}}_{1},{\hat {x}}_{2},\ldots ,{\hat {x}}_{j}\}$ with ${\hat {x}}_{j}=s_{i}$ that generates $Y=\{y_{1},y_{2},\ldots ,y_{j}\}$ . Each element of $T_{2}$ , $T_{2}[i,j]$ , stores ${\hat {x}}_{j-1}$ of the most likely path so far ${\hat {X}}=\{{\hat {x}}_{1},{\hat {x}}_{2},\ldots ,{\hat {x}}_{j-1},{\hat {x}}_{j}\}$ for $\forall j,2\leq j\leq T$

We fill entries of two tables $T_{1}[i,j],T_{2}[i,j]$ by increasing order of $K\cdot j+i$ .

T_{1}[i,j]=\max _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}

, and

T_{2}[i,j]=\arg \max _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}

   INPUT:  The observation space  $O=\{o_{1},o_{2},\dots ,o_{N}\}$ , 
           the state space  $S=\{s_{1},s_{2},\dots ,s_{K}\}$ , 
           a sequence of observations   $Y=\{y_{1},y_{2},\ldots ,y_{T}\}$  such that  $y_{t}==i$  if the 
             observation at time  $t$  is  $o_{i}$ ,
           transition matrix  $A$  of size  $K\cdot K$  such that  $A_{ij}$  stores the transition
             probability of transiting from state  $s_{i}$  to state  $s_{j}$ ,
           emission matrix  $B$  of size  $K\cdot N$  such that  $B_{ij}$  stores the probability of
             observing  $o_{j}$  from  state  $s_{i}$ , 
           an array of initial probabilities  $\pi$  of size  $K$  such that  $\pi _{i}$  stores the probability
             that  $x_{1}==s_{i}$ 
   OUTPUT: The most likely hidden state sequence  $X=\{x_{1},x_{2},\ldots ,x_{T}\}$ 
A01 function VITERBI( O, S,π,Y,A,B ) : X
A02     for each state s_i do
A03         T₁[i,1]←π_i $\cdot$ B_i $y_{1}$ 
A04         T₂[i,1]←0
A05     end for
A06     for i←2,3,...,T do
A07         for each state s_j do
A08             T₁[j,i]← $\max _{k}{(T_{1}[k,i-1]\cdot A_{kj}\cdot B_{jy_{i}})}$ 
A09             T₂[j,i]← $\arg \max _{k}{(T_{1}[k,i-1]\cdot A_{kj}\cdot B_{jy_{i}})}$ 
A10         end for
A11     end for
A12     z_T← $\arg \max _{k}{(T_{1}[k,T])}$ 
A13     x_T←s_{z_T}
A14     for i←T,T-1,...,2 do
A15         z_i-1←T₂[z_i,i]
A16         x_i-1←s_{z_i-1}
A17     end for
A18     return X
A19 end function

Example

Consider a primitive clinic in a village. People in the village have a very nice property that they are either healthy or have a fever. They can only tell if they have a fever by asking a doctor in the clinic. The wise doctor makes a diagnosis of fever by asking patients how they feel. Villagers only answer that they feel normal, dizzy, or cold.

Suppose a patient comes to the clinic each day and tells the doctor how she feels. The doctor believes that the health condition of this patient operates as a discrete Markov chain. There are two states, "Healthy" and "Fever", but the doctor cannot observe them directly, that is, they are hidden from him. On each day, there is a certain chance that the patient will tell the doctor he has one of the following feelings, depending on his health condition: "normal", "cold", or "dizzy". Those are the observations. The entire system is that of a hidden Markov model (HMM).

The doctor knows the villager's general health condition, and what symptoms patients complain of with or without fever on average. In other words, the parameters of the HMM are known. They can be represented as follows in the Python programming language:

states = ('Healthy', 'Fever')
 
observations = ('normal', 'cold', 'dizzy')
 
start_probability = {'Healthy': 0.6, 'Fever': 0.4}
 
transition_probability = {
   'Healthy' : {'Healthy': 0.7, 'Fever': 0.3},
   'Fever' : {'Healthy': 0.4, 'Fever': 0.6},
   }
 
emission_probability = {
   'Healthy' : {'normal': 0.5, 'cold': 0.4, 'dizzy': 0.1},
   'Fever' : {'normal': 0.1, 'cold': 0.3, 'dizzy': 0.6},
   }

In this piece of code, start_probability represents the doctor's belief about which state the HMM is in when the patient first visits (all he knows is that the patient tends to be healthy). The particular probability distribution used here is not the equilibrium one, which is (given the transition probabilities) approximately {'Healthy': 0.57, 'Fever': 0.43}. The transition_probability represents the change of the health condition in the underlying Markov chain. In this example, there is only a 30% chance that tomorrow the patient will have a fever if he is healthy today. The emission_probability represents how likely the patient is to feel on each day. If he is healthy, there is a 50% chance that he feels normal; if he has a fever, there is a 60% chance that he feels dizzy.

The patient visits three days in a row and the doctor discovers that on the first day he feels normal, on the second day he feels cold, on the third day he feels dizzy. The doctor has a question: what is the most likely sequence of health condition of the patient would explain these observations? This is answered by the Viterbi algorithm.

# Helps visualize the steps of Viterbi.
def print_dptable(V):
    print "    ",
    for i in range(len(V)): print "%7d" % i,
    print

    for y in V[0].keys():
        print "%.5s: " % y,
        for t in range(len(V)):
            print "%.7s" % ("%f" % V[t][y]),
        print

def viterbi(obs, states, start_p, trans_p, emit_p):
    V = [{}]
    path = {}

    # Initialize base cases (t == 0)
    for y in states:
        V[0][y] = start_p[y] * emit_p[y][obs[0]]
        path[y] = [y]

    # Run Viterbi for t > 0
    for t in range(1,len(obs)):
        V.append({})
        newpath = {}

        for y in states:
            (prob, state) = max([(V[t-1][y0] * trans_p[y0][y] * emit_p[y][obs[t]], y0) for y0 in states])
            V[t][y] = prob
            newpath[y] = path[state] + [y]

        # Don't need to remember the old paths
        path = newpath

    print_dptable(V)
    (prob, state) = max([(V[len(obs) - 1][y], y) for y in states])
    return (prob, path[state])

The function viterbi takes the following arguments: obs is the sequence of observations, e.g. ['normal', 'cold', 'dizzy']; states is the set of hidden states; start_p is the start probability; trans_p are the transition probabilities; and emit_p are the emission probabilities. For simplicity of code, we assume that the observation sequence obs is non-empty and that trans_p[i][j] and emit_p[i][j] is defined for all states i,j.

In the running example, the forward/Viterbi algorithm is used as follows:

def example():
    return viterbi(observations,
                   states,
                   start_probability,
                   transition_probability,
                   emission_probability)
print example()

This reveals that the observations ['normal', 'cold', 'dizzy'] were most likely generated by states ['Healthy', 'Healthy', 'Fever']. In other words, given the observed activities, the patient was most likely to have been healthy both on the first day when he felt normal as well as on the second day when he felt cold, and then he contracted a fever the third day.

The operation of Viterbi's algorithm can be visualized by means of a trellis diagram. The Viterbi path is essentially the shortest path through this trellis. The trellis for the clinic example is shown below; the corresponding Viterbi path is in bold:

Animation of the trellis diagram for the Viterbi algorithm. After Day 3, the most likely path is ['Healthy', 'Healthy', 'Fever'] — Animation of the trellis diagram for the Viterbi algorithm. After Day 3, the most likely path is `['Healthy', 'Healthy', 'Fever']`

When implementing Viterbi's algorithm, it should be noted that many languages use floating point arithmetic - as p is small, this may lead to underflow in the results. A common technique to avoid this is to take the logarithm of the probabilities and use it throughout the computation, the same technique used in the Logarithmic Number System. Once the algorithm has terminated, an accurate value can be obtained by performing the appropriate exponentiation.

Extensions

A generalization of the Viterbi algorithm, termed the max-sum algorithm (or max-product algorithm) can be used to find the most likely assignment of all or some subset of latent variables in a large number of graphical models, e.g. Bayesian networks, Markov random fields and conditional random fields. The latent variables need in general to be connected in a way somewhat similar to an HMM, with a limited number of connections between variables and some type of linear structure among the variables. The general algorithm involves message passing and is substantially similar to the belief propagation algorithm (which is the generalization of the forward-backward algorithm).

With the algorithm called iterative Viterbi decoding one can find the subsequence of an observation that matches best (on average) to a given HMM. This algorithm is proposed by Qi Wang, etc.^[3] to deal with turbo code. Iterative Viterbi decoding works by iteratively invoking a modified Viterbi algorithm, reestimating the score for a filler until convergence.

An alternative algorithm, the Lazy Viterbi algorithm, has been proposed recently.^[4] For many codes of practical interest,under reasonable noise conditions, the lazy decoder (using Lazy Viterbi algorithm) is much faster than the original Viterbi decoder (using Viterbi algorithm).^[5] This algorithm works by not expanding any nodes until it really needs to, and usually manages to get away with doing a lot less work (in software) than the ordinary Viterbi algorithm for the same result - however, it is not so easy to parallelize in hardware.

Moreover, the Viterbi algorithm has been extended to operate with a deterministic finite automaton in order to improve speed for use in stochastic letter-to-phoneme conversion.^[6]

Notes

^ 29 Apr 2005, G. David Forney Jr: The Viterbi Algorithm: A Personal History
^ Xing E, slide 11
^ Qi Wang (2002). "Iterative Viterbi Decoding, Trellis Shaping,and Multilevel Structure for High-Rate Parity-Concatenated TCM". IEEE TRANSACTIONS ON COMMUNICATIONS. 50: 48–55. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ A fast maximum-likelihood decoder for convolutional codes (PDF). Vehicular Technology Conference. 2002. pp. 371–375. doi:10.1109/VETECF.2002.1040367. {{cite conference}}: External link in |conferenceurl= (help); Unknown parameter |conferenceurl= ignored (|conference-url= suggested) (help); Unknown parameter |month= ignored (help)
^ A fast maximum-likelihood decoder for convolutional codes (PDF). Vehicular Technology Conference. 2002. p. 371. doi:10.1109/VETECF.2002.1040367. {{cite conference}}: External link in |conferenceurl= (help); Unknown parameter |conferenceurl= ignored (|conference-url= suggested) (help); Unknown parameter |month= ignored (help)
^ Luk, R.W.P. (1998). "Computational complexity of a fast Viterbi decoding algorithm for stochastic letter-phoneme transduction". IEEE Trans. Speech and Audio Processing. 6 (3): 217–225. doi:10.1109/89.668816. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

References

Viterbi AJ (1967). "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm". IEEE Transactions on Information Theory. 13 (2): 260–269. doi:10.1109/TIT.1967.1054010. {{cite journal}}: Unknown parameter |month= ignored (help) (note: the Viterbi decoding algorithm is described in section IV.) Subscription required.
Feldman J, Abou-Faycal I, Frigo M (2002). "A Fast Maximum-Likelihood Decoder for Convolutional Codes". Vehicular Technology Conference. 1: 371–375. doi:10.1109/VETECF.2002.1040367.{{cite journal}}: CS1 maint: multiple names: authors list (link)
Forney GD (1973). "The Viterbi algorithm". Proceedings of the IEEE. 61 (3): 268–278. doi:10.1109/PROC.1973.9030. {{cite journal}}: Unknown parameter |month= ignored (help) Subscription required.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 16.2. Viterbi Decoding". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Rabiner LR (1989). "A tutorial on hidden Markov models and selected applications in speech recognition". Proceedings of the IEEE. 77 (2): 257–286. doi:10.1109/5.18626. {{cite journal}}: Unknown parameter |month= ignored (help) (Describes the forward algorithm and Viterbi algorithm for HMMs).
Shinghal, R. and Godfried T. Toussaint, "Experiments in text recognition with the modified Viterbi algorithm," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-l, April 1979, pp. 184–193.
Shinghal, R. and Godfried T. Toussaint, "The sensitivity of the modified Viterbi algorithm to the source statistics," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-2, March 1980, pp. 181–185.

Implementations

External links

Implementations in Java, F#, Clojure, C# on Wikibooks
Tutorial on convolutional coding with viterbi decoding, by Chip Fleming
The history of the Viterbi Algorithm, by David Forney
A Gentle Introduction to Dynamic Programming and the Viterbi Algorithm
A tutorial for a Hidden Markov Model toolkit (implemented in C) that contains a description of the Viterbi algorithm

[1] 29 Apr 2005, G. David Forney Jr: The Viterbi Algorithm: A Personal History

[2] Xing E, slide 11

[3] Qi Wang (2002). "Iterative Viterbi Decoding, Trellis Shaping,and Multilevel Structure for High-Rate Parity-Concatenated TCM". IEEE TRANSACTIONS ON COMMUNICATIONS. 50: 48–55. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[4] A fast maximum-likelihood decoder for convolutional codes (PDF). Vehicular Technology Conference. 2002. pp. 371–375. doi:10.1109/VETECF.2002.1040367. {{cite conference}}: External link in |conferenceurl= (help); Unknown parameter |conferenceurl= ignored (|conference-url= suggested) (help); Unknown parameter |month= ignored (help)

[5] A fast maximum-likelihood decoder for convolutional codes (PDF). Vehicular Technology Conference. 2002. p. 371. doi:10.1109/VETECF.2002.1040367. {{cite conference}}: External link in |conferenceurl= (help); Unknown parameter |conferenceurl= ignored (|conference-url= suggested) (help); Unknown parameter |month= ignored (help)

[6] Luk, R.W.P. (1998). "Computational complexity of a fast Viterbi decoding algorithm for stochastic letter-phoneme transduction". IEEE Trans. Speech and Audio Processing. 6 (3): 217–225. doi:10.1109/89.668816. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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