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 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 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, speech synthesis, diarization, 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. The Viterbi algorithm is named after Andrew Viterbi, who proposed it in 1967 as a decoding algorithm for convolutional codes over noisy digital communication links. It has, however, a history of multiple invention, with at least seven independent discoveries, including those by Viterbi, Needleman and Wunsch, and Wagner and Fischer. 'Viterbi path' and 'Viterbi algorithm' have become standard terms for the application of dynamic programming algorithms to maximization problems involving probabilities.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 commonly called the 'Viterbi parse'. Another application is in target tracking, where the track is computed that assigns a maximum likelihood to a sequence of observations. 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 hidden Markov model. This algorithm is proposed by Qi Wang et al. 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. For many applications 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). While the original Viterbi algorithm calculates every node in the trellis of possible outcomes, the Lazy Viterbi algorithm maintains a prioritized list of nodes to evaluate in order, and the number of calculations required is typically fewer (and never more) than the ordinary Viterbi algorithm for the same result. However, it is not so easy to parallelize in hardware. This algorithm generates a path X = ( x 1 , x 2 , … , x T ) {displaystyle X=(x_{1},x_{2},ldots ,x_{T})} , which is a sequence of states x n ∈ S = { s 1 , s 2 , … , s K } {displaystyle x_{n}in S={s_{1},s_{2},dots ,s_{K}}} that generate the observations Y = ( y 1 , y 2 , … , y T ) {displaystyle Y=(y_{1},y_{2},ldots ,y_{T})} with y n ∈ O = { o 1 , o 2 , … , o N } {displaystyle y_{n}in O={o_{1},o_{2},dots ,o_{N}}} ( N {displaystyle N} being the count of observations (observation space, see below)). Two 2-dimensional tables of size K × T {displaystyle K imes T} are constructed: