Viterbi algorithm
Introduction
The Viterbi algorithm is a dynamic programming algorithm used 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 (HMMs). Named after its inventor, Andrew Viterbi, the algorithm is used in various applications such as speech recognition, bioinformatics, and wireless communication.
History and Development
The Viterbi algorithm was developed by Andrew Viterbi, an engineer and co-founder of Qualcomm Inc., in 1967. The algorithm was initially used in the field of telecommunications, specifically in decoding convolutional codes used in error detection and correction. Over time, the algorithm found its way into various other fields, including speech recognition, bioinformatics, and data science.
Theoretical Background
The Viterbi algorithm is based on the principles of dynamic programming and probability theory. It operates on a sequence of observed events and a set of possible states that could have resulted in these events. The algorithm calculates the most probable path of states that led to the observed events, given the probabilities of transitioning between states and the probabilities of each state emitting each observed event.
The algorithm operates on the assumption that the probability of being in a particular state depends only on the previous state (a property known as the Markov property) and that the observed events are conditionally independent given the sequence of states. These assumptions simplify the calculation of the most probable path, allowing the algorithm to operate efficiently even on large sequences.
Algorithm Description
The Viterbi algorithm operates in two main steps: initialization and recursion.
Initialization
In the initialization step, the algorithm sets up the data structures used in the computation. For each state, it calculates the initial probability of being in that state and emits the first observed event. This initial probability is typically given as part of the input to the algorithm.
Recursion
In the recursion step, the algorithm iteratively calculates the most probable path to each state at each time step, given the observed events up to that time. For each state and each time step, it calculates the maximum over all possible previous states of the probability of the most probable path ending in that previous state, times the probability of transitioning from that previous state to the current state, times the probability of emitting the current observed event given the current state.
After the recursion step, the algorithm has calculated the most probable path to each state at each time step. The most probable sequence of states—the Viterbi path—is then the sequence of states that has the maximum probability over all time steps.
Applications
The Viterbi algorithm has found wide application in various fields, thanks to its ability to efficiently calculate the most probable sequence of hidden states given a sequence of observed events.
Speech Recognition
In speech recognition, the observed events are the acoustic signals, and the hidden states are the words or phonemes that the speaker intended to say. The Viterbi algorithm is used to find the most probable sequence of words or phonemes that could have resulted in the observed acoustic signals.
Bioinformatics
In bioinformatics, the Viterbi algorithm is used in the analysis of biological sequences, such as DNA, RNA, and protein sequences. The observed events are the symbols in the sequence, and the hidden states are the biological features of interest, such as genes or protein-coding regions. The Viterbi algorithm is used to find the most probable annotation of the sequence, given the observed symbols and a model of the biological features.
Wireless Communication
In wireless communication, the Viterbi algorithm is used in the decoding of convolutional codes, which are used for error detection and correction. The observed events are the received symbols, and the hidden states are the transmitted symbols. The Viterbi algorithm is used to find the most probable sequence of transmitted symbols that could have resulted in the received symbols, given the probabilities of symbol transitions and the probabilities of symbol emissions.