Hidden Markov model

MarkovModel.png|right|thumb|300px| State transitions in a hidden Markov model (example)
x — hidden states
y — observable outputs
a — transition probabilities
b — output probabilities]] A hidden Markov model (HMM) is a statistical model where the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to determine the hidden parameters from the observable parameters. The extracted model parameters can then be used to perform further analysis, for example for pattern recognition applications.

In a regular Markov model, the state is directly visible to the observer, and therefore the state transition probabilities are the only parameters. In a hidden Markov model, the state is not directly visible, but variables influenced by the state are visible. Each state has a probability distribution over the possible output tokens. Therefore the sequence of tokens generated by an HMM gives some information about the sequence of states.

Hidden Markov models are especially known for their application in speech recognition and biological sequence analysis (bioinformatics).

Evolution of a Markov model

The preceding diagram emphasizes the state transitions of a HMM. It is also useful to explicitly represent the evolution of the model over time, with the states at different times t₁ and t₂ represented by different variables, x(t₁) and x(t₂).

In this diagram, it is understood that the time slices (x(t), y(t)) extend to previous and following times as needed. Typically the earliest slice is at time t=0 or time t=1.

Using Markov models

There are 3 canonical problems to solve with HMMs:

Given the model parameters, compute the probability of a particular output sequence. Solved by the forward algorithm.
Given the model parameters, find the most likely sequence of (hidden) states which could have generated a given output sequence. Solved by the Viterbi algorithm.
Given an output sequence, find the most likely set of state transition and output probabilities. Solved by the Baum-Welch algorithm or the Reversed Viterbi algorithm.

Another, more recent approach is to solve these problems by using the Junction tree algorithm.

A concrete example

This example is further elaborated in Viterbi algorithm page. g

Applications of hidden Markov models

speech recognition or optical character recognition
machine translation
bioinformatics and genomics
- prediction of protein-coding regions in genome sequences
- modelling families of related DNA or protein sequences
- prediction of secondary structure elements from protein primary sequences
and many more...

History

Hidden Markov Models were first described in a series of statistical papers by Leonard E. Baum and other authors in the second half of the 1960s. One of the first applications of HMMs was speech recognition, starting in the mid-1970s.Rabiner, p. 258

In the second half of the 1980s, HMMs began to be applied to the analysis of biological sequences, in particular DNA. Since then, they have become ubiquitous in the field of bioinformatics.Durbin et al.

Notes

References

Lawrence R. Rabiner, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77 (2), p. 257–286, February 1989.
Richard Durbin, Sean R. Eddy, Anders Krogh, Graeme Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1999. ISBN 0521629713.
Lior Pachter and Bernd Sturmfels. "Algebraic Statistics for Computational Biology" Cambridge University Press, 2005. ISBN 0521857007
Kristie Seymore, Andrew McCallum, and Roni Rosenfeld. Learning Hidden Markov Model Structure for Information Extraction. AAAI 99 Workshop on Machine Learning for Information Extraction, 1999. (also at CiteSeer: ^*)
http://www.comp.leeds.ac.uk/roger/HiddenMarkovModels/html_dev/main.html
J. Li, A. Najmi, R. M. Gray, Image classification by a two dimensional hidden Markov model, IEEE Transactions on Signal Processing, 48(2):517-33, February 2000.

External links

Hidden Markov Model (HMM) Toolbox for Matlab (by Kevin Murphy)
Hidden Markov Model Toolkit (HTK) (a portable toolkit for building and manipulating hidden Markov models)
Hidden Markov Models (an exposition using basic mathematics)
GHMM Library (home page of the GHMM Library project)
Jahmm Java Library (Java library and associated graphical application)
A step-by-step tutorial on HMMs (University of Leeds)
Software for Markov Models and Processes (TreeAge Software)

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