A hidden Markov model
) is a statistical model
in which 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. An HMM can be considered as the simplest dynamic Bayesian network
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 temporal pattern recognition such as speech, handwriting, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.
Architecture of a hidden Markov model
The diagram below shows the general architecture of an instantiated HMM. Each oval shape represents a random variable that can adopt a number of values. The random variable
is the hidden state at time
(with the model from the above diagram,
). The random variable
is the observation at time
). The arrows in the diagram (often called a trellis diagram
) denote conditional dependencies.
From the diagram, it is clear that the value of the hidden variable (at time ) only depends on the value of the hidden variable : the values at time and before have no influence. This is called the Markov property. Similarly, the value of the observed variable only depends on the value of the hidden variable (both at time ).
Probability of an observed sequence
The probability of observing a sequence of length is given by
where the sum runs over all possible hidden node sequences . Brute force calculation of is intractable for most real-life problems, as the number of possible hidden node sequences is typically extremely high. The calculation can however be sped up enormously using the forward algorithm or the equivalent backward algorithm.
Using hidden Markov models
There are three canonical problems associated with HMM:
- Given the parameters of the model, compute the probability of a particular output sequence, and the probabilities of the hidden state values given that output sequence. This problem is solved by the forward-backward algorithm.
- Given the parameters of the model, find the most likely sequence of hidden states that could have generated a given output sequence. This problem is solved by the Viterbi algorithm.
- Given an output sequence or a set of such sequences, find the most likely set of state transition and output probabilities. In other words, discover the parameters of the HMM given a dataset of sequences. This problem is solved by the Baum-Welch algorithm.
A concrete example
This example is further elaborated in the Viterbi algorithm page.
Applications of hidden Markov models
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.
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.
- 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 (1999). Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press.
- Lior Pachter and Bernd Sturmfels. "Algebraic Statistics for Computational Biology". Cambridge University Press, 2005. ISBN 0-521-85700-7.
- Olivier Cappé, Eric Moulines, Tobias Rydén. Inference in Hidden Markov Models, Springer, 2005. ISBN 0-387-40264-0.
- 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: ).
- Tutorial from University of Leeds
- 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.
- Y. Ephraim and N. Merhav, Hidden Markov processes, IEEE Trans. Inform. Theory, vol. 48, pp. 1518-1569, June 2002.
- B. Pardo and W. Birmingham. Modeling Form for On-line Following of Musical Performances AAAI-05 Proc., July 2005.
- Thad Starner, Alex Pentland. Visual Recognition of American Sign Language Using Hidden Markov Master's Thesis, MIT, Feb 1995, Program in Media Arts
- L.Satish and B.I.Gururaj. Use of hidden Markov models for partial discharge pattern classificationIEEE Transactions on Dielectrics and Electrical Insulation, Apr 1993.
The path-counting algorithm, an alternative to the Baum-Welch algorithm: