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Hidden Markov Models

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Book cover Markov Models for Pattern Recognition

Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

Abstract

In the field of pattern recognition, signals are frequently thought of as the product of a statistical generation process. The primary goal of analyzing these signals is to model their statistical properties as exactly as possible. However, the model to be determined should not only replicate the generation of certain data but also deliver useful information for segmenting the signals into meaningful units.

Hidden Markov models are able to account for both these modeling aspects. First, they define a generative statistical model that is able to generate data sequences according to rather complex probability distributions and that can be used for classifying sequential patterns. Second, information about the segmentation of the data considered can be derived from the two-stage stochastic process that is described by a hidden Markov model. Consequently, hidden Markov models possess the quite remarkable ability to treat segmentation and classification of patterns in an integrated framework.

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Notes

  1. 1.

    In the tradition of research at IBM, HMMs are described in a slightly different way. There outputs are generated during state transitions, i.e., on the edges of the model ([11], cf. e.g. [136, p. 17]). With respect to its expressive power this formulation is, however, completely equivalent to the one which is used in this book and also throughout the majority of the literature as is also shown in [136, pp. 35–36].

  2. 2.

    In [125, p. 150] analogously to the start probabilities also termination probabilities are defined. The HMM architectures used for analyzing biological sequences usually contain special non-emitting start and end states (see also Sect. 8.4).

  3. 3.

    The empirical distribution of a continuous random variable would assign non-zero density values only to known data points and, therefore, would be useless in practice. An approximation of the true density function by a Parzen estimate (cf. e.g. [65, Sect. 4.3]) would be possible in principle but require the storage of the complete data set.

  4. 4.

    According to Rabiner ([251], [252, p. 322]) the idea of characterizing the possible use cases of HMMs in the form of three fundamental problems paired with three corresponding algorithms for their solution goes back to Jack Ferguson, Institute for Defense Analyses [79].

  5. 5.

    Original quote from Ralph Waldo Emerson (American Philosopher, 1803–1882) from the essay “Self Reliance” (1841).

  6. 6.

    As the name indicates already, there exists a matching counterpart of the forward algorithm which is referred to as the backward algorithm. Taken together they constitute the so-called forward–backward algorithm which will be presented in its entirety during the description of the HMM parameter training in Sect. 5.7.

  7. 7.

    If specially marked end states are used (cf. e.g. [67, p. 51]), the maximization needs to be restricted to the appropriate set. Alternatively, also additional termination probabilities can be considered when computing the final path probabilities (cf. e.g. [125, p. 150]).

  8. 8.

    In general, the decision for the optimal predecessor state is not unique. Multiple sequences \(\boldsymbol{s}^{*}_{k}\) with identical scores maximizing Eq. (5.6) might exist. In practical applications, therefore, a rule for breaking up such ambiguities is necessary which might, e.g., be a preference for states with lower indices.

  9. 9.

    In practice this means that the chosen quality measure does not change any more within the scope of the available computational accuracy.

  10. 10.

    Of course the optimization of general continuous mixture densities is possible with all training methods. However, it always represents the most challenging part of the procedure.

  11. 11.

    The updated start probabilities \(\hat{\pi}_{i}\) can be considered as a special case of the transition probabilities \(\hat{a}_{ij}\) and can, therefore, be computed analogously.

  12. 12.

    The fact that the total output probability P(O∣λ) may be computed both via the forward and the backward algorithm can be exploited in practice to check the computations for consistency and accuracy.

  13. 13.

    Due to the close connection in their meaning and in order not to unnecessarily make the notation any more complex, the state probability is denoted with a single argument as γ t (i) and the state-transition probability with two arguments as γ t (i,j).

  14. 14.

    According to common opinion, HMMs do not perform a state transition into a specially marked end state when reaching the end of the observation sequence at time T. Therefore, the restriction to all prior points in time is necessary here.

  15. 15.

    For any given time t the symbol o k was either present in the observation sequence or not. Therefore, the probability P(S t =j,O t =o k ∣O,λ) either takes on the value zero or is equal to P(S t =j∣O,λ) which is simply γ t (j).

  16. 16.

    For time t=1 in Eq. (5.20) the term \(\sum_{i=1}^{N} \alpha_{t-1}(i) a_{ij}\) needs to be replaced by the respective start probability π j of the associated state.

  17. 17.

    Please note that in [209] and in subsequent works using this method, covariance modeling and state-transition probabilities hardly play any role. The update equation for covariances is, therefore, given in analogy to Eq. (5.24). Parameter updates for transition probabilities can be obtained in the same way as for discrete models (see Eq. (5.27)).

  18. 18.

    For reasons of efficiency it is rather obvious to use the k-means algorithm as vector quantization method as it achieves competitive results with only a single pass through the data. However, principally any algorithm for vector quantizer design or the unsupervised estimation of mixture densities could be applied.

  19. 19.

    The estimation of transition probabilities and discrete output probabilities is, for example, explained in [125, pp. 157–158].

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  1. Department of Computer Science, TU Dortmund University, Dortmund, Germany

    Gernot A. Fink

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Fink, G.A. (2014). Hidden Markov Models. In: Markov Models for Pattern Recognition. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-6308-4_5

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  • DOI: https://doi.org/10.1007/978-1-4471-6308-4_5

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