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980 results in Applied probability and stochastic networks

Page 20 of 49

  • Preface

  • John van der Hoek, University of South Australia, Robert J. Elliott, University of Calgary
  • Book:
    Introduction to Hidden Semi-Markov Models
    Published online:
    01 February 2018
    Print publication:
    08 February 2018, pp ix-x

  • Summary

    The purpose of this volume is to present the theory of Markov and semi-Markov processes in a discrete-time, finite-state framework. Given this background, hidden versions of these processes are introduced and related estimation and filtering results developed. The approach is similar to the earlier book, Elliott et al. (1995). That is, a central tool is the Radon–Nikodym theorem and related changes of probability measure. In the discrete-time, finite-state framework that we employ these have simple interpretations following from Bayes’ theorem.

    Markov chains and hidden Markov chains have found many applications in fields from finance, where the chains model different economic regimes, to genomics, where gene and protein structure is modelled as a hidden Markov model. Semi-Markov chains and hidden semi-Markov chains will have similar, possibly more realistic, applications. The genomics applications are modelled by discrete observations of these hidden chains.

    Recent books in the area include in particular Koski (2001) and Barbu and Limnios (2008). Koski includes many examples, not much theory and little on semi-Markov Models. Barbu and Limnios say that the estimation of discrete-time semi-Markov systems is almost absent from the literature. They present an alternative specification from the one adopted in this book and so we give alternative methods in a rigorous framework. They provide limited applications in genomics.

    This book carefully constructs relevant processes and proves required results. The filters and related parameter estimation methods we obtain for semi-Markov chains include new results. The occupation times in any state of a Markov chain are geometrically distributed; semi-Markov chains can have occupation times which are quite general and not necessarily geometrically distributed.

    Works on semi-Markov processes include Barbu and Limnios (2008), C， inlar (1975), Harlamov (2008), Howard (1971), Janssen and Manca (2010), and Koski (2001) from Chapter 11 onwards. C， inlar (1975) considers a countable state space.

    Hidden Markov models have found extensive applications in speech processing and genomics. References for these applications include Ferguson (1980), who considers more general occupation times. This problem was also investigated by Levinson (1986a,b), Ramesh and Wilpon (1992), and in the papers Guédon (1992) and Guédon and Cocozza-Thivent (1990). Genomic applications are treated in the thesis of Burge (1997) and the book Burge and Karlin (1997).

  • Appendix D - On Conditional Expectations

  • John van der Hoek, University of South Australia, Robert J. Elliott, University of Calgary
  • Book:
    Introduction to Hidden Semi-Markov Models
    Published online:
    01 February 2018
    Print publication:
    08 February 2018, pp 158-160

  • Summary

    Let X and Y be random variables with on some probability space. In fact, let us suppose that.

    By the Doob–Dynkin Lemma (see for example, Oksendal, 2010, Lemma 2.1.2)

    for some Borel function g : if Y takes values in.

    If Y takes countably many distinct values, then g can be described in more detail. The atoms of σ(Y), the σ-algebra generated by Y, are the disjoint sets

    This means that any element of σ(Y) is either ∅ or a countable union of atoms. The function g is characterized as follows: g is constant on each atom, and

    for all A ∈ σ(Y). In fact (D.1) is equivalent to the result holding for atoms A. This leads to

    where. If P, then

    But if, then (D.2) becomes

    as. This means that gk can be given any finite value. If we set, then

    So we can say then when, then.

    We could also note that

    and we can set on sets when and obtain a random variable equal to the conditional expected value with probability 1. As many authors use this convention, we shall also adopt it.

    Now let W be another random variable which takes countably many distinct values. If, then Then

    with probability 1. This is because

    with probability 1, since

    We could apply these ideas with, and and. If, then

    with probability 1. In the application above, if

    implies

    We could also discuss what is meant by P(A|B) when P(B) = 0. It should be the value of

    on the set B. Here.With our convention, we would set.

  • Appendix F - Earlier Applications of HiddenMarkov Chain Models

  • John van der Hoek, University of South Australia, Robert J. Elliott, University of Calgary
  • Book:
    Introduction to Hidden Semi-Markov Models
    Published online:
    01 February 2018
    Print publication:
    08 February 2018, pp 165-168

  • Summary

    Introduction

    In this appendix some earlier application methods are briefly described.

    Markov chain models can be used to provide probability models for sequences of symbols. This will aid in genome annotation. The types of questions that can be asked include the following: Does a particular sequence belong to a particular family and what can one say about its internal structure? How can one discriminate between two sequences?

    Some general reviews are given in (Durbin et al., 1998, Chapters 2 and 3), (Robin et al., 2005, Chapters 1 and 2), but a more detailed review of observed Markov chains is provided by (Koski, 2001, Chapter 9). We have added some extra details to Koski's treatment.

    A straightforward application of Markov chains to genome sequencing. This approach does not seem to work for the following reasons:

  • • The four bases A, T, G, C are not uniformly distributed in a sequence and the compositions vary within and between sequences.

  • • Various k-tuples of bases are not uniformly distributed. However, exons and introns are often separated on the basis of dinucleotide frequencies.

  • • It seems that higher-order chains need to be used as probabilities of a base in a particular location and then can depend not only on the immediately adjacent bases. In addition, the base composition can vary from one segment to another. The segmentation techniques for decomposing DNA sequences into homogeneous segments includes hidden Markov models.

    • Frame-dependent Markov chains. These use the GeneMark software; information can be found at

    http://genemark.biology.gatech.edu/GeneMark/gm_info.html

    Mixture transition distribution chain of order k. These are called MTD(k) models. For a Markov chain of order k with a state-space of size N, there are (N − 1)Nk entries in the transition matrix A to be estimated, (the column sums of A are 1), plus the initial probabilities. With N = 4 and k = 8, we have 3 ・ 48 = 196, 608 which is quite large. This has a further implication that we may not have enough data to calibrate all these entries in A. We comment on estimation using sparse data below.

  • 9 - Hidden Semi-Markov Models

  • John van der Hoek, University of South Australia, Robert J. Elliott, University of Calgary
  • Book:
    Introduction to Hidden Semi-Markov Models
    Published online:
    01 February 2018
    Print publication:
    08 February 2018, pp 110-124

  • Summary

    Introduction

    The chapter first presents a construction of a semi-Markov process X on its canonical probability space, the space of all sequences of elements of the state space. It is then supposed that the semi-Markov chain is not directly observed but that there is a second finite state process Y whose transitions depend on the state of the hidden process X.

    In Chapter 8 we described a process ﹛(Tn, Zn)﹜ with

    We also had

    With this decomposition, the parameters of the model are, with and and these are to be estimated.

    An alternative decomposition is

    With this decomposition, the parameters of the model that are to be estimated are and. In fact we shall estimate this second decomposition.

    The work of Ferguson (1980), Burge (1997), Burge and Karlin (1997), Bulla (2006), Bulla and Bulla (2006), Bulla et al. (2010), Gu#x00E9;don and Cocozza-Thivent (1990) and others have used the second specification but with Aj i not depending on m. In these works the model is simulated by first selecting X0 = Z0 according to an initial distribution. If X0 = ei, then a duration T1 − T0 is selected from the distribution given by \and then a change of state according to the distribution and so on. The specification of these authors is equivalent to assuming that (m) in the first formulation does not depend on When applying this restricted formulation, it is necessary to determine if the model is rich enough to model the application.

    Some authors give parametric forms to. This reduces the number of parameters of the model, but a suitable parametric model will need to be justified for each application. See Levinson (1986a,b), Ramesh and Wilpon (1992) Gu#x00E9;don (1992, 1999, 2003, 2007) and Gu#x00E9;don and Cocozza-Thivent (1990). A good review of various approaches is Yu (1986) but we shall use different notation and provide some alternative estimates.

Page 20 of 49