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927 results in Applied probability and stochastic networks

Page 18 of 47

  • 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.

Introduction to Hidden Semi-Markov Models
  • John van der Hoek, Robert J. Elliott
  • Published online:
    01 February 2018
    Print publication:
    08 February 2018
  • Markov chains and hidden Markov chains have applications in many areas of engineering and genomics. This book provides a basic introduction to the subject by first developing the theory of Markov processes in an elementary discrete time, finite state framework suitable for senior undergraduates and graduates. The authors then introduce semi-Markov chains and hidden semi-Markov chains, before developing related estimation and filtering results. Genomics applications are modelled by discrete observations of these hidden semi-Markov chains. This book contains new results and previously unpublished material not available elsewhere. The approach is rigorous and focused on applications.

  • Appendix B - Algorithm Design and Analysis

  • Tandy Warnow, University of Illinois, Urbana-Champaign
  • Book:
    Computational Phylogenetics
    Published online:
    26 October 2017
    Print publication:
    02 November 2017, pp 304-326

  • Summary

    This appendix provides necessary background in algorithm design and analysis, and in proving algorithms correct. It covers techniques for computing the running time of an algorithm, the standard “Big-O” notation, what it means for a problem to be NP-hard or NP-complete, and for a problem to be polynomial time. However, the material in this appendix is not meant as a substitute for undergraduate courses in discrete mathematics and algorithm design and analysis, but rather to provide some of the material that the reader needs to know. We recommend that readers without sufficient background in this area consult other textbooks for background material.

    Discrete Mathematics

    Graph Theory

    A graphG consists of a set V of vertices and a set E of edges, where the edges are unordered pairs of vertices; we often write graphs using notation G = (V,E). The edges of a simple graph are always distinct (thus, there are no self-loops, which are edges between a vertex and itself), and no parallel edges (two edges with the same endpoints). In this text, we will only discuss simple graphs.

    Two vertices that are connected by an edge are said to be adjacent and the edge that connects them is said to be incident to its vertex endpoints. If a vertex a is adjacent to vertex b, it is said to be a neighbor of b. The degree of a vertex is the number of edges that are incident with the vertex, which is the same as the number of vertices that are neighbors of the vertex.

    A path in a graph is a sequence of vertices v1, v2, …, vk so that vi is adjacent to vi+1 for each i = 1,2, …, k−1, with the assumption that no two vertices are the same except perhaps v1 and vk. A path in which any vertex can be repeated is called a walk. A graph for which every two vertices are connected by a path is said to be connected. The maximal connected subgraphs of a graph are called the components of the graph.

  • Appendix A - Primer on Biological Data and Evolution

  • Tandy Warnow, University of Illinois, Urbana-Champaign
  • Book:
    Computational Phylogenetics
    Published online:
    26 October 2017
    Print publication:
    02 November 2017, pp 299-303

  • Summary

    This chapter will provide a basic introduction to evolution, biological data, and standard approaches to biological data analysis.

    Phylogeny Estimation Pipeline

    Phylogeny estimation is a multi-step process, and understanding the process is helpful to developing and evaluating methods.

  • • The biologist identifies a question they wish to answer, and based on the question they select the species and genes they will analyze.

  • • Sequence data are collected for the species and genes. This may be done by accessing public databases, or by going into the field and gathering specimens. Typically, at the end of this step, for each selected gene and each species, at most one sequence is obtained. Often this is a DNA sequence, but in some cases it may be an RNA or amino acid sequence (also called a “protein sequence”).

  • • For each gene, a multiple sequence alignment of the different sequences is obtained. This process puts the sequences into a matrix so that the rows correspond to the different species, and the columns represent homologies (nucleotides having a common evolutionary history).

  • • For each gene, a phylogenetic tree is estimated based on the multiple sequence alignment computed in the previous step. This analysis is almost always based on a statistical model of sequence evolution, and most methods combine graph-theoretic methods with statistical estimation techniques. Furthermore, many methods attempt to solve NP-hard optimization problems, so heuristic techniques are often used.

  • • Statistical support for the individual branches of each gene tree is computed, typically using methods such as non-parametric bootstrapping. These support values let the biologist assess which aspects of the estimated evolutionary history are considered highly reliable, and which ones are not as reliable.

  • • Now we have a collection of trees and multiple sequence alignments, one for each gene. When the gene trees are either identical or very similar to each other, then the species tree is often estimated by concatenating the sequence alignments together, and then using standard techniques on the large “super-alignment” to compute a species tree. However, some biological phenomena cause gene trees to be different from each other, so that estimating the species tree can require different techniques, although the choice of technique depends on the cause for the discordance. When the discordance is due to horizontal gene transfer or hybrid speciation, then a phylogenetic network is needed.

  • 9 - Multiple Sequence Alignment

  • from PART II - MOLECULAR PHYLOGENETICS
  • Tandy Warnow, University of Illinois, Urbana-Champaign
  • Book:
    Computational Phylogenetics
    Published online:
    26 October 2017
    Print publication:
    02 November 2017, pp 178-233

  • Summary

    Introduction

    Phylogeny estimation generally begins by estimating a multiple sequence alignment on the set of sequences. Once the multiple sequence alignment is computed, a tree can then be computed on the alignment (Figure 9.1). Not surprisingly, errors in multiple sequence alignment estimation tend to produce errors in estimated trees (Ogden and Rosenberg, 2006; Nelesen et al., 2008; Liu et al., 2009a; Wang et al., 2012) and other downstream analyses. Hence, multiple sequence alignment is an important part of phylogeny estimation.

    As we have seen, there are many methods for estimating trees from gap-free data. However, because multiple sequence alignments almost always contain gaps, represented as dashes, phylogeny estimation methods must be modified to be able to analyze alignments with dashes. Typically this is performed by treating the dashes as missing data (i.e., missing data means there is an actual nucleotide or amino acid, but it is not known). Alternatively, the dashes are sometimes treated as an additional state in the sequence evolution model, thus producing five states for nucleotide alignments or 21 states for amino acid alignments. Finally, sometimes sites (i.e., columns in the multiple sequence alignment) containing dashes are eliminated from the alignment before a tree is computed. The different treatments of sequence alignments can result in quite different theoretical and empirical performance.

    Multiple sequence alignments are computed for different purposes, including phylogeny estimation and protein structure prediction, and the definition of what constitutes a correct alignment depends, at least in part, on the purpose for the alignment. For some biological datasets, curated alignments, typically based on experimentally confirmed structural features of the molecules (e.g., secondary structures or tertiary structures of RNAs and proteins), are used as benchmarks for evaluating alignment methods. Examples of such benchmarks for evaluating large amino acid alignments include HomFam (Sievers et al., 2011), BAliBASE (Thompson et al., 1999), and the 10AA collection (Nguyen et al., 2015b), while the Comparative Ribosomal Website (CRW) provides benchmarks for RNA alignment (Cannone et al., 2002). Evolutionary alignments, on the other hand, are defined by the evolutionary history relating the sequences.

  • 6 - Consensus and Agreement Trees

  • from PART I - BASIC TECHNIQUES
  • Tandy Warnow, University of Illinois, Urbana-Champaign
  • Book:
    Computational Phylogenetics
    Published online:
    26 October 2017
    Print publication:
    02 November 2017, pp 109-120

  • Summary

    Introduction

    In this chapter we discuss techniques for analyzing sets of trees, which are called profiles. Depending on how the set of trees was computed and the objective of the analysis, different techniques are needed.

    For example, when a maximum parsimony analysis is performed, many equally good trees may be found, all having the same “best” score (meaning that it is the best score found during the analysis). Similarly, when a Bayesian MCMC (see Section 8.7) analysis is performed, then a random sample of the trees is examined. Sometimes, many different types of methods are used on the same data, and for each analysis a set of trees is obtained. Then, from the full set of trees, each of which has been estimated on the same data, again some kind of point estimate is sought. In each of these cases, a “consensus method” is used to provide a point estimate of the tree from the full set of trees.

    An alternative objective might be to find those subsets of the taxon set on which all or most of the trees in the profile agree; this kind of approach does not produce a point estimate of the true tree, but can be used to identify the portion of the history on which all trees agree, and also (potentially) problematic taxa. We refer to this type of approach as an “agreement method.”

    In this chapter, we discuss consensus and agreement methods, which are methods for analyzing datasets of trees under the assumption that all the trees are estimated species trees with the same leafset S. We discuss supertree methods in Chapter 7, which extend the consensus methods in this chapter to allow for the taxon sets to be different between trees. Finally, methods for analyzing sets of gene trees that can differ from each other due to incomplete lineage sorting, horizontal gene transfer, or gene duplication and loss are discussed in Chapter 10. Many of the methods described in Chapter 10 can also be used as supertree methods.

    Consensus Trees

    In general, consensus methods are applied to unrooted trees (and we will define them in that context), but they can be modified so as to be applicable to rooted trees as well.

Page 18 of 47