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Algebraic Statistics for Computational Biology

Publisher: 
Cambridge University Press
Number of Pages: 
420
Price: 
60.00
ISBN: 
0-521-85700-7
Date Received: 
Thursday, October 20, 2005
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Lior Pachter and Bernd Sturmfels, editors
Publication Date: 
2005
Format: 
Hardcover
Category: 
Monograph

Preface page ix

Guide to the chapters xi

Acknowledgment of support xii

Part I Introduction to the four themes 1

1 Statistics L. Pachter and B. Sturmfels 3

1.1 Statistical models for discrete data 4

1.2 Linear models and toric models 9

1.3 Expectation Maximization 17

1.4 Markov models 24

1.5 Graphical models 33

2 Computation L. Pachter and B. Sturmfels 43

2.1 Tropical arithmetic and dynamic programming 44

2.2 Sequence alignment 49

2.3 Polytopes 59

2.4 Trees and metrics 67

2.5 Software 75

3 Algebra L. Pachter and B. Sturmfels 85

3.1 Varieties and Gr¨obner bases 86

3.2 Implicitization 94

3.3 Maximum likelihood estimation 102

3.4 Tropical geometry 109

3.5 The tree of life and other tropical varieties 117

4 Biology L. Pachter and B. Sturmfels 125

4.1 Genomes 126

4.2 The data 132

4.3 The problems 137

4.4 Statistical models for a biological sequence 141

4.5 Statistical models of mutation 147

v

© Cambridge University Press www.cambridge.org

Cambridge University Press

0521857007 - Algebraic Statistics for Computational Biology

Edited by Lior Pachter and Bernd Sturmfels

Table of Contents

More information

vi

Part II Studies on the four themes 161

5 Parametric Inference R. Mihaescu 165

5.1 Tropical sum-product decompositions 166

5.2 The polytope propagation algorithm 169

5.3 Algorithm complexity 173

5.4 Specialization of parameters 177

6 Polytope Propagation on Graphs M. Joswig 181

6.1 Polytopes from directed acyclic graphs 181

6.2 Specialization to hidden Markov models 185

6.3 An implementation in polymake 186

6.4 Returning to our example 191

7 Parametric Sequence Alignment

C. Dewey and K. Woods 193

7.1 Few alignments are optimal 193

7.2 Polytope propagation for alignments 195

7.3 Retrieving alignments from polytope vertices 199

7.4 Biologically correct alignments 202

8 Bounds for Optimal Sequence Alignment

S. Elizalde and F. Lam 206

8.1 Alignments and optimality 206

8.2 Geometric interpretation 208

8.3 Known bounds 211

8.4 The square root conjecture 212

9 Inference Functions S. Elizalde 215

9.1 What is an inference function? 215

9.2 The few inference functions theorem 217

9.3 Inference functions for sequence alignment 220

10 Geometry of Markov Chains E. Kuo 226

10.1 Viterbi sequences 226

10.2 Two- and three-state Markov chains 229

10.3 Markov chains with many states 231

10.4 Fully observed Markov models 233

11 Equations Defining Hidden Markov Models

N. Bray and J. Morton 237

11.1 The hidden Markov model 237

11.2 Gr¨obner bases 238

11.3 Linear algebra 240

11.4 Combinatorially described invariants 247

Contents

© Cambridge University Press www.cambridge.org

Cambridge University Press

0521857007 - Algebraic Statistics for Computational Biology

Edited by Lior Pachter and Bernd Sturmfels

Table of Contents

More information

vii

12 The EM Algorithm for Hidden Markov Models

I. B. Hallgr´ımsd´ottir, R. A. Milowski and J. Yu 250

12.1 The EM algorithm for hidden Markov models 250

12.2 An implementation of the Baum–Welch algorithm 254

12.3 Plots of the likelihood surface 257

12.4 The EM algorithm and the gradient of the likelihood 261

13 Homology Mapping with Markov Random Fields A. Caspi 264

13.1 Genome mapping 264

13.2 Markov random fields 267

13.3 MRFs in homology assignment 270

13.4 Tractable MAP inference in a subclass of MRFs 273

13.5 The Cystic Fibrosis Transmembrane Regulator 276

14 Mutagenetic Tree Models N. Beerenwinkel and M. Drton 278

14.1 Accumulative evolutionary processes 278

14.2 Mutagenetic trees 279

14.3 Algebraic invariants 282

14.4 Mixture models 287

15 Catalog of Small Trees

M. Casanellas, L. D. Garcia, and S. Sullivant 291

15.1 Notation and conventions 291

15.2 Fourier coordinates 295

15.3 Description of website features 297

15.4 Example 298

15.5 Using the invariants 303

16 The Strand Symmetric Model

M. Casanellas and S. Sullivant 305

16.1 Matrix-valued Fourier transform 306

16.2 Invariants for the 3-taxa tree 310

16.3 G-tensors 314

16.4 Extending invariants 318

16.5 Reduction to K1,3 319

17 Extending Tree Models to Splits Networks D. Bryant 322

17.1 Trees, splits and splits networks 322

17.2 Distance-based models for trees and splits graphs 325

17.3 A graphical model on a splits network 326

17.4 Group-based mutation models 327

17.5 Group-based models for trees and splits 330

17.6 A Fourier calculus for splits networks 332

Contents

© Cambridge University Press www.cambridge.org

Cambridge University Press

0521857007 - Algebraic Statistics for Computational Biology

Edited by Lior Pachter and Bernd Sturmfels

Table of Contents

More information

viii

18 Small Trees and Generalized Neighbor-Joining

M. Contois and D. Levy 335

18.1 From alignments to dissimilarity 335

18.2 From dissimilarity to trees 337

18.3 The need for exact solutions 342

18.4 Jukes–Cantor triples 344

19 Tree Construction using Singular Value Decomposition

N. Eriksson 347

19.1 The general Markov model 347

19.2 Flattenings and rank conditions 348

19.3 Singular Value Decomposition 351

19.4 Tree-construction algorithm 352

19.5 Performance analysis 355

20 Applications of Interval Methods to Phylogenetics

R. Sainudiin and R. Yoshida 359

20.1 Brief introduction to interval analysis 360

20.2 Enclosing the likelihood of a compact set of trees 366

20.3 Global optimization 366

20.4 Applications to phylogenetics 371

21 Analysis of Point Mutations in Vertebrate Genomes

J. Al-Aidroos and S. Snir 375

21.1 Estimating mutation rates 375

21.2 The ENCODE data 378

21.3 Synonymous substitutions 379

21.4 The rodent problem 381

22 Ultra-Conserved Elements in Vertebrate and Fly Genomes

M. Drton, N. Eriksson and G. Leung 387

22.1 The data 387

22.2 Ultra-conserved elements 390

22.3 Biology of ultra-conserved elements 392

22.4 Statistical significance of ultra-conservation 400

References 403

Index 418

 

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