Statistical methods in bioinformatics: an introduction
Gespeichert in:
| Format: | Elektronisch E-Book |
|---|---|
| Sprache: | German |
| Veröffentlicht: |
New York, NY
Springer
2005
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Statistics for biology and health
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| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (XX, 597 S.) graph. Darst. |
| ISBN: | 0387400826 9780387266480 |
| DOI: | 10.1007/b137845 |
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| 245 | 1 | 0 | |a Statistical methods in bioinformatics |b an introduction |c Warren J. Ewens ; Gregory R. Grant |
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Datensatz im Suchindex
| _version_ | 1804136422235439104 |
|---|---|
| adam_text | Contents
Preface vii
1 Probability Theory (i): One Random Variable 1
1.1 Introduction 1
1.2 Discrete Random Variables, Definitions 3
1.2.1 Probability Distributions and Parameters .... 3
1.2.2 Independence 6
1.3 Six Important Discrete Probability Distributions 8
1.3.1 One Bernoulli Trial 8
1.3.2 The Binomial Distribution 8
1.3.3 The Hypergeometric Distribution 10
1.3.4 The Uniform Distribution 13
1.3.5 The Geometric Distribution 14
1.3.6 The Negative Binomial and the Generalized
Geometric Distributions 16
1.3.7 The Poisson Distribution 17
1.3.8 Approximations 18
1.4 The Mean of a Discrete Random Variable 19
1.5 The Variance of a Discrete Random Variable 21
1.6 General Moments of a Probability Distribution 23
1.7 The Probability Generating Function 24
1.8 Continuous Random Variables 26
1.9 The Mean, Variance, and Median of a Continuous Random
Variable 28
xii Contents
1.9.1 Definitions 28
1.9.2 Chebyshev s Inequality 29
1.10 Five Important Continuous Distributions 30
1.10.1 The Uniform Distribution 30
1.10.2 The Normal Distribution 31
1.10.3 The Normal Approximation to a Discrete Distribu¬
tion 32
1.10.4 The Exponential Distribution 34
1.10.5 The Gamma Distribution 36
1.10.6 The Beta Distribution 37
1.11 The Moment Generating Function 38
1.12 Events 41
1.12.1 What Are Events? 41
1.12.2 Complements, Unions, and Intersections 42
1.12.3 Probabilities of Events 43
1.12.4 Conditional Probabilities 44
1.12.5 Independence of Events 46
1.13 The Memoryless Property of the Geometric and the
Exponential Distributions 48
1.14 Entropy and Related Concepts 49
1.14.1 Entropy 49
1.14.2 Relative Entropy 50
1.14.3 Scores and Support 50
1.15 Transformations 51
1.16 Empirical Methods 53
2 Probability Theory (ii): Many Random Variables 62
2.1 Introduction 62
2.2 The Independent Case 64
2.3 Generating Functions 66
2.3.1 Properties of Probability Generating Functions . 66
2.3.2 Properties of Moment Generating Functions ... 68
2.4 The Dependent Case 70
2.4.1 Covariance and Correlation 70
2.4.2 The Multinomial Distribution 71
2.4.3 The Multivariate Normal Distribution 72
2.5 Marginal Distributions 73
2.6 Conditional Distributions 75
2.7 Expected Values of Functions of Many Random Variables 80
2.8 Asymptotic Distributions 83
2.9 Indicator Random Variables 83
2.9.1 Definitions 83
2.9.2 Example: Sequencing EST Libraries 84
2.10 Derived Random Variables (i): Sums, Averages, and Min¬
ima 87
Contents xiii
2.10.1 Stuns and Averages 87
2.10.2 The Minimum of n Random Variables !)()
2.11 Derived Random Variables (ii): The Maximum of n
Random Variables 92
2.11.1 Distributional Properties: Continuous Random
Variables 92
2.11.2 Distributional Properties: Diserete Random Vari¬
ables 91
2.11.3 An Asymptotic Formula for the Distribution of
A max 97
2.12 Order Statistics 99
2.12.1 Definition 99
2.12.2 Example: The1 Uniform Distribution 101
2.12.3 The Sample Median 102
2.13 Transformations 103
3 Statistics (i): An Introduction to Statistical Inference 111
3.1 Introduction Ill
3.2 Classical and Bavesian Methods 112
3.3 Classical Estimation Methods 113
3.3.1 Unbiased Estimation Ill
3.3.2 Confidence Intervals Ill
3.3.3 Biased Estimators ll(i
3.4 Classical Hypothesis Testing 118
3.4.1 General Principles 11 S
3.4.2 P Values 122
3.4.3 Power Calculations 123
3.5 Hypothesis Testing: Examples 124
3.5.1 Example 1. Testing for a Mean: the One Sample
Case 124
3.5.2 Example1 2. The Two Sample / Test 125
3.5.3 Example 3. Tests on Variances 129
3.5.4 Example 4. Testing for the Parameters in a Multi¬
nomial Distribution 129
3.5.5 Example 5. Association tests 130
3.0 Likelihood Ratios. Information, and Support 133
3.7 Hypothesis Testing Using a Maximum as Test Statistic . 134
3.7.1 The Normal Distribution 134
3.7.2 P values for the Maximum of Geometric Random
Variables 130
3.8 Xonparametric Alternatives to the One Sample and Two
Sample / Tests 138
3.8.1 The Two Sample Permutation Test 139
3.8.2 The Mann Whitney Test 142
3.8.3 The Wilcoxon Signed Rank Test 143
xiv Contents
3.8.4 What is Assumed in a Non Parametric Test? . . . 145
3.9 The Bayesian Approach to Hypothesis Testing 146
3.10 The Bayesian Approach to Estimation 148
3.11 Multiple Testing 149
3.12 Combining the Results of Several Experiments 151
4 Stochastic Processes (i): Poisson Processes and Markov
Chains 155
4.1 The Homogeneous Poisson Process and the Poisson
Distribution 155
4.2 The Poisson and the Binomial Distributions 158
4.3 The Poisson and the Gamma Distributions 159
4.4 The Pure Birth Process 159
4.5 Introduction to Finite Markov Chains 161
4.6 Transition Probabilities and the Transition Probability
Matrix 162
4.7 Markov Chains with Absorbing States 164
4.8 Markov Chains with No Absorbing States 164
4.8.1 Stationary Distributions 165
4.8.2 Example 166
4.9 The Graphical Representation of a Markov Chain .... 167
4.10 Modeling 168
5 The Analysis of One DNA Sequence 174
5.1 Shotgun Sequencing 174
5.1.1 Introduction 174
5.1.2 Contigs 175
5.1.3 Anchored Contigs 179
5.2 Modeling DNA 183
5.3 Modeling Signals in DNA 185
5.3.1 Introduction 185
5.3.2 Weight Matrices: Independence 185
5.3.3 Markov Dependencies 186
5.3.4 Maximal Dependence Decomposition 187
5.4 Long Repeats 188
5.5 r Scans 192
5.6 The Analysis of Patterns 196
5.6.1 Introduction 196
5.6.2 Counting conventions 197
5.6.3 Notation and assumptions 198
5.7 Overlaps Counted 198
5.7.1 Number of Occurrences 198
5.7.2 Approximations to the Distribution of Yi(N) . . 201
5.7.3 Distance Between Occurrences 203
5.7.4 Beginning at the Origin 205
Contents xv
5.8 Overlaps Not Counted 207
5.8.1 General Comments 207
5.8.2 Distance Between Recurrences 208
5.8.3 Number of Recurrences 209
5.9 Motifs 211
6 The Analysis of Multiple DNA or Protein Sequences 220
6.1 Two Sequences: Frequency Comparisons 220
6.2 Alignments 222
6.3 Simple Tests for Significant Similarity in an Alignment . 223
6.4 Alignment Algorithms for Two Sequences 228
6.4.1 Introduction 228
6.4.2 Gapped Global Comparisons and Dynamic
Programming Algorithms 230
6.4.3 Fitting One Sequence into Another Using a Linear
Gap Model 233
6.4.4 Local Alignments with a Linear Gap Model . . . 234
6.4.5 Other Gap Models 235
6.4.6 Limitations of the Dynamic Programming Align¬
ment Algorithms 237
6.5 Protein Sequences and Substitution Matrices 238
6.5.1 Introduction 238
6.5.2 BLOSUM Substitution Matrices 239
6.5.3 PAM Substitution Matrices 244
6.5.4 A Simple Symmetric Evolutionary Matrix .... 248
6.6 Multiple Sequences 250
7 Stochastic Processes (ii): Random Walks 257
7.1 Introduction 257
7.2 The Simple Random Walk 259
7.2.1 Introduction 259
7.3 The Difference Equation Approach 260
7.3.1 Absorption Probabilities 260
7.3.2 Mean Number of Steps Taken Until the Walk Stops 261
7.4 The Moment Generating Function Approach 262
7.4.1 Introduction and Wald s identity 262
7.4.2 Absorption Probabilities 263
7.4.3 Mean Number of Steps Until the Walk Stops ... 264
7.4.4 An Asymptotic Case 265
7.5 General Walks 266
7.6 General Walks: Asymptotic Theory 268
7.6.1 Introduction 268
7.6.2 The Renewal Theorem 268
7.6.3 Unrestricted Walks 269
7.6.4 Restricted Walks 271
xvi Contents
8 Statistics (ii): Classical Estimation Theory 275
8.1 Introduction 275
8.2 Criteria for Good Estimators 275
8.3 Maximum Likelihood Estimation 277
8.3.1 The Discrete Case 277
8.3.2 The Continuous Case 279
8.3.3 Invariance Property 281
8.3.4 Asymptotic Properties 281
8.3.5 Many Parameters 285
8.4 Other Methods of Estimation 285
8.4.1 Introduction 285
8.4.2 The Method of Moments 285
8.4.3 Least Squares and Multiple Regression 287
8.5 Multivariate Methods 291
8.5.1 Introduction 291
8.5.2 Parameter Estimation 291
8.5.3 Principal Components 292
8.6 Bootstrap Methods: Estimation and Confidence Intervals 295
8.6.1 Introduction 295
8.6.2 The Plug in Concept 295
8.6.3 Classical Estimation Methods 296
8.6.4 Bootstrap Estimation Methods 297
8.6.5 Bootstrap Confidence Intervals 299
8.6.6 Examples 299
9 Statistics (ill): Classical Hypothesis Testing Theory 304
9.1 Introduction 304
9.2 Simple Fixed Sample Size Tests 304
9.2.1 The Likelihood Ratio 304
9.2.2 The Neyman Pearson Lemma 305
9.2.3 Example. Sequence matching 307
9.3 Classical Hypothesis Testing: Composite Fixed Sample
Size Tests 308
9.3.1 Introduction 308
9.3.2 Parameter Spaces and the Likelihood Ratio A . . 308
9.3.3 Example: t tests 309
9.4 The —2 log A Approximation 313
9.4.1 Theory 313
9.4.2 Examples 315
9.5 The Analysis of Variance (ANOVA) 318
9.5.1 Introduction 318
9.5.2 From t to F: Sums of Squares and the F Statistic 319
9.5.3 One way Fixed Effects ANOVA 320
9.5.4 The Two Way Fixed Effects ANOVA 323
9.5.5 Multi Way Fixed Effects ANOVAs 326
Contents xvii
9.5.6 The 2m Design, Confounding, and Fractional Repli¬
cation 326
9.5.7 Random and Mixed Effects Models 328
9.6 Multivariate Methods 329
9.6.1 Introduction 329
9.6.2 One sample T2 Tests 330
9.6.3 Two sample T2 Tests 331
9.6.4 Optimal Linear Functions: Discriminant Functions 332
9.7 ANOVA: the Repeated Measures Case 332
9.8 Bootstrap Methods: the Two sample t test 334
9.9 Sequential Analysis 336
9.9.1 The Sequential Probability Ratio Test 336
9.9.2 The Power Function for a Sequential Test .... 338
9.9.3 The Mean Sample Size 340
9.9.4 The Effect of Boundary Overshoot 342
10 BLAST 345
10.1 Introduction 345
10.2 The Comparison of Two Aligned Sequences 346
10.2.1 Introduction 346
10.2.2 The BLAST Random Walk 346
10.2.3 Parameter Calculations 347
10.2.4 The Choice of a Score 349
10.2.5 Bounds and Approximations for the BLAST P
Value 351
10.2.6 The Normalized and the Bit Scores 354
10.2.7 The Number of High Scoring Excursions 355
10.2.8 The Karlin Altschul Sum Statistic 356
10.3 The Comparison of Two Unaligned Sequences 358
10.3.1 Introduction 358
10.3.2 Theoretical and Empirical Background 359
10.3.3 Edge Effects 361
10.3.4 Multiple Testing 363
10.4 The Comparison of a Query Sequence Against a Database 364
10.5 Printouts 365
10.5.1 Example 366
10.5.2 A More Complicated Example 369
10.6 Minimum Significance Lengths 372
10.6.1 A Correct Choice of n 372
10.6.2 An Incorrect Choice of n 374
10.7 BLAST: A Parametric or a Non parametric Test? .... 375
10.8 Gapped BLAST and PSI BLAST 376
10.8.1 Gapped BLAST 376
10.8.2 PSI BLAST 378
10.9 Relation to Sequential Analysis 380
xviii Contents
11 Stochastic Processes (iii): Markov Chains 384
11.1 Introduction 384
11.2 Markov Chains with No Absorbing States IW )
11.2.1 Introduction 385
11.2.2 Convergence to the Stationary Distribution . . . . 585
11.2.. 5 Stationary Distributions: A Numerical Kxaniple . 386
11.2.4 Reversibility mid Detailed Balance 380
11.3 Higher Order Markov Dependence 389
11.3.1 Testing for Higher Order Markov Dependence . . 389
11.3.2 Testing for a Uniform Stationary Distribution . . 389
11.4 Patterns in Sequences with First Order Markov Depen¬
dence 390
11.5 Markov Chain Monte Carlo 392
11.5.1 The Hastings Metropolis Algorithm 392
11.5.2 Gibbs Sampling 393
11.5.3 Simulated Annealing 395
11.6 Markov Chains with Absorbing States 398
11.6.1 Theory 398
11.6.2 Examples 399
11.7 Continuous Time Markov Chains 403
11.7.1 Definitions 103
11.7.2 Time Dependent Solutions 404
11.7.3 The Stationary Distribution 404
11.7.4 Detailed Balance 405
11.7.5 Exponential Holding Times 405
11.7.6 The Embedded Chain 406
12 Hidden Markov Models 409
12.1 What is a Hidden Markov Model? 409
12.2 Three Algorithms 411
12.2.1 The Forward and Backward Algorithms 411
12.2.2 The Viterbi Algorithm 413
12.2.3 The Estimation Algorithms 414
12.3 Applications 417
12.3.1 Modeling Protein Families 417
12.3.2 Multiple Sequence Alignments 419
12.3.3 Pfam 421
12.3.4 Gene Finding 422
13 Gene Expression, Microarrays, and Multiple Testing 430
13.1 Introduction 430
13.1.1 Introduction to Microarrays 430
13.1.2 Microarray Data 432
13.1.3 Sources of Bias and Variation 438
13.2 The Statistical Analysis of Microarray Data: One Gene . 443
Contents xix
13.2.1 Introduction 443
13.2.2 Determining Whether a Gene is Expressed .... 443
13.2.3 Testing for Differential Expression 444
13.3 Differential Expression Multiple Genes 449
13.3.1 Introduction 449
13.3.2 Ranked lists 449
13.3.3 The Choice of Statistic 450
13.3.4 Confidence Measures 451
13.3.5 The Family Wise Error Rate (FWER) 454
13.3.6 The False Discovery Rate (FDR) 459
13.3.7 The ANOVA Approach: Many Genes 467
13.3.8 Comparing Two Groups by Discriminant Analysis 470
13.4 Principal Components and Microarrays 471
13.5 Clustering Methods 471
13.5.1 Hierarchical Clustering 471
13.5.2 Other Forms of Clustering 472
14 Evolutionary Models 475
14.1 Models of Nucleotide Substitution 475
14.2 Discrete Time Models 476
14.2.1 The Jukes Cantor Model 476
14.2.2 The Kimura Models 478
14.2.3 Further Generalizations of the Kimura Models . . 479
14.2.4 The Felsenstein Models 480
14.2.5 The HKY Model 482
14.2.6 Other Models 483
14.2.7 The Reversibility Criterion 483
14.2.8 The Simple Symmetric Ainino Acid Model .... 484
14.3 Continuous Time Models 484
14.3.1 The Continuous Time Jukes Cantor Model . . . 485
14.3.2 The Continuous Time Kimura Model 489
14.3.3 The Continuous Time Felsenstein Model 492
14.3.4 The Continuous Time HKY Model 493
14.3.5 Continuous Time Amino Acid Model 493
14.3.6 A Remark About Bias 494
15 Phylogenetic Tree Estimation 497
15.1 Introduction 497
15.2 Distances 499
15.3 Tree Reconstruction: The Ultrametric Case 501
15.4 Tree Reconstruction: the Neighbor Joining Approach . . 505
15.5 Inferred Distances 509
15.6 Tree Reconstruction: Parsimony 511
15.7 Tree Estimation: Maximum Likelihood 512
15.8 Example 517
xx Contents
15.9 Modeling, Estimation, and Hypothesis Testing 522
15.9.1 Estimation and Hypothesis Testing 522
15.9.2 Bootstrapping and Phylogenies 523
15.9.3 Assumptions and Problems 528
15.9.4 Phylogenetic Models and Hypothesis Testing . . . 531
Appendix A Basic Notions in Biology 537
Appendix B Mathematical Formulae and Results 540
B.I Numbers and Intervals 540
B.2 Sets and Set Notation 541
B.3 Factorials 541
B.4 Binomial Coefficients 542
B.5 The Binomial Theorem 542
B.6 Permutations and Combinations 542
B.7 Limits 543
B.8 Asymptotics 544
B.9 Stirling s Approximation 546
B.10 Entropy as Information 546
B.ll Infinite Series 547
B.12 Taylor Series 549
B.13 Uniqueness of Taylor Series 553
B.14 Laurent Series 553
B.15 Numerical Solutions of Equations 554
B.16 Statistical Differentials 554
B.17 The Gamma Function 555
B.18 Proof by Induction 556
B.19 Linear Algebra and Matrices 557
Appendix C Computational Aspects of the Binomial and
Generalized Geometric Distribution Functions 558
Appendix D BLAST: Sums of Normalized Scores 560
References 561
Author Index 581
Subject Index 586
|
| adam_txt |
Contents
Preface vii
1 Probability Theory (i): One Random Variable 1
1.1 Introduction 1
1.2 Discrete Random Variables, Definitions 3
1.2.1 Probability Distributions and Parameters . 3
1.2.2 Independence 6
1.3 Six Important Discrete Probability Distributions 8
1.3.1 One Bernoulli Trial 8
1.3.2 The Binomial Distribution 8
1.3.3 The Hypergeometric Distribution 10
1.3.4 The Uniform Distribution 13
1.3.5 The Geometric Distribution 14
1.3.6 The Negative Binomial and the Generalized
Geometric Distributions 16
1.3.7 The Poisson Distribution 17
1.3.8 Approximations 18
1.4 The Mean of a Discrete Random Variable 19
1.5 The Variance of a Discrete Random Variable 21
1.6 General Moments of a Probability Distribution 23
1.7 The Probability Generating Function 24
1.8 Continuous Random Variables 26
1.9 The Mean, Variance, and Median of a Continuous Random
Variable 28
xii Contents
1.9.1 Definitions 28
1.9.2 Chebyshev's Inequality 29
1.10 Five Important Continuous Distributions 30
1.10.1 The Uniform Distribution 30
1.10.2 The Normal Distribution 31
1.10.3 The Normal Approximation to a Discrete Distribu¬
tion 32
1.10.4 The Exponential Distribution 34
1.10.5 The Gamma Distribution 36
1.10.6 The Beta Distribution 37
1.11 The Moment Generating Function 38
1.12 Events 41
1.12.1 What Are Events? 41
1.12.2 Complements, Unions, and Intersections 42
1.12.3 Probabilities of Events 43
1.12.4 Conditional Probabilities 44
1.12.5 Independence of Events 46
1.13 The Memoryless Property of the Geometric and the
Exponential Distributions 48
1.14 Entropy and Related Concepts 49
1.14.1 Entropy 49
1.14.2 Relative Entropy 50
1.14.3 Scores and Support 50
1.15 Transformations 51
1.16 Empirical Methods 53
2 Probability Theory (ii): Many Random Variables 62
2.1 Introduction 62
2.2 The Independent Case 64
2.3 Generating Functions 66
2.3.1 Properties of Probability Generating Functions . 66
2.3.2 Properties of Moment Generating Functions . 68
2.4 The Dependent Case 70
2.4.1 Covariance and Correlation 70
2.4.2 The Multinomial Distribution 71
2.4.3 The Multivariate Normal Distribution 72
2.5 Marginal Distributions 73
2.6 Conditional Distributions 75
2.7 Expected Values of Functions of Many Random Variables 80
2.8 Asymptotic Distributions 83
2.9 Indicator Random Variables 83
2.9.1 Definitions 83
2.9.2 Example: Sequencing EST Libraries 84
2.10 Derived Random Variables (i): Sums, Averages, and Min¬
ima 87
Contents xiii
2.10.1 Stuns and Averages 87
2.10.2 The Minimum of n Random Variables !)()
2.11 Derived Random Variables (ii): The Maximum of n
Random Variables 92
2.11.1 Distributional Properties: Continuous Random
Variables 92
2.11.2 Distributional Properties: Diserete Random Vari¬
ables 91
2.11.3 An Asymptotic Formula for the Distribution of
A'max 97
2.12 Order Statistics 99
2.12.1 Definition 99
2.12.2 Example: The1 Uniform Distribution 101
2.12.3 The Sample Median 102
2.13 Transformations 103
3 Statistics (i): An Introduction to Statistical Inference 111
3.1 Introduction Ill
3.2 Classical and Bavesian Methods 112
3.3 Classical Estimation Methods 113
3.3.1 Unbiased Estimation Ill
3.3.2 Confidence Intervals Ill
3.3.3 Biased Estimators ll(i
3.4 Classical Hypothesis Testing 118
3.4.1 General Principles 11 S
3.4.2 P Values 122
3.4.3 Power Calculations 123
3.5 Hypothesis Testing: Examples 124
3.5.1 Example 1. Testing for a Mean: the One Sample
Case 124
3.5.2 Example1 2. The Two Sample / Test 125
3.5.3 Example 3. Tests on Variances 129
3.5.4 Example 4. Testing for the Parameters in a Multi¬
nomial Distribution 129
3.5.5 Example 5. Association tests 130
3.0 Likelihood Ratios. Information, and Support 133
3.7 Hypothesis Testing Using a Maximum as Test Statistic . 134
3.7.1 The Normal Distribution 134
3.7.2 P values for the Maximum of Geometric Random
Variables 130
3.8 Xonparametric Alternatives to the One Sample and Two
Sample / Tests 138
3.8.1 The Two Sample Permutation Test 139
3.8.2 The Mann Whitney Test 142
3.8.3 The Wilcoxon Signed Rank Test 143
xiv Contents
3.8.4 What is Assumed in a Non Parametric Test? . . . 145
3.9 The Bayesian Approach to Hypothesis Testing 146
3.10 The Bayesian Approach to Estimation 148
3.11 Multiple Testing 149
3.12 Combining the Results of Several Experiments 151
4 Stochastic Processes (i): Poisson Processes and Markov
Chains 155
4.1 The Homogeneous Poisson Process and the Poisson
Distribution 155
4.2 The Poisson and the Binomial Distributions 158
4.3 The Poisson and the Gamma Distributions 159
4.4 The Pure Birth Process 159
4.5 Introduction to Finite Markov Chains 161
4.6 Transition Probabilities and the Transition Probability
Matrix 162
4.7 Markov Chains with Absorbing States 164
4.8 Markov Chains with No Absorbing States 164
4.8.1 Stationary Distributions 165
4.8.2 Example 166
4.9 The Graphical Representation of a Markov Chain . 167
4.10 Modeling 168
5 The Analysis of One DNA Sequence 174
5.1 Shotgun Sequencing 174
5.1.1 Introduction 174
5.1.2 Contigs 175
5.1.3 Anchored Contigs 179
5.2 Modeling DNA 183
5.3 Modeling Signals in DNA 185
5.3.1 Introduction 185
5.3.2 Weight Matrices: Independence 185
5.3.3 Markov Dependencies 186
5.3.4 Maximal Dependence Decomposition 187
5.4 Long Repeats 188
5.5 r Scans 192
5.6 The Analysis of Patterns 196
5.6.1 Introduction 196
5.6.2 Counting conventions 197
5.6.3 Notation and assumptions 198
5.7 Overlaps Counted 198
5.7.1 Number of Occurrences 198
5.7.2 Approximations to the Distribution of Yi(N) . . 201
5.7.3 Distance Between Occurrences 203
5.7.4 Beginning at the Origin 205
Contents xv
5.8 Overlaps Not Counted 207
5.8.1 General Comments 207
5.8.2 Distance Between Recurrences 208
5.8.3 Number of Recurrences 209
5.9 Motifs 211
6 The Analysis of Multiple DNA or Protein Sequences 220
6.1 Two Sequences: Frequency Comparisons 220
6.2 Alignments 222
6.3 Simple Tests for Significant Similarity in an Alignment . 223
6.4 Alignment Algorithms for Two Sequences 228
6.4.1 Introduction 228
6.4.2 Gapped Global Comparisons and Dynamic
Programming Algorithms 230
6.4.3 Fitting One Sequence into Another Using a Linear
Gap Model 233
6.4.4 Local Alignments with a Linear Gap Model . . . 234
6.4.5 Other Gap Models 235
6.4.6 Limitations of the Dynamic Programming Align¬
ment Algorithms 237
6.5 Protein Sequences and Substitution Matrices 238
6.5.1 Introduction 238
6.5.2 BLOSUM Substitution Matrices 239
6.5.3 PAM Substitution Matrices 244
6.5.4 A Simple Symmetric Evolutionary Matrix . 248
6.6 Multiple Sequences 250
7 Stochastic Processes (ii): Random Walks 257
7.1 Introduction 257
7.2 The Simple Random Walk 259
7.2.1 Introduction 259
7.3 The Difference Equation Approach 260
7.3.1 Absorption Probabilities 260
7.3.2 Mean Number of Steps Taken Until the Walk Stops 261
7.4 The Moment Generating Function Approach 262
7.4.1 Introduction and Wald's identity 262
7.4.2 Absorption Probabilities 263
7.4.3 Mean Number of Steps Until the Walk Stops . 264
7.4.4 An Asymptotic Case 265
7.5 General Walks 266
7.6 General Walks: Asymptotic Theory 268
7.6.1 Introduction 268
7.6.2 The Renewal Theorem 268
7.6.3 Unrestricted Walks 269
7.6.4 Restricted Walks 271
xvi Contents
8 Statistics (ii): Classical Estimation Theory 275
8.1 Introduction 275
8.2 Criteria for "Good" Estimators 275
8.3 Maximum Likelihood Estimation 277
8.3.1 The Discrete Case 277
8.3.2 The Continuous Case 279
8.3.3 Invariance Property 281
8.3.4 Asymptotic Properties 281
8.3.5 Many Parameters 285
8.4 Other Methods of Estimation 285
8.4.1 Introduction 285
8.4.2 The Method of Moments 285
8.4.3 Least Squares and Multiple Regression 287
8.5 Multivariate Methods 291
8.5.1 Introduction 291
8.5.2 Parameter Estimation 291
8.5.3 Principal Components 292
8.6 Bootstrap Methods: Estimation and Confidence Intervals 295
8.6.1 Introduction 295
8.6.2 The "Plug in" Concept 295
8.6.3 Classical Estimation Methods 296
8.6.4 Bootstrap Estimation Methods 297
8.6.5 Bootstrap Confidence Intervals 299
8.6.6 Examples 299
9 Statistics (ill): Classical Hypothesis Testing Theory 304
9.1 Introduction 304
9.2 Simple Fixed Sample Size Tests 304
9.2.1 The Likelihood Ratio 304
9.2.2 The Neyman Pearson Lemma 305
9.2.3 Example. Sequence matching 307
9.3 Classical Hypothesis Testing: Composite Fixed Sample
Size Tests 308
9.3.1 Introduction 308
9.3.2 Parameter Spaces and the Likelihood Ratio A . . 308
9.3.3 Example: t tests 309
9.4 The —2 log A Approximation 313
9.4.1 Theory 313
9.4.2 Examples 315
9.5 The Analysis of Variance (ANOVA) 318
9.5.1 Introduction 318
9.5.2 From t to F: Sums of Squares and the F Statistic 319
9.5.3 One way Fixed Effects ANOVA 320
9.5.4 The Two Way Fixed Effects ANOVA 323
9.5.5 Multi Way Fixed Effects ANOVAs 326
Contents xvii
9.5.6 The 2m Design, Confounding, and Fractional Repli¬
cation 326
9.5.7 Random and Mixed Effects Models 328
9.6 Multivariate Methods 329
9.6.1 Introduction 329
9.6.2 One sample T2 Tests 330
9.6.3 Two sample T2 Tests 331
9.6.4 Optimal Linear Functions: Discriminant Functions 332
9.7 ANOVA: the Repeated Measures Case 332
9.8 Bootstrap Methods: the Two sample t test 334
9.9 Sequential Analysis 336
9.9.1 The Sequential Probability Ratio Test 336
9.9.2 The Power Function for a Sequential Test . 338
9.9.3 The Mean Sample Size 340
9.9.4 The Effect of Boundary Overshoot 342
10 BLAST 345
10.1 Introduction 345
10.2 The Comparison of Two Aligned Sequences 346
10.2.1 Introduction 346
10.2.2 The BLAST Random Walk 346
10.2.3 Parameter Calculations 347
10.2.4 The Choice of a Score 349
10.2.5 Bounds and Approximations for the BLAST P
Value 351
10.2.6 The Normalized and the Bit Scores 354
10.2.7 The Number of High Scoring Excursions 355
10.2.8 The Karlin Altschul Sum Statistic 356
10.3 The Comparison of Two Unaligned Sequences 358
10.3.1 Introduction 358
10.3.2 Theoretical and Empirical Background 359
10.3.3 Edge Effects 361
10.3.4 Multiple Testing 363
10.4 The Comparison of a Query Sequence Against a Database 364
10.5 Printouts 365
10.5.1 Example 366
10.5.2 A More Complicated Example 369
10.6 Minimum Significance Lengths 372
10.6.1 A Correct Choice of n 372
10.6.2 An Incorrect Choice of n 374
10.7 BLAST: A Parametric or a Non parametric Test? . 375
10.8 Gapped BLAST and PSI BLAST 376
10.8.1 Gapped BLAST 376
10.8.2 PSI BLAST 378
10.9 Relation to Sequential Analysis 380
xviii Contents
11 Stochastic Processes (iii): Markov Chains 384
11.1 Introduction 384
11.2 Markov Chains with No Absorbing States IW")
11.2.1 Introduction 385
11.2.2 Convergence to the Stationary Distribution . . . .'585
11.2.'5 Stationary Distributions: A Numerical Kxaniple . 386
11.2.4 Reversibility mid Detailed Balance 380
11.3 Higher Order Markov Dependence 389
11.3.1 Testing for Higher Order Markov Dependence . . 389
11.3.2 Testing for a Uniform Stationary Distribution . . 389
11.4 Patterns in Sequences with First Order Markov Depen¬
dence 390
11.5 Markov Chain Monte Carlo 392
11.5.1 The Hastings Metropolis Algorithm 392
11.5.2 Gibbs Sampling 393
11.5.3 Simulated Annealing 395
11.6 Markov Chains with Absorbing States 398
11.6.1 Theory 398
11.6.2 Examples 399
11.7 Continuous Time Markov Chains 403
11.7.1 Definitions 103
11.7.2 Time Dependent Solutions 404
11.7.3 The Stationary Distribution 404
11.7.4 Detailed Balance 405
11.7.5 Exponential Holding Times 405
11.7.6 The Embedded Chain 406
12 Hidden Markov Models 409
12.1 What is a Hidden Markov Model? 409
12.2 Three Algorithms 411
12.2.1 The Forward and Backward Algorithms 411
12.2.2 The Viterbi Algorithm 413
12.2.3 The Estimation Algorithms 414
12.3 Applications 417
12.3.1 Modeling Protein Families 417
12.3.2 Multiple Sequence Alignments 419
12.3.3 Pfam 421
12.3.4 Gene Finding 422
13 Gene Expression, Microarrays, and Multiple Testing 430
13.1 Introduction 430
13.1.1 Introduction to Microarrays 430
13.1.2 Microarray Data 432
13.1.3 Sources of Bias and Variation 438
13.2 The Statistical Analysis of Microarray Data: One Gene . 443
Contents xix
13.2.1 Introduction 443
13.2.2 Determining Whether a Gene is Expressed . 443
13.2.3 Testing for Differential Expression 444
13.3 Differential Expression Multiple Genes 449
13.3.1 Introduction 449
13.3.2 Ranked lists 449
13.3.3 The Choice of Statistic 450
13.3.4 Confidence Measures 451
13.3.5 The Family Wise Error Rate (FWER) 454
13.3.6 The False Discovery Rate (FDR) 459
13.3.7 The ANOVA Approach: Many Genes 467
13.3.8 Comparing Two Groups by Discriminant Analysis 470
13.4 Principal Components and Microarrays 471
13.5 Clustering Methods 471
13.5.1 Hierarchical Clustering 471
13.5.2 Other Forms of Clustering 472
14 Evolutionary Models 475
14.1 Models of Nucleotide Substitution 475
14.2 Discrete Time Models 476
14.2.1 The Jukes Cantor Model 476
14.2.2 The Kimura Models 478
14.2.3 Further Generalizations of the Kimura Models . . 479
14.2.4 The Felsenstein Models 480
14.2.5 The HKY Model 482
14.2.6 Other Models 483
14.2.7 The Reversibility Criterion 483
14.2.8 The Simple Symmetric Ainino Acid Model . 484
14.3 Continuous Time Models 484
14.3.1 The Continuous Time Jukes Cantor Model . . . 485
14.3.2 The Continuous Time Kimura Model 489
14.3.3 The Continuous Time Felsenstein Model 492
14.3.4 The Continuous Time HKY Model 493
14.3.5 Continuous Time Amino Acid Model 493
14.3.6 A Remark About Bias 494
15 Phylogenetic Tree Estimation 497
15.1 Introduction 497
15.2 Distances 499
15.3 Tree Reconstruction: The Ultrametric Case 501
15.4 Tree Reconstruction: the Neighbor Joining Approach . . 505
15.5 Inferred Distances 509
15.6 Tree Reconstruction: Parsimony 511
15.7 Tree Estimation: Maximum Likelihood 512
15.8 Example 517
xx Contents
15.9 Modeling, Estimation, and Hypothesis Testing 522
15.9.1 Estimation and Hypothesis Testing 522
15.9.2 Bootstrapping and Phylogenies 523
15.9.3 Assumptions and Problems 528
15.9.4 Phylogenetic Models and Hypothesis Testing . . . 531
Appendix A Basic Notions in Biology 537
Appendix B Mathematical Formulae and Results 540
B.I Numbers and Intervals 540
B.2 Sets and Set Notation 541
B.3 Factorials 541
B.4 Binomial Coefficients 542
B.5 The Binomial Theorem 542
B.6 Permutations and Combinations 542
B.7 Limits 543
B.8 Asymptotics 544
B.9 Stirling's Approximation 546
B.10 Entropy as Information 546
B.ll Infinite Series 547
B.12 Taylor Series 549
B.13 Uniqueness of Taylor Series 553
B.14 Laurent Series 553
B.15 Numerical Solutions of Equations 554
B.16 Statistical Differentials 554
B.17 The Gamma Function 555
B.18 Proof by Induction 556
B.19 Linear Algebra and Matrices 557
Appendix C Computational Aspects of the Binomial and
Generalized Geometric Distribution Functions 558
Appendix D BLAST: Sums of Normalized Scores 560
References 561
Author Index 581
Subject Index 586 |
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| index_date | 2024-07-02T17:06:40Z |
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| spelling | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant 2. ed. New York, NY Springer 2005 1 Online-Ressource (XX, 597 S.) graph. Darst. txt rdacontent c rdamedia cr rdacarrier Statistics for biology and health Statistik (DE-588)4056995-0 gnd rswk-swf Bioinformatik (DE-588)4611085-9 gnd rswk-swf Bioinformatik (DE-588)4611085-9 s Statistik (DE-588)4056995-0 s DE-604 Ewens, Warren J. Sonstige oth Grant, Gregory R. Sonstige oth https://doi.org/10.1007/b137845 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015579233&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Statistical methods in bioinformatics an introduction Statistik (DE-588)4056995-0 gnd Bioinformatik (DE-588)4611085-9 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4611085-9 |
| title | Statistical methods in bioinformatics an introduction |
| title_auth | Statistical methods in bioinformatics an introduction |
| title_exact_search | Statistical methods in bioinformatics an introduction |
| title_exact_search_txtP | Statistical methods in bioinformatics an introduction |
| title_full | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_fullStr | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_full_unstemmed | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_short | Statistical methods in bioinformatics |
| title_sort | statistical methods in bioinformatics an introduction |
| title_sub | an introduction |
| topic | Statistik (DE-588)4056995-0 gnd Bioinformatik (DE-588)4611085-9 gnd |
| topic_facet | Statistik Bioinformatik |
| url | https://doi.org/10.1007/b137845 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015579233&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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