Scan statistics:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London
Springer
2001
|
| Schriftenreihe: | Springer series in statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 333 - 366 |
| Beschreibung: | XV, 370 S. Ill. 24 cm |
| ISBN: | 038798819X |
Internformat
MARC
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| 100 | 1 | |a Glaz, Joseph |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Scan statistics |c Joseph Glaz ; Joseph Naus ; Sylvan Wallenstein |
| 264 | 1 | |a New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London |b Springer |c 2001 | |
| 300 | |a XV, 370 S. |b Ill. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in statistics | |
| 500 | |a Literaturverz. S. 333 - 366 | ||
| 650 | 4 | |a Statistiques ordonnées | |
| 650 | 4 | |a Order statistics | |
| 650 | 0 | 7 | |a Statistik |0 (DE-588)4056995-0 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Naus, Joseph |e Verfasser |4 aut | |
| 700 | 1 | |a Wallenstein, Sylvan |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804129159683768320 |
|---|---|
| adam_text | Contents
Preface vii
I Methods and Applications 1
1 Introduction 3
1.1 The Discrete Scan Statistic 5
1.2 Scan Statistics in Two Dimensions 6
1.3 Power of the Scan Statistic 7
1.4 Clusters and Intuition 8
2 Retrospective Scanning of Events Over Time 11
2.1 Conditional Case: Uniform Distribution of Events 11
2.1.1 Approximate Results for P(k: N, w) 12
2.2 The Scan Statistic on the Circle 18
2.3 The Ratchet Scan Statistic 20
2.4 Moments of Scan Statistics 21
2.4.1 Exact Values for Moments of the Scan Statistic Sw .... 21
2.4.2 The Expectation and Variance of the Wk, the Size of the
Smallest Interval 22
3 Prospective Scanning of Events Over Time 25
3.1 Poisson Distribution of Events 25
3.1.1 The Poisson Process 27
x Contents
3.2 Approximate Formula for P*(k;XT,w/T) 28
3.3 Handling Trends or Seasonality in Data 35
3.3.1 The Disjoint Week Procedure 36
3.3.2 Continuous 7 Day Scan 37
3.3.3 Comparison of Disjoint Week with 7 Day Scan for
a 52 Week Year 37
3.4 Moments of Scan Statistics 37
3.4.1 The Expectation and Variance of Wk, the Size of
the Smallest Interval 38
3.4.2 The Expected Waiting Time until a Cluster 38
3.5 The Distribution of the Number of Clusters in [0, T) 39
3.5.1 The Distribution of the Number of Nonoverlapping Clusters 40
3.5.2 The Distribution of the Number of Overlapping Clusters . 40
3.6 The Scan Statistic on the Circle 41
4 Success Scans in a Sequence of Trials 43
4.1 Binomial Distribution of Events: Discrete Time, Unconditional Case 43
4.2 A Null Model for the Unconditional Case: The Bernoulli Process . 44
4.2.1 Approximations for P (k m; N; p) with Applications . . 45
4.2.2 The Length of the Longest Quota 54
4.3 The Charge Problem 54
4.4 Binomial Distributed Events: Discrete Time, Conditional Case . . 56
4.5 Related Statistics 58
4.6 Longest Run of any Letter in a Sequence of r Letters 58
4.7 Moments of Scan Statistics 59
4.7.1 The Expectation and Variance of S m, the Size of
the Largest Cluster 59
4.7.2 The Expectation and Variance of W k, the Size of
the Smallest Interval 59
4.7.3 The Expected Waiting Time until a Cluster 60
5 Higher Dimensional Scans 61
5.1 Introduction 61
5.2 The Conditional Problem 63
5.2.1 Effect of the Shape of the Scanning Rectangle 65
5.3 The Unconditional Problem 75
5.4 Clustering on the Lattice 76
6 Scan Statistics in DNA and Protein Sequence Analysis 81
6.1 Introduction 81
6.2 Scanning for Clusters of Patterns 82
6.3 Matching in DNA Sequences 85
6.4 Matching in Multiple Random Letter Sequences 91
6.5 Sequencing Fragments to Reconstruct a Genome 94
6.6 Using Double Scans for More Effective Searches for Homologies 95
6.7 Correlated Descendant Sequences Scan Statistics 95
Contents xi
II Scan Distribution Theory and its Developments 97
7 Approaches Used for Derivations and Approximations 99
7.1 Introduction 99
7.2 Order Statistics and a Direct Integration Approach 99
7.3 Combinatorial Approach 101
7.3.1 The Karlin McGregor Theorem 103
7.4 Bonferroni Type Inequalities 104
7.5 The Q2IQ3 Approximation for Scan Statistics 107
7.6 Poisson and Compound Poisson Approximations 108
8 Scanning N Uniform Distributed Points: Exact Results 113
8.1 Introduction 113
8.2 The Direct Integration Approach 114
8.3 The Combinatorial Approach 115
8.4 The Derivation of P(k; N, w) for k (N + l)/2, w 1/2 . . . . 117
8.4.1 Finding P(B ) 117
8.4.2 Finding P{B n B2) for it N/2 119
8.5 A General Formula for P(it; N, /L) for Integer L 119
8.6 Simplifying Theorem 8.1 for the Special Case P(/t; A?, 1/3) ... 122
8.7 General Formula for P{k N,w) 123
8.8 Simplifying the General Formula 126
8.8.1 Finding P(A) 127
8.8.2 Finding P(B) 127
8.9 Generating the Piecewise Polynomials from the General Formulas 131
8.10 The Approach of Huffer and Lin 133
8.11 The Expectation and Variance of W/ , the Size of
the Smallest Interval 136
8.12 The Scan Statistic on the Circle 137
8.12.1 Derivation of Exact Results 138
8.12.2 Relation of Scan Statistic on Circle and
Coverage Problems 140
9 Scanning N Uniform Distributed Points: Bounds 141
9.1 Bounds Based on the Scanning Process Representation 141
9.2 Bounds Based on Spacings 147
9.3 Method of Moments Bounds Using Linear Programming 154
9.4 Bounds for E(SU.) 156
9.5 Bounds for the Distribution of a Scan Statistic on a Circle 158
10 Approximations for the Conditional Case 161
10.1 Spacings 162
10.2 Approximations Involving Probabilities 164
10.3 Declumping Techniques 165
10.4 The Method of Moments Based on Two Moments 166
xii Contents
10.4.1 Two Moment Markov Chain Approximation (MC2) . . 167
10.4.2 Two Moment Compound Poisson Approximations .... 168
10.4.3 Two Parameter Compound Poisson Geometric (CPG2) . . 169
10.4.4 Glaz s et al. Compound Poisson Approximation 169
10.4.5 Other Methods of Moments Approximations 170
10.5 Cell Occupancy Approximations: Introduction 170
10.5.1 Conditional Probabilities 171
10.5.2 Unconditional Probabilities 171
10.5.3 Joint Unconditional Probabilities 173
10.6 Markov Type Approximation (Naus, 1982) 174
10.7 Very Simple Approximations based on Multiples of b(k; N, w) . . 175
10.7.1 Derivation of Wallenstein Neff Approximations 176
10.7.2 Related Approximations 177
10.8 Approximations Based on Bounds 178
10.8.1 Approximation Based on Two Way Intersections 178
10.8.2 Approximation Based on Three Way Intersections .... 179
10.9 Other Methods 179
10.10 Comparisons 180
10.11 Circular Case 182
11 Scanning Points in a Poisson Process 185
11.1 Poisson Distribution of Events 185
11.1.1 The Poisson Process 185
11.2 Exact Results for P*(k; XT,w/T) 186
11.3 Bounds for the Distribution of the Scan Statistic 189
11.4 Approximations for the Distribution of the Scan Statistics 196
11.4.1 Aim s Approximation 197
11.4.2 Moments of Scan Statistics 199
12 The Generalized Birthday Problem 201
12.1 Binomial Distribution of Events: Discrete Time,
Unconditional Case 201
12.2 The Conditional Case: Exact Results 201
12.3 Bounds for the Conditional Scan Statistic 205
12.4 Approximations for the Conditional Scan Statistic 212
12.4.1 Approximations for i.i.d. 0 1 Bernoulli Trials 212
12.4.1.1 Product Type Approximations 212
12.4.1.2 Poisson Approximations 214
12.4.1.3 Compound Poisson Approximations 215
12.4.1.4 Approximations for the Expected Size and
Standard Derivation of the Scan Statistic . . . .217
12.4.2 Scan Statistics for Binomial and Poisson Distributions
Conditional on the Total Number of Events 218
12.4.2.1 Poisson Model 218
12.4.2.2 Binomial Model 218
Contents xiii
13 Scan Statistics for a Sequence of Discrete I.I.D. Variates 221
13.1 Binomial Distribution of Events. Discrete Time, Unconditional
Case. The Bernoulli Process 221
13.2 Exact Results for the Distribution of the Scan Statistics 222
13.2.1 The Distribution of the Length of the Longest Success
Run: P (k k N,p) 222
13.2.2 Exact Results for P (k m; N, p) 223
13.3 Bounds for S m for i.i.d. Integer Valued Random Variables 225
13.4 Approximations for P (k m;N,p) 231
13.4.1 The Special Case k = m: The Length of the Longest Run . 232
13.4.2 Erdos Renyi Laws 233
13.4.3 Approximating the Discrete Problem by the Continuous .235
13.5 The Charge Problem 236
13.5.1 Exact Results 236
13.5.2 Saperstein s Recursion for Gjt.m(2m) 236
13.5.3 The Case 2m N 3m 239
13.5.4 Approximate Results 239
13.5.5 Asymptotic Approximations 239
13.6 Longest Run of Any Letter in a Sequence of r Letters 240
13.7 Moments of Scan Statistics 241
13.7.1 The Expected Waiting Time till a Quota 241
14 Power 243
14.1 Introduction 243
14.1.1 Step Function Alternatives 244
14.2 Very Simple Approximations 246
14.2.1 Conditional Case 246
14.2.2 Unconditional Case 247
14.2.3 Discrete 247
14.3 Exact Results 247
14.4 Markovian Approximations for a Pulse of Width w 249
14.4.1 Approximation Applied in the General Context 249
14.4.2 Power when the Width of the Pulse and
Window are Identical 251
14.4.3 A Further Simplification 251
14.5 Intermediate Calculations: Exact Power when w/T = 1/2 or 1/3 . 252
14.5.1 One Way Probabilities 252
14.5.1.1 Continuous Case 252
14.5.2 Joint Probabilities 254
14.5.2.1 Continuous Case 254
14.6 Simple, Somewhat ad hoc Approximation for Power 256
14.6.1 A Very Simple Approximation to Power:
Unconditional Case 256
14.6.2 Conditional Continuous Case 257
14.6.3 Power for a Pulse Starting at b w 257
xiv Contents
14.6.4 Accuracy of Approximations 257
14.7 Power for Other Alternatives 258
14.8 Practical Implementation of Power: Conditional Case 258
15 Testing for Clustering 261
15.1 Introduction 261
15.2 Weinstock s Approach 262
15.3 An Optimal Test Statistic 263
15.3.1 Derivation of Approximate Significance Level 265
15.4 A Reasonable, but Nonoptimal Statistic 267
15.4.1 Proof of Approximation 267
15.5 Practical Implementation: Estimating fo(t) 269
15.5.1 Linearly Changing Density 270
15.6 Exact Distribution of a Statistic 270
16 Two Dimensional Scan Statistics 273
16.1 A Discrete Scan Statistic 273
16.1.1 Approximations and Inequalities for P(S mi mi = mxmj)
for i.i.d. Bernoulli Trials 275
16.1.2 A Product Type Approximation 277
16.1.3 A Bonferroni Type Inequality 279
16.1.4 Poisson Type Approximations 281
16.1.5 A Compound Poisson Approximation 282
16.1.6 A Markov Chain Embedding Method for the
Bernoulli Model 283
16.1.7 Approximations for the Expected Size and
Standard Deviation 286
16.1.8 A Multiple Scan Statistic 287
16.2 A Conditional Discrete Scan Statistic 287
16.2.1 Product Type Approximations 288
16.2.2 Poisson Approximations 290
16.2.3 A Compound Poisson Approximation 291
16.2.4 Bonferroni Type Inequalities 292
16.2.5 Approximations and Inequalities for the Expected Size
and Standard Deviation 293
16.2.6 Simulation Algorithms 295
16.2.6.1 Binomial Model 295
16.2.6.2 Poisson Model 296
16.3 A Scan Statistic for a Two Dimensional Poisson Process 297
16.4 A Scan Statistic for N Points 299
17 Number of Clusters: Ordered Spacings 301
17.1 Notation and Introduction 301
17.2 Local Declumping which Mimics the Global One 303
17.3 A Markovian Method of Declumping 304
Contents xv
17.4 Compound Poisson Approach 305
17.5 The Dandekar Modified Binomial 307
17.6 Another Approach Based on the Q2/Q3 Approximation 308
17.7 Summary of Results and Comparison of Procedures 309
17.7.1 Overlapping Clusters 309
17.7.2 Nonoverlapping Clusters 309
17.8 Exploratory Approach when r Cannot be Specified 309
17.8.1 Calculations with Fixed r 310
17.8.2 Calculations with Multiple Values of r Compared with
Experimental Results 311
17.8.3 Further Discussion of Example 312
18 Extensions of the Scan Statistic 313
18.1 The Double Scan Statistic 313
18.2 Clustering of Two Types of Letter Patterns 317
18.3 The Skip Scan Statistic 320
18.4 Unusually Small Scans 322
18.5 The Ratchet Scan 326
18.5.1 Definition of Ratchet Scan and Exact Distribution 326
18.5.2 Approximate Distributions 327
18.5.3 Equal Size Intervals, Combinatorial Type Bounds 327
18.5.4 Multivariate Normal Asymptotic for Circular Ratchet Scan
whenL= 12 329
18.6 Unknown Value of w 330
18.6.1 Other Statistics 332
References 333
Index 367
|
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| id | DE-604.BV014267252 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:00:43Z |
| institution | BVB |
| isbn | 038798819X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009785359 |
| oclc_num | 45392993 |
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| owner_facet | DE-739 DE-703 DE-384 DE-91G DE-BY-TUM DE-521 DE-83 DE-578 |
| physical | XV, 370 S. Ill. 24 cm |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Glaz, Joseph Verfasser aut Scan statistics Joseph Glaz ; Joseph Naus ; Sylvan Wallenstein New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London Springer 2001 XV, 370 S. Ill. 24 cm txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Literaturverz. S. 333 - 366 Statistiques ordonnées Order statistics Statistik (DE-588)4056995-0 gnd rswk-swf Statistik (DE-588)4056995-0 s DE-604 Naus, Joseph Verfasser aut Wallenstein, Sylvan Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009785359&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Glaz, Joseph Naus, Joseph Wallenstein, Sylvan Scan statistics Statistiques ordonnées Order statistics Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4056995-0 |
| title | Scan statistics |
| title_auth | Scan statistics |
| title_exact_search | Scan statistics |
| title_full | Scan statistics Joseph Glaz ; Joseph Naus ; Sylvan Wallenstein |
| title_fullStr | Scan statistics Joseph Glaz ; Joseph Naus ; Sylvan Wallenstein |
| title_full_unstemmed | Scan statistics Joseph Glaz ; Joseph Naus ; Sylvan Wallenstein |
| title_short | Scan statistics |
| title_sort | scan statistics |
| topic | Statistiques ordonnées Order statistics Statistik (DE-588)4056995-0 gnd |
| topic_facet | Statistiques ordonnées Order statistics Statistik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009785359&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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