Scan Statistics and Applications

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ISBN: 9780817640415 | 081764041X
Cover: Hardcover
Copyright: 8/1/1999

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Scan statistics are used in many areas of science and technology to analyze the occurence of observed clusters of events in time and space. The goal is to determine whether an observed cluster of events occurred by chance if it is assumed that the observed events follow a specified probability model. Scan Statistics and Applications is a comprehensive, edited survey that brings together the work of leading authorities to present the most current advances in theory and methodology for this new area of statistical research and application. The chapters contain broad coverage of theory and new analytical and computational methods and techniques in four categories: introductory survey, discrete scan statistics, continuous scan statistics, and applications. Features and Topics:* Comprehensive introductory survey chapter* Discrete scan statistics* Finite Markov chain imbedding* Continuous scan statistics* Spatial scan statistics* Applications in DNA sequence analysis* Monte Carlo approaches to testing order statistics and spacing The book is a valuable resource and state-of-the-art reference for all practitioners, researchers, and professionals in applied probability and statistics who use scan statistics in their work.

Preface

(1)

Contributors

xiii

(4)

List of Tables

xvii

(4)

List of Figures

xxi

PART I: INTRODUCTION AND PRELIMINARIES

(24)

1 Introduction to Scan Statistics

(24)

Joseph Glaz

N. Balakrishnan

1.1 Introduction

(1)

1.2 A Quick Glimpse

(1)

1.3 Order Statistics

(4)

1.4 Discrete Scan Statistics in the One-Dimensional Case

(3)

1.5 Discrete Scan Statistics in the Multidimensional Case

(1)

1.6 Continuous Scan Statistics in the One-Dimensional Case

(3)

1.7 Continuous Scan Statistics in the Multidimensional Case

(1)

References

(10)

PART II: DISCRETE SCAN STATISTICS

(86)

2 Approximations for the Distribution and the Moments of Discrete Scan Statistics

(40)

Jie Chen

Joseph Glaz

2.1 Introduction

(1)

2.2 Scan Statistic for i.i.d. Discrete Random Variables

(7)

2.2.1 Product-type approximations

(2)

2.2.2 Poisson approximations

(2)

2.2.3 Compound Poisson approximations

(2)

2.2.4 Approximations for the expected size and standard deviation of the scan statistic

(1)

2.2.5 A multiple scan statistic

(1)

2.3 Scan Statistic for i.i.d. Bernoulli Trials When the Number of Successes Is Known

(5)

2.3.1 Product-type approximations

(2)

2.3.2 Poisson approximations

(1)

2.3.3 Compound Poisson approximations

(2)

2.3.4 Approximations for the expected size and standard deviation of the scan statistic

(1)

2.4 Scan Statistics for Binomial and Poisson Distributions Conditional on the Total Number of Events

(2)

2.4.1 Poisson model

(1)

2.4.2 Binomial model

(1)

2.5 Two-Dimensional Scan Statistics

(4)

2.5.1 Product-type approximations

(1)

2.5.2 Poisson approximations

(1)

2.5.3 A compound Poisson approximation

(1)

2.5.4 Approximations for the expected size and standard deviation of the scan statistic

(1)

2.5.5 A multiple scan statistic

(1)

2.6 Numerical Examples

(1)

2.7 Concluding Remarks

(13)

References

(6)

3 Ratchet Scan and Disjoint Statistics

(30)

Joachim Krauth

3.1 Introduction

(2)

3.2 Disjoint Statistic

(12)

3.2.1 Definition

(1)

3.2.2 Results for the equiprobable case

(3)

3.2.3 Results for the general case

(2)

3.2.4 Power

(2)

3.2.5 Modifications and extensions

(4)

3.2.6 Applications

(1)

3.3 Linear Ratchet Scan Statistic

(4)

3.3.1 Definition

(1)

3.3.2 Results

(3)

3.3.3 The EMM procedure

(1)

3.3.4 Applications

(1)

3.4 Circular Ratchet Scan Statistic

(4)

3.4.1 Definition

(1)

3.4.2 Results

(2)

3.4.3 Modifications

(2)

3.4.4 Applications

(1)

3.5 Exact Bounds for the Upper Tail Probabilities of the Statistic M(m)

(2)

References

(6)

4 Scanning Multiple Sequences

(16)

Joseph I. Naus

4.1 Discrete Scan Statistic and Its Generalization to Multiple Sequences

(2)

4.2 Clusters in Multiple Sequences Over the Same Time Period

(3)

4.3 Matching in Multiple Random Letter Sequences

102

(4)

4.4 Lattice Problems Related to Common Words in Multiple Sequences

106

(1)

References

106

(7)

PART III: CONTINUOUS SCAN STATISTICS

113

(138)

5 Approximations of the Distributions of Scan Statistics of Poisson Processes

113

(28)

Sven Erick Alm

5.1 Introduction

113

(2)

5.2 Approximation in One Dimension

115

(5)

5.3 Approximation in Two Dimensions

120

(12)

5.3.1 Rectangular scanning sets

122

(4)

5.3.2 General scanning sets

126

(6)

5.4 Approximation in Three Dimensions

132

(3)

5.5 Approximation in Higher Dimensions

135

(1)

5.6 Comments on Simulations

135

(3)

References

138

(3)

6 An Approach to Computations Involving Spacings With Applications to the Scan Statistic

141

(24)

Fred W. Huffer

Chien-Tai Lin

6.1 Introduction

141

(2)

6.2 Properties of Spacings

143

(2)

6.3 The Basic Recursion

145

(2)

6.4 The Recursion for Exponential Variates

147

(1)

6.5 Binary Matrices With Two Rows

148

(2)

6.6 The Distribution of the Scan Statistic

150

(4)

6.7 The Marking Algorithm

154

(3)

6.8 The Moments of C(Omega)

157

(5)

References

162

(3)

7 Using Moments to Approximate the Distribution of the Scan Statistic

165

(26)

Fred W. Huffer

Chien-Tai Lin

7.1 Introduction

165

(1)

7.2 Bounds

166

(8)

7.2.1 The dual problem

169

(2)

7.2.2 Performance of bounds

171

(1)

7.2.3 Improving the bounds

171

(2)

7.2.4 Using more than four moments

173

(1)

7.3 Approximations

174

(15)

7.3.1 The approximation MC2

174

(3)

7.3.2 Compound Poisson approximations

177

(12)

References

189

(2)

8 Applying Ballot Problem Results to Compute Probabilities Required for a Generalization of the Scan Statistic

191

(12)

Sylvan Wallenstein

8.1 Introduction

191

(1)

8.2 Exact Distribution of a Statistic With Critical Values Changing on Each Interval of Length Omega

192

(4)

8.3 Finding the Probability for Step Function With 2L Steps

196

(2)

8.4 General Result

198

(3)

References

201

(2)

9 Scan Statistic and Multiple Scan Statistic

203

(22)

Chien-Tai Lin

9.1 Introduction

203

(2)

9.2 Methods of Evaluating the Tail Probabilities of S(Omega)

205

(6)

9.3 The Moments of C(Omega)

211

(2)

9.4 Method for Evaluating the Probabilities of C(Omega)

213

(2)

9.5 Approximations for P(C(Omega) is greater than or equal to 2)

215

(5)

References

220

(5)

10 On Poisson Approximation for Continuous Multiple Scan Statistics in Two Dimensions

225

(26)

Marianne Mansson

10.1 Introduction

225

(1)

10.2 Preliminaries

226

(1)

10.3 Historical Background

227

(3)

10.3.1 Multiple scan statistics

227

(2)

10.3.2 Scan statistics

229

(1)

10.4 The Probability of Covering a Given m-Subset

230

(4)

10.5 Poisson Approximation

234

(7)

10.5.1 Total variation distance

235

(4)

10.5.2 Simulations

239

(2)

10.6 Point Processes

241

(3)

10.7 Miscellaneous

244

(1)

10.7.1 Higher dimensions

244

(1)

10.7.2 The unconditional case

244

(1)

10.7.3 The torus convention

244

(1)

10.7.4 Compound Poisson approximation

244

(1)

References

245

(6)

PART IV: APPLICATIONS

251

(72)

11 A Start-Up Demonstration Test Using a Simple Scan-Based Statistic

251

(18)

Markos V. Koutras

N. Balakrishnan

11.1 Introduction

251

(2)

11.2 Probability Mass Function of T

253

(1)

11.3 Probability Generating Function of T

254

(3)

11.4 Mean, Variance, and Moments of T

257

(4)

11.5 Inference on p

261

(1)

11.6 Results for Markov-Dependent Start-Ups

262

(2)

11.7 Illustrative Example

264

(2)

References

266

(3)

12 Applications of the Scan Statistic in DNA Sequence Analysis

269

(18)

Ming-Ying Leung

Traci E. Yamashita

12.1 Introduction and Background

269

(2)

12.2 r-Scans and DNA Sequence Analysis

271

(6)

12.2.1 Duality

271

(1)

12.2.2 DNA sequence analysis

271

(3)

12.2.3 Application of A(r)

274

(3)

12.2.4 Biological implication of the significant clusters

277

(1)

12.3 Approximate Distributions of A(r)

277

(6)

12.3.1 The finite Poisson approximation

278

(2)

12.3.2 Declumping approximation

280

(1)

12.3.3 Compound Poisson approximation

281

(1)

12.3.4 Comparison with simulated probabilities

281

(2)

12.4 Concluding Remarks

283

(1)

References

284

(3)

13 On the Probability of Pattern Matching in Nonaligned DNA Sequences: A Finite Markov Chain Imbedding Approach

287

(16)

James C. Fu

W.Y. Wendy Lou

S.C. Chen

13.1 Introduction

287

(2)

13.2 The Matching Problem and Notation

289

(3)

13.3 Perfect Matching Probability

292

(2)

13.4 A Large Deviation Approximation

294

(2)

13.5 Numerical Results

296

(3)

13.6 Discussion

299

(1)

References

300

(3)

14 Spatial Scan Statistics: Models, Calculations, and Applications

303

(20)

Martin Kulldorff

14.1 Introduction

303

(1)

14.2 Models

304

(4)

14.2.1 A general model

304

(2)

14.2.2 Special cases

306

(1)

14.2.3 Related methods

307

(1)

14.3 Calculations

308

(7)

14.3.1 Probabilistic approximations

308

(1)

14.3.2 Monte Carlo-based hypothesis testing

308

(6)

14.3.3 Software

314

(1)

14.4 Applications

315

(4)

14.4.1 Epidemiology

315

(1)

14.4.2 Example: Brain cancer in New Mexico

316

(1)

14.4.3 Medical imaging

317

(1)

14.4.4 Astronomy

318

(1)

14.4.5 Archaeology and history

318

(1)

14.4.6 Urban and regional planning

318

(1)

14.4.7 Reconnaissance

319

(1)

14.4.8 Power

319

(1)

References

319

(4)

Index

323

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Scan Statistics and Applications

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