Mathematics Books

Book SynopsisA user-friendly introduction to metric and topological groups Topological Groups: An Introduction provides a self-contained presentation with an emphasis on important families of topological groups. The book uniquely provides a modern and balanced presentation by using metric groups to present a substantive introduction to topics such as duality, while also shedding light on more general results for topological groups. Filling the need for a broad and accessible introduction to the subject, the book begins with coverage of groups, metric spaces, and topological spaces before introducing topological groups. Since linear spaces, algebras, norms, and determinants are necessary tools for studying topological groups, their basic properties are developed in subsequent chapters. For concreteness, product topologies, quotient topologies, and compact-open topologies are first introduced as metric spaces before their open sets are characterized by topological properties.Trade Review"Topological Groups: an Introduction is an excellent book for advanced undergraduate and graduate-level courses on the topic. The book also serves as a valuable resource for professional working in the fields of mathematics, science, engineering, and physics." (Bulletin Bibliographique, 2011) "Recommended. Upper-division undergraduates through professionals." (Choice, 1 March 2011) Table of ContentsPreface. 1 Groups and Metrics. 1.1 Groups. 1.2 Metric and Topological Spaces. 1.3 Continuous Group Operations. 1.4 Subgroups and Their Quotient Spaces. 1.5 Compactness and Metric Groups. 2 Linear Spaces and Algebras. 2.1 Linear Structures on Groups. 2.2 Linear Functions. 2.3 Norms on Linear Spaces. 2.4 Continuous Linear Functions. 2.5 The Determinant Function. 3 The Subgroups of Rn. 3.1 Closed Subgroups. 3.2 Quotient Groups. 3.3 Dense Subgroups. 4 Matrix Groups. 4.1 General Linear Groups. 4.2 Orthogonal and Unitary Groups. 4.3 Triangular Groups. 4.4 One-Parameter Subgroups. 5 Connectedness of Topological Groups. 5.1 Connected Topological Spaces. 5.2 Connected Matrix Groups. 5.3 Compact Product Spaces. 5.4 Totally Disconnected Groups. 6 Metric Groups of Functions. 6.1 Real-Valued Functions. 6.2 The Compact-Open Topology. 6.3 Metric Groups of Isometries. 6.4 Metric Groups of Homeomorphisms. 6.5 Metric Groups of Homomorphisms. 7 Compact Groups. 7.1 Invariant Means. 7.2 Integral Equations. 7.3 Eigenfunctions. 7.4 Compact Abelian Groups. 7.5 Matrix Representations. 8 Character Groups. 8.1 Countable Discrete Abelian Groups. 8.2 The Duality Homomorphism. 8.3 Compactly Generated Abelian Groups. 8.4 A Duality Theorem. Bibliography. Index of Special Symbols. Index.

Read more less

Book SynopsisA comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and real-world applications With a sophisticated approach, Probability and Stochastic Processes successfully balances theory and applications in a pedagogical and accessible format.Table of ContentsList of Figures xvii List of Tables xx Preface xxi Acknowledgments xxiii Introduction 1 Part I Probability 1 Elements of Probability Measure 9 1.1 Probability Spaces 10 1.1.1 Null element of ℱ. Almost sure (a.s.) statements. Indicator of a set 21 1.2 Conditional Probability 22 1.3 Independence 29 1.4 Monotone Convergence Properties of Probability 31 1.5 Lebesgue Measure on the Unit Interval (0,1] 37 Problems 40 2 Random Variables 45 2.1 Discrete and Continuous Random Variables 48 2.2 Examples of Commonly Encountered Random Variables 52 2.3 Existence of Random Variables with Prescribed Distribution 65 2.4 Independence 68 2.5 Functions of Random Variables. Calculating Distributions 72 Problems 82 3 Applied Chapter: Generating Random Variables 87 3.1 Generating One-Dimensional Random Variables by Inverting the cdf 88 3.2 Generating One-Dimensional Normal Random Variables 91 3.3 Generating Random Variables. Rejection Sampling Method 94 3.4 Generating Random Variables. Importance Sampling 109 Problems 119 4 Integration Theory 123 4.1 Integral of Measurable Functions 124 4.2 Expectations 130 4.3 Moments of a Random Variable. Variance and the Correlation Coefficient 143 4.4 Functions of Random Variables. The Transport Formula 145 4.5 Applications. Exercises in Probability Reasoning 148 4.6 A Basic Central Limit Theorem: The DeMoivre–LaplaceTheorem: 150 Problems 152 5 Conditional Distribution and Conditional Expectation 157 5.1 Product Spaces 158 5.2 Conditional Distribution and Expectation. Calculation in Simple Cases 162 5.3 Conditional Expectation. General Definition 165 5.4 Random Vectors. Moments and Distributions 168 Problems 177 6 Moment Generating Function. Characteristic Function 181 6.1 Sums of Random Variables. Convolutions 181 6.2 Generating Functions and Applications 182 6.3 Moment Generating Function 188 6.4 Characteristic Function 192 6.5 Inversion and Continuity Theorems 199 6.6 Stable Distributions. Lvy Distribution 204 6.6.1 Truncated Lévy flight distribution 206 Problems 208 7 Limit Theorems 213 7.1 Types of Convergence 213 7.1.1 Traditional deterministic convergence types 214 7.1.2 Convergence in Lp 215 7.1.3 Almost sure (a.s.) convergence 216 7.1.4 Convergence in probability. Convergence in distribution 217 7.2 Relationships between Types of Convergence 221 7.2.1 A.S. and Lp 221 7.2.2 Probability, a.s., Lp convergence 223 7.2.3 Uniform Integrability 226 7.2.4 Weak convergence and all the others 228 7.3 Continuous Mapping Theorem. Joint Convergence. Slutsky’s Theorem 230 7.4 The Two Big Limit Theorems: LLN and CLT 232 7.4.1 A note on statistics 232 7.4.2 The order statistics 234 7.4.3 Limit theorems for the mean statistics 238 7.5 Extensions of CLT 245 7.6 Exchanging the Order of Limits and Expectations 251 Problems 252 8 Statistical Inference 259 8.1 The Classical Problems in Statistics 259 8.2 Parameter Estimation Problem 260 8.2.1 The case of the normal distribution, estimating mean when variance is unknown 262 8.2.2 The case of the normal distribution, comparing variances 264 8.3 Maximum Likelihood Estimation Method 265 8.3.1 The bisection method 267 8.4 The Method of Moments 276 8.5 Testing, the Likelihood Ratio Test 277 8.5.1 The likelihood ratio test 280 8.6 Confidence Sets 284 Problems 286 Part II Stochastic Processes 9 Introduction to Stochastic Processes 293 9.1 General Characteristics of Stochastic Processes 294 9.1.1 The index set I 294 9.1.2 The state space S 294 9.1.3 Adaptiveness, filtration, standard filtration 294 9.1.4 Pathwise realizations 296 9.1.5 The finite distribution of stochastic processes 296 9.1.6 Independent components 297 9.1.7 Stationary process 298 9.1.8 Stationary and independent increments 299 9.1.9 Other properties that characterize specific classes of stochastic processes 300 9.2 A Simple Process – The Bernoulli Process 301 Problems 304 10 The Poisson Process 307 10.1 Definitions 307 10.2 Inter-Arrival and Waiting Time for a Poisson Process 310 10.2.1 Proving that the inter-arrival times are independent 311 10.2.2 Memoryless property of the exponential distribution 315 10.2.3 Merging two independent Poisson processes 316 10.2.4 Splitting the events of the Poisson process into types 316 10.3 General Poisson Processes 317 10.3.1 Nonhomogenous Poisson process 318 10.3.2 The compound Poisson process 319 10.4 Simulation techniques. Constructing Poisson Processes 323 10.4.1 One-dimensional simple Poisson process 323 Problems 326 11 Renewal Processes 331 11.0.2 The renewal function 333 11.1 Limit Theorems for the Renewal Process 334 11.1.1 Auxiliary but very important results. Wald’s theorem. Discrete stopping time 336 11.1.2 An alternative proof of the elementary renewal theorem 340 11.2 Discrete Renewal Theory 344 11.3 The Key Renewal Theorem 349 11.4 Applications of the Renewal Theorems 350 11.5 Special cases of renewal processes 352 11.5.1 The alternating renewal process 353 11.5.2 Renewal reward process 358 11.6 The renewal Equation 359 11.7 Age-Dependent Branching processes 363 Problems 366 12 Markov Chains 371 12.1 Basic Concepts for Markov Chains 371 12.1.1 Definition 371 12.1.2 Examples of Markov chains 372 12.1.3 The Chapman– Kolmogorov equation 378 12.1.4 Communicating classes and class properties 379 12.1.5 Periodicity 379 12.1.6 Recurrence property 380 12.1.7 Types of recurrence 382 12.2 Simple Random Walk on Integers in d Dimensions 383 12.3 Limit Theorems 386 12.4 States in a MC. Stationary Distribution 387 12.4.1 Examples. Calculating stationary distribution 391 12.5 Other Issues: Graphs, First-Step Analysis 394 12.5.1 First-step analysis 394 12.5.2 Markov chains and graphs 395 12.6 A general Treatment of the Markov Chains 396 12.6.1 Time of absorption 399 12.6.2 An example 400 Problems 406 13 Semi-Markov and Continuous-time Markov Processes 411 13.1 Characterization Theorems for the General semi- Markov Process 413 13.2 Continuous-Time Markov Processes 417 13.3 The Kolmogorov Differential Equations 420 13.4 Calculating Transition Probabilities for a Markov Process General Approach 425 13.5 Limiting Probabilities for the Continuous-Time Markov Chain 426 13.6 Reversible Markov Process 429 Problems 432 14 Martingales 437 14.1 Definition and Examples 438 14.1.1 Examples of martingales 439 14.2 Martingales and Markov Chains 440 14.2.1 Martingales induced by Markov chains 440 14.3 Previsible Process. The Martingale Transform 442 14.4 Stopping Time. Stopped Process 444 14.4.1 Properties of stopping time 446 14.5 Classical Examples of Martingale Reasoning 449 14.5.1 The expected number of tosses until a binary pattern occurs 449 14.5.2 Expected number of attempts until a general pattern occurs 451 14.5.3 Gambler’s ruin probability – revisited 452 14.6 Convergence Theorems. L1 Convergence. Bounded Martingales in L2 456 Problems 458 15 Brownian Motion 465 15.1 History 465 15.2 Definition 467 15.2.1 Brownian motion as a Gaussian process 469 15.3 Properties of Brownian Motion 471 15.3.1 Hitting times. Reflection principle. Maximum value 474 15.3.2 Quadratic variation 476 15.4 Simulating Brownian Motions 480 15.4.1 Generating a Brownian motion path 480 15.4.2 Estimating parameters for a Brownian motion with drift 481 Problems 481 16 Stochastic Differential Equations 485 16.1 The Construction of the Stochastic Integral 487 16.1.1 Itȏ integral construction 490 16.1.2 An illustrative example 492 16.2 Properties of the Stochastic Integral 494 16.3 Itȏ lemma 495 16.4 Stochastic Differential Equations (SDEs) 499 16.4.1 A discussion of the types of solution for an SDE 501 16.5 Examples of SDEs 502 16.5.1 An analysis of Cox– Ingersoll– Ross (CIR) type models 507 16.5.2 Models similar to CIR 507 16.5.3 Moments calculation for the CIR model 509 16.5.4 Interpretation of the formulas for moments 511 16.5.5 Parameter estimation for the CIR model 511 16.6 Linear Systems of SDEs 513 16.7 A Simple Relationship between SDEs and Partial Differential Equations (PDEs) 515 16.8 Monte Carlo Simulations of SDEs 517 Problems 522 A Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations 527 A.1 Solving difference equations with constant coefficients 528 A.2 Generalized matrix inverse and pseudo-determinant 528 A.3 Connection between systems of differential equations and matrices 529 A.3.1 Writing a system of differential equations in matrix form 530 A.4 Linear Algebra results 533 A.4.1 Eigenvalues, eigenvectors of a square matrix 533 A.4.2 Matrix Exponential Function 534 A.4.3 Relationship between Exponential matrix and Eigenvectors 534 A.5 Finding fundamental solution of the homogeneous system 535 A.5.1 The case when all the eigenvalues are distinct and real 536 A.5.2 The case when some of the eigenvalues are complex 536 A.5.3 The case of repeated real eigenvalues 537 A.6 The nonhomogeneous system 538 A.6.1 The method of undetermined coefficients 538 A.6.2 The method of variation of parameters 539 A.7 Solving systems when P is non-constant 540 Bibliography 541 Index 547

Read more less

Book SynopsisThe material has been extensively class-tested for over ten years at both the author's own university and other institutions. The book is uniquely organized into two main sections, one on Fibonacci Numbers and one on Catalan Numbers, each containing subsections that explore related topics in intricate detail.Table of ContentsPreface xi Part One The Fibonacci Numbers 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Compositions and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for Fn 51 11. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The gcd Property for the Fibonacci Numbers 121 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 Part Two The Catalan Numbers 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Compositions, and Vertices and Arcs 183 23. Triangulating the Interior of a Convex Polygon 192 24. Some Examples from Graph Theory 195 25. Partial Orders, Total Orders, and Topological Sorting 205 26. Sequences and a Generating Tree 211 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219 28. The Catalan Numbers at Sporting Events 226 29. A Recurrence Relation for the Catalan Numbers 231 30. Triangulating the Interior of a Convex Polygon for the Second Time 236 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238 32. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions 250 33. The Narayana Numbers 268 34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers 282 35. Generalized Catalan Numbers 290 36. One Final Example? 296 Solutions for the Odd-Numbered Exercises 301 Index 355

Read more less

Book SynopsisQuantitative and Statistical Research Methods This user-friendly textbook teaches students to understand and apply procedural steps in completing quantitative studies. It explains statistics while progressing through the steps of the hypothesis-testing process from hypothesis to results. The research problems used in the book reflect statistical applications related to interesting and important topics. In addition, the book provides a Research Analysis and Interpretation Guide to help students analyze research articles. Designed as a hands-on resource, each chapter covers a single research problem and offers directions for implementing the research method from start to finish. Readers will learn how to: Pinpoint research questions and hypotheses Identify, classify, and operationally define the study variables Choose appropriate research designs Conduct power analysis Select an appropriate statistic for the problem <Table of ContentsTables and Figures ix Preface xvii The Authors xix Chapter 1 Introduction and Overview 1 Review of Foundational Research Concepts 3 Review of Foundational Statistical Information 6 The Normal Distribution 14 Chapter 2 Logical Steps of Conducting Quantitative Research: Hypothesis-Testing Process 29 Hypothesis-Testing Process 30 Chapter 3 Maximizing Hypothesis Decisions Using Power Analysis 39 Balance between Avoiding Type I and Type II Errors 41 Chapter 4 Research and Statistical Designs 53 Formulating Experimental Conditions 54 Reducing the Imprecision in Measurement 55 Controlling Extraneous Experimental Influences 57 Internal Validity and Experimental Designs 59 Choosing a Statistic to Use for an Analysis 67 Chapter 5 Introduction to IBM SPSS 20 77 The IBM SPSS 20 Data View Screen 80 Naming and Defining Variables in Variable View 80 Entering Data 86 Examples of Basic Analyses 87 Examples of Modifying Data Procedures 96 Chapter 6 Diagnosing Study Data for Inaccuracies and Assumptions 99 Research Example 100 Chapter 7 Randomized Design Comparing Two Treatments and a Control Using a One-Way Analysis of Variance 129 Research Problem 130 Study Variables 131 Research Design 133 Stating the Omnibus (Comprehensive) Research Question 135 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 136 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 137 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 138 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 Is True 143 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 144 Hypothesis Testing Step 6: Make Decision Regarding the H0 and Interpret Post Hoc Effect Sizes and Confidence Intervals 162 Formula Calculations of the Study Results 166 Chapter 8 Repeated-Treatment Design Using a Repeated-Measures Analysis of Variance 183 Research Problem 184 Study Variables 185 Research Design 186 Stating the Omnibus (Comprehensive) Research Question 189 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 190 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 191 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 192 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 Is True 195 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 196 Hypothesis Testing Step 6: Make Decision Regarding the H0 and Interpret Post Hoc Effect Sizes and Confidence Intervals 216 Formula Calculations of the Study Results 218 Chapter 9 Randomized Factorial Experimental Design Using a Factorial ANOVA 231 Research Problem 232 Study Variables 232 Research Design 233 Stating the Omnibus (Comprehensive) Research Questions 237 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 238 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 240 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 241 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 Is True 247 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 248 Hypothesis Testing Step 6: Make Decision Regarding the H0 and Interpret Post Hoc Effect Sizes and Confidence Intervals 271 Formula Calculations of the Study Results 278 Chapter 10 Analysis of Covariance 297 Research Problem 298 Study Variables 299 Research Design 300 Stating the Omnibus (Comprehensive) Research Question 301 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 301 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 302 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 302 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 is True 306 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 307 Hypothesis Testing Step 6: Make Decision Regarding the H0 and Interpret Post Hoc Effect Sizes and Confidence Intervals 324 Formula ANCOVA Calculations of the Study Results 327 ANCOVA Study Results 339 Chapter 11 Randomized Control Group and Repeated-Treatment Designs and Nonparametics 345 Research Problem 346 Study Variables 346 Research Design 347 Stating the Omnibus (Comprehensive) Research Question 349 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 349 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 350 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 350 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 is True 354 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 355 Hypothesis Testing Step 6: Make Decision Regarding the H0 and Interpret Post Hoc Effect Sizes 370 Formula Calculations 376 Nonparametric Research Problem Two: Friedman’s Rank Test for Correlated Samples and Wilcoxon’s Matched-Pairs Signed-Ranks Test 382 Chapter 12 Bivariate and Multivariate Correlation Methods Using Multiple Regression Analysis 401 Research Problem 402 Study Variables 402 Research Method 403 Stating the Omnibus (Comprehensive) Research Question 405 Hypothesis Testing Step 1: Establish the Alternative (Research) Hypothesis (Ha) 405 Hypothesis Testing Step 2: Establish the Null Hypothesis (H0) 406 Hypothesis Testing Step 3: Decide on a Risk Level (Alpha) of Rejecting the True H0 Considering Type I and II Errors and Power 406 Hypothesis Testing Step 4: Choose Appropriate Statistic and Its Sampling Distribution to Test the H0 Assuming H0 is True 407 Hypothesis Testing Step 5: Select Sample, Collect Data, Screen Data, Compute Statistic, and Determine Probability Estimates 407 Hand Calculations of Statistics 423 Chapter 13 Understanding Quantitative Literature and Research 439 Interpretation of a Quantitative Research Article 440 References 461 Index 465

Read more less

Book Synopsis* All Mathematica(R) commands used to solve and explain the book's theories and examples are provided both in the book and on a related Web site, allowing readers to modify these commands on their own to help solve their own problems.Trade Review"An accessible introduction to the theoretical and computational aspects of linear algebra using Maple(TM)." (TMCnet.com, 16 April 2011)Table of ContentsPreface. Conventions and Notations. 1. An Introduction to Mathematica. 1.1 The Very Basics. 1.2 Basic Arithmetic. 1.3 Lists and Matrices. 1.4 Expressions Versus Functions. 1.5 Plotting and Animations. 1.6 Solving Systems of Equations. 1.7 Basic Programming. 2. Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3. Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4. Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5. Determinants, Inverses, and Cramer’ Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column. 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer’s Rule. 6. Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 A Vector Projection. 7. A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 “Rolling” a Circle Along a Curve. 7.3 The TNB Frame. 8. Independence, Basis, and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9. Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear Maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10. The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area, and Arclength in R3. 10.4 Rotations, Reflections, and Rescalings in Three Dimensions. 10.5 Affine Maps. 11. Least-Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least-Squares Fits and Pseudoinverses. 12. Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First-Order Linear System if Differential Equations. 12.5 Basic Facts About Eigenvalues, Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Mathematica EigenFunction. Suggested Reading. Indices. Keyword Index. Index of Mathematica Commands.

Read more less

Book SynopsisThis concise book puts the focus on financial problem solving using readily accessible mathematical methods as tools for understanding. Selected formulae are used to illustrate and clarify the underlying logic of problem solving and to provide readers with additional opportunities to enhance their understanding of financial problems.Table of ContentsPreface xv UNIT I MATHEMATICAL INTRODUCTION 1 1 Numbers, Exponents, and Logarithms 3 1.1. Numbers, 3 1.2. Fractions, 3 1.3. Decimals, 5 1.4. Repetends, 6 1.5. Percentages, 7 1.6. Base Amount, Percentage Rate, and Percentage Amount, 8 1.7. Ratios, 9 1.8. Proportions, 10 1.9. Aliquots, 10 1.10. Exponents, 11 1.11. Laws of Exponents, 11 1.12. Exponential Function, 12 1.13. Natural Exponential Function, 13 1.14. Laws of Natural Exponents, 14 1.15. Scientific Notation, 15 1.16. Logarithms, 15 1.17. Laws of Logarithms, 16 1.18. Characteristic, Mantissa, and Antilogarithm, 16 1.19. Logarithmic Function, 18 2 Mathematical Progressions 20 2.1. Arithmetic Progression, 20 2.2. Geometric Progression, 23 2.3. Recursive Progression, 26 2.4. Infinite Geometric Progression, 28 2.5. Growth and Decay Curves, 29 2.6. Growth and Decay Functions with a Natural Logarithmic Base, 34 3 Statistical Measures 35 3.1. Basic Combinatorial Rules and Concepts, 35 3.2. Permutation, 37 3.3. Combination, 40 3.4. Probability, 41 3.5. Mathematical Expectation and Expected Value, 44 3.6. Variance, 46 3.7. Standard Deviation, 48 3.8. Covariance, 49 3.9. Correlation, 50 3.10. Normal Distribution, 52 Unit I Summary 54 List of Formulas 55 Exercises for Unit I 60 UNIT II MATHEMATICS OF THE TIME VALUE OF MONEY 63 Introduction 65 1 Simple Interest 67 1.1. Total Interest, 67 1.2. Rate of Interest, 67 1.3. Term of Maturity, 68 1.4. Current Value, 68 1.5. Future Value, 69 1.6. Finding n and r When the Current and Future Values are Both Known, 69 1.7. Simple Discount, 70 1.8. Calculating the Term in Days, 72 1.9. Ordinary Interest and Exact Interest, 73 1.10. Obtaining Ordinary Interest and Exact Interest in Terms of Each Other, 73 1.11. Focal Date and Equation of Value, 75 1.12. Equivalent Time: Finding an Average due Date, 78 1.13. Partial Payments, 80 1.14. Finding the Simple Interest Rate by the Dollar-Weighted Method, 81 2 Bank Discount 83 2.1. Finding FV Using the Discount Formula, 84 2.2. Finding the Discount Term and the Discount Rate, 84 2.3. Difference between a Simple Discount and a Bank Discount, 85 2.4. Comparing the Discount Rate to the Interest Rate, 87 2.5. Discounting a Promissory Note, 88 2.6. Discounting a Treasury Bill, 90 3 Compound Interest 93 3.1. The Compounding Formula, 94 3.2. Finding the Current Value, 97 3.3. Discount Factor, 98 3.4. Finding the Rate of Compound Interest, 100 3.5. Finding the Compounding Term, 100 3.6. The Rule of 72 and Other Rules, 101 3.7. Effective Interest Rate, 102 3.8. Types of Compounding, 104 3.9. Continuous Compounding, 105 3.10. Equations of Value for a Compound Interest, 106 3.11. Equated Time for a Compound Interest, 108 4 Annuities 110 4.1. Types of Annuities, 110 4.2. Future Value of an Ordinary Annuity, 111 4.3. Current Value of an Ordinary Annuity, 114 4.4. Finding the Payment of an Ordinary Annuity, 116 4.5. Finding the Term of an Ordinary Annuity, 118 4.6. Finding the Interest Rate of an Ordinary Annuity, 120 4.7. Annuity Due: Future and Current Values, 121 4.8. Finding the Payment of an Annuity Due, 123 4.9. Finding the Term of an Annuity Due, 124 4.10. Deferred Annuity, 126 4.11. Future and Current Values of a Deferred Annuity, 127 4.12. Perpetuities, 128 Unit II Summary 130 List of Formulas 132 Exercises for Unit II 138 UNIT III MATHEMATICS OF DEBT AND LEASING 145 1 Credit and Loans 147 1.1. Types of Debt, 147 1.2. Dynamics of Interest–Principal Proportions, 148 1.3. Premature Payoff, 152 1.4. Assessing Interest and Structuring Payments, 154 1.5. Cost of Credit, 158 1.6. Finance Charge and Average Daily Balance, 160 1.7. Credit Limit vs. Debt Limit, 162 2 Mortgage Debt 164 2.1. Analysis of Amortization, 164 2.2. Effects of Interest Rate, Term, and Down Payment on the Monthly Payment, 170 2.3. Graduated Payment Mortgage, 172 2.4. Mortgage Points and the Effective Rate, 176 2.5. Assuming a Mortgage Loan, 176 2.6. Prepayment Penalty on a Mortgage Loan, 177 2.7. Refinancing a Mortgage Loan, 178 2.8. Wraparound and Balloon Payment Loans, 180 2.9. Sinking Funds, 182 2.10. Comparing Amortization to Sinking Fund Methods, 187 3 Leasing 189 3.1. For the Lessee, 189 3.2. For the Lessor, 196 Unit III Summary 198 List of Formulas 199 Exercises for Unit III 202 UNIT IV MATHEMATICS OF CAPITAL BUDGETING AND DEPRECIATION 205 1 Capital Budgeting 207 1.1. Net Present Value, 207 1.2. Internal Rate of Return, 210 1.3. Profitability Index, 212 1.4. Capitalization and Capitalized Cost, 213 1.5. Other Capital Budgeting Methods, 216 2 Depreciation and Depletion 219 2.1. The Straight-Line Method, 220 2.2. The Fixed-Proportion Method, 223 2.3. The Sum-of-Digits Method, 226 2.4. The Amortization Method, 229 2.5. The Sinking Fund Method, 231 2.6. Composite Rate and Composite Life, 233 2.7. Depletion, 235 Unit IV Summary 239 List of Formulas 240 Exercises for Unit IV 243 UNIT V MATHEMATICS OF THE BREAK-EVEN POINT AND LEVERAGE 247 1 Break-Even Analysis 249 1.1. Deriving BEQ and BER, 249 1.2. BEQ and BER Variables, 251 1.3. Cash Break-Even Technique, 254 1.4. The Break-even Point and the Target Profit, 256 1.5. Algebraic Approach to the Break-Even Point, 257 1.6. The Break-Even Point When Borrowing, 261 1.7. Dual Break-Even Points, 264 1.8. Other Applications of the Break-Even Point, 267 1.9. BEQ and BER Sensitivity to their Variables, 272 1.10. Uses and Limitations of Break-Even Analysis, 272 2 Leverage 274 2.1. Operating Leverage, 274 2.2. Operating Leverage, Fixed Cost, and Business Risk, 277 2.3. Financial Leverage, 278 2.4. Total or Combined Leverage, 284 Unit V Summary 287 List of Formulas 289 Exercises for Unit V 291 UNIT VI MATHEMATICS OF INVESTMENT 295 1 Stocks 297 1.1. Buying and Selling Stocks, 298 1.2. Common Stock Valuation, 300 1.3. Cost of New Issues of Common Stock, 306 1.4. Stock Value with Two-Stage Dividend Growth, 307 1.5. Cost of Stock through the CAPM, 307 1.6. Other Methods for Common Stock Valuation, 308 1.7. Valuation of Preferred Stock, 309 1.8. Cost of Preferred Stock, 310 2 Bonds 311 2.1. Bond Valuation, 311 2.2. Premium and Discount Prices, 315 2.3. Premium Amortization, 317 2.4. Discount Accumulation, 319 2.5. Bond Purchase Price Between Interest Days, 321 2.6. Estimating the Yield Rate, 324 2.7. Duration, 328 3 Mutual Funds 330 3.1. Fund Evaluation, 331 3.2. Loads, 332 3.3. Performance Measures, 332 3.4. The Effect of Systematic Risk (β), 338 3.5. Dollar-Cost Averaging, 340 4 Options 341 4.1. Dynamics of Making Profits with Options, 343 4.2. Intrinsic Value of Calls and Puts, 344 4.3. Time Value of Calls and Puts, 347 4.4. The Delta Ratio, 348 4.5. Determinants of Option Value, 350 4.6. Option Valuation, 351 4.7. Combined Intrinsic Values of Options, 353 5 Cost of Capital and Ratio Analysis 357 5.1. Before- and After-Tax Cost of Capital, 357 5.2. Weighted-Average Cost of Capital, 358 5.3. Ratio Analysis, 359 5.4. The DuPont Model, 374 5.5. A Final Word about Ratios, 376 Unit VI Summary 377 List of Formulas 379 Exercises for Unit VI 384 UNIT VII MATHEMATICS OF RETURN AND RISK 387 1 Measuring Return and Risk 389 1.1. Expected Rate of Return, 389 1.2. Measuring the Risk, 390 1.3. Risk Aversion and Risk Premium, 394 1.4. Return and Risk at the Portfolio Level, 394 1.5. Markowitz’s Two-Asset Portfolio, 405 1.6. Lending and Borrowing at a Risk-Free Rate of Return, 408 1.7. Types of Risk, 409 2 The Capital Asset Pricing Model (CAPM) 411 2.1. The Financial Beta (β), 411 2.2. The CAPM Equation, 414 2.3. The Security Market Line, 416 2.4. SML Swing by Risk Aversion, 418 Unit VII Summary 422 List of Formulas 423 Exercises for Unit VII 425 UNIT VIII MATHEMATICS OF INSURANCE 429 1 Life Annuities 431 1.1. Mortality Table, 431 1.2. Commutation Terms, 436 1.3. Pure Endowment, 438 1.4. Types of Life Annuities, 439 2 Life Insurance 448 2.1. Whole Life Insurance Policy, 448 2.2. Annual Premium: Whole Life Basis, 449 2.3. Annual Premium: m-Payment Basis, 450 2.4. Deferred Whole Life Policy, 451 2.5. Deferred Annual Premium: Whole Life Basis, 452 2.6. Deferred Annual Premium: m-Payment Basis, 453 2.7. Term Life Insurance Policy, 454 2.8. Endowment Insurance Policy, 456 2.9. Annual Premium for the Endowment Policy, 457 2.10. Less than Annual Premiums, 458 2.11. Natural Premium vs. the Level Premium, 459 2.12. Reserve and Terminal Reserve Funds, 461 2.13. Benefits of the Terminal Reserve, 465 2.14. How Much Life Insurance Should You Buy?, 465 3 Property and Casualty Insurance 470 3.1. Deductibles and Co-Insurance, 472 3.2. Health Care Insurance, 473 3.3. Policy Limit, 476 Unit VIII Summary 477 List of Formulas 478 Exercises for Unit VIII 482 References 485 Appendix 487 Index 515

Read more less

Book SynopsisThis handbook provides a complete, but accessible compendium of all the major theorems, applications, and methodologies that are necessary for a clear understanding of probability. Each chapter is self-contained utilizing a common format. Algorithms and formulae are stressed when necessary and in an easy-to-locate fashion.Trade Review“On the whole, the book has two features that set it apart from similar books: the full solutions and the examples from finance. It is up to you to decide if that makes it worth your time checking it out.” (Mathematical Association of America, 1 November 2014) Table of ContentsList of Figures xv Preface xvii Introduction xix 1 Probability Space 1 1.1 Introduction/Purpose of the Chapter 1 1.2 Vignette/Historical Notes 2 1.3 Notations and Definitions 2 1.4 Theory and Applications 4 1.4.1 Algebras 4 1.4.2 Sigma Algebras 5 1.4.3 Measurable Spaces 7 1.4.4 Examples 7 1.4.5 The Borel _-Algebra 9 1.5 Summary 12 Exercises 12 2 Probability Measure 15 2.1 Introduction/Purpose of the Chapter 15 2.2 Vignette/Historical Notes 16 2.3 Theory and Applications 17 2.3.1 Definition and Basic Properties 17 2.3.2 Uniqueness of Probability Measures 22 2.3.3 Monotone Class 24 2.3.4 Examples 26 2.3.5 Monotone Convergence Properties of Probability 28 2.3.6 Conditional Probability 31 2.3.7 Independence of Events and _-Fields 39 2.3.8 Borel–Cantelli Lemmas 46 2.3.9 Fatou’s Lemmas 48 2.3.10 Kolmogorov’s Zero–One Law 49 2.4 Lebesgue Measure on the Unit Interval (01] 50 Exercises 52 3 Random Variables: Generalities 63 3.1 Introduction/Purpose of the Chapter 63 3.2 Vignette/Historical Notes 63 3.3 Theory and Applications 64 3.3.1 Definition 64 3.3.2 The Distribution of a Random Variable 65 3.3.3 The Cumulative Distribution Function of a Random Variable 67 3.3.4 Independence of Random Variables 70 Exercises 71 4 Random Variables: The Discrete Case 79 4.1 Introduction/Purpose of the Chapter 79 4.2 Vignette/Historical Notes 80 4.3 Theory and Applications 80 4.3.1 Definition and Basic Facts 80 4.3.2 Moments 84 4.4 Examples of Discrete Random Variables 89 4.4.1 The (Discrete) Uniform Distribution 89 4.4.2 Bernoulli Distribution 91 4.4.3 Binomial (n p) Distribution 92 4.4.4 Geometric (p) Distribution 95 4.4.5 Negative Binomial (r p) Distribution 101 4.4.6 Hypergeometric Distribution (N m n) 102 4.4.7 Poisson Distribution 104 Exercises 108 5 Random Variables: The Continuous Case 119 5.1 Introduction/Purpose of the Chapter 119 5.2 Vignette/Historical Notes 119 5.3 Theory and Applications 120 5.3.1 Probability Density Function (p.d.f.) 120 5.3.2 Cumulative Distribution Function (c.d.f.) 124 5.3.3 Moments 127 5.3.4 Distribution of a Function of the Random Variable 128 5.4 Examples 130 5.4.1 Uniform Distribution on an Interval [ab] 130 5.4.2 Exponential Distribution 133 5.4.3 Normal Distribution (_ _2) 136 5.4.4 Gamma Distribution 139 5.4.5 Beta Distribution 144 5.4.6 Student’s t Distribution 147 5.4.7 Pareto Distribution 149 5.4.8 The Log-Normal Distribution 151 5.4.9 Laplace Distribution 153 5.4.10 Double Exponential Distribution 155 Exercises 156 6 Generating Random Variables 177 6.1 Introduction/Purpose of the Chapter 177 6.2 Vignette/Historical Notes 178 6.3 Theory and Applications 178 6.3.1 Generating One-Dimensional Random Variables by Inverting the Cumulative Distribution Function (c.d.f.) 178 6.3.2 Generating One-Dimensional Normal Random Variables 183 6.3.3 Generating Random Variables. Rejection Sampling Method 186 6.3.4 Generating from a Mixture of Distributions 193 6.3.5 Generating Random Variables. Importance Sampling 195 6.3.6 Applying Importance Sampling 198 6.3.7 Practical Consideration: Normalizing Distributions 201 6.3.8 Sampling Importance Resampling 203 6.3.9 Adaptive Importance Sampling 204 6.4 Generating Multivariate Distributions with Prescribed Covariance Structure 205 Exercises 208 7 Random Vectors in Rn 210 7.1 Introduction/Purpose of the Chapter 210 7.2 Vignette/Historical Notes 210 7.3 Theory and Applications 211 7.3.1 The Basics 211 7.3.2 Marginal Distributions 212 7.3.3 Discrete Random Vectors 214 7.3.4 Multinomial Distribution 219 7.3.5 Testing Whether Counts are Coming from a Specific Multinomial Distribution 220 7.3.6 Independence 221 7.3.7 Continuous Random Vectors 223 7.3.8 Change of Variables. Obtaining Densities of Functions of Random Vectors 229 7.3.9 Distribution of Sums of Random Variables. Convolutions 231 Exercises 236 8 Characteristic Function 255 8.1 Introduction/Purpose of the Chapter 255 8.2 Vignette/Historical Notes 255 8.3 Theory and Applications 256 8.3.1 Definition and Basic Properties 256 8.3.2 The Relationship Between the Characteristic Function and the Distribution 260 8.4 Calculation of the Characteristic Function for Commonly Encountered Distributions 265 8.4.1 Bernoulli and Binomial 265 8.4.2 Uniform Distribution 266 8.4.3 Normal Distribution 267 8.4.4 Poisson Distribution 267 8.4.5 Gamma Distribution 268 8.4.6 Cauchy Distribution 269 8.4.7 Laplace Distribution 270 8.4.8 Stable Distributions. L´evy Distribution 271 8.4.9 Truncated L´evy Flight Distribution 274 Exercises 275 9 Moment-Generating Function 280 9.1 Introduction/Purpose of the Chapter 280 9.2 Vignette/Historical Notes 280 9.3 Theory and Applications 281 9.3.1 Generating Functions and Applications 281 9.3.2 Moment-Generating Functions. Relation with the Characteristic Functions 288 9.3.3 Relationship with the Characteristic Function 292 9.3.4 Properties of the MGF 292 Exercises 294 10 Gaussian Random Vectors 300 10.1 Introduction/Purpose of the Chapter 300 10.2 Vignette/Historical Notes 301 10.3 Theory and Applications 301 10.3.1 The Basics 301 10.3.2 Equivalent Definitions of a Gaussian Vector 303 10.3.3 Uncorrelated Components and Independence 309 10.3.4 The Density of a Gaussian Vector 313 10.3.5 Cochran’s Theorem 316 10.3.6 Matrix Diagonalization and Gaussian Vectors 319 Exercises 325 11 Convergence Types. Almost Sure Convergence. Lp-Convergence. Convergence in Probability 338 11.1 Introduction/Purpose of the Chapter 338 11.2 Vignette/Historical Notes 339 11.3 Theory and Applications: Types of Convergence 339 11.3.1 Traditional Deterministic Convergence Types 339 11.3.2 Convergence of Moments of an r.v.—Convergence in Lp 341 11.3.3 Almost Sure (a.s.) Convergence 342 11.3.4 Convergence in Probability 344 11.4 Relationships Between Types of Convergence 346 11.4.1 a.s. and Lp 347 11.4.2 Probability and a.s./Lp 351 11.4.3 Uniform Integrability 357 Exercises 359 12 Limit Theorems 372 12.1 Introduction/Purpose of the Chapter 372 12.2 Vignette/Historical Notes 372 12.3 Theory and Applications 375 12.3.1 Weak Convergence 375 12.3.2 The Law of Large Numbers 384 12.4 Central Limit Theorem 401 Exercises 409 13 Appendix A: Integration Theory. General Expectations 421 13.1 Integral of Measurable Functions 422 13.1.1 Integral of Simple (Elementary) Functions 422 13.1.2 Integral of Positive Measurable Functions 424 13.1.3 Integral of Measurable Functions 428 13.2 General Expectations and Moments of a Random Variable 429 13.2.1 Moments and Central Moments. Lp Space 430 13.2.2 Variance and the Correlation Coefficient 431 13.2.3 Convergence Theorems 433 14 Appendix B: Inequalities Involving Random Variables and Their Expectations 434 14.1 Functions of Random Variables. The Transport Formula 441 Bibliography 445 Index 447

Read more less

Book SynopsisNumeracy in Children''s Nursing and Healthcare is a handy, practical book which highlights the importance of numbers, numeracy and calculations in children''s nursing practice, instilling nursing students and qualified nurses with confidence and competence when working with numbers and calculating drug doses. This accessible guide covers all aspects of numeracy from basic skills through to complex drug administration, and provides case studies throughout enabling the reader to apply the theory to practice. Each chapter adopts the same accessible and easy-to-follow format, featuring learning outcomes, a case scenario, key numeracy information, hints and tips, activities and exercises, and a glossary of terms.Table of ContentsCONTRIBUTORS vi ACKNOWLEDGEMENTS vii ABOUT THE COMPANION WEBSITE ix GETTING STARTED: HOW TO USE THIS BOOK x 1 THE ROLE OF NUMERACY IN NURSING AND HEALTHCARE PRACTICE 1 2 COUNTING AND MEASURING 27 3 BASIC NUMERACY SKILLS UNDERPINNING CHILDREN AND YOUNG PEOPLE’S NURSING PRACTICE 65 4 ADVANCING ONWARDS: TAKING THE WHOLE NUMBER APART 97 5 PUTTING THE PIECES TOGETHER – A FORMULA FOR CHILDREN’S NURSES 131 6 ADMINISTERING MEDICINES AND MANAGING NUMBERS IN MORE COMPLEX SETTINGS – THE PHARMACIST AND NEONATAL NURSING PERSPECTIVES 161Gerard Donaghy and Lisa McCormack 7 CHILD DEVELOPMENT AND NUMBER SENSE 191 8 WHERE DO I GO FROM HERE? 221 ANSWERS 227 APPENDIX: FAMOUS MATHEMATICIANS 237 REFERENCES AND BIBLIOGRAPHY 241 INDEX 247

Read more less

Book SynopsisAn extensive update to a classic text Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences.Table of ContentsForeword to the first edition xiii From the preface to the first edition xvii Preface to the second edition xix Preface to the third edition xxi Notation xxiii 1 Mathematical foundations 1 1.1 Set theory 1 1.2 Topology in Euclidean spaces 3 1.3 Operations on subsets of Euclidean space 5 1.4 Mathematical morphology and image analysis 7 1.5 Euclidean isometries 9 1.6 Convex sets in Euclidean spaces 10 1.7 Functions describing convex sets 17 1.8 Polyconvex sets 24 1.9 Measure and integration theory 27 2 Point processes I: The Poisson point process 35 2.1 Introduction 35 2.2 The binomial point process 36 2.3 The homogeneous Poisson point process 41 2.4 The inhomogeneous and general Poisson point process 51 2.5 Simulation of Poisson point processes 53 2.6 Statistics for the homogeneous Poisson point process 55 3 Random closed sets I: The Boolean model 64 3.1 Introduction and basic properties 64 3.2 The Boolean model with convex grains 78 3.3 Coverage and connectivity 89 3.4 Statistics 95 3.5 Generalisations and variations 103 3.6 Hints for practical applications 106 4 Point processes II: General theory 108 4.1 Basic properties 108 4.2 Marked point processes 116 4.3 Moment measures and related quantities 120 4.4 Palm distributions 127 4.5 The second moment measure 139 4.6 Summary characteristics 143 4.7 Introduction to statistics for stationary spatial point processes 145 4.8 General point processes 156 5 Point processes III: Models 158 5.1 Operations on point processes 158 5.2 Doubly stochastic Poisson processes (Cox processes) 166 5.3 Neyman–Scott processes 171 5.4 Hard-core point processes 176 5.5 Gibbs point processes 178 5.6 Shot-noise fields 200 6 Random closed sets II: The general case 205 6.1 Basic properties 205 6.2 Random compact sets 213 6.3 Characteristics for stationary and isotropic random closed sets 216 6.4 Nonparametric statistics for stationary random closed sets 230 6.5 Germ–grain models 237 6.6 Other random closed set models 255 6.7 Stochastic reconstruction of random sets 276 7 Random measures 279 7.1 Fundamentals 279 7.2 Moment measures and related characteristics 284 7.3 Examples of random measures 286 8 Line, fibre and surface processes 297 8.1 Introduction 297 8.2 Flat processes 302 8.3 Planar fibre processes 314 8.4 Spatial fibre processes 330 8.5 Surface processes 333 8.6 Marked fibre and surface processes 339 9 Random tessellations, geometrical networks and graphs 343 9.1 Introduction and definitions 343 9.2 Mathematical models for random tessellations 346 9.3 General ideas and results for stationary planar tessellations 357 9.4 Mean-value formulae for stationary spatial tessellations 367 9.5 Poisson line and plane tessellations 370 9.6 STIT tessellations 375 9.7 Poisson-Voronoi and Delaunay tessellations 376 9.8 Laguerre tessellations 386 9.9 Johnson–Mehl tessellations 388 9.10 Statistics for stationary tessellations 390 9.11 Random geometrical networks 397 9.12 Random graphs 402 10 Stereology 411 10.1 Introduction 411 10.2 The fundamental mean-value formulae of stereology 413 10.3 Stereological mean-value formulae for germ–grain models 421 10.4 Stereological methods for spatial systems of balls 425 10.5 Stereological problems for nonspherical grains (shape-and-size problems) 436 10.6 Stereology for spatial tessellations 440 10.7 Second-order characteristics and directional distributions 444 References 453 Author index 507 Subject index 521

Read more less

Book SynopsisKendall's Advanced Theory of Statistics and Kendall's Library of Statistics The development of modern statistical theory is reflected in the history of the late Sir Maurice Kenfall's volumes, The Advanced Theory of Statistics. This landmark publication began life as a two-volume work and grew steadily as a single-authored work until the 1950s. In this edition, there is new material on skewness and kurtosis, hazard rate distribution, the bootstrap, the evaluation of the multivariate normal integral and ratios of quadratic forms. It also includes over 200 new references, 40 new exercises, and 20 further examples in the main text.Table of ContentsFrequency Distributions. Measures of Location and Dispersion. Moments and Cumulants. Characteristic Functions. Standard Distributions. Systems of Distributions. Multivariate Distributions. Probability and Statistical Inference. Random Sampling. Standard Errors. Exact Sampling Distributions. Cumulants of Sampling Distributions 1. Cumulants of Sampling Distributions 2. Order Statistics. The Multinormal Distribution and Quadratic Forms. Distributions Associated with the Normal. Appendix Tables. References. Matrix of Examples in Text. Indexes.

Read more less

Book SynopsisUnderstanding Biostatistics looks at the fundamentals of biostatistics, using elementary statistics to explore the nature of statistical tests. This book is intended to complement first-year statistics and biostatistics textbooks. The main focus here is on ideas, rather than on methodological details.Trade Review"Overall, the book is well-written . . . The topics are presented in a logical progression as is the level of their mathematical difficulty. Any biostatistician will find this a valuable complement to his/her favorite biostatistics textbook." (Journal of Biopharmaceutical Statistics, 2012)Table of ContentsPreface ix 1 Statistics and medical science 1 1.1 Introduction 1 1.2 On the nature of science 3 1.3 How the scientific method uses statistics 5 1.4 Finding an outcome variable to assess your hypothesis 7 1.5 How we draw medical conclusions from statistical results 8 1.6 A few words about probabilities 13 1.7 The need for honesty: the multiplicity issue 16 1.8 Prespecification and p-value history 19 1.9 Adaptive designs: controlling the risks in an experiment 21 1.10 The elusive concept of probability 23 1.11 Comments and further reading 26 References 27 2 Observational studies and the need for clinical trials 29 2.1 Introduction 29 2.2 Investigations of medical interventions and risk factors 29 2.3 Observational studies and confounders 33 2.4 The experimental study 39 2.5 Population risks and individual risks 42 2.6 Confounders, Simpson’s paradox and stratification 44 2.7 On incidence and prevalence in epidemiology 51 2.8 Comments and further reading 53 References 54 3 Study design and the bias issue 57 3.1 Introduction 57 3.2 What bias is all about 58 3.3 The need for a representative sample: on selection bias 58 3.4 Group comparability and randomization 61 3.5 Information bias in a cohort study 65 3.6 The study, or placebo, effect 68 3.7 The curse of missing values 70 3.8 Approaches to data analysis: avoiding self-inflicted bias 75 3.9 On meta-analysis and publication bias 79 3.10 Comments and further reading 81 References 82 4 The anatomy of a statistical test 85 4.1 Introduction 85 4.2 Statistical tests, medical diagnosis and Roman law 85 4.3 The risks with medical diagnosis 87 4.3.1 Medical diagnosis based on a single test 87 4.3.2 Bayes’ theorem and the use and misuse of screening tests 89 4.4 The law: a non-quantitative analogue 91 4.5 Risks in statistical testing 93 4.5.1 Does tonsillectomy increase the risk of Hodgkin’s lymphoma? 93 4.5.2 General discussion about statistical tests 98 4.6 Making statements about a binomial parameter 101 4.6.1 The frequentist approach 101 4.6.2 The Bayesian approach 104 4.7 The bell-shaped error distribution 109 4.8 Comments and further reading 112 References 113 4.A Appendix: The evolution of the central limit theorem 115 5 Learning about parameters, and some notes on planning 119 5.1 Introduction 119 5.2 Test statistics described by parameters 120 5.3 How we describe our knowledge about a parameter from an experiment 122 5.4 Statistical analysis of two proportions 127 5.4.1 Some ways to compare two proportions 127 5.4.2 Analysis of the group difference 130 5.5 Adjusting for confounders in the analysis 133 5.6 The power curve of an experiment 138 5.7 Some confusing aspects of power calculations 143 5.8 Comments and further reading 145 References 145 5.A Appendix: Some technical comments 146 5.A.1 The non-central hypergeometric distribution and 2 × 2 tables 146 5.A.2 The gamma and χ2 distributions 147 6 Empirical distribution functions 149 6.1 Introduction 149 6.2 How to describe the distribution of a sample 149 6.3 Describing the sample: descriptive statistics 153 6.4 Population distribution parameters 156 6.5 Confidence in the CDF and its parameters 158 6.6 Analysis of paired data 162 6.7 Bootstrapping 163 6.8 Meta-analysis and heterogeneity 166 6.9 Comments and further reading 169 References 170 6.A Appendix: Some technical comments 171 6.A.1 The extended family of the univariate Gaussian distributions 171 6.A.2 The Wiener process and its bridge 173 6.A.3 Confidence regions for the CDF and the Kolmogorov–Smirnov test 174 7 Correlation and regression in bivariate distributions 177 7.1 Introduction 177 7.2 Bivariate distributions and correlation 178 7.3 On baseline corrections and other covariates 183 7.4 Bivariate Gaussian distributions 186 7.5 Regression to the mean 189 7.6 Statistical analysis of bivariate Gaussian data 195 7.7 Simultaneous analysis of two binomial proportions 199 7.8 Comments and further reading 203 References 203 7.A Appendix: Some technical comments 205 7.A.1 The regression to the mode equation 205 7.A.2 Analysis of data from the multivariate Gaussian distribution 206 7.A.3 On the geometric approach to univariate confidence limits 207 8 How to compare the outcome in two groups 209 8.1 Introduction 209 8.2 Simple models that compare two distributions 210 8.3 Comparison done the horizontal way 212 8.4 Analysis done the vertical way 216 8.5 Some ways to compute p-values 224 8.6 The discrete Wilcoxon test 226 8.7 The two-period crossover trial 229 8.8 Multivariate analysis and analysis of covariance 232 8.9 Comments and further reading 239 References 240 8.A Appendix: About U-statistics 241 9 Least squares, linear models and beyond 245 9.1 Introduction 245 9.2 The purpose of mathematical models 246 9.3 Different ways to do least squares 250 9.4 Logistic regression, with variations 252 9.5 The two-step modeling approach 257 9.6 The effect of missing covariates 260 9.7 The exponential family of distributions 263 9.8 Generalized linear models 269 9.9 Comments and further reading 270 References 270 10 Analysis of dose response 273 10.1 Introduction 273 10.2 Dose–response relationship 274 10.3 Relative dose potency and therapeutic ratio 278 10.4 Subject-specific and population averaged dose response 279 10.5 Estimation of the population averaged dose–response relationship 281 10.6 Estimating subject-specific dose responses 285 10.7 Comments and further reading 288 References 288 11 Hazards and censored data 289 11.1 Introduction 289 11.2 Censored observations: incomplete knowledge 290 11.3 Hazard models from a population perspective 291 11.4 The impact of competing risks 296 11.5 Heterogeneity in survival analysis 300 11.6 Recurrent events and frailty 304 11.7 The principles behind the analysis of censored data 306 11.8 The Kaplan–Meier estimator of the CDF 309 11.9 Comments and further reading 312 References 313 11.A Appendix: On the large-sample approximations of counting processes 314 12 From the log-rank test to the Cox proportional hazards model 317 12.1 Introduction 317 12.2 Comparing hazards between two groups 318 12.3 Nonparametric tests for hazards 319 12.4 Parameter estimation in hazard models 324 12.5 The accelerated failure time model 328 12.6 The Cox proportional hazards model 331 12.7 On omitted covariates and stratification in the log-rank test 336 12.8 Comments and further reading 338 References 339 12.A Appendix: Comments on interval-censored data 341 13 Remarks on some estimation methods 343 13.1 Introduction 343 13.2 Estimating equations and the robust variance estimate 344 13.3 From maximum likelihood theory to generalized estimating equations 351 13.4 The analysis of recurrent events 355 13.5 Defining and estimating mixed effects models 360 13.6 Comments and further reading 366 References 367 13.A Appendix: Formulas for first-order bias 368 Index 371

Read more less

Book SynopsisIn 1989 the first edition of this book set out Gittins'' pioneering index solution to the multi-armed bandit problem and his subsequent investigation of a wide of sequential resource allocation and stochastic scheduling problems. Since then there has been a remarkable flowering of new insights, generalizations and applications, to which Glazebrook and Weber have made major contributions. This second edition brings the story up to date. There are new chapters on the achievable region approach to stochastic optimization problems, the construction of performance bounds for suboptimal policies, Whittle''s restless bandits, and the use of Lagrangian relaxation in the construction and evaluation of index policies. Some of the many varied proofs of the index theorem are discussed along with the insights that they provide. Many contemporary applications are surveyed, and over 150 new references are included. Over the past 40 years the Gittins index has helped theoreticians and practTable of ContentsForeword. Foreword to the first edition. Preface. Preface to the first edition. 1 Introduction or Exploration. Exercises. 2 Main Ideas: Gittins Index. 2.1 Introduction. 2.2 Decision processes. 2.3 Simple families of alternative bandit processes. 2.4 Dynamic programming. 2.5 Gittins index theorem. 2.6 Gittins index. 2.7 Proof of the index theorem by interchanging bandit portions. 2.8 Continuous-time bandit processes. 2.9 Proof of the index theorem by induction and interchange argument. 2.10 Calculation of Gittins indices. 2.11 Monotonicity conditions. 2.12 History of the index theorem. 2.13 Some decision process theory. Exercises. 3 Necessary Assumptions for Indices. 3.1 Introduction. 3.2 Jobs. 3.3 Continuous-time jobs. 3.4 Necessary assumptions. 3.5 Beyond the necessary assumptions. Exercises. 4 Superprocesses, Precedence Constraints and Arrivals. 4.1 Introduction. 4.2 Bandit superprocesses. 4.3 The index theorem for superprocesses. 4.4 Stoppable bandit processes. 4.5 Proof of the index theorem by freezing and promotion rules. 4.6 The index theorem for jobs with precedence constraints. 4.7 Precedence constraints forming an out-forest. 4.8 Bandit processes with arrivals. 4.9 Tax problems. 4.10 Near optimality of nearly index policies. Exercises. 5 The Achievable Region Methodology. 5.1 Introduction. 5.2 A simple example. 5.3 Proof of the index theorem by greedy algorithm. 5.4 Generalized conservation laws and indexable systems. 5.5 Performance bounds for policies for branching bandits. 5.6 Job selection and scheduling problems. 5.7 Multi-armed bandits on parallel machines. Exercises. 6 Restless Bandits and Lagrangian Relaxation. 6.1 Introduction. 6.2 Restless bandits. 6.3 Whittle indices for restless bandits. 6.4 Asymptotic optimality. 6.5 Monotone policies and simple proofs of indexability. 6.6 Applications to multi-class queuing systems. 6.7 Performance bounds for the Whittle index policy. 6.8 Indices for more general resource configurations. Exercises. 7 Multi-Population Random Sampling (Theory). 7.1 Introduction. 7.2 Jobs and targets. 7.3 Use of monotonicity properties. 7.4 General methods of calculation: use of invariance properties. 7.5 Random sampling times. 7.6 Brownian reward processes. 7.7 Asymptotically normal reward processes. 7.8 Diffusion bandits. Exercises. 8 Multi-Population Random Sampling (Calculations). 8.1 Introduction. 8.2 Normal reward processes (known variance). 8.3 Normal reward processes (mean and variance both unknown). 8.4 Bernoulli reward processes. 8.5 Exponential reward processes. 8.6 Exponential target process. 8.7 Bernoulli/exponential target process. Exercises. 9 Further Exploitation. 9.1 Introduction. 9.2 Website morphing. 9.3 Economics. 9.4 Value of information. 9.5 More on job-scheduling problems. 9.6 Military applications. References. Tables. Index.

Read more less

Book SynopsisThis comprehensive text takes a critical look at the modern development of the theory of survey sampling as well as the foundations of survey sampling, and explains how to put this theory into practice.Trade Review"A task for the current, and future, generation is the research and development of methods for integrating data from multiple sources by explicitly addressing the different measurement errors. Those who read this book and address its challenges will be well placed to deal with the research opportunities ahead—both foreseen and yet to be identified."—Carl M. O'Brien, Lowestoft Laboratory, International Statistical Review (2020) doi:10.1111/insr.12420Table of ContentsList of Figures xiii List of Tables xvii List of Algorithms xix Preface xxi Preface to the First French Edition xxiii Table of Notations xxv 1 A History of Ideas in Survey Sampling Theory 1 1.1 Introduction 1 1.2 Enumerative Statistics During the 19th Century 2 1.3 Controversy on the use of Partial Data 4 1.4 Development of a Survey Sampling Theory 5 1.5 The US Elections of 1936 6 1.6 The Statistical Theory of Survey Sampling 6 1.7 Modeling the Population 8 1.8 Attempt to a Synthesis 9 1.9 Auxiliary Information 9 1.10 Recent References and Development 10 2 Population, Sample, and Estimation 13 2.1 Population 13 2.2 Sample 14 2.3 Inclusion Probabilities 15 2.4 Parameter Estimation 17 2.5 Estimation of a Total 18 2.6 Estimation of a Mean 19 2.7 Variance of the Total Estimator 20 2.8 Sampling with Replacement 22 Exercises 24 3 Simple and Systematic Designs 27 3.1 Simple Random Sampling without Replacement with Fixed Sample Size 27 3.1.1 Sampling Design and Inclusion Probabilities 27 3.1.2 The Expansion Estimator and its Variance 28 3.1.3 Comment on the Variance–Covariance Matrix 31 3.2 Bernoulli Sampling 32 3.2.1 Sampling Design and Inclusion Probabilities 32 3.2.2 Estimation 34 3.3 Simple Random Sampling with Replacement 36 3.4 Comparison of the Designs with and Without Replacement 38 3.5 Sampling with Replacement and Retaining Distinct Units 38 3.5.1 Sample Size and Sampling Design 38 3.5.2 Inclusion Probabilities and Estimation 41 3.5.3 Comparison of the Estimators 44 3.6 Inverse Sampling with Replacement 45 3.7 Estimation of Other Functions of Interest 47 3.7.1 Estimation of a Count or a Proportion 47 3.7.2 Estimation of a Ratio 48 3.8 Determination of the Sample Size 50 3.9 Implementation of Simple Random Sampling Designs 51 3.9.1 Objectives and Principles 51 3.9.2 Bernoulli Sampling 51 3.9.3 Successive Drawing of the Units 52 3.9.4 Random Sorting Method 52 3.9.5 Selection–Rejection Method 53 3.9.6 The Reservoir Method 54 3.9.7 Implementation of Simple Random Sampling with Replacement 56 3.10 Systematic Sampling with Equal Probabilities 57 3.11 Entropy for Simple and Systematic Designs 58 3.11.1 Bernoulli Designs and Entropy 58 3.11.2 Entropy and Simple Random Sampling 60 3.11.3 General Remarks 61 Exercises 61 4 Stratification 65 4.1 Population and Strata 65 4.2 Sample, Inclusion Probabilities, and Estimation 66 4.3 Simple Stratified Designs 68 4.4 Stratified Design with Proportional Allocation 70 4.5 Optimal Stratified Design for the Total 71 4.6 Notes About Optimality in Stratification 74 4.7 Power Allocation 75 4.8 Optimality and Cost 76 4.9 Smallest Sample Size 76 4.10 Construction of the Strata 77 4.10.1 General Comments 77 4.10.2 Dividing a Quantitative Variable in Strata 77 4.11 Stratification Under Many Objectives 79 Exercises 80 5 Sampling with Unequal Probabilities 83 5.1 Auxiliary Variables and Inclusion Probabilities 83 5.2 Calculation of the Inclusion Probabilities 84 5.3 General Remarks 85 5.4 Sampling with Replacement with Unequal Inclusion Probabilities 86 5.5 Nonvalidity of the Generalization of the Successive Drawing without Replacement 88 5.6 Systematic Sampling with Unequal Probabilities 89 5.7 Deville’s Systematic Sampling 91 5.8 Poisson Sampling 92 5.9 Maximum Entropy Design 95 5.10 Rao–Sampford Rejective Procedure 98 5.11 Order Sampling 100 5.12 Splitting Method 101 5.12.1 General Principles 101 5.12.2 Minimum Support Design 103 5.12.3 Decomposition into Simple Random Sampling Designs 104 5.12.4 Pivotal Method 107 5.12.5 Brewer Method 109 5.13 Choice of Method 110 5.14 Variance Approximation 111 5.15 Variance Estimation 114 Exercises 115 6 Balanced Sampling 119 6.1 Introduction 119 6.2 Balanced Sampling: Definition 120 6.3 Balanced Sampling and Linear Programming 122 6.4 Balanced Sampling by Systematic Sampling 123 6.5 Methode of Deville, Grosbras, and Roth 124 6.6 Cube Method 125 6.6.1 Representation of a Sampling Design in the form of a Cube 125 6.6.2 Constraint Subspace 126 6.6.3 Representation of the Rounding Problem 127 6.6.4 Principle of the Cube Method 130 6.6.5 The Flight Phase 130 6.6.6 Landing Phase by Linear Programming 133 6.6.7 Choice of the Cost Function 134 6.6.8 Landing Phase by Relaxing Variables 135 6.6.9 Quality of Balancing 135 6.6.10 An Example 136 6.7 Variance Approximation 137 6.8 Variance Estimation 140 6.9 Special Cases of Balanced Sampling 141 6.10 Practical Aspects of Balanced Sampling 141 Exercise 142 7 Cluster and Two-stage Sampling 143 7.1 Cluster Sampling 143 7.1.1 Notation and Definitions 143 7.1.2 Cluster Sampling with Equal Probabilities 146 7.1.3 Sampling Proportional to Size 147 7.2 Two-stage Sampling 148 7.2.1 Population, Primary, and Secondary Units 149 7.2.2 The Expansion Estimator and its Variance 151 7.2.3 Sampling with Equal Probability 155 7.2.4 Self-weighting Two-stage Design 156 7.3 Multi-stage Designs 157 7.4 Selecting Primary Units with Replacement 158 7.5 Two-phase Designs 161 7.5.1 Design and Estimation 161 7.5.2 Variance and Variance Estimation 162 7.6 Intersection of Two Independent Samples 163 Exercises 165 8 Other Topics on Sampling 167 8.1 Spatial Sampling 167 8.1.1 The Problem 167 8.1.2 Generalized Random Tessellation Stratified Sampling 167 8.1.3 Using the Traveling Salesman Method 169 8.1.4 The Local Pivotal Method 169 8.1.5 The Local Cube Method 169 8.1.6 Measures of Spread 170 8.2 Coordination in Repeated Surveys 172 8.2.1 The Problem 172 8.2.2 Population, Sample, and Sample Design 173 8.2.3 Sample Coordination and Response Burden 174 8.2.4 Poisson Method with Permanent Random Numbers 175 8.2.5 Kish and Scott Method for Stratified Samples 176 8.2.6 The Cotton and Hesse Method 176 8.2.7 The Rivière Method 177 8.2.8 The Netherlands Method 178 8.2.9 The Swiss Method 178 8.2.10 Coordinating Unequal Probability Designs with Fixed Size 181 8.2.11 Remarks 181 8.3 Multiple Survey Frames 182 8.3.1 Introduction 182 8.3.2 Calculating Inclusion Probabilities 183 8.3.3 Using Inclusion Probability Sums 184 8.3.4 Using a Multiplicity Variable 185 8.3.5 Using a Weighted Multiplicity Variable 186 8.3.6 Remarks 187 8.4 Indirect Sampling 187 8.4.1 Introduction 187 8.4.2 Adaptive Sampling 188 8.4.3 Snowball Sampling 188 8.4.4 Indirect Sampling 189 8.4.5 The Generalized Weight Sharing Method 190 8.5 Capture–Recapture 191 9 Estimation with a Quantitative Auxiliary Variable 195 9.1 The Problem 195 9.2 Ratio Estimator 196 9.2.1 Motivation and Definition 196 9.2.2 Approximate Bias of the Ratio Estimator 197 9.2.3 Approximate Variance of the Ratio Estimator 198 9.2.4 Bias Ratio 199 9.2.5 Ratio and Stratified Designs 199 9.3 The Difference Estimator 201 9.4 Estimation by Regression 202 9.5 The Optimal Regression Estimator 204 9.6 Discussion of the Three Estimation Methods 205 Exercises 208 10 Post-Stratification and Calibration on Marginal Totals 209 10.1 Introduction 209 10.2 Post-Stratification 209 10.2.1 Notation and Definitions 209 10.2.2 Post-Stratified Estimator 211 10.3 The Post-Stratified Estimator in Simple Designs 212 10.3.1 Estimator 212 10.3.2 Conditioning in a Simple Design 213 10.3.3 Properties of the Estimator in a Simple Design 214 10.4 Estimation by Calibration on Marginal Totals 217 10.4.1 The Problem 217 10.4.2 Calibration on Marginal Totals 218 10.4.3 Calibration and Kullback–Leibler Divergence 220 10.4.4 Raking Ratio Estimation 221 10.5 Example 221 Exercises 224 11 Multiple Regression Estimation 225 11.1 Introduction 225 11.2 Multiple Regression Estimator 226 11.3 Alternative Forms of the Estimator 227 11.3.1 Homogeneous Linear Estimator 227 11.3.2 Projective Form 228 11.3.3 Cosmetic Form 228 11.4 Calibration of the Multiple Regression Estimator 229 11.5 Variance of the Multiple Regression Estimator 230 11.6 Choice of Weights 231 11.7 Special Cases 231 11.7.1 Ratio Estimator 231 11.7.2 Post-stratified Estimator 231 11.7.3 Regression Estimation with a Single Explanatory Variable 233 11.7.4 Optimal Regression Estimator 233 11.7.5 Conditional Estimation 235 11.8 Extension to Regression Estimation 236 Exercise 236 12 Calibration Estimation 237 12.1 Calibrated Methods 237 12.2 Distances and Calibration Functions 239 12.2.1 The Linear Method 239 12.2.2 The Raking Ratio Method 240 12.2.3 Pseudo Empirical Likelihood 242 12.2.4 Reverse Information 244 12.2.5 The Truncated Linear Method 245 12.2.6 General Pseudo-Distance 246 12.2.7 The Logistic Method 249 12.2.8 Deville Calibration Function 249 12.2.9 Roy and Vanheuverzwyn Method 251 12.3 Solving Calibration Equations 252 12.3.1 Solving by Newton’s Method 252 12.3.2 Bound Management 253 12.3.3 Improper Calibration Functions 254 12.3.4 Existence of a Solution 254 12.4 Calibrating on Households and Individuals 255 12.5 Generalized Calibration 256 12.5.1 Calibration Equations 256 12.5.2 Linear Calibration Functions 257 12.6 Calibration in Practice 258 12.7 An Example 259 Exercises 260 13 Model-Based approach 263 13.1 Model Approach 263 13.2 The Model 263 13.3 Homoscedastic Constant Model 267 13.4 Heteroscedastic Model 1 Without Intercept 267 13.5 Heteroscedastic Model 2 Without Intercept 269 13.6 Univariate Homoscedastic Linear Model 270 13.7 Stratified Population 271 13.8 Simplified Versions of the Optimal Estimator 273 13.9 Completed Heteroscedasticity Model 276 13.10 Discussion 277 13.11 An Approach that is Both Model- and Design-based 277 14 Estimation of Complex Parameters 281 14.1 Estimation of a Function of Totals 281 14.2 Variance Estimation 282 14.3 Covariance Estimation 283 14.4 Implicit Function Estimation 283 14.5 Cumulative Distribution Function and Quantiles 284 14.5.1 Cumulative Distribution Function Estimation 284 14.5.2 Quantile Estimation: Method 1 284 14.5.3 Quantile Estimation: Method 2 285 14.5.4 Quantile Estimation: Method 3 287 14.5.5 Quantile Estimation: Method 4 288 14.6 Cumulative Income, Lorenz Curve, and Quintile Share Ratio 288 14.6.1 Cumulative Income Estimation 288 14.6.2 Lorenz Curve Estimation 289 14.6.3 Quintile Share Ratio Estimation 289 14.7 Gini Index 290 14.8 An Example 291 15 Variance Estimation by Linearization 295 15.1 Introduction 295 15.2 Orders of Magnitude in Probability 295 15.3 Asymptotic Hypotheses 300 15.3.1 Linearizing a Function of Totals 301 15.3.2 Variance Estimation 303 15.4 Linearization of Functions of Interest 303 15.4.1 Linearization of a Ratio 303 15.4.2 Linearization of a Ratio Estimator 304 15.4.3 Linearization of a Geometric Mean 305 15.4.4 Linearization of a Variance 305 15.4.5 Linearization of a Covariance 306 15.4.6 Linearization of a Vector of Regression Coefficients 307 15.5 Linearization by Steps 308 15.5.1 Decomposition of Linearization by Steps 308 15.5.2 Linearization of a Regression Coefficient 308 15.5.3 Linearization of a Univariate Regression Estimator 309 15.5.4 Linearization of a Multiple Regression Estimator 309 15.6 Linearization of an Implicit Function of Interest 310 15.6.1 Estimating Equation and Implicit Function of Interest 310 15.6.2 Linearization of a Logistic Regression Coefficient 311 15.6.3 Linearization of a Calibration Equation Parameter 313 15.6.4 Linearization of a Calibrated Estimator 313 15.7 Influence Function Approach 314 15.7.1 Function of Interest, Functional 314 15.7.2 Definition 315 15.7.3 Linearization of a Total 316 15.7.4 Linearization of a Function of Totals 316 15.7.5 Linearization of Sums and Products 317 15.7.6 Linearization by Steps 318 15.7.7 Linearization of a Parameter Defined by an Implicit Function 318 15.7.8 Linearization of a Double Sum 319 15.8 Binder’s Cookbook Approach 321 15.9 Demnati and Rao Approach 322 15.10 Linearization by the Sample Indicator Variables 324 15.10.1 The Method 324 15.10.2 Linearization of a Quantile 326 15.10.3 Linearization of a Calibrated Estimator 327 15.10.4 Linearization of a Multiple Regression Estimator 328 15.10.5 Linearization of an Estimator of a Complex Function with Calibrated Weights 329 15.10.6 Linearization of the Gini Index 330 15.11 Discussion on Variance Estimation 331 Exercises 331 16 Treatment of Nonresponse 333 16.1 Sources of Error 333 16.2 Coverage Errors 334 16.3 Different Types of Nonresponse 334 16.4 Nonresponse Modeling 335 16.5 Treating Nonresponse by Reweighting 336 16.5.1 Nonresponse Coming from a Sample 336 16.5.2 Modeling the Nonresponse Mechanism 337 16.5.3 Direct Calibration of Nonresponse 339 16.5.4 Reweighting by Generalized Calibration 341 16.6 Imputation 342 16.6.1 General Principles 342 16.6.2 Imputing From an Existing Value 342 16.6.3 Imputation by Prediction 342 16.6.4 Link Between Regression Imputation and Reweighting 343 16.6.5 Random Imputation 345 16.7 Variance Estimation with Nonresponse 347 16.7.1 General Principles 347 16.7.2 Estimation by Direct Calibration 348 16.7.3 General Case 349 16.7.4 Variance for Maximum Likelihood Estimation 350 16.7.5 Variance for Estimation by Calibration 353 16.7.6 Variance of an Estimator Imputed by Regression 356 16.7.7 Other Variance Estimation Techniques 357 17 Summary Solutions to the Exercises 359 Bibliography 379 Author Index 405 Subject Index 411

Read more less

Book SynopsisThe statistical methods used to evaluate and compare different methods of measurement are a vital common component of all methods of scientific research. This book provides a practically orientated guide to the statistical models used in the evaluation of measurement errors with a wide variety of illustrative examples taken from across the sciences. After introducing basic concepts, such as precision, reproducibility and reliability, a detailed discussion of the sources of variability of measurements and associated variance components models is provided. The central chapters deal with the design and analysis of method comparison studies (concentrating primarily on quantitative measurements) ranging from simple paired comparisons to more complex studies involving three or more methods. This leads on to a review of methods for categorical measures.

Read more less

Book SynopsisStatistical pattern recognition relates to the use of statistical techniques for analysing data measurements in order to extract information and make justified decisions. It is a very active area of study and research, which has seen many advances in recent years. Applications such as data mining, web searching, multimedia data retrieval, face recognition, and cursive handwriting recognition, all require robust and efficient pattern recognition techniques. This third edition provides an introduction to statistical pattern theory and techniques, with material drawn from a wide range of fields, including the areas of engineering, statistics, computer science and the social sciences. The book has been updated to cover new methods and applications, and includes a wide range of techniques such as Bayesian methods, neural networks, support vector machines, feature selection and feature reduction techniques.Technical descriptions and motivations are provided, and the techniques are illustTrade Review“In the end I must add that this book is so appealing that I often found myself lost in the reading, pausing the overview of the manuscript in order to look more into some presented subject, and not being able to continue until I had finished seeing all about it.” (Zentralblatt MATH, 1 December 2012)Table of ContentsPreface xix Notation xxiii 1 Introduction to Statistical Pattern Recognition 1 1.1 Statistical Pattern Recognition 1 1.1.1 Introduction 1 1.1.2 The Basic Model 2 1.2 Stages in a Pattern Recognition Problem 4 1.3 Issues 6 1.4 Approaches to Statistical Pattern Recognition 7 1.5 Elementary Decision Theory 8 1.5.1 Bayes’ Decision Rule for Minimum Error 8 1.5.2 Bayes’ Decision Rule for Minimum Error – Reject Option 12 1.5.3 Bayes’ Decision Rule for Minimum Risk 13 1.5.4 Bayes’ Decision Rule for Minimum Risk – Reject Option 15 1.5.5 Neyman–Pearson Decision Rule 15 1.5.6 Minimax Criterion 18 1.5.7 Discussion 19 1.6 Discriminant Functions 20 1.6.1 Introduction 20 1.6.2 Linear Discriminant Functions 21 1.6.3 Piecewise Linear Discriminant Functions 23 1.6.4 Generalised Linear Discriminant Function 24 1.6.5 Summary 26 1.7 Multiple Regression 27 1.8 Outline of Book 29 1.9 Notes and References 29 Exercises 31 2 Density Estimation – Parametric 33 2.1 Introduction 33 2.2 Estimating the Parameters of the Distributions 34 2.2.1 Estimative Approach 34 2.2.2 Predictive Approach 35 2.3 The Gaussian Classifier 35 2.3.1 Specification 35 2.3.2 Derivation of the Gaussian Classifier Plug-In Estimates 37 2.3.3 Example Application Study 39 2.4 Dealing with Singularities in the Gaussian Classifier 40 2.4.1 Introduction 40 2.4.2 Na¨ive Bayes 40 2.4.3 Projection onto a Subspace 41 2.4.4 Linear Discriminant Function 41 2.4.5 Regularised Discriminant Analysis 42 2.4.6 Example Application Study 44 2.4.7 Further Developments 45 2.4.8 Summary 46 2.5 Finite Mixture Models 46 2.5.1 Introduction 46 2.5.2 Mixture Models for Discrimination 48 2.5.3 Parameter Estimation for Normal Mixture Models 49 2.5.4 Normal Mixture Model Covariance Matrix Constraints 51 2.5.5 How Many Components? 52 2.5.6 Maximum Likelihood Estimation via EM 55 2.5.7 Example Application Study 60 2.5.8 Further Developments 62 2.5.9 Summary 63 2.6 Application Studies 63 2.7 Summary and Discussion 66 2.8 Recommendations 66 2.9 Notes and References 67 Exercises 67 3 Density Estimation – Bayesian 70 3.1 Introduction 70 3.1.1 Basics 72 3.1.2 Recursive Calculation 72 3.1.3 Proportionality 73 3.2 Analytic Solutions 73 3.2.1 Conjugate Priors 73 3.2.2 Estimating the Mean of a Normal Distribution with Known Variance 75 3.2.3 Estimating the Mean and the Covariance Matrix of a Multivariate Normal Distribution 79 3.2.4 Unknown Prior Class Probabilities 85 3.2.5 Summary 87 3.3 Bayesian Sampling Schemes 87 3.3.1 Introduction 87 3.3.2 Summarisation 87 3.3.3 Sampling Version of the Bayesian Classifier 89 3.3.4 Rejection Sampling 89 3.3.5 Ratio of Uniforms 90 3.3.6 Importance Sampling 92 3.4 Markov Chain Monte Carlo Methods 95 3.4.1 Introduction 95 3.4.2 The Gibbs Sampler 95 3.4.3 Metropolis–Hastings Algorithm 103 3.4.4 Data Augmentation 107 3.4.5 Reversible Jump Markov Chain Monte Carlo 108 3.4.6 Slice Sampling 109 3.4.7 MCMC Example – Estimation of Noisy Sinusoids 111 3.4.8 Summary 115 3.4.9 Notes and References 116 3.5 Bayesian Approaches to Discrimination 116 3.5.1 Labelled Training Data 116 3.5.2 Unlabelled Training Data 117 3.6 Sequential Monte Carlo Samplers 119 3.6.1 Introduction 119 3.6.2 Basic Methodology 121 3.6.3 Summary 125 3.7 Variational Bayes 126 3.7.1 Introduction 126 3.7.2 Description 126 3.7.3 Factorised Variational Approximation 129 3.7.4 Simple Example 131 3.7.5 Use of the Procedure for Model Selection 135 3.7.6 Further Developments and Applications 136 3.7.7 Summary 137 3.8 Approximate Bayesian Computation 137 3.8.1 Introduction 137 3.8.2 ABC Rejection Sampling 138 3.8.3 ABC MCMC Sampling 140 3.8.4 ABC Population Monte Carlo Sampling 141 3.8.5 Model Selection 142 3.8.6 Summary 143 3.9 Example Application Study 144 3.10 Application Studies 145 3.11 Summary and Discussion 146 3.12 Recommendations 147 3.13 Notes and References 147 Exercises 148 4 Density Estimation – Nonparametric 150 4.1 Introduction 150 4.1.1 Basic Properties of Density Estimators 150 4.2 k-Nearest-Neighbour Method 152 4.2.1 k-Nearest-Neighbour Classifier 152 4.2.2 Derivation 154 4.2.3 Choice of Distance Metric 157 4.2.4 Properties of the Nearest-Neighbour Rule 159 4.2.5 Linear Approximating and Eliminating Search Algorithm 159 4.2.6 Branch and Bound Search Algorithms: kd-Trees 163 4.2.7 Branch and Bound Search Algorithms: Ball-Trees 170 4.2.8 Editing Techniques 174 4.2.9 Example Application Study 177 4.2.10 Further Developments 178 4.2.11 Summary 179 4.3 Histogram Method 180 4.3.1 Data Adaptive Histograms 181 4.3.2 Independence Assumption (Naïve Bayes) 181 4.3.3 Lancaster Models 182 4.3.4 Maximum Weight Dependence Trees 183 4.3.5 Bayesian Networks 186 4.3.6 Example Application Study – Naïve Bayes Text Classification 190 4.3.7 Summary 193 4.4 Kernel Methods 194 4.4.1 Biasedness 197 4.4.2 Multivariate Extension 198 4.4.3 Choice of Smoothing Parameter 199 4.4.4 Choice of Kernel 201 4.4.5 Example Application Study 202 4.4.6 Further Developments 203 4.4.7 Summary 203 4.5 Expansion by Basis Functions 204 4.6 Copulas 207 4.6.1 Introduction 207 4.6.2 Mathematical Basis 207 4.6.3 Copula Functions 208 4.6.4 Estimating Copula Probability Density Functions 209 4.6.5 Simple Example 211 4.6.6 Summary 212 4.7 Application Studies 213 4.7.1 Comparative Studies 216 4.8 Summary and Discussion 216 4.9 Recommendations 217 4.10 Notes and References 217 Exercises 218 5 Linear Discriminant Analysis 221 5.1 Introduction 221 5.2 Two-Class Algorithms 222 5.2.1 General Ideas 222 5.2.2 Perceptron Criterion 223 5.2.3 Fisher’s Criterion 227 5.2.4 Least Mean-Squared-Error Procedures 228 5.2.5 Further Developments 235 5.2.6 Summary 235 5.3 Multiclass Algorithms 236 5.3.1 General Ideas 236 5.3.2 Error-Correction Procedure 237 5.3.3 Fisher’s Criterion – Linear Discriminant Analysis 238 5.3.4 Least Mean-Squared-Error Procedures 241 5.3.5 Regularisation 246 5.3.6 Example Application Study 246 5.3.7 Further Developments 247 5.3.8 Summary 248 5.4 Support Vector Machines 249 5.4.1 Introduction 249 5.4.2 Linearly Separable Two-Class Data 249 5.4.3 Linearly Nonseparable Two-Class Data 253 5.4.4 Multiclass SVMs 256 5.4.5 SVMs for Regression 257 5.4.6 Implementation 259 5.4.7 Example Application Study 262 5.4.8 Summary 263 5.5 Logistic Discrimination 263 5.5.1 Two-Class Case 263 5.5.2 Maximum Likelihood Estimation 264 5.5.3 Multiclass Logistic Discrimination 266 5.5.4 Example Application Study 267 5.5.5 Further Developments 267 5.5.6 Summary 268 5.6 Application Studies 268 5.7 Summary and Discussion 268 5.8 Recommendations 269 5.9 Notes and References 270 Exercises 270 6 Nonlinear Discriminant Analysis – Kernel and Projection Methods 274 6.1 Introduction 274 6.2 Radial Basis Functions 276 6.2.1 Introduction 276 6.2.2 Specifying the Model 278 6.2.3 Specifying the Functional Form 278 6.2.4 The Positions of the Centres 279 6.2.5 Smoothing Parameters 281 6.2.6 Calculation of the Weights 282 6.2.7 Model Order Selection 284 6.2.8 Simple RBF 285 6.2.9 Motivation 286 6.2.10 RBF Properties 288 6.2.11 Example Application Study 288 6.2.12 Further Developments 289 6.2.13 Summary 290 6.3 Nonlinear Support Vector Machines 291 6.3.1 Introduction 291 6.3.2 Binary Classification 291 6.3.3 Types of Kernel 292 6.3.4 Model Selection 293 6.3.5 Multiclass SVMs 294 6.3.6 Probability Estimates 294 6.3.7 Nonlinear Regression 296 6.3.8 Example Application Study 296 6.3.9 Further Developments 297 6.3.10 Summary 298 6.4 The Multilayer Perceptron 298 6.4.1 Introduction 298 6.4.2 Specifying the MLP Structure 299 6.4.3 Determining the MLP Weights 300 6.4.4 Modelling Capacity of the MLP 307 6.4.5 Logistic Classification 307 6.4.6 Example Application Study 310 6.4.7 Bayesian MLP Networks 311 6.4.8 Projection Pursuit 313 6.4.9 Summary 313 6.5 Application Studies 314 6.6 Summary and Discussion 316 6.7 Recommendations 317 6.8 Notes and References 318 Exercises 318 7 Rule and Decision Tree Induction 322 7.1 Introduction 322 7.2 Decision Trees 323 7.2.1 Introduction 323 7.2.2 Decision Tree Construction 326 7.2.3 Selection of the Splitting Rule 327 7.2.4 Terminating the Splitting Procedure 330 7.2.5 Assigning Class Labels to Terminal Nodes 332 7.2.6 Decision Tree Pruning – Worked Example 332 7.2.7 Decision Tree Construction Methods 337 7.2.8 Other Issues 339 7.2.9 Example Application Study 340 7.2.10 Further Developments 341 7.2.11 Summary 342 7.3 Rule Induction 342 7.3.1 Introduction 342 7.3.2 Generating Rules from a Decision Tree 345 7.3.3 Rule Induction Using a Sequential Covering Algorithm 345 7.3.4 Example Application Study 350 7.3.5 Further Developments 351 7.3.6 Summary 351 7.4 Multivariate Adaptive Regression Splines 351 7.4.1 Introduction 351 7.4.2 Recursive Partitioning Model 351 7.4.3 Example Application Study 355 7.4.4 Further Developments 355 7.4.5 Summary 356 7.5 Application Studies 356 7.6 Summary and Discussion 358 7.7 Recommendations 358 7.8 Notes and References 359 Exercises 359 8 Ensemble Methods 361 8.1 Introduction 361 8.2 Characterising a Classifier Combination Scheme 362 8.2.1 Feature Space 363 8.2.2 Level 366 8.2.3 Degree of Training 368 8.2.4 Form of Component Classifiers 368 8.2.5 Structure 369 8.2.6 Optimisation 369 8.3 Data Fusion 370 8.3.1 Architectures 370 8.3.2 Bayesian Approaches 371 8.3.3 Neyman–Pearson Formulation 373 8.3.4 Trainable Rules 374 8.3.5 Fixed Rules 375 8.4 Classifier Combination Methods 376 8.4.1 Product Rule 376 8.4.2 Sum Rule 377 8.4.3 Min, Max and Median Combiners 378 8.4.4 Majority Vote 379 8.4.5 Borda Count 379 8.4.6 Combiners Trained on Class Predictions 380 8.4.7 Stacked Generalisation 382 8.4.8 Mixture of Experts 382 8.4.9 Bagging 385 8.4.10 Boosting 387 8.4.11 Random Forests 389 8.4.12 Model Averaging 390 8.4.13 Summary of Methods 396 8.4.14 Example Application Study 398 8.4.15 Further Developments 399 8.5 Application Studies 399 8.6 Summary and Discussion 400 8.7 Recommendations 401 8.8 Notes and References 401 Exercises 402 9 Performance Assessment 404 9.1 Introduction 404 9.2 Performance Assessment 405 9.2.1 Performance Measures 405 9.2.2 Discriminability 406 9.2.3 Reliability 413 9.2.4 ROC Curves for Performance Assessment 415 9.2.5 Population and Sensor Drift 419 9.2.6 Example Application Study 421 9.2.7 Further Developments 422 9.2.8 Summary 423 9.3 Comparing Classifier Performance 424 9.3.1 Which Technique is Best? 424 9.3.2 Statistical Tests 425 9.3.3 Comparing Rules When Misclassification Costs are Uncertain 426 9.3.4 Example Application Study 428 9.3.5 Further Developments 429 9.3.6 Summary 429 9.4 Application Studies 429 9.5 Summary and Discussion 430 9.6 Recommendations 430 9.7 Notes and References 430 Exercises 431 10 Feature Selection and Extraction 433 10.1 Introduction 433 10.2 Feature Selection 435 10.2.1 Introduction 435 10.2.2 Characterisation of Feature Selection Approaches 439 10.2.3 Evaluation Measures 440 10.2.4 Search Algorithms for Feature Subset Selection 449 10.2.5 Complete Search – Branch and Bound 450 10.2.6 Sequential Search 454 10.2.7 Random Search 458 10.2.8 Markov Blanket 459 10.2.9 Stability of Feature Selection 460 10.2.10 Example Application Study 462 10.2.11 Further Developments 462 10.2.12 Summary 463 10.3 Linear Feature Extraction 463 10.3.1 Principal Components Analysis 464 10.3.2 Karhunen–Lo`eve Transformation 475 10.3.3 Example Application Study 481 10.3.4 Further Developments 482 10.3.5 Summary 483 10.4 Multidimensional Scaling 484 10.4.1 Classical Scaling 484 10.4.2 Metric MDS 486 10.4.3 Ordinal Scaling 487 10.4.4 Algorithms 490 10.4.5 MDS for Feature Extraction 491 10.4.6 Example Application Study 492 10.4.7 Further Developments 493 10.4.8 Summary 493 10.5 Application Studies 493 10.6 Summary and Discussion 495 10.7 Recommendations 495 10.8 Notes and References 496 Exercises 497 11 Clustering 501 11.1 Introduction 501 11.2 Hierarchical Methods 502 11.2.1 Single-Link Method 503 11.2.2 Complete-Link Method 506 11.2.3 Sum-of-Squares Method 507 11.2.4 General Agglomerative Algorithm 508 11.2.5 Properties of a Hierarchical Classification 508 11.2.6 Example Application Study 509 11.2.7 Summary 509 11.3 Quick Partitions 510 11.4 Mixture Models 511 11.4.1 Model Description 511 11.4.2 Example Application Study 512 11.5 Sum-of-Squares Methods 513 11.5.1 Clustering Criteria 514 11.5.2 Clustering Algorithms 515 11.5.3 Vector Quantisation 520 11.5.4 Example Application Study 530 11.5.5 Further Developments 530 11.5.6 Summary 531 11.6 Spectral Clustering 531 11.6.1 Elementary Graph Theory 531 11.6.2 Similarity Matrices 534 11.6.3 Application to Clustering 534 11.6.4 Spectral Clustering Algorithm 535 11.6.5 Forms of Graph Laplacian 535 11.6.6 Example Application Study 536 11.6.7 Further Developments 538 11.6.8 Summary 538 11.7 Cluster Validity 538 11.7.1 Introduction 538 11.7.2 Statistical Tests 539 11.7.3 Absence of Class Structure 540 11.7.4 Validity of Individual Clusters 541 11.7.5 Hierarchical Clustering 542 11.7.6 Validation of Individual Clusterings 542 11.7.7 Partitions 543 11.7.8 Relative Criteria 543 11.7.9 Choosing the Number of Clusters 545 11.8 Application Studies 546 11.9 Summary and Discussion 549 11.10 Recommendations 551 11.11 Notes and References 552 Exercises 553 12 Complex Networks 555 12.1 Introduction 555 12.1.1 Characteristics 557 12.1.2 Properties 557 12.1.3 Questions to Address 559 12.1.4 Descriptive Features 560 12.1.5 Outline 560 12.2 Mathematics of Networks 561 12.2.1 Graph Matrices 561 12.2.2 Connectivity 562 12.2.3 Distance Measures 562 12.2.4 Weighted Networks 563 12.2.5 Centrality Measures 563 12.2.6 Random Graphs 564 12.3 Community Detection 565 12.3.1 Clustering Methods 565 12.3.2 Girvan–Newman Algorithm 568 12.3.3 Modularity Approaches 570 12.3.4 Local Modularity 571 12.3.5 Clique Percolation 573 12.3.6 Example Application Study 574 12.3.7 Further Developments 575 12.3.8 Summary 575 12.4 Link Prediction 575 12.4.1 Approaches to Link Prediction 576 12.4.2 Example Application Study 578 12.4.3 Further Developments 578 12.5 Application Studies 579 12.6 Summary and Discussion 579 12.7 Recommendations 580 12.8 Notes and References 580 Exercises 580 13 Additional Topics 581 13.1 Model Selection 581 13.1.1 Separate Training and Test Sets 582 13.1.2 Cross-Validation 582 13.1.3 The Bayesian Viewpoint 583 13.1.4 Akaike’s Information Criterion 583 13.1.5 Minimum Description Length 584 13.2 Missing Data 585 13.3 Outlier Detection and Robust Procedures 586 13.4 Mixed Continuous and Discrete Variables 587 13.5 Structural Risk Minimisation and the Vapnik–Chervonenkis Dimension 588 13.5.1 Bounds on the Expected Risk 588 13.5.2 The VC Dimension 589 References 591 Index 637

Read more less

Book SynopsisStatistical pattern recognition relates to the use of statistical techniques for analysing data measurements in order to extract information and make justified decisions. It is a very active area of study and research, which has seen many advances in recent years.Trade Review"In the end I must add that this book is so appealing that I often found myself lost in the reading, pausing the overview of the manuscript in order to look more into some presented subject, and not being able to continue until I had finished seeing all about it.” (Zentralblatt MATH, 1 December 2012)Table of ContentsPreface xix Notation xxiii 1 Introduction to Statistical Pattern Recognition 1 1.1 Statistical Pattern Recognition 1 1.1.1 Introduction 1 1.1.2 The Basic Model 2 1.2 Stages in a Pattern Recognition Problem 4 1.3 Issues 6 1.4 Approaches to Statistical Pattern Recognition 7 1.5 Elementary Decision Theory 8 1.5.1 Bayes’ Decision Rule for Minimum Error 8 1.5.2 Bayes’ Decision Rule for Minimum Error – Reject Option 12 1.5.3 Bayes’ Decision Rule for Minimum Risk 13 1.5.4 Bayes’ Decision Rule for Minimum Risk – Reject Option 15 1.5.5 Neyman–Pearson Decision Rule 15 1.5.6 Minimax Criterion 18 1.5.7 Discussion 19 1.6 Discriminant Functions 20 1.6.1 Introduction 20 1.6.2 Linear Discriminant Functions 21 1.6.3 Piecewise Linear Discriminant Functions 23 1.6.4 Generalised Linear Discriminant Function 24 1.6.5 Summary 26 1.7 Multiple Regression 27 1.8 Outline of Book 29 1.9 Notes and References 29 Exercises 31 2 Density Estimation – Parametric 33 2.1 Introduction 33 2.2 Estimating the Parameters of the Distributions 34 2.2.1 Estimative Approach 34 2.2.2 Predictive Approach 35 2.3 The Gaussian Classifier 35 2.3.1 Specification 35 2.3.2 Derivation of the Gaussian Classifier Plug-In Estimates 37 2.3.3 Example Application Study 39 2.4 Dealing with Singularities in the Gaussian Classifier 40 2.4.1 Introduction 40 2.4.2 Na¨ive Bayes 40 2.4.3 Projection onto a Subspace 41 2.4.4 Linear Discriminant Function 41 2.4.5 Regularised Discriminant Analysis 42 2.4.6 Example Application Study 44 2.4.7 Further Developments 45 2.4.8 Summary 46 2.5 Finite Mixture Models 46 2.5.1 Introduction 46 2.5.2 Mixture Models for Discrimination 48 2.5.3 Parameter Estimation for Normal Mixture Models 49 2.5.4 Normal Mixture Model Covariance Matrix Constraints 51 2.5.5 How Many Components? 52 2.5.6 Maximum Likelihood Estimation via EM 55 2.5.7 Example Application Study 60 2.5.8 Further Developments 62 2.5.9 Summary 63 2.6 Application Studies 63 2.7 Summary and Discussion 66 2.8 Recommendations 66 2.9 Notes and References 67 Exercises 67 3 Density Estimation – Bayesian 70 3.1 Introduction 70 3.1.1 Basics 72 3.1.2 Recursive Calculation 72 3.1.3 Proportionality 73 3.2 Analytic Solutions 73 3.2.1 Conjugate Priors 73 3.2.2 Estimating the Mean of a Normal Distribution with Known Variance 75 3.2.3 Estimating the Mean and the Covariance Matrix of a Multivariate Normal Distribution 79 3.2.4 Unknown Prior Class Probabilities 85 3.2.5 Summary 87 3.3 Bayesian Sampling Schemes 87 3.3.1 Introduction 87 3.3.2 Summarisation 87 3.3.3 Sampling Version of the Bayesian Classifier 89 3.3.4 Rejection Sampling 89 3.3.5 Ratio of Uniforms 90 3.3.6 Importance Sampling 92 3.4 Markov Chain Monte Carlo Methods 95 3.4.1 Introduction 95 3.4.2 The Gibbs Sampler 95 3.4.3 Metropolis–Hastings Algorithm 103 3.4.4 Data Augmentation 107 3.4.5 Reversible Jump Markov Chain Monte Carlo 108 3.4.6 Slice Sampling 109 3.4.7 MCMC Example – Estimation of Noisy Sinusoids 111 3.4.8 Summary 115 3.4.9 Notes and References 116 3.5 Bayesian Approaches to Discrimination 116 3.5.1 Labelled Training Data 116 3.5.2 Unlabelled Training Data 117 3.6 Sequential Monte Carlo Samplers 119 3.6.1 Introduction 119 3.6.2 Basic Methodology 121 3.6.3 Summary 125 3.7 Variational Bayes 126 3.7.1 Introduction 126 3.7.2 Description 126 3.7.3 Factorised Variational Approximation 129 3.7.4 Simple Example 131 3.7.5 Use of the Procedure for Model Selection 135 3.7.6 Further Developments and Applications 136 3.7.7 Summary 137 3.8 Approximate Bayesian Computation 137 3.8.1 Introduction 137 3.8.2 ABC Rejection Sampling 138 3.8.3 ABC MCMC Sampling 140 3.8.4 ABC Population Monte Carlo Sampling 141 3.8.5 Model Selection 142 3.8.6 Summary 143 3.9 Example Application Study 144 3.10 Application Studies 145 3.11 Summary and Discussion 146 3.12 Recommendations 147 3.13 Notes and References 147 Exercises 148 4 Density Estimation – Nonparametric 150 4.1 Introduction 150 4.1.1 Basic Properties of Density Estimators 150 4.2 k-Nearest-Neighbour Method 152 4.2.1 k-Nearest-Neighbour Classifier 152 4.2.2 Derivation 154 4.2.3 Choice of Distance Metric 157 4.2.4 Properties of the Nearest-Neighbour Rule 159 4.2.5 Linear Approximating and Eliminating Search Algorithm 159 4.2.6 Branch and Bound Search Algorithms: kd-Trees 163 4.2.7 Branch and Bound Search Algorithms: Ball-Trees 170 4.2.8 Editing Techniques 174 4.2.9 Example Application Study 177 4.2.10 Further Developments 178 4.2.11 Summary 179 4.3 Histogram Method 180 4.3.1 Data Adaptive Histograms 181 4.3.2 Independence Assumption (Na¨ive Bayes) 181 4.3.3 Lancaster Models 182 4.3.4 Maximum Weight Dependence Trees 183 4.3.5 Bayesian Networks 186 4.3.6 Example Application Study – Na¨ive Bayes Text Classification 190 4.3.7 Summary 193 4.4 Kernel Methods 194 4.4.1 Biasedness 197 4.4.2 Multivariate Extension 198 4.4.3 Choice of Smoothing Parameter 199 4.4.4 Choice of Kernel 201 4.4.5 Example Application Study 202 4.4.6 Further Developments 203 4.4.7 Summary 203 4.5 Expansion by Basis Functions 204 4.6 Copulas 207 4.6.1 Introduction 207 4.6.2 Mathematical Basis 207 4.6.3 Copula Functions 208 4.6.4 Estimating Copula Probability Density Functions 209 4.6.5 Simple Example 211 4.6.6 Summary 212 4.7 Application Studies 213 4.7.1 Comparative Studies 216 4.8 Summary and Discussion 216 4.9 Recommendations 217 4.10 Notes and References 217 Exercises 218 5 Linear Discriminant Analysis 221 5.1 Introduction 221 5.2 Two-Class Algorithms 222 5.2.1 General Ideas 222 5.2.2 Perceptron Criterion 223 5.2.3 Fisher’s Criterion 227 5.2.4 Least Mean-Squared-Error Procedures 228 5.2.5 Further Developments 235 5.2.6 Summary 235 5.3 Multiclass Algorithms 236 5.3.1 General Ideas 236 5.3.2 Error-Correction Procedure 237 5.3.3 Fisher’s Criterion – Linear Discriminant Analysis 238 5.3.4 Least Mean-Squared-Error Procedures 241 5.3.5 Regularisation 246 5.3.6 Example Application Study 246 5.3.7 Further Developments 247 5.3.8 Summary 248 5.4 Support Vector Machines 249 5.4.1 Introduction 249 5.4.2 Linearly Separable Two-Class Data 249 5.4.3 Linearly Nonseparable Two-Class Data 253 5.4.4 Multiclass SVMs 256 5.4.5 SVMs for Regression 257 5.4.6 Implementation 259 5.4.7 Example Application Study 262 5.4.8 Summary 263 5.5 Logistic Discrimination 263 5.5.1 Two-Class Case 263 5.5.2 Maximum Likelihood Estimation 264 5.5.3 Multiclass Logistic Discrimination 266 5.5.4 Example Application Study 267 5.5.5 Further Developments 267 5.5.6 Summary 268 5.6 Application Studies 268 5.7 Summary and Discussion 268 5.8 Recommendations 269 5.9 Notes and References 270 Exercises 270 6 Nonlinear Discriminant Analysis – Kernel and Projection Methods 274 6.1 Introduction 274 6.2 Radial Basis Functions 276 6.2.1 Introduction 276 6.2.2 Specifying the Model 278 6.2.3 Specifying the Functional Form 278 6.2.4 The Positions of the Centres 279 6.2.5 Smoothing Parameters 281 6.2.6 Calculation of the Weights 282 6.2.7 Model Order Selection 284 6.2.8 Simple RBF 285 6.2.9 Motivation 286 6.2.10 RBF Properties 288 6.2.11 Example Application Study 288 6.2.12 Further Developments 289 6.2.13 Summary 290 6.3 Nonlinear Support Vector Machines 291 6.3.1 Introduction 291 6.3.2 Binary Classification 291 6.3.3 Types of Kernel 292 6.3.4 Model Selection 293 6.3.5 Multiclass SVMs 294 6.3.6 Probability Estimates 294 6.3.7 Nonlinear Regression 296 6.3.8 Example Application Study 296 6.3.9 Further Developments 297 6.3.10 Summary 298 6.4 The Multilayer Perceptron 298 6.4.1 Introduction 298 6.4.2 Specifying the MLP Structure 299 6.4.3 Determining the MLP Weights 300 6.4.4 Modelling Capacity of the MLP 307 6.4.5 Logistic Classification 307 6.4.6 Example Application Study 310 6.4.7 Bayesian MLP Networks 311 6.4.8 Projection Pursuit 313 6.4.9 Summary 313 6.5 Application Studies 314 6.6 Summary and Discussion 316 6.7 Recommendations 317 6.8 Notes and References 318 Exercises 318 7 Rule and Decision Tree Induction 322 7.1 Introduction 322 7.2 Decision Trees 323 7.2.1 Introduction 323 7.2.2 Decision Tree Construction 326 7.2.3 Selection of the Splitting Rule 327 7.2.4 Terminating the Splitting Procedure 330 7.2.5 Assigning Class Labels to Terminal Nodes 332 7.2.6 Decision Tree Pruning – Worked Example 332 7.2.7 Decision Tree Construction Methods 337 7.2.8 Other Issues 339 7.2.9 Example Application Study 340 7.2.10 Further Developments 341 7.2.11 Summary 342 7.3 Rule Induction 342 7.3.1 Introduction 342 7.3.2 Generating Rules from a Decision Tree 345 7.3.3 Rule Induction Using a Sequential Covering Algorithm 345 7.3.4 Example Application Study 350 7.3.5 Further Developments 351 7.3.6 Summary 351 7.4 Multivariate Adaptive Regression Splines 351 7.4.1 Introduction 351 7.4.2 Recursive Partitioning Model 351 7.4.3 Example Application Study 355 7.4.4 Further Developments 355 7.4.5 Summary 356 7.5 Application Studies 356 7.6 Summary and Discussion 358 7.7 Recommendations 358 7.8 Notes and References 359 Exercises 359 8 Ensemble Methods 361 8.1 Introduction 361 8.2 Characterising a Classifier Combination Scheme 362 8.2.1 Feature Space 363 8.2.2 Level 366 8.2.3 Degree of Training 368 8.2.4 Form of Component Classifiers 368 8.2.5 Structure 369 8.2.6 Optimisation 369 8.3 Data Fusion 370 8.3.1 Architectures 370 8.3.2 Bayesian Approaches 371 8.3.3 Neyman–Pearson Formulation 373 8.3.4 Trainable Rules 374 8.3.5 Fixed Rules 375 8.4 Classifier Combination Methods 376 8.4.1 Product Rule 376 8.4.2 Sum Rule 377 8.4.3 Min, Max and Median Combiners 378 8.4.4 Majority Vote 379 8.4.5 Borda Count 379 8.4.6 Combiners Trained on Class Predictions 380 8.4.7 Stacked Generalisation 382 8.4.8 Mixture of Experts 382 8.4.9 Bagging 385 8.4.10 Boosting 387 8.4.11 Random Forests 389 8.4.12 Model Averaging 390 8.4.13 Summary of Methods 396 8.4.14 Example Application Study 398 8.4.15 Further Developments 399 8.5 Application Studies 399 8.6 Summary and Discussion 400 8.7 Recommendations 401 8.8 Notes and References 401 Exercises 402 9 Performance Assessment 404 9.1 Introduction 404 9.2 Performance Assessment 405 9.2.1 Performance Measures 405 9.2.2 Discriminability 406 9.2.3 Reliability 413 9.2.4 ROC Curves for Performance Assessment 415 9.2.5 Population and Sensor Drift 419 9.2.6 Example Application Study 421 9.2.7 Further Developments 422 9.2.8 Summary 423 9.3 Comparing Classifier Performance 424 9.3.1 Which Technique is Best? 424 9.3.2 Statistical Tests 425 9.3.3 Comparing Rules When Misclassification Costs are Uncertain 426 9.3.4 Example Application Study 428 9.3.5 Further Developments 429 9.3.6 Summary 429 9.4 Application Studies 429 9.5 Summary and Discussion 430 9.6 Recommendations 430 9.7 Notes and References 430 Exercises 431 10 Feature Selection and Extraction 433 10.1 Introduction 433 10.2 Feature Selection 435 10.2.1 Introduction 435 10.2.2 Characterisation of Feature Selection Approaches 439 10.2.3 Evaluation Measures 440 10.2.4 Search Algorithms for Feature Subset Selection 449 10.2.5 Complete Search – Branch and Bound 450 10.2.6 Sequential Search 454 10.2.7 Random Search 458 10.2.8 Markov Blanket 459 10.2.9 Stability of Feature Selection 460 10.2.10 Example Application Study 462 10.2.11 Further Developments 462 10.2.12 Summary 463 10.3 Linear Feature Extraction 463 10.3.1 Principal Components Analysis 464 10.3.2 Karhunen–Lo`eve Transformation 475 10.3.3 Example Application Study 481 10.3.4 Further Developments 482 10.3.5 Summary 483 10.4 Multidimensional Scaling 484 10.4.1 Classical Scaling 484 10.4.2 Metric MDS 486 10.4.3 Ordinal Scaling 487 10.4.4 Algorithms 490 10.4.5 MDS for Feature Extraction 491 10.4.6 Example Application Study 492 10.4.7 Further Developments 493 10.4.8 Summary 493 10.5 Application Studies 493 10.6 Summary and Discussion 495 10.7 Recommendations 495 10.8 Notes and References 496 Exercises 497 11 Clustering 501 11.1 Introduction 501 11.2 Hierarchical Methods 502 11.2.1 Single-Link Method 503 11.2.2 Complete-Link Method 506 11.2.3 Sum-of-Squares Method 507 11.2.4 General Agglomerative Algorithm 508 11.2.5 Properties of a Hierarchical Classification 508 11.2.6 Example Application Study 509 11.2.7 Summary 509 11.3 Quick Partitions 510 11.4 Mixture Models 511 11.4.1 Model Description 511 11.4.2 Example Application Study 512 11.5 Sum-of-Squares Methods 513 11.5.1 Clustering Criteria 514 11.5.2 Clustering Algorithms 515 11.5.3 Vector Quantisation 520 11.5.4 Example Application Study 530 11.5.5 Further Developments 530 11.5.6 Summary 531 11.6 Spectral Clustering 531 11.6.1 Elementary Graph Theory 531 11.6.2 Similarity Matrices 534 11.6.3 Application to Clustering 534 11.6.4 Spectral Clustering Algorithm 535 11.6.5 Forms of Graph Laplacian 535 11.6.6 Example Application Study 536 11.6.7 Further Developments 538 11.6.8 Summary 538 11.7 Cluster Validity 538 11.7.1 Introduction 538 11.7.2 Statistical Tests 539 11.7.3 Absence of Class Structure 540 11.7.4 Validity of Individual Clusters 541 11.7.5 Hierarchical Clustering 542 11.7.6 Validation of Individual Clusterings 542 11.7.7 Partitions 543 11.7.8 Relative Criteria 543 11.7.9 Choosing the Number of Clusters 545 11.8 Application Studies 546 11.9 Summary and Discussion 549 11.10 Recommendations 551 11.11 Notes and References 552 Exercises 553 12 Complex Networks 555 12.1 Introduction 555 12.1.1 Characteristics 557 12.1.2 Properties 557 12.1.3 Questions to Address 559 12.1.4 Descriptive Features 560 12.1.5 Outline 560 12.2 Mathematics of Networks 561 12.2.1 Graph Matrices 561 12.2.2 Connectivity 562 12.2.3 Distance Measures 562 12.2.4 Weighted Networks 563 12.2.5 Centrality Measures 563 12.2.6 Random Graphs 564 12.3 Community Detection 565 12.3.1 Clustering Methods 565 12.3.2 Girvan–Newman Algorithm 568 12.3.3 Modularity Approaches 570 12.3.4 Local Modularity 571 12.3.5 Clique Percolation 573 12.3.6 Example Application Study 574 12.3.7 Further Developments 575 12.3.8 Summary 575 12.4 Link Prediction 575 12.4.1 Approaches to Link Prediction 576 12.4.2 Example Application Study 578 12.4.3 Further Developments 578 12.5 Application Studies 579 12.6 Summary and Discussion 579 12.7 Recommendations 580 12.8 Notes and References 580 Exercises 580 13 Additional Topics 581 13.1 Model Selection 581 13.1.1 Separate Training and Test Sets 582 13.1.2 Cross-Validation 582 13.1.3 The Bayesian Viewpoint 583 13.1.4 Akaike’s Information Criterion 583 13.1.5 Minimum Description Length 584 13.2 Missing Data 585 13.3 Outlier Detection and Robust Procedures 586 13.4 Mixed Continuous and Discrete Variables 587 13.5 Structural Risk Minimisation and the Vapnik–Chervonenkis Dimension 588 13.5.1 Bounds on the Expected Risk 588 13.5.2 The VC Dimension 589 References 591 Index 637

Read more less

Book SynopsisThe latest tools and techniques for pricing and risk management This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications.Table of ContentsPreface ix 1 Correlation Risk in Finance 1 1.1 Correlation Risk in Pricing and Risk Management 1 1.2 Implied vs Realized Correlation 3 1.3 Bottom-up vs Top-down Models 4 1.4 Copula Functions 4 1.5 Spatial and Temporal Dependence 5 1.6 Long-range Dependence 5 1.7 Multivariate GARCH Models 7 1.8 Copulas and Convolution 8 2 Copula Functions: The State of the Art 11 2.1 Copula Functions: The Basic Recipe 11 2.2 Market Co-movements 14 2.3 Delta Hedging Multivariate Digital Products 16 2.4 Linear Correlation 19 2.5 Rank Correlation 20 2.6 Multivariate Spearman’s Rho 22 2.7 Survival Copulas and Radial Symmetry 23 2.8 Copula Volume and Survival Copulas 24 2.9 Tail Dependence 27 2.10 Long/Short Correlation 27 2.11 Families of Copulas 29 2.11.1 Elliptical Copulas 29 2.11.2 Archimedean Copulas 31 2.12 Kendall Function 33 2.13 Exchangeability 34 2.14 Hierarchical Copulas 35 2.15 Conditional Probability and Factor Copulas 39 2.16 Copula Density and Vine Copulas 42 2.17 Dynamic Copulas 45 2.17.1 Conditional Copulas 45 2.17.2 Pseudo-copulas 46 3 Copula Functions and Asset Price Dynamics 49 3.1 The Dynamics of Speculative Prices 49 3.2 Copulas and Markov Processes: The DNO approach 51 3.2.1 The * and _ Product Operators 52 3.2.2 Product Operators and Markov Processes 55 3.2.3 Self-similar Copulas 58 3.2.4 Simulating Markov Chains with Copulas 62 3.3 Time-changed Brownian Copulas 63 3.3.1 CEV Clock Brownian Copulas 64 3.3.2 VG Clock Brownian Copulas 65 3.4 Copulas and Martingale Processes 66 3.4.1 C-Convolution 67 3.4.2 Markov Processes with Independent Increments 75 3.4.3 Markov Processes with Dependent Increments 78 3.4.4 Extracting Dependent Increments in Markov Processes 81 3.4.5 Martingale Processes 83 3.5 Multivariate Processes 86 3.5.1 Multivariate Markov Processes 86 3.5.2 Granger Causality and the Martingale Condition 88 4 Copula-based Econometrics of Dynamic Processes 91 4.1 Dynamic Copula Quantile Regressions 91 4.2 Copula-based Markov Processes: Non-linear Quantile Autoregression 93 4.3 Copula-based Markov Processes: Semi-parametric Estimation 99 4.4 Copula-based Markov Processes: Non-parametric Estimation 108 4.5 Copula-based Markov Processes: Mixing Properties 110 4.6 Persistence and Long Memory 113 4.7 C-convolution-based Markov Processes: The Likelihood Function 116 5 Multivariate Equity Products 121 5.1 Multivariate Equity Products 121 5.1.1 European Multivariate Equity Derivatives 122 5.1.2 Path-dependent Equity Derivatives 125 5.2 Recursions of Running Maxima and Minima 126 5.3 The Memory Feature 130 5.4 Risk-neutral Pricing Restrictions 132 5.5 Time-changed Brownian Copulas 133 5.6 Variance Swaps 135 5.7 Semi-parametric Pricing of Path-dependent Derivatives 136 5.8 The Multivariate Pricing Setting 137 5.9 H-Condition and Granger Causality 137 5.10 Multivariate Pricing Recursion 138 5.11 Hedging Multivariate Equity Derivatives 141 5.12 Correlation Swaps 144 5.13 The Term Structure of Multivariate Equity Derivatives 147 5.13.1 Altiplanos 148 5.13.2 Everest 150 5.13.3 Spread Options 150 6 Multivariate Credit Products 153 6.1 Credit Transfer Finance 153 6.1.1 Univariate Credit Transfer Products 154 6.1.2 Multivariate Credit Transfer Products 155 6.2 Credit Information: Equity vs CDS 158 6.3 Structural Models 160 6.3.1 Univariate Model: Credit Risk as a Put Option 160 6.3.2 Multivariate Model: Gaussian Copula 161 6.3.3 Large Portfolio Model: Vasicek Formula 163 6.4 Intensity-based Models 164 6.4.1 Univariate Model: Poisson and Cox Processes 165 6.4.2 Multivariate Model: Marshall–Olkin Copula 165 6.4.3 Homogeneous Model: Cuadras Aug´e Copula 167 6.5 Frailty Models 170 6.5.1 Multivariate Model: Archimedean Copulas 170 6.5.2 Large Portfolio Model: Sch¨onbucher Formula 171 6.6 Granularity Adjustment 171 6.7 Credit Portfolio Analysis 172 6.7.1 Semi-unsupervised Cluster Analysis: K-means 172 6.7.2 Unsupervised Cluster Analysis: Kohonen Self-organizing Maps 174 6.7.3 (Semi-)unsupervised Cluster Analysis: Hierarchical Correlation Model 175 6.8 Dynamic Analysis of Credit Risk Portfolios 176 7 Risk Capital Management 181 7.1 A Review of Value-at-Risk and Other Measures 181 7.2 Capital Aggregation and Allocation 185 7.2.1 Aggregation: C-Convolution 187 7.2.2 Allocation: Level Curves 189 7.2.3 Allocation with Constraints 191 7.3 Risk Measurement of Managed Portfolios 193 7.3.1 Henriksson–Merton Model 195 7.3.2 Semi-parametric Analysis of Managed Funds 200 7.3.3 Market-neutral Investments 201 7.4 Temporal Aggregation of Risk Measures 202 7.4.1 The Square-root Formula 203 7.4.2 Temporal Aggregation by C-convolution 203 8 Frontier Issues 207 8.1 Levy Copulas 207 8.2 Pareto Copulas 210 8.3 Semi-martingale Copulas 212 A Elements of Probability 215 A.1 Elements of Measure Theory 215 A.2 Integration 216 A.2.1 Expected Values and Moments 217 A.3 The Moment-generating Function or Laplace Transform 218 A.4 The Characteristic Function 219 A.5 Relevant Probability Distributions 219 A.6 Random Vectors and Multivariate Distributions 224 A.6.1 The Multivariate Normal Distribution 225 A.7 Infinite Divisibility 226 A.8 Convergence of Sequences of Random Variables 228 A.81 The Strong Law of Large Numbers 229 A.9 The Radon–Nikodym Derivative 229 A.10 Conditional Expectation 229 B Elements of Stochastic Processes Theory 231 B.1 Stochastic Processes 231 B.1.1 Filtrations 231 B.1.2 Stopping Times 232 B.2 Martingales 233 B.3 Markov Processes 234 B.4 L´evy Processes 237 B.4.1 Subordinators 240 B.5 Semi-martingales 240 References 245 Extra Reading 251 Index 259

Read more less

Book SynopsisDiscusses analytical issues and intricate financial instruments in a way that it is accessible to postgraduate students with or without a previous background in probability theory and finance. This title covers an overview of MATLAB and the various components that will be used alongside it throughout the textbook.Trade Review“The book can be warmly recommended to readers who wish to learn the main methods of quantitative finance without delving into its mathematical foundations.” (Zentralblatt MATH, 1 December 2012) Table of ContentsPreface xi 1 An Introduction to Probability Theory 1 1.1 The Notion of a Set and a Sample Space 1 1.2 Sigma Algebras or Field 2 1.3 Probability Measure and Probability Space 2 1.4 Measurable Mapping 3 1.5 Cumulative Distribution Functions 4 1.6 Convergence in Distribution 5 1.7 Random Variables 5 1.8 Discrete Random Variables 6 1.9 Example of Discrete Random Variables: The Binomial Distribution 6 1.10 Hypergeometric Distribution 7 1.11 Poisson Distribution 8 1.12 Continuous Random Variables 9 1.13 Uniform Distribution 9 1.14 The Normal Distribution 9 1.15 Change of Variable 11 1.16 Exponential Distribution 12 1.17 Gamma Distribution 12 1.18 Measurable Function 13 1.19 Cumulative Distribution Function and Probability Density Function 13 1.20 Joint, Conditional and Marginal Distributions 17 1.21 Expected Values of Random Variables and Moments of a Distribution 19 2 Stochastic Processes 25 2.1 Stochastic Processes 25 2.2 Martingales Processes 26 2.3 Brownian Motions 29 2.4 Brownian Motion and the Reflection Principle 32 2.5 Geometric Brownian Motions 35 3 Ito Calculus and Ito Integral 37 3.1 Total Variation and Quadratic Variation of Differentiable Functions 37 3.2 Quadratic Variation of Brownian Motions 39 3.3 The Construction of the Ito Integral 40 3.4 Properties of the Ito Integral 41 3.5 The General Ito Stochastic Integral 42 3.6 Properties of the General Ito Integral 43 3.7 Construction of the Ito Integral with Respect to Semi-Martingale Integrators 44 3.8 Quadratic Variation of a General Bounded Martingale 46 4 The Black and Scholes Economy 55 4.1 Introduction 55 4.2 Trading Strategies and Martingale Processes 55 4.3 The Fundamental Theorem of Asset Pricing 56 4.4 Martingale Measures 58 4.5 Girsanov Theorem 59 4.6 Risk-Neutral Measures 62 5 The Black and Scholes Model 67 5.1 Introduction 67 5.2 The Black and Scholes Model 67 5.3 The Black and Scholes Formula 68 5.4 Black and Scholes in Practice 70 5.5 The Feynman–Kac Formula 71 6 Monte Carlo Methods 79 6.1 Introduction 79 6.2 The Data Generating Process (DGP) and the Model 79 6.3 Pricing European Options 80 6.4 Variance Reduction Techniques 81 7 Monte Carlo Methods and American Options 91 7.1 Introduction 91 7.2 Pricing American Options 91 7.3 Dynamic Programming Approach and American Option Pricing 92 7.4 The Longstaff and Schwartz Least Squares Method 93 7.5 The Glasserman and Yu Regression Later Method 95 7.6 Upper and Lower Bounds and American Options 96 8 American Option Pricing: The Dual Approach 101 8.1 Introduction 101 8.2 A General Framework for American Option Pricing 101 8.3 A Simple Approach to Designing Optimal Martingales 104 8.4 Optimal Martingales and American Option Pricing 104 8.5 A Simple Algorithm for American Option Pricing 105 8.6 Empirical Results 106 8.7 Computing Upper Bounds 107 8.8 Empirical Results 109 9 Estimation of Greeks using Monte Carlo Methods 113 9.1 Finite Difference Approximations 113 9.2 Pathwise Derivatives Estimation 114 9.3 Likelihood Ratio Method 116 9.4 Discussion 118 10 Exotic Options 121 10.1 Introduction 121 10.2 Digital Options 121 10.3 Asian Options 122 10.4 Forward Start Options 123 10.5 Barrier Options 123 10.5.1 Hedging Barrier Options 125 11 Pricing and Hedging Exotic Options 129 11.1 Introduction 129 11.2 Monte Carlo Simulations and Asian Options 129 11.3 Simulation of Greeks for Exotic Options 130 11.4 Monte Carlo Simulations and Forward Start Options 131 11.5 Simulation of the Greeks for Exotic Options 132 11.6 Monte Carlo Simulations and Barrier Options 132 12 Stochastic Volatility Models 137 12.1 Introduction 137 12.2 The Model 137 12.3 Square Root Diffusion Process 138 12.4 The Heston Stochastic Volatility Model (HSVM) 139 12.5 Processes with Jumps 143 12.6 Application of the Euler Method to Solve SDEs 143 12.7 Exact Simulation Under SV 144 12.8 Exact Simulation of Greeks Under SV 146 13 Implied Volatility Models 151 13.1 Introduction 151 13.2 Modelling Implied Volatility 152 13.3 Examples 153 14 Local Volatility Models 157 14.1 An Overview 157 14.2 The Model 159 14.3 Numerical Methods 161 15 An Introduction to Interest Rate Modelling 167 15.1 A General Framework 167 15.2 Affine Models (AMs) 169 15.3 The Vasicek Model 171 15.4 The Cox, Ingersoll and Ross (CIR) Model 173 15.5 The Hull and White (HW) Model 174 15.6 The Black Formula and Bond Options 175 16 Interest Rate Modelling 177 16.1 Some Preliminary Definitions 177 16.2 Interest Rate Caplets and Floorlets 178 16.3 Forward Rates and Numeraire 180 16.4 Libor Futures Contracts 181 16.5 Martingale Measure 183 17 Binomial and Finite Difference Methods 185 17.1 The Binomial Model 185 17.2 Expected Value and Variance in the Black and Scholes and Binomial Models 186 17.3 The Cox–Ross–Rubinstein Model 187 17.4 Finite Difference Methods 188 Appendix 1 An Introduction to MATLAB 191 A1.1 What is MATLAB? 191 A1.2 Starting MATLAB 191 A1.3 Main Operations in MATLAB 192 A1.4 Vectors and Matrices 192 A1.5 Basic Matrix Operations 194 A1.6 Linear Algebra 195 A1.7 Basics of Polynomial Evaluations 196 A1.8 Graphing in MATLAB 196 A1.9 Several Graphs on One Plot 197 A1.10 Programming in MATLAB: Basic Loops 199 A1.11 M-File Functions 200 A1.12 MATLAB Applications in Risk Management 200 A1.13 MATLAB Programming: Application in Financial Economics 202 Appendix 2 Mortgage Backed Securities 205 A2.1 Introduction 205 A2.2 The Mortgage Industry 206 A2.3 The Mortgage Backed Security (MBS) Model 207 A2.4 The Term Structure Model 208 A2.5 Preliminary Numerical Example 210 A2.6 Dynamic Option Adjusted Spread 210 A2.7 Numerical Example 212 A2.8 Practical Numerical Examples 213 A2.9 Empirical Results 214 A2.10 The Pre-Payment Model 215 Appendix 3 Value at Risk 217 A3.1 Introduction 217 A3.2 Value at Risk (VaR) 217 A3.3 The Main Parameters of a VaR 218 A3.4 VaR Methodology 219 A3.5 Empirical Applications 222 A3.6 Fat Tails and VaR 224 Bibliography 227 References 229 Index 233

Read more less

Book SynopsisThe risk discipline is young and there are a number of ideas, perspectives and conceptions of risk out there. A number of such common conceptions of risk are examined in the book, related to the risk concept, risk assessments, uncertainty analyses, risk perception, the precautionary principle, risk management and decision making under uncertainty.Trade Review"Therefore it is enjoyably readable by a wide audience, by virtue of the efficacy of a simple - even if accurate and rigorous - treatment of conceptually advanced issues." (Zentralblatt MATH, 2011) Table of ContentsPreface. Acknowledgements. 1 Risk is Equal to the Expected Value. 2 Risk is a Probability or Probability Distribution. 3 Risk Equals a Probability Distribution Quantile (Value-at-Risk). 4 Risk Equals Uncertainty. 5 Risk is Equal to an Event. 6 Risk Equals Expected Disutility. 7 Risk is Restricted to the Case of Objective Probabilities. 8 Risk is the Same as Risk Perception. 9 Risk Relates to Negative Consequences Only. 10 Risk is Determined by the Historical Data. 11 Risk Assessments Produce an Objective Risk Picture. 12 There are Large Inherent Uncertainties in Risk Analyses. 13 Model Uncertainty Should be Quantified. 14 It is Meaningful and Useful to Distinguish between Stochastic and Epistemic Uncertainties. 15 Bayesian Analysis is Based on the Use of Probability Models and Bayesian Updating. 16 Sensitivity Analysis is a Type of Uncertainty Analysis. 17 The Main Objective of Risk Management is Risk Reduction. 18 Decision-Making Under Uncertainty Should be Based on Science (Analysis). 19 The Precautionary Principle and Risk Management Cannot be Meaningfully Integrated. 20 Conclusions. Index.

Read more less

Book SynopsisThe Encyclopaedic Companion to Medical Statistics, contains a readable accounts of almost 400 statistical topics central to current medical research. Each entry has been written by an individual chosen for both their expertise in the field and their ability to communicate statistical concepts successfully to medical researchers.Trade Review“I would happily recommend this book to all medical libraries … Public libraries should try to counteract this, in the interests of public health and safety, and might find this book to be a useful back-up tool for the purpose, though it is really designed for a more specialised readership.” (Reference Reviews Journal, 2011) "About 400 entries, updated and revised for this edition, are provided, including 30 new ones, ranging from active control equivalence studies to multiple linear regression to z-score." (Booknews, 1 April 2011) Table of ContentsForeword. Preface to the Second Edition. Preface. Biographical Information on the Editors. List of Contributors. Abbreviations and Acronyms. Encyclopaedic Companion to Medical Statistics A–Z.

Read more less

Book SynopsisWe live in a world of measurements. Measurements, be they of length, speed, weight, temperature, intelligence, income, endurance, greed, gross domestic product, quality of life, unemployment or skill at a job, are all numerical manifestations of the extent of some underlying attribute.Table of Contents1. Introduction. 2. The nature of measurement. 3. The process of measurement. 4. Accuracy of measurement. 5. Measurement in psychology. 6. Measurement in medicine. 7. Measurement in the physical sciences. 8. Measurement in economics and the social science. 9. Measurement in other areas.

Read more less

Book SynopsisThe complete guide to statistical modelling with GENSTAT Focusing on solving practical problems and using real datasets collected during research of various sorts, Statistical Modelling Using GENSTAT emphasizes developing and understanding statistical tools. Throughout the text, these statistical tools are applied to answer the very questions the original researchers sought to answer. GENSTAT, the powerful statistical software, is introduced early in the book and practice problems are carried out using the software, in the process helping students to understand the application of statistical methods to real-world data.

Read more less

Book SynopsisTable of ContentsPreface to Second Edition xiPreface to First Edition xiiGlossary of Abbreviations xiii1 The Bayesian Method 12 Inference and Decisions 223 General Principles and Theory 614 Subjective Probability 945 Non-subjective Theories 1166 Subjective Prior Distributions 1357 Model Comparison 1638 Robustness and Model Criticism 2009 Computation 23310 Markov Chain Monte Carlo 26211 The Linear Model 30512 Discrete Data Models 24013 Nonparametric Models 37414 Other Standard Models 40215 Short Case Studies 423References 441Index of Examples in Text 471Index 473

Read more less

Book Synopsis

Read more less

Book SynopsisThe use of census microdata - which is the extremely detailed information collected through the 'ultracensusing' of a small number of households at the time of the last census - forms an important part of many research activities in subject areas as diverse as geography, demography, sociology, economics, politics and statistics. It is vital that the researchers know how the data was collected, which statistical techniques are useful and why, and how to model the data in order to draw inferences about the whole population. Analyzing Census Microdata meets and is relevant to a wide and international market. Analyzing Census Microdata - written by some of the leading authorities in the field - provides the first guide to the analysis of census microdata, with the basic statistical summary techniques presented in a clear and concise way. It contains a large number of up-to-date examples, drawing on data from the USA, the UK and a wide variety of other international contexts. I concludes wi

Read more less

Book SynopsisThe development of statistical theory in the past fifty years is faithfully reflected in the history of the late Sir Maurice Kendall's volumes The Advanced Theory of Statistics. The Advanced Theory began life as a two volume work (Volume 1, 1943; Volume 2, 1946) and grew steadily, as a single authored work until the late fifties.Table of ContentsPreface to the Sixth Edition. List of Examples. Glossary of Abbreviations. 17. Estimation and Sufficiency. 18. Estimation: Maximum Likelihood and Other Methods. 19. Interval Estimation. 20. Tests of Hypotheses: Simple Null Hypotheses. 21. Tests of Hypotheses: Composite Hypotheses. 22. Likelihood Ratio Tests and Test Efficiency. 23. Invariance and Equivariance. 24. Sequential Methods. 25. Tests of Fit. 26. Comparative Statistical Inference. 27. Statistical Relationship: Linear Regression and Correlation. 28. Partial and Multiple Correlation. 29. The General Linear Model. 30. Fixed Effects Analysis of Variance. 31. Other Analysis of Variance Models. 32. Analysis and Diagnostics for the Linear Model. Appendix Tables. References. Index of Examples in Text. Author Index. Subject Index.

Read more less

Book SynopsisTable of ContentsIntroduction. The analysis of segregation and population frequency. The analysis of genetic linkage. The analysis of allelic associations. The analysis of continuous and quasi-continuous channel. References. Index

Read more less

Book SynopsisOffering a comprehensive overview of statistics in market research Statistics in Market Research details the various ways that statistical analyses can be applied to real-world questions that arise in the field of marketing. This overview provides those in the field with the basic information they'll need to solve marketing problems and can be applied to promotion, branding, advertising, segmenting, sales forecasting, and a host of other marketing issues. The book covers an overview of the use of statistics in marketing and then moves on to specific topics, including interdependence techniques, dependence techniques, and more advanced statistics techniques. Taken together, the reader is provided with an excellent learning tool and resource for future problem solving.Table of Contents1. Statistics and marketing. 2. Factor analysis. 3. Correspondence analysis. 4. Cluster analysis. 5. Multiple regression analysis. 6. Discriminant analysis. 7. Conjoint analysis. 8. Path analysis and structural equation models. 9. Data mining.

Read more less

Book SynopsisThis bookisa completely revised and updated version of this invaluable text which allows science students to extend necessary skills and techniques, with the topics being developed through examples in science which are easily understood by students from a range of disciplines. The introductory approach eases students into the subject, progressing to cover topics relevant to first and second year study and support data analysis for final year projects. The revision of the material in the book has been matched, on the accompanying website, with the extensive use of video, providing worked answers to over 200 questions in the book plus additional tutorial support. The second edition has also improved the learning approach for key topic areas to make it even more accessible and user-friendly, making it a perfect resource for students of all abilities. The expanding website provides a wide range of support material, providing a study environment within which students can develop their iTable of ContentsPreface xi On-line Learning Support xv 1 Mathematics and Statistics in Science 1 1.1 Data and Information 2 1.2 Experimental Variation and Uncertainty 2 1.3 Mathematical Models in Science 4 2 Scientific Data 7 2.1 Scientific Numbers 8 2.2 Scientific Quantities 15 2.3 Chemical Quantities 20 2.4 Angular Measurements 31 3 Equations in Science 41 3.1 Basic Techniques 41 3.2 Rearranging Simple Equations 53 3.3 Symbols 63 3.4 Further Equations 68 3.5 Quadratic and Simultaneous Equations 78 4 Linear Relationships 87 4.1 Straight Line Graph 89 4.2 Linear Regression 99 4.3 Linearization 107 5 Logarithmic and Exponential Functions 113 5.1 Mathematics of e, ln and log 114 5.2 Exponential Growth and Decay 128 6 Rates of Change 145 6.1 Rate of Change 145 6.2 Differentiation 152 7 Statistics for Science 161 7.1 Analysing Replicate Data 162 7.2 Describing and Estimating 168 7.3 Frequency Statistics 176 7.4 Probability 190 7.5 Factorials, Permutations and Combinations 203 8 Distributions and Uncertainty 211 8.1 Normal Distribution 212 8.2 Uncertainties in Measurement 217 8.3 Presenting Uncertainty 224 8.4 Binomial and Poisson Distributions 230 9 Scientific Investigation 243 9.1 Scientific Systems 243 9.2 The ‘Scientific Method’ 245 9.3 Decision Making with Statistics 246 9.4 Hypothesis Testing 250 9.5 Selecting Analyses and Tests 256 10 t-tests and F-tests 261 10.1 One-sample t-tests 262 10.2 Two-sample t-tests 267 10.3 Paired t-tests 272 10.4 F-tests 274 11 ANOVA – Analysis of Variance 279 11.1 One-way ANOVA 279 11.2 Two-way ANOVA 286 11.3 Two-way ANOVA with Replication 290 11.4 ANOVA Post Hoc Testing 296 12 Non-parametric Tests for Medians 299 12.1 One-sample Wilcoxon Test 301 12.2 Two-sample Mann–Whitney U-test 305 12.3 Paired Wilcoxon Test 308 12.4 Kruskal–Wallis and Friedman Tests 311 13 Correlation and Regression 315 13.1 Linear Correlation 316 13.2 Statistics of Correlation and Regression 320 13.3 Uncertainty in Linear Calibration 324 14 Frequency and Proportion 331 14.1 Chi-squared Contingency Table 332 14.2 Goodness of Fit 340 14.3 Tests for Proportion 343 15 Experimental Design 349 15.1 Principal Techniques 349 15.2 Planning a Research Project 357 Appendix I: Microsoft Excel 359 Appendix II: Cumulative z-areas for Standard Normal Distribution 363 Appendix III: Critical Values: t-statistic and Chi-squared, χ2 365 Appendix IV: Critical F-values at 0.05 (95 %) Significance 367 Appendix V: Critical Values at 0.05 (95 %) Significance for: Pearson’s Correlation Coefficient, r, Spearman’s Rank Correlation Coefficient, rS , and Wilcoxon Lower Limit, WL 369 Appendix VI: Mann–Whitney Lower Limit, UL, at 0.05 (95 %) Significance 371 Short Answers to ‘Q’ Questions 373 Index 379

Read more less

Book SynopsisIntroduces optimal experimental design, in an accessible style that means very minimal mathematical background knowledge is needed. Provides guidelines for practitioners to increase the efficiency of their designs, demonstrating how optimal designs can reduce a study's costs and sample sizes.Table of ContentsPreface xi Acknowledgements xiii 1 Introduction to designs 1 1.1 Introduction 1 1.2 Stages of the research process 4 1.2.1 Choice of a ‘good’ design 5 1.3 Research design 6 1.3.1 Choice of independent variables and levels 6 1.3.2 Units of analysis 6 1.3.3 Variables 7 1.3.4 Replication 8 1.4 Types of research designs 8 1.5 Requirements for a ‘good’ design 9 1.5.1 Statistical conclusion validity 10 1.5.2 Internal validity 12 1.5.3 Control of (unwanted) variation 13 1.6 Ethical aspects of design choice 16 1.7 Exact versus approximate designs 17 1.8 Examples 19 1.8.1 Radiation dosage example 19 1.8.2 Designs for the Poggendorff and Ponzo illusion experiments 20 1.8.3 Uncertainty about best fitting regression models 22 1.8.4 Designs for a priori contrasts among composite faces 23 1.8.5 Designs for calibration of item parameters in item response theory models 24 1.9 Summary 26 2 Designs for simple linear regression 27 2.1 Design problem for a linear model 27 2.1.1 The design 28 2.1.2 The linear regression model 31 2.1.3 Estimation of parameters and efficiency 32 2.2 Designs for radiation-dosage example 35 2.3 Relative efficiency and sample size 36 2.4 Simultaneous inference 37 2.5 Optimality criteria 39 2.5.1 D-optimality criterion 40 2.5.2 A-optimality criterion 41 2.5.3 G-optimality criterion 41 2.5.4 E-optimality criterion 43 2.5.5 Number of distinct design points 43 2.6 Relative efficiency 44 2.7 Matrix formulation of designs for linear regression 44 2.8 Summary 49 3 Designs for multiple linear regression analysis 51 3.1 Design problem for multiple linear regression 51 3.1.1 The design 52 3.1.2 The multiple linear regression model 54 3.1.3 Estimation of parameters and efficiency 54 3.2 Designs for vocabulary-growth study 56 3.3 Relative efficiency and sample size 60 3.4 Simultaneous inference 61 3.5 Optimality criteria for a subset of parameters 62 3.6 Relative efficiency 64 3.7 Designs for polynomial regression model 65 3.7.1 Exact D-optimal designs for a quadratic regression model 69 3.7.2 Scale dependency of A- and E-optimality criteria 71 3.8 The Poggendorff and Ponzo illusion study 71 3.9 Uncertainty about best fitting regression models 76 3.10 Matrix notation of designs for multiple regression models 79 3.10.1 Design for regression models with two independent variables 80 3.10.2 Design for regression models with two non-additive independent variables 82 3.11 Summary 85 4 Designs for analysis of variance models 87 4.1 A typical design problem for an analysis of variance model 87 4.1.1 The design 89 4.1.2 The analysis of variance model 90 4.1.3 Formulation of an ANOVA model as a regression model 91 4.2 Estimation of parameters and efficiency 95 4.2.1 Measures of uncertainty 96 4.3 Simultaneous inference and optimality criteria 97 4.4 Designs for groups under stress study 98 4.4.1 A priori planned unequal sample sizes 99 4.4.2 Not planned unequal sample sizes 100 4.5 Specific hypotheses and contrasts 101 4.5.1 Loss of efficiency and power 103 4.6 Designs for the composite faces study 106 4.7 Balanced designs versus unbalanced designs 109 4.8 Matrix notation for Groups under Stress study 109 4.9 Summary 111 5 Designs for logistic regression models 113 5.1 Design problem for logistic regression 113 5.2 The design 114 5.3 The logistic regression model 115 5.3.1 Design for a single dichotomous independent variable 116 5.3.2 Design for multiple qualitative independent variables 122 5.3.3 Design for a single quantitative independent variable 125 5.3.4 Design for two independent quantitative variables 130 5.4 Approaches to deal with local optimality 133 5.5 Designs for calibration of item parameters in item response theory models 134 5.6 Matrix formulation of designs for logistic regression 137 5.6.1 Hours of practice experiment 138 5.6.2 Problem solving study 140 5.7 Summary 141 6 Designs for multilevel models 143 6.1 Design problem for multilevel models 143 6.1.1 The design 144 6.1.2 Validity considerations 146 6.2 The multilevel regression model 147 6.2.1 Cluster randomization of treatment 147 6.2.2 Subject randomization of treatment 149 6.3 Cluster versus subject randomization 151 6.4 Cost function 153 6.5 Example: Nursing home study 155 6.5.1 Cluster randomization 157 6.5.2 Subject randomization 159 6.6 Optimal design and power 160 6.6.1 Power for cluster randomized design 162 6.6.2 Power for multi-center design 164 6.6.3 Increase of efficiency and power by including covariates 165 6.6.4 Unequal sample sizes 165 6.7 Design effect in multilevel surveys 166 6.7.1 Values of intra-class correlation ρ 168 6.7.2 Cluster randomized sampling versus simple random sampling 168 6.8 Matrix formulation of the multilevel model 169 6.8.1 Cluster randomization of treatment 170 6.8.2 Subject randomization of treatment 172 6.9 Summary 174 7 Longitudinal designs for repeated measurement models 175 7.1 Design problem for repeated measurements 175 7.2 The design 179 7.3 Analysis techniques for repeated measures 180 7.4 The linear mixed effects model for repeated measurement data 181 7.4.1 Random intercept model 182 7.4.2 Random intercept and slope model 183 7.5 Variance–covariance structures 184 7.5.1 Compound symmetry structure 184 7.5.2 Auto-correlation structure 185 7.6 Estimation of parameters and efficiency 187 7.6.1 Small sample behaviour of estimators 188 7.7 Bone mineral density example 189 7.7.1 Improvement of the longitudinal design 194 7.8 Cost function 196 7.9 D-optimal designs for linear mixed effects models with autocorrelated errors 200 7.10 Miscellanea 207 7.10.1 Homoscedasticity 207 7.10.2 Uninformative dropout 208 7.11 Matrix formulation of the linear mixed effects model 208 7.12 Summary 211 8 Two-treatment crossover designs 213 8.1 Design problem for crossover studies 213 8.2 The design 216 8.3 Confounding treatment effects with nuisance effects 218 8.4 The linear model for crossover designs 221 8.5 Estimation of parameters and efficiency 223 8.6 Cost and efficiency of the crossover design 223 8.6.1 Cost function 226 8.7 Optimal crossover designs for two treatments 229 8.7.1 Some further observations 231 8.8 Matrix formulation of the mixed model for crossover designs 232 8.9 Summary 235 9 Alternative optimal designs for linear models 237 9.1 Introduction 237 9.2 Information matrix 238 9.3 DA- or Ds-optimal designs 239 9.4 Extrapolation optimal design 241 9.5 L-optimal designs 242 9.6 Bayesian optimal designs 244 9.7 Minimax optimal design 247 9.8 Multiple-objective optimal designs 250 9.8.1 Constrained optimal design 251 9.8.2 Compound optimal design 253 9.9 Summary 255 10 Optimal designs for nonlinear models 257 10.1 Introduction 257 10.2 Linear models versus nonlinear models 258 10.2.1 The Arrhenius equation 258 10.2.2 The compartmental model 259 10.2.3 The Michaelis–Menten model 260 10.2.4 The Emax model 261 10.3 Design issues for nonlinear models 261 10.3.1 Local optimality 262 10.4 Alternative optimal designs with examples 265 10.4.1 DA or Ds-optimal design 265 10.4.2 Extrapolation optimal design 266 10.4.3 Optimal design for estimating percentiles 266 10.5 Bayesian optimal designs 267 10.6 Minimax optimal design 269 10.7 Multiple-objective optimal designs 271 10.8 Optimal design for model discrimination 273 10.9 Summary 275 11 Resources for the construction of optimal designs 277 11.1 Introduction 277 11.2 Sequential construction of optimal designs 278 11.3 Exchange of design points 283 11.3.1 Exchange algorithms 283 11.4 Other algorithms 284 11.5 Optimal design software 285 11.6 A web site for finding optimal designs 286 11.6.1 Optimal designs for the Michaelis–Menten and Emax models 288 11.6.2 Optimal designs for discriminating among toxicological models 290 11.7 Summary 294 References 295 Author Index 313 Subject Index 319

Read more less

Book SynopsisThis book focuses on computational methods for large-scale statistical inverse problems and provides an introduction to statistical Bayesian and frequentist methodologies. Recent research advances for approximation methods are discussed, along with Kalman filtering methods and optimization-based approaches to solving inverse problems.Table of Contents1 Introduction 1.1 Introduction 1.2 Statistical Methods 1.3 Approximation Methods 1.4 Kalman Filtering 1.5 Optimization 2 A Primer of Frequentist and Bayesian Inference in Inverse Problems 2.1 Introduction 2.2 Prior Information and Parameters: What do you know, and what do you want to know? 2.3 Estimators: What can you do with what you measure? 2.4 Performance of estimators: How well can you do? 2.5 Frequentist performance of Bayes estimators for a BNM 2.6 Summary Bibliography 3 Subjective Knowledge or Objective Belief? An Oblique Look to Bayesian Methods 3.1 Introduction 3.2 Belief, information and probability 3.3 Bayes' formula and updating probabilities 3.4 Computed examples involving hypermodels 3.5 Dynamic updating of beliefs 3.6 Discussion Bibliography 4 Bayesian and Geostatistical Approaches to Inverse Problems 4.1 Introduction 4.2 The Bayesian and Frequentist Approaches 4.3 Prior Distribution 4.4 A Geostatistical Approach 4.5 Concluding Bibliography 5 Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference 5.1 Introduction 5.2 Bayesian Model Formulation 5.3 Application: Cosmic Microwave Background 5.4 Discussion Bibliography 6 Bayesian Partition Models for Subsurface Characterization 6.1 Introduction 6.2 Model equations and problem setting 6.3 Approximation of the response surface using the Bayesian Partition Model and two-stage MCMC 6.4 Numerical results 6.5 Conclusions Bibliography 7 Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems 7.1 Introduction 7.2 Reducing the computational cost of solving statistical inverse problems 7.3 General formulation 7.4 Model reduction 7.5 Stochastic spectral methods 7.6 Illustrative example 7.7 Conclusions Bibliography 8 Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation 8.1 Introduction 8.2 Linear Parabolic Equations 8.3 Bayesian Parameter Estimation 8.4 Concluding Remarks Bibliography 9 Calibration and Uncertainty Analysis for Computer Simulations with Multivariate Output 9.1 Introduction 9.2 Gaussian Process Models 9.3 Bayesian Model Calibration 9.4 Case Study: Thermal Simulation of Decomposing Foam 9.5 Conclusions Bibliography 10 Bayesian Calibration of Expensive Multivariate Computer Experiments 10.1 Calibration of computer experiments 10.2 Principal component emulation 10.3 Multivariate calibration 10.4 Summary Bibliography 11 The Ensemble Kalman Filter and Related Filters 11.1 Introduction 11.2 Model Assumptions 11.3 The Traditional Kalman Filter (KF) 11.4 The Ensemble Kalman Filter (EnKF) 11.5 The Randomized Maximum Likelihood Filter (RMLF) 11.6 The Particle Filter (PF) 11.7 Closing Remarks 11.8 Appendix A: Properties of the EnKF Algorithm 11.9 Appendix B: Properties of the RMLF Algorithm Bibliography 12 Using the ensemble Kalman Filter for history matching and uncertainty quantification of complex reservoir models 12.1 Introduction 12.2 Formulation and solution of the inverse problem 12.3 EnKF history matching workflow 12.4 Field Case 12.5 Conclusion Bibliography 13 Optimal Experimental Design for the Large-Scale Nonlinear Ill-posed Problem of Impedance Imaging 13.1 Introduction 13.2 Impedance Tomography 13.3 Optimal Experimental Design - Background 13.4 Optimal Experimental Design for Nonlinear Ill-Posed Problems 13.5 Optimization Framework 13.6 Numerical Results 13.7 Discussion and Conclusions Bibliography 14 Solving Stochastic Inverse Problems: A Sparse Grid Collocation Approach 14.1 Introduction 14.2 Mathematical developments 14.3 Numerical Examples 14.4 Summary Bibliography 15 Uncertainty analysis for seismic inverse problems: two practical examples 15.1 Introduction 15.2 Traveltime inversion for velocity determination. 15.3 Prestack stratigraphic inversion 15.4 Conclusions Bibliography 16 Solution of inverse problems using discrete ODE adjoints 16.1 Introduction 16.2 Runge-Kutta Methods 16.3 Adaptive Steps 16.4 Linear Multistep Methods 16.5 Numerical Results 16.6 Application to Data Assimilation 16.7 Conclusions Bibliography TBD

Read more less

Book SynopsisIn the spatial or spatio-temporal context, specifying the correct covariance function is fundamental to obtain efficient predictions, and to understand the underlying physical process of interest. This book focuses on covariance and variogram functions, their role in prediction, and appropriate choice of these functions in applications. Both recent and more established methods are illustrated to assess many common assumptions on these functions, such as, isotropy, separability, symmetry, and intrinsic correlation. After an extensive introduction to spatial methodology, the book details the effects of common covariance assumptions and addresses methods to assess the appropriateness of such assumptions for various data structures. Key features: An extensive introduction to spatial methodology including a survey of spatial covariance functions and their use in spatial prediction (kriging) is given. Explores methodology for assessing the appropriatenTable of ContentsPreface xi 1 Introduction 1 1.1 Stationarity 4 1.2 The effect of correlation in estimation and prediction 5 1.2.1 Estimation 5 1.2.2 Prediction 12 1.3 Texas tidal data 14 2 Geostatistics 21 2.1 A model for optimal prediction and error assessment 23 2.2 Optimal prediction (kriging) 25 2.2.1 An example: phosphorus prediction 28 2.2.2 An example in the power family of variogram functions 32 2.3 Prediction intervals 34 2.3.1 Predictions and prediction intervals for lognormal observations 35 2.4 Universal kriging 38 2.4.1 Optimal prediction in universal kriging 39 2.5 The intuition behind kriging 40 2.5.1 An example: the kriging weights in the phosphorus data 41 3 Variogram and covariance models and estimation 45 3.1 Empirical estimation of the variogram or covariance function 45 3.1.1 Robust estimation 46 3.1.2 Kernel smoothing 47 3.2 On the necessity of parametric variogram and covariance models 47 3.3 Covariance and variogram models 48 3.3.1 Spectral methods and the Matérn covariance model 51 3.4 Convolution methods and extensions 55 3.4.1 Variogram models where no covariance function exists 56 3.4.2 Jumps at the origin and the nugget effect 56 3.5 Parameter estimation for variogram and covariance models 57 3.5.1 Estimation with a nonconstant mean function 62 3.6 Prediction for the phosphorus data 63 3.7 Nonstationary covariance models 69 4 Spatial models and statistical inference 71 4.1 Estimation in the Gaussian case 74 4.1.1 A data example: model fitting for the wheat yield data 75 4.2 Estimation for binary spatial observations 78 4.2.1 Edge effects 83 4.2.2 Goodness of model fit 84 5 Isotropy 87 5.1 Geometric anisotropy 91 5.2 Other types of anisotropy 92 5.3 Covariance modeling under anisotropy 93 5.4 Detection of anisotropy: the rose plot 94 5.5 Parametric methods to assess isotropy 96 5.6 Nonparametric methods of assessing anisotropy 97 5.6.1 Regularly spaced data case 97 5.6.2 Irregularly spaced data case 101 5.6.3 Choice of spatial lags for assessment of isotropy 104 5.6.4 Test statistics 105 5.6.5 Numerical results 107 5.7 Assessment of isotropy for general sampling designs 111 5.7.1 A stochastic sampling design 111 5.7.2 Covariogram estimation and asymptotic properties 112 5.7.3 Testing for spatial isotropy 113 5.7.4 Numerical results for general spatial designs 115 5.7.5 Effect of bandwidth and block size choice 117 5.8 An assessment of isotropy for the longleaf pine sizes 120 6 Space–time data 123 6.1 Space–time observations 123 6.2 Spatio-temporal stationarity and spatio-temporal prediction 124 6.3 Empirical estimation of the variogram, covariance models, and estimation 125 6.3.1 Space–time symmetry and separability 126 6.4 Spatio-temporal covariance models 127 6.4.1 Nonseparable space–time covariance models 128 6.5 Space–time models 130 6.6 Parametric methods of assessing full symmetry and space–time separability 132 6.7 Nonparametric methods of assessing full symmetry and space–time separability 133 6.7.1 Irish wind data 139 6.7.2 Pacific Ocean wind data 141 6.7.3 Numerical experiments based on the Irish wind data 142 6.7.4 Numerical experiments on the test for separability for data on a grid 144 6.7.5 Taylor’s hypothesis 145 6.8 Nonstationary space–time covariance models 147 7 Spatial point patterns 149 7.1 The Poisson process and spatial randomness 150 7.2 Inhibition models 156 7.3 Clustered models 158 8 Isotropy for spatial point patterns 167 8.1 Some large sample results 169 8.2 A test for isotropy 170 8.3 Practical issues 171 8.4 Numerical results 173 8.4.1 Poisson cluster processes 173 8.4.2 Simple inhibition processes 176 8.5 An application to leukemia data 177 9 Multivariate spatial and spatio-temporal models 181 9.1 Cokriging 183 9.2 An alternative to cokriging 186 9.2.1 Statistical model 187 9.2.2 Model fitting 188 9.2.3 Prediction 191 9.2.4 Validation 192 9.3 Multivariate covariance functions 194 9.3.1 Variogram function or covariance function? 195 9.3.2 Intrinsic correlation, separable models 196 9.3.3 Coregionalization and kernel convolution models 197 9.4 Testing and assessing intrinsic correlation 198 9.4.1 Testing procedures for intrinsic correlation and symmetry 201 9.4.2 Determining the order of a linear model of coregionalization 202 9.4.3 Covariance estimation 204 9.5 Numerical experiments 205 9.5.1 Symmetry 205 9.5.2 Intrinsic correlation 207 9.5.3 Linear model of coregionalization 209 9.6 A data application to pollutants 209 9.7 Discussion 213 10 Resampling for correlated observations 215 10.1 Independent observations 218 10.1.1 U-statistics 218 10.1.2 The jackknife 220 10.1.3 The bootstrap 221 10.2 Other data structures 224 10.3 Model-based bootstrap 225 10.3.1 Regression 225 10.3.2 Time series: autoregressive models 227 10.4 Model-free resampling methods 228 10.4.1 Resampling for stationary dependent observations 230 10.4.2 Block bootstrap 232 10.4.3 Block jackknife 233 10.4.4 A numerical experiment 233 10.5 Spatial resampling 236 10.5.1 Model-based resampling 237 10.5.2 Monte Carlo maximum likelihood 238 10.6 Model-free spatial resampling 240 10.6.1 A spatial numerical experiment 244 10.6.2 Spatial bootstrap 246 10.7 Unequally spaced observations 246 Bibliography 251 Index 263

Read more less

Book SynopsisHow to Be a Quantitative Ecologist is comprised of two equal parts on mathematics and statistics with emphasis on quantitative skills. A major component of this guide is computer implementation techniques, accompanied by computer practicals using the language R.Trade Review“For those looking through R books for something a bit more technical, this book will be an essential accomplice to mastering R.” (British Ecological Society, 1 April 2013) “The book is written in a style that is easy to read and for which one quickly forgets that the examples are essentially mathematical in nature. If you are an ecologist who has shied away from quantitative ecology in the past then this may be the text to convince you that there is much to be learnt from quantitative ecology. I thoroughly recommend this book and trust that you enjoy reading it as much as I did.” (International Statistical Review, 2012) "After a course of one or two semesters using this textbook, he says, students should have the absolute minimum of knowledge about quantitative research that ecologists need, but can provide a foundation for students who want to move further in that direction." (Book News, 1 August 2011) Table of ContentsHow I chose to write this book, and why you might choose to read it. Preface. 0. How to start a meaningful relationship with your computer. Introduction to R. 0.1 What is R? 0.2 Why use R for this book? 0.3 Computing with a scientific package like R. 0.4 Installing and interacting with R. 0.5 Style conventions. 0.6 Valuable R accessories. 0.7 Getting help. 0.8 Basic R usage. 0.9 Importing data from a spreadsheet. 0.10 Storing data in data frames. 0.11 Exporting data from R. 0.12 Quitting R. 1. How to make mathematical statements. Numbers, equations and functions. 1.1 Qualitative and quantitative scales. 1.2 Numbers. 1.3 Symbols. 1.4 Logical operations. 1.5 Algebraic operations. 1.6 Manipulating numbers. 1.7 Manipulating units. 1.8 Manipulating expressions. 1.9 Polynomials. 1.10 Equations. 1.11 First order polynomial equations. 1.12 Proportionality and scaling: a special kind of first order polynomial equation. 1.13 Second and higher order polynomial equations. 1.14 Systems of polynomial equations. 1.15 Inequalities. 1.16 Coordinate systems. 1.17 Complex numbers. 1.18 Relations and functions. 1.19 The graph of a function. 1.20 First order polynomial functions. 1.21 Higher order polynomial functions. 1.22 The relationship between equations and functions. 1.23 Other useful functions. 1.24 Inverse functions. 1.25 Functions of more than one variable. 2. How to describe regular shapes and patterns. Geometry and trigonometry. 2.1 Primitive elements. 2.2 Axioms of Euclidean geometry. 2.3 Propositions. 2.4 Distance between two points. 2.5 Areas and volumes. 2.6 Measuring angles. 2.7 The trigonometric circle. 2.8 Trigonometric functions. 2.9 Polar coordinates. 2.10 Graphs of trigonometric functions. 2.11 Trigonometric identities. 2.12 Inverses of trigonometric functions. 2.13 Trigonometric equations. 2.14 Modifying the basic trigonometric graphs. 2.15 Superimposing trigonometric functions. 2.16 Spectral analysis. 2.17 Fractal geometry. 3. How to change things, one step at a time. Sequences, difference equations and logarithms. 3.1 Sequences. 3.2 Difference equations. 3.3 Higher order difference equations. 3.4 Initial conditions and parameters. 3.5 Solutions of a difference equation. 3.6 Equilibrium solutions. 3.7 Stable and unstable equilibria. 3.8 Investigating stability. 3.9 Chaos. 3.10 Exponential function. 3.11 Logarithmic function. 3.12 Logarithmic equations. 4. How to change things, continuously. Derivatives and their applications. 4.1 Average rate of change. 4.2 Instantaneous rate of change. 4.3 Limits. 4.4 The derivative of a function. 4.5 Differentiating polynomials. 4.6 Differentiating other functions. 4.7 The chain rule. 4.8 Higher order derivatives. 4.9 Derivatives of functions of many variables. 4.10 Optimisation. 4.11 Local stability for difference equations. 4.12 Series expansions. 5. How to work with accumulated change. Integrals and their applications. 5.1 Antiderivatives. 5.2 Indefinite integrals. 5.3 Three analytical methods of integration. 5.4 Summation. 5.5 Area under a curve. 5.6 Definite integrals. 5.7 Some properties of definite integrals. 5.8 Improper integrals. 5.9 Differential equations. 5.10 Solving differential equations. 5.11 Stability analysis for differential equations. 6. How to keep stuff organised in tables. Matrices and their applications. 6.1 Matrices. 6.2 Matrix operations. 6.3 Geometric interpretation of vectors and square matrices. 6.4 Solving systems of equations with matrices. 6.5 Markov chains. 6.6 Eigenvalues and eigenvectors. 6.7 Leslie matrix models. 6.8 Analysis of linear dynamical systems. 6.9 Analysis of nonlinear dynamical systems. 7. How to visualise and summarise data. Descriptive statistics. 7.1 Overview of statistics. 7.2 Statistical variables. 7.3 Populations and samples. 7.4 Single-variable samples. 7.5 Frequency distributions. 7.6 Measures of centrality. 7.7 Measures of spread. 7.8 Skewness and kurtosis. 7.9 Graphical summaries. 7.10 Data sets with more than one variable. 7.11 Association between qualitative variables. 7.12 Association between quantitative variables. 7.13 Joint frequency distributions. 8. How to put a value on uncertainty. Probability. 8.1 Random experiments and event spaces. 8.2 Events. 8.3 Frequentist probability. 8.4 Equally likely events. 8.5 The union of events. 8.6 Conditional probability. 8.7 Independent events. 8.8 Total probability. 8.9 Bayesian probability. 9. How to identify different kinds of randomness. Probability distributions. 9.1 Probability distributions. 9.2 Discrete probability distributions. 9.3 Continuous probability distributions. 9.4 Expectation. 9.5 Named distributions. 9.6 Equally likely events: the uniform distribution. 9.7 Hit or miss: the Bernoulli distribution. 9.8 Count of occurrences in a given number of trials: the binomial distribution. 9.9 Counting different types of occurrences: the multinomial distribution. 9.10 Number of occurrences in a unit of time or space: the Poisson distribution. 9.11 The gentle art of waiting: geometric, negative binomial, exponential and gamma distributions. 9.12 Assigning probabilities to probabilities: the beta and Dirichlet distributions. 9.13 Perfect symmetry: the normal distribution. 9.14 Because it looks right: using probability distributions empirically. 9.15 Mixtures, outliers and the t-distribution. 9.16 Joint, conditional and marginal probability distributions. 9.17 The bivariate normal distribution. 9.18 Sums of random variables: the central limit theorem. 9.19 Products of random variables: the log-normal distribution. 9.20 Modelling residuals: the chi-square distribution. 9.21 Stochastic simulation. 10. How to see the forest from the trees. Estimation and testing. 10.1 Estimators and their properties. 10.2 Normal theory. 10.3 Estimating the population mean. 10.4 Estimating the variance of a normal population. 10.5 Confidence intervals. 10.6 Inference by bootstrapping. 10.7 More general estimation methods. 10.8 Estimation by least squares. 10.9 Estimation by maximum likelihood. 10.10 Bayesian estimation. 10.11 Link between maximum likelihood and Bayesian estimation. 10.12 Hypothesis testing: rationale. 10.13 Tests for the population mean. 10.14 Tests comparing two different means. 10.15 Hypotheses about qualitative data. 10.16 Hypothesis testing debunked. 11. How to separate the signal from the noise. Statistical modelling. 11.1 Comparing the means of several populations. 11.2 Simple linear regression. 11.3 Prediction. 11.4 How good is the best-fit line? 11.5 Multiple linear regression. 11.6 Model selection. 11.7 Generalised linear models. 11.8 Evaluation, diagnostics and model selection for GLMs. 11.9 Modelling dispersion. 11.10 Fitting more complicated models to data: polynomials, interactions, nonlinear regression. 11.11 Letting the data suggest more complicated models: smoothing. 11.12 Partitioning variation: mixed effects models. 12. How to measure similarity. Multivariate methods 12.1 The problem with multivariate data. 12.2 Ordination in general. 12.3 Principal components analysis. 12.4 Clustering in general. 12.5 Agglomerative hierarchical clustering. 12.6 Nonhierarchical clustering: k means analysis. 12.7 Classification in general. 12.8 Logistic regression: two classes. 12.9 Logistic regression: many classes. Further reading. References. Appendix: Formulae. R Index. Index.

Read more less

Book SynopsisHow to Be a Quantitative Ecologist is comprised of two equal parts on mathematics and statistics with emphasis on quantitative skills. A major component of this guide is computer implementation techniques, accompanied by computer practicals using the language R.Trade Review“For those looking through R books for something a bit more technical, this book will be an essential accomplice to mastering R.” (British Ecological Society, 1 April 2013) “The book is written in a style that is easy to read and for which one quickly forgets that the examples are essentially mathematical in nature. If you are an ecologist who has shied away from quantitative ecology in the past then this may be the text to convince you that there is much to be learnt from quantitative ecology. I thoroughly recommend this book and trust that you enjoy reading it as much as I did.” (International Statistical Review, 2012) "After a course of one or two semesters using this textbook, he says, students should have the absolute minimum of knowledge about quantitative research that ecologists need, but can provide a foundation for students who want to move further in that direction." (Book News, 1 August 2011) Table of ContentsHow I chose to write this book, and why you might choose to read it. Preface. 0. How to start a meaningful relationship with your computer. Introduction to R. 0.1 What is R? 0.2 Why use R for this book? 0.3 Computing with a scientific package like R. 0.4 Installing and interacting with R. 0.5 Style conventions. 0.6 Valuable R accessories. 0.7 Getting help. 0.8 Basic R usage. 0.9 Importing data from a spreadsheet. 0.10 Storing data in data frames. 0.11 Exporting data from R. 0.12 Quitting R. 1. How to make mathematical statements. Numbers, equations and functions. 1.1 Qualitative and quantitative scales. 1.2 Numbers. 1.3 Symbols. 1.4 Logical operations. 1.5 Algebraic operations. 1.6 Manipulating numbers. 1.7 Manipulating units. 1.8 Manipulating expressions. 1.9 Polynomials. 1.10 Equations. 1.11 First order polynomial equations. 1.12 Proportionality and scaling: a special kind of first order polynomial equation. 1.13 Second and higher order polynomial equations. 1.14 Systems of polynomial equations. 1.15 Inequalities. 1.16 Coordinate systems. 1.17 Complex numbers. 1.18 Relations and functions. 1.19 The graph of a function. 1.20 First order polynomial functions. 1.21 Higher order polynomial functions. 1.22 The relationship between equations and functions. 1.23 Other useful functions. 1.24 Inverse functions. 1.25 Functions of more than one variable. 2. How to describe regular shapes and patterns. Geometry and trigonometry. 2.1 Primitive elements. 2.2 Axioms of Euclidean geometry. 2.3 Propositions. 2.4 Distance between two points. 2.5 Areas and volumes. 2.6 Measuring angles. 2.7 The trigonometric circle. 2.8 Trigonometric functions. 2.9 Polar coordinates. 2.10 Graphs of trigonometric functions. 2.11 Trigonometric identities. 2.12 Inverses of trigonometric functions. 2.13 Trigonometric equations. 2.14 Modifying the basic trigonometric graphs. 2.15 Superimposing trigonometric functions. 2.16 Spectral analysis. 2.17 Fractal geometry. 3. How to change things, one step at a time. Sequences, difference equations and logarithms. 3.1 Sequences. 3.2 Difference equations. 3.3 Higher order difference equations. 3.4 Initial conditions and parameters. 3.5 Solutions of a difference equation. 3.6 Equilibrium solutions. 3.7 Stable and unstable equilibria. 3.8 Investigating stability. 3.9 Chaos. 3.10 Exponential function. 3.11 Logarithmic function. 3.12 Logarithmic equations. 4. How to change things, continuously. Derivatives and their applications. 4.1 Average rate of change. 4.2 Instantaneous rate of change. 4.3 Limits. 4.4 The derivative of a function. 4.5 Differentiating polynomials. 4.6 Differentiating other functions. 4.7 The chain rule. 4.8 Higher order derivatives. 4.9 Derivatives of functions of many variables. 4.10 Optimisation. 4.11 Local stability for difference equations. 4.12 Series expansions. 5. How to work with accumulated change. Integrals and their applications. 5.1 Antiderivatives. 5.2 Indefinite integrals. 5.3 Three analytical methods of integration. 5.4 Summation. 5.5 Area under a curve. 5.6 Definite integrals. 5.7 Some properties of definite integrals. 5.8 Improper integrals. 5.9 Differential equations. 5.10 Solving differential equations. 5.11 Stability analysis for differential equations. 6. How to keep stuff organised in tables. Matrices and their applications. 6.1 Matrices. 6.2 Matrix operations. 6.3 Geometric interpretation of vectors and square matrices. 6.4 Solving systems of equations with matrices. 6.5 Markov chains. 6.6 Eigenvalues and eigenvectors. 6.7 Leslie matrix models. 6.8 Analysis of linear dynamical systems. 6.9 Analysis of nonlinear dynamical systems. 7. How to visualise and summarise data. Descriptive statistics. 7.1 Overview of statistics. 7.2 Statistical variables. 7.3 Populations and samples. 7.4 Single-variable samples. 7.5 Frequency distributions. 7.6 Measures of centrality. 7.7 Measures of spread. 7.8 Skewness and kurtosis. 7.9 Graphical summaries. 7.10 Data sets with more than one variable. 7.11 Association between qualitative variables. 7.12 Association between quantitative variables. 7.13 Joint frequency distributions. 8. How to put a value on uncertainty. Probability. 8.1 Random experiments and event spaces. 8.2 Events. 8.3 Frequentist probability. 8.4 Equally likely events. 8.5 The union of events. 8.6 Conditional probability. 8.7 Independent events. 8.8 Total probability. 8.9 Bayesian probability. 9. How to identify different kinds of randomness. Probability distributions. 9.1 Probability distributions 300 9.2 Discrete probability distributions 301 9.3 Continuous probability distributions 304 9.4 Expectation 306 9.5 Named distributions 309 9.6 Equally likely events: the uniform distribution. 9.7 Hit or miss: the Bernoulli distribution. 9.8 Count of occurrences in a given number of trials: the binomial distribution. 9.9 Counting different types of occurrences: the multinomial distribution. 9.10 Number of occurrences in a unit of time or space: the Poisson distribution. 9.11 The gentle art of waiting: geometric, negative binomial, exponential and gamma distributions. 9.12 Assigning probabilities to probabilities: the beta and Dirichlet distributions. 9.13 Perfect symmetry: the normal distribution. 9.14 Because it looks right: using probability distributions empirically. 9.15 Mixtures, outliers and the t-distribution. 9.16 Joint, conditional and marginal probability distributions. 9.17 The bivariate normal distribution. 9.18 Sums of random variables: the central limit theorem. 9.19 Products of random variables: the log-normal distribution. 9.20 Modelling residuals: the chi-square distribution. 9.21 Stochastic simulation. 10. How to see the forest from the trees. Estimation and testing. 10.1 Estimators and their properties. 10.2 Normal theory. 10.3 Estimating the population mean. 10.4 Estimating the variance of a normal population. 10.5 Confidence intervals. 10.6 Inference by bootstrapping. 10.7 More general estimation methods. 10.8 Estimation by least squares. 10.9 Estimation by maximum likelihood. 10.10 Bayesian estimation. 10.11 Link between maximum likelihood and Bayesian estimation. 10.12 Hypothesis testing: rationale. 10.13 Tests for the population mean. 10.14 Tests comparing two different means. 10.15 Hypotheses about qualitative data. 10.16 Hypothesis testing debunked. 11. How to separate the signal from the noise. Statistical modelling. 11.1 Comparing the means of several populations. 11.2 Simple linear regression. 11.3 Prediction. 11.4 How good is the best-fit line? 11.5 Multiple linear regression. 11.6 Model selection. 11.7 Generalised linear models. 11.8 Evaluation, diagnostics and model selection for GLMs. 11.9 Modelling dispersion 409 11.10 Fitting more complicated models to data: polynomials, interactions, nonlinear regression. 11.11 Letting the data suggest more complicated models: smoothing. 11.12 Partitioning variation: mixed effects models. 12. How to measure similarity. Multivariate methods 12.1 The problem with multivariate data. 12.2 Ordination in general. 12.3 Principal components analysis. 12.4 Clustering in general. 12.5 Agglomerative hierarchical clustering. 12.6 Nonhierarchical clustering: k means analysis. 12.7 Classification in general. 12.8 Logistic regression: two classes. 12.9 Logistic regression: many classes. Further reading. References. Appendix: Formulae. R Index. Index.

Read more less

Book SynopsisA cluster randomization trial is one in which intact social units, or clusters of individuals, are randomized to different intervention groups. Trials randomizing clusters have become particularly widespread in the evaluation of non-therapeutic interventions, including lifestyle modification, educational programmes and innovations in the provision of health care. The increasing popularity of this design among health researchers over the past two decades has led to an extensive body of methodology on the subject. This is the first book to present a systematic and united treatment of this topic; it contains distinctive chapters on the history of cluster randomized trials, ethical issues and reporting guidelines.Table of ContentsAcknowledgements. Preface. 1. Introduction. 1.1 Why randomize clusters? 1.2 What is the impact of cluster randomization on the design and analysis of a trial? 1.3 Quantifying the effect of clustering. 1.4 Randomized versus non-randomized comparisons. 1.5 The unit of inference. 1.6 Terminology: what’s in a name? 2. The historical development of cluster randomized trials. 2.1 Randomized trials before 1950. 2.2 Cluster randomized trials between 1950 and 1978. 2.3 Cluster randomized trails since 1978. 3. Issues arising in the planning of cluster randomization trials. 3.1 Selecting interventions. 3.2 Setting eligibility criteria. 3.3 Measuring subject response. 3.4 The most commonly used experimental designs. 3.5 Factorial and crossover designs. 3.6 Selecting an experimental design. 3.7 The importance of cluster-level replication. 3.8 Strategies for conducting successful trials. 4. The role of informed consent and other ethical issues. 4.1 The risk of harm. 4.2 Informed consent. 4.3 Subject blindness and informed consent. 4.4 Randomized consent designs. 4.5 Ethical issues and trial monitoring. 5. Sample size estimation for cluster randomization designs. 5.1 General issues of sample size estimation. 5.2 The completely randomized design. 5.3 The matched-pair design. 5.4 The stratified design. 5.5 Issues involving losses to follow-up. 5.6 Strategies for achieving desired power. 6. Analysis of binary outcomes. 6.1 Selecting the unit of analysis. 6.2 The completely randomized design. 6.3 The matched-pair design. 6.4 The stratified design. 7. Analysis of quantitative outcomes. 7.1 The completely randomized design. 7.2 The matched-pair design. 7.3 The stratified design. 8. Analysis of count, time to event and categorical outcomes. 8.1 Count and time to event data. 8.2 Categorical data. 9. Reporting of cluster randomization trials. 9.1 Reporting of study design. 9.2 Reporting of study results. References. Index.

Read more less

Book SynopsisStatisticians rely heavily on making models of 'causal situations' in order to fully explain and predict events. Modelling therefore plays a vital part in all applications of statistics and is a component of most undergraduate programmes.Table of ContentsSeries preface. Preface. 1. Introduction. 1.1 Models in data analysis. 1.2 Populations and samples. 1.3 Variables and factors. 1.4 Observational and experimental data. 1.5 Statistical models. 2. Distributions and inference. 2.1 Random variables and probability distributions. 2.2 Probability distributions as models. 2.3 Some common distributions. 2.4 Sampling distributions. 2.5 Inference. 2.6 Postscript. 3. Normal response and quantitative explanatory variables: regression. 3.1 Motivation. 3.2 Simple regression. 3.3 Multiple regression. 3.4 Model building. 3.5 Model validation and criticism. 3.6 Comparison of regressions. 3.7 Non-linear models. 4. Normal response and qualitative explanatory variables: analysis of variance. 4.1 Motivation. 4.2 One-way arrangements. 4.3 Cross-classifications. 4.4 Nested classifications. 4.5 A general approach via multiple regression. 4.6 Analysis of covariance. 5. Non-normality: the theory of generalized linear models. 5.1 Introduction. 5.2 The generalized linear model. 5.3 Fitting the model. 5.4 Assessing the fit of a model: deviance. 5.5 Comparing models: analysis of deviance. 5.6 Normal models. 5.7 Inspecting and checking models. 5.8 Software. 6. Binomial response variables: logistic regression and related method. 6.1 Binary response data. 6.2 Modelling binary response probabilities. 6.3 Logistic regression. 6.4 Related methods. 6.5 Ordered polytomous data. 7. Tables of counts and log-linear models. 7.1 Introduction. 7.2 Data mechanisms and distributions. 7.3 Log-linear models for means. 7.4 Models for contingency tables. 7.5 Analysis. 7.6 Applications. 8. Further topics. 8.1 Introduction. 8.2 Continuous non-normal responses. 8.3 Quasi-likelihood. 8.4 Overdispersion. 8.5 Non-parametric models. 8.6 Conclusion: the art of model building. References. Index.

Read more less

Book SynopsisDecision-theoretic ideas can structure the process of inference together with the decision-making that inference supports. Statistical decision theory is the sub-discipline of statistics which explores and develops this structure. Typically, discusion of decision theory within one discipline does not recognise that other disciplines may have considered the same or similar problems. This text, Volume 9 in the prestigious Kendall's Library of Statistics, provides an overview of the main ideas and concepts of statistical decision theory and sets it within the broader concept of decision theory, decision analysis and decision support as they are practised in many disciplines beyond statistics - including artificial intelligence, economics, operational research, philosophy and psychology.

Read more less

Book SynopsisProviding a general survey of the theory of measurement error models, including the functional, structural, and ultrastructural models, this book is written in the of the Kendall and Stuart Advanced Theory of Statistics set and, like that series, includes exercises at the end of the chapters. The goal is to emphasize the ideas and practical implications of the theory in a style that does not concentrate on the theorem-proof format.Table of ContentsPreface. 1. Introduction to Linear Measurement Error Models. 1.1 Preliminaries. 1.2 Elementary Properties of Measurement Error Models. 1.3 Maximum Likelihood Estimation in the Univariate Normal Measurement Error Model. 1.4 The ME Model with Correlated Errors. 1.5 The Equation Error Model. 1.6 The Berkson Model. 1.7 Maximum Likelihood Estimation of Transformed Data: Elimination of Nuisance Parameters. 1.8 Bibliographic Notes and Discussion. 1.9 Exercises. 2. Properties of Estimate and Predictors. 2.1 Asymptotic Properties of ME Model Parameter Estimates. 2.2 Asymptotic Properties of Equation Error Model Estimates. 2.3 Finite-Sample Properties. 2.4 Implications Regarding Confidence Regions. 2.5 Prediction and Calibration under Measurement Error Models. 2.6 Bibliographic Notes and Discussion. 2.7 Exercises. 2.8 Research Problems. 3. Comparing Model Assumptions and Modifying Least Squares. 3.1 Issues Facing Users of ME Models. 3.2 A Unified Approach to the Functional, Structural, and Ultrastructural Relationships. 3.3 Identifiability Assumptions and the Equation Error Model. 3.4 Generalized Least Squares. 3.5 Modified Least Squares. 3.6 Bibliographic Notes and Discussion. 3.7 Exercises. 4. Alternative Approaches to the Measurement Error Model. 4.1 Introduction and Overview. 4.2 Instrumental Variable Estimators. 4.3 Grouping Methods. 4.4 Methods Based on Ranks. 4.5 Methods of Higher-order Moments and Product Cumulants. 4.6 Bibliographic Notes and Discussion. 4.7 Exercises. 5. Linear Measurement Error Model with Vector Explanatory Variables. 5.1 Introduction. 5.2 Identifiability 5.3 The Equation Error Model. 5.4 Maximum Likelihood for the No-Equation-Error Model. 5.5 Alternative Approaches to Estimating the Parameters. 5.6 Asymptotic Properties of the Estimates. 5.7 Bibliographic Notes and Discussion. 5.8 Exercises. 5.9 Research Problem. 6. Polynomial Measurement Error Models. 6.1 Introduction. 6.2 The Nonlinear Structural Model. 6.3 Identifiability in Nonlinear ME Models. 6.4 Polynomial Model with Equation Error. 6.5 The Polynomial Functional Relationship without Equation Error. 6.6 Polynomial Berkson Model. 6.7 Bibliographic Notes and Discussion. 6.8 Exercises. 7. Robust Estimation in Measurement Error Models. 7.1 Introduction. 7.2 Robust Orthogonal Regression. 7.3 Robust Measurement Error Model Estimation via Robust Covariance Matrices. 7.4 Computational Methods for Robust Orthogonal Regression. 7.5 Bibliographic Notes and Discussion. 7.7 Exercises. 7.8 Research Problem. 8. Additional Topics. 8.1 Estimation of the True Variables. 8.2 Obtaining Identifiability Assumption Information. 8.3 Conclusions. 8.4 Relations to Other Latent Variables Models. 8.5 The factor analysis model. 8.6 Terminology. 8.7 Exercises. Appendix A. Identification in Measurement Error Models. A.1 Overview. A.2 Structural Model. A.3 Functional Model. A.4 Identiability and Consistent Estimation. Bibliography. Author Index. Subject Index.

Read more less

Book SynopsisThe design and analysis of experiments is typically taught as part of a second level course in statistics. Many different types and levels of students will require this information in order to progress with their studies and research. This text is thus offered as an introduction to this wide ranging and important subject. It has the advantage of explaining in an accessible way the basic principles behind good experimental thinking, planning and action. The authors have used their experience in teaching related courses to separate out what seem to be the essential basic contents for everyone, and to combine with these some of the most useful additional topics in biological, industrial, medical, and environmental experimentation.Table of ContentsPreface. 1 Collecting data by experiments. 1.1 Introduction. 1.2 Experiments. 1.3 Measurements of yield or response. 1.4 Natural variation in data. 1.5 Initial data analysis. 1.6 General applications of experimentation. 1.7 Exercises. 2 Basic statistical methods: the normal distribution. 2.1 Statistical inference for one sample of normally distributed data. 2.2 Hypothesis test. 2.3 Comparison of two samples of normally distributed data. 2.4 The F-test for comparing two estimated variances. 2.5 Confidence interval for the difference between two means. 2.6 'Paired data' t-test when samples are not independent. 2.7 Linear functions of normally distributed variables. 2.8 Linear models including normal random variation. 2.9 Exercises. 3 Principles of experimental design. 3.1 Introduction. 3.2 Treatment structure. 3.3 Changing background conditions – the need for comparison. 3.4 Replication. 3.5 Randomization. 3.6 Blocking. 3.7 Sources of variation. 3.8 Planning the size of an experiment. 3.9 Exercises. 4 The analysis of data from orthogonal designs. 4.1 Introduction. 4.2 Comparing treatments. 4.3 Confidence intervals. 4.4 Homogeneity of variance. 4.5 The randomized complete block. 4.6 Duncan's multiple range test. 4.7 Extra replication of important treatments. 4.8 Contrasts among treatments. 4.9 Latin squares and other orthogonal designs. 4.10 Graeco-Latin squares. 4.11 Two fallacies. 4.12 Assumptions in analysis: using residuals to examine them. 4.13 Transformations. 4.14 Theory of variance stabilization. 4.15 Missing data in block designs. 4.16 Exercises. Appendix 4A Cochran's Theorem on Quadratic Forms. 5 Factorial experiments. 5.1 Introduction. 5.2 Notation for factors at two levels. 5.3 Definition of main effect and interaction. 5.4 Three factors each at two levels. 5.5 A single factor at more than two levels. 5.6 General method for computing coefficients for orthogonal polynomials. 5.7 Exercises. 6 Experiments with many factors: confounding and fractional replication. 6.1 Introduction. 6.2 The principal block in confounding. 6.3 Single replicate. 6.4 Small experiments: partial confounding. 6.5 Very large experiments: fractional replication. 6.6 Replicates smaller than half size. 6.7 Confounding with fractional replication. 6.8 Confounding three-level factors. 6.9 Fractional replication in 3-level experiments. 6.10 Exercises. Appendix 6A Methods of confounding in 2p factorial experiments. 7 Confounding main effects – split-plot designs. 7.1 Introduction. 7.2 Linear model and analysis. 7.3 Studying interactions. 7.4 Repeated splitting. 7.5 Confounding in split-plot experiments. 7.6 Other designs for main plots. 7.7 Criss-cross design. 7.8 Exercises. 8 Industrial experimentation. 8.1 Introduction. 8.2 Taguchi methods in statistical quality control. 8.3 Loss functions. 8.4 Sources of variation. 8.5 Orthogonal arrays. 8.6 Choice of design. 9 Response surfaces and mixture designs. 9.1 Introduction. 9.2 Are experimental conditions ‘constant’? 9.3 Response surfaces. 9.4 Experiments with three factors, x1, x2 and x3. 9.5 Second-order surfaces. 9.6 Contour diagrams in analysis. 9.7 Transformations. 9.8 Mixture designs. 9.9 Other types of response surface. 9.10 Exercises. 10 The analysis of covariance. 10.1 Introduction. 10.2 Analysis for a design in randomized blocks: general theory. 10.3 Individual contrasts. 10.4 Dummy covariance. 10.5 Systematic trend not removed by blocking. 10.6 Accidents in recording. 10.7 Assumptions in covariance analysis. 10.8 Missing values. 10.9 Double covariance. 10.10 Exercises. 11 Balanced incomplete blocks and general non-orthogonal block designs. 11.1 Introduction. 11.2 Definition and existence of a balanced incomplete block. 11.3 Methods of construction. 11.4 Linear model and analysis. 11.5 Row and column design: the Youden square. 11.6 General block designs. 11.7 Linear model and analysis. 11.8 Generalized inverse. 11.9 Application to designs with special patterns. 11.10 Exercises. Appendix 11A Generalized inverse matrix by spectral decomposition. Appendix 11B Natural contrasts and effective replication. 12 More advanced designs. 12.1 Introduction. 12.2 Crossover designs. 12.3 Lattices. 12.4 Alpha designs. 12.5 Partially balanced incomplete blocks (PBIBs). 13 Random effects models: variance components and sampling schemes. 13.1 Introduction. 13.2 Two stages of sampling: between and within units. 13.3 Assessing alternative sampling schemes. 13.4 Using variance components in planning when sampling costs are given. 13.5 Three levels of variation. 13.6 Costs in a three-stage scheme. 13.7 Example where one estimate is negative. 13.8 Exercises. 14 Computer output using SAS. Bibliography and references. Tables. Index.

Read more less

Book SynopsisPsychologists need to know about statistics but they can easily get bogged down in unfamiliar jargon. This new dictionary is an invaluable reference for all undergraduate and postgraduate students of psychology. It provides definitions of the statistical terms most commonly encountered in psychological literature.

Read more less

Book SynopsisDecision making in all spheres of activity involves uncertainty. If rational decisions have to be made, they have to be based on the past observations of the phenomenon in question. Data collection, model building and inference from the data collected, validation of the model and refinement of the model are the key steps or building blocks involved in any rational decision making process. Stochastic processes are widely used for model building in the social, physical, engineering, and life sciences as well as in financial economics. Statistical inference for stochastic processes is of great importance from the theoretical as well as from applications point of view in model building. During the past twenty years, there has been a large amount of progress in the study of inferential aspects for continuous as well as discrete time stochastic processes. Diffusion type processes are a large class of continuous time processes which are widely used for stochastic modelling. the book aims to b

Read more less

Book SynopsisProgress in engineering and the physical sciences, agriculture and the biological sciences, and to some extent social science, depends on experiments. The design of such experiments is crucial. If they are poorly designed they will be inefficient and may lead to misleading conclusions.Table of ContentsPreface. Notation. Website for data sets. Sources of data. 1 Guide. 2 Descriptive statistics and plotting. 3 Single sample experiments. 4 Comparison proportions. 5 Comparing two treatments. 6 Comparison of several means. 7 Factorial experiments with factors at two levels. 8 Fractional factorial experiments with factors at two levels. 9 Response surfaces. 10 Hill climbing. 11 Robust design. 12 Hierarchical (nested) designs. 13 Two factors at several levels. 14 Crossed and nested factors, and split-plot designs. 15 Mixture designs. 16 Discrete response. Post-script. Authors. Glossary. References. Index.

Read more less

Book SynopsisMultivariate analysis plays an important role in the understanding of complex data sets requiring simultaneous examination of all variables. Breaking through the apparent disorder of the information, it provides the means for both describing and exploring data, aiming to extract the underlying patterns and structure. This intermediate-level textbook introduces the reader to the variety of methods by which multivariate statistical analysis may be undertaken. Now in its 2nd edition, ''Applied Multivariate Data Analysis'' has been fully expanded and updated, including major chapter revisions as well as new sections on neural networks and random effects models for longitudinal data. Maintaining the easy-going style of the first edition, the authors provide clear explanations of each technique, as well as supporting figures and examples, and minimal technical jargon. With extensive exercises following every chapter, ''Applied Multivariate Data Analysis'' is a valuable resource for students Table of Contents1 Multivariate data and multivariate statistics. 1.1 Introduction. 1.2 Types of data. 1.3 Basic multivariate statistics. 1.4 The aims of multivariate analysis. 2 Exploring multivariate data graphically. 2.1 Introduction. 2.2 The scatterplot. 2.3 The scatterplot matrix. 2.4 Enhancing the scatterplot. 2.5 Coplots and trellis graphics. 2.6 Checking distributional assumptions using probability plots. 2.7 Summary. Exercises. 3 Principal components analysis. 3.1 Introduction. 3.2 Algebraic basics of principal components. 3.3 Rescaling principal components. 3.4 Calculating principal component scores. 3.5 Choosing the number of components. 3.6 Two simple examples of principal components analysis. 3.7 More complex examples of the application of principal components analysis. 3.8 Using principal components analysis to select a subset of variables. 3.9 Using the last few principal components. 3.10 The biplot. 3.11 Geometrical interpretation of principal components analysis. 3.12 Projection pursuit. 3.13 Summary. Exercises. 4 Correspondence analysis. 4.1 Introduction. 4.2 A simple example of correspondence analysis. 4.3 Correspondence analysis for two-dimensional contingency tables. 4.4 Three applications of correspondence analysis. 4.5 Multiple correspondence analysis. 4.6 Summary Exercises. 5 Multidimensional scaling. 5.1 Introduction. 5.2 Proximity matrices and examples of multidimensional scaling. 5.4 Metric least-squares multidimensional scaling. 5.5 Non-metric multidimensional scaling. 5.6 Non-Euclidean metrics. 5.7 Three-way multidimensional scaling. 5.8 Inference in multidimensional scaling. 5.9 Summary. Exercises. 6 Cluster analysis. 6.1 Introduction. 6.2 Agglomerative hierarchical clustering techniques. 6.3 Optimization methods. 6.4 Finite mixture models for cluster analysis. 6.5 Summary. Exercises. 7 The generalized linear model. 7.1 Linear models. 7.2 Non-linear models. 7.3 Link functions and error distributions in the generalized linear model. 7.4 Summary. Exercises. 8 Regression and the analysis of variance. 8.1 Introduction. 8.2 Least-squares estimation for regression and analysis of variance models. 8.3 Direct and indirect effects. 8.4 Summary. Exercises. 9 Log-linear and logistic models for categorical multivariate data. 9.1 Introduction. 9.2 Maximum likelihood estimation for log-linear and linear-logistic models. 9.3 Transition models for repeated binary response measures. 9.4 Summary. Exercises. 10 Models for multivariate response variables. 10.1 Introduction. 10.2 Repeated quantitative measures. 10.3 Multivariate tests. 10.4 Random effects models for longitudinal data. 10.5 Logistic models for multivariate binary responses. 10.6 Marginal models for repeated binary response measures. 10.7 Marginal modelling using generalized estimating equations. 10.8 Random effects models for multivariate repeated binary response measures. 10.9 Summary. Exercises. 11 Discrimination, classification and pattern recognition. 11.1 Introduction. 11.2 A simple example. 11.3 Some examples of allocation rules. 11.4 Fisher's linear discriminant function. 11.5 Assessing the performance of a discriminant function. 11.6 Quadratic discriminant functions. 11.7 More than two groups. 11.8 Logistic discrimination. 11.9 Selecting variables. 11.10 Other methods for deriving classification rules. 11.11 Pattern recognition and neural networks. 11.12 Summary. Exercises. 12 Exploratory factor analysis. 12.1 Introduction. 12.2 The basic factor analysis model. 12.3 Estimating the parameters in the factor analysis model. 12.4 Rotation of factors. 12.5 Some examples of the application of factor analysis. 12.6 Estimating factor scores. 12.7 Factor analysis with categorical variables. 12.8 Factor analysis and principal components analysis compared. 12.9 Summary. Exercises. 13 Confirmatory factor analysis and covariance structure models. 13.1 Introduction. 13.2 Path analysis and path diagrams. 13.3 Estimation of the parameters in structural equation models. 13.4 A simple covariance structure model and identification. 13.5 Assessing the fit of a model. 13.6 Some examples of fitting confirmatory factor analysis models. 13.7 Structural equation models. 13.8 Causal models and latent variables: myths and realities. 13.9 Summary. Exercises. Appendices. A Software packages. A.1 General-purpose packages. A.2 More specialized packages. B Missing values. C Answers to selected exercises. References. Index.

Read more less

Book SynopsisIt is difficult to imagine that the statistical analysis of compositional data has been a major issue of concern for more than 100 years. It is even more difficult to realize that so many statisticians and users of statistics are unaware of the particular problems affecting compositional data, as well as their solutions.Table of ContentsPreface xvii List of Contributors xix Part I Introduction 1 1 A Short History of Compositional Data Analysis 3 John Bacon-Shone 1.1 Introduction 3 1.2 Spurious Correlation 3 1.3 Log and Log-Ratio Transforms 4 1.4 Subcompositional Dependence 5 1.5 alr, clr, ilr: Which Transformation to Choose? 5 1.6 Principles, Perturbations and Back to the Simplex 6 1.7 Biplots and Singular Value Decompositions 7 1.8 Mixtures 7 1.9 Discrete Compositions 8 1.10 Compositional Processes 8 1.11 Structural, Counting and Rounded Zeros 8 1.12 Conclusion 9 Acknowledgement 9 References 9 2 Basic Concepts and Procedures 12 Juan Jos´e Egozcue and Vera Pawlowsky-Glahn 2.1 Introduction 12 2.2 Election Data and Raw Analysis 13 2.3 The Compositional Alternative 15 2.3.1 Scale Invariance: Vectors with Proportional Positive Components Represent the Same Composition 15 2.3.2 Subcompositional Coherence: Analyses Concerning a Subset of Parts Must Not Depend on Other Non-Involved Parts 16 2.3.3 Permutation Invariance: The Conclusions of a Compositional Analysis Should Not Depend on the Order of the Parts 17 2.4 Geometric Settings 17 2.5 Centre and Variability 22 2.6 Conclusion 27 Acknowledgements 27 References 27 Part II Theory – Statistical Modelling 29 3 The Principle of Working on Coordinates 31 Glòria Mateu-Figueras, Vera Pawlowsky-Glahn and Juan José Egozcue 3.1 Introduction 31 3.2 The Role of Coordinates in Statistics 32 3.3 The Simplex 33 3.3.1 Basis of the Simplex 34 3.3.2 Working on Orthonormal Coordinates 35 3.4 Move or Stay in the Simplex 38 3.5 Conclusions 40 Acknowledgements 41 References 41 4 Dealing with Zeros 43 Josep Antoni Martún-Fernández, Javier Palarea-Albaladejo and Ricardo Antonio Olea 4.1 Introduction 43 4.2 Rounded Zeros 44 4.2.1 Non-Parametric Replacement of Rounded Zeros 45 4.2.2 Parametric Modified EM Algorithm for Rounded Zeros 47 4.3 Count Zeros 50 4.4 Essential Zeros 53 4.5 Difficulties, Troubles and Challenges 55 Acknowledgements 57 References 57 5 Robust Statistical Analysis 59 Peter Filzmoser and Karel Hron 5.1 Introduction 59 5.2 Elements of Robust Statistics from a Compositional Point of View 60 5.3 Robust Methods for Compositional Data 63 5.3.1 Multivariate Outlier Detection 64 5.3.2 Principal Component Analysis 64 5.3.3 Discriminant Analysis 65 5.4 Case Studies 66 5.4.1 Multivariate Outlier Detection 66 5.4.2 Principal Component Analysis 68 5.4.3 Discriminant Analysis 68 5.5 Summary 70 Acknowledgement 71 References 71 6 Geostatistics for Compositions 73 Raimon Tolosana-Delgado, Karl Gerald van den Boogaart and Vera Pawlowsky-Glahn 6.1 Introduction 73 6.2 A Brief Summary of Geostatistics 74 6.3 Cokriging of Regionalised Compositions 76 6.4 Structural Analysis of Regionalised Composition 76 6.5 Dealing with Zeros: Replacement Strategies and Simplicial Indicator Cokriging 78 6.6 Application 79 6.6.1 Delimiting the Body: Simplicial Indicator Kriging 81 6.6.2 Interpolating the Oil–Brine–Solid Content 82 6.7 Conclusions 84 Acknowledgements 84 References 84 7 Compositional VARIMA Time Series 87 Carles Barceló-Vidal, Lucúa Aguilar and Josep Antoni Martún-Fernández 7.1 Introduction 87 7.2 The Simplex SD as a Compositional Space 89 7.2.1 Basic Concepts and Notation 89 7.2.2 The Covariance Structure on the Simplex 90 7.3 Compositional Time Series Models 91 7.3.1 C-Stationary Processes 92 7.3.2 C-VARIMA Processes 93 7.4 CTS Modelling: An Example 94 7.4.1 Expenditure Shares in the UK 94 7.4.2 Model Selection 95 7.4.3 Estimation of Parameters 96 7.4.4 Interpretation and Comparison 96 7.5 Discussion 99 Acknowledgements 99 References 100 Appendix 102 8 Compositional Data and Correspondence Analysis 104 Michael Greenacre 8.1 Introduction 104 8.2 Comparative Technical Definitions 105 8.3 Properties and Interpretation of LRA and CA 107 8.4 Application to Fatty Acid Compositional Data 107 8.5 Discussion and Conclusions 111 Acknowledgements 112 References 112 9 Use of Survey Weights for the Analysis of Compositional Data 114 Monique Graf 9.1 Introduction 114 9.2 Elements of Survey Design 115 9.2.1 Randomization 115 9.2.2 Design-Based Estimation 118 9.3 Application to Compositional Data 122 9.3.1 Weighted Arithmetic and Geometric Means 123 9.3.2 Closed Arithmetic Mean of Amounts 123 9.3.3 Centred Log-Ratio of the Geometric Mean Composition 124 9.3.4 Closed Geometric Mean Composition 124 9.3.5 Example: Swiss Earnings Structure Survey (SESS) 125 9.4 Discussion 126 References 126 10 Notes on the Scaled Dirichlet Distribution 128 Gianna Serafina Monti, Glòria Mateu-Figueras and Vera Pawlowsky-Glahn 10.1 Introduction 128 10.2 Genesis of the Scaled Dirichlet Distribution 129 10.3 Properties of the Scaled Dirichlet Distribution 131 10.3.1 Graphical Comparison 131 10.3.2 Membership in the Exponential Family 133 10.3.3 Measures of Location and Variability 134 10.4 Conclusions 136 Acknowledgements 137 References 137 Part III Theory – Algebra and Calculus 139 11 Elements of Simplicial Linear Algebra and Geometry 141 Juan José Egozcue, Carles Barceló-Vidal, Josep Antoni Martún-Fernández, Eusebi Jarauta-Bragulat, José Luis Dúaz-Barrero and Glòria Mateu-Figueras 11.1 Introduction 141 11.2 Elements of Simplicial Geometry 142 11.2.1 n-Part Simplex 142 11.2.2 Vector Space 143 11.2.3 Centred Log-Ratio Representation 146 11.2.4 Metrics 147 11.2.5 Orthonormal Basis and Coordinates 149 11.3 Linear Functions 151 11.3.1 Linear Functions Defined on the Simplex 152 11.3.2 Simplicial Linear Function Defined on a Real Space 153 11.3.3 Simplicial Linear Function Defined on the Simplex 154 11.4 Conclusions 156 Acknowledgements 156 References 156 12 Calculus of Simplex-Valued Functions 158 Juan José Egozcue, Eusebi Jarauta-Bragulat and José Luis Díaz-Barrero 12.1 Introduction 158 12.2 Limits, Continuity and Differentiability 161 12.2.1 Limits and Continuity 161 12.2.2 Differentiability 163 12.2.3 Higher Order Derivatives 169 12.3 Integration 171 12.3.1 Antiderivatives. Indefinite Integral 171 12.3.2 Integration of Continuous SV Functions 172 12.4 Conclusions 174 Acknowledgements 175 References 175 13 Compositional Differential Calculus on the Simplex 176 Carles Barceló-Vidal, Josep Antoni Martún-Fernández and Glòria Mateu-Figueras 13.1 Introduction 176 13.2 Vector-Valued Functions on the Simplex 177 13.2.1 Scale-Invariant Vector-Valued Functions on Rn + 177 13.2.2 Vector-Valued Functions on Sn 178 13.3 C-Derivatives on the Simplex 178 13.3.1 Derivative of a Scale-Invariant Vector-Valued Function on Rn + 178 13.3.2 Directional C-Derivatives 180 13.3.3 C-Derivative 182 13.3.4 C-Gradient 184 13.3.5 Critical Points of a C-Differentiable Real-Valued Function on Sn 184 13.4 Example: Experiments with Mixtures 185 13.4.1 Polynomial of Degree One 185 13.4.2 Polynomial of Degree Two 186 13.4.3 Polynomial of Degree One in Logarithms 187 13.4.4 A numerical Example 188 13.5 Discussion 189 Acknowledgements 190 References 190 Part IV Applications 191 14 Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right? 193 David Lovell, Warren Müller, Jen Taylor, Alec Zwart and Chris Helliwell 14.1 Introduction 193 14.2 The Omics Imp and Two Bioscience Experiment Paradigms 194 14.3 The Impact of Compositional Constraints in the Omics 197 14.3.1 Univariate Impact of Compositional Constraints 197 14.3.2 Impact of Compositional Constraints on Multivariate Distance Metrics 199 14.4 Impact of Compositional Constraints on Correlation and Covariance 201 14.4.1 Compositional Constraints, Covariance, Correlation and Log-Transformed Data 202 14.4.2 A Simulation Approach to Understanding the Impact of Closure 202 14.5 Implications 204 14.5.1 Gathering Information to Infer Absolute Abundance 204 14.5.2 Analysing Compositional Omics Data Appropriately 205 Acknowledgements 206 References 206 15 Hardy–Weinberg Equilibrium: A Nonparametric Compositional Approach 208 Jan Graffelman and Juan José Egozcue 15.1 Introduction 208 15.2 Genetic Data Sets 209 15.3 Classical Tests for HWE 210 15.4 A Compositional Approach 210 15.5 Example 214 15.6 Conclusion and Discussion 215 Acknowledgements 215 References 215 16 Compositional Analysis in Behavioural and Evolutionary Ecology 218 Michele Edoardo Raffaele Pierotti and Josep Antoni Martún-Fernández 16.1 Introduction 218 16.2 CODA in Population Genetics 219 16.3 CODA in Habitat Choice 222 16.4 Multiple Choice and Individual Variation in Preferences 224 16.5 Ecological Specialization 228 16.6 Time Budgets: More on Specialization 229 16.7 Conclusions 231 Acknowledgements 231 References 231 17 Flying in Compositional Morphospaces: Evolution of Limb Proportions in Flying Vertebrates 235 Luis Azevedo Rodrigues, Josep Daunis-i-Estadella, Glòria Mateu-Figueras and Santiago Thi´o-Henestrosa 17.1 Introduction 235 17.2 Flying Vertebrates – General Anatomical and Functional Characteristics 236 17.3 Materials 236 17.4 Methods 238 17.5 Aitchison Distance Disparity Metrics 239 17.5.1 Intragroup Aitchison Distance 239 17.5.2 Intergroup Aitchison Distance 240 17.6 Statistical Tests 243 17.7 Biplots 244 17.7.1 Chiroptera 244 17.7.2 Pterosauria 245 17.8 Balances 246 17.9 Size Effect 249 17.10 Final Remarks 249 17.10.1 All Groups 250 17.10.2 Aves 250 17.10.3 Pterosauria 250 17.10.4 Chiroptera 251 Acknowledgements 252 References 252 18 Natural Laws Governing the Distribution of the Elements in Geochemistry: The Role of the Log-Ratio Approach 255 Antonella Buccianti 18.1 Introduction 255 18.2 Geochemical Processes and Log-Ratio Approach 256 18.3 Log-Ratio Approach and Water Chemistry 258 18.4 Log-Ratio Approach and Volcanic Gas Chemistry 261 18.5 Log-Ratio Approach and Subducting Sediment Composition 263 18.6 Conclusions 265 Acknowledgements 265 References 265 19 Compositional Data Analysis in Planetology: The Surfaces of Mars and Mercury 267 Helmut Lammer, Peter Wurz, Josep Antoni Martún-Fernández and Herbert Iwo Maria Lichtenegger 19.1 Introduction 267 19.1.1 Mars 267 19.1.2 Mercury 269 19.1.3 Analysis of Surface Composition 270 19.2 Compositional Analysis of Mars’ Surface 270 19.3 Compositional Analysis of Mercury’s Surface 274 19.4 Conclusion 278 Acknowledgement 278 References 278 20 Spectral Analysis of Compositional Data in Cyclostratigraphy 282 Eulogio Pardo-Igúzquiza and Javier Heredia 20.1 Introduction 282 20.2 The Method 283 20.3 Case Study 285 20.4 Discussion 287 20.5 Conclusions 288 Acknowledgement 288 References 288 21 Multivariate Geochemical Data Analysis in Physical Geography 290 Jennifer McKinley and Christopher David Lloyd 21.1 Introduction 290 21.2 Context 291 21.3 Data 293 21.4 Analysis 295 21.5 Discussion 299 21.6 Conclusion 300 Acknowledgement 300 References 300 22 Combining Isotopic and Compositional Data: A Discrimination of Regions Prone to Nitrate Pollution 302 Roger Puig, Raimon Tolosana-Delgado, Neus Otero and Albert Folch 22.1 Introduction 302 22.2 Study Area 303 22.2.1 Maresme 304 22.2.2 Osona 305 22.2.3 Lluc¸an`es 305 22.2.4 Empord`a 306 22.2.5 Selva 306 22.3 Analytical Methods 306 22.4 Statistical Treatment 307 22.4.1 Data Scaling 307 22.4.2 Linear Discriminant Analysis 309 22.4.3 Discriminant Biplots 310 22.5 Results and Discussion 311 22.6 Conclusions 314 Acknowledgements 315 References 315 23 Applications in Economics 318 Tim Fry 23.1 Introduction 318 23.2 Consumer Demand Systems 319 23.3 Miscellaneous Applications 322 23.4 Compositional Time Series 323 23.5 New Directions 323 23.6 Conclusion 325 References 325 Part V Software 327 24 Exploratory Analysis Using CoDaPack 3D 329 Santiago Thió-Henestrosa and Josep Daunis-i-Estadella 24.1 CoDaPack 3D Description 329 24.2 Data Set Description 331 24.3 Exploratory Analysis 333 24.3.1 Numerical Analysis 333 24.3.2 Biplot 334 24.3.3 The Ternary Diagram 335 24.3.4 Principal Component Analysis 336 24.3.5 Balance-Dendrogram 336 24.3.6 By Groups Description 338 24.4 Summary and Conclusions 339 Acknowledgements 340 References 340 25 robCompositions: An R-package for Robust Statistical Analysis of Compositional Data 341 Matthias Templ, Karel Hron and Peter Filzmoser 25.1 General Information on the R-package robCompositions 341 25.1.1 Data Sets Included in the Package 342 25.1.2 Design Principles 343 25.2 Expressing Compositional Data in Coordinates 343 25.3 Multivariate Statistical Methods for Compositional Data Containing Outliers 345 25.3.1 Multivariate Outlier Detection 345 25.3.2 Principal Component Analysis and the Robust Compositional Biplot 347 25.3.3 Discriminant Analysis 350 25.4 Robust Imputation of Missing Values 351 25.5 Summary 354 References 354 26 Linear Models with Compositions in R 356 Raimon Tolosana-Delgado and Karl Gerald van den Boogaart 26.1 Introduction 356 26.2 The Illustration Data Set 357 26.2.1 The Data 357 26.2.2 Descriptive Analysis of Compositional Characteristics 358 26.3 Explanatory Binary Variable 360 26.4 Explanatory Categorical Variable 363 26.5 Explanatory Continuous Variable 365 26.6 Explanatory Composition 367 26.7 Conclusions 370 Acknowledgement 371 References 371 Index 373

Read more less

Book SynopsisThis book provides a thorough analysis of internal rating systems. Two case studies are devoted to building and validating statistical-based models for borrowers' ratings, using SPSS-PASW and SAS statistical packages. Mainstream approaches to building and validating models for assigning counterpart ratings to small and medium enterprises are discussed, together with their implications on lending strategy. Key Features: Presents an accessible framework for bank managers, students and quantitative analysts, combining strategic issues, management needs, regulatory requirements and statistical bases. Discusses available methodologies to build, validate and use internal rate models. Demonstrates how to use statistical packages for building statistical-based credit rating systems. Evaluates sources of model risks and strategic risks when using statistical-based rating systems in lending. This book will prove to be of great value to banTable of ContentsPreface xi About the authors xiii 1 The emergence of credit ratings tools 1 2 Classifications and key concepts of credit risk 5 2.1 Classification 5 2.1.1 Default mode and value-based valuations 5 2.1.2 Default risk 6 2.1.3 Recovery risk 7 2.1.4 Exposure risk 8 2.2 Key concepts 8 2.2.1 Expected losses 8 2.2.2 Unexpected losses, VAR, and concentration risk 9 2.2.3 Risk adjusted pricing 13 3 Rating assignment methodologies 17 3.1 Introduction 17 3.2 Experts-based approaches 19 3.2.1 Structured experts-based systems 19 3.2.2 Agencies’ ratings 22 3.2.3 From borrower ratings to probabilities of default 26 3.2.4 Experts-based internal ratings used by banks 31 3.3 Statistical-based models 32 3.3.1 Statistical-based classification 32 3.3.2 Structural approaches 34 3.3.3 Reduced form approaches 38 3.3.4 Statistical methods: linear discriminant analysis 41 3.3.5 Statistical methods: logistic regression 54 3.3.6 From partial ratings modules to the integrated model 58 3.3.7 Unsupervised techniques for variance reduction and variables’ association 60 3.3.8 Cash flow simulations 73 3.3.9 A synthetic vision of quantitative-based statistical models 76 3.4 Heuristic and numerical approaches 77 3.4.1 Expert systems 78 3.4.2 Neural networks 81 3.4.3 Comparison of heuristic and numerical approaches 85 3.5 Involving qualitative information 86 4 Developing a statistical-based rating system 93 4.1 The process 93 4.2 Setting the model’s objectives and generating the dataset 96 4.2.1 Objectives and nature of data to be collected 96 4.2.2 The time frame of data 96 4.3 Case study: dataset and preliminary analysis 97 4.3.1 The dataset: an overview 97 4.3.2 Duplicate cases analysis 103 4.3.3 Missing values analysis 104 4.3.4 Missing value treatment 107 4.3.5 Other preliminary overviews 109 4.4 Defining an analysis sample 114 4.4.1 Rationale for splitting the dataset into an analysis sample and a validation sample 114 4.4.2 How to split the dataset into an analysis sample and a validation sample 114 4.5 Univariate and bivariate analyses 116 4.5.1 Indicators’ economic meanings, working hypotheses and structural monotonicity 117 4.5.2 Empirical assessment of working hypothesis 130 4.5.3 Normality and homogeneity of variance 137 4.5.4 Graphical analysis 140 4.5.5 Discriminant power 145 4.5.6 Empirical monotonicity 157 4.5.7 Correlations 160 4.5.8 Analysis of outliers 162 4.5.9 Transformation of indicators 164 4.5.10 Summary table of indicators and short listing 177 4.6 Estimating a model and assessing its discriminatory power 184 4.6.1 Steps and case study simplifications 184 4.6.2 Linear discriminant analysis 185 4.6.3 Logistic regression 210 4.6.4 Refining models 216 4.7 From scores to ratings and from ratings to probabilities of default 229 5 Validating rating models 237 5.1 Validation profiles 237 5.2 Roles of internal validation units 239 5.3 Qualitative and quantitative validation 241 5.3.1 Qualitative validation 242 5.3.2 Quantitative validation 249 6 Case study: Validating PanAlp Bank’s statistical-based rating system for financial institutions 257 6.1 Case study objectives and context 257 6.2 The ‘Development report’ for the validation unit 258 6.2.1 Shadow rating approach for financial institutions 258 6.2.2 Missing value analysis 259 6.2.3 Interpreting financial ratios for financial institutions and setting working hypotheses 260 6.2.4 Monotonicity 263 6.2.5 Analysis of means 263 6.2.6 Assessing normality of distributions: histograms and normal Q–Q plots 263 6.2.7 Box plots analysis 266 6.2.8 Normality tests 267 6.2.9 Homogeneity of variance tests 269 6.2.10 F-ratio and F-Test 270 6.2.11 ROC curves 270 6.2.12 Correlations 270 6.2.13 Outliers 270 6.2.14 Short listing and linear discriminant analysis 272 6.3 The ‘Validation report’ by the validation unit 274 7 Ratings usage opportunities and warnings 285 7.1 Internal ratings: critical to credit risk management 285 7.2 Internal ratings assignment trends 289 7.3 Statistical-based ratings and regulation: conflicting objectives? 291 7.4 Statistical-based ratings and customers: needs and fears 295 7.5 Limits of statistical-based ratings 298 7.6 Statistical-based ratings and the theory of financial intermediation 305 7.7 Statistical-based ratings usage: guidelines 310 Bibliography 315 Index 321

Read more less

Book SynopsisMost of the intriguing social phenomena of our time, such as international terrorism, social inequality, and urban ethnic segregation, are consequences of complex forms of agent interaction that are difficult to observe methodically and experimentally. This book looks at a new research stream that makes use of advanced computer simulation modelling techniques to spotlight agent interaction that allows us to explain the emergence of social patterns. It presents a method to pursue analytical sociology investigations that look at relevant social mechanisms in various empirical situations, such as markets, urban cities, and organisations. This book: Provides a comprehensive introduction to epistemological, theoretical and methodological features of agent-based modelling in sociology through various discussions and examples. Presents the pros and cons of using agent-based models in sociology. Explores agent-based models in combining quantitative and Trade Review“This book should be inserted into all sociological libraries as a vanguard for the rest of us - if it not torn to shreds by enraged sociologists it will very usefully inform them. Newcomers to ABM and even old hands, but especially those who have to survive within sociology, will find it a very valuable asset.” (Journal of Artificial Societies and Social Simulation, 1 January 2013) Table of ContentsPreface ix 1 What is agent-based computational sociology all about? 1 1.1 Predecessors and fathers 3 1.2 The main ideas of agent-based computational sociology 9 1.2.1 The primacy of models 9 1.2.2 The generative approach 11 1.2.3 The micro–macro link 13 1.2.4 Process and change 15 1.2.5 The unexcluded middle 16 1.2.6 Trans-disciplinarity 17 1.3 What are ABMs? 18 1.4 A classification of ABM use in social research 20 References 26 2 Cooperation, coordination and social norms 33 2.1 Direct reciprocity and the persistence of interaction 36 2.2 Strong reciprocity and social sanctions 42 2.3 Disproportionate prior exposure 49 2.4 Partner selection 54 2.5 Reputation 62 2.6 The emergence of conventions 69 References 78 3 Social influence 85 3.1 Segregation dynamics 88 3.2 Threshold behavior and opinions 97 3.3 Culture dynamics and diversity 103 3.4 Social reflexivity 109 References 122 4 The methodology 131 4.1 The method 134 4.2 Replication 140 4.2.1 The querelle about segregation 144 4.2.2 The querelle about trust and mobility 147 4.3 Multi-level empirical validation 151 References 159 5 Conclusions 165 References 172 Appendix A 175 A. 1 Research centers 175 A. 2 Scientific associations 177 A. 3 Journals 178 A. 4 Simulation tools 179 References 179 Appendix B 181 B. 1 Example I: Partner selection and dynamic networks (Boero, Bravo and Squazzoni 2010) 182 B. 2 Example II: Reputation (Boero et al. 2010) 211 References 234 Index 235

Read more less