Probability and statistics Books

Book SynopsisLearning a computer language like R can be either frustrating, fun or boring. Having fun requires challenges that wake up the learner's curiosity but also provide an emotional reward for overcoming them. The book is designed so that it includes smaller and bigger challenges, in what I call playgrounds, in the hope that all readers will enjoy their path to R fluency. Fluency in the use of a language is a skill that is acquired through practice and exploration. For students and professionals in the biological sciences, humanities and many applied fields, recognizing the parallels between R and natural languages should help them feel at home with R. The approach I use is similar to that of a travel guide, encouraging exploration and describing the available alternatives and how to reach them. The intention is to guide the reader through the R landscape of 2024 and beyond.What is new in the second edition?Text expanded by more than 25% to include additional R features

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Book SynopsisThe goal of this unique text is to provide an experience that would facilitate a better transition for mathematics majors to the advanced proof-based courses required for their major.If you feel like you love mathematics but hate proofs, this book is for you. The change from example-based courses such as Introductory Calculus to the proof-based courses in the major is often abrupt, and some students are left with the unpleasant feeling that a subject they loved has turned into material they find hard to understand.The book exposes students and readers to some fundamental content and essential methods of constructing mathematical proofs in the context of four main courses required for the mathematics major probability, linear algebra, real analysis, and abstract algebra.Following an optional foundational chapter on background material, four short chapters, each focusing on a particular course, provide a slow-paced but rigorous introduction. Students get

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisProbabilistic numerical computation formalises the connection between machine learning and applied mathematics. Numerical algorithms approximate intractable quantities from computable ones. They estimate integrals from evaluations of the integrand, or the path of a dynamical system described by differential equations from evaluations of the vector field. In other words, they infer a latent quantity from data. This book shows that it is thus formally possible to think of computational routines as learning machines, and to use the notion of Bayesian inference to build more flexible, efficient, or customised algorithms for computation. The text caters for Masters'' and PhD students, as well as postgraduate researchers in artificial intelligence, computer science, statistics, and applied mathematics. Extensive background material is provided along with a wealth of figures, worked examples, and exercises (with solutions) to develop intuition.Trade Review'Computational methods for solving numerical problems lie at the heart of many of the technological advances in science and engineering over the last five decades, and underpin fields as diverse as artificial intelligence, climate modelling, and epidemiology. This impressive text rethinks numerical problems through the lens of probabilistic inference and decision making. This fresh perspective opens up a new chapter in this field, and suggests new and highly efficient methods. A landmark achievement!' Zoubin Ghahramani, University of Cambridge'This beautiful book is both timely and important with deep roots in powerful early exposition in numerical analysis. In this stunning and comprehensive new book, early developments from Kac and Larkin have been comprehensively built upon, formalised and extended by including modern day machine learning, numerical analysis and the formal Bayesian statistical methodology. Probabilistic Numerical methodology is of enormous importance for this age of data-centric science and Hennig, Osborne and Kersting are to be congratulated in providing us with this definitive volume.' Mark Girolami, University of Cambridge and The Alan Turing Institute'Numerical analysis is at the very heart of digital computing: every result of a computation on a digital computer is a only finite-precision representation of the true mathematical quantity where the precision is the tradeoff between computation time and accuracy. This book presents an in-depth overview of both the past and present of the newly emerging area of probabilistic numerics, where recent advances in probabilistic machine learning are used to develop principled improvements which are both faster and more accurate than classical numerical analysis algorithms. A must-read for every algorithm developer and practitioner in optimization!' Ralf Herbrich, Hasso Plattner Institute'Probabilistic Numerics spans from the intellectual fireworks of the dawn of a new field to its practical algorithmic consequences. It is precise but accessible and rich in wide-ranging, principled examples. This convergence of ideas from diverse fields in lucid style is the very fabric of good science.' Carl Edward Rasmussen, University of Cambridge'An important read for anyone who has thought about uncertainty in numerical methods; an essential read for anyone who hasn't …' John Cunningham, Columbia University'This is a rare example of a textbook that essentially founds a new field, re-casting numerics on stronger, more general foundations. A tour de force.' David Duvenaud, University of Toronto'The idea of applying probabilistic inference to the problem of numerical analysis must appear bold, possibly outrageous, even to an entrenched Bayesian statistician. Many in machine learning are now familiar with the application of Bayesian methods to problems that involve randomness, say, the estimation of quantities from noisy data. But to apply the 'calculus of uncertainty' to unknown mathematical facts, where the uncertainty arises only from our lack of knowledge, opens up a universe of new possibilities. This elegant idea is at the core of Probabilistic Numerics, and the authors succeed in demonstrating its potential to transform the way we think about computation itself. And that's not even considering what would happen if we were to apply probabilistic numerics to the numerical problems that arise from probabilistic numerics itself!' Thore Graepel, Senior Vice President, Altos Labs'… the machine learning background of the authors comes through clearly in the book … I thoroughly recommend it.' Chris J. Oates, SIAM ReviewTable of ContentsIntroduction; 1. Mathematical background; 2. Integration; 3. Linear algebra; 4. Local optimisation; 5. Global optimisation; 6. Solving ordinary differential equations; 7. The frontier; Solutions to exercises; References; Index.

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Representation of data; 2. Measures of central tendency; 3. Measures of variation; 4. Probability; 5. Permutations and combinations; 6. Probability distributions; 7. The binomial and geometric distributions; 8. The normal distribution; Answers

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Hypothesis testing; 2. The Poisson distribution; 3. Linear combinations of random variables; 4. Continuous random variables; 5. Sampling; 6. Estimation; Answers

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020. The coursebook provides clear explanations and practice exercises to help students master maths skills. This edition comes with a subscription to the Probability & Statistics 1 Cambridge Online Mathematics component. With all of the materials found within the coursebook, including answers, Cambridge Online Mathematics offers students the facility to demonstrate their working, as well as opportunities for self-assessment, and allows teachers to set classroom and homework exercises to individual students or classes, with the ability to track progress. The online resource requires internet access. For more information on how to use Cambridge Online Mathematics, please see inside the front cover.

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Book SynopsisDo you want to know what a parametric test is and when not to perform one? Do you get confused between odds ratios and relative risks? Want to understand the difference between sensitivity and specificity? Would like to find out what the fuss is about Bayes'' theorem? Then this book is for you! Physicians need to understand the principles behind medical statistics. They don''t need to learn the formula. The software knows it already! This book explains the fundamental concepts of medical statistics so that the learner will become confident in performing the most commonly used statistical tests. Each chapter is rich in anecdotes, illustrations, questions, and answers. Not enough? There is more material online with links to free statistical software, webpages, multimedia content, a practice dataset to get hands-on with data analysis, and a Single Best Answer questionnaire for the exam.Trade Review'An accessible book by a practising doctor, aimed at other doctors, which explains key statistical concepts in words and pictures. An excellent foundation for those seeking to understand the numbers in medical journal articles and quantitative reports.' Professor Trish Greenhalgh, University of Oxford, UK'Statistics forms the starting point for evidence based medicine, though most medics would argue that their own statistical awareness is still near the starting point! This book eases you into the awesome, exciting, exhilarating world of statistics, and makes you understand just how cool it really is. It will unleash your inner statistician that no-one knew existed - especially you!' Professor Dan Perry, Children's Orthopaedic Surgeon and Fellow of Wolfson College, University of Oxford, UK'The book provides a light-hearted introduction to the basic concepts in medical statistics. A couple of hundred pages long with short chapters, the book delivers with clear focus the key statistical concepts alongside some general knowledge to lighten what is sometimes a very arid subject. The description of concepts with graphs and figures support the visual learner. I thoroughly enjoyed the quick questions presented alongside the description of concepts to test understanding, with the answers at the end of chapter which linked to bullet point summaries, help to consolidate the concepts covered. I thought it was an excellent way for someone to start on their path to understanding this area. Finally, I particularly appreciated the last chapter with its focus on the work by our dear friend Doug Altman.' Professor Rafael Perera, Professor of Medical Statistics, University of Oxford, UK'This is an excellent introductory book for medical statistics. It's well written, easy to read, with some great examples of statistics in everyday clinical practice. The question and answer format is especially useful in reinforcing key concepts discussed in the chapter. There are lots of additional learning material included in the online resource for those seeking a more detailed understanding of the topic. The author is to be congratulated on making an important but difficult subject appear relatively straightforward and interesting to even the non-expert.' Professor Paul Banaszkiewicz, Consultant Orthopaedic Surgeon North East NHS Surgical Centre (NENSC), Gateshead, UK, and Visiting Professor Northumbria University, Newcastle-upon-Tyne, UK 'The modern practice in orthopaedic and trauma surgery is a completely different practice to that which many of us grew up with and is now firmly founded on research and evidence. And this evidence itself is built around scientific method and statistical analysis. This excellent book provides a comprehensive guide to biostatistics for the orthopaedic surgeon and aspiring clinician scientist. Through clear explanations of complex concepts the author succeeds in simplifying the difficult and I am sure this will become an essential reference for all involved in orthopaedic surgery.' Professor Ben Ollivere, Professor of Orthopaedic Trauma, University of Nottingham, UKTable of ContentsPreface; Acknowledgements; How to get the best out of this book; 1. Medicine and numbers: what is the connection?; 2. Measuring a variable: what is the difference between eye colour and height; 3. Summarising data: communicating easily; 4. Why average and range is not always enough: standard deviation and standard error; 5. The normal distribution: what's so 'normal' about it?; 6. Confidence interval: what is your guesstimate?; 7. Innocent until proven guilty! The null hypothesis; 8. Errors in hypothesis tests: learn your α from your β; 9. The randomized controlled trial: why does it have to be random?; 10. Choosing a statistical test: to t or not to t?; 11. Finding the odd one out: the ANOVA test; 12. Categorically different? The Chi-Squared test; 13. If the line fits: correlation and linear regression; 14. Hindsight is 20/20: logistic regression; 15. Don't risk the odds. risk vs odds as the outcome measure; 16. I will survive! Time to event data analysis; 17. High-ceiling or low threshold? Accuracy of a diagnostic test; 18. Apples or oranges? Meta-analysis of selected studies; 19. Lies, damned lies and statistics: untangling facts from fiction!; Glossary; Appendix 1. Are you ready to test yourself? Single Best Answer questionnaire; Appendix 2. Software and practice problems; Appendix 3. Practice database; Appendix 4. References and further reading; Appendix 5. Further resources.

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Book SynopsisBayesian Statistics is the school of thought that combines prior beliefs with the likelihood of a hypothesis to arrive at posterior beliefs. The first edition of Peter Lee s book appeared in 1989, but the subject has moved ever onwards, with increasing emphasis on Monte Carlo based techniques.Trade Review“As a lifelong non-statistician and sporadic “user” of statistics, I have not come across another advanced statistics book (as I would characterize this one) that offers so much to the non-expert and, I’ll bet, to the expert as well. The book has my highest recommendation.” (Computing Reviews, 7 January 2013)Table of ContentsPreface xix Preface to the First Edition xxi 1 Preliminaries 1 1.1 Probability and Bayes’ Theorem 1 1.1.1 Notation 1 1.1.2 Axioms for probability 2 1.1.3 ‘Unconditional’ probability 5 1.1.4 Odds 6 1.1.5 Independence 7 1.1.6 Some simple consequences of the axioms; Bayes’ Theorem 7 1.2 Examples on Bayes’ Theorem 9 1.2.1 The Biology of Twins 9 1.2.2 A political example 10 1.2.3 A warning 10 1.3 Random variables 12 1.3.1 Discrete random variables 12 1.3.2 The binomial distribution 13 1.3.3 Continuous random variables 14 1.3.4 The normal distribution 16 1.3.5 Mixed random variables 17 1.4 Several random variables 17 1.4.1 Two discrete random variables 17 1.4.2 Two continuous random variables 18 1.4.3 Bayes’ Theorem for random variables 20 1.4.4 Example 21 1.4.5 One discrete variable and one continuous variable 21 1.4.6 Independent random variables 22 1.5 Means and variances 23 1.5.1 Expectations 23 1.5.2 The expectation of a sum and of a product 24 1.5.3 Variance, precision and standard deviation 25 1.5.4 Examples 25 1.5.5 Variance of a sum; covariance and correlation 27 1.5.6 Approximations to the mean and variance of a function of a random variable 28 1.5.7 Conditional expectations and variances 29 1.5.8 Medians and modes 31 1.6 Exercises on Chapter 1 31 2 Bayesian inference for the normal distribution 36 2.1 Nature of Bayesian inference 36 2.1.1 Preliminary remarks 36 2.1.2 Post is prior times likelihood 36 2.1.3 Likelihood can be multiplied by any constant 38 2.1.4 Sequential use of Bayes’ Theorem 38 2.1.5 The predictive distribution 39 2.1.6 A warning 39 2.2 Normal prior and likelihood 40 2.2.1 Posterior from a normal prior and likelihood 40 2.2.2 Example 42 2.2.3 Predictive distribution 43 2.2.4 The nature of the assumptions made 44 2.3 Several normal observations with a normal prior 44 2.3.1 Posterior distribution 44 2.3.2 Example 46 2.3.3 Predictive distribution 47 2.3.4 Robustness 47 2.4 Dominant likelihoods 48 2.4.1 Improper priors 48 2.4.2 Approximation of proper priors by improper priors 49 2.5 Locally uniform priors 50 2.5.1 Bayes’ postulate 50 2.5.2 Data translated likelihoods 52 2.5.3 Transformation of unknown parameters 52 2.6 Highest density regions 54 2.6.1 Need for summaries of posterior information 54 2.6.2 Relation to classical statistics 55 2.7 Normal variance 55 2.7.1 A suitable prior for the normal variance 55 2.7.2 Reference prior for the normal variance 58 2.8 HDRs for the normal variance 59 2.8.1 What distribution should we be considering? 59 2.8.2 Example 59 2.9 The role of sufficiency 60 2.9.1 Definition of sufficiency 60 2.9.2 Neyman’s factorization theorem 61 2.9.3 Sufficiency principle 63 2.9.4 Examples 63 2.9.5 Order statistics and minimal sufficient statistics 65 2.9.6 Examples on minimal sufficiency 66 2.10 Conjugate prior distributions 67 2.10.1 Definition and difficulties 67 2.10.2 Examples 68 2.10.3 Mixtures of conjugate densities 69 2.10.4 Is your prior really conjugate? 71 2.11 The exponential family 71 2.11.1 Definition 71 2.11.2 Examples 72 2.11.3 Conjugate densities 72 2.11.4 Two-parameter exponential family 73 2.12 Normal mean and variance both unknown 73 2.12.1 Formulation of the problem 73 2.12.2 Marginal distribution of the mean 75 2.12.3 Example of the posterior density for the mean 76 2.12.4 Marginal distribution of the variance 77 2.12.5 Example of the posterior density of the variance 77 2.12.6 Conditional density of the mean for given variance 77 2.13 Conjugate joint prior for the normal distribution 78 2.13.1 The form of the conjugate prior 78 2.13.2 Derivation of the posterior 80 2.13.3 Example 81 2.13.4 Concluding remarks 82 2.14 Exercises on Chapter 2 82 3 Some other common distributions 85 3.1 The binomial distribution 85 3.1.1 Conjugate prior 85 3.1.2 Odds and log-odds 88 3.1.3 Highest density regions 90 3.1.4 Example 91 3.1.5 Predictive distribution 92 3.2 Reference prior for the binomial likelihood 92 3.2.1 Bayes’ postulate 92 3.2.2 Haldane’s prior 93 3.2.3 The arc-sine distribution 94 3.2.4 Conclusion 95 3.3 Jeffreys’ rule 96 3.3.1 Fisher’s information 96 3.3.2 The information from several observations 97 3.3.3 Jeffreys’ prior 98 3.3.4 Examples 98 3.3.5 Warning 100 3.3.6 Several unknown parameters 100 3.3.7 Example 101 3.4 The Poisson distribution 102 3.4.1 Conjugate prior 102 3.4.2 Reference prior 103 3.4.3 Example 104 3.4.4 Predictive distribution 104 3.5 The uniform distribution 106 3.5.1 Preliminary definitions 106 3.5.2 Uniform distribution with a fixed lower endpoint 107 3.5.3 The general uniform distribution 108 3.5.4 Examples 110 3.6 Reference prior for the uniform distribution 110 3.6.1 Lower limit of the interval fixed 110 3.6.2 Example 111 3.6.3 Both limits unknown 111 3.7 The tramcar problem 113 3.7.1 The discrete uniform distribution 113 3.8 The first digit problem; invariant priors 114 3.8.1 A prior in search of an explanation 114 3.8.2 The problem 114 3.8.3 A solution 115 3.8.4 Haar priors 117 3.9 The circular normal distribution 117 3.9.1 Distributions on the circle 117 3.9.2 Example 119 3.9.3 Construction of an HDR by numerical integration 120 3.9.4 Remarks 122 3.10 Approximations based on the likelihood 122 3.10.1 Maximum likelihood 122 3.10.2 Iterative methods 123 3.10.3 Approximation to the posterior density 123 3.10.4 Examples 124 3.10.5 Extension to more than one parameter 126 3.10.6 Example 127 3.11 Reference posterior distributions 128 3.11.1 The information provided by an experiment 128 3.11.2 Reference priors under asymptotic normality 130 3.11.3 Uniform distribution of unit length 131 3.11.4 Normal mean and variance 132 3.11.5 Technical complications 134 3.12 Exercises on Chapter 3 134 4 Hypothesis testing 138 4.1 Hypothesis testing 138 4.1.1 Introduction 138 4.1.2 Classical hypothesis testing 138 4.1.3 Difficulties with the classical approach 139 4.1.4 The Bayesian approach 140 4.1.5 Example 142 4.1.6 Comment 143 4.2 One-sided hypothesis tests 143 4.2.1 Definition 143 4.2.2 P-values 144 4.3 Lindley’s method 145 4.3.1 A compromise with classical statistics 145 4.3.2 Example 145 4.3.3 Discussion 146 4.4 Point (or sharp) null hypotheses with prior information 146 4.4.1 When are point null hypotheses reasonable? 146 4.4.2 A case of nearly constant likelihood 147 4.4.3 The Bayesian method for point null hypotheses 148 4.4.4 Sufficient statistics 149 4.5 Point null hypotheses for the normal distribution 150 4.5.1 Calculation of the Bayes’ factor 150 4.5.2 Numerical examples 151 4.5.3 Lindley’s paradox 152 4.5.4 A bound which does not depend on the prior distribution 154 4.5.5 The case of an unknown variance 155 4.6 The Doogian philosophy 157 4.6.1 Description of the method 157 4.6.2 Numerical example 157 4.7 Exercises on Chapter 4 158 5 Two-sample problems 162 5.1 Two-sample problems – both variances unknown 162 5.1.1 The problem of two normal samples 162 5.1.2 Paired comparisons 162 5.1.3 Example of a paired comparison problem 163 5.1.4 The case where both variances are known 163 5.1.5 Example 164 5.1.6 Non-trivial prior information 165 5.2 Variances unknown but equal 165 5.2.1 Solution using reference priors 165 5.2.2 Example 167 5.2.3 Non-trivial prior information 167 5.3 Variances unknown and unequal (Behrens–Fisher problem) 168 5.3.1 Formulation of the problem 168 5.3.2 Patil’s approximation 169 5.3.3 Example 170 5.3.4 Substantial prior information 170 5.4 The Behrens–Fisher controversy 171 5.4.1 The Behrens–Fisher problem from a classical standpoint 171 5.4.2 Example 172 5.4.3 The controversy 173 5.5 Inferences concerning a variance ratio 173 5.5.1 Statement of the problem 173 5.5.2 Derivation of the F distribution 174 5.5.3 Example 175 5.6 Comparison of two proportions; the 2 × 2 table 176 5.6.1 Methods based on the log-odds ratio 176 5.6.2 Example 177 5.6.3 The inverse root-sine transformation 178 5.6.4 Other methods 178 5.7 Exercises on Chapter 5 179 6 Correlation, regression and the analysis of variance 182 6.1 Theory of the correlation coefficient 182 6.1.1 Definitions 182 6.1.2 Approximate posterior distribution of the correlation coefficient 184 6.1.3 The hyperbolic tangent substitution 186 6.1.4 Reference prior 188 6.1.5 Incorporation of prior information 189 6.2 Examples on the use of the correlation coefficient 189 6.2.1 Use of the hyperbolic tangent transformation 189 6.2.2 Combination of several correlation coefficients 189 6.2.3 The squared correlation coefficient 190 6.3 Regression and the bivariate normal model 190 6.3.1 The model 190 6.3.2 Bivariate linear regression 191 6.3.3 Example 193 6.3.4 Case of known variance 194 6.3.5 The mean value at a given value of the explanatory variable 194 6.3.6 Prediction of observations at a given value of the explanatory variable 195 6.3.7 Continuation of the example 195 6.3.8 Multiple regression 196 6.3.9 Polynomial regression 196 6.4 Conjugate prior for the bivariate regression model 197 6.4.1 The problem of updating a regression line 197 6.4.2 Formulae for recursive construction of a regression line 197 6.4.3 Finding an appropriate prior 199 6.5 Comparison of several means – the one way model 200 6.5.1 Description of the one way layout 200 6.5.2 Integration over the nuisance parameters 201 6.5.3 Derivation of the F distribution 203 6.5.4 Relationship to the analysis of variance 203 6.5.5 Example 204 6.5.6 Relationship to a simple linear regression model 206 6.5.7 Investigation of contrasts 207 6.6 The two way layout 209 6.6.1 Notation 209 6.6.2 Marginal posterior distributions 210 6.6.3 Analysis of variance 212 6.7 The general linear model 212 6.7.1 Formulation of the general linear model 212 6.7.2 Derivation of the posterior 214 6.7.3 Inference for a subset of the parameters 215 6.7.4 Application to bivariate linear regression 216 6.8 Exercises on Chapter 6 217 7 Other topics 221 7.1 The likelihood principle 221 7.1.1 Introduction 221 7.1.2 The conditionality principle 222 7.1.3 The sufficiency principle 223 7.1.4 The likelihood principle 223 7.1.5 Discussion 225 7.2 The stopping rule principle 226 7.2.1 Definitions 226 7.2.2 Examples 226 7.2.3 The stopping rule principle 227 7.2.4 Discussion 228 7.3 Informative stopping rules 229 7.3.1 An example on capture and recapture of fish 229 7.3.2 Choice of prior and derivation of posterior 230 7.3.3 The maximum likelihood estimator 231 7.3.4 Numerical example 231 7.4 The likelihood principle and reference priors 232 7.4.1 The case of Bernoulli trials and its general implications 232 7.4.2 Conclusion 233 7.5 Bayesian decision theory 234 7.5.1 The elements of game theory 234 7.5.2 Point estimators resulting from quadratic loss 236 7.5.3 Particular cases of quadratic loss 237 7.5.4 Weighted quadratic loss 238 7.5.5 Absolute error loss 238 7.5.6 Zero-one loss 239 7.5.7 General discussion of point estimation 240 7.6 Bayes linear methods 240 7.6.1 Methodology 240 7.6.2 Some simple examples 241 7.6.3 Extensions 243 7.7 Decision theory and hypothesis testing 243 7.7.1 Relationship between decision theory and classical hypothesis testing 243 7.7.2 Composite hypotheses 245 7.8 Empirical Bayes methods 245 7.8.1 Von Mises’ example 245 7.8.2 The Poisson case 246 7.9 Exercises on Chapter 7 247 8 Hierarchical models 253 8.1 The idea of a hierarchical model 253 8.1.1 Definition 253 8.1.2 Examples 254 8.1.3 Objectives of a hierarchical analysis 257 8.1.4 More on empirical Bayes methods 257 8.2 The hierarchical normal model 258 8.2.1 The model 258 8.2.2 The Bayesian analysis for known overall mean 259 8.2.3 The empirical Bayes approach 261 8.3 The baseball example 262 8.4 The Stein estimator 264 8.4.1 Evaluation of the risk of the James–Stein estimator 267 8.5 Bayesian analysis for an unknown overall mean 268 8.5.1 Derivation of the posterior 270 8.6 The general linear model revisited 272 8.6.1 An informative prior for the general linear model 272 8.6.2 Ridge regression 274 8.6.3 A further stage to the general linear model 275 8.6.4 The one way model 276 8.6.5 Posterior variances of the estimators 277 8.7 Exercises on Chapter 8 277 9 The Gibbs sampler and other numerical methods 281 9.1 Introduction to numerical methods 281 9.1.1 Monte Carlo methods 281 9.1.2 Markov chains 282 9.2 The EM algorithm 283 9.2.1 The idea of the EM algorithm 283 9.2.2 Why the EM algorithm works 285 9.2.3 Semi-conjugate prior with a normal likelihood 287 9.2.4 The EM algorithm for the hierarchical normal model 288 9.2.5 A particular case of the hierarchical normal model 290 9.3 Data augmentation by Monte Carlo 291 9.3.1 The genetic linkage example revisited 291 9.3.2 Use of R 291 9.3.3 The genetic linkage example in R 292 9.3.4 Other possible uses for data augmentation 293 9.4 The Gibbs sampler 294 9.4.1 Chained data augmentation 294 9.4.2 An example with observed data 296 9.4.3 More on the semi-conjugate prior with a normal likelihood 299 9.4.4 The Gibbs sampler as an extension of chained data augmentation 301 9.4.5 An application to change-point analysis 302 9.4.6 Other uses of the Gibbs sampler 306 9.4.7 More about convergence 309 9.5 Rejection sampling 311 9.5.1 Description 311 9.5.2 Example 311 9.5.3 Rejection sampling for log-concave distributions 311 9.5.4 A practical example 313 9.6 The Metropolis–Hastings algorithm 317 9.6.1 Finding an invariant distribution 317 9.6.2 The Metropolis–Hastings algorithm 318 9.6.3 Choice of a candidate density 320 9.6.4 Example 321 9.6.5 More realistic examples 322 9.6.6 Gibbs as a special case of Metropolis–Hastings 322 9.6.7 Metropolis within Gibbs 323 9.7 Introduction to WinBUGS and OpenBUGS 323 9.7.1 Information about WinBUGS and OpenBUGS 323 9.7.2 Distributions in WinBUGS and OpenBUGS 324 9.7.3 A simple example using WinBUGS 324 9.7.4 The pump failure example revisited 327 9.7.5 DoodleBUGS 327 9.7.6 coda 329 9.7.7 R2WinBUGS and R2OpenBUGS 329 9.8 Generalized linear models 332 9.8.1 Logistic regression 332 9.8.2 A general framework 334 9.9 Exercises on Chapter 9 335 10 Some approximate methods 340 10.1 Bayesian importance sampling 340 10.1.1 Importance sampling to find HDRs 343 10.1.2 Sampling importance re-sampling 344 10.1.3 Multidimensional applications 344 10.2 Variational Bayesian methods: simple case 345 10.2.1 Independent parameters 347 10.2.2 Application to the normal distribution 349 10.2.3 Updating the mean 350 10.2.4 Updating the variance 351 10.2.5 Iteration 352 10.2.6 Numerical example 352 10.3 Variational Bayesian methods: general case 353 10.3.1 A mixture of multivariate normals 353 10.4 ABC: Approximate Bayesian Computation 356 10.4.1 The ABC rejection algorithm 356 10.4.2 The genetic linkage example 358 10.4.3 The ABC Markov Chain Monte Carlo algorithm 360 10.4.4 The ABC Sequential Monte Carlo algorithm 362 10.4.5 The ABC local linear regression algorithm 365 10.4.6 Other variants of ABC 366 10.5 Reversible jump Markov chain Monte Carlo 367 10.5.1 RJMCMC algorithm 367 10.6 Exercises on Chapter 10 369 Appendix A Common statistical distributions 373 A.1 Normal distribution 374 A.2 Chi-squared distribution 375 A.3 Normal approximation to chi-squared 376 A.4 Gamma distribution 376 A.5 Inverse chi-squared distribution 377 A.6 Inverse chi distribution 378 A.7 Log chi-squared distribution 379 A.8 Student’s t distribution 380 A.9 Normal/chi-squared distribution 381 A.10 Beta distribution 382 A.11 Binomial distribution 383 A.12 Poisson distribution 384 A.13 Negative binomial distribution 385 A.14 Hypergeometric distribution 386 A.15 Uniform distribution 387 A.16 Pareto distribution 388 A.17 Circular normal distribution 389 A.18 Behrens’ distribution 391 A.19 Snedecor’s F distribution 393 A.20 Fisher’s z distribution 393 A.21 Cauchy distribution 394 A.22 The probability that one beta variable is greater than another 395 A.23 Bivariate normal distribution 395 A.24 Multivariate normal distribution 396 A.25 Distribution of the correlation coefficient 397 Appendix B Tables 399 B.1 Percentage points of the Behrens–Fisher distribution 399 B.2 Highest density regions for the chi-squared distribution 402 B.3 HDRs for the inverse chi-squared distribution 404 B.4 Chi-squared corresponding to HDRs for log chi-squared 406 B.5 Values of F corresponding to HDRs for log F 408 Appendix C R programs 430 Appendix D Further reading 436 D.1 Robustness 436 D.2 Nonparametric methods 436 D.3 Multivariate estimation 436 D.4 Time series and forecasting 437 D.5 Sequential methods 437 D.6 Numerical methods 437 D.7 Bayesian networks 437 D.8 General reading 438 References 439 Index 455

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Book SynopsisFundamentals of Statistical Reasoning in Education 4th Edition, like the first three editions, is written largely with students of education in mind. Accordingly, Theodore Coladarci and Casey D. Cobb have drawn primarily on examples and issues found in school settings, such as those having to do with instruction, learning, motivation, and assessment. The emphasis on educational applications notwithstanding, the authors are confident that readers will find Fundamentals of Statistical Reasoning in Education 4th Edition of general relevance to other disciplines in the behavioral sciences as well. The 4th Edition of Fundamentals is still designed as a one semester book. The authors intentionally sidestep topics that few introductory courses cover (e.g., factorial analysis of variance, repeated measures analysis of variance, multiple regression). At the same time, effect size and confidence intervals are incorporated throughout, which today are regarded Trade Review"This book, like the first three editions, is written largely with students of education in mind. Accordingly, the authors have drawn primarily on examples and issues found in school settings, such as those having to do with instruction, learning, motivation, and assessment. The emphasis on educational applications notwithstanding, the authors are confident that readers will find this book of general relevance to other disciplines in the behavioral sciences as well." (Zentralblatt MATH 2016)Table of ContentsChapter 1 Introduction 1 1.1 Why Statistics? 1 1.2 Descriptive Statistics 2 1.3 Inferential Statistics 3 1.4 The Role of Statistics in Educational Research 4 1.5 Variables and Their Measurement 5 1.6 Some Tips on Studying Statistics 8 PART 1 DESCRIPTIVE STATISTICS 13 Chapter 2 Frequency Distributions 14 2.1 Why Organize Data? 14 2.2 Frequency Distributions for Quantitative Variables 14 2.3 Grouped Scores 15 2.4 Some Guidelines for Forming Class Intervals 17 2.5 Constructing a Grouped-Data Frequency Distribution 18 2.6 The Relative Frequency Distribution 19 2.7 Exact Limits 21 2.8 The Cumulative Percentage Frequency Distribution 22 2.9 Percentile Ranks 23 2.10 Frequency Distributions for Qualitative Variables 25 2.11 Summary 26 Chapter 3 Graphic Representation 34 3.1 Why Graph Data? 34 3.2 Graphing Qualitative Data: The Bar Chart 34 3.3 Graphing Quantitative Data: The Histogram 35 3.4 Relative Frequency and Proportional Area 39 3.5 Characteristics of Frequency Distributions 41 3.6 The Box Plot 44 3.7 Summary 45 Chapter 4 Central Tendency 52 4.1 The Concept of Central Tendency 52 4.2 The Mode 52 4.3 The Median 53 4.4 The Arithmetic Mean 54 4.5 Central Tendency and Distribution Symmetry 57 4.6 Which Measure of Central Tendency to Use? 59 4.7 Summary 59 Chapter 5 Variability 66 5.1 Central Tendency Is Not Enough: The Importance of Variability 66 5.2 The Range 67 5.3 Variability and Deviations From the Mean 68 5.4 The Variance 69 5.5 The Standard Deviation 70 5.6 The Predominance of the Variance and Standard Deviation 71 5.7 The Standard Deviation and the Normal Distribution 72 5.8 Comparing Means of Two Distributions: The Relevance of Variability 73 5.9 In the Denominator: n Versus n −1 75 5.10 Summary 76 Chapter 6 Normal Distributions and Standard Scores 81 6.1 A Little History: Sir Francis Galton and the Normal Curve 81 6.2 Properties of the Normal Curve 82 6.3 More on the Standard Deviation and the Normal Distribution 82 6.4 z Scores 84 6.5 The Normal Curve Table 87 6.6 Finding Area When the Score Is Known 88 6.7 Reversing the Process: Finding Scores When the Area Is Known 91 6.8 Comparing Scores From Different Distributions 93 6.9 Interpreting Effect Size 94 6.10 Percentile Ranks and the Normal Distribution 96 6.11 Other Standard Scores 97 6.12 Standard Scores Do Not “Normalize” a Distribution 98 6.13 The Normal Curve and Probability 98 6.14 Summary 99 Chapter 7 Correlation 106 7.1 The Concept of Association 106 7.2 Bivariate Distributions and Scatterplots 106 7.3 The Covariance 111 7.4 The Pearson r 117 7.5 Computation of r: The Calculating Formula 118 7.6 Correlation and Causation 120 7.7 Factors Influencing Pearson r 122 7.8 Judging the Strength of Association: r2 125 7.9 Other Correlation Coefficients 127 7.10 Summary 127 Chapter 8 Regression and Prediction 134 8.1 Correlation Versus Prediction 134 8.2 Determining the Line of Best Fit 135 8.3 The Regression Equation in Terms of Raw Scores 138 8.4 Interpreting the Raw-Score Slope 141 8.5 The Regression Equation in Terms of z Scores 141 8.6 Some Insights Regarding Correlation and Prediction 142 8.7 Regression and Sums of Squares 145 8.8 Residuals and Unexplained Variation 147 8.9 Measuring the Margin of Prediction Error: The Standard Error of Estimate 148 8.10 Correlation and Causality (Revisited) 152 8.11 Summary 153 PART 2 INFERENTIAL STATISTICS 163 Chapter 9 Probability and Probability Distributions 164 9.1 Statistical Inference: Accounting for Chance in Sample Results 164 9.2 Probability: The Study of Chance 165 9.3 Definition of Probability 166 9.4 Probability Distributions 168 9.5 The OR/addition Rule 169 9.6 The AND/Multiplication Rule 171 9.7 The Normal Curve as a Probability Distribution 172 9.8 “So What?”—Probability Distributions as the Basis for Statistical Inference 174 9.9 Summary 175 Chapter 10 Sampling Distributions 179 10.1 From Coins to Means 179 10.2 Samples and Populations 180 10.3 Statistics and Parameters 181 10.4 Random Sampling Model 181 10.5 Random Sampling in Practice 183 10.6 Sampling Distributions of Means 184 10.7 Characteristics of a Sampling Distribution of Means 185 10.8 Using a Sampling Distribution of Means to Determine Probabilities 188 10.9 The Importance of Sample Size (n) 191 10.10 Generality of the Concept of a Sampling Distribution 193 10.11 Summary 193 Chapter 11 Testing Statistical Hypotheses About μ When σ Is Known: The One-Sample z Test 199 11.1 Testing a Hypothesis About μ: Does “Homeschooling” Make a Difference? 199 11.2 Dr. Meyer’s Problem in a Nutshell 200 11.3 The Statistical Hypotheses: H0 and H1 201 11.4 The Test Statistic z 202 11.5 The Probability of the Test Statistic: The p Value 203 11.6 The Decision Criterion: Level of Significance (α) 204 11.7 The Level of Significance and Decision Error 207 11.8 The Nature and Role of H0 and H1 209 11.9 Rejection Versus Retention of H0 209 11.10 Statistical Significance Versus Importance 210 11.11 Directional and Nondirectional Alternative Hypotheses 212 11.12 The Substantive Versus the Statistical 214 11.13 Summary 215 Chapter 12 Estimation 222 12.1 Hypothesis Testing Versus Estimation 222 12.2 Point Estimation Versus Interval Estimation 223 12.3 Constructing an Interval Estimate of μ 224 12.4 Interval Width and Level of Confidence 226 12.5 Interval Width and Sample Size 227 12.6 Interval Estimation and Hypothesis Testing 228 12.7 Advantages of Interval Estimation 230 12.8 Summary 230 Chapter 13 Testing Statistical Hypotheses About μ When σ Is Not Known: The One-Sample t Test 235 13.1 Reality: σ Often Is Unknown 235 13.2 Estimating the Standard Error of the Mean 236 13.3 The Test Statistic t 237 13.4 Degrees of Freedom 238 13.5 The Sampling Distribution of Student’s t 239 13.6 An Application of Student’s t 242 13.7 Assumption of Population Normality 244 13.8 Levels of Significance Versus p Values 244 13.9 Constructing a Confidence Interval for μ When σ Is Not Known 246 13.10 Summary 247 Chapter 14 Comparing the Means of Two Populations: Independent Samples 253 14.1 From One Mu (μ) to Two 253 14.2 Statistical Hypotheses 254 14.3 The Sampling Distribution of Differences Between Means 255 14.4 Estimating σx̄1-x̄2 257 14.5 The t Test for Two Independent Samples 259 14.6 Testing Hypotheses About Two Independent Means: An Example 260 14.7 Interval Estimation of μ1 − μ2 262 14.8 Appraising the Magnitude of a Difference: Measures of Effect Size for − 264 14.9 How Were Groups Formed? The Role of Randomization 268 14.10 Statistical Inferences and Nonstatistical Generalizations 269 14.11 Summary 270 Chapter 15 Comparing the Means of Dependent Samples 278 15.1 The Meaning of “Dependent” 278 15.2 Standard Error of the Difference Between Dependent Means 279 15.3 Degrees of Freedom 281 15.4 The t Test for Two Dependent Samples 281 15.5 Testing Hypotheses About Two Dependent Means: An Example 283 15.6 Interval Estimation of μD 286 15.7 Summary 287 Chapter 16 Comparing the Means of Three or More Independent Samples: One-Way Analysis of Variance 294 16.1 Comparing More Than Two Groups: Why Not Multiplet Tests? 294 16.2 The Statistical Hypotheses in One-Way ANOVA 295 16.3 The Logic of One-Way ANOVA: An Overview 296 16.4 Alison’s Reply to Gregory 299 16.5 Partitioning the Sums of Squares 300 16.6 Within-Groups and Between- Groups Variance Estimates 303 16.7 The F Test 304 16.8 Tukey’s “HSD” Test 306 16.9 Interval Estimation of μi − μj 308 16.10 One-Way ANOVA: Summarizing the Steps 309 16.11 Estimating the Strength of the Treatment Effect: Effect Size (ω2) 311 16.12 ANOVA Assumptions (and Other Considerations) 312 16.13 Summary 313 Chapter 17 Inferences about the Pearson Correlation Coefficient 322 17.1 From μ to ρ 322 17.2 The Sampling Distribution of r When ρ = 0 322 17.3 Testing the Statistical Hypothesis That ρ = 0 324 17.4 An Example 324 17.5 In Brief: Student’s t Distribution and the Regression Slope (b) 326 17.6 Table E 326 17.7 The Role of n in the Statistical Significance of r 328 17.8 Statistical Significance Versus Importance (Again) 329 17.9 Testing Hypotheses Other Than ρ = 0 329 17.10 Interval Estimation of ρ 330 17.11 Summary 332 Chapter 18 Making Inferences From Frequency Data 338 18.1 Frequency Data Versus Score Data 338 18.2 A Problem Involving Frequencies: The One-Variable Case 339 18.3 χ2: A Measure of Discrepancy Between Expected and Observed Frequencies 340 18.4 The Sampling Distribution of χ2 341 18.5 Completion of the Voter Survey Problem: The χ2 Goodness-of-Fit Test 343 18.6 The χ2 Test of a Single Proportion 344 18.7 Interval Estimate of a Single Proportion 345 18.8 When There Are Two Variables: The χ2 Test of Independence 347 18.9 Finding Expected Frequencies in the Two-Variable Case 348 18.10 Calculating the Two-Variable χ2 350 18.11 The χ2 Test of Independence: Summarizing the Steps 351 18.12 The 2 × 2 Contingency Table 352 18.13 Testing a Difference Between Two Proportions 353 18.14 The Independence of Observations 353 18.15 χ2 and Quantitative Variables 354 18.16 Other Considerations 355 18.17 Summary 355 Chapter 19 Statistical “Power” (and How to Increase It) 363 19.1 The Power of a Statistical Test 363 19.2 Power and Type II Error 364 19.3 Effect Size (Revisited) 365 19.4 Factors Affecting Power: The Effect Size 366 19.5 Factors Affecting Power: Sample Size 367 19.6 Additional Factors Affecting Power 368 19.7 Significance Versus Importance 369 19.8 Selecting an Appropriate Sample Size 370 19.9 Summary 373 Epilogue A Note on (Almost) Assumption-Free Tests 379 References 380 Appendix A Review of Basic Mathematics 382 A.1 Introduction 382 A.2 Symbols and Their Meaning 382 A.3 Arithmetic Operations Involving Positive and Negative Numbers 383 A.4 Squares and Square Roots 383 A.5 Fractions 384 A.6 Operations Involving Parentheses 385 A.7 Approximate Numbers, Computational Accuracy, and Rounding 386 Appendix B Answers to Selected End-of-Chapter Problems 387 Appendix C Statistical Tables 408 Glossary 421 Index 427 Useful Formulas 433

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Book SynopsisMathematics and Statistics for Financial Risk Management is a practical guide to modern financial risk management for both practitioners and academics. Now in its second edition with more topics, more sample problems and more real world examples, this popular guide to financial risk management introduces readers to practical quantitative techniques for analyzing and managing financial risk. In a concise and easy-to-read style, each chapter introduces a different topic in mathematics or statistics.As different techniques are introduced, sample problems and application sections demonstrate how these techniques can be applied to actual risk management problems. Exercises at the end of each chapter and the accompanying solutions at the end of the book allow readers to practice the techniques they are learning and monitor their progress.A companion Web site includes interactive Excel spreadsheet examples and templates. Mathematics and Statistics for FinanciTable of ContentsPreface ix What’s New in the Second Edition xi Acknowledgments xiii Chapter 1 Some Basic Math 1 Logarithms 1 Log Returns 2 Compounding 3 Limited Liability 4 Graphing Log Returns 5 Continuously Compounded Returns 6 Combinatorics 8 Discount Factors 9 Geometric Series 9 Problems 14 Chapter 2 Probabilities 15 Discrete Random Variables 15 Continuous Random Variables 15 Mutually Exclusive Events 21 Independent Events 22 Probability Matrices 22 Conditional Probability 24 Problems 26 Chapter 3 Basic Statistics 29 Averages 29 Expectations 34 Variance and Standard Deviation 39 Standardized Variables 41 Covariance 42 Correlation 43 Application: Portfolio Variance and Hedging 44 Moments 47 Skewness 48 Kurtosis 51 Coskewness and Cokurtosis 53 Best Linear Unbiased Estimator (BLUE) 57 Problems 58 Chapter 4 Distributions 61 Parametric Distributions 61 Uniform Distribution 61 Bernoulli Distribution 63 Binomial Distribution 65 Poisson Distribution 68 Normal Distribution 69 Lognormal Distribution 72 Central Limit Theorem 73 Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 76 Chi-Squared Distribution 77 Student’s t Distribution 78 F-Distribution 79 Triangular Distribution 81 Beta Distribution 82 Mixture Distributions 83 Problems 86 Chapter 5 Multivariate Distributions and Copulas 89 Multivariate Distributions 89 Copulas 97 Problems 111 Chapter 6 Bayesian Analysis 113 Overview 113 Bayes’ Theorem 113 Bayes versus Frequentists 119 Many-State Problems 120 Continuous Distributions 124 Bayesian Networks 128 Bayesian Networks versus Correlation Matrices 130 Problems 132 Chapter 7 Hypothesis Testing and Confidence Intervals 135 Sample Mean Revisited 135 Sample Variance Revisited 137 Confidence Intervals 137 Hypothesis Testing 139 Chebyshev’s Inequality 142 Application: VaR 142 Problems 152 Chapter 8 Matrix Algebra 155 Matrix Notation 155 Matrix Operations 156 Application: Transition Matrices 163 Application: Monte Carlo Simulations Part II: Cholesky Decomposition 165 Problems 168 Chapter 9 Vector Spaces 169 Vectors Revisited 169 Orthogonality 172 Rotation 177 Principal Component Analysis 181 Application: The Dynamic Term Structure of Interest Rates 185 Application: The Structure of Global Equity Markets 191 Problems 193 Chapter 10 Linear Regression Analysis 195 Linear Regression (One Regressor) 195 Linear Regression (Multivariate) 203 Application: Factor Analysis 208 Application: Stress Testing 211 Problems 212 Chapter 11 Time Series Models 215 Random Walks 215 Drift-Diffusion Model 216 Autoregression 217 Variance and Autocorrelation 222 Stationarity 223 Moving Average 227 Continuous Models 228 Application: GARCH 230 Application: Jump-Diffusion Model 232 Application: Interest Rate Models 232 Problems 234 Chapter 12 Decay Factors 237 Mean 237 Variance 243 Weighted Least Squares 244 Other Possibilities 245 Application: Hybrid VaR 245 Problems 247 Appendix A Binary Numbers 249 Appendix B Taylor Expansions 251 Appendix C Vector Spaces 253 Appendix D Greek Alphabet 255 Appendix E Common Abbreviations 257 Appendix F Copulas 259 Answers 263 References 303 About the Author 305 About the Companion Website 307 Index 309

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Book SynopsisTable of ContentsPreface xiii Part One — Matrices 1 Basic properties of vectors and matrices 3 1 Introduction 3 2 Sets 3 3 Matrices: addition and multiplication 4 4 The transpose of a matrix 6 5 Square matrices 6 6 Linear forms and quadratic forms 7 7 The rank of a matrix 9 8 The inverse 10 9 The determinant 10 10 The trace 11 11 Partitioned matrices 12 12 Complex matrices 14 13 Eigenvalues and eigenvectors 14 14 Schur’s decomposition theorem 17 15 The Jordan decomposition 18 16 The singular-value decomposition 20 17 Further results concerning eigenvalues 20 18 Positive (semi)definite matrices 23 19 Three further results for positive definite matrices 25 20 A useful result 26 21 Symmetric matrix functions 27 Miscellaneous exercises 28 Bibliographical notes 30 2 Kronecker products, vec operator, and Moore-Penrose inverse 31 1 Introduction 31 2 The Kronecker product 31 3 Eigenvalues of a Kronecker product 33 4 The vec operator 34 5 The Moore-Penrose (MP) inverse 36 6 Existence and uniqueness of the MP inverse 37 7 Some properties of the MP inverse 38 8 Further properties 39 9 The solution of linear equation systems 41 Miscellaneous exercises 43 Bibliographical notes 45 3 Miscellaneous matrix results 47 1 Introduction 47 2 The adjoint matrix 47 3 Proof of Theorem 3.1 49 4 Bordered determinants 51 5 The matrix equation AX = 0 51 6 The Hadamard product 52 7 The commutation matrix Kmn 54 8 The duplication matrix Dn 56 9 Relationship between Dn+1 and Dn, I 58 10 Relationship between Dn+1 and Dn, II 59 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) 63 13 The bordered Gramian matrix 65 14 The equations X1A + X2B′ = G1,X1B = G2 67 Miscellaneous exercises 69 Bibliographical notes 70 Part Two — Differentials: the theory 4 Mathematical preliminaries 73 1 Introduction 73 2 Interior points and accumulation points 73 3 Open and closed sets 75 4 The Bolzano-Weierstrass theorem 77 5 Functions 78 6 The limit of a function 79 7 Continuous functions and compactness 80 8 Convex sets 81 9 Convex and concave functions 83 Bibliographical notes 86 5 Differentials and differentiability 87 1 Introduction 87 2 Continuity 88 3 Differentiability and linear approximation 90 4 The differential of a vector function 91 5 Uniqueness of the differential 93 6 Continuity of differentiable functions 94 7 Partial derivatives 95 8 The first identification theorem 96 9 Existence of the differential, I 97 10 Existence of the differential, II 99 11 Continuous differentiability 100 12 The chain rule 100 13 Cauchy invariance 102 14 The mean-value theorem for real-valued functions 103 15 Differentiable matrix functions 104 16 Some remarks on notation 106 17 Complex differentiation 108 Miscellaneous exercises 110 Bibliographical notes 110 6 The second differential 111 1 Introduction 111 2 Second-order partial derivatives 111 3 The Hessian matrix 112 4 Twice differentiability and second-order approximation, I 113 5 Definition of twice differentiability 114 6 The second differential 115 7 Symmetry of the Hessian matrix 117 8 The second identification theorem 119 9 Twice differentiability and second-order approximation, II 119 10 Chain rule for Hessian matrices 121 11 The analog for second differentials 123 12 Taylor’s theorem for real-valued functions 124 13 Higher-order differentials 125 14 Real analytic functions 125 15 Twice differentiable matrix functions 126 Bibliographical notes 127 7 Static optimization 129 1 Introduction 129 2 Unconstrained optimization 130 3 The existence of absolute extrema 131 4 Necessary conditions for a local minimum 132 5 Sufficient conditions for a local minimum: first-derivative test 134 6 Sufficient conditions for a local minimum: second-derivative test 136 7 Characterization of differentiable convex functions 138 8 Characterization of twice differentiable convex functions 141 9 Sufficient conditions for an absolute minimum 142 10 Monotonic transformations 143 11 Optimization subject to constraints 144 12 Necessary conditions for a local minimum under constraints 145 13 Sufficient conditions for a local minimum under constraints 149 14 Sufficient conditions for an absolute minimum under constraints 154 15 A note on constraints in matrix form 155 16 Economic interpretation of Lagrange multipliers 155 Appendix: the implicit function theorem 157 Bibliographical notes 159 Part Three — Differentials: the practice 8 Some important differentials 163 1 Introduction 163 2 Fundamental rules of differential calculus 163 3 The differential of a determinant 165 4 The differential of an inverse 168 5 Differential of the Moore-Penrose inverse 169 6 The differential of the adjoint matrix 172 7 On differentiating eigenvalues and eigenvectors 174 8 The continuity of eigenprojections 176 9 The differential of eigenvalues and eigenvectors: symmetric case 180 10 Two alternative expressions for dλ 183 11 Second differential of the eigenvalue function 185 Miscellaneous exercises 186 Bibliographical notes 189 9 First-order differentials and Jacobian matrices 191 1 Introduction 191 2 Classification 192 3 Derisatives 192 4 Derivatives 194 5 Identification of Jacobian matrices 196 6 The first identification table 197 7 Partitioning of the derivative 197 8 Scalar functions of a scalar 198 9 Scalar functions of a vector 198 10 Scalar functions of a matrix, I: trace 199 11 Scalar functions of a matrix, II: determinant 201 12 Scalar functions of a matrix, III: eigenvalue 202 13 Two examples of vector functions 203 14 Matrix functions 204 15 Kronecker products 206 16 Some other problems 208 17 Jacobians of transformations 209 Bibliographical notes 210 10 Second-order differentials and Hessian matrices 211 1 Introduction 211 2 The second identification table 211 3 Linear and quadratic forms 212 4 A useful theorem 213 5 The determinant function 214 6 The eigenvalue function 215 7 Other examples 215 8 Composite functions 217 9 The eigenvector function 218 10 Hessian of matrix functions, I 219 11 Hessian of matrix functions, II 219 Miscellaneous exercises 220 Part Four — Inequalities 11 Inequalities 225 1 Introduction 225 2 The Cauchy-Schwarz inequality 226 3 Matrix analogs of the Cauchy-Schwarz inequality 227 4 The theorem of the arithmetic and geometric means 228 5 The Rayleigh quotient 230 6 Concavity of λ1 and convexity of λn 232 7 Variational description of eigenvalues 232 8 Fischer’s min-max theorem 234 9 Monotonicity of the eigenvalues 236 10 The Poincar´e separation theorem 236 11 Two corollaries of Poincar´e’s theorem 237 12 Further consequences of the Poincar´e theorem 238 13 Multiplicative version 239 14 The maximum of a bilinear form 241 15 Hadamard’s inequality 242 16 An interlude: Karamata’s inequality 242 17 Karamata’s inequality and eigenvalues 244 18 An inequality concerning positive semidefinite matrices 245 19 A representation theorem for ( ∑api )1/p 246 20 A representation theorem for (trAp)1/p 247 21 Hölder’s inequality 248 22 Concavity of log|A| 250 23 Minkowski’s inequality 251 24 Quasilinear representation of |A|1/n 253 25 Minkowski’s determinant theorem 255 26 Weighted means of order p 256 27 Schlömilch’s inequality 258 28 Curvature properties of Mp(x, a) 259 29 Least squares 260 30 Generalized least squares 261 31 Restricted least squares 262 32 Restricted least squares: matrix version 264 Miscellaneous exercises 265 Bibliographical notes 269 Part Five — The linear model 12 Statistical preliminaries 273 1 Introduction 273 2 The cumulative distribution function 273 3 The joint density function 274 4 Expectations 274 5 Variance and covariance 275 6 Independence of two random variables 277 7 Independence of n random variables 279 8 Sampling 279 9 The one-dimensional normal distribution 279 10 The multivariate normal distribution 280 11 Estimation 282 Miscellaneous exercises 282 Bibliographical notes 283 13 The linear regression model 285 1 Introduction 285 2 Affine minimum-trace unbiased estimation 286 3 The Gauss-Markov theorem 287 4 The method of least squares 290 5 Aitken’s theorem 291 6 Multicollinearity 293 7 Estimable functions 295 8 Linear constraints: the case M(R′) ⊂M(X′) 296 9 Linear constraints: the general case 300 10 Linear constraints: the case M(R′) ∩M(X′) = {0} 302 11 A singular variance matrix: the case M(X) ⊂M(V ) 304 12 A singular variance matrix: the case r(X′V +X) = r(X) 305 13 A singular variance matrix: the general case, I 307 14 Explicit and implicit linear constraints 307 15 The general linear model, I 310 16 A singular variance matrix: the general case, II 311 17 The general linear model, II 314 18 Generalized least squares 315 19 Restricted least squares 316 Miscellaneous exercises 318 Bibliographical notes 319 14 Further topics in the linear model 321 1 Introduction 321 2 Best quadratic unbiased estimation of σ2 322 3 The best quadratic and positive unbiased estimator of σ2 322 4 The best quadratic unbiased estimator of σ2 324 5 Best quadratic invariant estimation of σ2 326 6 The best quadratic and positive invariant estimator of σ2 327 7 The best quadratic invariant estimator of σ2 329 8 Best quadratic unbiased estimation: multivariate normal case 330 9 Bounds for the bias of the least-squares estimator of σ2, I 332 10 Bounds for the bias of the least-squares estimator of σ2, II 333 11 The prediction of disturbances 335 12 Best linear unbiased predictors with scalar variance matrix 336 13 Best linear unbiased predictors with fixed variance matrix, I 338 14 Best linear unbiased predictors with fixed variance matrix, II 340 15 Local sensitivity of the posterior mean 341 16 Local sensitivity of the posterior precision 342 Bibliographical notes 344 Part Six — Applications to maximum likelihood estimation 15 Maximum likelihood estimation 347 1 Introduction 347 2 The method of maximum likelihood (ML) 347 3 ML estimation of the multivariate normal distribution 348 4 Symmetry: implicit versus explicit treatment 350 5 The treatment of positive definiteness 351 6 The information matrix 352 7 ML estimation of the multivariate normal distribution: distinct means 354 8 The multivariate linear regression model 354 9 The errors-in-variables model 357 10 The nonlinear regression model with normal errors 359 11 Special case: functional independence of mean and variance parameters 361 12 Generalization of Theorem 15.6 362 Miscellaneous exercises 364 Bibliographical notes 365 16 Simultaneous equations 367 1 Introduction 367 2 The simultaneous equations model 367 3 The identification problem 369 4 Identification with linear constraints on B and Γ only 371 5 Identification with linear constraints on B, Γ, and ∑ 371 6 Nonlinear constraints 373 7 FIML: the information matrix (general case) 374 8 FIML: asymptotic variance matrix (special case) 376 9 LIML: first-order conditions 378 10 LIML: information matrix 381 11 LIML: asymptotic variance matrix 383 Bibliographical notes 388 17 Topics in psychometrics 389 1 Introduction 389 2 Population principal components 390 3 Optimality of principal components 391 4 A related result 392 5 Sample principal components 393 6 Optimality of sample principal components 395 7 One-mode component analysis 395 8 One-mode component analysis and sample principal components 398 9 Two-mode component analysis 399 10 Multimode component analysis 400 11 Factor analysis 404 12 A zigzag routine 407 13 A Newton-Raphson routine 408 14 Kaiser’s varimax method 412 15 Canonical correlations and variates in the population 414 16 Correspondence analysis 417 17 Linear discriminant analysis 418 Bibliographical notes 419 Part Seven — Summary 18 Matrix calculus: the essentials 423 1 Introduction 423 2 Differentials 424 3 Vector calculus 426 4 Optimization 429 5 Least squares 431 6 Matrix calculus 432 7 Interlude on linear and quadratic forms 434 8 The second differential 434 9 Chain rule for second differentials 436 10 Four examples 438 11 The Kronecker product and vec operator 439 12 Identification 441 13 The commutation matrix 442 14 From second differential to Hessian 443 15 Symmetry and the duplication matrix 444 16 Maximum likelihood 445 Further reading 448 Bibliography 449 Index of symbols 467 Subject index 471

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Book SynopsisSince the first edition of this book published, Bayesian networks have become even more important for applications in a vast array of fields. This second edition includes new material on influence diagrams, learning from data, value of information, cybersecurity, debunking bad statistics, and much more. Focusing on practical real-world problem-solving and model building, as opposed to algorithms and theory, it explains how to incorporate knowledge with data to develop and use (Bayesian) causal models of risk that provide more powerful insights and better decision making than is possible from purely data-driven solutions.Features Provides all tools necessary to build and run realistic Bayesian network models Supplies extensive example models based on real risk assessment problems in a wide range of application domains provided; for example, finance, safety, systems reliability, law, forensics, cybersecurity and more IntrodTrade ReviewPraise for the first edition: "By offering many attractive examples of Bayesian networks and by making use of software that allows one to play with the networks, readers will definitely get a feel for what can be done with Bayesian networks. … the power and also uniqueness of the book stem from the fact that it is essentially practice oriented, but with a clear aim of equipping the developer of Bayesian networks with a clear understanding of the underlying theory. Anyone involved in everyday decision making looking for a better foundation of what is now mainly based on intuition will learn something from the book."—Peter J.F. Lucas, Journal of Statistical Theory and Practice, Vol. 8, March 2014 "… very useful to practitioners, professors, students, and anyone interested in understanding the application of Bayesian networks to risk assessment and decision analysis. Having many years of experience in the area, I highly recommend the book."—William E. Vesely, International Journal of Performability Engineering, July 2013 "Risk Assessment and Decision Analysis with Bayesian Networks is a brilliant book. Being a non-mathematician, I’ve found all of the other books on BNs to be an impenetrable mass of mathematical gobble-de-gook. This, in my view, has slowed the uptake of BNs in many disciplines because people simply cannot understand why you would use them and how you can use them. This book finally makes BNs comprehensible, and I plan to develop a risk assessment course at the University of Queensland using this book as the recommended textbook."—Carl Smith, School of Agriculture and Food Sciences, The University of Queensland "… although there have been several excellent books dedicated to Bayesian networks and related methods, these books tend to be aimed at readers who already have a high level of mathematical sophistication … . As such they are not accessible to readers who are not already proficient in those subjects. This book is an exciting development because it addresses this problem. … it should be understandable by any numerate reader interested in risk assessment and decision making. The book provides sufficient motivation and examples (as well as the mathematics and probability where needed from scratch) to enable readers to understand the core principles and power of Bayesian networks. However, the focus is on ensuring that readers can build practical Bayesian network models … readers are provided with a tool that performs the propagation, so they will be able to build their own models to solve real-world risk assessment problems."—From the Foreword by Judea Pearl, UCLA Computer Science Department and 2011 Turing Award winner "Let's be honest, most risk assessment methodologies are guesses, and not very good ones at that. People collect statistics about what they can see and then assume it tells them something about what they can't. The problem is that people assume the world follows nice distributions embedded in the world's fabric and that we simply need a little data to get the parameters right. Fenton and Neil take readers on an excellent journey through a more modern and appropriate way to make sense of uncertainty by leveraging prior beliefs and emerging evidence. Along the way they provide a wakeup call for the classic statistical views of risk and eloquently show the biases, fallacies and misconceptions that exist in such a view, and how dangerous they are for those making decisions.The book is not condescending to those without a mathematical background and is not too simple for those who do. It sets a nice tone which focuses more on how readers should think about risk and uncertainty and then uses a wealth of practical examples to show them how Bayesian methods can deliver powerful insights.After reading this book, you should be in no doubt that not only is it possible to model risk from the perspective of understanding how it behaves, but also that is necessarily the only sensible way to do so if you want to do something useful with your model and make correct decisions from it.Anyone aspiring to work, or already working, in the field of risk is well advised to read this book and put it into practice."—Neil Cantle, Milliman "The lovely thing about Risk Assessment and Decision Analysis with Bayesian Networks is that it holds your hand while it guides you through this maze of statistical fallacies, p-values, randomness and subjectivity, eventually explaining how Bayesian networks work and how they can help to avoid mistakes. There are loads of vivid examples (for instance, one explaining the Monty Hall problem), and it doesn’t skim over any of the technical details …"—Angela Saini (MIT Knight Science Journalism Fellow 2012-2013) on her blog, December 2012 "As computational chip size and product development cycle time approach zero, survival in the software industry becomes predicated on three related capabilities: prediction, diagnosis, and causality. These are the competitive advantages in 21st century software design testing. Fenton and Neil not only make a compelling case for Bayesian inference, but they also meticulously and patiently guide software engineers previously untrained in probability theory toward competence in mathematics. We have been waiting for decades for the last critical component that will make Bayesian a household word in industry: the incredible combination of an accessible software tool and an accompanying and brilliantly written textbook. Now software testers have the math, the algorithms, the tool, and the book. We no longer have any excuses for not dramatically raising our technology game to meet that challenge of continuous testing. Fenton and Neil came to our rescue, and just in the nick of time. Thanks, guys."—Michael Corning, Microsoft Corporation "This is an awesome book on using Bayesian networks for risk assessment and decision analysis. What makes this book so great is both its content and style. Fenton and Neil explain how the Bayesian networks work and how they can be built and applied to solve various decision-making problems in different areas. Even more importantly, the authors very clearly demonstrate motivations and advantages for using Bayesian networks over other modelling techniques. The core ideas are illustrated by lots of examples—from toy models to real-world applications. In contrast with many other books, this one is very easy to follow and does not require a strong mathematical or statistical background. I highly recommend this book to all researchers, students and practitioners who would like to go beyond traditional statistics or automated data mining techniques and incorporate expert knowledge in their models."—Dr. Lukasz Radlinski, Szczecin University "It is the first book that takes the art and science of developing Bayesian network models for actual problems as seriously as the underlying mathematics. The reader will obtain a good understanding of the methods as they are introduced through well-motivated and intuitive examples and attractive case studies. The authors do this in such a way that readers with little previous exposure to probability theory and statistics will be able to grasp and appreciate the power of Bayesian networks. While this in itself is already a major achievement, the authors go far beyond this by providing very close and pragmatic links between model building and the required techniques. It, thus, shares insights that are mostly missing from other textbooks, making this book also of interest to advanced readers, lecturers and researchers in the area."—Prof.dr. Peter Lucas, Institute for Computing and Information Sciences, Radboud University Nijmegen, and Leiden Institute of Advanced Computer Science, Leiden University "This book gives a thorough account of Bayesian networks, one of the most widely used frameworks for reasoning with uncertainty, and their application in domains as diverse as system reliability modelling and legal reasoning. The book's central premise is that ‘essentially, all models are wrong, but some are useful’ (G.E.P. Box), and the book distinguishes itself by focusing on the art of building useful models for risk assessment and decision analysis rather than on delving into mathematical detail of the models that are built. The authors are renowned for their ability to put Bayesian network technology into practical use, and it is therefore no surprise that the book is filled to the brim with motivating and relevant examples. With the accompanying evaluation copy of the excellent AgenaRisk software, readers can easily play around with the examples and gain valuable insights of how the models behave ‘at work.’ I believe this book should be of interest to practitioners working with risk assessment and decision making and also as a valuable textbook in undergraduate courses on probability and risk."—Helge Langseth, Norwegian University of Science and Technology "Bayesian networks are revolutionizing the way experts assess and manage uncertainty. This is the first book to explain this powerful new tool to a non-specialist audience. It takes us on a compelling journey from the basics of probability to sophisticated networks of system design, finance and crime. This trip is greatly supported by free software, allowing readers to explore and develop Bayesian networks for themselves. The style is accessible and entertaining, without sacrificing conceptual or mathematical rigor. This book is a must-read for anyone wanting to learn about Bayesian networks; it provides the know-how and software so that we can all share this adventure into risk and uncertainty."—David Lagnado, Senior Lecturer in Cognitive and Decision Sciences, University College London "This is the book I have wanted to see for many years. Whilst we are entitled to see appropriate duty of care in any risk management scenario, ill-informed practice is in fact prevalent in industry and society. There is little real excuse for this as classical decision theory has a long established history, and it can now be operationalized in complex scenarios using the Bayesian network technology that is a core theme of this book. The problem has been that most books on Bayesian networks and decision theory focus in depth on the technical foundations, and provide little in the way of practical guidance on how to use the technology to support real-world risk assessment and decision making.In contrast, Norman Fenton and Martin Neil have provided a clearly written and highly readable book that is packed with informative and insightful examples. I had fun reading it, but there is also sufficient technical detail so that one can obtain a deep understanding of the subject from studying the book. It is a joy, and one that I keep dipping back into."—Paul Krause, Professor of Software Engineering, University of Surrey "Given the massive uncertainties managers now need to operate within, this book is both vital and timely. Fenton and Neil’s explanation of how to create practical models that simulate real-life strategic scenarios gives hard-pressed managers a new tool that they can use to understand potential impacts and opportunities. This book should be required reading for anyone involved in strategy, business planning, or significant decision-making."—Rob Wirszycz, Celaton Limited Praise for the first edition: "By offering many attractive examples of Bayesian networks and by making use of software that allows one to play with the networks, readers will definitely get a feel for what can be done with Bayesian networks. … the power and also uniqueness of the book stem from the fact that it is essentially practice oriented, but with a clear aim of equipping the developer of Bayesian networks with a clear understanding of the underlying theory. Anyone involved in everyday decision making looking for a better foundation of what is now mainly based on intuition will learn something from the book."—Peter J.F. Lucas, Journal of Statistical Theory and Practice, Vol. 8, March 2014 "… very useful to practitioners, professors, students, and anyone interested in understanding the application of Bayesian networks to risk assessment and decision analysis. Having many years of experience in the area, I highly recommend the book."—William E. Vesely, International Journal of Performability Engineering, July 2013 "Risk Assessment and Decision Analysis with Bayesian Networks is a brilliant book. Being a non-mathematician, I’ve found all of the other books on BNs to be an impenetrable mass of mathematical gobble-de-gook. This, in my view, has slowed the uptake of BNs in many disciplines because people simply cannot understand why you would use them and how you can use them. This book finally makes BNs comprehensible, and I plan to develop a risk assessment course at the University of Queensland using this book as the recommended textbook."—Carl Smith, School of Agriculture and Food Sciences, The University of Queensland "… although there have been several excellent books dedicated to Bayesian networks and related methods, these books tend to be aimed at readers who already have a high level of mathematical sophistication … . As such they are not accessible to readers who are not already proficient in those subjects. This book is an exciting development because it addresses this problem. … it should be understandable by any numerate reader interested in risk assessment and decision making. The book provides sufficient motivation and examples (as well as the mathematics and probability where needed from scratch) to enable readers to understand the core principles and power of Bayesian networks. However, the focus is on ensuring that readers can build practical Bayesian network models … readers are provided with a tool that performs the propagation, so they will be able to build their own models to solve real-world risk assessment problems."—From the Foreword by Judea Pearl, UCLA Computer Science Department and 2011 Turing Award winner "Let's be honest, most risk assessment methodologies are guesses, and not very good ones at that. People collect statistics about what they can see and then assume it tells them something about what they can't. The problem is that people assume the world follows nice distributions embedded in the world's fabric and that we simply need a little data to get the parameters right. Fenton and Neil take readers on an excellent journey through a more modern and appropriate way to make sense of uncertainty by leveraging prior beliefs and emerging evidence. Along the way they provide a wakeup call for the classic statistical views of risk and eloquently show the biases, fallacies and misconceptions that exist in such a view, and how dangerous they are for those making decisions.The book is not condescending to those without a mathematical background and is not too simple for those who do. It sets a nice tone which focuses more on how readers should think about risk and uncertainty and then uses a wealth of practical examples to show them how Bayesian methods can deliver powerful insights.After reading this book, you should be in no doubt that not only is it possible to model risk from the perspective of understanding how it behaves, but also that is necessarily the only sensible way to do so if you want to do something useful with your model and make correct decisions from it.Anyone aspiring to work, or already working, in the field of risk is well advised to read this book and put it into practice."—Neil Cantle, Milliman "The lovely thing about Risk Assessment and Decision Analysis with Bayesian Networks is that it holds your hand while it guides you through this maze of statistical fallacies, p-values, randomness and subjectivity, eventually explaining how Bayesian networks work and how they can help to avoid mistakes. There are loads of vivid examples (for instance, one explaining the Monty Hall problem), and it doesn’t skim over any of the technical details …"—Angela Saini (MIT Knight Science Journalism Fellow 2012-2013) on her blog, December 2012 "As computational chip size and product development cycle time approach zero, survival in the software industry becomes predicated on three related capabilities: prediction, diagnosis, and causality. These are the competitive advantages in 21st century software design testing. Fenton and Neil not only make a compelling case for Bayesian inference, but they also meticulously and patiently guide software engineers previously untrained in probability theory toward competence in mathematics. We have been waiting for decades for the last critical component that will make Bayesian a household word in industry: the incredible combination of an accessible software tool and an accompanying and brilliantly written textbook. Now software testers have the math, the algorithms, the tool, and the book. We no longer have any excuses for not dramatically raising our technology game to meet that challenge of continuous testing. Fenton and Neil came to our rescue, and just in the nick of time. Thanks, guys."—Michael Corning, Microsoft Corporation "This is an awesome book on using Bayesian networks for risk assessment and decision analysis. What makes this book so great is both its content and style. Fenton and Neil explain how the Bayesian networks work and how they can be built and applied to solve various decision-making problems in different areas. Even more importantly, the authors very clearly demonstrate motivations and advantages for using Bayesian networks over other modelling techniques. The core ideas are illustrated by lots of examples—from toy models to real-world applications. In contrast with many other books, this one is very easy to follow and does not require a strong mathematical or statistical background. I highly recommend this book to all researchers, students and practitioners who would like to go beyond traditional statistics or automated data mining techniques and incorporate expert knowledge in their models."—Dr. Lukasz Radlinski, Szczecin University "It is the first book that takes the art and science of developing Bayesian network models for actual problems as seriously as the underlying mathematics. The reader will obtain a good understanding of the methods as they are introduced through well-motivated and intuitive examples and attractive case studies. The authors do this in such a way that readers with little previous exposure to probability theory and statistics will be able to grasp and appreciate the power of Bayesian networks. While this in itself is already a major achievement, the authors go far beyond this by providing very close and pragmatic links between model building and the required techniques. It, thus, shares insights that are mostly missing from other textbooks, making this book also of interest to advanced readers, lecturers and researchers in the area."—Prof.dr. Peter Lucas, Institute for Computing and Information Sciences, Radboud University Nijmegen, and Leiden Institute of Advanced Computer Science, Leiden University "This book gives a thorough account of Bayesian networks, one of the most widely used frameworks for reasoning with uncertainty, and their application in domains as diverse as system reliability modelling and legal reasoning. The book's central premise is that ‘essentially, all models are wrong, but some are useful’ (G.E.P. Box), and the book distinguishes itself by focusing on the art of building useful models for risk assessment and decision analysis rather than on delving into mathematical detail of the models that are built. The authors are renowned for their ability to put Bayesian network technology into practical use, and it is therefore no surprise that the book is filled to the brim with motivating and relevant examples. With the accompanying evaluation copy of the excellent AgenaRisk software, readers can easily play around with the examples and gain valuable insights of how the models behave ‘at work.’ I believe this book should be of interest to practitioners working with risk assessment and decision making and also as a valuable textbook in undergraduate courses on probability and risk."—Helge Langseth, Norwegian University of Science and Technology "Bayesian networks are revolutionizing the way experts assess and manage uncertainty. This is the first book to explain this powerful new tool to a non-specialist audience. It takes us on a compelling journey from the basics of probability to sophisticated networks of system design, finance and crime. This trip is greatly supported by free software, allowing readers to explore and develop Bayesian networks for themselves. The style is accessible and entertaining, without sacrificing conceptual or mathematical rigor. This book is a must-read for anyone wanting to learn about Bayesian networks; it provides the know-how and software so that we can all share this adventure into risk and uncertainty."—David Lagnado, Senior Lecturer in Cognitive and Decision Sciences, University College London "This is the book I have wanted to see for many years. Whilst we are entitled to see appropriate duty of care in any risk management scenario, ill-informed practice is in fact prevalent in industry and society. There is little real excuse for this as classical decision theory has a long established history, and it can now be operationalized in complex scenarios using the Bayesian network technology that is a core theme of this book. The problem has been that most books on Bayesian networks and decision theory focus in depth on the technical foundations, and provide little in the way of practical guidance on how to use the technology to support real-world risk assessment and decision making.In contrast, Norman Fenton and Martin Neil have provided a clearly written and highly readable book that is packed with informative and insightful examples. I had fun reading it, but there is also sufficient technical detail so that one can obtain a deep understanding of the subject from studying the book. It is a joy, and one that I keep dipping back into."—Paul Krause, Professor of Software Engineering, University of Surrey "Given the massive uncertainties managers now need to operate within, this book is both vital and timely. Fenton and Neil’s explanation of how to create practical models that simulate real-life strategic scenarios gives hard-pressed managers a new tool that they can use to understand potential impacts and opportunities. This book should be required reading for anyone involved in strategy, business planning, or significant decision-making."—Rob Wirszycz, Celaton Limited Table of ContentsThere Is More to Assessing Risk Than Statistics. The Need for Causal, Explanatory Models in Risk Assessment. Measuring Uncertainty: The Inevitability of Subjectivity. The Basics of Probability. Bayes’ Theorem and Conditional Probability. From Bayes’ Theorem to Bayesian Networks. Defining the Structure of Bayesian Networks. Building and Eliciting Node Probability Tables. Numeric Variables and Continuous Distribution Functions. Hypothesis Testing and Confidence Intervals. Modeling Operational Risk. Systems Reliability Modeling. Bayes and the Law. Learning Bayesian Networks. Decision making, Influence Diagrams and Value of information. Bayesian networks in forensics. Using Bayesian networks to debunk bad statistics. Bayesian networks for football prediction. Appendix A: The Basics of Counting. Appendix B: The Algebra of Node Probability Tables. Appendix C: Junction Tree Algorithm. Appendix D: Dynamic Discretization. Appendix E: Statistical Distributions.

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Book SynopsisAbout our authors David F. Groebner is Professor Emeritus of Production Management in the College of Business and Economics at Boise State University. He has bachelor's and master's degrees in engineering and a Ph.D. in business administration. After working as an engineer, he has taught statistics and related subjects for 27 years. In addition to writing textbooks and academic papers, Groebner has worked extensively with both small and large organizations, including Hewlett-Packard, Boise Cascade, Albertson's, and Ore-Ida. He has worked with numerous government agencies, including Boise City and the U.S. Air Force. Patrick W. Shannon, Ph.D. is Dean and Professor of Supply Chain Operations Management in the College of Business and Economics at Boise State University. In addition to his administrative responsibilities, he has taught graduate and undergraduate courses in business statistics, quality management, and production and operaTable of Contents The Where, Why, and How of Data Graphs, Charts, and Tables: Describing Your Data Describing Data Using Numerical Measures 1 - 3 SPECIAL REVIEW SECTION Introduction to Probability Discrete Probability Distributions Introduction to Continuous Probability Distributions Introduction to Sampling Distributions Estimating Single Population Parameters Introduction to Hypothesis Testing Estimation and Hypothesis Testing for Two Population Parameters Hypothesis Tests and Estimation for Population Variances Analysis of Variance 8 - 12 SPECIAL REVIEW SECTION Goodness-of-Fit Tests and Contingency Analysis Introduction to Linear Regression and Correlation Analysis Multiple Regression Analysis and Model Building Analyzing and Forecasting Time-Series Data Introduction to Nonparametric Statistics Introducing Business Analytics Introduction to Decision Analysis (Online) Introduction to Quality and Statistical Process Control (Online) APPENDICES A to P

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Book SynopsisMark Triola, MD, FACP is the Associate Dean for Educational Informatics at NYU School of Medicine, the founding director of the NYU Langone Medical Center Institute for Innovations in Medical Education (IIME), and an Associate Professor of Medicine. Dr. Triola's research focuses on precision education and the use of AI tools to efficiently personalize medical education for individual learners and give new insights to their programs and coaches. His lab develops new learning technologies and AI-driven educational interventions and also works to define educationally sensitive patient and system outcomes that can be used to assess the impact of training. Dr. Triola and IIME have been funded by the National Institutes of Health, the Josiah Macy Jr. Foundation, the Department of Education, the Department of Defense, and the American Medical Association's Accelerating Change in Medical Education program. Mario F. Triola is a ProfeTable of Contents INTRODUCTION TO STATISTICS 1-1 Statistical and Critical Thinking 1-2 Types of Data 1-3 Collecting Sample Data 1-4 Ethics in Statistics (download only) EXPLORING DATA WITH TABLES AND GRAPHS 2-1 Frequency Distributions for Organizing and Summarizing Data 2-2 Histograms 2-3 Graphs That Enlighten and Graphs That Deceive 2-4 Scatterplots, Correlation, and Regression DESCRIBING, EXPLORING, AND COMPARING DATA 3-1 Measures of Center 3-2 Measures of Variation 3-3 Measures of Relative Standing and Boxplots PROBABILITY 4-1 Basic Concepts of Probability 4-2 Addition Rule and Multiplication Rule 4-3 Complements, Conditional Probability, and Bayes' Theorem 4-4 Risks and Odds 4-5 Rates of Mortality, Fertility, and Morbidity 4-6 Counting DISCRETE PROBABILITY DISTRIBUTIONS 5-1 Probability Distributions 5-2 Binomial Probability Distributions 5-3 Poisson Probability Distributions NORMAL PROBABILITY DISTRIBUTIONS 6-1 The Standard Normal Distribution 6-2 Real Applications of Normal Distributions 6-3 Sampling Distributions and Estimators 6-4 The Central Limit Theorem 6-5 Assessing Normality 6-6 Normal as Approximation to Binomial (download only) ESTIMATING PARAMETERS AND DETERMINING SAMPLE SIZES 7-1 Estimating a Population Proportion 7-2 Estimating a Population Mean 7-3 Estimating a Population Standard Deviation or Variance 7-4 Bootstrapping: Using Technology for Estimates HYPOTHESIS TESTING 8-1 Basics of Hypothesis Testing 8-2 Testing a Claim About a Proportion 8-3 Testing a Claim About a Mean 8-4 Testing a Claim About a Standard Deviation or Variance 8-5 Resampling: Using Technology for Hypothesis Testing INFERENCES FROM TWO SAMPLES 9-1 Two Proportions 9-2 Two Means: Independent Samples 9-3 Matched Pairs 9-4 Two Variances or Standard Deviations 9-5 Resampling: Using Technology for Inferences CORRELATION AND REGRESSION 10-1 Correlation 10-2 Regression 10-3 Prediction Intervals and Variation 10-4 Multiple Regression 10-5 Dummy Variables and Logistic Regression GOODNESS-OF-FIT AND CONTINGENCY TABLES 11-1 Goodness-of-Fit 11-2 Contingency Tables ANALYSIS OF VARIANCE 12-1 One-Way ANOVA 12-2 Two-Way ANOVA NONPARAMETRIC TESTS 13-1 Basics of Nonparametric Tests 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks Test for Matched Pairs 13-4 Wilcoxon Rank-Sum Test for Two Independent Samples 13-5 Kruskal-Wallis Test for Three or More Samples 13-6 Rank Correlation SURVIVAL ANALYSIS 14-1 Life Tables 14-2 Kaplan-Meier Survival Analysis APPENDICES A: Tables and Formulas B: Data Sets C: Websites and Bibliography of Books D: Answers to Odd-Numbered Section Exercises (and all Quick Quizzes, all Review Exercises, and all Cumulative Review Exercises) Subject Index

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Book SynopsisA high school student can create deep Q-learning code to control her robot, without any understanding of the meaning of ''deep'' or ''Q'', or why the code sometimes fails. This book is designed to explain the science behind reinforcement learning and optimal control in a way that is accessible to students with a background in calculus and matrix algebra. A unique focus is algorithm design to obtain the fastest possible speed of convergence for learning algorithms, along with insight into why reinforcement learning sometimes fails. Advanced stochastic process theory is avoided at the start by substituting random exploration with more intuitive deterministic probing for learning. Once these ideas are understood, it is not difficult to master techniques rooted in stochastic control. These topics are covered in the second part of the book, starting with Markov chain theory and ending with a fresh look at actor-critic methods for reinforcement learning.Trade Review'Control Systems and Reinforcement Learning is a densely packed book with a vivid, conversational style. It speaks both to computer scientists interested in learning about the tools and techniques of control engineers and to control engineers who want to learn about the unique challenges posed by reinforcement learning and how to address these challenges. The author, a world-class researcher in control and probability theory, is not afraid of strong and perhaps controversial opinions, making the book entertaining and attractive for open-minded readers. Everyone interested in the "why" and "how" of RL will use this gem of a book for many years to come.' Csaba Szepesvári, Canada CIFAR AI Chair, University of Alberta, and Head of the Foundations Team at DeepMind'This book is a wild ride, from the elements of control through to bleeding-edge topics in reinforcement learning. Aimed at graduate students and very good undergraduates who are willing to invest some effort, the book is a lively read and an important contribution.' Shane G. Henderson, Charles W. Lake, Jr. Chair in Productivity, Cornell University'Reinforcement learning, now the de facto workhorse powering most AI-based algorithms, has deep connections with optimal control and dynamic programing. Meyn explores these connections in a marvelous manner and uses them to develop fast, reliable iterative algorithms for solving RL problems. This excellent, timely book from a leading expert on stochastic optimal control and approximation theory is a must-read for all practitioners in this active research area.' Panagiotis Tsiotras, David and Andrew Lewis Chair and Professor, Guggenheim School of Aerospace Engineering, Georgia Institute of TechnologyTable of Contents1. Introduction; Part I. Fundamentals Without Noise: 2. Control crash course; 3. Optimal control; 4. ODE methods for algorithm design; 5. Value function approximations; Part II. Reinforcement Learning and Stochastic Control: 6. Markov chains; 7. Stochastic control; 8. Stochastic approximation; 9. Temporal difference methods; 10. Setting the stage, return of the actors; A. Mathematical background; B. Markov decision processes; C. Partial observations and belief states; References; Glossary of Symbols and Acronyms; Index.

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Book SynopsisNew 2017 Cambridge A Level Maths and Further Maths resources to help students with learning and revision. Written for the AQA AS/A Level Further Mathematics specification for first teaching from 2017, this print Student Book and Cambridge Elevate edition covers the Statistics content for AS and A Level. It balances accessible exposition with a wealth of worked examples, exercises and opportunities to test and consolidate learning, providing a clear and structured pathway for progressing through the course. It is underpinned by a strong pedagogical approach, with an emphasis on skills development and the synoptic nature of the course. Available online and on tablet devices through the Cambridge Elevate app. Includes answers to aid independent study.

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Book SynopsisMake some headway in the notoriously tough subject of business statistics Business Statistics For Dummies helps you understand the core concepts and principles of business statistics, and how they relate to the business world. This book tracks to a typical introductory course offered at the undergraduate, so you know you'll find all the content you need to pass your class and get your degree. You'll get an introduction to statistical problems and processes common to the world of global business and economics. Written in clear and simple language, Business Statistics For Dummies gives you an introduction to probability, sampling techniques and distributions, and drawing conclusions from data. You'll also discover how to use charts and graphs to visualize the most important properties of a data set. Grasp the core concepts, principles, and methods of business statistics Learn tricky concepts with simplified explanations and illustrative graphsTable of ContentsIntroduction 1 Part 1: Getting Started with Business Statistics 5 Chapter 1: The Art and Science of Business Statistics 7 Chapter 2: Pictures Tell the Story: Graphical Representations of Data 21 Chapter 3: Identifying the Center of a Data Set 35 Chapter 4: Measuring Variation in a Data Set 53 Chapter 5: Measuring How Data Sets Are Related to Each Other 71 Part 2: Probability Theory and Probability Distributions 95 Chapter 6: Probability Theory: Measuring the Likelihood of Events 97 Chapter 7: Probability Distributions and Random Variables 115 Chapter 8: The Binomial and Poisson Distributions 127 Chapter 9: The Normal Distribution: So Many Possibilities! 145 Chapter 10: Sampling Techniques and Distributions 165 Part 3: Drawing Conclusions from Samples 185 Chapter 11: Confidence Intervals and the Student’s t-Distribution 187 Chapter 12: Testing Hypotheses about the Population Mean 205 Chapter 13: Applications of the Chi-Square Distribution 245 Chapter 14: Applications of the F-Distribution 273 Part 4: More Advanced Techniques: Regression Analysis and Spreadsheet Modeling 287 Chapter 15: Simple Regression Analysis 289 Chapter 16: Key Statistical Techniques in Excel 317 Part 5: The Part of Tens 343 Chapter 17: Ten Common Errors That Arise in Statistical Analysis 345 Chapter 18: (Almost) Ten Key Categories of Formulas for Business Statistics 353 Index 363

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Book SynopsisChoosing and Using Statistics remains an invaluable guide for students using a computer package to analyse data from research projects and practical class work. The text takes a pragmatic approach to statistics with a strong focus on what is actually needed. There are chapters giving useful advice on the basics of statistics and guidance on the presentation of data. The book is built around a key to selecting the correct statistical test and then gives clear guidance on how to carry out the test and interpret the output from four commonly used computer packages: SPSS, Minitab, Excel, and (new to this edition) the free program, R. Only the basics of formal statistics are described and the emphasis is on jargon-free English but any unfamiliar words can be looked up in the extensive glossary. This new 3rd edition of Choosing and Using Statistics is a must for all students who use a computer package to apply statistics in practical and project work. Features neTrade Review"Written in a concise and direct style, this book presents a selection of some of the most widely used statistical tests and data exploration techniques." (Biological Conservation, 1 March 2012) "Written in a concise and direct style, this book presents a selection of some of the most widely used statistical tests and data exploration techniques ... In general, this book is a very good primer for students with no statistical expertise." (Biological Conservation Reviews, 2011) "This book makes everything so easy. Complicated tests are effortlessly condensed, and the instructions are almost too easy to follow. Diagrams and sample data sets are used frequently so you can practise using tests before applying them to your own data sets, whilst the logical layout guides you toward the correct test for both your data, and what you want to prove (or disprove)." (Animals & Men, February 2011)Table of ContentsPreface xiii The third edition xiv How to use this book xiv Packages used xv Example data xv Acknowledgements for the first edition xv Acknowledgements for the second edition xv Acknowledgements for the third edition xvi 1 Eight steps to successful data analysis 1 2 The basics 2 Observations 2 Hypothesis testing 2 P-values 3 Sampling 3 Experiments 4 Statistics 4 Descriptive statistics 5 Tests of difference 5 Tests of relationships 5 Tests for data investigation 6 3 Choosing a test: a key 7 Remember: eight steps to successful data analysis 7 The art of choosing a test 7 A key to assist in your choice of statistical test 8 4 Hypothesis testing, sampling and experimental design 23 Hypothesis testing 23 Acceptable errors 23 P-values 24 Sampling 25 Choice of sample unit 25 Number of sample units 26 Positioning of sample units to achieve a random sample 26 Timing of sampling 27 Experimental design 27 Control 28 Procedural controls 28 Temporal control 28 Experimental control 29 Statistical control 29 Some standard experimental designs 29 5 Statistics, variables and distributions 32 What are statistics? 32 Types of statistics 33 Descriptive statistics 33 Parametric statistics 33 Non-parametric statistics 33 What is a variable? 33 Types of variables or scales of measurement 34 Measurement variables 34 Continuous variables 34 Discrete variables 35 How accurate do I need to be? 35 Ranked variables 35 Attributes 35 Derived variables 36 Types of distribution 36 Discrete distributions 36 The Poisson distribution 36 The binomial distribution 37 The negative binomial distribution 39 The hypergeometric distribution 39 Continuous distributions 40 The rectangular distribution 40 The normal distribution 40 The standardized normal distribution 40 Convergence of a Poisson distribution to a normal distribution 41 Sampling distributions and the 'central limit theorem' 41 Describing the normal distribution further 41 Skewness 41 Kurtosis 43 Is a distribution normal? 43 Transformations 43 An example 44 The angular transformation 44 The logit transformation 45 The t-distribution 46 Confidence intervals 47 The chi-square distribution 47 The exponential distribution 47 Non-parametric 'distributions' 48 Ranking, quartiles and the interquartile range 48 Box and whisker plots 48 6 Descriptive and presentational techniques 49 General advice 49 Displaying data: summarizing a single variable 49 Box and whisker plot (box plot) 49 Displaying data: showing the distribution of a single variable 50 Bar chart: for discrete data 50 Histogram: for continuous data 51 Pie chart: for categorical data or attribute data 52 Descriptive statistics 52 Statistics of location or position 52 Arithmetic mean 53 Geometric mean 53 Harmonic mean 53 Median 53 Mode 53 Statistics of distribution, dispersion or spread 55 Range 55 Interquartile range 55 Variance 55 Standard deviation (SD) 55 Standard error (SE) 56 Confidence intervals (CI) or confidence limits 56 Coefficient of variation 56 Other summary statistics 56 Skewness 57 Kurtosis 57 Using the computer packages 57 General 57 Displaying data: summarizing two or more variables 62 Box and whisker plots (box plots) 62 Error bars and confidence intervals 63 Displaying data: comparing two variables 63 Associations 63 Scatterplots 64 Multiple scatterplots 64 Trends, predictions and time series 65 Lines 65 Fitted lines 67 Confidence intervals 67 Displaying data: comparing more than two variables 68 Associations 68 Three-dimensional scatterplots 68 Multiple trends, time series and predictions 69 Multiple fitted lines 69 Surfaces 70 7 The tests 1: tests to look at differences 72 Do frequency distributions differ? 72 Questions 72 G-test 72 An example 73 Chi-square test 75 An example 76 Kolmogorov–Smirnov test 86 An example 87 Anderson–Darling test 89 Shapiro–Wilk test 90 Graphical tests for normality 90 Do the observations from two groups differ? 92 Paired data 92 Paired t-test 92 Wilcoxon signed ranks test 96 Sign test 99 Unpaired data 103 t-test 103 One-way ANOVA 111 Mann–Whitney U 119 Do the observations from more than two groups differ? 123 Repeated measures 123 Friedman test (for repeated measures) 123 Repeated-measures ANOVA 127 Independent samples 128 One-way ANOVA 129 Post hoc testing: after one-way ANOVA 138 Kruskal–Wallis test 142 Post hoc testing: after the Kruskal–Wallis test 145 There are two independent ways of classifying the data 145 One observation for each factor combination (no replication) 146 Friedman test 146 Two-way ANOVA (without replication) 152 More than one observation for each factor combination (with replication) 160 Interaction 160 Two-way ANOVA (with replication) 163 An example 164 Scheirer–Ray–Hare test 175 An example 175 There are more than two independent ways to classify the data 182 Multifactorial testing 182 Three-way ANOVA (without replication) 183 Three-way ANOVA (with replication) 184 An example 184 Multiway ANOVA 191 Not all classifications are independent 192 Non-independent factors 192 Nested factors 192 Random or fixed factors 193 Nested or hierarchical designs 193 Two-level nested-design ANOVA 193 An example 193 8 The tests 2: tests to look at relationships 199 Is there a correlation or association between two variables? 199 Observations assigned to categories 199 Chi-square test of association 199 An example 200 Cramér coefficient of association 208 Phi coefficient of association 209 Observations assigned a value 209 'Standard' correlation (Pearson's product-moment correlation) 210 An example 210 Spearman's rank-order correlation 214 An example 215 Kendall rank-order correlation 218 An example 218 Regression 219 An example 220 Is there a cause-and-effect relationship between two variables? 220 Questions 220 'Standard' linear regression 221 Prediction 221 Interpreting r2 222 Comparison of regression and correlation 222 Residuals 222 Confidence intervals 222 Prediction interval 223 An example 223 Kendall robust line-fit method 230 Logistic regression 230 An example 231 Model II regression 235 Polynomial, cubic and quadratic regression 235 Tests for more than two variables 236 Tests of association 236 Questions 236 Correlation 236 Partial correlation 237 Kendall partial rank-order correlation 237 Cause(s) and effect(s) 237 Questions 237 Regression 237 Analysis of covariance (ANCOVA) 238 Multiple regression 242 Stepwise regression 242 Path analysis 243 9 The tests 3: tests for data exploration 244 Types of data 244 Observation, inspection and plotting 244 Principal component analysis (PCA) and factor analysis 244 An example 245 Canonical variate analysis 251 Discriminant function analysis 251 An example 251 Multivariate analysis of variance (MANOVA) 256 An example 256 Multivariate analysis of covariance (MANCOVA) 259 Cluster analysis 259 DECORANA and TWINSPAN 263 Symbols and letters used in statistics 264 Greek letters 264 Symbols 264 Upper-case letters 265 Lower-case letters 266 Glossary 267 Assumptions of the tests 282 What if the assumptions are violated? 284 Hints and tips 285 Using a computer 285 Sampling 286 Statistics 286 Displaying the data 287 A table of statistical tests 289 Index 291

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Book SynopsisNeed to learn statistics as part of your job, or want help passing a statistics course? Statistics in a Nutshell is a clear and concise introduction and reference for anyone who's new to the subject. This book gives you a solid understanding of statistics without being too simple, yet without the numbing complexity of most college texts.

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Book SynopsisPraise for the First Edition:. . . the book serves as an excellent tutorial on the R language, providing examples that illustrate programming concepts in the context of practical computational problems. The book will be of great interest for all specialists working on computational statistics and Monte Carlo methods for modeling and simulation. Tzvetan Semerdjiev, Zentralblatt MathComputational statistics and statistical computing are two areas within statistics that may be broadly described as computational, graphical, and numerical approaches to solving statistical problems. Like its bestselling predecessor, Statistical Computing with R, Second Edition covers the traditional core material of these areas with an emphasis on using the R language via an examples-based approach. The new edition is up-to-date with the many advances that have been made in recent years. Features Provides an overview of cTrade ReviewPraise for the First Edition:"… an excellent tutorial on the R language, providing examples that illustrate programming concepts in the context of practical computational problems. The book will be of great interest for all specialists working on computational statistics and Monte Carlo methods for modeling and simulation." —Tzvetan Semerdjiev, Zentralblatt Math, 2008, Vol. 1137 "Statistical computing and computational statistics are two areas of statistics described as computational, graphical, and numerical approaches to solving statistical problems. Statistical Computing with R comprises, thorough and examples-based approach, the conventional core material of computational statistics with an emphasis on R... This book includes standard statistical computing topics using the R language... All examples in the text are realised in R. Software is actively maintained, it has good connectivity to various types of data and other systems, and it is versatile. In addition, R is very stable and reliable... The book also includes exercises and applications in all chapters, as well as coverage of recent advances including R Studio. Many examples are included, fully implemented in the R statisticalcomputing environment, and the R code for the examples can be downloaded from the author’s website. Most examples and exercises apply datasets accessible in the R distribution or simulated data. The author, Maria L. Rizzo, is a Full Professor at the Department of Mathematics and Statistics of Bowling Green State University (US) and is an expert on Applied Statistics, Statistical Computing, and Energy Statistics... After finishing the book, I feel that it is a well-written text useful for biostatisticians and graduate teachers, principally because it is written by a leading expert who is engaged in statistical modelling and methodological developments and applications in the real world. In my opinion, the book is a must-have for the interested biostatistician audience."- Luca Bertolaccini, ISCB December 2019 Praise for the First Edition:"… an excellent tutorial on the R language, providing examples that illustrate programming concepts in the context of practical computational problems. The book will be of great interest for all specialists working on computational statistics and Monte Carlo methods for modeling and simulation." —Tzvetan Semerdjiev, Zentralblatt Math, 2008, Vol. 1137 "Statistical computing and computational statistics are two areas of statistics described as computational, graphical, and numerical approaches to solving statistical problems. Statistical Computing with R comprises, thorough and examples-based approach, the conventional core material of computational statistics with an emphasis on R... This book includes standard statistical computing topics using the R language... All examples in the text are realised in R. Software is actively maintained, it has good connectivity to various types of data and other systems, and it is versatile. In addition, R is very stable and reliable... The book also includes exercises and applications in all chapters, as well as coverage of recent advances including R Studio. Many examples are included, fully implemented in the R statistical computing environment, and the R code for the examples can be downloaded from the author’s website. Most examples and exercises apply datasets accessible in the R distribution or simulated data. The author, Maria L. Rizzo, is a Full Professor at the Department of Mathematics and Statistics of Bowling Green State University (US) and is an expert on Applied Statistics, Statistical Computing, and Energy Statistics... After finishing the book, I feel that it is a well-written text useful for biostatisticians and graduate teachers, principally because it is written by a leading expert who is engaged in statistical modelling and methodological developments and applications in the real world. In my opinion, the book is a must-have for the interested biostatistician audience."- Luca Bertolaccini, ISCB December 2019 "...This book tries to keep a balance between theory and practice, with more focus on the latter...also provides plenty of R codes to help the readers practice what they learned from the book. As stated in the preface, the targeted readers of this book are graduate students and advanced undergraduates with preparation in the relevant mathematics foundations. From this point of view, the content of the book fits well to the anticipated audience...I really appreciate the section on “finding source code” in Chapter 15. A lot of the libraries in R are written in C or Fortran. Occasionally, we need to dig into those codes and make changes to suit our needs. It is very helpful in our daily research to be able to find the source code and compile the changes...Finally, I would like to give credit to the author on making their code available on github. This makes it convenient for readers to try the code themselves without lots of typing. It also allows the authors to easily make updated code available to readers."- Ling Leng, JASA, September 2020 Table of ContentsIntroduction. Probability and Statistics Review. Methods for Generating Random Variables. Visualization of Multivariate Data. Monte Carlo Integration and Variance Reduction. Monte Carlo Methods in Inference. Bootstrap and Jackknife. Permutation Tests. Markov Chain Monte Carlo Methods. Probability Density Estimation. Smoothing and Nonparametric Regression. High Dimensional Data. Numerical Methods in R. Optimization.

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Book SynopsisExam Board: MEILevel: A-levelSubject: MathematicsFirst Teaching: September 2017First Exam: June 2018An OCR endorsed textbookHelp students to develop their knowledge and apply their reasoning to mathematical problems with textbooks that draw on the well-known MEI (Mathematics in Education and Industry) series, updated and tailored to the 2017 OCR (MEI) specification and developed by subject experts and MEI.- Ensure targeted development of reasoning and problem-solving skills with plenty of practice questions and structured exercises that build mathematical skills and techniques.- Build connections between topics, using real-world contexts to help develop mathematical modelling skills, thus providing a fuller and more coherent understanding of mathematical concepts.- Help students to overcome misconceptions and develop insight into problem solving with annotated worked examples.- Develop u

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Book SynopsisPacked with insights, Lorenzo Bergomi's Stochastic Volatility Modeling explains how stochastic volatility is used to address issues arising in the modeling of derivatives, including: Which trading issues do we tackle with stochastic volatility? How do we design models and assess their relevance? How do we tell which models are usable and when does calibration make sense? This manual covers the practicalities of modeling local volatility, stochastic volatility, local-stochastic volatility, and multi-asset stochastic volatility. In the course of this exploration, the author, Risk's 2009 Quant of the Year and a leading contributor to volatility modeling, draws on his experience as head quant in Société Générale's equity derivatives division. Clear and straightforward, the book takes readers through various modeling challenges, all originating in actual trading/hedging issTrade Review"With this book, Bergomi has actually offered a precious gift to the whole quant community: his very rich and concrete experience on volatility modelling organized in 500 pages and 12 chapters full of insights; and to the academic community as well: new ideas, points of view, and questions that could well feed their research for years."- Julien Guyon, Quantitative Finance"[Stochastic Volatility Modeling] should be read by practitioners, as it is the only one providing a strong quantitative framework to the (Delta and Vega) hedging of Equity derivatives. It should also be read by academics who will benefit from practical insights. It should finally be read by (motivated) students, who will definitely find areas to dig deeper in, both theoretically and numerically […] This book should be seen as a strong case for the need of a deeper understanding of derivatives' modelling (and their risks). Lorenzo Bergomi provides us here with new tools (variance curve models, metrics such as the At-The-Money Forward Skew and the Skew Stickiness Ratio) as well as new results on hedging and P&L computations of actual trading strategies, which have been so far too much overlooked in mathematical finance research. Welcome to the new era of Derivatives Modelling!"- Antoine Jacquier, Newsletter of the Bachelier Finance Society, November 2017Table of ContentsIntroduction. Local volatility. Forward-start options. Stochastic volatility: introduction. Variance swaps. An example of one-factor dynamics: the Heston model. Forward variance models. The smile of stochastic volatility models. Linking static and dynamic properties of stochastic volatility models. What causes equity smiles? Multi-asset stochastic volatility. Local-stochastic volatility models.

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Book SynopsisBaseball is not the only sport to use "moneyball." American football fans, teams, and gamblers are increasingly using data to gain an edge against the competition. This concise book provides a clear introduction to using statistical models to analyze football data.

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Book SynopsisContains Nearly 100 Pages of New MaterialThe recent financial crisis has shown that credit risk in particular and finance in general remain important fields for the application of mathematical concepts to real-life situations. While continuing to focus on common mathematical approaches to model credit portfolios, Introduction to Credit Risk Modeling, Second Edition presents updates on model developments that have occurred since the publication of the best-selling first edition.New to the Second Edition An expanded section on techniques for the generation of loss distributions Introductory sections on new topics, such as spectral risk measures, an axiomatic approach to capital allocation, and nonhomogeneous Markov chains Updated sections on the probability of default, exposure-at-default, loss-given-default, and regulatory capital A new section on multi-period models Recent developments in structured credit The financial crisis illustrated the importance of effectively communicating model outcomes and ensuring that the variation in results is clearly understood by decision makers. The crisis also showed that more modeling and more analysis are superior to only one model. This accessible, self-contained book recommends using a variety of models to shed light on different aspects of the true nature of a credit risk problem, thereby allowing the problem to be viewed from different angles. Trade Review… this is a concise book for exploring the limitations of credit risk models and, to a lesser degree, asset valuation models. Read this book for a companionable journey through some of the limiting assumptions that make the models tractable. … it may be the first one [book] that wastes no time in getting to the point, and moving on.—Annals of Actuarial Science, Vol. 5, June 2011Bluhm, Overbeck, and Wagner offer help to mathematicians and physicists leaving the academy to work as risk or portfolio managers. For this introduction, they focus on main themes rather than details, and on portfolio rather than single obligor risk. … this second [edition] takes account of problems in the banking industry [from] 2007-09.—SciTech Book News, February 2011Having a valid and up-to-date credit risk model (or models) is one of the most important aspects in today’s risk management. The models require quite a bit of technical as well as practical know-how. Introduction to Credit Risk Modeling serves this purpose well. … it would best fit the practitioner’s needs. For students it can also be of great use, as an introductory course for credit risk models. A great first step into credit risk modeling. … The book provides a nice coherent overview of the methods used in capital allocation. … The book is written in a mixture of theorem-proof and applied styles. … I find this rather pleasing, as it gives the reader the edge of theoretical exposition, which is extremely important. … One really useful side of the book is that it provides step-by-step guide to methods presented. This should be really appreciated in industry and among students. …—MAA Reviews, January 2011Praise for the First EditionThis is an outstanding book on the default models that are used internally by financial institutions. This practical book delves into the mathematics, the assumptions and the approximations that practitioners apply to make these models work.—Glyn A. Holton, Contingency AnalysisThere are so many financial tools available today and numbers are likely to grow in the future. If you work in this field of credit risk modeling, it is worth looking at the theoretical background, and this book is a well-rounded introduction.—Journal of the Operational Research SocietyAs an introductory survey, it does an admirable job. … this book is an important guide into the field of credit risk models. Mainly for the practitioner … It is well written, fairly easy to follow.—Horst Behncke, Zentralblatt MATH… this is a concise book for exploring the limitations of credit risk models and, to a lesser degree, asset valuation models. Read this book for a companionable journey through some of the limiting assumptions that make the models tractable. … it may be the first one [book] that wastes no time in getting to the point, and moving on.—Annals of Actuarial Science, Vol. 5, June 2011Bluhm, Overbeck, and Wagner offer help to mathematicians and physicists leaving the academy to work as risk or portfolio managers. For this introduction, they focus on main themes rather than details, and on portfolio rather than single obligor risk. … this second [edition] takes account of problems in the banking industry [from] 2007-09.—SciTech Book News, February 2011Having a valid and up-to-date credit risk model (or models) is one of the most important aspects in today’s risk management. The models require quite a bit of technical as well as practical know-how. Introduction to Credit Risk Modeling serves this purpose well. … it would best fit the practitioner’s needs. For students it can also be of great use, as an introductory course for credit risk models. A great first step into credit risk modeling. … The book provides a nice coherent overview of the methods used in capital allocation. … The book is written in a mixture of theorem-proof and applied styles. … I find this rather pleasing, as it gives the reader the edge of theoretical exposition, which is extremely important. … One really useful side of the book is that it provides step-by-step guide to methods presented. This should be really appreciated in industry and among students. …—MAA Reviews, January 2011Praise for the First EditionThis is an outstanding book on the default models that are used internally by financial institutions. This practical book delves into the mathematics, the assumptions and the approximations that practitioners apply to make these models work.—Glyn A. Holton, Contingency AnalysisThere are so many financial tools available today and numbers are likely to grow in the future. If you work in this field of credit risk modelling it is worth looking at the theoretical background, and this book is a well-rounded introduction.—Journal of the Operational Research SocietyAs an introductory survey it does an admirable job. … this book is an important guide into the field of credit risk models. Mainly for the practitioner … It is well written, fairly easy to follow.—Horst Behncke, Zentralblatt MATHTable of ContentsThe Basics of Credit Risk Management. Modeling Correlated Defaults. Asset Value Models. The CreditRisk+ Model. Risk Measures and Capital Allocation. Term Structure of Default Probability. Credit Derivatives. Collateralized Debt Obligations. References. Index.

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Book SynopsisSurvey Weights: A Step-by-Step Guide to Calculation is the first guide geared toward Stata users that systematically covers the major steps taken in creating survey weights. These weights are used to project a sample to some larger population and can be computed for either probability or nonprobability samples. Sample designs can range from simple, single-stage samples to more complex, multistage samples, each of which may use specialized steps in weighting to account for selection probabilities, nonresponse, inaccurate coverage of a population by a sample, and auxiliary data to improve precision and compensate for coverage errors. The authors provide many examples with Stata code.Table of ContentsOverview of weighting. Initial steps in weighting probability samples. Adjustments for nonresponse. Calibration and other uses of auxiliary data in weighting. Use of weights in variance estimation. Nonprobability samples. Weighting for some special cases. Quality of survey weights.

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Book SynopsisIntroduction to Time Series Using Stata, Revised Edition provides a step-by-step guide to essential time-series techniques–from the incredibly simple to the quite complex– and, at the same time, demonstrates how these techniques can be applied in the Stata statistical package. The emphasis is on an understanding of the intuition underlying theoretical innovations and an ability to apply them. Real-world examples illustrate the application of each concept as it is introduced, and care is taken to highlight the pitfalls, as well as the power, of each new tool. The Revised Edition has been updated for Stata 16.Table of ContentsJust enough Stata Getting startedAll about dataLooking at dataStatisticsOdds and endsMaking a dateTyping dates and date variablesLooking aheadJust enough statistics Random variables and their momentsHypothesis testsLinear regressionMultiple-equation modelsTime seriesFiltering time-series dataPreparing to analyze a time seriesThe four components of a time seriesSome simple filtersAdditional filtersPoints to rememberA first pass at forecastingForecast fundamentalsFilters that forecastPoints to rememberLooking aheadAutocorrelated disturbancesAutocorrelationRegression models with autocorrelated disturbancesTesting for autocorrelationEstimation with first-order autocorrelated dataEstimating the mortgage rate equation Points to rememberUnivariate time-series modelsThe general linear processLag polynomials: Notation or prestidigitations?The ARMA modelStationarity and invertibilityWhat can ARMA models do?Points to rememberLooking aheadModeling a real-world time seriesGetting ready to model a time seriesThe Box-Jenkins approachSpecifying an ARMA modelEstimationLooking for trouble: Model diagnostic checkingForecasting with ARIMA modelsComparing forecastsPoints to rememberWhat have we learned so far?Looking aheadTime-varying volatilityExamples of time-varying volatilityARCH: A model of time-varying volatility Extensions to the ARCH modelPoints to rememberModel of multiple time seriesVector autoregressionsA VAR of the U.S. macroeconomyWho’s on first?SVARsPoints to rememberLooking aheadModels of nonstationary times seriesTrend and unit rootsTesting for unit rootsCointegration: Looking for a long-term relationshipCointegrating relationships and VECMFrom intuition to VECM: An examplePoints to rememberLooking aheadClosing observationsMaking sense of it allWhat did we miss?FarewellReferences

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Book SynopsisAn Introduction to Stata for Health Researchers, Fifth Edition updates this classic book that has become a standard reference for health researchers. As with previous editions, readers will learn to work effectively in Stata to perform data management, compute descriptive statistics, create meaningful graphs, fit regression models, and perform survival analysis. The fifth edition adds examples of performing power, precision, and sample-size analysis; working with Unicode characters; managing data with ICD-9 and ICD-10 codes; and creating customized tables.With many worked examples and downloadable datasets, this text is the ideal resource for hands-on learning, whether for students in a statistics course or for researchers in fields such as epidemiology, biostatistics, and public health who are learning to use Stata's tools for health research.Table of ContentsI The basics 1. Getting started 2. Getting help—and more 3. Command syntax II Data management 4. Variables 5. Getting data in and out of Stata 6. Adding explanatory text to data 7. Calculations 8. Commands affecting data structure 9. Taking good care of your data III Analysis 10. Description and simple analysis 11. Regression analysis 12. Time-to-event data 13. Power, precision, and sample-size analysis 14. Measurement and diagnosis 15. Miscellaneous IV Graphs 16. Graphs V Advanced topics 17. Advanced topics

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Book SynopsisInterpreting and Visualizing Regression Models Using Stata, Second Edition provides clear and simple examples illustrating how to interpret and visualize a wide variety of regression models. Including over 200 figures, the book illustrates linear models with continuous predictors (modeled linearly, using polynomials, and piecewise), interactions of continuous predictors, categorical predictors, interactions of categorical predictors, and interactions of continuous and categorical predictors. The book also illustrates how to interpret and visualize results from multilevel models, models where time is a continuous predictor, models with time as a categorical predictor, nonlinear models (such as logistic or ordinal logistic regression), and models involving complex survey data. The examples illustrate the use of the margins, marginsplot, contrast, and pwcompare commands. This new edition reflects new and enhanced features added to Stata, most importantly the ability to label statistical output using value labels associated with factor variables. As a result, output regarding marital status is labeled using intuitive labels like Married and Unmarried instead of using numeric values such as 1 and 2. All the statistical output in this new edition capitalizes on this new feature, emphasizing the interpretation of results based on variables labeled using intuitive value labels. Additionally, this second edition illustrates other new features, such as using transparency in graphics to more clearly visualize overlapping confidence intervals and using small sample-size estimation with mixed models. If you ever find yourself wishing for simple and straightforward advice about how to interpret and visualize regression models using Stata, this book is for you.Table of ContentsIntroduction; Continuous predictors: Linear; Continuous predictors: Polynomials; Continuous predictors: Piecewise models; Continuous by continuous interactions; Continuous by continuous by continuous interactions; Categorical predictors; Categorical by categorical interactions; Categorical by categorical by categorical interactions; Linear by categorical interactions; Polynomial by categorical interactions; Piecewise by categorical interactions; Continuous by continuous by categorical interactions; Continuous by categorical by categorical interactions; Multilevel models; Time as a continuous predictor; Time as a categorical predictor; Nonlinear models; Complex survey data

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Book SynopsisStrange happenstances and chance encounters have puzzled us for centuries. This fun and fascinating book takes readers on a journey through the mathematics behind coincidences both famous and never-before-examined. From peculiar patterns in geometry and calculus to the famous Waring Problem, and other astonishing numerical curiosities, The Call of Coincidence begins by examining the mathematical properties that underpin everything there is. Next, author Owen O’Shea – along with fictional guides Charlie Chance and the enigmatic Dr. Moogle – reveals surprising connections and correlations throughout history, including numerical coincidences behind the reign of King Richard III, the sinking of the SS Edmund Fitzgerald, the 1996 FIFA World Cup, and much, much more. By investigating the properties, puzzles, and problems within, you will gain a newfound appreciation for the beautiful simplicity of mathematics in its many forms. Featuring surprising trivia gems alongside serious questions like why there is something rather than nothing, readers will be enriched by this exploration of remarkable number coincidences and the mathematics that make them possible – and probable.

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Book SynopsisThis book provides an undergraduate introduction to analysing data for data science, computer science, and quantitative social science students. It uniquely combines a hands-on approach to data analysis – supported by numerous real data examples and reusable [R] code – with a rigorous treatment of probability and statistical principles. Where contemporary undergraduate textbooks in probability theory or statistics often miss applications and an introductory treatment of modern methods (bootstrapping, Bayes, etc.), and where applied data analysis books often miss a rigorous theoretical treatment, this book provides an accessible but thorough introduction into data analysis, using statistical methods combining the two viewpoints. The book further focuses on methods for dealing with large data-sets and streaming-data and hence provides a single-course introduction of statistical methods for data science.Trade Review“Having taught data analytics at the introductory graduate level, I welcome the authors’ textbook as an essential resource for training well-grounded entry-level data scientists. … A data scientist shall provide competent data science professional services to a client. … Training in both the theory and practice of data analytics is a requirement for such competence. The authors’ textbook definitely provides a valuable resource for such training.” (Harry J. Foxwell, Computing Reviews, July 7, 2022)Table of Contents1 A First Look at Data.- 2 Sampling Plans and Estimates.- 3 Probability Theory.- 4 Random Variables and Distributions.- 5 Estimation.- 6 Multiple Random Variables.- 7 Making Decisions in Uncertainty.- 8 Bayesian Statistics.

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Book SynopsisThis book introduces descriptive statistics and covers a broad range of topics of interest to students and researchers in various applied science disciplines. This includes measures of location, spread, skewness, and kurtosis; absolute and relative measures; and classification of spread, skewness, and kurtosis measures, L-moment based measures, van Zwet ordering of kurtosis, and multivariate kurtosis. Several novel topics are discussed including the recursive algorithm for sample variance; simplification of complicated summation expressions; updating formulas for sample geometric, harmonic and weighted means; divide-and-conquer algorithms for sample variance and covariance; L-skewness; spectral kurtosis, etc. A large number of exercises are included in each chapter that are drawn from various engineering fields along with examples that are illustrated using the R programming language. Basic concepts are introduced before moving on to computational aspects. Some applications in bioinformatics, finance, metallurgy, pharmacokinetics (PK), solid mechanics, and signal processing are briefly discussed. Every analyst who works with numeric data will find the discussion very illuminating and easy to follow.Table of ContentsDescriptive Statistics.- Measures of Location.- Measures of Spread.- Measures of Skewness and Kurtosis.

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Book SynopsisUncertainty in Optimization.- Information Structures and Feasibility.- Modeling the Objective Function.- Scenario Tree Generation, With Michal Kaut and Jamie Fairbrother.- High-Dimentional Dependent Randomness. With Zhaoxia Guo and Michal Kaut.- Service Network Design, With Arnt-Gunnar Lium and Teodor Gabriel Crainic.- A Multi-dimensional Newsboy Problem with Substitution, With Hajnalka Vaagen.- References.- Index.

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Book SynopsisThis is the fifth edition of a very successful textbook on clinical trials methodology, written by recognized leaders who have long and extensive experience in all areas of clinical trials. The three authors of the first four editions have been joined by two others who add great expertise. A chapter on regulatory issues has been included and the chapter on data monitoring has been split into two and expanded. Many contemporary clinical trial examples have been added. There is much new material on adverse events, adherence, issues in analysis, electronic data, data sharing and international trials.This book is intended for the clinical researcher who is interested in designing a clinical trial and developing a protocol. It is also of value to researchers and practitioners who must critically evaluate the literature of published clinical trials and assess the merits of each trial and the implications for the care and treatment of patients. The authors use numerous examples of published clinical trials to illustrate the fundamentals.The text is organized sequentially from defining the question to trial closeout. One chapter is devoted to each of the critical areas to aid the clinical trial researcher. These areas include pre-specifying the scientific questions to be tested and appropriate outcome measures, determining the organizational structure, estimating an adequate sample size, specifying the randomization procedure, implementing the intervention and visit schedules for participant evaluation, establishing an interim data and safety monitoring plan, detailing the final analysis plan and reporting the trial results according to the pre-specified objectives.Although a basic introductory statistics course is helpful in maximizing the benefit of this book, a researcher or practitioner with limited statistical background would still find most if not all the chapters understandable and helpful. While the technical material has been kept to a minimum, the statistician may still find the principles and fundamentals presented in this text useful. Trade Review“This book aims to assist investigators in improving the quality of their clinical trials and protocols by discussing fundamental concepts with examples and in-depth review of the literature. … This is a valuable resource for students, clinicians, and researchers who are interested in designing a clinical trial or in critically appraising the published literature on clinical trials.” (Pooja Sethi, Doody’s Book Reviews, December, 2015)Table of ContentsIntroduction to Clinical Trials.- Ethical Issues.- What is the Question?.- Study Population.- Basic Study Design.- The Randomization Process.- Blinding.- Sample Size.- Baseline Assessment.- Recruitment of Study Participants.- Data Collection and Quality Control.- Assessment and Reporting of Harm.- Assessment of Health Related Quality of Life.- Participant Adherence.- Survival Analysis.- Monitoring Committee Structure & Function.- Statistical Methods Used in Interim Monitoring.- Issues in Data Analysis.- Closeout.- Reporting and Interpreting of Results.- Multicenter Trials.- Regulatory Issues.

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Book SynopsisThis textbook on practical data analytics unites fundamental principles, algorithms, and data. Algorithms are the keystone of data analytics and the focal point of this textbook. Clear and intuitive explanations of the mathematical and statistical foundations make the algorithms transparent. But practical data analytics requires more than just the foundations. Problems and data are enormously variable and only the most elementary of algorithms can be used without modification. Programming fluency and experience with real and challenging data is indispensable and so the reader is immersed in Python and R and real data analysis. By the end of the book, the reader will have gained the ability to adapt algorithms to new problems and carry out innovative analyses.This book has three parts:(a) Data Reduction: Begins with the concepts of data reduction, data maps, and information extraction. The second chapter introduces associative statistics, the mathematical foundation of scalable algorithms and distributed computing. Practical aspects of distributed computing is the subject of the Hadoop and MapReduce chapter.(b) Extracting Information from Data: Linear regression and data visualization are the principal topics of Part II. The authors dedicate a chapter to the critical domain of Healthcare Analytics for an extended example of practical data analytics. The algorithms and analytics will be of much interest to practitioners interested in utilizing the large and unwieldly data sets of the Centers for Disease Control and Prevention's Behavioral Risk Factor Surveillance System.(c) Predictive Analytics Two foundational and widely used algorithms, k-nearest neighbors and naive Bayes, are developed in detail. A chapter is dedicated to forecasting. The last chapter focuses on streaming data and uses publicly accessible data streams originating from the Twitter API and the NASDAQ stock market in the tutorials.This book is intended for a one- or two-semester course in data analytics for upper-division undergraduate and graduate students in mathematics, statistics, and computer science. The prerequisites are kept low, and students with one or two courses in probability or statistics, an exposure to vectors and matrices, and a programming course will have no difficulty. The core material of every chapter is accessible to all with these prerequisites. The chapters often expand at the close with innovations of interest to practitioners of data science. Each chapter includes exercises of varying levels of difficulty. The text is eminently suitable for self-study and an exceptional resource for practitioners.Trade Review“This 430-page book contains an excellent collection of information on the subject of practical algorithms used in data science. The discussion of each algorithm starts with some basic concepts, followed by a tutorial with real datasets and detailed code examples in Python or R. Each chapter has a set of exercise problems so readers can practice the concepts learned in the chapter. … a good reference for practitioners, or a good textbook for graduate or upper-class undergraduate students.” (Xiannong Meng, Computing Reviews, September, 2017)“This textbook on practical data analytics unites fundamental principles, algorithms, and data. … this book is devoted to upper-division undergraduate and graduate students in mathematics, statistics, and computer science. It is intended for a one- or two-semester course in data analytics and reflects the authors’ research experience in data science concepts and the teaching skills in various areas. … The text is eminently suitable for self-study and an exceptional resource for practitioners.” (Krzysztof J. Szajowski, zbMATH 1367.62005, 2017) Table of ContentsIntroduction.- Data Mapping and Data Dictionaries.- Scalable Algorithms and Associative Statistics.- Hadoop and MapReduce.- Data Visualization.- Linear Regression Methods.- Healthcare Analytics.- Cluster Analysis.- k-Nearest Neighbor Prediction Functions.- The Multinomial Naive Bayes Prediction Function.- Forecasting.- Real-time Analytics.

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Book SynopsisStochastic processes are tools used widely by statisticians and researchers working in the mathematics of finance. This book for self-study provides a detailed treatment of conditional expectation and probability, a topic that in principle belongs to probability theory, but is essential as a tool for stochastic processes. The book centers on exercises as the main means of explanation.Trade ReviewThis book fulfils its aim of providing good and interesting material for advanced undergraduate study. The Times Higher Education Supplement This is probably one of the best books to begin learning about the sometimes complex topic of stochastic calculus and stochastic processes from a more mathematical approach. Some literature are often accused of unnecessarily complicating the subject when applied to areas of finance. With this book you are allowed to explore the rigorous side of stochastic calculus, yet maintain a physical insight of what is going on. The authors have concentrated on the most important and useful topics that are encountered in common physical and financial systems www.quantnotes.com Table of Contents1. Review of Probability.- 1.1 Events and Probability.- 1.2 Random Variables.- 1.3 Conditional Probability and Independence.- 1.4 Solutions.- 2. Conditional Expectation.- 2.1 Conditioning on an Event.- 2.2 Conditioning on a Discrete Random Variable.- 2.3 Conditioning on an Arbitrary Random Variable.- 2.4 Conditioning on a ?-Field.- 2.5 General Properties.- 2.6 Various Exercises on Conditional Expectation.- 2.7 Solutions.- 3. Martingales in Discrete.- 3.1 Sequences of Random Variables.- 3.2 Filtrations.- 3.3 Martingales.- 3.4 Games of Chance.- 3.5 Stopping Times.- 3.6 Optional Stopping Theorem.- 3.7 Solutions.- 4. Martingale Inequalities and Convergence.- 4.1 Doob’s Martingale Inequalities.- 4.2 Doob’s Martingale Convergence Theorem.- 4.3 Uniform Integrability and L1 Convergence of Martingales.- 4.4 Solutions.- 5. Markov Chains.- 5.1 First Examples and Definitions.- 5.2 Classification of States.- 5.3 Long-Time Behaviour of Markov Chains: General Case.- 5.4 Long-Time Behaviour of Markov Chains with Finite State Space.- 5.5 Solutions.- 6. Stochastic Processes in Continuous Time.- 6.1 General Notions.- 6.2 Poisson Process.- 6.2.1 Exponential Distribution and Lack of Memory.- 6.2.2 Construction of the Poisson Process.- 6.2.3 Poisson Process Starts from Scratch at Time t.- 6.2.4 Various Exercises on the Poisson Process.- 6.3 Brownian Motion.- 6.3.1 Definition and Basic Properties.- 6.3.2 Increments of Brownian Motion.- 6.3.3 Sample Paths.- 6.3.4 Doob’s Maximal L2 Inequality for Brownian Motion.- 6.3.5 Various Exercises on Brownian Motion.- 6.4 Solutions.- 7. Itô Stochastic Calculus.- 7.1 Itô Stochastic Integral: Definition.- 7.2 Examples.- 7.3 Properties of the Stochastic Integral.- 7.4 Stochastic Differential and Itô Formula.- 7.5 Stochastic Differential Equations.- 7.6 Solutions.

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Book SynopsisThis book shows you how to analyze data sets systematically and to use Excel 2019 to extract information from data almost effortlessly. Both are (not) an art!The statistical methods are presented and discussed using a single data set. This makes it clear how the methods build on each other and gradually more and more information can be extracted from the data. The Excel functions used are explained in detail - the procedure can be easily transferred to other data sets. Various didactic elements facilitate orientation and working with the book: At the checkpoints, the most important aspects from each chapter are briefly summarized. In the freak knowledge section, more advanced aspects are addressed to whet the appetite for more. All examples are calculated with hand and Excel. Numerous applications and solutions as well as further data sets are available on the author's internet platform. This book is a translation of the original German 2nd edition Statistik angewandt mit Excel by Franz Kronthaler, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2021. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.Table of ContentsPart 1 - Basic knowledge and tools to apply statistics.- Statistics is fun.- Excel: A brief introduction and the statistical possibilities.- Part 2 - Describe, nothing but describe.- Mean values: How people and objects behave on average.- Scatter: The deviation from average behavior.- Graphs: The possibility to represent data visually.- Correlation: Of the correlation.- Ratio and index numbers: The chance to generate new things from old knowledge.- Part 3 - From Few to All.- Of Data and the Truth.- Hypotheses: Just a specification of the question.- Normal distribution and other test distributions.- Hypothesis testing: What is Valid?.- Part 4 - Procedures for Testing Hypotheses.- The Mean Test.- The Test for Difference of Means in Independent Samples.- The Test for Difference of Means in Dependent Samples.- The Analysis of Variance for Testing for Group Differences in More than Two Groups.- The Test for Correlation in Metric, Ordinal, and Nominal Data.- Further Test Procedures for Nominal Variables.- Summary Part IV - Overview of testing procedures.- Part 5 - Regression analysis.- The linear single regression.- The multiple regression analysis.- Part 6 - What's next.- Brief report on a research question.- Further statistical procedures.- Interesting and further statistics books.- Another data set to practice on - Intern of a company.- Appendix.

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Book SynopsisThe book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered. Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time models. The basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations. Central results of stochastic analysis such as the Itô formula, Girsanov's theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models. Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics.Advanced knowledge of probability theory is assumed, in particular of discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, Lévy processes, processes with independent increments, Markov processes). The book is thus suitable for advanced students as a companion reading and for instructors as a basis for their own courses.This book is a translation of the original German 1st edition Stochastische Prozesse und Finanzmathematik by Ludger Rüschendorf, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com) and in a subsequent editing, improved by the author. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.Table of ContentsOption pricing in models in discrete time.- Scorohod's embedding theorem and Donsker's theorem.- Stochastic integration.- Elements of stochastic analysis.- Option pricing in complete and incomplete markets.- Utility optimization, minimum distance martingales, and utility indiff.- Variance-minimum hedging.

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Book SynopsisThis 2nd edition compendium contains and explains essential statistical formulas within an economic context. Expanded by more than 100 pages compared to the 1st edition, the compendium has been supplemented with numerous additional practical examples, which will help readers to better understand the formulas and their practical applications. This statistical formulary is presented in a practice-oriented, clear, and understandable manner, as it is needed for meaningful and relevant application in global business, as well as in the academic setting and economic practice.The topics presented include, but are not limited to: statistical signs and symbols, descriptive statistics, empirical distributions, ratios and index figures, correlation analysis, regression analysis, inferential statistics, probability calculation, probability distributions, theoretical distributions, statistical estimation methods, confidence intervals, statistical testing methods, the Peren-Clement index, and the usual statistical tables.Given its scope, the book offers an indispensable reference guide and is a must-read for undergraduate and graduate students, as well as managers, scholars, and lecturers in business, politics, and economics.Table of ContentsStatistical Signs and Symbols.- Descriptive Statistics.- Inferential Statistics.- Probability Calculation.- Statistical Tables.- Bibliography.- Index.

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Book SynopsisModelling with the Itô integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory.This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black-Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Itô calculus and/or stochastic finance.Trade Review"This book under review can be determined as a very successful work ... the author's choice of the material is done with good taste and expertise ... It can be strongly recommended to graduate students and practitioners in the field of finance and economics." Mathematics Abstracts, 2000 "... this is a well-written book, which makes the difficult object of mathematical finance easy to understand also for non-mathematicians. It might be useful for economics students and all practitioners in the field of finance who are interested in the mathematical methodology behind the Black-Scholes model." Statistical Papers, 2000Table of ContentsPreliminaries - basic concepts from probability theory; stochastic processes; Brownian motion; conditional expectation; Martingales; the stochastic integral - the Riemann and Riemann-Stieltjes; integrals; the Ito integral; the Ito lemma; the Stratonovich and other integrals; stochastic differential equations - deterministic differential equations; Ito stochastic differential equations; the general linear differential equation; numerical solution; applications of stochastic calculus in finance - the Black-Scholes option-pricing formula; a useful technique - change of measure. Appendices: modes of convergence; inequalities; non-differentiability and unbounded variation of Brownian sample paths; proof of the existence of the general Ito stochastic integral; the Radon-Nikodym theorem; proof of the existence and uniqueness of the conditional expectation.

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Book SynopsisThe main objective of this text is to facilitate a student's smooth learning transition from a course on probability to its applications in various areas. To achieve this goal, students are encouraged to experiment numerically with problems requiring computer solutions.

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Book SynopsisThis book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.Table of ContentsProbability Background.- Gambling Problems.- Random Walks.- Discrete-Time Markov Chains.- First Step Analysis.- Classification of States.- Long-Run Behavior of Markov Chains.- Branching Processes.- Continuous-Time Markov Chains.- Discrete-Time Martingales.- Spatial Poisson Processes.- Reliability Theory.

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