TEXTBOOK
Part of Institute of Mathematical Statistics Monographs
PUBLICATION PLANNED FOR: June 2021
AVAILABILITY: Not yet published - available from June 2021FORMAT:
Paperback ISBN: 9781108823418
The twenty-first century has seen a breathtaking expansion of statistical methodology, both in scope and influence. 'Data science' and 'machine learning' have become familiar terms in the news, as statistical methods are brought to bear upon the enormous data sets of modern science and commerce. How did we get here? And where are we going? How does it all fit together? Now in paperback and fortified with exercises, this book delivers a concentrated course in modern statistical thinking. Beginning with classical inferential theories - Bayesian, frequentist, Fisherian - individual chapters take up a series of influential topics: survival analysis, logistic regression, empirical Bayes, the jackknife and bootstrap, random forests, neural networks, Markov Chain Monte Carlo, inference after model selection, and dozens more. The distinctly modern approach integrates methodology and algorithms with statistical inference. Each chapter ends with class-tested exercises, and the book concludes with speculation on the future direction of statistics and data science.
Now in paperback and fortified with exercises, this book provides a course in modern statistical thinking written by two world-leading researchers
130 class-tested exercises covering theory, methods, and computation help students make the link to scientific knowledge (and uncertainty)
Clarifies both traditional methods and current, popular algorithms (e.g. neural nets, random forests), giving students a broad and modern appreciation of the topic
Copyright Year 2022
ISBN 9780367686949
June 1, 2021 Forthcoming by Chapman and Hall/CRC
136 Pages 19 B/W Illustrations
Format : Hardback
Combinatorial Nullstellensatz is a novel theorem in algebra introduced by Noga Alon to tackle combinatorial problems in diverse areas of mathematics. This book focuses on the applications of this theorem to graph colouring. A key step in the applications of Combinatorial Nullstellensatz is to show that the coefficient of a certain monomial in the expansion of a polynomial is nonzero. The major part of the book concentrates on three methods for calculating the coefficients:
Alon-Tarsi orientation: The task is to show that a graph has an orientation with given maximum out-degree and for which the number of even Eulerian sub-digraphs is different from the number of odd Eulerian sub-digraphs. In particular, this method is used to show that a graph whose edge set decomposes into a Hamilton cycle and vertex-disjoint triangles is 3-choosable, and that every planar graph has a matching whose deletion results in a 4-choosable graph.
Interpolation formula for the coefficient: This method is in particular used to show that toroidal grids of even order are 3-choosable, r-edge colourable r-regular planar graphs are r-edge choosable, and complete graphs of order p+1, where p is a prime, are p-edge choosable.
Coefficients as the permanents of matrices: This method is in particular used in the study of the list version of vertex-edge weighting and to show that every graph is (2,3)-choosable.
It is suited as a reference book for a graduate course in mathematics.
Some definitions and notations. Combinatorial Nullstellensatz. Introduction. An application of CNS to additive number theory. Application of CNS to geometry. CNS and a subgraph problem. 0-1 vectors in a hyperplane. Alon-Tarsi Theorem and its Applications. Alon-Tarsi Theorem. Bipartite graphs and acyclic orientations. The Cartesian product of a path and an odd cycle. A solution to a problem of Erd?os. Bound for AT (G) in terms of degree. Planar graphs. Planar graph minus a matching. Discharging method. Hypergraph colouring. Paintability of graphs. Generalizations of CNS and applications. Number of nonzero points. Multisets. Coefficient of a highest degree monomial. Calculation of NS(a). Alon-Tarsi number of K2*n and cycle powers. Alon-Tarsi numbers of toroidal grids. List colouring of line graphs. r-regular planar graphs. Complete graphs Kp+1 for odd prime p. Jaegerfs Conjecture. Permanent and vertex-edge weighting. Permanent as the coefficient. Edge weighting and total weighting. Polynomial associated to total weighting. Permanent index. Trees with an even number of edges. Complete graphs. Every graph is (2, 3)-choosable.
Copyright Year 2022
ISBN 9780367893446
July 5, 2021 Forthcoming by Chapman and Hall/CRC
344 Pages 51 B/W Illustrations
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Malliavin Calculus in Finance: Theory and Practice aims to introduce the study of stochastic volatility (SV) models via Malliavin Calculus.
Malliavin calculus has had a profound impact on stochastic analysis. Originally motivated by the study of the existence of smooth densities of certain random variables, it has proved to be a useful tool in many other problems. In particular, it has found applications in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks.
The objective of this book is to offer a bridge between theory and practice. It shows that Malliavin calculus is an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results.
Intermediate-advanced level text on quantitative finance, oriented to practitioners with a basic background in stochastic analysis, which could also be useful for researchers and students in quantitative finance
Includes examples on concrete models such as the Heston, the SABR and rough volatilities, as well as several numerical experiments and the corresponding Python scripts
Covers applications on vanillas, forward start options, and options on the VIX.
The book also has a Github repository with the Python library corresponding to the numerical examples in the text. The library has been implemented so that the users can re-use the numerical code for building their examples. The repository can be accessed here: https://bit.ly/2KNex2Y.
I. A primer on option pricing and volatility modeling. 1. The option pricing problem. 1.1. Derivatives. 1.2. Non-arbitrage prices and the Black-Scholes formula. 1.3. The Black-Scholes model. 1.4. The Black-Scholes implied volatility and the non-constant volatility case. 1.5. Chapter's digest. 2. The volatility process. 2.1. The estimation of the integrated and the spot volatility. 2.2. Local volatilities. 2.3. Stochastic volatilities. 2.4. Stochastic-local volatilities 2.5. Models based on the fractional Brownian motion and rough volatilities. 2.6. Volatility derivatives. 2.7. Chapterfs Digest. II. Mathematical tools. 3. A primer on Malliavin Calculus. 3.1. Definitions and basic properties. 3.2. Computation of Malliavin Derivatives. 3.3. Malliavin derivatives for general SV models. 3.4. Chapter's digest. 4. Key tools in Malliavin Calculus. 4.1. The Clark-Ocone-Haussman formula. 4.2. The integration by parts formula. 4.3. The anticipating It^o's formula. 4.4. Chapterfs Digest. 5. Fractional Brownian motion and rough volatilities. 5.1. The fractional Brownian motion. 5.2. The Riemann-Liouville fractional Brownian motion. 5.3. Stochastic integration with respect to the fBm. 5.4. Simulation methods for the fBm and the RLfBm. 5.5. The fractional Brownian motion in finance. 5.6. The Malliavin derivative of fractional volatilities. 5.7. Chapter's digest. III. Applications of Malliavin Calculus to the study of the implied volatility surface. 6. The ATM short time level of the implied volatility. 6.1. Basic definitions and notation. 6.2. The classical Hull and White formula. An extension of the Hull and White formula from the anticipating Ito's formula. 6.4. Decomposition formulas for implied volatilities. 6.5. The ATM short-time level of the implied volatility. 6.6. Chapter's digest. 7. The ATM short-time skew. 7.1. The term structure of the empirical implied volatility surface. 7.2. The main problem and notations. 7.3. The uncorrelated case. 7.4. The correlated case. 7.5. The short-time limit of implied volatility skew. 7.6. Applications. 7.7. Is the volatility long-memory, short memory, or both?. 7.8. A comparison with jump-diffusion models: the Bates model. 7.9. Chapter's digest. 8.0. The ATM short-time curvature. 8.1. Some empirical facts. 8.2. The uncorrelated case. 8.3. The correlated case. 8.4. Examples. 8.5. Chapter's digest. IV. The implied volatility of non-vanilla options. 9. Options with random strikes and the forward smile. 9.1. A decomposition formula for random strike options. 9.2. Forward start options as random strike options. 9.3. Forward-Start options and the decomposition formula. 9.4. The ATM short-time limit of the implied volatility. 9.5. At-the-money skew. 9.6. At-the-money curvature. 9.7. Chapter's digest. 10. Options on the VIX. 10.1. The ATM short time level and skew of the implied volatility. 10.2. VIX options. 10.3. Chapter's digest. Bibliography. Index.
Copyright Year 2022
ISBN 9780367457754
July 5, 2021 Forthcoming by Chapman and Hall/CRC
246 Pages 54 B/W Illustrations
Numerical Methods for Unsteady Compressible Flow Problems is written to give both mathematicians and engineers an overview of the state of the art in the field, as well as of new developments. The focus is on methods for the compressible Navier-Stokes equations, the solutions of which can exhibit shocks, boundary layers and turbulence. The idea of the text is to explain the important ideas to the reader, while giving enough detail and pointers to literature to facilitate implementation of methods and application of concepts.
The book covers high order methods in space, such as Discontinuous Galerkin methods, and high order methods in time, in particular implicit ones. A large part of the text is reserved to discuss iterative methods for the arising large nonlinear and linear equation systems. Ample space is given to both state-of-the-art multigrid and preconditioned Newton-Krylov schemes.
Applications to aerospace, high-speed vehicles, heat transfer, and more besides
Suitable as a textbook for graduate-level courses in CFD, or as a reference for practitioners in the field
Preface. 1. Introduction. 1.1. The method of lines. 1.2. Hardware. 1.3. Notation. 1.4. Outline. 2. The Governing Equation. 2.1. The Navier-Stokes Equations. 2.2. Nondimensionalization. 2.3. Source terms. 2.4. Simplifications of the Navier-Stokes equations. 2.5. The Euler Equations. 2.6. Solution theory. 2.7. Boundary layers. 2.8. Boundary layers. 2.9. Laminar and turbulent flows. 3. The Space discretization. 3.1. Structured and unstructured Grids. 3.2. Finite Volume Methods. 3.3. The Line Integrals and Numerical Flux Functions. 3.4 Convergence theory for finite volume methods. 3.5. Source Terms. 3.6. Finite volume methods of higher order. 3.7. Discontinuous Galerkin methods. 3.8. Convergence theory for DG methods. 3.9. Boundary Conditions. 3.10. Spatial Adaptation. 4. Time Integration Schemes. 4.1. Order of convergence and order of consistency. 4.2 Stability. 4.3. Stiff problems. 4.4. Backward Differentiation formulas. 4.5. Runge-Kutta methods. 4.6. Rosenbrock-type methods. 4.7. Adaptive time step size selection. 4.8. Operator Splittings. 4.9. Alternatives to the method of lines. 4.10. Parallelization in time. 5. Solving equation systems. 5.1. The nonlinear systems. 5.2. The linear systems. 5.3. Rate of convergence and error. 5.4. Termination criteria. 5.5. Fixed Point methods. 5.6. Multigrid methods. 5.7. Newton's method. 5.8. Krylov subspace methods. 5.9. Jacobian Free Newton-Krylov methods. 5.10. Comparison of GMRES and BiCGSTAB. 5.11. Comparison of variants of Newton's method. 6. Preconditioning linear systems. 6.1. Preconditioning for JFNK schemes. 6.2. Specific preconditioners. 6.3. Preconditioning in parallel. 6.4. Sequences of linear systems. 6.5. Discretization for the preconditioner. 7. The final schemes. 7.1. DIRK scheme. 7.2. Rosenbrock scheme. 7.3. Parallelization. 7.4. Efficiency of Finite Volume schemes. 7.5. Efficiency of Discontinuous Galerkin schemes. 8. Thermal Fluid Structure Interaction. 8.1. Gas Quenching. 8.2. The mathematical model. 8.3. Space discretization. 8.4. Coupled time integration. 8.5. Dirichlet-Neumann iteration. 8.6. Alternative solvers. 8.7. Numerical Results. A. Test problems. A.1. Shu-Vortex. A.2. Supersonic Flow around a cylinder. A.3. Wind Turbine. A.4. Vortex shedding behind a sphere. B. Coefficients of time integration methods. Bibliography. Index.
Copyright Year 2022
ISBN 9780367366513
July 9, 2021 Forthcoming by Chapman and Hall/CRC
272 Pages
Bayesian Networks: With Examples in R, Second Edition introduces Bayesian networks using a hands-on approach. Simple yet meaningful examples illustrate each step of the modelling process and discuss side-by-side the underlying theory and its application using R code. The examples start from the simplest notions and gradually increase in complexity. In particular, this new edition contains significant new material on topics from modern machine learning practice: dynamic networks, networks with heterogeneous variables, and model validation.
The first three chapters explain the whole process of Bayesian network modelling, from structure learning to parameter learning to inference. These chapters cover discrete, Gaussian, and conditional Gaussian Bayesian networks. The following two chapters delve into dynamic networks (to model temporal data) and into networks including arbitrary random variables (using Stan). The book then gives a concise but rigorous treatment of the fundamentals of Bayesian networks and offers an introduction to causal Bayesian networks. It also presents an overview of R packages and other software implementing Bayesian networks. The final chapter evaluates two real-world examples: a landmark causal protein-signaling network published in Science and a probabilistic graphical model for predicting the composition of different body parts.
Covering theoretical and practical aspects of Bayesian networks, this book provides you with an introductory overview of the field. It gives you a clear, practical understanding of the key points behind this modelling approach and, at the same time, it makes you familiar with the most relevant packages used to implement real-world analyses in R. The examples covered in the book span several application fields, data-driven models and expert systems, probabilistic and causal perspectives, thus giving you a starting point to work in a variety of scenarios.
Online supplementary materials include the data sets and the code used in the book, which will all be made available from https://www.bnlearn.com/book-crc-2ed/
Preface to the Second Edition
Preface to the First Edition
1. The Discrete Case: Multinomial Bayesian Networks
Introductory Example: Train Use Survey
Graphical Representation
Probabilistic Representation
Estimating the Parameters: Conditional Probability Tables
Learning the DAG Structure: Tests and Scores
Conditional Independence Tests
Network Scores
Using Discrete Bayesian Networks
Using the DAG Structure
Using the Conditional Probability Tables
Exact Inference
Approximate Inference
Plotting Discrete Bayesian Networks
Plotting DAGs
Plotting Conditional Probability Distributions
Further Reading
2. The Continuous Case: Gaussian Bayesian Networks
Introductory Example: Crop Analysis
Graphical Representation
Probabilistic Representation
Estimating the Parameters: Correlation Coefficients
Learning the DAG Structure: Tests and Scores
Conditional Independence Tests
Network Scores
Using Gaussian Bayesian Networks
Exact Inference
Approximate Inference
Plotting Gaussian Bayesian Networks
Plotting DAGs
Plotting Conditional Probability Distributions
More Properties
Further Reading
3. The Mixed Case: Conditional Gaussian Bayesian Networks
Introductory Example: Healthcare Costs
Graphical and Probabilistic Representation
Estimating the Parameters: Mixtures of Regressions
Learning the DAG Structure: Tests and Scores
Using Conditional Gaussian Bayesian Networks
Further Reading
4. Time Series: Dynamic Bayesian Networks
Introductory Example: Domotics
Graphical Representation
Probabilistic Representation
Learning a Dynamic Bayesian Network
Using Dynamic Bayesian Networks
Plotting Dynamic Bayesian Networks
Further Reading
5. More Complex Cases: General Bayesian Networks
Introductory Example: A&E Waiting Times
Graphical and Probabilistic Representation
Building the Model in Stan
Generating Data
Exploring the Variables
Estimating the Parameters in Stan
Further Reading
6. Theory and Algorithms for Bayesian Networks
Conditional Independence and Graphical Separation
Bayesian Networks
Markov Blankets
Moral Graphs
Bayesian Network Learning
Structure Learning
Constraint-based Algorithms
Score-based Algorithms
Hybrid Algorithms
Parameter Learning
Bayesian Network Inference
Probabilistic Reasoning and Evidence
Algorithms for Belief Updating
Exact Inference Algorithms
Approximate Inference Algorithms
Causal Bayesian Networks
Evaluating a Bayesian Network
Further Reading
7. Software for Bayesian Networks
An Overview of R Packages
The deal Package
The catnet Package
The pcalg Package
The abn Package
Stan and BUGS Software Packages
Stan: a Feature Overview
Inference Based on MCMC Sampling
Other Software Packages
BayesiaLab
Hugin
GeNIe
8. Real-World Applications of Bayesian Networks
Learning Protein-Signalling Networks
A Gaussian Bayesian Network
Discretising Gene Expressions
Model Averaging
Choosing the Significance Threshold
Handling Interventional Data
Querying the Network
Predicting the Body Composition
Aim of the Study
Designing the Predictive Approach
Assessing the Quality of a Predictor
The Saturated BN
Convenient BNs
Looking for Candidate BNs
Further Reading
A Graph Theory
A Graphs, Nodes and Arcs
A The Structure of a Graph
A Further Reading
B Probability Distributions
B General Features
B Marginal and Conditional Distributions
B Discrete Distributions
B Binomial Distribution
B Multinomial Distribution
B Other Common Distributions
B Bernoulli Distribution
B Poisson Distribution
B Continuous Distributions
B Normal Distribution
B Multivariate Normal Distribution
B Other Common Distributions
B Chi-square Distribution
B Studentfs t Distribution
B Beta Distribution
B Dirichlet Distribution
B Conjugate Distributions
B Further Reading
C A Note about Bayesian Networks
C Bayesian Networks and Bayesian Statistics
ISBN: 978-1-119-65493-3
August 2021
Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics
Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations as well, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more.
Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized Ordinary Differential Equations and impulsive differential equations, functional differential equations, measure differential equations and dynamic equations on time scales. The book?s descriptions will be of use in many mathematical contexts, as well as in the social and natural sciences. Readers will also benefit from the inclusion of:
A thorough introduction to regulated functions, including their basic properties, equiregulated sets, uniform convergence, and relatively compact sets
An exploration of the Kurzweil integral, including its definitions and basic properties
A discussion of measure functional differential equations, including impulsive measure FDEs
The interrelationship between generalized ODEs and measure FDEs
A treatment of the basic properties of generalized ODEs, including the existence and uniqueness of solutions, and prolongation and maximal solutions
Perfect for researchers and graduate students in Differential Equations and Dynamical Systems, Generalized Ordinary Differential Equations in Abstract Spaces and Applications will also earn a place in the libraries of advanced undergraduate students taking courses in the subject and hoping to move onto graduate studies.