Multivariate Kernel Smoothing and Its Applications: (Chapman & Hall/CRC Monographs on Statistics & Applied Probability)

Kernel smoothing has greatly evolved since its inception to become an essential methodology in the data science tool kit for the 21st century. Its widespread adoption is due to its fundamental role for multivariate exploratory data analysis, as well as the crucial role it plays in composite solutions to complex data challenges. Multivariate Kernel Smoothing and Its Applications offers a comprehensive overview of both aspects. It begins with a thorough exposition of the approaches to achieve the two basic goals of estimating probability density functions and their derivatives. The focus then turns to the applications of these approaches to more complex data analysis goals, many with a geometric/topological flavour, such as level set estimation, clustering (unsupervised learning), principal curves, and feature significance. Other topics, while not direct applications of density (derivative) estimation but sharing many commonalities with the previous settings, include classification (supervised learning), nearest neighbour estimation, and deconvolution for data observed with error. For a data scientist, each chapter contains illustrative Open data examples that are analysed by the most appropriate kernel smoothing method. The emphasis is always placed on an intuitive understanding of the data provided by the accompanying statistical visualisations. For a reader wishing to investigate further the details of their underlying statistical reasoning, a graduated exposition to a unified theoretical framework is provided. The algorithms for efficient software implementation are also discussed. José E. Chacón is an associate professor at the Department of Mathematics of the Universidad de Extremadura in Spain. Tarn Duong is a Senior Data Scientist for a start-up which provides short distance carpooling services in France. Both authors have made important contributions to kernel smoothing research over the last couple of decades.

Table of Contents:
Preface List of Figures List of Tables List of Algorithms Introduction Exploratory data analysis with density estimation Exploratory data analysis with density derivatives estimation Clustering/Unsupervised learning Classification/Supervised learning Suggestions on how to read this monograph Density estimation Histogram density estimation Kernel density estimation Probability contours as multivariate quantiles Contour colour scales Gains from unconstrained bandwidth matrices Advice for practical bandwidth selection Squared error analysis Asymptotic squared error formulas Optimal bandwidths Convergence of density estimators Further mathematical analysis of density estimators Asymptotic expansion of the MISE Asymptotically optimal bandwidth Vector versus vector half parametrisations Bandwidth selectors for density estimation Normal scale bandwidths Maximal smoothing bandwidths Normal mixture bandwidths Unbiased cross validation bandwidths Biased cross validation bandwidths Plug in bandwidths Smoothed cross validation bandwidths Empirical comparison of bandwidth selectors Theoretical comparison of bandwidth selectors Further mathematical analysis of bandwidth selectors Relative convergence rates of bandwidth selectors Optimal pilot bandwidth selectors Convergence rates with data-based bandwidths Modified density estimation Variable bandwidth density estimators Balloon density estimators Sample point density estimators Bandwidth selectors for variable kernel estimation Transformation density estimators Boundary kernel density estimators Beta boundary kernels Linear boundary kernels Kernel choice Higher order kernels Further mathematical analysis of modified density estimators Asymptotic error for sample point variable bandwidth estimators Asymptotic error for linear boundary estimators Density derivative estimation Kernel density derivative estimators Density gradient estimators Density Hessian estimators General density derivative estimators Gains from unconstrained bandwidth matrices Advice for practical bandwidth selection Empirical comparison of bandwidths of different derivative orders Squared error analysis Bandwidth selection for density derivative estimators Normal scale bandwidths Normal mixture bandwidths Unbiased cross validation bandwidths Plug in bandwidths Smoothed cross validation bandwidths Convergence rates of bandwidth selectors Case study: the normal density Exact MISE Curvature matrix Asymptotic MISE Normal scale bandwidth Asymptotic MSE for curvature estimation Further mathematical analysis Taylor expansions for vector-valued functions Relationship between multivariate normal moments Applications related to density and density derivative estimation Level set estimation Modal region and bump estimation Density support estimation Density-based clustering Stable/unstable manifolds Mean shift clustering Choice of the normalising matrix in the mean shift Density ridge estimation Feature significance Supplementary topics in data analysis Density difference estimation and significance testing Classification Density estimation for data measured with error Classical density deconvolution estimation Weighted density deconvolution estimation Manifold estimation Nearest neighbour estimation Further mathematical analysis Squared error analysis for deconvolution kernel density estimators Optimal selection of the number of nearest neighbours Computational algorithms R implementation Approximate binned estimation Approximate density estimation Approximate density derivative and functional estimation Recursive normal density derivatives Recursive normal functionals Numerical optimisation over matrix spaces

About the Author :
José E. Chacón is an associate professor at the Department of Mathematics of the Universidad de Extremadura in Spain. Tarn Duong is a Senior Data Scientist for a start-up which provides short distance carpooling services in France. Both authors have made important contributions to kernel smoothing research over the last couple of decades.

Review :
"I am very impressed with this book. It addresses issues that are not discussed in any detail in any other book on density estimation. Furthermore, it is very well-written and contains a wealth of interesting examples. In fact, this is probably one of the best books I have seen on density estimation. Some topics in this book that are not covered in detail in any other book include: multivariate bandwidth matrices, details of the asymptotic MSE for general bandwidth matrices, derivative estimation, level sets, density clustering and significance testing for modal regions. This makes the book unique. The authors have written the book in such a way that it can be used by two different types of readers: data analysts who are not interested in the mathematical details, and students/researchers who do want the details. The `how to read this monograph' is very useful." ~Larry Wasserman, Carnegie Mellon University