| Description |
1 online resource (249 pages) |
| Series |
Chapman and Hall/CRC Monographs on Statistics and Applied Probability Ser |
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Chapman and Hall/CRC Monographs on Statistics and Applied Probability Ser
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| Contents |
Cover; Half Title; MONOGRAPHS ON STATISTICS ANDAPPLIED PROBABILITY; Title; Copyright; Dedication; Contents; Preface; List of Figures; List of Tables; List of Algorithms; Chapter 1 Introduction; 1.1 Exploratory data analysis with density estimation; 1.2 Exploratory data analysis with density derivatives estimation; 1.3 Clustering/unsupervised learning; 1.4 Classification/supervised learning; 1.5 Suggestions on how to read this monograph; Chapter 2 Density estimation; 2.1 Histogram density estimation; 2.2 Kernel density estimation; 2.2.1 Probability contours as multivariate quantiles |
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2.2.2 Contour colour scales2.3 Gains from unconstrained bandwidth matrices; 2.4 Advice for practical bandwidth selection; 2.5 Squared error analysis; 2.6 Asymptotic squared error formulas; 2.7 Optimal bandwidths; 2.8 Convergence of density estimators; 2.9 Further mathematical analysis of density estimators; 2.9.1 Asymptotic expansion of the mean integrated squared error; 2.9.2 Asymptotically optimal bandwidth; 2.9.3 Vector versus vector half parametrisations; Chapter 3 Bandwidth selectors for density estimation; 3.1 Normal scale bandwidths; 3.2 Maximal smoothing bandwidths |
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3.3 Normal mixture bandwidths3.4 Unbiased cross validation bandwidths; 3.5 Biased cross validation bandwidths; 3.6 Plug-in bandwidths; 3.7 Smoothed cross validation bandwidths; 3.8 Empirical comparison of bandwidth selectors; 3.9 Theoretical comparison of bandwidth selectors; 3.10 Further mathematical analysis of bandwidth selectors; 3.10.1 Relative convergence rates of bandwidth selectors; 3.10.2 Optimal pilot bandwidth selectors; 3.10.3 Convergence rates with data-based bandwidths; Chapter 4 Modified density estimation; 4.1 Variable bandwidth density estimators |
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4.1.1 Balloon density estimators4.1.2 Sample point density estimators; 4.1.3 Bandwidth selectors for variable kernel estimation; 4.2 Transformation density estimators; 4.3 Boundary kernel density estimators; 4.3.1 Beta boundary kernels; 4.3.2 Linear boundary kernels; 4.4 Kernel choice; 4.5 Higher order kernels; 4.6 Further mathematical analysis of modified density estimators; 4.6.1 Asymptotic error for sample point variable band-width estimators; 4.6.2 Asymptotic error for linear boundary estimators; Chapter 5 Density derivative estimation; 5.1 Kernel density derivative estimators |
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5.1.1 Density gradient estimators5.1.2 Density Hessian estimators; 5.1.3 General density derivative estimators; 5.2 Gains from unconstrained bandwidth matrices; 5.3 Advice for practical bandwidth selection; 5.4 Empirical comparison of bandwidths of different derivative orders; 5.5 Squared error analysis; 5.6 Bandwidth selection for density derivative estimators; 5.6.1 Normal scale bandwidths; 5.6.2 Normal mixture bandwidths; 5.6.3 Unbiased cross validation bandwidths; 5.6.4 Plug-in bandwidths; 5.6.5 Smoothed cross validation bandwidths; 5.7 Relative convergence rates of bandwidth selectors |
| Summary |
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. 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 |
| Notes |
5.8 Case study: The normal density |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Smoothing
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Mathematical statistics.
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Kernel functions.
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Kernel functions
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Mathematical statistics
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| Form |
Electronic book
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| Author |
Duong, Tarn
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| ISBN |
9780429939143 |
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0429939140 |
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9780429485572 |
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0429485573 |
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