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This bookpresents material on both the analysis of the classical concepts of correlation and on the development of their robust versions, as well as discussing the related concepts of correlation matrices, partial correlation, canonical correlation, rank correlations, with the corresponding robust and non-robust estimation procedures. Every chapter contains a set of examples with simulated and real-life data.
Key features:
* Makes modern and robust correlation methods readily available and understandable to practitioners, specialists, and consultants working in various fields.
* Focuses on implementation of methodology and application of robust correlation with R.
* Introduces the main approaches in robust statistics, such as Huber's minimax approach and Hampel's approach based on influence functions.
* Explores various robust estimates of the correlation coefficient including the minimax variance and bias estimates as well as the most B- and V-robust estimates.
* Contains applications of robust correlation methods to exploratory data analysis, multivariate statistics, statistics of time series, and to real-life data.
* Includes an accompanying website featuring computer code and datasets
* Features exercises and examples throughout the text using both small and large data sets.
Theoretical and applied statisticians, specialists in multivariate statistics, robust statistics, robust time series analysis, data analysis and signal processing will benefit from this book. Practitioners who use correlation based methods in their work as well as postgraduate students in statistics will also find this book useful.
List of contents
Preface xv
Acknowledgements xvii
About the Companion Website xix
1 Introduction 1
1.1 Historical Remarks 1
1.2 Ontological Remarks 4
1.2.1 Forms of data representation 5
1.2.2 Types of data statistics 5
1.2.3 Principal aims of statistical data analysis 6
1.2.4 Prior information about data distributions and related approaches to statistical data analysis 6
References 8
2 Classical Measures of Correlation 10
2.1 Preliminaries 10
2.2 Pearson's Correlation Coefficient: Definitions and Interpretations 12
2.2.1 Introductory remarks 13
2.2.2 Correlation via regression 13
2.2.3 Correlation via the coefficient of determination 16
2.2.4 Correlation via the variances of the principal components 18
2.2.5 Correlation via the cosine of the angle between the variable vectors 21
2.2.6 Correlation via the ratio of two means 22
2.2.7 Pearson's correlation coefficient between random events 23
2.3 Nonparametric Measures of Correlation 24
2.3.1 Introductory remarks 24
2.3.2 The quadrant correlation coefficient 26
2.3.3 The Spearman rank correlation coefficient 27
2.3.4 The Kendall tau-rank correlation coefficient 28
2.3.5 Concluding remark 29
2.4 Informational Measures of Correlation 29
2.5 Summary 31
References 31
3 Robust Estimation of Location 33
3.1 Preliminaries 33
3.2 Huber's Minimax Approach 35
3.2.1 Introductory remarks 35
3.2.2 Minimax variance M-estimates of location 36
3.2.3 Minimax bias M-estimates of location 43
3.2.4 L-estimates of location 44
3.2.5 R-estimates of location 45
3.2.6 The relations between M-, L- and R-estimates of location 46
3.2.7 Concluding remarks 47
3.3 Hampel's Approach Based on Influence Functions 47
3.3.1 Introductory remarks 47
3.3.2 Sensitivity curve 47
3.3.3 Influence function and its properties 49
3.3.4 Local measures of robustness 51
3.3.5 B- and V-robustness 52
3.3.6 Global measure of robustness: the breakdown point 52
3.3.7 Redescending M-estimates 53
3.3.8 Concluding remark 56
3.4 Robust Estimation of Location: A Sequel 56
3.4.1 Introductory remarks 56
3.4.2 Huber's minimax variance approach in distribution density models of a non-neighborhood nature 57
3.4.3 Robust estimation of location in distribution models with a bounded variance 62
3.4.4 On the robustness of robust solutions: stability of least informative distributions 69
3.4.5 Concluding remark 73
3.5 Stable Estimation 73
3.5.1 Introductory remarks 73
3.5.2 Variance sensitivity 74
3.5.3 Estimation stability 76
3.5.4 Robustness of stable estimates 78
3.5.5 Maximin stable redescending M-estimates 83
3.5.6 Concluding remarks 84
3.6 Robustness Versus Gaussianity 85
3.6.1 Introductory remarks 85
3.6.2 Derivations of the Gaussian distribution 87
3.6.3 Properties of the Gaussian distribution 92
3.6.4 Huber's minimax approach and Gaussianity 100
3.6.5 Concluding remarks 101
3.7 Summary 102
References 102
4 Robust Estimation of Scale 107
4.1 Preliminaries 107
4.1.1 Introductory remarks 107
4.1.2 Estimation of scale in data analysis 108
4.1.3 Measures of scale defined by functionals 110
4.2 M- and L-
About the author
Georgy L. Shevlyakov, Department of Applied Mathematics, St. Petersburg State Polytechnic University, Russia
Hannu Oja, School of Health Sciences, University of Tampere, Finland
Summary
This bookpresents material on both the analysis of the classical concepts of correlation and on the development of their robust versions, as well as discussing the related concepts of correlation matrices, partial correlation, canonical correlation, rank correlations, with the corresponding robust and non-robust estimation procedures.
