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Robust Correlation : Theory and Applications.

By: Contributor(s): Material type: TextTextSeries: Wiley Series in Probability and Statistics SeriesPublisher: Newark : John Wiley & Sons, Incorporated, 2016Copyright date: ©2016Edition: 1st edDescription: 1 online resource (353 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781119264538
Subject(s): Genre/Form: Additional physical formats: Print version:: Robust CorrelationDDC classification:
  • 519.537
LOC classification:
  • QA278.2
Online resources:
Contents:
Cover -- Title Page -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgements -- About the Companion Website -- Chapter 1 Introduction -- 1.1 Historical Remarks -- 1.2 Ontological Remarks -- 1.2.1 Forms of data representation -- 1.2.2 Types of data statistics -- 1.2.3 Principal aims of statistical data analysis -- 1.2.4 Prior information about data distributions and related approaches to statistical data analysis -- References -- Chapter 2 Classical Measures of Correlation -- 2.1 Preliminaries -- 2.2 Pearson's Correlation Coefficient: Definitions and Interpretations -- 2.2.1 Introductory remarks -- 2.2.2 Correlation via regression -- 2.2.3 Correlation via the coefficient of determination -- 2.2.4 Correlation via the variances of the principal components -- 2.2.5 Correlation via the cosine of the angle between the variable vectors -- 2.2.6 Correlation via the ratio of two means -- 2.2.7 Pearson's correlation coefficient between random events -- 2.3 Nonparametric Measures of Correlation -- 2.3.1 Introductory remarks -- 2.3.2 The quadrant correlation coefficient -- 2.3.3 The Spearman rank correlation coefficient -- 2.3.4 The Kendall -rank correlation coefficient -- 2.3.5 Concluding remark -- 2.4 Informational Measures of Correlation -- 2.5 Summary -- References -- Chapter 3 Robust Estimation of Location -- 3.1 Preliminaries -- 3.2 Huber's Minimax Approach -- 3.2.1 Introductory remarks -- 3.2.2 Minimax variance M-estimates of location -- 3.2.3 Minimax bias M-estimates of location -- 3.2.4 L-estimates of location -- 3.2.5 R-estimates of location -- 3.2.6 The relations between M-, L- and R-estimates of location -- 3.2.7 Concluding remarks -- 3.3 Hampel's Approach Based on Influence Functions -- 3.3.1 Introductory remarks -- 3.3.2 Sensitivity curve -- 3.3.3 Influence function and its properties -- 3.3.4 Local measures of robustness.
3.3.5 B- and V-robustness -- 3.3.6 Global measure of robustness: the breakdown point -- 3.3.7 Redescending M-estimates -- 3.3.8 Concluding remark -- 3.4 Robust Estimation of Location: A Sequel -- 3.4.1 Introductory remarks -- 3.4.2 Huber's minimax variance approach in distribution density models of a non-neighborhood nature -- 3.4.3 Robust estimation of location in distribution models with a bounded variance -- 3.4.4 On the robustness of robust solutions: stability of least informative distributions -- 3.4.5 Concluding remark -- 3.5 Stable Estimation -- 3.5.1 Introductory remarks -- 3.5.2 Variance sensitivity -- 3.5.3 Estimation stability -- 3.5.4 Robustness of stable estimates -- 3.5.5 Maximin stable redescending M-estimates -- 3.5.6 Concluding remarks -- 3.6 Robustness Versus Gaussianity -- 3.6.1 Introductory remarks -- 3.6.2 Derivations of the Gaussian distribution -- 3.6.3 Properties of the Gaussian distribution -- 3.6.4 Huber's minimax approach and Gaussianity -- 3.6.5 Concluding remarks -- 3.7 Summary -- References -- Chapter 4 Robust Estimation of Scale -- 4.1 Preliminaries -- 4.1.1 Introductory remarks -- 4.1.2 Estimation of scale in data analysis -- 4.1.3 Measures of scale defined by functionals -- 4.2 M- and L-Estimates of Scale -- 4.2.1 M-estimates of scale -- 4.2.2 L-estimates of scale -- 4.3 Huber Minimax Variance Estimates of Scale -- 4.3.1 Introductory remarks -- 4.3.2 The least informative distribution -- 4.3.3 Minimax variance M- and L-estimates of scale -- 4.4 Highly Efficient Robust Estimates of Scale -- 4.4.1 Introductory remarks -- 4.4.2 The median of absolute deviations and its properties -- 4.4.3 The quartile of pair-wise absolute differences Qn estimate and its properties -- 4.4.4 M-estimate approximations to the Qn estimate: MQ n, FQ n, and FQn estimates of scale -- 4.5 Monte Carlo Experiment.
4.5.1 A remark on the Monte Carlo experiment accuracy -- 4.5.2 Monte Carlo experiment: distribution models -- 4.5.3 Monte Carlo experiment: estimates of scale -- 4.5.4 Monte Carlo experiment: characteristics of performance -- 4.5.5 Monte Carlo experiment: results -- 4.5.6 Monte Carlo experiment: discussion -- 4.5.7 Concluding remarks -- 4.6 Summary -- References -- Chapter 5 Robust Estimation of Correlation Coefficients -- 5.1 Preliminaries -- 5.2 Main Groups of Robust Estimates of the Correlation Coefficient -- 5.2.1 Introductory remarks -- 5.2.2 Direct robust counterparts of Pearson's correlation coefficient -- 5.2.3 Robust correlation via nonparametric measures of correlation -- 5.2.4 Robust correlation via robust regression -- 5.2.5 Robust correlation via robust principal component variances -- 5.2.6 Robust correlation via two-stage procedures -- 5.2.7 Concluding remarks -- 5.3 Asymptotic Properties of the Classical Estimates of the Correlation Coefficient -- 5.3.1 Pearson's sample correlation coefficient -- 5.3.2 The maximum likelihood estimate of the correlation coefficient at the normal -- 5.4 Asymptotic Properties of Nonparametric Estimates of Correlation -- 5.4.1 Introductory remarks -- 5.4.2 The quadrant correlation coefficient -- 5.4.3 The Kendall rank correlation coefficient -- 5.4.4 The Spearman rank correlation coefficient -- 5.5 Bivariate Independent Component Distributions -- 5.5.1 Definition and properties -- 5.5.2 Independent component and Tukey gross-error distribution models -- 5.6 Robust Estimates of the Correlation Coefficient Based on Principal Component Variances -- 5.7 Robust Minimax Bias and Variance Estimates of the Correlation Coefficient -- 5.7.1 Introductory remarks -- 5.7.2 Minimax property -- 5.7.3 Concluding remarks -- 5.8 Robust Correlation via Highly Efficient Robust Estimates of Scale -- 5.8.1 Introductory remarks.
5.8.2 Asymptotic bias and variance of generalized robust estimates of the correlation coefficient -- 5.8.3 Concluding remarks -- 5.9 Robust M-Estimates of the Correlation Coefficient in Independent Component Distribution Models -- 5.9.1 Introductory remarks -- 5.9.2 The maximum likelihood estimate of the correlation coefficient in independent component distribution models -- 5.9.3 M-estimates of the correlation coefficient -- 5.9.4 Asymptotic variance of M-estimators -- 5.9.5 Minimax variance M-estimates of the correlation coefficient -- 5.9.6 Concluding remarks -- 5.10 Monte Carlo Performance Evaluation -- 5.10.1 Introductory remarks -- 5.10.2 Monte Carlo experiment set-up -- 5.10.3 Discussion -- 5.10.4 Concluding remarks -- 5.11 Robust Stable Radical M-Estimate of the Correlation Coefficient of the Bivariate Normal Distribution -- 5.11.1 Introductory remarks -- 5.11.2 Asymptotic characteristics of the stable radical estimate of the correlation coefficient -- 5.11.3 Concluding remarks -- 5.12 Summary -- References -- Chapter 6 Classical Measures of Multivariate Correlation -- 6.1 Preliminaries -- 6.2 Covariance Matrix and Correlation Matrix -- 6.3 Sample Mean Vector and Sample Covariance Matrix -- 6.4 Families of Multivariate Distributions -- 6.4.1 Construction of multivariate location-scatter models -- 6.4.2 Multivariate symmetrical distributions -- 6.4.3 Multivariate normal distribution -- 6.4.4 Multivariate elliptical distributions -- 6.4.5 Independent component model -- 6.4.6 Copula models -- 6.5 Asymptotic Behavior of Sample Covariance Matrix and Sample Correlation Matrix -- 6.6 First Uses of Covariance and Correlation Matrices -- 6.7 Working with the Covariance Matrix-Principal Component Analysis -- 6.7.1 Principal variables -- 6.7.2 Interpretation of principal components -- 6.7.3 Asymptotic behavior of the eigenvectors and eigenvalues.
6.8 Working with Correlations-Canonical Correlation Analysis -- 6.8.1 Canonical variates and canonical correlations -- 6.8.2 Testing for independence between subvectors -- 6.9 Conditionally Uncorrelated Components -- 6.10 Summary -- References -- Chapter 7 Robust Estimation of Scatter and Correlation Matrices -- 7.1 Preliminaries -- 7.2 Multivariate Location and Scatter Functionals -- 7.3 Influence Functions and Asymptotics -- 7.4 M-functionals for Location and Scatter -- 7.5 Breakdown Point -- 7.6 Use of Robust Scatter Matrices -- 7.6.1 Ellipticity assumption -- 7.6.2 Robust correlation matrices -- 7.6.3 Principal component analysis -- 7.6.4 Canonical correlation analysis -- 7.7 Further Uses of Location and Scatter Functionals -- 7.8 Summary -- References -- Chapter 8 Nonparametric Measures of Multivariate Correlation -- 8.1 Preliminaries -- 8.2 Univariate Signs and Ranks -- 8.3 Marginal Signs and Ranks -- 8.4 Spatial Signs and Ranks -- 8.5 Affine Equivariant Signs and Ranks -- 8.6 Summary -- References -- Chapter 9 Applications to Exploratory Data Analysis: Detection of Outliers -- 9.1 Preliminaries -- 9.2 State of the Art -- 9.2.1 Univariate boxplots -- 9.2.2 Bivariate boxplots -- 9.3 Problem Setting -- 9.4 A New Measure of Outlier Detection Performance -- 9.4.1 Introductory remarks -- 9.4.2 H-mean: motivation, definition and properties -- 9.5 Robust Versions of the Tukey Boxplot with Their Application to Detection of Outliers -- 9.5.1 Data generation and performance measure -- 9.5.2 Scale and shift contamination -- 9.5.3 Real-life data results -- 9.5.4 Concluding remarks -- 9.6 Robust Bivariate Boxplots and Their Performance Evaluation -- 9.6.1 Bivariate FQ-boxplot -- 9.6.2 Bivariate FQ-boxplot performance -- 9.6.3 Measuring the elliptical deviation from the convex hull -- 9.7 Summary -- References.
Chapter 10 Applications to Time Series Analysis: Robust Spectrum Estimation.
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Cover -- Title Page -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgements -- About the Companion Website -- Chapter 1 Introduction -- 1.1 Historical Remarks -- 1.2 Ontological Remarks -- 1.2.1 Forms of data representation -- 1.2.2 Types of data statistics -- 1.2.3 Principal aims of statistical data analysis -- 1.2.4 Prior information about data distributions and related approaches to statistical data analysis -- References -- Chapter 2 Classical Measures of Correlation -- 2.1 Preliminaries -- 2.2 Pearson's Correlation Coefficient: Definitions and Interpretations -- 2.2.1 Introductory remarks -- 2.2.2 Correlation via regression -- 2.2.3 Correlation via the coefficient of determination -- 2.2.4 Correlation via the variances of the principal components -- 2.2.5 Correlation via the cosine of the angle between the variable vectors -- 2.2.6 Correlation via the ratio of two means -- 2.2.7 Pearson's correlation coefficient between random events -- 2.3 Nonparametric Measures of Correlation -- 2.3.1 Introductory remarks -- 2.3.2 The quadrant correlation coefficient -- 2.3.3 The Spearman rank correlation coefficient -- 2.3.4 The Kendall -rank correlation coefficient -- 2.3.5 Concluding remark -- 2.4 Informational Measures of Correlation -- 2.5 Summary -- References -- Chapter 3 Robust Estimation of Location -- 3.1 Preliminaries -- 3.2 Huber's Minimax Approach -- 3.2.1 Introductory remarks -- 3.2.2 Minimax variance M-estimates of location -- 3.2.3 Minimax bias M-estimates of location -- 3.2.4 L-estimates of location -- 3.2.5 R-estimates of location -- 3.2.6 The relations between M-, L- and R-estimates of location -- 3.2.7 Concluding remarks -- 3.3 Hampel's Approach Based on Influence Functions -- 3.3.1 Introductory remarks -- 3.3.2 Sensitivity curve -- 3.3.3 Influence function and its properties -- 3.3.4 Local measures of robustness.

3.3.5 B- and V-robustness -- 3.3.6 Global measure of robustness: the breakdown point -- 3.3.7 Redescending M-estimates -- 3.3.8 Concluding remark -- 3.4 Robust Estimation of Location: A Sequel -- 3.4.1 Introductory remarks -- 3.4.2 Huber's minimax variance approach in distribution density models of a non-neighborhood nature -- 3.4.3 Robust estimation of location in distribution models with a bounded variance -- 3.4.4 On the robustness of robust solutions: stability of least informative distributions -- 3.4.5 Concluding remark -- 3.5 Stable Estimation -- 3.5.1 Introductory remarks -- 3.5.2 Variance sensitivity -- 3.5.3 Estimation stability -- 3.5.4 Robustness of stable estimates -- 3.5.5 Maximin stable redescending M-estimates -- 3.5.6 Concluding remarks -- 3.6 Robustness Versus Gaussianity -- 3.6.1 Introductory remarks -- 3.6.2 Derivations of the Gaussian distribution -- 3.6.3 Properties of the Gaussian distribution -- 3.6.4 Huber's minimax approach and Gaussianity -- 3.6.5 Concluding remarks -- 3.7 Summary -- References -- Chapter 4 Robust Estimation of Scale -- 4.1 Preliminaries -- 4.1.1 Introductory remarks -- 4.1.2 Estimation of scale in data analysis -- 4.1.3 Measures of scale defined by functionals -- 4.2 M- and L-Estimates of Scale -- 4.2.1 M-estimates of scale -- 4.2.2 L-estimates of scale -- 4.3 Huber Minimax Variance Estimates of Scale -- 4.3.1 Introductory remarks -- 4.3.2 The least informative distribution -- 4.3.3 Minimax variance M- and L-estimates of scale -- 4.4 Highly Efficient Robust Estimates of Scale -- 4.4.1 Introductory remarks -- 4.4.2 The median of absolute deviations and its properties -- 4.4.3 The quartile of pair-wise absolute differences Qn estimate and its properties -- 4.4.4 M-estimate approximations to the Qn estimate: MQ n, FQ n, and FQn estimates of scale -- 4.5 Monte Carlo Experiment.

4.5.1 A remark on the Monte Carlo experiment accuracy -- 4.5.2 Monte Carlo experiment: distribution models -- 4.5.3 Monte Carlo experiment: estimates of scale -- 4.5.4 Monte Carlo experiment: characteristics of performance -- 4.5.5 Monte Carlo experiment: results -- 4.5.6 Monte Carlo experiment: discussion -- 4.5.7 Concluding remarks -- 4.6 Summary -- References -- Chapter 5 Robust Estimation of Correlation Coefficients -- 5.1 Preliminaries -- 5.2 Main Groups of Robust Estimates of the Correlation Coefficient -- 5.2.1 Introductory remarks -- 5.2.2 Direct robust counterparts of Pearson's correlation coefficient -- 5.2.3 Robust correlation via nonparametric measures of correlation -- 5.2.4 Robust correlation via robust regression -- 5.2.5 Robust correlation via robust principal component variances -- 5.2.6 Robust correlation via two-stage procedures -- 5.2.7 Concluding remarks -- 5.3 Asymptotic Properties of the Classical Estimates of the Correlation Coefficient -- 5.3.1 Pearson's sample correlation coefficient -- 5.3.2 The maximum likelihood estimate of the correlation coefficient at the normal -- 5.4 Asymptotic Properties of Nonparametric Estimates of Correlation -- 5.4.1 Introductory remarks -- 5.4.2 The quadrant correlation coefficient -- 5.4.3 The Kendall rank correlation coefficient -- 5.4.4 The Spearman rank correlation coefficient -- 5.5 Bivariate Independent Component Distributions -- 5.5.1 Definition and properties -- 5.5.2 Independent component and Tukey gross-error distribution models -- 5.6 Robust Estimates of the Correlation Coefficient Based on Principal Component Variances -- 5.7 Robust Minimax Bias and Variance Estimates of the Correlation Coefficient -- 5.7.1 Introductory remarks -- 5.7.2 Minimax property -- 5.7.3 Concluding remarks -- 5.8 Robust Correlation via Highly Efficient Robust Estimates of Scale -- 5.8.1 Introductory remarks.

5.8.2 Asymptotic bias and variance of generalized robust estimates of the correlation coefficient -- 5.8.3 Concluding remarks -- 5.9 Robust M-Estimates of the Correlation Coefficient in Independent Component Distribution Models -- 5.9.1 Introductory remarks -- 5.9.2 The maximum likelihood estimate of the correlation coefficient in independent component distribution models -- 5.9.3 M-estimates of the correlation coefficient -- 5.9.4 Asymptotic variance of M-estimators -- 5.9.5 Minimax variance M-estimates of the correlation coefficient -- 5.9.6 Concluding remarks -- 5.10 Monte Carlo Performance Evaluation -- 5.10.1 Introductory remarks -- 5.10.2 Monte Carlo experiment set-up -- 5.10.3 Discussion -- 5.10.4 Concluding remarks -- 5.11 Robust Stable Radical M-Estimate of the Correlation Coefficient of the Bivariate Normal Distribution -- 5.11.1 Introductory remarks -- 5.11.2 Asymptotic characteristics of the stable radical estimate of the correlation coefficient -- 5.11.3 Concluding remarks -- 5.12 Summary -- References -- Chapter 6 Classical Measures of Multivariate Correlation -- 6.1 Preliminaries -- 6.2 Covariance Matrix and Correlation Matrix -- 6.3 Sample Mean Vector and Sample Covariance Matrix -- 6.4 Families of Multivariate Distributions -- 6.4.1 Construction of multivariate location-scatter models -- 6.4.2 Multivariate symmetrical distributions -- 6.4.3 Multivariate normal distribution -- 6.4.4 Multivariate elliptical distributions -- 6.4.5 Independent component model -- 6.4.6 Copula models -- 6.5 Asymptotic Behavior of Sample Covariance Matrix and Sample Correlation Matrix -- 6.6 First Uses of Covariance and Correlation Matrices -- 6.7 Working with the Covariance Matrix-Principal Component Analysis -- 6.7.1 Principal variables -- 6.7.2 Interpretation of principal components -- 6.7.3 Asymptotic behavior of the eigenvectors and eigenvalues.

6.8 Working with Correlations-Canonical Correlation Analysis -- 6.8.1 Canonical variates and canonical correlations -- 6.8.2 Testing for independence between subvectors -- 6.9 Conditionally Uncorrelated Components -- 6.10 Summary -- References -- Chapter 7 Robust Estimation of Scatter and Correlation Matrices -- 7.1 Preliminaries -- 7.2 Multivariate Location and Scatter Functionals -- 7.3 Influence Functions and Asymptotics -- 7.4 M-functionals for Location and Scatter -- 7.5 Breakdown Point -- 7.6 Use of Robust Scatter Matrices -- 7.6.1 Ellipticity assumption -- 7.6.2 Robust correlation matrices -- 7.6.3 Principal component analysis -- 7.6.4 Canonical correlation analysis -- 7.7 Further Uses of Location and Scatter Functionals -- 7.8 Summary -- References -- Chapter 8 Nonparametric Measures of Multivariate Correlation -- 8.1 Preliminaries -- 8.2 Univariate Signs and Ranks -- 8.3 Marginal Signs and Ranks -- 8.4 Spatial Signs and Ranks -- 8.5 Affine Equivariant Signs and Ranks -- 8.6 Summary -- References -- Chapter 9 Applications to Exploratory Data Analysis: Detection of Outliers -- 9.1 Preliminaries -- 9.2 State of the Art -- 9.2.1 Univariate boxplots -- 9.2.2 Bivariate boxplots -- 9.3 Problem Setting -- 9.4 A New Measure of Outlier Detection Performance -- 9.4.1 Introductory remarks -- 9.4.2 H-mean: motivation, definition and properties -- 9.5 Robust Versions of the Tukey Boxplot with Their Application to Detection of Outliers -- 9.5.1 Data generation and performance measure -- 9.5.2 Scale and shift contamination -- 9.5.3 Real-life data results -- 9.5.4 Concluding remarks -- 9.6 Robust Bivariate Boxplots and Their Performance Evaluation -- 9.6.1 Bivariate FQ-boxplot -- 9.6.2 Bivariate FQ-boxplot performance -- 9.6.3 Measuring the elliptical deviation from the convex hull -- 9.7 Summary -- References.

Chapter 10 Applications to Time Series Analysis: Robust Spectrum Estimation.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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