Robust Correlation Theory and Applications 1st Edition by Georgy Shevlyakov, Hannu Oja ISBN 978-1118493458 1118493451
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Product details:
ISBN 10: 1118493451
ISBN 13: 978-1118493458
Author: Georgy L. Shevlyakov, Hannu Oja
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.
Table of contents:
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
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 r-rank correlation coefficient
2.3.5 Concluding remark
2.4 Informational Measures of Correlation
2.5 Summary
References
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
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 Q estimate and its properties
4.4.4 M-estimate approximations to the Q, estimate: MQ, FQ, and FQ 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
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
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
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
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
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
10 Applications to Time Series Analysis: Robust Spectrum Estimation
10.1 Preliminaries
10.2 Classical Estimation of a Power Spectrum
10.2.1 Introductory remarks
10.2.2 Classical nonparametric estimation of a power spectrum
10.2.3 Parametric estimation of a power spectrum
10.3 Robust Estimation of a Power Spectrum
10.3.1 Introductory remarks
10.3.2 Robust analogs of the discrete Fourier transform
10.3.3 Robust nonparametric estimation
10.3.4 Robust estimation of power spectrum through the Yule-Walker equations
10.3.5 Robust estimation through robust filtering
10.4 Performance Evaluation
10.4.1 Robustness of the median Fourier transform power spectra
10.4.2 Additive outlier contamination model
10.4.3 Disorder contamination model
10.4.4 Concluding remarks
10.5 Summary
References
11 Applications to Signal Processing: Robust Detection
11.1 Preliminaries
11.1.1 Classical approach to detection
11.1.2 Robust minimax approach to hypothesis testing
11.1.3 Asymptotically optimal robust detection of a weak signal
11.2 Robust Minimax Detection Based on a Distance Rule
11.2.1 Introductory remarks
11.2.2 Asymptotic robust minimax detection of a known constant signal with the p-distance rule
11.2.3 Detection performance in asymptotics and on finite samples
11.2.4 Concluding remarks
11.3 Robust Detection of a Weak Signal with RedescendingM-Estimates
11.3.1 Introductory remarks
11.3.2 Detection error sensitivity and stability
11.3.3 Performance evaluation: a comparative study
11.3.4 Concluding remarks
11.4 A Unified Neyman-Pearson Detection of Weak Signals in a Fusion Model with Fading Channels and Non-Gaussian Noises
11.4.1 Introductory remarks
11.4.2 Problem setting-an asymptotic fusion rule
11.4.3 Asymptotic performance analysis
11.4.4 Numerical results
11.4.5 Concluding remarks
11.5 Summary
References
12 Final Remarks
12.1 Points of Growth: Open Problems in Multivariate Statistics
12.2 Points of Growth: Open Problems in Applications
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Tags: Georgy Shevlyakov, Hannu Oja, Robust Correlation, Theory and Applications