Rank-Based Methods for Shrinkage and Selection With Application to Machine Learning
A. K. Md. Ehsanes Saleh Carleton University, Ottawa, Canada
Mohammad Arashi Ferdowsi University of Mashhad, Mashhad, Iran
Resve A. Saleh University of British Columbia, Vancouver, Canada
Mina Norouzirad Center for Mathematics and Application of NOVA University Lisbon, Lisbon, Portugal
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Library of Congress Cataloging-in-Publication Data
ISBN 9781119625391 Cover image: [Production Editor to insert] Cover design by [Production Editor to insert] Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
We dedicate this book to
Shahidara Saleh
Reihaneh Soleimani, Elena Arashi
Lynn Hilchie Saleh
Abbas Ali Norouzirad, Fereshteh Arefian
Contents 1
Cover
8
Foreword
9
Preface
10 1 Introduction to Rank-based Regression1.1 Introduction1.2 Robustness of the Median1.2.1 Mean vs. Median1.2.2 Breakdown Point1.2.3 Order and Rank Statistics1.3 Simple Linear Regression1.3.1 Least Squares Estimator (LSE)1.3.2 Theil’s Estimator1.3.3 Belgium Telephone Data Set1.3.4 Estimation and Standard Error Comparison1.4 Outliers and their Detection1.4.1 Outlier Detection1.5 Motivation for Rank-based Methods1.5.1 Effect of a Single Outlier1.5.2 Using Rank for the Location Model1.5.3 Using Rank for the Slope1.6 The Rank Dispersion Function1.6.1 Ranking and Scoring Details1.6.2 Detailed Procedure for R-estimation1.7 Shrinkage Estimation and Subset Selection1.7.1 Multiple Linear Regression using Rank1.7.2 Penalty Functions1.7.3 Shrinkage Estimation1.7.4 Subset Selection1.7.5 Blended Approaches1.8 Summary1.9 Problems 11 2 Characteristics of Rank-based Penalty Estimators2.1 Introduction2.2 Motivation for Penalty Estimators2.3 Multivariate Linear Regression2.3.1 Multivariate Least Squares Estimation2.3.2 Multivariate R-estimation2.3.3 Multicollinearity2.4 Ridge Regression2.4.1 Ridge Applied to Least Squares Estimation2.4.2 Ridge Applied to Rank Estimation2.5 Example: Swiss Fertility Data Set2.5.1 Estimation and Standard Errors2.5.2 Parameter Variance using Bootstrap2.5.3 Reducing Variance using Ridge2.5.4 Ridge Traces2.6 Selection of Ridge Parameter λ22.6.1 Quadratic Risk2.6.2 K-fold Cross-validation Scheme2.7 LASSO and aLASSO2.7.1 Subset Selection2.7.2 Least Squares
