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Constrained Principal Component Analysis and Related Techniques

By Yoshio Takane

Chapman and Hall/CRC – 2013 – 251 pages

Series: Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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    978-1-46-655666-9
    October 23rd 2013

Description

In multivariate data analysis, regression techniques predict one set of variables from another while principal component analysis (PCA) finds a subspace of minimal dimensionality that captures the largest variability in the data.

  • How can regression analysis and PCA be combined in a beneficial way?
  • Why and when is it a good idea to combine them?
  • What kind of benefits are we getting from them?

Addressing these questions, Constrained Principal Component Analysis and Related Techniques shows how constrained PCA (CPCA) offers a unified framework for these approaches.

The book begins with four concrete examples of CPCA that provide readers with a basic understanding of the technique and its applications. It gives a detailed account of two key mathematical ideas in CPCA: projection and singular value decomposition. The author then describes the basic data requirements, models, and analytical tools for CPCA and their immediate extensions. He also introduces techniques that are special cases of or closely related to CPCA and discusses several topics relevant to practical uses of CPCA. The book concludes with a technique that imposes different constraints on different dimensions (DCDD), along with its analytical extensions. MATLAB® programs for CPCA and DCDD as well as data to create the book’s examples are available on the author’s website.

Contents

Introduction

Analysis of Mezzich’s Data

Analysis of Food and Cancer Data

Analysis of Greenacre’s Data

Analysis of Tocher’s Data

A Summary of the Analyses in This Chapter

Mathematical Foundation

Preliminaries

Projection Matrices

Singular Value Decomposition (SVD)

Constrained Principal Component Analysis (CPCA)

Data Requirements

CPCA: Method

Generalizations

Special Cases and Related Methods

Pre- and Postprocessings

Redundancy Analysis (RA)

Canonical Correlation Analysis (CANO)

Canonical Discriminant Analysis (CDA)

Multidimensional Scaling (MDS)

Correspondence Analysis (CA)

Constrained CA

Nonsymmetric CA (NSCA)

Multiple-Set CANO (GCANO)

Multiple Correspondence Analysis (MCA)

Vector Preference Models

Two-Way CANDELINC

Growth Curve Models (GCM)

Extended Growth Curve Models (ExGCM)

Seemingly Unrelated Regression (SUR)

Wedderburn–Guttman Decomposition

Multilevel RA (MLRA)

Weighted Low Rank Approximations (WLRA)

Orthogonal Procrustes Rotation

PCA of Image Data Matrices

Related Topics of Interest

Dimensionality Selection

Reliability Assessment

Determining the Value of δ

Missing Data

Robust Estimations

Data Transformations

Biplot

Probabilistic PCA

Different Constraints on Different Dimensions (DCDD)

Model and Algorithm

Additional Constraints

Example 1

Example 2

Residual Analysis

Graphical Display of Oblique Components

Extended Redundancy Analysis (ERA)

Generalized Structured Component Analysis (GSCA)

Epilogue

Appendix

Bibliography

Index

Author Bio

Yoshio Takane is an emeritus professor at McGill University and an adjunct professor at the University of Victoria. He is a former president of the Psychometric Society and a recipient of a Career Award from the Behaviormetric Society of Japan and a Special Award from the Japanese Psychological Association. His recent interests include regularization techniques for multivariate data analysis, acceleration methods for iterative model fitting, the development of structural equation models for analyzing brain connectivity, and various kinds of singular value decompositions. He earned his DL from the University of Tokyo and PhD from the University of North Carolina at Chapel Hill.

Name: Constrained Principal Component Analysis and Related Techniques (Hardback)Chapman and Hall/CRC 
Description: By Yoshio Takane. In multivariate data analysis, regression techniques predict one set of variables from another while principal component analysis (PCA) finds a subspace of minimal dimensionality that captures the largest variability in the data. How can regression...
Categories: Statistical Theory & Methods, Mathematics & Statistics for Engineers, Regression Analysis and Multivariate Statistics