Nonlinear Principal Component Analysis and Its Applications

Authors: Yuichi Mori, Masahiro Kuroda, Naomichi Makino



Shows that PCA, nonlinear PCA, and MCA can be integrated as a single formulation, which can easily be extended to several applications

Provides an acceleration algorithm that speeds up the convergent sequences generated by the alternating least squares and is a remedy for computational cost

Introduces applications related to nonlinear PCA: variable selection for mixed measurement levels data, sparse multiple correspondence analysis, and joint dimension reduction and clustering

This book expounds the principle and related applications of nonlinear principal component analysis (PCA), which is useful method to analyze mixed measurement levels data. In the part dealing with the principle, after a brief introduction of ordinary PCA, a PCA for categorical data (nominal and ordinal) is introduced as nonlinear PCA, in which an optimal scaling technique is used to quantify the categorical variables. The alternating least squares (ALS) is the main algorithm in the method. Multiple correspondence analysis (MCA), a special case of nonlinear PCA, is also introduced. All formulations in these methods are integrated in the same manner as matrix operations. Because any measurement levels data can be treated consistently as numerical data and ALS is a very powerful tool for estimations, the methods can be utilized in a variety of fields such as biometrics, econometrics, psychometrics, and sociology. In the applications part of the book, four applications are introduced: variable selection for mixed measurement levels data, sparse MCA, joint dimension reduction and clustering methods for categorical data, and acceleration of ALS computation. The variable selection methods in PCA that originally were developed for numerical data can be applied to any types of measurement levels by using nonlinear PCA. Sparseness and joint dimension reduction and clustering for nonlinear data, the results of recent studies, are extensions obtained by the same matrix operations in nonlinear PCA. Finally, an acceleration algorithm is proposed to reduce the problem of computational cost in the ALS iteration in nonlinear multivariate methods. This book thus presents the usefulness of nonlinear PCA which can be applied to different measurement levels data in diverse fields. As well, it covers the latest topics including the extension of the traditional statistical method, newly proposed nonlinear methods, and computational efficiency in the methods.

Table of contents

Front Matter

Introduction

Nonlinear Principal Component Analysis
• Front Matter
• Nonlinear Principal Component Analysis
• Multiple Correspondence Analysis

Applications and Related Topics
• Front Matter
• Variable Selection in Nonlinear Principal Component Analysis
• Sparse Multiple Correspondence Analysis
• Joint Dimension Reduction and Clustering
• Acceleration of Convergence of the Alternating Least Squares Algorithm for Nonlinear Principal Component Analysis

Back Matter

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