This is the first book that equally details both factor analysis (FA) and principal component analysis (PCA) with the goal of showing a hierarchy formed by FA and PCA procedures. Here, the term “hierarchy” is related to constrainedness; procedures A and B forming a hierarchy means that A is a constrained variant of B or vice versa. It is also a unique feature of this book to detail two FA procedures that were presented in this century and differ from a prevalent FA procedure. 

The FA and PCA procedures treated in this book are the following seven: classic FA (CLFA), matrix decomposition FA (MDFA), completely decomposed FA (CDFA), minimum rank FA (MRFA), PCA, its random variant (RPCA), and probabilistic PCA (PPCA). Here, MDFA and CDFA are the FA procedures presented in this century and CLFA is the prevalent one.

A hierarchy of MDFA, CDFA, and PCA is derived from the non-random version of a comprehensive model which is a hybrid of FA and PCA. Another hierarchy of MRFA, RPCA, CLFA, and PPCA is derived from the random version of the comprehensive model. The goal of this book is unifying the above two hierarchies into a single one. This hierarchy allows readers to completely capture interrelationships among the FA and PCA procedures. How the hierarchy can be used in practices of FA and PCA is also described.

This book is most appropriate for researchers specialized in multivariate data analysis, particularly, in FA and PCA. It is also of great help for graduate students to study FA and PCA. Readers should have a basic knowledge of matrix algebra, mathematical statistics, and multivariate data analysis.

The author is the professor of statistics in Faculty of Data Science, Kyoto Women’s University, since 2024 April. Before it, he was the professor of the statistics for behavioral science in Graduate School of Human Sciences, Osaka University. He is an elected member of International Statistical Institute and used to be the president of the Japanese Society of Computational Statistics. He was also the chairman of the local organizing committee of the conference “Data Science, Statistics and Visualisation 2019 (DSSV 2019)”. 

The author published a number of the papers on the matrix-intensive approaches to multivariate analysis in journals for computational statistics and psychometrics. Thanks to evaluations of those papers, he won the outstanding carrier prizes from the Behaviormetric Society, the Japanese Society of Computational Statistics, and the Japanese Classification Society. 

From Springer, the author published the book “Matrix-Based Introduction to Multivariate Data Analysis” in 2016 and its second edition in 2020. This second edition includes the chapters for a matrix decomposition formulation of factor analysis (FA), comparisons between principal component analysis (PCA) and (FA), and sparse estimations in FA, in addition of the chapters for PCA and a prevalent formulation of FA that were included in the first edition.