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.
