By Yuichi Mori, Masahiro Kuroda, Naomichi Makino
This e-book expounds the main and similar purposes of nonlinear vital part research (PCA), that's necessary option to study combined size degrees facts. within the half facing the primary, after a quick advent of standard PCA, a PCA for specific information (nominal and ordinal) is brought as nonlinear PCA, during which an optimum scaling approach is used to quantify the specific variables. The alternating least squares (ALS) is the most set of rules within the strategy. a number of correspondence research (MCA), a unique case of nonlinear PCA, is additionally brought. All formulations in those tools are built-in within the similar demeanour as matrix operations. simply because any size degrees information could be handled continually as numerical info and ALS is the most important instrument for estimations, the tools can be used in a number of fields resembling biometrics, econometrics, psychometrics, and sociology. within the functions a part of the publication, 4 purposes are brought: variable choice for combined size degrees information, sparse MCA, joint measurement relief and clustering tools for express facts, and acceleration of ALS computation. The variable choice tools in PCA that initially have been constructed for numerical information may be utilized to any forms of dimension degrees through the use of nonlinear PCA. Sparseness and joint size relief and clustering for nonlinear facts, the result of contemporary stories, are extensions bought by means of an analogous matrix operations in nonlinear PCA. eventually, an acceleration set of rules is proposed to lessen the matter of computational expense within the ALS generation in nonlinear multivariate tools. This booklet therefore offers the usefulness of nonlinear PCA which are utilized to diversified dimension degrees information in different fields. to boot, it covers the newest subject matters together with the extension of the conventional statistical technique, newly proposed nonlinear equipment, and computational potency within the methods.
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Additional resources for Nonlinear Principal Component Analysis and Its Applications
4 Algorithm of Rank-Restricted MCA 25 for each variable. 1) in the current and previous steps is less than ε. The rank-restricted MCA algorithm is summarized as follows: Step 1: Initial values are chosen for Ai , Z, and Ai . 3). Values for Z can be chosen randomly, which should satisfy the constraints. Then, Ai is given as A = n −1 Qi Gi Z for each variable. Step 2: Update the quantification parameters by solving the orthogonal Procrustes problems. Step 3: Update the MCA model parameters by solving the orthogonal Procrustes and regression problems.
2, was proposed in Mori et al. (1997). Computations were performed in Kuroda et al. (2011) as an application of accelerated computation for nonlinear PCA (acceleration is introduced in Chap. 7). PCA to select a subset of variables from mixed measurement level data. PCA (Tanaka and Mori 1997) is intended to derive PCs that are computed using only a selected subset but represent all of the variables, including those not selected. 2 Modified PCA for Mixed Measurement Level Data 33 to apply practically, although it is obtained from small numbers of variables.
An analysis and synthesis of multiple correspondence analysis, optimal scaling, dualscaling, homogeneity analysis and other methods for quantifying categorical multivariate data. Psychom. : Quantitative analysis of qualitative data. Psychom. 46, 357–388 (1981) Part II Applications and Related Topics The second part of this book consists of four chapters, which examine applications of nonlinear principal component analysis (PCA). Four applications are introduced: variable selection for mixed measurement level data, sparse multiple correspondence analysis (MCA), reduced k-means clustering, and acceleration of alternating least squares (ALS) computation.