Alternating least squares in nonlinear principal components

Masahiro Kuroda, Yuichi Mori, Iizuka Masaya, Michio Sakakihara

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)


    Principal components analysis (PCA) is probably the most popular descriptive multivariate method for analyzing quantitative data with ratio and interval scale measures. When applying PCA to nominal and ordinal data, the data are processed by a method such as optimal scaling, which nonlinearly transforms nominal and ordinal data into quantitative data. Therefore, PCA with optimal scaling is called nonlinear PCA. Nonlinear PCA reveals nonlinear relationships among variables with different measurement levels and therefore presents a more flexible alternative to ordinary PCA. The alternating least squares algorithm is utilized for nonlinear PCA. The algorithm alternates between optimal scaling for quantifying nominal and ordinal data and ordinary PCA for analyzing optimally scaled data. This article discusses two nonlinear PCA algorithms, namely, PRINCIPALS and PRINCALS.

    Original languageEnglish
    Pages (from-to)456-464
    Number of pages9
    JournalWiley Interdisciplinary Reviews: Computational Statistics
    Issue number6
    Publication statusPublished - Nov 2013


    • Alternating least squares algorithm
    • Homogeneity
    • Optimal scaling
    • Quantification

    ASJC Scopus subject areas

    • Statistics and Probability


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