2. Nonlinear Principal Component Analysis

Principal components analysis (PCA) is a commonly used descriptive multivariate method for handling quantitative data and can be extended to deal with mixed measurement level data. For the extended PCA with such a mixture of quantitative and qualitative data, we require the quantification of qualitative data in order to obtain optimal scaling data. PCA with optimal scaling is referred to as nonlinear PCA, (Gifi, Nonlinear Multivariate Analysis. Wiley, Chichester, 1990). Nonlinear PCA including optimal scaling alternates between estimating the parameters of PCA and quantifying qualitative data. The alternating least squares (ALS) algorithm is used as the algorithm for nonlinear PCA and can find least squares solutions by minimizing two types of loss functions: a low-rank approximation and homogeneity analysis with restrictions. PRINCIPALS of Young et al. (Principal components of mixed measurement level multivariate data: an alternating least squares method with optimal scaling features 43:279–281, 1978) and PRINCALS of Gifi (Nonlinear Multivariate Analysis. Wiley, Chichester, 1990) are used for the computation.