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. WIREs Comput Stat 2013, 5:456–464. doi: 10.1002/wics.1279This article is categorized under: Algorithms and Computational Methods > Algorithms Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Algorithms and Computational Methods > Numerical Methods Statistical Models > Nonlinear Models Algorithms and Computational Methods > Least Squares