Multivariate Prediction with Nonlinear Principal Components Analysis: Theory

We propose the notion of multivariate predictability as a measure of goodness-of-fit in data reduction techniques which are useful for visualizing and screening data. For quantitative variables this leads to the usual sums-of-squares and variance accounted for criteria. For categorical variables we show how to predict the category-levels of all variables associated with every point (case). The proportion of predictions which agree with the true categories gives the measure of fit. The ideas are very general; as an illustration we use nonlinear principal components analysis (NLPCA) in association with ordered categorical variables. A detailed example using data from the International Social Survey Program (ISSP) will be given in Blasius and Gower (quality and quantity, 39, to appear). It will be shown that the predictability criterion suggests that the fits are rather better than is indicated by “percentage of variance accounted for”.This article was written while John Gower was a visiting professor at the ZA-Eurolab, at the Zentralarchiv für Empirische Sozialforschung, University of Cologne, Germany. The ZA is a Large Scale Facility funded by the Training and Mobility of Researchers program of the European Union.