Multivariate prediction with nonlinear principal components analysis (methodology)

Gower and Blasius (Quality and Quantity, 39, 2005) proposed the notion of multivariate predictability as a measure of goodness-of-fit in data reduction techniques which is 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, and in particular for ordered categorical variables, they showed how to predict the levels of all variables associated with every point (case). The proportion of predictions which agree with the true category-levels gives the measure of fit. The ideas are very general; as an illustration they used nonlinear principal components analysis. An example of the method is described in this paper using data drawn from 23 countries participating in the International Social Survey Program (1995), paying special attention to two sets of variables concerned with Regional and National Identity. It turns out that the predictability criterion suggests that the fits are rather better than is indicated by “percentage of variance accounted for”