ANALYSIS OF ADDITIVE DEPENDENCIES AND CONCURVITIES USING SMALLEST ADDITIVE PRINCIPAL COMPONENTS

Additive principal components are a nonlinear generalization of linear principal components. Their distinguishing feature is that linear for ms Sigma(i) alpha(i)X(i) are replaced with additive functions Sigma(i) phi(i)(X(i)). A considerable amount of flexibility for fitting data i s gained when linear methods are replaced with additive ones. Our inte rest is in the smallest principal components, which is somewhat uncomm on. Smallest additive principal components amount to data descriptions in terms of approximate implicit equations: Sigma(i) phi(i)(X(i)) app roximate to 0. Estimation of such equations is a data-analytic method in its own right, competing in some cases with the more customary regr ession approaches. It is also a diagnostic tool in additive regression for detection of so-called ''concurvity.'' This term describes degene racies among predictor variables in additive regression models, simila r to collinearity in linear regression models. Concurvity may lead to statistically unstable contributions of variables to additive models. As an example, we show in a reanalysis of the ozone data from Breiman and Friedman that concurvity does indeed exist in this particular data , a fact which may impact the interpretation of the additive fits. In the second half of this paper, we give some second-order theory, inclu ding the description of null situations and eigenexpansions derived fr om associated eigenproblems. We show how ACE and additive principal co mponents are related, and we outline some analytical methods for theor etical calculations of additive principal components. Lastly we consid er methods of estimation and computation. Additive principal component s have had a long tradition in psychometric research and correspondenc e analysis. They have started receiving attention by statisticians onl y in recent years.