A faster algorithm for ridge regression of reduced rank data

Regression data sets typically have many more cases than variables, but this is not always the case. Some current problems in chemometrics-for example fitting quantitative structure activity relationships-may involve fitting linear models to data sets in which the number of predictors far exceeds the number of cases. Ridge regression is an approach that has some theoretical foundation and has performed well in comparison with alternatives such as PLS and subset regression. Direct implementation of the regression formulation leads to a O(np² + p³) calculation, which is substantial if p is large. We show that ridge regression may be performed in a O(np²) computation-a potentially large saving when p is larger than n. The algorithm lends itself to the use of case weights, to robust bounded influence fitting, and cross-validation. The method is illustrated with a chemometric data set with 255 predictors, but only 18 cases, a ratio not unusual in QSAR problems.