Principal component analysis with boundary constraints

Observed data often belong to some specific intervals of values (for instance in case of percentages or proportions) or are higher (lower) than pre-specified values (for instance, chemical concentrations are higher than zero). The use of classical principal component analysis (PCA) may lead to extract components such that the reconstructed data take unfeasible values. In order to cope with this problem, a Constrained generalization of PCA is proposed. The new technique, called bounded principal component analysis (B-PCA), detects components such that the reconstructed data are constrained to belong to some pre-specified bounds. This is done by implementing a row-wise alternating least squares (ALS) algorithm, which exploits the potentialities of the least squares with inequality (LSI) algorithm. The results of a simulation study and two applications to bounded data are discussed for evaluating how the method and the algorithm for solving it work in practice. Copyright (C) 2007 John Wiley & Sons, Ltd.