Methodology and Theory for Nonnegative-score Principal Component Analysis

We develop nonparametric methods, and theory, for analysing data on a random p-vector Y represented as a linear form in a p-vector X, say Y = AX, where the components of X are nonnegative and uncorrelated. Problems of this nature are motivated by a wide range of applications in which physical considerations deny the possibility that X can have negative components. Our approach to inference is founded on a necessary and sufficient condition for the existence of unique, nonnegative-score principal components. The condition replaces an earlier, sufficient constraint given in the engineering literature, and is related to a notion of sparsity that arises frequently in nonnegative principal component analysis. We discuss theoretical aspects of our estimators of the transformation that produces nonnegative-score principal components, showing that the estimators have optimal properties.