Statistica Sinica 23 (2013), 963-988

METHODOLOGY AND THEORY FOR

NONNEGATIVE-SCORE PRINCIPAL COMPONENT

ANALYSIS

Peter Bajorski$^{1}$, Peter Hall$^2$ and Hyam Rubinstein$^2$

$^1$Rochester Institute of Technology and $^2$The University of Melbourne

Abstract: 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 $Y= {\bf A} X$, 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.

Key words and phrases: Correlation, independent component analysis, nonparametric statistics, permutation, principal component analysis, rate of convergence, rotation.