Penalized Likelihood Principal Component Rotation for Enhanced Interpretation

Principal component analysis provides a ready exploratory tool for multivariate data when a priori models are unavailable. The objective is to assign interpretations to the components that provide intuition and serve as a guide for further exploration. However, principal components based on small samples are subject to high sampling variation that can obscure straightforward interpretations. Ad hoc techniques like Varimax rotation can enhance interpretability, at the expense of possibly losing fidelity to the data. These problems are alleviated if rotation criteria are instead used as penalty functions in a maximum penalized likelihood setting. Advantageous features of this approach include a smooth continuum of possible rotations, preferential rotation of components that are poorly defined, and a way to measure fidelity of rotated components to the data. Some computational challenges inherent in this technique have been alleviated by recent developments in algorithms for optimization subject to orthogonality constraints.