Orthogonal simple component analysis: A new, exploratory approach

TL;DR

This paper introduces an exploratory method for interpreting principal components by identifying simple, orthogonal axes with small integer elements, providing visual tools to aid user interpretation without enforcing a unique solution.

Contribution

The paper proposes a new approach and algorithm for exploratory analysis of principal components focusing on simple, interpretable axes close to the original components, enhancing interpretability.

Findings

Provides an automated visual display of solutions ordered by simplicity and accuracy.

Identifies sets of orthogonal axes with small integer elements close to principal components.

Supports user-driven interpretation without enforcing a unique solution.

Abstract

Combining principles with pragmatism, a new approach and accompanying algorithm are presented to a longstanding problem in applied statistics: the interpretation of principal components. Following Rousson and Gasser [53 (2004) 539--555] @p250pt@ the ultimate goal is not to propose a method that leads automatically to a unique solution, but rather to develop tools for assisting the user in his or her choice of an interpretable solution. Accordingly, our approach is essentially exploratory. Calling a vector 'simple' if it has small integer elements, it poses the open question: @p250pt@ What sets of simply interpretable orthogonal axes---if any---are angle-close to the principal components of interest? its answer being presented in summary form as an automated visual display of the solutions found, ordered in terms of overall measures of simplicity, accuracy and star quality, from which the user may choose. Here, 'star quality' refers to striking overall patterns in the sets of axes found, deserving to be especially drawn to the user's attention precisely because they have emerged from the data, rather than being imposed on it by (implicitly) adopting a model. Indeed, other things being equal, explicit models can be checked by seeing if their fits occur in our exploratory analysis, as we illustrate. Requiring orthogonality, attractive visualization and dimension reduction features of principal component analysis are retained.