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.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS374 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
