  How does one find dimensions in multivariate data that are reliably expressed
across repetitions? For example, in a brain imaging study one may want to
identify combinations of neural signals that are reliably expressed across
multiple trials or subjects. For a behavioral assessment with multiple ratings,
one may want to identify an aggregate score that is reliably reproduced across
raters. Correlated Components Analysis (CorrCA) addresses this problem by
identifying components that are maximally correlated between repetitions (e.g.
trials, subjects, raters). Here we formalize this as the maximization of the
ratio of between-repetition to within-repetition covariance. We show that this
criterion maximizes repeat-reliability, defined as mean over variance across
repeats, and that it leads to CorrCA or to multi-set Canonical Correlation
Analysis, depending on the constraints. Surprisingly, we also find that CorrCA
is equivalent to Linear Discriminant Analysis for zero-mean signals, which
provides an unexpected link between classic concepts of multivariate analysis.
We present an exact parametric test of statistical significance based on the
F-statistic for normally distributed independent samples, and present and
validate shuffle statistics for the case of dependent samples. Regularization
and extension to non-linear mappings using kernels are also presented. The
algorithms are demonstrated on a series of data analysis applications, and we
provide all code and data required to reproduce the results.
