  We propose a new data-driven method to select the optimal number of relevant
components in Principal Component Analysis (PCA). This new method applies to
correlation matrices whose time autocorrelation function decays more slowly
than an exponential, giving rise to long memory effects. In comparison with
other available methods present in the literature, our procedure does not rely
on subjective evaluations and is computationally inexpensive. The underlying
basic idea is to use a suitable factor model to analyse the residual memory
after sequentially removing more and more components, and stopping the process
when the maximum amount of memory has been accounted for by the retained
components. We validate our methodology on both synthetic and real financial
data, and find in all cases a clear and computationally superior answer
entirely compatible with available heuristic criteria, such as cumulative
variance and cross-validation.
