# A memory-based method to select the number of relevant components in Principal Component Analysis

## TL;DR

This paper introduces a data-driven, computationally efficient method for selecting the optimal number of components in PCA, especially suited for data with long memory effects, validated on synthetic and financial datasets.

## Contribution

It presents a novel, objective approach that analyzes residual memory to determine the number of relevant PCA components, outperforming existing methods in computational efficiency.

## Key findings

- Method accurately identifies the number of components in synthetic data.
- Method provides results consistent with heuristic criteria in financial data.
- Approach is computationally inexpensive and objective.

## Abstract

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.