Uncertainties in Signal Recovery from Heterogeneous and Convoluted Time Series with Principal Component Analysis

TL;DR

This paper analyzes how heterogeneities, noise, and temporal convolutions affect PCA's ability to recover low-dimensional structures in time series data, providing analytical predictions and simulations.

Contribution

It introduces a theoretical framework using the replica method to predict PCA performance under realistic data heterogeneities and convolutions.

Findings

Sample variability harms trajectory reconstruction but aids mode inference.

Temporal convolution fluctuations hinder perfect mode recovery.

Predictions are validated through extensive simulations.

Abstract

Principal Component Analysis (PCA) is one of the most used tools for extracting low-dimensional representations of data, in particular for time series. Performances are known to strongly depend on the quality (amount of noise) and the quantity of data. We here investigate the impact of heterogeneities, often present in real data, on the reconstruction of low-dimensional trajectories and of their associated modes. We focus in particular on the effects of sample-to-sample fluctuations and of component-dependent temporal convolution and noise in the measurements. We derive analytical predictions for the error on the reconstructed trajectory and the confusion between the modes using the replica method in a high-dimensional setting, in which the number and the dimension of the data are comparable. We find in particular that sample-to-sample variability, is deleterious for the reconstruction of the signal trajectory, but beneficial for the inference of the modes, and that the fluctuations in the temporal convolution kernels prevent perfect recovery of the latent modes even for very weak measurement noise. Our predictions are corroborated by simulations with synthetic data for a variety of control parameters.