Theoretical analysis of phase-rectified signal averaging (PRSA) algorithm

TL;DR

This paper provides a theoretical analysis of the PRSA algorithm, revealing its asymptotic behavior with oscillatory signals and Gaussian processes, and highlighting the need for careful interpretation of its outputs.

Contribution

It offers the first theoretical insights into PRSA's behavior, including asymptotic forms for oscillatory inputs and a central limit theorem for Gaussian processes.

Findings

PRSA output asymptotically resembles a sum of sinusoidal components.

A central limit theorem describes PRSA output distribution for Gaussian inputs.

Results suggest cautious interpretation of PRSA in scientific applications.

Abstract

Phase-rectified signal averaging (PRSA) is a widely used algorithm to analyze nonstationary biomedical time series. The method operates by identifying hinge points in the time series according to prescribed rules, extracting segments centered at these points (with overlap permitted), and then averaging the segments. The resulting output is intended to capture the underlying quasi-oscillatory pattern of the signal, which can subsequently serve as input for further scientific analysis. However, a theoretical analysis of PRSA is lacking. In this paper, we investigate PRSA under two settings. First, when the input consists of a superposition of two oscillatory components, $\cos(2\pi t)+A\cos(2\pi (\xi t+\phi))$, where $A>0$, $\xi\in (0,1)$ and $\phi\in [0,1)$, we show that, asymptotically when the sample size $n\to \infty$, the PRSA output takes the form $A'\sin(2\pi t)+B'\sin(2\pi \xi t)$, where $A',B'\neq 0$. Second, when the input is a stationary Gaussian random process, we establish a central limit theorem: under mild regularity conditions, the averaged vector produced by PRSA converges in distribution to a Gaussian random vector as $n\to \infty$ with mean determined by the covariance structure of the random process. These results indicate that caution is warranted when interpreting PRSA outputs for scientific applications.