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

TL;DR

This work provides a rigorous theoretical framework for phase-rectified signal averaging (PRSA) under two standard models: a deterministic two-harmonic signal and a stationary Gaussian process. In the deterministic case, PRSA outputs converge to a sum of sine components, revealing how hinge-point selection reshapes the signal; in the stochastic case, PRSA obeys a law of large numbers and a central limit theorem, with the limiting mean tied to the covariance structure and the threshold. Collectively, the results show that PRSA can produce nontrivial quasi-periodic-like outputs even from simple signals or pure noise, urging cautious interpretation in biomedical applications. The paper also develops technical tools, including Weyl equidistribution arguments, cumulant control, and explicit CLT covariances, and outlines future extensions to non-Gaussian, nonstationary, and multivariate settings. These insights lay groundwork for a more reliable theoretical foundation of PRSA-based analyses in practice.

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, $\cos(2πt)+A\cos(2π(ξt+φ))$, where $A>0$, $ξ\in (0,1)$ and $φ\in [0,1)$, we show that, asymptotically when the sample size $n\to \infty$, the PRSA output takes the form $A'\sin(2πt)+B'\sin(2πξt)$, 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.

Figures (3)

• Figure 1: Illustration of PRSA with $c=0$ of deterministic signals sampled at 200 Hz satisfying the two harmonics model with different $(A,\xi)$ pairs. Left column: a 20 seconds segment of the signal is shown in gray, with the hinge points marked in red. The $A$ and $\xi$ are shown. Right column: the low frequency component is shown in gray, the PRSA result, $z_{n,L}$ is shown in black, and the theoretically predicted output is superimposed as the dashed red curve. The vertical blue line indicated the middle point of $z_{n,L}$.
• Figure 2: Illustration of PRSA with different $c\in \mathbb{R}$ of deterministic signals $f(t)=\cos(2\pi t)+0.7\cos(2\pi\times 0.2 t+\phi)$, where $\phi\in [0,2\pi)$, sampled at 10 Hz. Left column: a 20 seconds segment of the signal is shown in gray, with the hinge points marked in red. Middle column: the PRSA result, $z_{n,L}$ is shown in black, and the theoretically predicted output is superimposed as the dashed red curve. The vertical blue line indicated the middle point of $z_{n,L}$. Right column: the magnitude of the Fourier transform of $z_{n,L}$ to indicate how the PRSA output depends on $c$.
• Figure 3: PRSA of stationary stochastic signals. Left column: a realization of a random process indicated in the upper left corner is shown in gray, with the points $w_n>0$ marked in red; that is, $c=0$. Middle column: the predicted law of large number of PRSA with $c=0$ is the red curve, and the PRSA output $z_{n,L}$ with $c=0$ is superimposed as the dark gray curve. The vertical dashed blue line indicated the middle point of $z_{n,L}$. Right column: the PRSA outputs $z_{n,L}$ with $c=-3,0,3$ are superimposed as the black, dark gray and light gray curves, respectively. The vertical dashed blue line indicated the middle point of $z_{n,L}$. As predicted by the theorem, these curves differ only by a constant scaling factor.

Theorems & Definitions (25)

• Remark
• Remark
• Theorem 3.1
• proof
• Remark
• Lemma 4.1
• proof
• Remark
• Proposition 4.2
• proof