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

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.