Enhanced scaling crossover detection in long-range correlated time series

Abstract

Various time series, such as biological signals and stock prices, exhibit long-range correlations and a fractal nature characterized by the power-law scaling in the low-frequency range of the power spectrum. Instead of the power spectral analysis, scaling analysis methods such as Detrended Fluctuation Analysis (DFA) and Detrending Moving-Average Analysis (DMA) have been employed to estimate the scaling exponent. Scaling analysis results often uncover crossover phenomena, highlighting two distinct scaling regions—a short- and a long-range exponent. Estimating the two respective scaling exponents and the crossover point is crucial, as they can provide insight into different underlying mechanisms or dynamics operating at various scales. However, DFA and DMA with higher-order detrending tend to distort the time scales, and methods for accurately estimating the crossover have not been thoroughly investigated. This study addresses scale distortions in higher-order DFA and DMA by leveraging the relationship between the fluctuation function and the power spectrum of time series. We propose a method for crossover estimation using the Savitzky–Golay differentiation filter. Applying this method to numerical experiments with an autoregressive process exhibiting crossovers demonstrated that our proposed technique estimates crossovers more accurately than conventional segmented regression approaches used in DFA and DMA. We applied the proposed scaling crossover estimation method to the time series of the postural center of pressure (CoP), showcasing its practical applications in studying long-range correlations in empirical time series.