Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem
Sunia Tanweer, Firas A. Khasawneh
TL;DR
The paper tackles the problem of distinguishing diffusive stochastic processes from deterministic signals using a single discrete time series. It introduces a nonparametric diffusion test based on excursion theory, linking the observed excursion count $N_\varepsilon$ to the quadratic variation $[X]_T$ via $\lim_{\varepsilon\to0} \varepsilon^2 N_\varepsilon(X) = [X]_T/2$, and uses the ratio $K(\varepsilon) = N_\varepsilon^{\mathrm{emp}} / N_\varepsilon^{\mathrm{theory}}$ with a log-log slope $s$ to classify dynamics. The method is model-free and applies to continuous semimartingales with finite quadratic variation, including nonlinear Ito diffusions, while sharply distinguishing deterministic systems. It is validated on canonical diffusions, chaotic maps, and real-world data (finance and audio), demonstrating robust performance and practical utility as a theory-driven diagnostic tool. The approach offers a universal, interpretable alternative to entropy or recurrence-based methods for separating stochasticity from determinism in diverse domains.
Abstract
We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.