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Sampling Theorem

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Related Experiment Video

Updated: May 15, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Detecting stochasticity in discrete signals via nonparametric excursion theorem.

Sunia Tanweer¹, Firas A Khasawneh¹

¹Department of Computational Mathematics, Sciences and Engineering, Michigan State University, Michigan 48824, USA.

Chaos (Woodbury, N.Y.)

|May 13, 2026

Summary

This study introduces a novel framework to differentiate diffusive stochastic processes from deterministic signals in time series data. The method reliably distinguishes between these dynamics using excursion counts and quadratic variation, offering a theoretically certified approach.

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A Method for Tracking the Time Evolution of Steady-State Evoked Potentials

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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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A Method for Tracking the Time Evolution of Steady-State Evoked Potentials

A Method for Tracking the Time Evolution of Steady-State Evoked Potentials

Published on: May 25, 2019

Area of Science:

Stochastic processes
Time series analysis
Nonlinear dynamics

Background:

Distinguishing between stochastic diffusion and deterministic signals is crucial in many scientific fields.
Current methods often rely on subjective measures like entropy or recurrence, lacking theoretical certification.
A robust, model-free approach is needed for accurate classification.

Purpose of the Study:

To develop a practical and theoretically certified framework for distinguishing diffusive stochastic processes from deterministic signals.
To establish a data-driven diffusion test based on excursion theory.
To provide a reliable method for analyzing time series data without prior model assumptions.

Main Methods:

Utilizing classical excursion and crossing theorems for continuous semimartingales.
Correlating the number of excursions (Nε) with the quadratic variation ([X]T) of the process.
Developing a data-driven diffusion test by comparing empirical excursion counts to theoretical expectations and analyzing the resulting ratio K(ε) via a log-log slope deviation.

Main Results:

Demonstrated that a universal scaling law based on excursion counts holds for continuous semimartingales but fails for deterministic systems.
Developed a robust diffusion test that classifies time series as diffusion-like or not based on the ε-2 law.
Successfully applied the method to various canonical stochastic and deterministic systems, including chaotic maps and the stochastic Duffing system.

Conclusions:

The proposed framework offers a theoretically certified and robust method for distinguishing diffusive stochastic processes from deterministic signals.
The nonparametric and model-free approach relies on the universal small-scale structure of continuous semimartingales.
This method provides a significant advancement over existing subjective techniques for time series analysis.