A combinatorial view of stochastic processes: White noise.

Area of Science:

• Stochastic Processes
• Time Series Analysis
• Statistical Physics

Background:

• White noise is a foundational concept in stochastic processes and time series modeling.
• Existing methods for analyzing white noise may not fully capture its complex characteristics.

Purpose of the Study:

• To present a new perspective on white noise analysis using combinatorial methods.
• To introduce ordinal pattern analysis for abstracting time series into permutations.
• To develop a functional to classify permutations based on asymmetry.

Main Methods:

• Incorporation of ordinal pattern analysis to represent time series as sequences of permutations.
• Introduction of a functional to partition permutations into asymmetry classes.
• Computation of the exact probability mass function (p.m.f.) for the functional over the symmetric group.
• Approximation of the p.m.f. using a Gaussian probability density function.

Main Results:

• The study provides an exact p.m.f. for the asymmetry functional in infinite white noise realizations.
• A Gaussian approximation to the p.m.f. is established, simplifying statistical analysis.
• The method is demonstrated on spatial increment data from 3D gold nanoparticle diffusion tracks.

Conclusions:

• Ordinal pattern analysis offers a powerful, combinatorially grounded framework for understanding white noise.
• The developed functional and its p.m.f. provide novel tools for both theoretical and applied time series analysis.
• This approach facilitates convenient statistical analysis, as shown in the nanoparticle diffusion example.