What is it about?

Similar to the sign of a permutation, we introduce a new permutation statistic that allows us to make a combinatorial analysis of white noise via delay embeddings, by ranking the entries of the embedding vectors and computing the newly introduced functional and its corresponding distribution. We show that the new permutation statistic is asymptotically normally distributed (infinite delay embedding dimension), and this allows us to create simple statistical tests of time independence in single time series, i.e., white noise character.

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Why is it important?

Although the analysis of time series is a fairly well-developed area of research in applied mathematics, the most common approaches rely on the availability of batches of time series, i.e. ensembles. Useful approaches for cases where only one or few samples in time are available are, in general, less explored and harder to develop, and here we present one such advance.


This research presents a deep study on stochastic processes by taking fundamental concepts from combinatorics and more specifically, combinatorial stochastic processes to deliver practical methods for time series analysis. I think one of the key perspectives provided by this article is that it highlights the importance of looking for different approaches to make statistical analysis, particularly, the combinatorial properties of time series.

Alvaro Diaz-Ruelas
Max-Planck-Gesellschaft zur Forderung der Wissenschaften

Read the Original

This page is a summary of: A combinatorial view of stochastic processes: White noise, Chaos An Interdisciplinary Journal of Nonlinear Science, December 2022, American Institute of Physics,
DOI: 10.1063/5.0097187.
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