Matching of observations of dynamical systems, with applications to sequence matching

TL;DR

This paper analyzes the statistical distribution of the closest encounters between observations of dynamical systems, revealing a Gumbel distribution influenced by trajectory length, dimensions, and divergence tendencies, with applications to sequence matching.

Contribution

It introduces a theoretical framework linking dynamical system observations to extreme value theory and sequence matching, including formulas for chaotic maps and numerical illustrations.

Findings

Distribution of closest encounters follows Gumbel type.

Extremal Index indicates divergence tendency of observations.

Distributional results for strongly mixing processes.

Abstract

We study the statistical distribution of the closest encounter between generic smooth observations computed along different trajectories of a rapidly mixing dynamical system. At the limit of large trajectories, we obtain a distribution of Gumbel type that depends on both the length of the trajectories and on the Generalized Dimensions of the image measure. It is also modulated by an Extremal Index, that informs on the tendency of nearby observations to diverge along with the evolution of the dynamics. We give a formula for this quantity for a class of chaotic maps of the interval and regular observations. We present diverse numerical applications illustrating the theory and discuss the implications of these results for the study of physical systems. Finally, we discuss the connection between this problem and the problem of the longest matching block common to different sequences of symbols. In particular, we obtain a distributional result for strongly mixing processes.