Retrodicting Chaotic Systems: An Algorithmic Information Theory Approach

TL;DR

This paper explores retrodicting past states in chaotic systems using algorithmic information theory, proposing methods that identify the most plausible initial conditions based on simplicity and density, with mixed success across different maps.

Contribution

It introduces two novel algorithmic information theory-based methods for retrodiction in chaotic systems and demonstrates their application and limitations on various well-known maps.

Findings

Methods outperform random guessing in some cases

Approach is effective for certain maps and parameters

Open problems include computational cost and noise sensitivity

Abstract

Making accurate inferences about data is a key task in science and mathematics. Here we study the problem of \emph{retrodiction}, inferring past values of a series, in the context of chaotic dynamical systems. Specifically, we are interested in inferring the starting value $x_0$ in the series $x_0,x_1,x_2,\dots,x_n$ given the value of $x_n$, and the associated function $f$ which determines the series as $f(x_i)=x_{i+1}$. Even in the deterministic case this is a challenging problem, due to mixing and the typically exponentially many candidate past values in the pre-image of any given value $x_n$ (e.g., a current observation). We study this task from the perspective of algorithmic information theory, which motivates two approaches: One to search for the `simplest' value in the set of candidates, and one to look for the value in the lowest density region of the candidates. We test these methods numerically on the logistic map, Tent map, Bernoulli map, and Julia/Mandelbrot map, which are well-studied maps in chaos theory. The methods aid in retrodiction by assigning low ranks to candidates which are more likely to be the true starting value. Our approach works well in some parameter and map cases, and outperforms several other retrodiction techniques (each of which fails to outperform random guessing). Nonetheless, the approach is not effective in all cases, and several open problems remain including computational cost and sensitivity to noise. All of these methods are unified through a Gaussian Process (GP) perspective, motivating complexity-based priors for GPs.