Retrodicting Chaotic Systems: An Algorithmic Information Theory Approach

TL;DR

The paper tackles exact retrodiction in deterministic chaotic maps by inferring $x_0$ from $x_n$, an ill-posed task due to exponentially many preimages. It develops two complementary approaches rooted in algorithmic information theory: ranking candidates by Kolmogorov-complexity approximations and ranking by low local density, both aimed at reducing the baseline uncertainty of $\log_2 m$ bits. The methods are tested on classic chaotic maps (Logistic, Tent, Bernoulli, Julia) and show that, in several settings, the true $x_0$ can be identified or significantly prioritized, outperforming several baselines; however, performance is map- and parameter-dependent and faces computational and noise-related challenges. The authors also unify these approaches within a Gaussian Process perspective, introducing complexity-based priors and outlining how GP and AIT viewpoints connect via shadowing ideas, with clear directions for future work including efficiency improvements and extensions to more complex systems. Overall, the work advances exact retrodiction in chaotic dynamics by leveraging information-theoretic priors and sheds light on the interplay between compression, density structure, and backward reconstruction.

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.

Figures (8)

• Figure 1: Retrodiction with digit complexity method ($D\leq 2$). (top left) logistic map; (top right) Tent map; (bottom left) Bernoulli map; (bottom right) Julia map. In each panel, the mean number of distinct candidates $m$ rises exponentially (blue), typically as $m=2^n$ (error bars $\pm$1 std. dev.). In contrast, the mean log rank $r$ (red) of the true starting value has a much lower value and does not continue to grow. Hence, $\log_2(r)\ll \log_2(m)$, typically, for large enough $n$, and the true starting value is in a small subset of the candidates.
• Figure 2: Bernoulli map retrodiction with higher decimal places, $D$. (top left) $D=4$; (top right) $D=6$; (bottom left) $D=8$; (bottom right) $D=10$. As $D$ gets larger, typical complexity values increase, hence attaining accurate retrodiction requires larger and larger $n$ values. Each data point is an average over 50 samples ($\pm$ 1 std. dev.).
• Figure 3: Illustration of the shadowing property. A pseudo-orbit $\{y_t\}$ (red, dashed) with small step errors is shadowed by a true orbit $\{x_t\}$ (blue, solid), which remains within distance $\varepsilon$ of it for all $t$.
• Figure 4: A selection of comparison retrodiction methods do no better than mere random guessing as a method of retrodiction. All three AIT approaches outperform the comparison methods. Bar heights are mean rank ($\pm$ 1 std. dev.). The top panel is for $n=5$ iterations, and bottom panel is for $n=10$ iterations.
• Figure 5: Effects of noisy trajectories on retrodiction accuracy for the Bernoulli map. Increasing the noise level decreases the accuracy, especially for larger $n$. For very small noise levels, and small $n$ partial retrodiction is still possible.