arXiv subjects

Kamal Dingle

Publications and source records attributed to Kamal Dingle.

2 recordsLinked to original sources

arXiv2026

Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective

Spurious correlations are common in time-series analysis because simple, low-complexity patterns can produce high Pearson correlations even between unrelated series. We argue that Kolmogorov complexity, interpreted as resistance to compression, provides a principled safeguard against such false positives. Using effective Hausdorff dimension, we show that the probability of accidental correlation between two independent series decays exponentially with their complexity, while noise can inflate observed complexity and must therefore be accounted for in practice. We illustrate these ideas with coupled logistic maps and multivariate fractional Brownian motion (mfBm), where the Hurst parameter $H$ controls both complexity and Hausdorff dimension $(\dim_H = 2 - H)$. Both models show that false positives are much more common among low-complexity series than among high-complexity ones. We introduce the joint complexity indicator \[ J_{\rm LZ} = \sqrt{\widetilde{C}_{\rm LZ}(x)\widetilde{C}_{\rm LZ}(y)}, \] which captures joint high complexity rather than simple similarity between individual complexities. Its threshold can be calibrated from the mfBm false-positive curve. In logistic maps, $J_{\rm LZ}$ also anticipates the collapse of individual complexity just before synchronization. We recommend establishing stationarity first, then reporting $J_{\rm LZ}$ alongside $ρ$, and treating high correlation among low-complexity series with skepticism.

math.DS↗

arXiv2025

Retrodicting Chaotic Systems: An Algorithmic Information Theory Approach

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.

math.DS↗