Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective

2026-03-30

AI Summary

This study addresses the problem of spurious high correlations arising from low-complexity patterns in time series, which can lead to erroneous inference. Building on effective Hausdorff dimension and Kolmogorov complexity as measures of incompressibility, the authors theoretically demonstrate that the probability of accidental correlation between independent sequences decays exponentially with their complexity. To leverage this insight, they propose a joint Lempel-Ziv complexity metric, denoted \( J_{LZ} \), and calibrate its significance threshold using multivariate fractional Brownian motion (mfBm). This framework enables reliable detection of genuine associations among high-complexity sequences and provides early warning of complexity collapse preceding synchronization. Experimental results confirm that the method substantially suppresses false correlations in high-complexity regimes and achieves superior performance in both synchronization prediction and false positive control.

Abstract

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.

Problem

• spurious correlations
• time series
• Kolmogorov complexity
• Hausdorff dimension
• false positives

Innovation

• Kolmogorov complexity
• Hausdorff dimension
• spurious correlation
• joint complexity indicator
• time series analysis