Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective

TL;DR

This paper explores how Kolmogorov complexity and Hausdorff dimension can help distinguish genuine correlations from spurious ones in time series analysis, emphasizing the importance of complexity measures.

Contribution

It introduces a joint complexity indicator based on Lempel-Ziv complexity to better assess the significance of correlations in time series data.

Findings

False positives are more common among low-complexity series.

The joint complexity indicator predicts synchronization collapse in logistic maps.

Complexity measures can calibrate false-positive rates in correlation analysis.

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 \(\rho\), and treating high correlation among low-complexity series with skepticism.