Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective
This addresses a common issue in time-series analysis for researchers and practitioners by providing a principled method to avoid false positives, though it is incremental in applying existing complexity measures to this domain.
The paper tackles the problem of spurious correlations in time-series analysis by using Kolmogorov complexity and Hausdorff dimension to show that false positives decay exponentially with series complexity, and introduces a joint complexity indicator to detect high-complexity correlations, validated on models like logistic maps and multivariate fractional Brownian motion.
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 \(Ï\), and treating high correlation among low-complexity series with skepticism.