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.

翻译：时间序列分析中常出现伪相关现象，因为即便在无关序列之间，简单低复杂度模式也可能产生高Pearson相关系数。我们提出，将Kolmogorov复杂性解释为抗压缩性，能为此类误判提供原则性防御。利用有效Hausdorff维数，我们证明两个独立序列间偶然相关的概率随其复杂性指数衰减，而噪声会人为增加观测复杂度，因此需在实践中加以考量。通过耦合逻辑斯蒂映射与多元分数布朗运动(mfBm)模型验证上述思想，其中Hurst参数\(H\)同时控制复杂度与Hausdorff维数\((\dim_H = 2 - H)\)。两个模型均表明，低复杂度序列的伪相关发生率显著高于高复杂度序列。我们引入联合复杂度指标\[ J_{\rm LZ} = \sqrt{\widetilde{C}_{\rm LZ}(x)\widetilde{C}_{\rm LZ}(y)} \]，该指标捕捉的是联合高复杂度而非个体复杂度的简单相似性，其阈值可通过mfBm伪相关曲线标定。在逻辑斯蒂映射中，\(J_{\rm LZ}\)还能在同步发生前预警个体复杂度的骤降。建议首先检验序列平稳性，然后同时报告\(J_{\rm LZ}\)与\(ρ\)，并对低复杂度序列中的高相关系数保持审慎态度。