Correlation coefficients play a pivotal role in quantifying linear relationships between random variables. Yet, their application to time series data is very challenging due to temporal dependencies. This paper introduces a novel approach to estimate the statistical significance of correlation coefficients in time series data, addressing the limitations of traditional methods based on the concept of effective degrees of freedom (or effective sample size, ESS). These effective degrees of freedom represent the independent sample size that would yield comparable test statistics under the assumption of no temporal correlation. We propose to assume a parametric Gaussian form for the autocorrelation function. We show that this assumption, motivated by a Laplace approximation, enables a simple estimator of the ESS that depends only on the temporal derivatives of the time series. Through numerical experiments, we show that the proposed approach yields accurate statistics while significantly reducing computational overhead. In addition, we evaluate the adequacy of our approach on real physiological signals, for assessing the connectivity measures in electrophysiology and detecting correlated arm movements in motion capture data. Our methodology provides a simple tool for researchers working with time series data, enabling robust hypothesis testing in the presence of temporal dependencies.

翻译：相关系数在量化随机变量之间的线性关系中起着关键作用。然而，由于时间依赖性的存在，将其应用于时间序列数据极具挑战性。本文提出了一种估计时间序列数据中相关系数统计显著性的新方法，该法基于有效自由度（或有效样本量，ESS）概念，克服了传统方法的局限性。有效自由度代表在无时间相关性假设下能产生可比检验统计量的独立样本量。我们提出采用自相关函数的参数化高斯形式，并证明这种基于拉普拉斯近似的假设能够构建出仅依赖于时间序列时间导数的ESS简易估计量。通过数值实验表明，该方法在显著降低计算开销的同时能获得精确的统计量。此外，我们还在真实生理信号中评估了该方法的适用性，用于电生理学中的连接性度量检测以及运动捕捉数据中相关手臂运动的识别。本方法为处理时间序列数据的研究人员提供了简易工具，使其能够在存在时间依赖性的情况下进行稳健的假设检验。