  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.
