Learning Time-Varying Correlation Networks with FDR Control via Time-Varying P-values

This paper presents a systematic framework for controlling false discovery rate in learning time-varying correlation networks from high-dimensional, non-linear, non-Gaussian and non-stationary time series with an increasing number of potential abrupt change points in means. We propose a bootstrap-assisted approach to derive dependent and time-varying P-values from a robust estimate of time-varying correlation functions, which are not sensitive to change points. Our procedure is based on a new high-dimensional Gaussian approximation result for the uniform approximation of P-values across time and different coordinates. Moreover, we establish theoretically guaranteed Benjamini–Hochberg and Benjamini–Yekutieli procedures for the dependent and time-varying P-values, which can achieve uniform false discovery rate control. The proposed methods are supported by rigorous mathematical proofs and simulation studies. We also illustrate the real-world application of our framework using both brain electroencephalogram and financial time series data.

💡 Research Summary

This paper addresses the critical challenge of learning time-varying correlation networks from complex, high-dimensional time series data while rigorously controlling the False Discovery Rate (FDR). The data under consideration are particularly challenging: they can be non-linear, non-Gaussian, non-stationary, and may contain an increasing number of abrupt change-points in their mean functions—characteristics common in fields like neuroscience (e.g., EEG signals) and finance.

The authors propose a comprehensive three-stage framework. First, to handle potential jumps in the mean trend, they employ a difference-based local linear estimator for the time-varying correlation functions. This estimator is robust to discontinuities, effectively isolating the underlying dependent error processes. Second, and most innovatively, the framework generates “time-varying P-values” for testing the null hypothesis of zero correlation at each time point for each pair of variables. This is achieved by approximating the estimation error process via a high-dimensional Gaussian approximation theorem specifically developed for the product of two locally stationary time series. A multiplier bootstrap scheme is then used to simulate the distribution of these errors, yielding valid P-values that are inherently dependent across both time and variable pairs.

Third, the framework incorporates these streams of dependent P-values into a multiple testing procedure with uniform FDR control across the entire time continuum. The authors establish the theoretical validity of both the Benjamini-Hochberg (B-H) and the more conservative Benjamini-Yekutieli (B-Y) procedures in this context. Under mild regularity conditions, they prove that their method provides asymptotic uniform FDR control at a pre-specified level α for all time points within the interior of the observation interval.

The theoretical backbone of the work is a novel Gaussian approximation result for high-dimensional, locally stationary processes observed in hyper-rectangles, which generalizes existing literature and is essential for proving the simultaneous validity of the myriad time-varying P-values. Extensive simulation studies demonstrate that the proposed method successfully controls the FDR near the target level under various data-generating models while maintaining high statistical power. Practical utility is showcased through applications to real-world EEG data, revealing dynamic brain connectivity patterns, and financial time series, illustrating evolving market linkages. This work provides a statistically sound and practically useful toolbox for discovering dynamic network structures from noisy, non-stationary observational data.