Uncertainty quantification of synchrosqueezing transform under complicated nonstationary noise

We propose a bootstrapping framework to quantify uncertainty in time-frequency representations (TFRs) generated by the short-time Fourier transform (STFT) and the STFT-based synchrosqueezing transform (SST) for oscillatory signals with time-varying amplitude and frequency contaminated by complex nonstationary noise. To this end, we leverage a recent high-dimensional Gaussian approximation technique to establish a sequential Gaussian approximation for nonstationary processes under mild assumptions. This result is of independent interest and provides a theoretical basis for characterizing the approximate Gaussianity of STFT-induced TFRs as random fields. Building on this foundation, we establish the robustness of SST-based signal decomposition in the presence of nonstationary noise. Furthermore, assuming locally stationary noise, we develop a Gaussian autoregressive bootstrap for uncertainty quantification of SST-based TFRs and provide theoretical justification. We validate the proposed methods with simulations and illustrate their practical utility by analyzing spindle activity in electroencephalogram recordings. Our work bridges time-frequency analysis in signal processing and nonlinear spectral analysis of time series in statistics.

💡 Research Summary

The paper introduces a comprehensive bootstrap framework for quantifying uncertainty in time‑frequency representations (TFRs) produced by the short‑time Fourier transform (STFT) and the STFT‑based synchrosqueezing transform (SST) when the underlying oscillatory signal is corrupted by complex, non‑stationary noise. The authors address three major methodological challenges.

First, they establish that the discrete STFT of a non‑Gaussian, non‑stationary process can be uniformly approximated by a complex Gaussian random field. To achieve this, they combine a recent high‑dimensional Gaussian approximation technique (building on Chernozhukov, Chetverikov, and Kato) with a blocking strategy, yielding a sequential Gaussian approximation. Under mild moment and dependence conditions on the noise (Assumption 2.2), and with sufficiently large window length, the weighted sums that constitute each time‑frequency point converge in distribution to a complex Gaussian variable. This result is formalized in Theorem 6.2 and provides a rigorous justification for treating the STFT‑induced TFR as approximately Gaussian even in realistic, non‑stationary settings.

Second, the paper proves the robustness of the SST‑based signal reconstruction formula for signals that satisfy the adaptive harmonic model (AHM). The AHM assumes slowly varying amplitude and instantaneous frequency, with sufficient separation between components. While prior work demonstrated SST robustness only for stationary Gaussian noise, the authors extend the analysis to the same filtration‑based non‑stationary noise that generates the observations. By carefully controlling the discretization error between the continuous and discrete STFT, and by analyzing the nonlinear reassignment rule of SST, they show that the reconstructed signal deviates from the true AHM signal by at most O(ε) (Theorem 6.4). This establishes that SST remains a reliable tool for extracting intrinsic mode functions even under complex noise structures.

Third, recognizing that practitioners often lack detailed knowledge of the noise structure, the authors propose a bootstrap scheme based on locally stationary approximations of the noise. Assuming the noise is locally stationary, they fit a time‑varying autoregressive (tvAR) model to capture its evolving dynamics. The tvAR model is then used to generate Gaussian bootstrap replicates of the noise, which are added back to the estimated signal and processed through the SST pipeline. The authors prove that the error introduced by the tvAR approximation does not inflate the uncertainty beyond a controllable bound, and that the bootstrapped SST TFRs consistently approximate the distribution of the original TFR (Theorem 6.5). This method can be viewed as an extension of the AR‑sieve bootstrap to a non‑stationary, nonlinear frequency‑domain context.

The theoretical contributions are complemented by extensive simulations and a real‑world application to electroencephalogram (EEG) spindle detection. Simulations explore various non‑stationary noise regimes (including piecewise locally stationary processes with abrupt changes) and demonstrate that the Gaussian approximation and bootstrap confidence intervals achieve the nominal coverage rates predicted by the theory. In the EEG case study, the proposed bootstrap yields reliable confidence bands for spindle power and frequency estimates, illustrating practical utility in biomedical signal analysis where noise is often non‑Gaussian and time‑varying.

Overall, the paper bridges signal‑processing techniques (STFT, SST) with modern high‑dimensional probability and time‑series bootstrap methodology. Its three main contributions—(1) a sequential Gaussian approximation for discrete STFT under non‑stationary noise, (2) a robustness proof for SST reconstruction in the same setting, and (3) a tvAR‑based bootstrap for uniform uncertainty quantification—constitute the first uniform statistical inference results for nonlinear time‑frequency analysis. The framework opens the door to rigorous hypothesis testing and model validation in a wide range of applications, from neuroscience to climate science and finance, where reliable time‑frequency analysis under complex noise is essential.