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Uncertainty quantification of synchrosqueezing transform under complicated nonstationary noise

Hau-Tieng Wu, Zhou Zhou

TL;DR

This work develops a principled framework for uncertainty quantification of time-frequency representations produced by STFT and SST under complex nonstationary noise. By leveraging a high-dimensional sequential Gaussian approximation, it establishes that the discrete STFT-based TF representations are well-approximated by complex Gaussian fields, enabling rigorous UQ for SST, including a robust reconstruction error bound. It introduces a tvAR-based bootstrap under local stationarity to quantify uncertainty, construct thresholds, and detect oscillatory components with high reliability. The theoretical results are complemented by simulations and a sleep spindle EEG application, demonstrating practical utility in biomedical signal analysis and bridging nonlinear time-series spectral analysis with statistical inference. The work provides new guarantees for nonlinear TF tools and offers a scalable approach to uncertainty quantification in nonstationary TF analysis.

Abstract

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.

Uncertainty quantification of synchrosqueezing transform under complicated nonstationary noise

TL;DR

This work develops a principled framework for uncertainty quantification of time-frequency representations produced by STFT and SST under complex nonstationary noise. By leveraging a high-dimensional sequential Gaussian approximation, it establishes that the discrete STFT-based TF representations are well-approximated by complex Gaussian fields, enabling rigorous UQ for SST, including a robust reconstruction error bound. It introduces a tvAR-based bootstrap under local stationarity to quantify uncertainty, construct thresholds, and detect oscillatory components with high reliability. The theoretical results are complemented by simulations and a sleep spindle EEG application, demonstrating practical utility in biomedical signal analysis and bridging nonlinear time-series spectral analysis with statistical inference. The work provides new guarantees for nonlinear TF tools and offers a scalable approach to uncertainty quantification in nonstationary TF analysis.

Abstract

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.

Paper Structure

This paper contains 29 sections, 14 theorems, 149 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Adaptive harmonic model
  4. Non-stationary noise (NSN) model
  5. Algorithm
  6. discrete short-time Fourier transform (STFT)
  7. Continuous STFT and its discretization
  8. Synchrosqueezing transform (SST)
  9. Signal reconstruction by SST
  10. Numerical implementation
  11. Uncertainty quantification of SST by bootstrapping
  12. Bootstrap the noise
  13. Application to uncertainty quantification of SST
  14. Numerical results
  15. Non-null case
  16. ...and 14 more sections

Key Result

Theorem 6.1

Suppose the HDNS time series $\{z_i\}_{i=1}^n$ satisfy Assumption model assumption 4 with $A > \sqrt{\chi} + 1$. Further, assume that $\|z_i\|_p\leq B_d$ for some $B_d>0$. On a sufficiently rich probability space, there exist $\{\hat{z}_i\}_{i=1}^n$ so that $\hat{z}_i \stackrel{\mathcal{D}}{=} z_i$, where the implied constant depends on $p$ and the dependence and moment of $z_i$.

Figures (13)

  • Figure 1: An 8s segment of the noisy signal and reconstruction results. (a) one realization of the noisy signal (gray) and the corresponding true deterministic signal (gray). (b) reconstructed deterministic signal (gray) and the true deterministic signal (gray). (c) reconstructed noise (gray) and the true realized noise (gray).
  • Figure 2: (a) the true tvAR coefficients, and the estimated coefficients of the approximate tvAR process. (b) one realization of $x_i$ and the reconstructed $x_i$. (c) the bootstrapped $x_i$.
  • Figure 3: Thresholding results. (a) The TFR of a nonnull signal determined by STFT. (b) 99% percentile of the bootstrapping as the threshold, with interpolation to the whole grid. (c) Thresholded (a) by (b). (d) The TFR of a nonnull signal determined by SST. (e) 99% percentile of the bootstrapping as the threshold, with interpolation to the whole grid. (f) Thresholded (d) by (e).
  • Figure 4: UQ of TFR. Panels (a)-(b) and (c)-(d) jointly depict the pointwise 95% confidence intervals of the TFRs obtained from STFT and SST respectively, using $5000$ realizations of the random process model. Panels (a) and (c) show the 2.5% percentile maps, whereas (b) and (d) are the corresponding 97.5% percentiles. Panels (e)-(f) and (g)-(h) present analogous 95% confidence intervals derived from bootstrap resampling of the noise and reconstructed signal (5 000 repetitions). Panels (e) and (g) show the 2.5% percentile maps, and (f) and (h) the 97.5% percentile maps.
  • Figure 5: The rejection rate of the STFT-SCR and SST-SCR over a series of simulated signals, with the signal amplitude $a$ ranging from 0 to 2. The dashed (solid resp.) curves are based on the STFT-SCR (SST-SCR resp.) with different bootstrapping algorithms, where DZ, HK, and Po indicates bootstrapping algorithm from ding2023autoregressive, HafnerKirch2017, and Poskitt2025.
  • ...and 8 more figures

Theorems & Definitions (32)

  • Remark
  • Example
  • Remark
  • Theorem 6.1
  • Lemma 6.1
  • Theorem 6.2
  • Theorem 6.4
  • Theorem 6.5
  • Theorem SI.7.1
  • Lemma SI.7.1: Rosenthal inequality
  • ...and 22 more