- Date: 29 May. (Thu.)
- Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
- Time: 16:50–18:20
- Speaker : Hau-Tieng Wu (New York University, Courant)
- Title: Quantifying Uncertainty in Time-Frequency Analysis under Nonstationary Noise
- Abstract:
Time-frequency (TF) analysis provides a flexible framework for studying nonstationary time series, with wide applications across the sciences and engineering. Yet in real-world data, noise is ubiquitous and often nonstationary, and our ability to quantify uncertainty in TF representations remains limited. In this talk, we present recent progress on understanding and addressing this challenge. We show that the short-time Fourier transform (STFT) of a broad class of nonstationary noise processes that is defined via filtrations satisfying mild moment and dependence conditions can be approximated in L2 by a Gaussian process. This result arises from a sequential Gaussian approximation theorem, which may be of independent interest. In the presence of signal (the nonnull setup), we prove that the reconstruction formula underlying the synchrosqueezing transform (SST) remains stable under such noise, with uniform error bounds. When the noise satisfies a stronger local stationarity property, we develop a bootstrap procedure with theoretical guarantee to quantify uncertainty in both the STFT and SST, which relies on a time-varying autoregressive approximation of the noise. We conclude by demonstrating the practical value of this framework through an application to airflow signals recorded during sleep.
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