Abstract
Time-series analysis is critical for a diversity of applications in science and engineering. By leveraging the strengths of modern gradient descent algorithms, the Fourier transform, multi-resolution analysis, and Bayesian spectral analysis, we propose a data-driven approach to time-frequency analysis that circumvents many of the shortcomings of classic approaches, including the extraction of nonstationary signals with discontinuities in their behavior. The method introduced is equivalent to a {\em nonstationary Fourier mode decomposition} (NFMD) for nonstationary and nonlinear temporal signals, allowing for the accurate identification of instantaneous frequencies and their amplitudes. The method is demonstrated on a diversity of time-series data, including on data from cantilever-based electrostatic force microscopy to quantify the time-dependent evolution of charging dynamics at the nanoscale.
Highlights
Time series data analysis is ubiquitous and foundational in scientific analysis and engineering model design [1]
By integrating elements of modern gradient descent algorithms, the Fourier transform, multi-resolution analysis, and Bayesian spectral analysis [7], we can train an interpretable Fourier modebased model for analyzing nonstationary signals with periodic components, circumventing the challenges normally associated with nonstationary processes and allowing for accurate identification of instantaneous frequencies and their amplitudes
The method results in a superior time-frequency analysis to the Hilbert-Huang transform (HHT) for nonstationary signals, and improves both temporal and spatial resolutions compared to the short time Fourier transform (STFT)
Summary
Time series data analysis is ubiquitous and foundational in scientific analysis and engineering model design [1]. In a typical scientific workflow, observations are made on a system and fit to a time series model, which can include classical methods from statistics, such as ARIMA (autoregressive integrated moving average) and its variants [1], or more recent neural network based approaches [2], [3], such as LSTM [4] (long-term, shortterm memory), GRU (gated recurrent units) [5], and echostate networks [6] These diverse mathematical strategies regress to models fit to historical training data, often making assumptions that the data is generated from a stationary process with Gaussian distributed statistics. By integrating elements of modern gradient descent algorithms, the Fourier transform, multi-resolution analysis, and Bayesian spectral analysis [7], we can train an interpretable Fourier modebased model for analyzing nonstationary signals with periodic components, circumventing the challenges normally associated with nonstationary processes and allowing for accurate identification of instantaneous frequencies and their amplitudes
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