🤖 AI Summary
Nonparametric spectral density estimation for continuous-time processes observed under irregular spatiotemporal sampling remains challenging due to aliasing and the severe ill-posedness of conventional nonuniform Fourier inversion.
Method: This paper proposes the Weighted Nonuniform Fourier Sum (WUNFS) estimator, which introduces a high-accuracy adaptive window function to construct a nonuniform Fourier quadrature framework—thereby suppressing aliasing at its source and circumventing the ill-conditioning inherent in standard nonuniform Fourier inversion.
Contribution/Results: We derive a theoretical bias bound for WUNFS and demonstrate its natural extensibility to multivariate settings. Experiments show that WUNFS significantly outperforms both the periodogram and the Lomb–Scargle periodogram (LSP), especially for spectra with slow decay and under multidimensional irregular sampling, achieving substantial gains in estimation accuracy and robustness.
📝 Abstract
We introduce a nonparametric spectral density estimator for continuous-time and continuous-space processes measured at fully irregular locations. Our estimator is constructed using a weighted nonuniform Fourier sum whose weights yield a high-accuracy quadrature rule with respect to a user-specified window function. The resulting estimator significantly reduces the aliasing seen in periodogram approaches and least squares spectral analysis, sidesteps the dangers of ill-conditioning of the nonuniform Fourier inverse problem, and can be adapted to a wide variety of irregular sampling settings. After a discussion of methods for computing the necessary weights and a theoretical analysis of sources of bias, we close with demonstrations of the method's efficacy, including for processes that exhibit very slow spectral decay and for processes in multiple dimensions.