Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space

Abstract

We present a novel method for stochastic interpolation of sparsely sampled time signals based on a superstatistical random process generated from a multivariate Gaussian scale mixture. In comparison to other stochastic interpolation methods such as Gaussian process regression, our method possesses strong multifractal properties and is thus applicable to a broad range of real-world time series, e.g. from solar wind or atmospheric turbulence. Furthermore, we provide a sampling algorithm in terms of a mixing procedure that consists of generating a 1 + 1-dimensional field u(t, {\xi}), where each Gaussian component u{\xi}(t) is synthesized with identical underlying noise but different covariance function C{\xi}(t,s) parameterized by a log-normally distributed parameter {\xi}. Due to the Gaussianity of each component u{\xi}(t), we can exploit standard sampling alogrithms such as Fourier or wavelet methods and, most importantly, methods to constrain the process on the sparse measurement points. The scale mixture u(t) is then initialized by assigning each point in time t a {\xi}(t) and therefore a specific value from u(t, {\xi}), where the time-dependent parameter {\xi}(t) follows a log-normal process with a large correlation time scale compared to the correlation time of u(t, {\xi}). We juxtapose Fourier and wavelet methods and show that a multiwavelet-based hierarchical approximation of the interpolating paths, which produce a sparse covariance structure, provide an adequate method to locally interpolate large and sparse datasets.