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 ξ . Due to the Gaussianity of each component $u_\xi(t)$ , we can exploit standard sampling algorithms 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.

Keywords

• conditional simulation
• intermittency
• multifractal
• superstatistics
• circulant embedding
• multiwavelets