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.
<p>The transport of cosmic rays in turbulent magnetic fields is commonly investigated by solving the Newton-Lorentz equation of test particles in synthetic turbulence fields. These fields are typically generated from superpositions of Fourier modes with prescribed power spectrum and uncorrelated random phases, bringing the advantage of covering a wide range of turbulence scales at manageable computational effort. However, almost all of these models to date only account for second-order Gaussian statistics and thus fail to include intermittent features. Recent observations of the solar wind suggest that astrophysical magnetic fields are strongly non-Gaussian, and the question of how such higher-order statistics impact cosmic ray transport has only received limited attention. To address this, we present an algorithm for generating synthetic turbulence based on Kolmogorov&#8217;s log-normal model of intermittency. It generates a divergence-free magnetic field by computing the curl of a vector potential, which in turn is obtained from an inverse wavelet transform of a continuous log-normal cascade process. We investigate the statistics of the generated fields, show that anomalous scaling properties are accurately reproduced and discuss implications on cosmic ray transport. *Supported by DFG (SFB 1491)</p>
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, ξ), where each Gaussian component u ξ (t) is synthesized with identical underlying noise but different covariance function C ξ (t, s) parameterized by a log-normally distributed parameter ξ. Due to the Gaussianity of each component u ξ (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 ξ(t) and therefore a specific value from u(t, ξ), where the time-dependent parameter ξ(t) follows a log-normal process with a large correlation time scale compared to the correlation time of u(t, ξ). 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.
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