Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space
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 s...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2023-01-01
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| Series: | Journal of Physics: Complexity |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/2632-072X/acb128 |
| _version_ | 1797844584610398208 |
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| author | Jeremiah Lübke Jan Friedrich Rainer Grauer |
| author_facet | Jeremiah Lübke Jan Friedrich Rainer Grauer |
| author_sort | Jeremiah Lübke |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-09T17:25:41Z |
| format | Article |
| id | doaj.art-30208537cebb4cf692c9c5817d9bdfa8 |
| institution | Directory Open Access Journal |
| issn | 2632-072X |
| language | English |
| last_indexed | 2024-04-09T17:25:41Z |
| publishDate | 2023-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Journal of Physics: Complexity |
| spelling | doaj.art-30208537cebb4cf692c9c5817d9bdfa82023-04-18T13:51:39ZengIOP PublishingJournal of Physics: Complexity2632-072X2023-01-014101500510.1088/2632-072X/acb128Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet spaceJeremiah Lübke0https://orcid.org/0000-0001-6338-9728Jan Friedrich1https://orcid.org/0000-0002-9862-6268Rainer Grauer2https://orcid.org/0000-0003-0622-071XInstitute for Theoretical Physics I, Ruhr-University Bochum , Universitätsstr. 150, Bochum 44801, GermanyForWind, Institute of Physics, University of Oldenburg , Küpkersweg 70, 26129 Oldenburg, GermanyInstitute for Theoretical Physics I, Ruhr-University Bochum , Universitätsstr. 150, Bochum 44801, GermanyWe 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.https://doi.org/10.1088/2632-072X/acb128conditional simulationintermittencymultifractalsuperstatisticscirculant embeddingmultiwavelets |
| spellingShingle | Jeremiah Lübke Jan Friedrich Rainer Grauer Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space Journal of Physics: Complexity conditional simulation intermittency multifractal superstatistics circulant embedding multiwavelets |
| title | Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space |
| title_full | Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space |
| title_fullStr | Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space |
| title_full_unstemmed | Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space |
| title_short | Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space |
| title_sort | stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in fourier and wavelet space |
| topic | conditional simulation intermittency multifractal superstatistics circulant embedding multiwavelets |
| url | https://doi.org/10.1088/2632-072X/acb128 |
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