Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator
With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the differ...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/5/517 |
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| author | Leonardo Rydin Gorjão Dirk Witthaut Klaus Lehnertz Pedro G. Lind |
| author_facet | Leonardo Rydin Gorjão Dirk Witthaut Klaus Lehnertz Pedro G. Lind |
| author_sort | Leonardo Rydin Gorjão |
| collection | DOAJ |
| description | With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data. |
| first_indexed | 2024-03-10T12:00:37Z |
| format | Article |
| id | doaj.art-257cf28df669423f809dd2ccd3ee34aa |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T12:00:37Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-257cf28df669423f809dd2ccd3ee34aa2023-11-21T16:57:11ZengMDPI AGEntropy1099-43002021-04-0123551710.3390/e23050517Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal OperatorLeonardo Rydin Gorjão0Dirk Witthaut1Klaus Lehnertz2Pedro G. Lind3Forschungszentrum Jülich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, GermanyForschungszentrum Jülich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, GermanyDepartment of Epileptology, University Hospital Bonn, Venusberg Campus 1, 53127 Bonn, GermanyDepartment of Computer Science, OsloMet—Oslo Metropolitan University, P.O. Box 4 St. Olavs plass, N-0130 Oslo, NorwayWith the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.https://www.mdpi.com/1099-4300/23/5/517stochastic processesKramers–Moyal equationKramers–Moyal coefficientsFokker–Planck equationarbitrary-order approximationsnon-parametric estimators |
| spellingShingle | Leonardo Rydin Gorjão Dirk Witthaut Klaus Lehnertz Pedro G. Lind Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator Entropy stochastic processes Kramers–Moyal equation Kramers–Moyal coefficients Fokker–Planck equation arbitrary-order approximations non-parametric estimators |
| title | Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator |
| title_full | Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator |
| title_fullStr | Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator |
| title_full_unstemmed | Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator |
| title_short | Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator |
| title_sort | arbitrary order finite time corrections for the kramers moyal operator |
| topic | stochastic processes Kramers–Moyal equation Kramers–Moyal coefficients Fokker–Planck equation arbitrary-order approximations non-parametric estimators |
| url | https://www.mdpi.com/1099-4300/23/5/517 |
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