Abstract
In this chapter we present the steps of reconstruction procedure for writing down the Langevin and jump diffusion stochastic dynamical equations for uni- and bivariate time series, sampled with time intervals \(\tau \).
[Type][CrossLinking]The original version of this chapter was revised: Belated correction has been incorporated. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-18472-8_24
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The finite \(\tau \) expansions of the KM conditional moments \(K^{(1)}_i\) and \(K^{(2)}_{ij}\) with known \(D_i^{(1)}(\mathbf{x},t)\) and \(D_{ij}^{(2)}(\mathbf{x},t)\) are
$$\begin{aligned} K^{(1)}_i= & {} \tau D^{(1)}_i + \frac{\tau ^2}{2} \left( \sum _k D^{(1)}_k \partial _{x_k} D^{(1)}_i \right. \\&\quad \left. + \sum _{kl} D^{(2)}_{kl} \partial _{x_l}\partial _{x_k} D^{(1)}_i\right) + \mathcal {O}(\tau ^3) \end{aligned}$$$$\begin{aligned} K^{(2)}_{ij}= & {} 2\tau D^{(2)}_{ij} + \ \tau ^2 \Bigg [ D^{(1)}_i D^{(1)}_j \\&\quad \quad + \sum _k \left( D^{(2)}_{jk} \partial _{x_k} D^{(1)}_i + D^{(2)}_{ik} \partial _{x_k} D^{(1)}_j \right) \\&\quad \quad + \sum _k D^{(1)}_k \partial _{x_k} D^{(2)}_{ij} + \sum _{kl} D^{(2)}_{kl} \partial _{x_l}\partial _{x_k} D^{(2)}_{ij} \Bigg ] \\&+ \ \mathcal {O}(\tau ^3). \end{aligned}$$where
$$\begin{aligned} K^{(1)}_i(\mathbf{x},t,\tau ) = \left\langle \left[ \mathbf{x}(t+\tau ) - \mathbf{x}(t)\right] _i \right\rangle |_{\mathbf{x}(t) = \mathbf{x}} \end{aligned}$$$$\begin{aligned} K^{(2)} _{ij}(\mathbf{x},t,\tau ) = \left\langle \left[ \mathbf{x}(t+\tau ) - \mathbf{x}(t)\right] _i \left[ \mathbf{x}(t+\tau ) - \mathbf{x}(t)\right] _j \right\rangle |_{\mathbf{x}(t) = \mathbf{x}} \end{aligned}$$where, \(\partial _{x_i} = \partial / \partial x_i\).
- 2.
with \(a^2 - 2 a b + b^2 + 4 c^2 \ne 0\), \(ab-c^2 \ge 0\) and \(c\ne 0\). There are 9 other real solutions for h, l amd k, depending on the values of a, b and c.
- 3.
Now let us consider a discrete stationary process x(t), with unit Markov–Einstein time scale \(t_M=1\) (in units of the data lag). If k is the number of bins needed to precisely represent, or evaluate, the PDF of x(t), the data are partitioned into k bins, with each bin having an equal number of data points. Each bin is represented by a node of the equivalent complex network of the series, with k being the number of the nodes in the network. Nodes i and j are linked if as the time increases the value of x(t) in bin i changes to that in bin j in one time step. A weight \(w_{ij}\) is attributed to a link ij, which is the number of times that a given value of x(t) changes from its value in bin i to bin j in one time step, and is normalized at each node. The transition matrix \(\mathbf{W}=[w_{ij}]\) is, in general, not symmetric. Thus, the network is both directed and weighted. For some time series with a finite Markov–Einstein time scale \(t_M>1\), one can construct the transition matrix with entries given by \(p[x(t)|x(t-t_M)]\), and attribute to each node a set of arrays of data with length \(t_M\), see [16] for more details.
References
L. Rydin Gorjão, F. Meirinhos, Python KM, a Python package to calculate Kramers–Moyal coefficients, https://github.com/LRydin/KramersMoyal
J. Prusseit, K. Lehnertz, Phys. Rev. E 77, 041914 (2008)
H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989)
F. Lenz, Statistical analysis and stochastic modelling of foraging Bumblebees, Doctoral thesis, Queen Mary University of London
A. Bahraminasab, D. Kenwright, A. Stefanovska, F. Ghasemi, P.V.E. McClintock, IET Syst. Biol. 2, 48 (2008)
N. Schaudinnus, Stochastic modeling of biomolecular systems using the data-driven Langevin equation, Doctoral thesis, Albert-Ludwigs-Universität
F. Nikakhtar, M. Ayromlou, S. Baghram, S. Rahvar, M.R. Rahimi Tabar, R.K. Sheth, Mon. Not. R. Astron. Soc. 478, 5296 (2018)
M. Anvari, K. Lehnertz, M.R. Rahimi Tabar, J. Peinke, Sci. Rep. 6, 35435 (2016)
L.R. Gorjão, J. Heysel, K. Lehnertz, M.R. Rahimi Tabar, arXiv:1907.05371
J. Peinke, M.R. Rahimi Tabar, M. Wächter, Annu. Rev. Condens. Matter Phys. 10 (2019)
A. Nawroth, J. Peinke, Phys. Lett. A 360, 234 (2006)
R. Stresing, J. Peinke, New J. Phys. 12, 103046 (2010)
A. Nawroth, R. Friedrich, J. Peinke, New J. Phys. 12, 083021 (2010)
A. Hadjihosseini, M. Wächter, N.P. Hoffmann, J. Peinke, New J. Phys. 18, 013017 (2016)
A.H. Shirazi, G.R. Jafari, J. Davoudi, J. Peinke, M.R. Rahimi Tabar, M. Sahimi, J. Stat. Mech. P07046 (2009)
J. Zhang, M. Small, Phys. Rev. Lett. 96, 238701 (2006)
A. Shreim, P. Grassberger, W. Nadler, B. Samuelsson, J.E.S. Socolar, M. Paczuski, Phys. Rev. Lett. 98, 198701 (2007)
L. Lacasa, B. Luque, F. Ballesteros, J. Luque, J.C. Nu\(\tilde{n}\)o, Proc. Nat. Acad. Sci. 105, 4972 (2008)
P. Manshour, M.R. Rahimi Tabar, J. Peinke, J. Stat. Mech. 8, P08031 (2015)
R.-Q. Su, W.-X. Wang, X. Wang, Y.-C. Lai, R. Soc. Open Sci. 3, 150577 (2016)
Z. Gao, M. Small, J. Kurths, Europhys. Lett. 116, 50001 (2016)
M. Jiang, X. Gao, H. An, H. Li, B. Sun, Sci. Rep. 7, 10486 (2017)
A. Campanharo, M. Sirer, R. Malmgren, F. Ramos, L. Nunes Amaral, PLoS ONE 6, e23378 (2011)
S. Karimi, A.H. Darooneh, Phys. A 392, 287 (2013)
A.M. Nuñez, L. Lacasa, J.P. Gomez, B. Luque, Visibility algorithms: a short review, in New Frontiers in Graph Theory ed. by Y. Zhang (InTech, Rijeka, Croatia, 2012), pp. 119–152
C. Ma, H.S. Chen, Y.C. Lai, H.F. Zhang, Phys. Rev. E 97, 022301 (2018)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Tabar, M.R.R. (2019). Reconstruction Procedure for Writing Down the Langevin and Jump-Diffusion Dynamics from Empirical Uni- and Bivariate Time Series. In: Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-18472-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-18472-8_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-18471-1
Online ISBN: 978-3-030-18472-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)