Langevin

CRAN: Langevin citation info To cite the 'Langevin' package in publications use: Rinn P, Lind P, Wächter M, Peinke J (2016). “The Langevin Approach: An R Package for Modeling Markov Processes.” Journal of Open Research Software , 4 (1), e34. ISSN 2049-9647, doi:10.5334/jors.123 . To cite the 'Langevin Approach' in publications use: Friedrich R, Peinke J, Sahimi M, Reza Rahimi Tabar M (2011). “Approaching complexity by stochastic methods: From biological systems to turbulence.” Physics Reports , 506 (5), 87–162. ISSN 0370-1573, doi:10.1016/j.physrep.2011.05.003 . Corresponding BibTeX entries: @Article{Rinn2016, title = {The Langevin Approach: An R Package for Modeling Markov Processes}, author = {Philip Rinn and Pedro G. Lind and Matthias Wächter and Joachim Peinke}, journal = {Journal of Open Research Software}, year = {2016}, number = {1}, pages = {e34}, volume = {4}, doi = {10.5334/jors.123}, issn = {2049-9647}, publisher = {Ubiquity Press}, } @Article{Friedrich2011, title = {Approaching complexity by stochastic methods: From biological systems to turbulence}, author = {Rudolf Friedrich and Joachim Peinke and Muhammad Sahimi and Mohammed {Reza Rahimi Tabar}}, journal = {Physics Reports}, year = {2011}, number = {5}, pages = {87--162}, volume = {506}, doi = {10.1016/j.physrep.2011.05.003}, issn = {0370-1573}, publisher = {Elsevier BV}, }

README code{white-space: pre-wrap;} span.smallcaps{font-variant: small-caps;} span.underline{text-decoration: underline;} div.column{display: inline-block; vertical-align: top; width: 50%;} div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;} ul.task-list{list-style: none;} pre > code.sourceCode { white-space: pre; position: relative; } pre > code.sourceCode > span { display: inline-block; line-height: 1.25; } pre > code.sourceCode > span:empty { height: 1.2em; } .sourceCode { overflow: visible; } code.sourceCode > span { color: inherit; text-decoration: inherit; } div.sourceCode { margin: 1em 0; } pre.sourceCode { margin: 0; } @media screen { div.sourceCode { overflow: auto; } } @media print { pre > code.sourceCode { white-space: pre-wrap; } pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; } } pre.numberSource code { counter-reset: source-line 0; } pre.numberSource code > span { position: relative; left: -4em; counter-increment: source-line; } pre.numberSource code > span > a:first-child::before { content: counter(source-line); position: relative; left: -1em; text-align: right; vertical-align: baseline; border: none; display: inline-block; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; padding: 0 4px; width: 4em; color: #aaaaaa; } pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; } div.sourceCode { } @media screen { pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; } } code span.al { color: #ff0000; font-weight: bold; } /* Alert */ code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */ code span.at { color: #7d9029; } /* Attribute */ code span.bn { color: #40a070; } /* BaseN */ code span.bu { color: #008000; } /* BuiltIn */ code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */ code span.ch { color: #4070a0; } /* Char */ code span.cn { color: #880000; } /* Constant */ code span.co { color: #60a0b0; font-style: italic; } /* Comment */ code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */ code span.do { color: #ba2121; font-style: italic; } /* Documentation */ code span.dt { color: #902000; } /* DataType */ code span.dv { color: #40a070; } /* DecVal */ code span.er { color: #ff0000; font-weight: bold; } /* Error */ code span.ex { } /* Extension */ code span.fl { color: #40a070; } /* Float */ code span.fu { color: #06287e; } /* Function */ code span.im { color: #008000; font-weight: bold; } /* Import */ code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */ code span.kw { color: #007020; font-weight: bold; } /* Keyword */ code span.op { color: #666666; } /* Operator */ code span.ot { color: #007020; } /* Other */ code span.pp { color: #bc7a00; } /* Preprocessor */ code span.sc { color: #4070a0; } /* SpecialChar */ code span.ss { color: #bb6688; } /* SpecialString */ code span.st { color: #4070a0; } /* String */ code span.va { color: #19177c; } /* Variable */ code span.vs { color: #4070a0; } /* VerbatimString */ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */ Langevin Analysis in One and Two Dimensions The Langevin package provides R functions to estimate drift and diffusion functions from time series and generate synthetic time series from given drift and diffusion coefficients. Documentation All functions of the Langevin package have corresponding help files. Additionally the package ships a pdf vignette which corresponds to a paper published in the Journal of Open Research Software. Citation To cite the Langevin package and/or the mathematical concept behind it correctly see citation("Langevin") for details. Examples The help files for each function contain usage examples, additionally the package repository ships a script with examples that reproduce the figures from the vignette. Installation Released and tested versions of the Langevin package are available at CRAN and can be installed from within R via install.packages ( "Langevin" ) The development version of the Langevin package can be installed from within R via install.packages ( "devtools" ) devtools :: install_git ( "https://gitlab.uni-oldenburg.de/TWiSt/Langevin.git" ) Authors Philip Rinn, Pedro G. Lind and David Bastine License GPL (>= 2)

Help for package Langevin const macros = { "\\R": "\\textsf{R}", "\\mbox": "\\text", "\\code": "\\texttt"}; function processMathHTML() { var l = document.getElementsByClassName('reqn'); for (let e of l) { katex.render(e.textContent, e, { throwOnError: false, macros }); } return; } Package {Langevin} Contents Langevin-package Langevin1D Langevin2D plot.Langevin print.Langevin summary.Langevin timeseries1D timeseries2D Type: Package Title: Langevin Analysis in One and Two Dimensions Version: 1.3.3 Date: 2025-10-12 Description: Estimate drift and diffusion functions from time series and generate synthetic time series from given drift and diffusion coefficients. Encoding: UTF-8 License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] URL: https://gitlab.uni-oldenburg.de/TWiSt/Langevin LazyLoad: yes ByteCompile: yes NeedsCompilation: yes Depends: R (≥ 3.3.0) Imports: Rcpp (≥ 1.0.12) LinkingTo: Rcpp, RcppArmadillo (≥ 15.0.2-2) RoxygenNote: 7.3.1 Packaged: 2025-10-12 17:27:29 UTC; philip Author: Philip Rinn [aut, cre], Pedro G. Lind [aut], David Bastine [ctb] Maintainer: Philip Rinn <philip.rinn@uni-oldenburg.de> Repository: CRAN Date/Publication: 2025-10-12 18:30:02 UTC An R package for stochastic data analysis Description The Langevin package provides functions to estimate drift and diffusion functions from data sets. Details This package was developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg (Germany). Mathematical Background A wide range of dynamic systems can be described by a stochastic differential equation, the Langevin equation. The time derivative of the system trajectory \dot{X}(t) can be expressed as a sum of a deterministic part D^{(1)} and the product of a stochastic force \Gamma(t) and a weight coefficient D^{(2)} . The stochastic force \Gamma(t) is \delta -correlated Gaussian white noise. For stationary continuous Markov processes Siegert et al. and Friedrich et al. developed a method to reconstruct drift D^{(1)} and diffusion D^{(2)} directly from measured data. \dot{X}(t) = D^{(1)}(X(t),t) + \sqrt{D^{(2)}(X(t),t)}\,\Gamma(t)\quad \mathrm{with} D^{(n)}(x,t) = \lim_{\tau \rightarrow 0} \frac{1}{\tau} M^{(n)}(x,t,\tau)\quad \mathrm{and} M^{(n)}(x,t,\tau) = \frac{1}{n!} \langle (X(t+\tau) - x)^n \rangle |_{X(t) = x} The Langevin equation should be interpreted in the way that for every time t_i where the system meets an arbitrary but fixed point x in phase space, X(t_i+\tau) is defined by the deterministic function D^{(1)}(x) and the stochastic function \sqrt{D^{(2)}(x)}\Gamma(t_i) . Both, D^{(1)}(x) and D^{(2)}(x) are constant for fixed x . One can integrate drift and diffusion numerically over small intervals. If the system is at time t in the state x = X(t) the drift can be calculated for small \tau by averaging over the difference of the system state at t+\tau and the state at t . The average has to be taken over the whole ensemble or in the stationary case over all t = t_i with X(t_i) = x . Diffusion can be calculated analogously. Author(s) Philip Rinn References A review of the Langevin method can be found at: Friedrich, R.; et al. (2011) Approaching Complexity by Stochastic Methods: From Biological Systems to Turbulence . Physics Reports, 506(5), 87–162. For further reading: Risken, H. (1996) The Fokker-Planck equation . Springer. Siegert, S.; et al. (1998) Analysis of data sets of stochastic systems . Phys. Lett. A. Friedrich, R.; et al. (2000) Extracting model equations from experimental data . Phys. Lett. A. Honisch, C.; Friedrich, R. (2011). Estimation of Kramers-Moyal coefficients at low sampling rates. . Physical Review E, 83(6), 066701. See Also Useful links: https://gitlab.uni-oldenburg.de/TWiSt/Langevin Calculate the Drift and Diffusion of one-dimensional stochastic processes Description Langevin1D calculates the Drift and Diffusion vectors (with errors) for a given time series. Usage Langevin1D( data, bins, steps, sf = ifelse(is.ts(data), frequency(data), 1), bin_min = 100, reqThreads = -1, kernel = FALSE, h ) Arguments data a vector containing the time series or a time-series object. bins a scalar denoting the number of bins to calculate the conditional moments on. steps a vector giving the \tau steps to calculate the conditional moments (in samples (= \tau * sf )). Only used if kernel is FALSE . sf a scalar denoting the sampling frequency (optional if data is a time-series object). bin_min a scalar denoting the minimal number of events per bin . Defaults to 100 . reqThreads a scalar denoting how many threads to use. Defaults to -1 which means all available cores. Only used if kernel is FALSE . kernel a logical denoting if the kernel based Nadaraya-Watson estimator should be used to calculate drift and diffusion vectors. h a scalar denoting the bandwidth of the data. Defaults to Scott's variation of Silverman's rule of thumb. Only used if kernel is TRUE . Value Langevin1D returns a list with thirteen (six if kernel is TRUE ) components: D1 a vector of the Drift coefficient for each bin . eD1 a vector of the error of the Drift coefficient for each bin . D2 a vector of the Diffusion coefficient for each bin . eD2 a vector of the error of the Diffusion coefficient for each bin . D4 a vector of the fourth Kramers-Moyal coefficient for each bin . mean_bin a vector of the mean value per bin . density a vector of the number of events per bin . If kernel is FALSE . M1 a matrix of the first conditional moment for each \tau . Rows correspond to bin , columns to \tau . If kernel is FALSE . eM1 a matrix of the error of the first conditional moment for each \tau . Rows correspond to bin , columns to \tau . If kernel is FALSE . M2 a matrix of the second conditional moment for each \tau . Rows correspond to bin , columns to \tau . If kernel is FALSE . eM2 a matrix of the error of the second conditional moment for each \tau . Rows correspond to bin , columns to \tau . If kernel is FALSE . M4 a matrix of the fourth conditional moment for each \tau . Rows correspond to bin , columns to \tau . If kernel is FALSE . U a vector of the bin borders. If kernel is FALSE . Author(s) Philip Rinn See Also Langevin2D Examples # Set number of bins, steps and the sampling frequency bins <- 20 steps <- c(1:5) sf <- 1000 #### Linear drift, constant diffusion # Generate a time series with linear D^1 = -x and constant D^2 = 1 x <- timeseries1D(N = 1e6, d11 = -1, d20 = 1, sf = sf) # Do the analysis est <- Langevin1D(data = x, bins = bins, steps = steps, sf = sf) # Plot the result and add the theoretical expectation as red line plot(est$mean_bin, est$D1) lines(est$mean_bin, -est$mean_bin, col = "red") plot(est$mean_bin, est$D2) abline(h = 1, col = "red") #### Cubic drift, constant diffusion # Generate a time series with cubic D^1 = x - x^3 and constant D^2 = 1 x <- timeseries1D(N = 1e6, d13 = -1, d11 = 1, d20 = 1, sf = sf) # Do the analysis est <- Langevin1D(data = x, bins = bins, steps = steps, sf = sf) # Plot the result and add the theoretical expectation as red line plot(est$mean_bin, est$D1) lines(est$mean_bin, est$mean_bin - est$mean_bin^3, col = "red") plot(est$mean_bin, est$D2) abline(h = 1, col = "red") Calculate the Drift and Diffusion of two-dimensional stochastic processes Description Langevin2D calculates the Drift (with error) and Diffusion matrices for given time series. Usage Langevin2D( data, bins, steps, sf = ifelse(is.mts(data), frequency(data), 1), bin_min = 100, reqThreads = -1 ) Arguments data a matrix containing the time series as columns or a time-series object. bins a scalar denoting the number of bins to calculate Drift and Diffusion on. steps a vector giving the \tau steps to calculate the moments (in samples). sf a scalar denoting the sampling frequency (optional if data is a time-series object). bin_min a scalar denoting the minimal number of events per bin . Defaults to 100 . reqThreads a scalar denoting how many threads to use. Defaults to -1 which mea