kramersmoyal: Kramers--Moyal coefficients for stochastic processes

Abstract

A general problem for evaluating Markovian stochastic processes is the retrieval of the moments or the Kramers-Moyal coefficients M from data or time-series. The Kramers-Moyal coefficients are derived from an Taylor expansion of the master equation that describes the probability evolution of a Markovian stochastic process.Given a set of stochastic data, ergodic or quasi-stationary, the extensive literature of stochastic processes awards a set of measures, such as the Kramers-Moyal coefficients or its moments, which link stochastic processes to a probabilistic description of the process or of the family of processes (Risken, 1996). Most commonly known is the Fokker-Planck equation or truncated forward Kolmogorov equation, partial differential equations, obtained from the Taylor expansion of the master equation.Of particular relevance is the growing evidence that real-world data displays higher-order (n > 2) Kramers-Moyal coefficients, which has a two-fold consequence: The common truncation at third order of the forward Kolmogorov equation, giving rise to the Fokker-Planck equation, is no longer valid. The existence of higher-order (n > 2) Kramers-Moyal coefficients in recorded data thus invalidates the aforementioned common argument for truncation, thus rendering the Fokker-Planck description insufficient (Tabar, 2019). A clear and common example is the presence of discontinuous jumps in data (Aït-Sahalia, 2002;Anvari, Tabar, Peinke, & Lehnertz, 2016), which can give rise to higher-order Kramers-Moyal coefficients, as are evidenced in Gorjão, Heysel, Lehnertz, & Tabar (2019) and references within.Calculating the moments or Kramers-Moyal coefficients strictly from data can be computationally heavy for long data series and is prone to innaccuracy especially where the density of data points is scarce, for example, usually at the boundaries on the domain of the process. The most straightforward approach is to perform a histogram-based estimation to evaluate the moments of the system at hand. This has two main drawbacks: it requires a discrete space of examination of the process and is shown to be less accurate than using kernel-based estimators (Lamouroux & Lehnertz, 2009). This library is based on a kernel-based estimation, i.e., the Nadaraya-Watson kernel estimator (Nadaraya, 1964;Watson, 1964), which allows for more robust results given both a wider range of possible kernel shapes to perform the calculation, as well as retrieving the results in a nonbinned coordinate space, unlike histogram regressions (Silverman, 2018). It further employs a convolution of the time series with the selected kernel, circumventing the computational issue of sequential array summation, the most common bottleneck in integration time and computer memory.

Non-Markovian modeling of protein folding

“…For the underdamped LE, the relation between the second-order velocity KMC and the friction profile reads ( 7 ) For the overdamped LE, follows from the second-order position KMC as ( SI Appendix , section 9 ). For the numerical computation of the KMCs, we use kernel-density estimators ( 45 ) ( SI Appendix , section 13 ). In Fig.…”

Open data base analysis of scaling and spatio-temporal properties of power grid frequencies

“…A more detailed derivation is provided in Supplementary Note 2 and discussed in [38,39]. And a technical discussion of extracting the aggregated noise amplitude is presented in [45]. We note that the scaling law has to be modified if the noise at the nodes is not Gaussian [38].…”

Change of Persistence in European Electricity Spot Prices

“…In principle, the Kramers-Moyal coefficients are defined in the limit τ → 0. In practical applications the resolution is limited, such that we resort to the smallest increments in the data set [72][73][74].…”