jumpdiff

jumpdiff is a python library with non-parametric Nadaraya─Watson estimators to extract the parameters of jump-diffusion processes. With jumpdiff one can extract the parameters of a jump-diffusion process from one-dimensional timeseries, employing both a kernel-density estimation method combined with a set on second-order corrections for a precise retrieval of the parameters for short timeseries.

Installation

To install jumpdiff, run

   pip install jumpdiff

Then on your favourite editor just use

   import jumpdiff as jd

Dependencies

The library parameter estimation depends on numpy and scipy solely. The mathematical formulae depend on sympy. It stems from kramersmoyal project, but functions independently from it³.

Documentation

You can find the documentation here.

Jump-diffusion processes

The theory

Jump-diffusion processes¹, as the name suggest, are a mixed type of stochastic processes with a diffusive and a jump term. One form of these processes which is mathematically traceable is given by the Stochastic Differential Equation

which has 4 main elements: a drift term , a diffusion term , and jump amplitude term , which is given by a Gaussian distribution, and finally a jump rate . You can find a good review on this topic in Ref. 2.

Integrating a jump-diffusion process

Let us use the functions in jumpdiff to generate a jump-difussion process, and subsequently retrieve the parameters. This is a good way to understand the usage of the integrator and the non-parametric retrieval of the parameters.

First we need to load our library. We will call it jd

import jumpdiff as jd

Let us thus define a jump-diffusion process and use jd_process to integrate it. Do notice here that we need the drift and diffusion as functions.

# integration time and time sampling
t_final = 10000
delta_t = 0.001

# A drift function
def a(x):
    return -0.5*x

# and a (constant) diffusion term
def b(x):
    return 0.75

# Now define a jump amplitude and rate
xi = 2.5
lamb = 1.75

# and simply call the integration function
X = jd.jd_process(t_final, delta_t, a=a, b=b, xi=xi, lamb=lamb)

This will generate a jump diffusion process X of length int(10000/0.001) with the given parameters.

Using jumpdiff to retrieve the parameters

Moments and Kramers─Moyal coefficients

Take the timeseries X and use the function moments to retrieve the conditional moments of the process. For now let us focus on the shortest time lag, so we can best approximate the Kramers─Moyal coefficients. For this case we can simply employ

edges, moments = jd.moments(timeseries = X)

In the array edges are the limits of our space, and in our array moments are recorded all 6 powers/order of our conditional moments. Let us take a look at these before we proceed, to get acquainted with them.

We can plot the first moment with any conventional plotter, so lets use here plotly from matplotlib

import matplotlib.plotly as plt

# we want the first power, so we need 'moments[1,...]'
plt.plot(edges, moments[1,...])

The first moment here (i.e., the first Kramers─Moyal coefficient) is given solely by the drift term that we have selected -0.5*x

And the second moment (i.e., the second Kramers─Moyal coefficient) is a mixture of both the contributions of the diffusive term and the jump terms and .

You have this stored in moments[2,...].

Retrieving the jump-related terms

Naturally one of the most pertinent questions when addressing jump-diffusion processes is the possibility of recovering these same parameters from data. For the given jump-diffusion process we can use the jump_amplitude and jump_rate functions to non-parametrically estimate the jump amplitude and jump rate terms.

After having the moments in hand, all we need is

# first estimate the jump amplitude
xi_est = jd.jump_amplitude(moments = moments)

# and now estimated the jump rate
lamb_est = jd.jump_rate(moments = moments)

which resulted in our case in (xi_est) ξ = 2.43 ± 0.17 and (lamb_est) λ = 1.744 * delta_t (don't forget to divide lamb_est by delta_t)!

Other functions and options

Include in this package is also the Milstein scheme of integration, particularly important when the diffusion term has some spacial x dependence. moments can actually calculate the conditional moments for different lags, using the parameter lag.

In formulae the set of formulas needed to calculate the second order corrections are given (in sympy).

Contributions

We welcome reviews and ideas from everyone. If you want to share your ideas, upgrades, doubts, or simply report a bug, open an issue here on GitHub, or contact us directly. If you need help with the code, the theory, or the implementation, drop us an email. We abide to a Conduct of Fairness.

Changelog

• Version 0.4 - Designing a set of self-consistency checks, the documentation, examples, and a trial code. Code at PyPi.
• Version 0.3 - Designing a straightforward procedure to retrieve the jump amplitude and jump rate functions, alongside with a easy sympy displaying the correction.
• Version 0.2 - Introducing the second-order corrections to the moments
• Version 0.1 - Design an implementation of the moments functions, generalising kramersmoyal km.

Literature and Support

History

This project was started in 2017 at the neurophysik by Leonardo Rydin Gorjão, Jan Heysel, Klaus Lehnertz, and M. Reza Rahimi Tabar, and separately by Pedro G. Lind, at the Department of Computer Science, Oslo Metropolitan University. From 2019 to 2021, Pedro G. Lind, Leonardo Rydin Gorjão, and Dirk Witthaut developed a set of corrections and an implementation for python, presented here.

Funding

Helmholtz Association Initiative Energy System 2050 - A Contribution of the Research Field Energy and the grant No. VH-NG-1025 and STORM - Stochastics for Time-Space Risk Models project of the Research Council of Norway (RCN) No. 274410.

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Bibliography

¹ Tabar, M. R. R. Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Springer, International Publishing (2019), Chapter Stochastic Processes with Jumps and Non-vanishing Higher-Order Kramers–Moyal Coefficients.

² Friedrich, R., Peinke, J., Sahimi, M., Tabar, M. R. R. Approaching complexity by stochastic methods: From biological systems to turbulence, Physics Reports 506, 87–162 (2011).

³ Rydin Gorjão, L., Meirinhos, F. kramersmoyal: Kramers–Moyal coefficients for stochastic processes. Journal of Open Source Software, 4(44) (2019).

Extended Literature

You can find further reading on SDE, non-parametric estimatons, and the general principles of the Fokker–Planck equation, Kramers–Moyal expansion, and related topics in the classic (physics) books

• Risken, H. The Fokker–Planck equation. Springer, Berlin, Heidelberg (1989).
• Gardiner, C.W. Handbook of Stochastic Methods. Springer, Berlin (1985).

And an extensive review on the subject here