kramersmoyal is a python package designed to obtain the Kramers–Moyal coefficients, or conditional moments, from stochastic data of any dimension. It employs kernel density estimations, instead of a histogram approach, to ensure better results for low number of points as well as allowing better fitting of the results.

The paper is now officially published on JOSS. The paper is also available here, or you can find it in the ArXiv.

To install kramersmoyal, just use pip

pip install kramersmoyal

Then on your favourite editor just use

from kramersmoyal import km

The library depends on numpy and scipy.

A Jupyter notebook with this example can be found here

Take, for example, the well-documented one-dimension Ornstein–Uhlenbeck process, also known as Vašíček process, see here. This process is governed by two main parameters: the mean-reverting parameter θ and the diffusion parameter σ

which can be solved in various ways. For our purposes, recall that the drift coefficient, i.e., the first-order Kramers–Moyal coefficient, is given by and the second-order Kramers–Moyal coefficient is , i.e., the diffusion.

Generate an exemplary Ornstein–Uhlenbeck process with your favourite integrator, e.g., the Euler–Maruyama or with a more powerful tool from JiTCSDE found on GitHub. For this example let's take θ=.3 and σ=.1, over a total time of 500 units, with a sampling of 1000 Hertz, and from the generated data series retrieve the two parameters, the drift -θy(t) and diffusion σ.

Here is a short code on generating a Ornstein–Uhlenbeck stochastic trajectory with a simple Euler–Maruyama integration method

# integration time and time sampling
t_final = 500
delta_t = 0.001

# The parameters theta and sigma
theta = 0.3
sigma = 0.1

# The time array of the trajectory
time = np.arange(0, t_final, delta_t)

# Initialise the array y
y = np.zeros(time.size)

# Generate a Wiener process
dw = np.random.normal(loc=0, scale=np.sqrt(delta_t), size=time.size)

# Integrate the process
for i in range(1,time.size):
    y[i] = y[i-1] - theta*y[i-1]*delta_t + sigma*dw[i]

From here we have a plain example of an Ornstein–Uhlenbeck process, always drifting back to zero, due to the mean-reverting drift θ. The effect of the noise can be seen across the whole trajectory.

Take the timeseries y and let's study the Kramers–Moyal coefficients. For this let's look at the drift and diffusion coefficients of the process, i.e., the first and second Kramers–Moyal coefficients, with an epanechnikov kernel

# The kmc holds the results, where edges holds the binning space
kmc, edges = km(y, powers=2)

This results in

Notice here that to obtain the Kramers–Moyal coefficients you need to divide kmc by the timestep delta_t. This normalisation stems from the Taylor-like approximation, i.e., the Kramers–Moyal expansion (delta t → 0).

A Jupyter notebook with this example can be found here

A two-dimensional diffusion process is a stochastic process that comprises two and allows for a mixing of these noise terms across its two dimensions.

where we will select a set of state-dependent parameters obeying

with and .

As an example, let's take the following set of parameters for the drift vector and diffusion matrix

# integration time and time sampling
t_final = 2000
delta_t = 0.001

# Define the drift vector N
N = np.array([2.0, 1.0])

# Define the diffusion matrix g
g = np.array([[0.5, 0.0], [0.0, 0.5]])

# The time array of the trajectory
time = np.arange(0, t_final, delta_t)

Integrating the previous stochastic trajectory with a simple Euler–Maruyama integration method

# Initialise the array y
y = np.zeros([time.size, 2])

# Generate two Wiener processes with a scale of np.sqrt(delta_t)
dW = np.random.normal(loc=0, scale=np.sqrt(delta_t), size=[time.size, 2])

# Integrate the process (takes about 20 secs)
for i in range(1, time.size):
    y[i,0] = y[i-1,0]  -  N[0] * y[i-1,0] * delta_t + g[0,0]/(1 + np.exp(y[i-1,0]**2)) * dW[i,0]  +  g[0,1] * dW[i,1]
    y[i,1] = y[i-1,1]  -  N[1] * y[i-1,1] * delta_t + g[1,0] * dW[i,0]  +  g[1,1]/(1 + np.exp(y[i-1,1]**2)) * dW[i,1]

The stochastic trajectory in 2 dimensions for 10 time units (10000 data points)

First notice that all the results now will be two-dimensional surfaces, so we will need to plot them as such

# Choose the size of your target space in two dimensions
bins = [100, 100]

# Introduce the desired orders to calculate, but in 2 dimensions
powers = np.array([[0,0], [1,0], [0,1], [1,1], [2,0], [0,2], [2,2]])
# insert into kmc:   0      1      2      3      4      5      6

# Notice that the first entry in [,] is for the first dimension, the
# second for the second dimension...

# Choose a desired bandwidth bw
bw = 0.1

# Calculate the Kramers−Moyal coefficients
kmc, edges = km(y, bw=bw, bins=bins, powers=powers)

# The K−M coefficients are stacked along the first dim of the
# kmc array, so kmc[1,...] is the first K−M coefficient, kmc[2,...]
# is the second. These will be 2-dimensional matrices

Now one can visualise the Kramers–Moyal coefficients (surfaces) in green and the respective theoretical surfaces in black. (Don't forget to normalise: kmc / delta_t).

We welcome reviews and ideas from everyone. If you want to share your ideas or 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, do not hesitate to contact us, we are here to help. We abide to a Conduct of Fairness.

Next on the list is

• Include more kernels
• Work through the documentation carefully

• Version 0.4.1 - Changing CI. Correcting kmc[0,:] normalisation. Various Simplifications. Bins as ints, powers as ints.
• Version 0.4.0 - Added the documentation, first testers, and the Conduct of Fairness
• Version 0.3.2 - Adding 2 kernels: triagular and quartic and extending the documentation and examples.
• Version 0.3.1 - Corrections to the fft triming after convolution.
• Version 0.3.0 - The major breakthrough: Calculates the Kramers–Moyal coefficients for data of any dimension.
• Version 0.2.0 - Introducing convolutions and gaussian and uniform kernels. Major speed up in the calculations.
• Version 0.1.0 - One and two dimensional Kramers–Moyal coefficients with an epanechnikov kernel.

The study of stochastic processes from a data-driven approach is grounded in extensive mathematical work. From the applied perspective there are several references to understand stochastic processes, the Fokker–Planck equations, and the Kramers–Moyal expansion

• Tabar, M. R. R. (2019). Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Springer, International Publishing
• Risken, H. (1989). The Fokker–Planck equation. Springer, Berlin, Heidelberg.
• Gardiner, C.W. (1985). Handbook of Stochastic Methods. Springer, Berlin.

You can find and extensive review on the subject here¹

This project was started in 2017 at the neurophysik by Leonardo Rydin Gorjão, Jan Heysel, Klaus Lehnertz, and M. Reza Rahimi Tabar. Francisco Meirinhos later devised the hard coding to python. The project is now supported by Dirk Witthaut and the Institute of Energy and Climate Research Systems Analysis and Technology Evaluation.

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.

¹ Friedrich, R., Peinke, J., Sahimi, M., Tabar, M. R. R. Approaching complexity by stochastic methods: From biological systems to turbulence, Phys. Rep. 506, 87–162 (2011).