kramersmoyal 0.4.1
Calculate Kramers-Moyal coefficients for stochastic process of any dimension, up to any order.
Requires Python: >=3.7

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# KramersMoyal
`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](https://doi.org/10.21105/joss.01693) is now officially published on [JOSS](https://joss.theoj.org/). The paper is also available [here](/paper/paper.pdf), or you can find it in the [ArXiv](https://arxiv.org/abs/1912.09737).

# Installation
To install `kramersmoyal`, just use `pip`

```
pip install kramersmoyal
```
Then on your favourite editor just use
```python
from kramersmoyal import km
```

## Dependencies
The library depends on `numpy` and `scipy`.

# A one-dimensional stochastic process

A Jupyter notebook with this example can be found [here](/examples/kmc.ipynb)

## The theory
Take, for example, the well-documented one-dimension Ornstein–Uhlenbeck process, also known as Va&#353;&#237;&#269;ek process, see [here](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process). This process is governed by two main parameters: the mean-reverting parameter &theta; and the diffusion parameter &sigma;

<img src="/other/OU_eq.png" title="Ornstein–Uhlenbeck process" height="25"/>

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 ![](/other/inline_KM_1.png) and the second-order Kramers–Moyal coefficient is ![](/other/inline_KM_2.png), i.e., the diffusion.

Generate an exemplary Ornstein–Uhlenbeck process with your favourite integrator, e.g., the [Euler–Maruyama](https://en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method) or with a more powerful tool from [`JiTCSDE`](https://github.com/neurophysik/jitcsde) found on GitHub.
For this example let's take &theta;=.3 and &sigma;=.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 -&theta;y(t) and diffusion &sigma;.

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

```python
# 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 &theta;. The effect of the noise can be seen across the whole trajectory.

<img src="/other/fig1.png" title="Ornstein–Uhlenbeck process" height="200"/>

## Using `kramersmoyal`
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
```python

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

This results in

<img src="/other/fig2.png" title="Drift and diffusion terms of an Ornstein–Uhlenbeck process" height="200"/>

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` &rarr; 0).

# A two-dimensional diffusion process

A Jupyter notebook with this example can be found [here](/examples/kmc.ipynb)

## Theory

A two-dimensional diffusion process is a stochastic process that comprises two ![](/other/inline_W.png) and allows for a mixing of these noise terms across its two dimensions.

<img src="/other/2D-diffusion.png" alt="2D-diffusion" title="A 2-dimensional diffusion process" height="60" />

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

<img src="/other/parameters_2D-diffusion.png" alt="2D-diffusion" title="Specific parameters for the diffusion process" height="70" />

with ![](/other/inline_parameters_2D-diffusion_1.png) and ![](/other/inline_parameters_2D-diffusion_2.png).

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

```python
# 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 a 2-dimensional process
Integrating the previous stochastic trajectory with a simple Euler–Maruyama integration method

```python
# 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)

<img src="/other/fig3.png" alt="2D-diffusion" title="2-dimensional trajectory" height="280" />

## Back to `kramersmoyal` and the Kramers–Moyal coefficients
First notice that all the results now will be two-dimensional surfaces, so we will need to plot them as such

```python
# 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`).

<img src="/other/fig4.png" alt="2D-diffusion" title="2-dimensional Kramers–Moyal surfaces (green) and the theoretical surfaces (black)" height="480" />

# Contributions
We welcome reviews and ideas from everyone. If you want to share your ideas or report a bug, open an [issue](https://github.com/LRydin/KramersMoyal/issues) 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](contributions.md).

# TODOs
Next on the list is
- Include more kernels
- Work through the documentation carefully

# Changelog
- 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.

# Literature and Support

### Literature
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 [here](http://sharif.edu/~rahimitabar/pdfs/80.pdf)<sup>1</sup>

### History
This project was started in 2017 at the [neurophysik](https://www.researchgate.net/lab/Klaus-Lehnertz-Lab-2) 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](https://www.fz-juelich.de/iek/iek-ste/EN/Home/home_node.html).

### 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.

---

<sup>1</sup> 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)](https://doi.org/10.1016/j.physrep.2011.05.003).