landaupy

A simple and pure Python implementation¹ of the Landau distribution, since no common package (Scipy, Numpy, etc.) provides this. The algorithm to calculate the Landau distribution was adapted from the Root implementation which is documented here.

¹ The only extra requirements are numpy and scipy, which are trivial to install in my experience (you probably already have them). There is no need for any strange thing like C/C++→Python compilers/bridges, etc. that are hard to install and configure.

Installation

Installing with Git:

pip install git+https://github.com/SengerM/landaupy

Manual installation:

1. Download the repository as a .zip file.
2. Decompress the .zip file wherever you like, for example /wherever/I/like.
3. Run pip install <path_to_decompressed>, for example pip install /wherever/I/like/landaupy.

Usage

See the examples.

Landau distribution

A simple example of the Landau distribution:

from landaupy import landau
import numpy as np

pdf = landau.pdf(x=1,x_mpv=2,xi=3) # Calculate in a single point.
print(pdf)

pdf = landau.pdf(
	x = np.linspace(-11,111,9999), # Calculate along a numpy array.
	x_mpv = 2, # `x_mpv` is also vectorized, you can put a numpy array here.
	xi = 3 # `xi` is also vectorized, you can put a numpy array here.
)
print(pdf)

The Landau cumulative distribution function is also available:

from landaupy import landau
import numpy as np

cdf = landau.cdf(x=1,x_mpv=2,xi=3) # Calculate in a single point.
print(cdf)

cdf = landau.cdf(
	x = np.linspace(-11,111,999), # Calculate along a numpy array.
	x_mpv = 2,
	xi = 3
)
print(cdf)

Random samples can also be generated:

from landaupy import landau

samples = landau.sample(x_mpv=0, xi=3, n_samples=9999)

For more, see the examples.

Langauss distribution

I also implemented the so called langauss distribution which is the convolution of a Gaussian and a Landau, useful when working with real life particle detectors. The usage is similar: A simple example of the Landau distribution:

from landaupy import langauss
import numpy as np

pdf = langauss.pdf(x=1,landau_x_mpv=2,landau_xi=3,gauss_sigma=3) # Calculate in a single point.
print(pdf)

pdf = langauss.pdf(
	x = np.linspace(-11,111,999), # Calculate along a numpy array.
	landau_x_mpv = 2,
	landau_xi = 3,
	gauss_sigma = 3
)
print(pdf)

The Landau cumulative distribution function is also available:

from landaupy import langauss
import numpy as np

cdf = langauss.cdf(x=1,landau_x_mpv=2,landau_xi=3,gauss_sigma=3) # Calculate in a single point.
print(cdf)

cdf = langauss.cdf(
	x = np.linspace(-11,111,99), # Calculate along a numpy array.
	landau_x_mpv = 2,
	landau_xi = 3,
	gauss_sigma = 3
)
print(cdf)

Random samples can also be generated:

from landaupy import langauss

samples = langauss.sample(landau_x_mpv=0, landau_xi=1, gauss_sigma=2, n_samples=9999)

For more, see the examples.

Differences WRT the Root version

Despite this implementation is based in the original from Root, I made a few changes to be more consistent with the rest of the world.

Normalization

One of the things I changed is the normalization of the langauss.pdf distribution such that it integrates to 1. All the PDFs in this package integrate to 1. You can verify the normalization with the following code, which integrates the PDFs:

from landaupy import landau
from landaupy import langauss
import numpy as np
import scipy.integrate as integrate
import warnings

with warnings.catch_warnings():
	warnings.simplefilter("ignore")
	# Silence warnings since the numerical integrations between -inf and +inf are prompt to show some warnings.
	for x_mpv in [0,10,55]:
		for xi in [1,2,10]:
			# Integrate to verify normalization:
			print(f'x_mpv={x_mpv}, xi={xi} → integral(landau.pdf) = {integrate.quad(lambda x: landau.pdf(x, x_mpv=x_mpv, xi=xi), -float("inf"), float("inf"))[0]}')
			for sigma in [.1,1,5,10]:
				print(f'x_mpv={x_mpv}, xi={xi}, sigma={sigma} → integral(langauss.pdf) = {integrate.quad(lambda x: langauss.pdf(x, landau_x_mpv=x_mpv, landau_xi=xi, gauss_sigma=sigma), -float("inf"), float("inf"))[0]}')

The MPV argument

One of the arguments of the landau.pdf function is x_mpv. This is really the x position of the Most Probable Value (MPV), which is usually what one is interested in. In the original implementation from Root the parameter is called x0 and it is close to the MPV but it is not the MPV. The relation between them is given by x0 = x_mpv + 0.22278298*xi and I stole it from the Root implementation of the langauss function.

Naming

I renamed langau to langauss, otherwise a simple typo can change the distribution you intend to use. Also renamed some of the parameters with more meaningful names.

Landau distribution in SciPy 🙂

There is a pull request that integrates the Landau distribution in the standard SciPy package, making it trivially available for all Python users in the world. If you find this useful, go to the pull request and give it a 👍.