Introduction

In this section, we will briefly explain what is Random Matrix Theory (RMT), and we will introduce scikit-rmt, a library that provides RMT tools to mathematicians, engineers, physicists, economists or any interested scientist in the Python scientific ecosystem.

# Author: Alejandro Santorum Varela
# License: BSD 3-Clause

What is Random Matrix Theory?

Random mathematical objects, such as graphs or tensors, have always been deeply studied because of their applications in economics, physics and engineering. The growing computational capacity allows to work and simulate this class of objects efficiently to improve in many disciplines.

In particular, two-dimensional random tensors are known as random matrices. Random Matrix Theory is the is the branch of statistics that studies the properties of matrices whose inputs are random variables. Many random matrices have similar behaviour and properties, especially if we analyze their eigenvalue spectrum, that is why they are usually classified into ensembles of random matrices, which are sets of them that share common features.

What is scikit-rmt?

scikit-rmt is a Python library containing classes and functions that allow you to perform RMT tasks. Using it you can:

• Sample many types of random matrices of the main ensembles: Gaussian ensemble, Wishart ensemble, Manova ensemble and Circular ensemble.

• Plot and analyze random matrix eigenvalue spectrum of the sampled ensembles.

• Calculate eigenvalue joint probability density function of the sampled random matrices.

• Plot and study main spectrum laws: Wigner Semicircle Law, Marchenko-Pastur Law and Tracy-Widom Law.

• Estimation of covariance matrices with several methods, such as non-lineal shrinkage analytical estimator.

As an example, the following code shows how to sample the most known ensemble of random matrices: Gaussian Orthogonal Ensemble (GOE).

from skrmt.ensemble import GaussianEnsemble

goe = GaussianEnsemble(beta=1, n=5)
print(goe.matrix)

[[-0.95210124 -0.32681968 -0.3741851   0.81073791  0.30625307]
 [-0.32681968  0.56846157 -0.98495991 -0.46675197 -0.42483888]
 [-0.3741851  -0.98495991 -1.21556006  0.35919924 -1.67950253]
 [ 0.81073791 -0.46675197  0.35919924  0.47188183  0.87782606]
 [ 0.30625307 -0.42483888 -1.67950253  0.87782606  0.52622123]]

To study its eigenvalue spectrum, its size should be a little bit larger.

from skrmt.ensemble import GaussianEnsemble

goe = GaussianEnsemble(beta=1, n=1000)
goe.plot_eigval_hist(bins=80)