scikit-rmt 0.8.1

Random Matrix Theory Python package

scikit-rmt: Random Matrix Theory Python package

Random Matrix Theory, or RMT, is the field of Statistics that analyses matrices that their entries are random variables.

This package offers classes, methods and functions to give support to RMT in Python. Includes a wide range of utils to work with different random matrix ensembles, random matrix spectral laws and estimation of covariance matrices. See documentation or visit the https://github.com/AlejandroSantorum/scikit-rmt of the project for further information on the features included in the package.

---

Documentation

The documentation is available at https://scikit-rmt.readthedocs.io/en/latest/, which includes detailed information of the different modules, classes and methods of the package, along with several examples showing different funcionalities.

---

Installation

Using a virtual environment is recommended to minimize the chance of conflicts. However, the global installation should work properly as well.

Local installation using venv (recommended)

Navigate to your project directory.

cd MyProject

Create a virtual environment (you can change the name "env").

python3 -m venv env

Activate the environment "env".

source env/bin/activate

Install using pip.

pip install scikit-rmt

You may need to use pip3.

pip3 install scikit-rmt

Global installation

Just install it using pip or pip3.

pip install scikit-rmt

Requirements

scikit-rmt depends on the following packages:

• numpy - The fundamental package for scientific computing with Python
• matplotlib - Plotting with Python
• scipy - Scientific computation in Python

Check the pinned versions in the requirements.txt file.

---

Main features

scikit-rmt provides support to analyze, study and simulate Random Matrix Theory properties and results:

• Sampling of the following random matrix ensembles:
  • Gaussian Ensemble (GOE, GUE and GSE; for beta=1, 2 and 4 respectively): class GaussianEnsemble.
  • Wishart Ensemble (WRE, WCE and WQE; for beta=1, 2 and 4 respectively): class WishartEnsemble.
  • Manova Ensemble (MRE, MCE and MQE; for beta=1, 2 and 4 respectively): class ManovaEnsemble.
  • Circular Ensemble (COE, CUE, CSE; for beta=1, 2 and 4 respectively): class CircularEnsemble.
• Eigenvalue computation for the previous ensembles using the class method eigvals.
• Computation of the joint eigenvalue probability density function given the random matrix eigenvalues, using the class method joint_eigval_pdf.
• Computation of the empirical spectral histogram by executing the class method eigval_hist.
• Plotting of the empirical spectral histogram using the class method plot_eigval_hist.
• Computation and plotting of the following spectral laws (spectral_law sub-module), including the calculation and plotting of the corresponding probability density functions (PDFs) and cumulative distribution functions (CDFs):
  • Wigner Semicircle law: describes the limiting distribution of the eigenvalues of Wigner matrices (in particular, random matrices from the Gaussian Ensemble). Implemented by class WignerSemicircleDistribution.
  • Tracy-Widom law: describes the limiting distribution of the largest eigenvalue of Wigner matrices (in particular, random matrices from the Gaussian Ensemble). Implemented by class TracyWidomDistribution.
  • Marchenko-Pastur law: describes the limiting distribution of the eigenvalues of Wishart matrices (random matrices from the Wishart Ensemble). Implemented by class MarchenkoPasturDistribution.
  • Manova Spectrum law: introduced and proved by K. W. Wachter (1980), it describes the limiting distribution of the eigenvalues of Manova matrices (random matrices from the Manova Ensemble). Implemented by class ManovaSpectrumDistribution.
• Covariance matrix estimation using different matrix smoothing and estimation methods (covariance module).
  • Estimation of sample covariance matrix using sample_estimator function.
  • Finite-sample Optimal (FSOpt) estimator by fsopt_estimator function.
  • Empirical Bayesian estimator by empirical_bayesian_estimator function. Haff in "Estimation of a covariance matrix under Stein’s loss" (1985).
  • Minimax estimator by minimax_estimator function
  • Linear shrinkage estimator by linear_shrinkage_estimator function. Ledoit and Wolf in "A well-conditioned estimator for large-dimensional covariance matrices" (2004).
  • Analytical shrinkage estimator by analytical_shrinkage_estimator function. Ledoit and Wolf in "Analytical nonlinear shrinkage of large-dimensional covariance matrices" (2020).

---

A brief tutorial

First of all, several random matrix ensembles can be sampled: Gaussian Ensembles, Wishart Ensembles, Manova Ensembles and Circular Ensembles. As an example, the following code shows how to sample a Gaussian Orthogonal Ensemble (GOE) random matrix.

from skrmt.ensemble.gaussian_ensemble import GaussianEnsemble
# sampling a GOE (beta=1) matrix of size 3x3
goe = GaussianEnsemble(beta=1, n=3)
print(goe.matrix)

[[ 0.34574696 -0.10802385  0.38245343]
 [-0.10802385 -0.60113963  0.28624612]
 [ 0.38245343  0.28624612 -0.96503739]]

Its spectral density can be easily plotted:

# sampling a GOE matrix of size 1000x1000
goe = GaussianEnsemble(beta=1, n=1000)
# plotting its spectral distribution in the interval (-2,2)
goe.plot_eigval_hist(bins=80, interval=(-2,2), density=True)

If we sample a non-symmetric/non-hermitian random matrix, its eigenvalues do not need to be real, so a 2D complex histogram has been implemented in order to study spectral density of these type of random matrices. It would be the case, for example, of Circular Symplectic Ensemble (CSE).

from skrmt.ensemble.circular_ensemble import CircularEnsemble

# sampling a CSE (beta=4) matrix of size 2000x2000
cse = CircularEnsemble(beta=4, n=1000)
cse.plot_eigval_hist(bins=80)

We can boost histogram representation using the results described by A. Edelman and I. Dumitriu in Matrix Models for Beta Ensembles and by J. Albrecht, C. Chan, and A. Edelman in Sturm Sequences and Random Eigenvalue Distributions (check references). Sampling certain random matrices (Gaussian Ensemble and Wishart Ensemble matrices) in its tridiagonal form we can speed up histogramming procedure. The following graphical simulation using GOE matrices tries to illustrate it.

On the other hand, for all the supported ensembles, the joint eigenvalue probability density function can be computed by using the class method joint_eigval_pdf(eigvals=None). By default, the method computes the joint eigenvalue PDF of the eigenvalues of the sampled random matrix. However, the method can be called using a pre-computed array of eigenvalues, and the joint eigenvalue PDF of these eigenvalues is returned. AS an example, we can simulate sampling and histogramming many eigenvalues of GOE random matrices of size 2x2, as well as illustrating the joint eigenvalue PDF for GOE matrices 2x2. This simulation is shown in the figure below.

In addition, several spectral laws can be analyzed using this library, such as Wigner's Semicircle Law, Marchenko-Pastur Law and Tracy-Widom Law. The analytical probability density function can also be plotted by using the plot_law_pdf=True argument.

Plot of Wigner's Semicircle Law, sampling 100000 independent eigenvalues of a GOE matrix:

from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

wsd = WignerSemicircleDistribution(beta=1)
wsd.plot_empirical_pdf(sample_size=100000, bins=80, density=True)

from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

wsd = WignerSemicircleDistribution(beta=1)
wsd.plot_empirical_pdf(sample_size=100000, bins=80, density=True, plot_law_pdf=True)

Plot of Marchenko-Pastur Law, sampling 100000 independent eigenvalues of a WRE matrix:

from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
mpd.plot_empirical_pdf(sample_size=100000, bins=80, density=True)

from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
mpd.plot_empirical_pdf(sample_size=100000, bins=80, density=True, plot_law_pdf=True)

Plot of Tracy-Widom Law, sampling 30000 maximum eigenvalues of GOE matrices:

from skrmt.ensemble.spectral_law import TracyWidomDistribution

twd = TracyWidomDistribution(beta=1)
twd.plot_empirical_pdf(sample_size=30000, bins=80, density=True)

from skrmt.ensemble.spectral_law import TracyWidomDistribution

twd = TracyWidomDistribution(beta=1)
twd.plot_empirical_pdf(sample_size=30000, bins=80, density=True, plot_law_pdf=True)

The proposed package scikit-rmt also provides support for computing and analyzing the probability density function (PDF) and cumulative distribution function (CDF) of the main random matrix ensembles. In particular, the following classes are implemented in the ensemble.spectral_law module:

• WignerSemicircleDistribution in skrmt.ensemble.spectral_law.
• MarchenkoPasturDistribution in skrmt.ensemble.spectral_law.
• TracyWidomDistribution in skrmt.ensemble.spectral_law.
• ManovaSpectrumDistribution in skrmt.ensemble.spectral_law.

For example, WignerSemicircleDistribution can be used to plot the Wigner's Semicircle Law PDF:

import numpy as np
import matplotlib.pyplot as plt
from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

x1 = np.linspace(-5, 5, num=1000)
x2 = np.linspace(-10, 10, num=2000)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12,4))

for sigma in [0.5, 1.0, 2.0, 4.0]:
    wsd = WignerSemicircleDistribution(beta=1, center=0.0, sigma=sigma)

    y1 = wsd.pdf(x1)
    y2 = wsd.pdf(x2)

    ax1.plot(x1, y1, label=f"$\sigma$ = {sigma} (R = ${wsd.radius}$)")
    ax2.plot(x2, y2, label=f"$\sigma$ = {sigma} (R = ${wsd.radius}$)")

ax1.legend()
ax1.set_xlabel("x", fontweight="bold")
ax1.set_xticks([-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5])
ax1.set_ylabel("density", fontweight="bold")

ax2.legend()
ax2.set_xlabel("x", fontweight="bold")
ax2.set_xticks([-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10])
ax2.set_ylabel("density", fontweight="bold")

fig.suptitle("Wigner Semicircle probability density function (PDF)", fontweight="bold")
plt.show()

Similarly, MarchenkoPasturDistribution can be used to plot the Marchenko-Pastur Law PDF:

import numpy as np
import matplotlib.pyplot as plt
from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

x1 = np.linspace(0, 4, num=1000)
x2 = np.linspace(0, 5, num=2000)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12,4))

for ratio in [0.2, 0.4, 0.6, 1.0, 1.4]:
    mpl = MarchenkoPasturDistribution(beta=1, ratio=ratio, sigma=1.0)

    y1 = mpl.pdf(x1)
    y2 = mpl.pdf(x2)

    ax1.plot(x1, y1, label=f"$\lambda$ = {ratio} ")
    ax2.plot(x2, y2, label=f"$\lambda$ = {ratio} ")

ax1.legend()
ax1.set_ylim(0, 1.4)
ax1.set_xlabel("x", fontweight="bold")
ax1.set_ylabel("density", fontweight="bold")

ax2.legend()
ax2.set_ylim(0, 1.4)
ax2.set_xlim(0, 1)
ax2.set_xlabel("x", fontweight="bold")
ax2.set_ylabel("density", fontweight="bold")

fig.suptitle("Marchenko-Pastur probability density function (PDF)", fontweight="bold")
plt.show()

In the following example, we show how we can plot the PDF and CDF of the Tracy-Widom distribution using the class TracyWidomDistribution:

import numpy as np
import matplotlib.pyplot as plt
from skrmt.ensemble.spectral_law import TracyWidomDistribution

x = np.linspace(-5, 2, num=1000)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12,4))

for beta in [1,2,4]:
    twd = TracyWidomDistribution(beta=beta)

    y_pdf = twd.pdf(x)
    y_cdf = twd.cdf(x)

    ax1.plot(x, y_pdf, label=f"$\\beta$ = {beta}")
    ax2.plot(x, y_cdf, label=f"$\\beta$ = {beta}")

ax1.legend()
ax1.set_xlabel("x", fontweight="bold")
ax1.set_ylabel("density", fontweight="bold")
ax1.set_title("Probability density function")

ax2.legend()
ax2.set_xlabel("x", fontweight="bold")
ax2.set_ylabel("distribution", fontweight="bold")
ax2.set_title("Cumulative distribution function")

fig.suptitle("Tracy Widom Law", fontweight="bold")
plt.show()

The other module of this library implements several covariance matrix estimators:

• Sample estimator.
• Finite-sample optimal estimator (FSOpt estimator).
• Non-linear shrinkage analytical estimator (Ledoit & Wolf, 2020).
• Linear shrinkage estimator (Ledoit & Wolf, 2004).
• Empirical Bayesian estimator (Haff, 1980).
• Minimax estimator (Stain, 1982).

For certain problems, sample covariance matrix is not the best estimation for the population covariance matrix.

The following code illustrates the usage of the estimators.

from skrmt.covariance import analytical_shrinkage_estimator

# load dataset with your own/favorite function (such as pandas.read_csv)
X = load_dataset('dataset_file.data')

# get estimation
Sigma = analytical_shrinkage_estimator(X)

# ... Do something with Sigma. For example, PCA.

For more information or insight about the usage of the library, you can visit the official documentation https://scikit-rmt.readthedocs.io/en/latest/ or the directory notebooks, that contains several Python notebooks with tutorials and plenty of examples.

---

License

The package is licensed under the BSD 3-Clause License. A copy of the license can be found along with the code.

---

Main references

• James Albrecht, Cy Chan, and Alan Edelman, "Sturm Sequences and Random Eigenvalue Distributions", Foundations of Computational Mathematics, vol. 9 iss. 4 (2009), pp 461-483. [pdf] [doi]

• Ioana Dumitriu and Alan Edelman, "Matrix Models for Beta Ensembles", Journal of Mathematical Physics, vol. 43 no. 11 (2002), pp. 5830-5547 arXiv:math-ph/0206043

• Rowan Killip and Rostyslav Kozhan, "Matrix Models and Eigenvalue Statistics for Truncations of Classical Ensembles of Random Unitary Matrices", Communications in Mathematical Physics, vol. 349 (2017) pp. 991-1027. arxiv.org/pdf/1501.05160.pdf

• Olivier Ledoit and Michael Wolf, "Analytical Nonlinear Shrinkage of Large-dimensional Covariance Matrices", Annals of Statistics, vol. 48, no. 5 (2020) pp. 3043–3065. [pdf]

• Olivier Ledoit and Michael Wolf, "A Well-conditioned Estimator for Large-dimensional Covariance Matrices", Journal of Multivariate Analysis, vol. 88 (2004) pp. 365–411. [pdf]

---

Attribution

This project has been developed by Alejandro Santorum Varela (2021) as part of the final degree project in Computer Science (Autonomous University of Madrid), supervised by Alberto Suárez González.

If you happen to use scikit-rmt in your work or research, please cite its GitHub repository:

A. Santorum, "scikit-rmt", https://github.com/AlejandroSantorum/scikit-rmt, 2021. GitHub repository.

The corresponding BibTex entry is

@misc{Santorum2021,
  author = {A. Santorum},
  title = {scikit-rmt},
  year = {2021},
  howpublished = {\url{https://github.com/AlejandroSantorum/scikit-rmt}},
  note = {GitHub repository}
}