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        :ref:`Go to the end <sphx_glr_download_auto_tutorial_plot_0_introduction.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_tutorial_plot_0_introduction.py:


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.

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.. code-block:: Python


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








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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.

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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.

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

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.. code-block:: Python


    from skrmt.ensemble import GaussianEnsemble

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





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 .. code-block:: none

    [[-0.04372412  1.31157883  0.95599519 -0.83875624 -0.79408893]
     [ 1.31157883 -2.00560375  0.06793482  0.13091436  0.70730579]
     [ 0.95599519  0.06793482  2.53381583  0.29490057  0.77931517]
     [-0.83875624  0.13091436  0.29490057  2.080257   -0.11735385]
     [-0.79408893  0.70730579  0.77931517 -0.11735385 -1.68784083]]




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

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.. code-block:: Python


    from skrmt.ensemble import GaussianEnsemble

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



.. image-sg:: /auto_tutorial/images/sphx_glr_plot_0_introduction_001.png
   :alt: Eigenvalue histogram
   :srcset: /auto_tutorial/images/sphx_glr_plot_0_introduction_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.209 seconds)


.. _sphx_glr_download_auto_tutorial_plot_0_introduction.py:

.. only:: html

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