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.. only:: html

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

.. _sphx_glr_auto_tutorial_plot_3_eigenvalue_independence.py:


Eigenvalue independence
=======================

In this tutorial, we highlight the fact that eigenvalues of a random matrix
sample are *not* independent. However, random numbers sampled using the PDF
of a spectral law (e.g. Wigner's Semicircle law) are actually drawn independently.

.. GENERATED FROM PYTHON SOURCE LINES 10-14

.. code-block:: Python


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








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Independent random samples vs. eigenvalues of a random matrix sample
--------------------------------------------------------------------

Using **scikit-rmt** to simulate the behaviour of the ensembles, it is possible
to illustrate how the eigenvalues of a random matrix sample are not independent,
since they are draw from the same matrix sample. However, random variates sampled
using the PDF of a spectral law (for example, Wigner's Semicircle law) are drawn
independently.

We can observe this phenomenon by plotting the **histogram of a random matrix**
alongside the PDF of the corresponding spectral law. This is easily done by using
the function ``plot_spectral_hist_and_law`` in ``skrmt.ensemble.utils``:

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.. code-block:: Python


    from skrmt.ensemble.gaussian_ensemble import GaussianEnsemble
    from skrmt.ensemble.utils import plot_spectral_hist_and_law

    goe = GaussianEnsemble(beta=1, n=3000, tridiagonal_form=True)
    plot_spectral_hist_and_law(ensemble=goe, bins=60)




.. image-sg:: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_001.png
   :alt: Ensemble spectral histogram vs law PDF
   :srcset: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_001.png
   :class: sphx-glr-single-img





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In the previous example, the histogram of the spectrum of a sample from
the Gaussian Ensemble was plotted next to the PDF of the Wigner's
Semicircle law.

It can be observed that the eigenvalues of a single sample of a random matrix 
are *not* independent since the fluctuations of the histogram compared to
Wigner's Semicircle PDF are really small. In contrast, if we compare independent
random samples from the Wigner Semicircle law and the actual PDF we observe
higher fluctuations for the *same sample size* (in this example, 3000).

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.. code-block:: Python


    from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

    wsd = WignerSemicircleDistribution(beta=1)
    wsd.plot_empirical_pdf(
        sample_size=3000,
        bins=60,
        density=True,
        plot_law_pdf=True
    )




.. image-sg:: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_002.png
   :alt: Wigner Semicircle Law - Eigenvalue histogram
   :srcset: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_002.png
   :class: sphx-glr-single-img





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Similarly, this can be seen for other ensembles. For example, with the
Wishart Ensemble.

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.. code-block:: Python


    from skrmt.ensemble.wishart_ensemble import WishartEnsemble
    from skrmt.ensemble.utils import plot_spectral_hist_and_law

    wre = WishartEnsemble(beta=1, p=1000, n=3000, tridiagonal_form=True)
    plot_spectral_hist_and_law(ensemble=wre, bins=60)




.. image-sg:: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_003.png
   :alt: Ensemble spectral histogram vs law PDF
   :srcset: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_003.png
   :class: sphx-glr-single-img





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The fluctuations with respect the PDF of the Marchenko-Pastur law are larger
if we directly draw independent random samples from that distribution:

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.. code-block:: Python


    from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

    mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
    mpd.plot_empirical_pdf(
        sample_size=1000,
        bins=60,
        density=True,
        plot_law_pdf=True
    )



.. image-sg:: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_004.png
   :alt: Marchenko-Pastur Law - Eigenvalue histogram
   :srcset: /auto_tutorial/images/sphx_glr_plot_3_eigenvalue_independence_004.png
   :class: sphx-glr-single-img






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   **Total running time of the script:** (0 minutes 0.641 seconds)


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