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.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/distributions/multivariate/plot_multivariate_normal.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_distributions_multivariate_plot_multivariate_normal.py>`
        to download the full example code or to run this example in your browser via Binder

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

.. _sphx_glr_auto_examples_distributions_multivariate_plot_multivariate_normal.py:


Multivariate normal distribution
==================================

This examples shows the use of the multivariate normal distributions class. In particular:

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- How to define one of the univariate distributions supported by UQpy
- How to plot the pdf of the distribution
- How to extract the moments of the distribution
- How to draw random samples from the distribution

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Initially we have to import the necessary modules.

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.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt
    from UQpy.distributions.collection.MultivariateNormal import MultivariateNormal








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Example of a multivariate normal distribution
---------------------------------------------
Note that multivariate normal distribution can facilitate any number of dimensions.
In order to define it two arguments are necessary, specifically, a list containing the
mean values for each one of the dimensions and a covariance matrix with shape
(ndimensions, ndimensions)

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.. code-block:: default


    print(MultivariateNormal.__bases__)
    dist = MultivariateNormal(mean=[1., 2.], cov=[[4., -0.2], [-0.2, 1.]])





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    (<class 'UQpy.distributions.baseclass.DistributionND.DistributionND'>,)




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Plot the two-dimensional pdf of the distribution.
-------------------------------------------------


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.. code-block:: default


    fig, ax = plt.subplots(ncols=1, figsize=(10, 4))
    x = np.arange(-6.0, 6.0, 0.1)
    y = np.arange(-6.0, 6.0, 0.1)
    X, Y = np.meshgrid(x, y)
    Z = dist.pdf(x=np.concatenate([X.reshape((-1, 1)), Y.reshape((-1, 1))], axis=1))
    CS = ax.contour(X, Y, Z.reshape(X.shape))
    ax.clabel(CS, inline=1, fontsize=10)
    ax.set_xlabel('dimension 1')
    ax.set_ylabel('dimension 2')
    ax.set_title('Contour plot of pdf')
    ax.set_xlim([-4, 6])
    ax.set_ylim([-2, 6])
    plt.show()




.. image-sg:: /auto_examples/distributions/multivariate/images/sphx_glr_plot_multivariate_normal_001.png
   :alt: Contour plot of pdf
   :srcset: /auto_examples/distributions/multivariate/images/sphx_glr_plot_multivariate_normal_001.png
   :class: sphx-glr-single-img





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Print the multivariate moments of the distribution.
---------------------------------------------------
Providing a single or multiple consecutive initials of the following four distributions moments

- 'm': mean
- 'v': variance
- 's': skewness
- 'k': kurtosis

allows the user to obtain the respective moments from all underlying univariate distributions.
In the following examples providing the string 'mv' to the moments function, returns the respective means and
variances.

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.. code-block:: default

    print(dist.moments())
    print(dist.moments(moments2return='mv'))





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ([1.0, 2.0], [[4.0, -0.2], [-0.2, 1.0]])
    ([1.0, 2.0], [[4.0, -0.2], [-0.2, 1.0]])




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Generate 1000 random samples from the binomial distribution.
-------------------------------------------------------------

Important: the output of rvs is a (nsamples, 1) ndarray.

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.. code-block:: default


    data = dist.rvs(nsamples=1000)

    fig, ax = plt.subplots(ncols=1, figsize=(10, 4))
    ax.scatter(data[:, 0], data[:, 1], alpha=0.2)
    ax.set_xlabel('dimension 1')
    ax.set_ylabel('dimension 2')
    ax.set_title('random samples')
    ax.set_xlim([-4, 6])
    ax.set_ylim([-2, 6])




.. image-sg:: /auto_examples/distributions/multivariate/images/sphx_glr_plot_multivariate_normal_002.png
   :alt: random samples
   :srcset: /auto_examples/distributions/multivariate/images/sphx_glr_plot_multivariate_normal_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (-2.0, 6.0)




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

   **Total running time of the script:** ( 0 minutes  0.240 seconds)


.. _sphx_glr_download_auto_examples_distributions_multivariate_plot_multivariate_normal.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/SURGroup/UQpy/master?urlpath=lab/tree/notebooks/auto_examples/distributions/multivariate/plot_multivariate_normal.ipynb
        :alt: Launch binder
        :width: 150 px



    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_multivariate_normal.py <plot_multivariate_normal.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_multivariate_normal.ipynb <plot_multivariate_normal.ipynb>`


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