.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/stochastic_processes/spectral/spectral_nd_1d.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_stochastic_processes_spectral_spectral_nd_1d.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_stochastic_processes_spectral_spectral_nd_1d.py:


N-dimensional & one-variable
=================================================================

In this example, the Spectral Representation Method is used to generate stochastic processes from a prescribed Power
Spectrum. This example illustrates how to use the :class:`.SpectralRepresentation` class for 'n' dimensional and one
variable case and compare the statistics of the generated stochastic processes with the expected values.

.. GENERATED FROM PYTHON SOURCE LINES 13-15

Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to
import the :class:`.SpectralRepresentation` class from the StochasticProcesses module of UQpy.

.. GENERATED FROM PYTHON SOURCE LINES 18-25

.. code-block:: default


    from UQpy.stochastic_process import SpectralRepresentation
    import numpy as np
    import matplotlib.pyplot as plt
    from pylab import *
    plt.style.use('seaborn')


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The input parameters necessary for the generating of the stochastic processes are given below:

.. GENERATED FROM PYTHON SOURCE LINES 30-50

.. code-block:: default


    n_sim = 1000 # Num of samples

    n = 2 # Num of dimensions

    m = 1 # Num of variables

    T = 10
    nt = 200
    dt = T/nt
    t = np.linspace(0, T-dt, nt)

    # Frequency
    W = np.array([1.0, 1.5])
    nw = 100
    dw = W / nw
    x_list = [np.linspace(0, W[i] - dw[i], nw) for i in range(n)]
    xy_list = np.array(np.meshgrid(*x_list, indexing='ij'))



.. GENERATED FROM PYTHON SOURCE LINES 51-52

Defining the Power Spectral Density(S)

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.. code-block:: default


    S_nd_1v = 125 / 4 * np.linalg.norm(xy_list, axis=0) ** 2 * np.exp(-5 * np.linalg.norm(xy_list, axis=0))


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Make sure that the input parameters are in order to prevent aliasing

.. GENERATED FROM PYTHON SOURCE LINES 63-85

.. code-block:: default


    t_u = 2*np.pi/2/W

    if dt>t_u.all():
        print('Error')

    SRM_object = SpectralRepresentation(n_sim, S_nd_1v, [dt, dt], dw, [nt, nt], [nw, nw], random_state=128)
    samples_nd_1v = SRM_object.samples

    t_list = [t for _ in range(n)]
    tt_list = np.array(np.meshgrid(*t_list, indexing='ij'))

    fig1 = plt.figure()
    plt.title('2d random field with a prescribed Power Spectrum')
    pcm = pcolor(tt_list[0], tt_list[1], samples_nd_1v[0, 0], cmap='RdBu_r', vmin=-6, vmax=6)
    plt.colorbar(pcm, extend='both', orientation='vertical')
    plt.xlabel('$X_{1}$')
    plt.ylabel('$X_{2}$')
    plt.show()

    print('The mean of the samples is ', np.mean(samples_nd_1v), 'whereas the expected mean is 0.000')
    print('The variance of the samples is ', np.var(samples_nd_1v), 'whereas the expected variance is ',
          np.sum(S_nd_1v)*np.prod(dw)*(2**n))

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

   **Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_stochastic_processes_spectral_spectral_nd_1d.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/stochastic_processes/spectral/spectral_nd_1d.ipynb
        :alt: Launch binder
        :width: 150 px



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

      :download:`Download Python source code: spectral_nd_1d.py <spectral_nd_1d.py>`

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

      :download:`Download Jupyter notebook: spectral_nd_1d.ipynb <spectral_nd_1d.ipynb>`


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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_