.. 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_1d_mv.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_1d_mv.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_1d_mv.py:


One-dimensional & multiple variables
=================================================================

In this example, the Spectral Representation Method is used to generate stochastic processes from a prescribed Power
Spectrum and associated Cross Spectral Density. This example illustrates how to use the SRM class for a one dimensional
and 'm' 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 :class:`stochastic_processes` module of UQpy.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: default


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


.. GENERATED FROM PYTHON SOURCE LINES 25-26

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

.. GENERATED FROM PYTHON SOURCE LINES 29-47

.. code-block:: default


    n_sim = 10000  # Num of samples

    n = 1 # Num of dimensions

    m = 3 # Num of variables

    T = 10  # Time(1 / T = dw)
    nt = 256  # Num.of Discretized Time
    F = 1 / T * nt / 2  # Frequency.(Hz)
    nf = 128  # Num of Discretized Freq.

    # # Generation of Input Data(Stationary)
    dt = T / nt
    t = np.linspace(0, T - dt, nt)
    df = F / nf
    f = np.linspace(0, F - df, nf)


.. GENERATED FROM PYTHON SOURCE LINES 48-49

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

.. GENERATED FROM PYTHON SOURCE LINES 52-58

.. code-block:: default


    t_u = 2*np.pi/2/F

    if dt>t_u:
        print('Error')


.. GENERATED FROM PYTHON SOURCE LINES 59-60

Defining the Power Spectral Density Function (S) and the Cross Spectral Density (g)

.. GENERATED FROM PYTHON SOURCE LINES 63-107

.. code-block:: default


    S_11 = 38.3 / (1 + 6.19 * f) ** (5 / 3)
    S_22 = 43.4 / (1 + 6.98 * f) ** (5 / 3)
    S_33 = 135 / (1 + 21.8 * f) ** (5 / 3)

    g_12 = np.exp(-0.1757 * f)
    g_13 = np.exp(-3.478 * f)
    g_23 = np.exp(-3.392 * f)

    S_list = np.array([S_11, S_22, S_33])
    g_list = np.array([g_12, g_13, g_23])

    # Assembly of S_jk
    S_sqrt = np.sqrt(S_list)
    S_jk = np.einsum('i...,j...->ij...', S_sqrt, S_sqrt)
    # Assembly of g_jk
    g_jk = np.zeros_like(S_jk)
    counter = 0
    for i in range(m):
        for j in range(i + 1, m):
            g_jk[i, j] = g_list[counter]
            counter = counter + 1
    g_jk = np.einsum('ij...->ji...', g_jk) + g_jk

    for i in range(m):
        g_jk[i, i] = np.ones_like(S_jk[0, 0])
    S = S_jk * g_jk


    SRM_object = SpectralRepresentation(n_sim, S, dt, df, nt, nf)
    samples = SRM_object.samples

    fig, ax = plt.subplots()
    plt.title('Realisation of the Spectral Representation Method')
    plt.plot(t, samples[0, 0], label='1st dimension')
    plt.plot(t, samples[0, 1], label='2nd dimension')
    plt.plot(t, samples[0, 2], label='3rd dimension')
    ax.yaxis.grid(True)
    ax.xaxis.grid(True)
    plt.legend()
    plt.show()

    print('The mean of the samples is ', np.mean(samples), 'whereas the expected mean is 0.000')
    print('The variance of the samples is ', np.var(samples), 'whereas the expected variance is ',
          np.sum(S_list)*np.prod(df)*2/m)

.. 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_1d_mv.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_1d_mv.ipynb
        :alt: Launch binder
        :width: 150 px



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

      :download:`Download Python source code: spectral_1d_mv.py <spectral_1d_mv.py>`

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

      :download:`Download Jupyter notebook: spectral_1d_mv.ipynb <spectral_1d_mv.ipynb>`


.. only:: html

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

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