.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_userdefined_spectral_model.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_userdefined_spectral_model.py:


Create a custom spectral model
==============================

This use case illustrates how the User can define his own density
spectral function from parametric models. The library allows it thanks to
the object *UserDefinedSpectralModel* defined from:

-   a frequency grid :math:`(-f_c, \dots, f_c)` with step :math:`\delta f`, stored
    in the object *RegularGrid*,
-   a collection of hermitian matrices :math:`\in \mathbb{M}_d(\mathbb{C})`
    stored in the object *HermitianMatrixCollection*, which are the
    images of each point of the frequency grid through the density
    spectral function.


The library builds a constant piecewise function on :math:`[-f_c,f_c]`, where
the intervals where the density spectral function is constant are
centered on the points of the frequency grid, of length :math:`\delta f`.
Then, it is possible to evaluate the spectral density function for a
given frequency thanks to the method [*computeSpectralDensity*]{}: if
the frequency is not inside the interval :math:`[-f_c,f_c]`, an exception is thrown.
Otherwise, it returns the hermitian matrix of the
subinterval of :math:`[-f_c,f_c]` that contains the given frequency.

In the following script, we illustrate how to create a modified low
pass model of dimension :math:`d=1` with exponential decrease defined by:
:math:`S: \mathbb{R} \rightarrow  \mathbb{R}` where

-   Frequency value :math:`f` should be positive,
-   for :math:`f < 5 Hz`, the spectral density function is constant: :math:`S(f)=1.0`,
-   for :math:`f > 5 Hz`, the spectral density function is equal to :math:`S(f) = \exp \left[- 2.0 (f - 5.0)^2 \right]`.


The frequency grid is :math:`]0, f_c] = ]0,10]` with :math:`\delta f = 0.2` Hz.
The figure draws the spectral density.



.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    import math as m
    ot.Log.Show(ot.Log.NONE)








Create the frequency grid:


.. code-block:: default

    fmin = 0.1
    df = 0.5
    N = int((10.0 - fmin)/ df)
    fgrid =  ot.RegularGrid(fmin, df, N)









Define the spectral function:


.. code-block:: default

    def s(f):
        if(f <= 5.0):
            return 1.0
        else:
            x = f - 5.0
            return m.exp(-2.0 * x * x)









Create the collection of HermitianMatrix:


.. code-block:: default

    coll = ot.HermitianMatrixCollection()
    for k in range(N):
        frequency = fgrid.getValue(k)
        matrix = ot.HermitianMatrix(1)
        matrix[0, 0] = s(frequency)
        coll.add(matrix)








Create the spectral model:


.. code-block:: default

    spectralModel = ot.UserDefinedSpectralModel(fgrid, coll)

    # Get the spectral density function computed at first frequency values
    firstFrequency = fgrid.getStart()
    frequencyStep = fgrid.getStep()
    firstHermitian = spectralModel(firstFrequency)

    # Get the spectral function at frequency + 0.3 * frequencyStep
    spectralModel(frequency + 0.3 * frequencyStep)






.. raw:: html

    <p>[[ (2.50622e-15,0) ]]</p>
    <br />
    <br />

Draw the spectral density


.. code-block:: default


    # Create the curve of the spectral function
    x = ot.Sample(N, 2)
    for k in range(N):
        frequency = fgrid.getValue(k)
        x[k, 0] = frequency
        value = spectralModel(frequency)
        x[k, 1] = value[0, 0].real

    # Create the graph
    graph = ot.Graph('Spectral user-defined model', 'Frequency', 'Spectral density value', True)
    curve = ot.Curve(x, 'UserSpectral')
    graph.add(curve)
    graph.setLegendPosition('topright')
    view = viewer.View(graph)
    plt.show()



.. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_userdefined_spectral_model_001.png
    :alt: Spectral user-defined model
    :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.073 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_userdefined_spectral_model.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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