.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_data_analysis/estimate_stochastic_processes/plot_estimate_multivariate_arma.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_data_analysis_estimate_stochastic_processes_plot_estimate_multivariate_arma.py>`
        to download the full example code.

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

.. _sphx_glr_auto_data_analysis_estimate_stochastic_processes_plot_estimate_multivariate_arma.py:


Estimate a multivariate ARMA process
====================================

.. GENERATED FROM PYTHON SOURCE LINES 7-33

The objective of the Use Case is to estimate a multivariate ARMA model
from a stationary time series using the maximum likelihood estimator
and a centered normal white noise.
The data can be a unique time series or several time series collected
in a process sample.

We estimate :math:`(\underline{\beta}, \sigma^2)` thanks to the
class:`~openturns.ARMALikelihoodFactory` object and its method *build*, acting on a
time series or on a sample of time series. It produces a result of
type *ARMA*.
Note that no evaluation of selection criteria such as *AIC* and *BIC*
is done.

The synthetic data is generated from the 2-d ARMA model:

.. math::
   \begin{aligned}
   X_{0,t} - 0.5 X_{0,t-1} - 0.1 X_{1,t-1} = E_{0,t} - 0.4 E_{0,t-1} \\
   X_{1,t} - 0.4 X_{0,t-1} - 0.5 X_{1,t-1} - 0.25 X_{0,t-2} = E_{1,t} - 0.4 E_{1,t-1}
   \end{aligned}

with E the white noise:

.. math::
    E \sim \mathcal{N} ([0,0], [0.1,0.2])


.. GENERATED FROM PYTHON SOURCE LINES 35-39

.. code-block:: Python

    import openturns as ot

    ot.Log.Show(ot.Log.NONE)








.. GENERATED FROM PYTHON SOURCE LINES 40-41

Create a 2-d ARMA process

.. GENERATED FROM PYTHON SOURCE LINES 41-66

.. code-block:: Python

    p = 2
    q = 1
    dim = 2

    # Tmin , Tmax and N points for TimeGrid
    dt = 1.0
    size = 400
    timeGrid = ot.RegularGrid(0.0, dt, size)

    # white noise
    cov = ot.CovarianceMatrix([[0.1, 0.0], [0.0, 0.2]])
    whiteNoise = ot.WhiteNoise(ot.Normal([0.0] * dim, cov), timeGrid)

    # AR/MA coefficients
    ar = ot.ARMACoefficients(p, dim)
    ar[0] = ot.SquareMatrix([[-0.5, -0.1], [-0.4, -0.5]])
    ar[1] = ot.SquareMatrix([[0.0, 0.0], [-0.25, 0.0]])

    ma = ot.ARMACoefficients(q, dim)
    ma[0] = ot.SquareMatrix([[-0.4, 0.0], [0.0, -0.4]])

    # ARMA model creation
    arma = ot.ARMA(ar, ma, whiteNoise)
    arma






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class= ARMA timeGrid=class=RegularGrid name=Unnamed start=0 step=1 n=400 coefficients AR=class=ARMACoefficients, shift=0, value=class=SquareMatrix dimension=2 implementation=class=MatrixImplementation name=Unnamed rows=2 columns=2 values=[-0.5,-0.4,-0.1,-0.5], shift=1, value=class=SquareMatrix dimension=2 implementation=class=MatrixImplementation name=Unnamed rows=2 columns=2 values=[0,-0.25,0,0] coefficients MA=class=ARMACoefficients, shift=0, value=class=SquareMatrix dimension=2 implementation=class=MatrixImplementation name=Unnamed rows=2 columns=2 values=[-0.4,0,0,-0.4] noiseDistribution= class=Normal name=Normal dimension=2 mean=class=Point name=Unnamed dimension=2 values=[0,0] sigma=class=Point name=Unnamed dimension=2 values=[0.316228,0.447214] correlationMatrix=class=CorrelationMatrix dimension=2 implementation=class=MatrixImplementation name=Unnamed rows=2 columns=2 values=[1,0,0,1] state= class= ARMAState x= class=Sample name=Unnamed implementation=class=SampleImplementation name=Unnamed size=2 dimension=2 data=[[-0.420367,-1.68311],[0.0105101,-1.12405]] epsilon= class=Sample name=Unnamed implementation=class=SampleImplementation name=Unnamed size=1 dimension=2 data=[[0.29469,-0.47215]]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 67-68

Create a realization

.. GENERATED FROM PYTHON SOURCE LINES 68-70

.. code-block:: Python

    timeSeries = ot.TimeSeries(arma.getRealization())








.. GENERATED FROM PYTHON SOURCE LINES 71-72

Estimate the process from the previous realization

.. GENERATED FROM PYTHON SOURCE LINES 72-77

.. code-block:: Python

    factory = ot.ARMALikelihoodFactory(p, q, dim)
    factory.setInitialConditions(ar, ma, cov)

    arma_est = ot.ARMA(factory.build(timeSeries))
    print("Estimated ARMA= ", arma_est)




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

 .. code-block:: none

    Estimated ARMA=  ARMA(X_{0,t} - 0.383177 X_{0,t-1} - 0.249709 X_{1,t-1} + 0.158317 X_{0,t-2} + 0.0745476 X_{1,t-2} = E_{0,t} - 0.298018 E_{0,t-1} - 0.154134 E_{1,t-1}
    X_{1,t} - 0.350859 X_{0,t-1} - 0.614683 X_{1,t-1} - 0.274685 X_{0,t-2} + 0.0364205 X_{1,t-2} = E_{1,t} - 0.106077 E_{0,t-1} - 0.493617 E_{1,t-1}, E_t ~ Normal(mu = [0,0], sigma = [0.297786,0.425897], R = [[  1         -0.0794717 ]
     [ -0.0794717  1         ]]))





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

   **Total running time of the script:** (0 minutes 4.095 seconds)


.. _sphx_glr_download_auto_data_analysis_estimate_stochastic_processes_plot_estimate_multivariate_arma.py:

.. only:: html

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

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

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

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

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

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_estimate_multivariate_arma.zip <plot_estimate_multivariate_arma.zip>`