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

.. only:: html

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

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

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

.. _sphx_glr_auto_reliability_reliability_processes_plot_estimate_probability_monte_carlo_process.py:


Estimate a process-based event probability
==========================================

.. GENERATED FROM PYTHON SOURCE LINES 7-63

The objective of this example is to evaluate the probability of an
event based on a stochastic process, using the Monte Carlo estimator.

Let :math:`X: \Omega \times \mathcal{D} \rightarrow \mathbb{R}^d` be a stochastic process
of dimension :math:`d`, where :math:`\mathcal{D} \in \mathbb{R}^n` is discretized on the mesh
:math:`\mathcal{M}`.

We define the event :math:`\mathcal{E}` as:

.. math::
   \begin{aligned}
     \displaystyle \mathcal{E}(X) = \bigcup_{\underline{t}\in \mathcal{M}}\left\{X_{\underline{t}}  \in \mathcal{A}  \right\}
   \end{aligned}

where :math:`\mathcal{A}` is a domain of :math:`\mathbb{R}^d`.

We estimate the probabilty :math:`p=\mathbb{P}\left(\mathcal{E}(X)\right)` with the Monte Carlo
estimator.

The Monte Carlo algorithm is manipulated the same way as in the case
where the event is based on a random variable independent of time.

We illustrate the algorithm on the example of the bidimensionnal white
noise process :math:`\varepsilon: \Omega \times \mathcal{D} \rightarrow \mathbb{R}^2` where
:math:`\mathcal{D}\in \mathbb{R}`, distributed according to the bidimensionnal standard
normal distribution (with zero mean, unit variance and independent
marginals).

We consider the domain :math:`\mathcal{A} =  [1,2] \times [1,2]`. Then the event :math:`\mathcal{E}` writes:

.. math::
   \begin{aligned}
     \displaystyle \mathcal{E}(\varepsilon) = \bigcup_{\underline{t}\in \mathcal{M}}\left\{\varepsilon_{t}  \in \mathcal{A}  \right\}
   \end{aligned}

For all time stamps :math:`t \in \mathcal{M}`, the probability :math:`p_1` that the process
enters into the domain :math:`\mathcal{A}` at time :math:`t` writes, using the independence
property of the marginals:

.. math::
   \begin{aligned}
     p_1 = \mathbb{P}\left(\varepsilon_t  \in \mathcal{A}\right) = (\Phi(2) - \Phi(1))^2
   \end{aligned}

with :math:`\Phi` the cumulative distribution function of the scalar standard *Normal* distribution.

As the process is discretized on a time grid of size :math:`N` and using the
independence property of the white noise between two different time
stamps and the fact that the white noise follows the same distribution
at each time :math:`t`, the final probability :math:`p` writes:

.. math::
   p = \mathbb{P}\left(\mathcal{E}(\varepsilon)\right) = 1 - (1 - p_1)^{N}

With :math:`K=10^4` realizations, using the Monte Carlo estimator, we obtain :math:`p_K = 0.1627`,
to be compared to the exact value :math:`p=0.17008` for a time grid of size :math:`N=10`.

.. GENERATED FROM PYTHON SOURCE LINES 65-68

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as otv








.. GENERATED FROM PYTHON SOURCE LINES 69-70

Create a time grid

.. GENERATED FROM PYTHON SOURCE LINES 70-75

.. code-block:: Python

    tMin = 0.0
    timeStep = 0.1
    n = 100
    tgrid = ot.RegularGrid(tMin, timeStep, n)








.. GENERATED FROM PYTHON SOURCE LINES 76-77

Create a normal process

.. GENERATED FROM PYTHON SOURCE LINES 77-82

.. code-block:: Python

    amplitude = [5.0]
    scale = [3.0]
    model = ot.ExponentialModel(scale, amplitude)
    process = ot.GaussianProcess(model, tgrid)








.. GENERATED FROM PYTHON SOURCE LINES 83-84

Create the 1-d domain A: [2.,5.]

.. GENERATED FROM PYTHON SOURCE LINES 84-88

.. code-block:: Python

    lowerBound = [2.0]
    upperBound = [5.0]
    domain = ot.Interval(lowerBound, upperBound)








.. GENERATED FROM PYTHON SOURCE LINES 89-90

Create an event from a Process and a Domain

.. GENERATED FROM PYTHON SOURCE LINES 90-92

.. code-block:: Python

    event = ot.ProcessEvent(process, domain)








.. GENERATED FROM PYTHON SOURCE LINES 93-94

Create the Monte-Carlo algorithm

.. GENERATED FROM PYTHON SOURCE LINES 94-111

.. code-block:: Python

    montecarlo = ot.ProbabilitySimulationAlgorithm(event)

    # Define the maximum number of simulations
    montecarlo.setMaximumOuterSampling(1000)

    # Define the block size
    montecarlo.setBlockSize(100)

    # Define the maximum coefficient of variation
    montecarlo.setMaximumCoefficientOfVariation(0.0025)

    # Run the algorithm
    montecarlo.run()

    # Get the result
    montecarlo.getResult()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    probabilityEstimate=9.295122e-01 varianceEstimate=5.380082e-06 standard deviation=2.32e-03 coefficient of variation=2.50e-03 confidenceLength(0.95)=9.09e-03 outerSampling=123 blockSize=100
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 112-115

.. code-block:: Python

    graph = montecarlo.drawProbabilityConvergence(0.95)
    view = otv.View(graph)




.. image-sg:: /auto_reliability/reliability_processes/images/sphx_glr_plot_estimate_probability_monte_carlo_process_001.svg
   :alt: ProbabilitySimulationAlgorithm convergence graph at level 0.95
   :srcset: /auto_reliability/reliability_processes/images/sphx_glr_plot_estimate_probability_monte_carlo_process_001.svg
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 116-117

.. code-block:: Python

    otv.View.ShowAll()








.. _sphx_glr_download_auto_reliability_reliability_processes_plot_estimate_probability_monte_carlo_process.py:

.. only:: html

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

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

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

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

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

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