.. 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_non_stationary_covariance_model.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_data_analysis_estimate_stochastic_processes_plot_estimate_non_stationary_covariance_model.py: Estimate a non stationary covariance function ============================================= .. GENERATED FROM PYTHON SOURCE LINES 7-29 The objective of this use case is to estimate :math:`C` from several fields generated by the process :math:`X`. We suppose that the process is not stationary. In the following example, we illustrate the estimation of the non stationary covariance model :math:`C : \mathcal{D} \times [-4, 4] \rightarrow \mathbb{R}` defined by: .. math:: \begin{aligned} \displaystyle C(s,t) = \exp\left(-\dfrac{4|s-t|}{1+s^2+t^2}\right) \end{aligned} The domain :math:`\mathcal{D}` is discretized on a mesh :math:`\mathcal{M}` which is a time grid with 64 points. We build a normal process :math:`X: \Omega \times [-4, 4] \rightarrow \mathbb{R}` with zero mean and :math:`C` as covariance function. We discretize the covariance model :math:`C` using :math:`C(t_k, t_\ell)` for each :math:`(t_k, t_\ell)\in \mathcal{M} \times \mathcal{M}`. We get a :math:`N=10^3` fields from the process :math:`X` from which we estimate the covariance model :math:`C`. We use the object *NonStationaryCovarianceModelFactory* which creates a *UserDefinedCovarianceModel*. .. GENERATED FROM PYTHON SOURCE LINES 31-38 .. code-block:: Python import math as m import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 39-40 Create the time grid .. GENERATED FROM PYTHON SOURCE LINES 40-47 .. code-block:: Python t0 = -4.0 tmax = 4.0 N = 64 dt = (tmax - t0) / N tgrid = ot.RegularGrid(t0, dt, N) .. GENERATED FROM PYTHON SOURCE LINES 48-49 Create the covariance function at (s,t) .. GENERATED FROM PYTHON SOURCE LINES 49-53 .. code-block:: Python def C(s, t): return m.exp(-4.0 * abs(s - t) / (1 + (s * s + t * t))) .. GENERATED FROM PYTHON SOURCE LINES 54-55 Draw... .. GENERATED FROM PYTHON SOURCE LINES 55-67 .. code-block:: Python def f(X): s, t = X return [C(s, t)] func = ot.PythonFunction(2, 1, f) func.setDescription([":math:`s`", ":math:`t`", ":math:`cov`"]) graph = func.draw([t0] * 2, [tmax] * 2) graph.setTitle("Original covariance model") graph.setLegendPosition("") view = viewer.View(graph) .. image-sg:: /auto_data_analysis/estimate_stochastic_processes/images/sphx_glr_plot_estimate_non_stationary_covariance_model_001.png :alt: Original covariance model :srcset: /auto_data_analysis/estimate_stochastic_processes/images/sphx_glr_plot_estimate_non_stationary_covariance_model_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 68-69 Create data from a non stationary normal process Omega * [0,T]--> R .. GENERATED FROM PYTHON SOURCE LINES 69-80 .. code-block:: Python # Create the collection of HermitianMatrix covariance = ot.CovarianceMatrix(N) for k in range(N): s = tgrid.getValue(k) for ll in range(k + 1): t = tgrid.getValue(ll) covariance[k, ll] = C(s, t) covmodel = ot.UserDefinedCovarianceModel(tgrid, covariance) .. GENERATED FROM PYTHON SOURCE LINES 81-83 Create the normal process with that covariance model based on the mesh tgrid .. GENERATED FROM PYTHON SOURCE LINES 83-89 .. code-block:: Python process = ot.GaussianProcess(covmodel, tgrid) # Get a sample of fields from the process N = 1000 sample = process.getSample(N) .. GENERATED FROM PYTHON SOURCE LINES 90-91 The covariance model factory .. GENERATED FROM PYTHON SOURCE LINES 91-96 .. code-block:: Python factory = ot.NonStationaryCovarianceModelFactory() # Estimation on a sample estimatedModel = factory.build(sample) .. GENERATED FROM PYTHON SOURCE LINES 97-102 .. code-block:: Python graph = estimatedModel.draw(0, 0, t0, tmax, 256, False) graph.setTitle("Estimated covariance model") graph.setLegendPosition("") view = viewer.View(graph) plt.show() .. image-sg:: /auto_data_analysis/estimate_stochastic_processes/images/sphx_glr_plot_estimate_non_stationary_covariance_model_002.png :alt: Estimated covariance model :srcset: /auto_data_analysis/estimate_stochastic_processes/images/sphx_glr_plot_estimate_non_stationary_covariance_model_002.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_data_analysis_estimate_stochastic_processes_plot_estimate_non_stationary_covariance_model.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_non_stationary_covariance_model.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_estimate_non_stationary_covariance_model.py `