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

.. only:: html

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

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

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

.. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_draw_covariance_models.py:


Kriging : draw covariance models
================================

.. GENERATED FROM PYTHON SOURCE LINES 5-11

.. code-block:: Python


    import openturns as ot
    import openturns.viewer as otv
    from matplotlib import pylab as plt
    import pylab as pl








.. GENERATED FROM PYTHON SOURCE LINES 12-21

Abstract
--------

Gaussian processes are a common fixture in `UQ`.
They are defined by their covariance function and the library implements several of them.
In this example we plot covariance functions and modify their parameters
for two families of models: the generalized exponential model and the Matérn models.

For visualization sake, we limit ourselves to the dimension 1.

.. GENERATED FROM PYTHON SOURCE LINES 21-23

.. code-block:: Python

    dimension = 1








.. GENERATED FROM PYTHON SOURCE LINES 24-25

We set the lower bound to zero for stationary kernels

.. GENERATED FROM PYTHON SOURCE LINES 25-28

.. code-block:: Python

    ot.ResourceMap.SetAsScalar("CovarianceModel-DefaultTMin", 0.0)









.. GENERATED FROM PYTHON SOURCE LINES 29-36

The generalized exponential model
---------------------------------

The :class:`~openturns.GeneralizedExponential` class implements a generalized exponential with a
parameter :math:`p < 0 \leq 2` exponent. The case :math:`p=1` is the standard exponential model
while :math:`p=2` is the squared exponential.


.. GENERATED FROM PYTHON SOURCE LINES 38-44

Various parameters p and a fixed correlation length of 0.1
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In this part we set the correlation length to :math:`\theta = 0.1` and study three different models
with parameters :math:`p=0.25`, :math:`p=1` and :math:`p=2` and trajectories from Gaussian processes
based on these models.

.. GENERATED FROM PYTHON SOURCE LINES 46-47

We define the :math:`p = 0.25` generalized exponential model :

.. GENERATED FROM PYTHON SOURCE LINES 47-49

.. code-block:: Python

    covarianceModel = ot.GeneralizedExponential([0.1], 0.25)








.. GENERATED FROM PYTHON SOURCE LINES 50-51

We define the :math:`p = 1` generalized exponential model :

.. GENERATED FROM PYTHON SOURCE LINES 51-53

.. code-block:: Python

    covarianceModel2 = ot.GeneralizedExponential([0.1], 1.0)








.. GENERATED FROM PYTHON SOURCE LINES 54-55

We define the :math:`p = 2` generalized exponential model :

.. GENERATED FROM PYTHON SOURCE LINES 55-57

.. code-block:: Python

    covarianceModel3 = ot.GeneralizedExponential([0.1], 2.0)








.. GENERATED FROM PYTHON SOURCE LINES 58-59

We draw the covariance models :

.. GENERATED FROM PYTHON SOURCE LINES 59-68

.. code-block:: Python

    graphModel = covarianceModel.draw()
    graphModel.add(covarianceModel2.draw())
    graphModel.add(covarianceModel3.draw())
    graphModel.setColors(["green", "orange", "blue"])
    graphModel.setXTitle(r"$\tau = \|s-t\|$")
    graphModel.setYTitle(r"$C(\tau)$")
    graphModel.setLegends([r"$p = 0.25$", r"$p = 1$", r"$p = 2$"])









.. GENERATED FROM PYTHON SOURCE LINES 69-72

For each covariance model we build a Gaussian process and generate a random trajectory of
on :math:`[-1,1]`.
We first build a discretization of this interval with a regular grid with step 0.01.

.. GENERATED FROM PYTHON SOURCE LINES 72-78

.. code-block:: Python

    xmin = -1.0
    step = 0.01
    n = 200
    grid1D = ot.RegularGrid(xmin, step, n + 1)
    nbTrajectories = 1








.. GENERATED FROM PYTHON SOURCE LINES 79-80

We define the first Gaussian process and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 80-83

.. code-block:: Python

    process = ot.GaussianProcess(covarianceModel, grid1D)
    sample = process.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 84-85

then the second one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 85-88

.. code-block:: Python

    process2 = ot.GaussianProcess(covarianceModel2, grid1D)
    sample2 = process2.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 89-90

and finally the third one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 90-93

.. code-block:: Python

    process3 = ot.GaussianProcess(covarianceModel3, grid1D)
    sample3 = process3.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 94-95

We draw the trajectories :

.. GENERATED FROM PYTHON SOURCE LINES 95-104

.. code-block:: Python

    graphTraj = sample.drawMarginal(0)
    graphTraj.add(sample2.drawMarginal(0))
    graphTraj.add(sample3.drawMarginal(0))
    graphTraj.setXTitle(r"$x$")
    graphTraj.setYTitle(r"$GP_{\nu}(x)$")
    graphTraj.setTitle("Random realization from the covariance model")
    graphTraj.setColors(["green", "orange", "blue"])
    graphTraj.setLegends([r"$p = 0.25$", r"$p = 1$", r"$p = 2$"])








.. GENERATED FROM PYTHON SOURCE LINES 105-106

We present each covariance model and the corresponding trajectory side by side.

.. GENERATED FROM PYTHON SOURCE LINES 106-113

.. code-block:: Python

    fig = pl.figure(figsize=(12, 4))
    ax_pdf = fig.add_subplot(1, 2, 1)
    _ = otv.View(graphModel, figure=fig, axes=[ax_pdf])
    ax_cdf = fig.add_subplot(1, 2, 2)
    _ = otv.View(graphTraj, figure=fig, axes=[ax_cdf])
    _ = fig.suptitle(r"Generalized Exponential Model : influence of the p parameter")




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_001.png
   :alt: Generalized Exponential Model : influence of the p parameter
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 114-116

The blue trajectory corresponding to the parameter :math:`p=2` is smooth as expected as compared with
the :math:`p=0.25` process which is less regular.

.. GENERATED FROM PYTHON SOURCE LINES 119-125

The exponential model (:math:`p=1`)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In the case of the exponential model (:math:`p=1`) we show the influence of the correlation length on
the trajectories.


.. GENERATED FROM PYTHON SOURCE LINES 127-128

with correlation length :math:`\theta = 0.01`  :

.. GENERATED FROM PYTHON SOURCE LINES 128-130

.. code-block:: Python

    covarianceModel = ot.GeneralizedExponential([0.01], 1.0)








.. GENERATED FROM PYTHON SOURCE LINES 131-132

with correlation length :math:`\theta = 0.1` :

.. GENERATED FROM PYTHON SOURCE LINES 132-134

.. code-block:: Python

    covarianceModel2 = ot.GeneralizedExponential([0.1], 1.0)








.. GENERATED FROM PYTHON SOURCE LINES 135-136

with correlation length :math:`\theta = 1.0`

.. GENERATED FROM PYTHON SOURCE LINES 136-138

.. code-block:: Python

    covarianceModel3 = ot.GeneralizedExponential([1.0], 1.0)








.. GENERATED FROM PYTHON SOURCE LINES 139-140

We draw the covariance models :

.. GENERATED FROM PYTHON SOURCE LINES 140-149

.. code-block:: Python

    graphModel = covarianceModel.draw()
    graphModel.add(covarianceModel2.draw())
    graphModel.add(covarianceModel3.draw())
    graphModel.setColors(["green", "orange", "blue"])
    graphModel.setXTitle(r"$\tau = \|s-t\|$")
    graphModel.setYTitle(r"$C(\tau)$")
    graphModel.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1$"])









.. GENERATED FROM PYTHON SOURCE LINES 150-153

For each covariance model we build a Gaussian process and generate a random trajectory of
on :math:`[-1,1]`.
We first build a discretization of this interval with a regular grid with step 0.01.

.. GENERATED FROM PYTHON SOURCE LINES 153-159

.. code-block:: Python

    xmin = -1.0
    step = 0.01
    n = 200
    grid1D = ot.RegularGrid(xmin, step, n + 1)
    nbTrajectories = 1








.. GENERATED FROM PYTHON SOURCE LINES 160-161

We define the first Gaussian process and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 161-164

.. code-block:: Python

    process = ot.GaussianProcess(covarianceModel, grid1D)
    sample = process.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 165-166

then the second one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 166-169

.. code-block:: Python

    process2 = ot.GaussianProcess(covarianceModel2, grid1D)
    sample2 = process2.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 170-171

and finally the third one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 171-174

.. code-block:: Python

    process3 = ot.GaussianProcess(covarianceModel3, grid1D)
    sample3 = process3.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 175-176

We draw the trajectories :

.. GENERATED FROM PYTHON SOURCE LINES 176-185

.. code-block:: Python

    graphTraj = sample.drawMarginal(0)
    graphTraj.add(sample2.drawMarginal(0))
    graphTraj.add(sample3.drawMarginal(0))
    graphTraj.setXTitle(r"$x$")
    graphTraj.setYTitle(r"$GP_{\theta}(x)$")
    graphTraj.setTitle("Random realization from the covariance model")
    graphTraj.setColors(["green", "orange", "blue"])
    graphTraj.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1$"])








.. GENERATED FROM PYTHON SOURCE LINES 186-187

We present each covariance model and the corresponding tracjectory side by side.

.. GENERATED FROM PYTHON SOURCE LINES 187-194

.. code-block:: Python

    fig = pl.figure(figsize=(12, 4))
    ax_pdf = fig.add_subplot(1, 2, 1)
    _ = otv.View(graphModel, figure=fig, axes=[ax_pdf])
    ax_cdf = fig.add_subplot(1, 2, 2)
    _ = otv.View(graphTraj, figure=fig, axes=[ax_cdf])
    _ = fig.suptitle(r"Exponential Model : influence of correlation length $\theta$")




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_002.png
   :alt: Exponential Model : influence of correlation length $\theta$
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 195-196

We observe a smoother trajectory with a high correlation value.

.. GENERATED FROM PYTHON SOURCE LINES 199-206

The squared exponential (:math:`p=2`)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^


In the case of the squared exponential model (:math:`p=2`) we show the influence of the correlation length on
the trajectories.


.. GENERATED FROM PYTHON SOURCE LINES 208-209

with correlation length :math:`\theta = 0.01`  :

.. GENERATED FROM PYTHON SOURCE LINES 209-211

.. code-block:: Python

    covarianceModel = ot.GeneralizedExponential([0.01], 2.0)








.. GENERATED FROM PYTHON SOURCE LINES 212-213

with correlation length :math:`\theta = 0.1` :

.. GENERATED FROM PYTHON SOURCE LINES 213-215

.. code-block:: Python

    covarianceModel2 = ot.GeneralizedExponential([0.1], 2.0)








.. GENERATED FROM PYTHON SOURCE LINES 216-217

with correlation length :math:`\theta = 1.0`

.. GENERATED FROM PYTHON SOURCE LINES 217-219

.. code-block:: Python

    covarianceModel3 = ot.GeneralizedExponential([1.0], 2.0)








.. GENERATED FROM PYTHON SOURCE LINES 220-221

We draw the covariance models :

.. GENERATED FROM PYTHON SOURCE LINES 221-230

.. code-block:: Python

    graphModel = covarianceModel.draw()
    graphModel.add(covarianceModel2.draw())
    graphModel.add(covarianceModel3.draw())
    graphModel.setColors(["green", "orange", "blue"])
    graphModel.setXTitle(r"$\tau = \|s-t\|$")
    graphModel.setYTitle(r"$C(\tau)$")
    graphModel.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1$"])









.. GENERATED FROM PYTHON SOURCE LINES 231-234

For each covariance model we build a Gaussian process and generate a random trajectory of
on :math:`[-1,1]`.
We first build a discretization of this interval with a regular grid with step 0.01.

.. GENERATED FROM PYTHON SOURCE LINES 234-240

.. code-block:: Python

    xmin = -1.0
    step = 0.01
    n = 200
    grid1D = ot.RegularGrid(xmin, step, n + 1)
    nbTrajectories = 1








.. GENERATED FROM PYTHON SOURCE LINES 241-242

We define the first Gaussian process and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 242-245

.. code-block:: Python

    process = ot.GaussianProcess(covarianceModel, grid1D)
    sample = process.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 246-247

then the second one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 247-250

.. code-block:: Python

    process2 = ot.GaussianProcess(covarianceModel2, grid1D)
    sample2 = process2.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 251-252

and finally the third one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 252-255

.. code-block:: Python

    process3 = ot.GaussianProcess(covarianceModel3, grid1D)
    sample3 = process3.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 256-257

We draw the trajectories :

.. GENERATED FROM PYTHON SOURCE LINES 257-266

.. code-block:: Python

    graphTraj = sample.drawMarginal(0)
    graphTraj.add(sample2.drawMarginal(0))
    graphTraj.add(sample3.drawMarginal(0))
    graphTraj.setXTitle(r"$x$")
    graphTraj.setYTitle(r"$GP_{\theta}(x)$")
    graphTraj.setTitle("Random realization from the covariance model")
    graphTraj.setColors(["green", "orange", "blue"])
    graphTraj.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1$"])








.. GENERATED FROM PYTHON SOURCE LINES 267-268

We present each covariance model and the corresponding tracjectory side by side.

.. GENERATED FROM PYTHON SOURCE LINES 268-278

.. code-block:: Python


    fig = pl.figure(figsize=(12, 4))
    ax_pdf = fig.add_subplot(1, 2, 1)
    _ = otv.View(graphModel, figure=fig, axes=[ax_pdf])
    ax_cdf = fig.add_subplot(1, 2, 2)
    _ = otv.View(graphTraj, figure=fig, axes=[ax_cdf])
    _ = fig.suptitle(
        r"Squared exponential model : influence of correlation length $\theta$"
    )




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_003.png
   :alt: Squared exponential model : influence of correlation length $\theta$
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 279-281

Except for very small values of the correlation length, trajectories are usually smooth. It is the
main effect of the squared exponential model which leads to smooth processes.

.. GENERATED FROM PYTHON SOURCE LINES 284-290

The Matérn covariance model
---------------------------

The :class:`~openturns.MaternModel` class implements the Matern model of parameter :math:`\nu`.
This parameter controls the smoothness of the process : for any :math:`\nu = n + \frac{1}{2}` the
process is :math:`n` times continuously differentiable.

.. GENERATED FROM PYTHON SOURCE LINES 292-299

Influence of the regularity
^^^^^^^^^^^^^^^^^^^^^^^^^^^

In this paragraph we represent three models with different regularity and generate the
corresponding random trajectories. We shall use :math:`\nu = 0.5`, :math:`\nu = 1.5` and
:math:`\nu = 2.5` and observe the regularity.


.. GENERATED FROM PYTHON SOURCE LINES 301-302

We define the :math:`\nu = 0.5` Matern model :

.. GENERATED FROM PYTHON SOURCE LINES 302-304

.. code-block:: Python

    covarianceModel = ot.MaternModel([1.0], 0.5)








.. GENERATED FROM PYTHON SOURCE LINES 305-306

We define the :math:`\nu = 1.5` Matern model :

.. GENERATED FROM PYTHON SOURCE LINES 306-308

.. code-block:: Python

    covarianceModel2 = ot.MaternModel([1.0], 1.5)








.. GENERATED FROM PYTHON SOURCE LINES 309-310

We define the :math:`\nu = 2.5` Matern model :

.. GENERATED FROM PYTHON SOURCE LINES 310-312

.. code-block:: Python

    covarianceModel3 = ot.MaternModel([1.0], 2.5)








.. GENERATED FROM PYTHON SOURCE LINES 313-314

We draw the covariance models :

.. GENERATED FROM PYTHON SOURCE LINES 314-323

.. code-block:: Python

    graphModel = covarianceModel.draw()
    graphModel.add(covarianceModel2.draw())
    graphModel.add(covarianceModel3.draw())
    graphModel.setColors(["green", "orange", "blue"])
    graphModel.setXTitle(r"$\tau = \|s-t\|$")
    graphModel.setYTitle(r"$C(\tau)$")
    graphModel.setLegends([r"$\nu = 1/2$", r"$\nu = 3/2$", r"$\nu = 5/2$"])









.. GENERATED FROM PYTHON SOURCE LINES 324-327

For each covariance model we build a Gaussian process and generate a random trajectory of
on :math:`[-1,1]`.
We first build a discretization of this interval with a regular grid with step 0.001.

.. GENERATED FROM PYTHON SOURCE LINES 327-333

.. code-block:: Python

    xmin = -5.0
    step = 0.01
    n = 1000
    grid1D = ot.RegularGrid(xmin, step, n + 1)
    nbTrajectories = 1








.. GENERATED FROM PYTHON SOURCE LINES 334-335

We define the first Gaussian process and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 335-338

.. code-block:: Python

    process = ot.GaussianProcess(covarianceModel, grid1D)
    sample = process.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 339-340

then the second one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 340-343

.. code-block:: Python

    process2 = ot.GaussianProcess(covarianceModel2, grid1D)
    sample2 = process2.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 344-345

and finally the third one and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 345-348

.. code-block:: Python

    process3 = ot.GaussianProcess(covarianceModel3, grid1D)
    sample3 = process3.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 349-350

We draw the trajectories :

.. GENERATED FROM PYTHON SOURCE LINES 350-359

.. code-block:: Python

    graphTraj = sample.drawMarginal(0)
    graphTraj.add(sample2.drawMarginal(0))
    graphTraj.add(sample3.drawMarginal(0))
    graphTraj.setXTitle(r"$x$")
    graphTraj.setYTitle(r"$GP_{\nu}(x)$")
    graphTraj.setTitle("Random realization from the covariance model")
    graphTraj.setColors(["green", "orange", "blue"])
    graphTraj.setLegends([r"$\nu = 1/2$", r"$\nu = 3/2$", r"$\nu = 5/2$"])








.. GENERATED FROM PYTHON SOURCE LINES 360-361

We present each covariance model and the corresponding tracjectory side by side.

.. GENERATED FROM PYTHON SOURCE LINES 361-368

.. code-block:: Python

    fig = pl.figure(figsize=(12, 4))
    ax_pdf = fig.add_subplot(1, 2, 1)
    _ = otv.View(graphModel, figure=fig, axes=[ax_pdf])
    ax_cdf = fig.add_subplot(1, 2, 2)
    _ = otv.View(graphTraj, figure=fig, axes=[ax_cdf])
    _ = fig.suptitle(r"Matern model : influence of the regularity $\nu$ parameter")




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_004.png
   :alt: Matern model : influence of the regularity $\nu$ parameter
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 369-371

The red trajectory is the least regular (:math:`nu = 0.5`) as it is only continuous. We see that the
the blue trajectory is more smooth as expected.

.. GENERATED FROM PYTHON SOURCE LINES 374-382

Variation of the correlation length
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In this paragraph we fix the regularity by choosing :math:`\nu = 1.5` so we expect a continuously
differentiable realization.
We then use three different correlation lengths :math:`\theta = 0.01`, :math:`\theta = 0.1` and
:math:`\theta = 1.0` and observe the impact on realizations of Gaussian processes based on these
covariance models.

.. GENERATED FROM PYTHON SOURCE LINES 384-385

We define the Matern model with :math:`\theta = 0.01` :

.. GENERATED FROM PYTHON SOURCE LINES 385-387

.. code-block:: Python

    covarianceModel = ot.MaternModel([0.01], 1.5)








.. GENERATED FROM PYTHON SOURCE LINES 388-389

We define the Matern model with :math:`\theta = 0.1` :

.. GENERATED FROM PYTHON SOURCE LINES 389-391

.. code-block:: Python

    covarianceModel2 = ot.MaternModel([0.1], 1.5)








.. GENERATED FROM PYTHON SOURCE LINES 392-393

We define the Matern model with :math:`\theta = 1.0` :

.. GENERATED FROM PYTHON SOURCE LINES 393-395

.. code-block:: Python

    covarianceModel3 = ot.MaternModel([1.0], 1.5)








.. GENERATED FROM PYTHON SOURCE LINES 396-397

We draw the covariance models :

.. GENERATED FROM PYTHON SOURCE LINES 397-406

.. code-block:: Python

    graphModel = covarianceModel.draw()
    graphModel.add(covarianceModel2.draw())
    graphModel.add(covarianceModel3.draw())
    graphModel.setColors(["green", "orange", "blue"])
    graphModel.setXTitle(r"$\tau = \|s-t\|$")
    graphModel.setYTitle(r"$C(\tau)$")
    graphModel.setTitle("Matern covariance model with \nu = 3/2")
    graphModel.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1.0$"])








.. GENERATED FROM PYTHON SOURCE LINES 407-410

For each covariance model we build a Gaussian process and generate a random trajectory of
on :math:`[-1,1]`.
We build a discretization of this interval with a regular grid with step 0.01.

.. GENERATED FROM PYTHON SOURCE LINES 410-416

.. code-block:: Python

    xmin = -1.0
    step = 0.01
    n = 200
    grid1D = ot.RegularGrid(xmin, step, n + 1)
    nbTrajectories = 1








.. GENERATED FROM PYTHON SOURCE LINES 417-418

We define the first Gaussian process and its trajectory :

.. GENERATED FROM PYTHON SOURCE LINES 418-421

.. code-block:: Python

    process = ot.GaussianProcess(covarianceModel, grid1D)
    sample = process.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 422-423

then the second process :

.. GENERATED FROM PYTHON SOURCE LINES 423-426

.. code-block:: Python

    process2 = ot.GaussianProcess(covarianceModel2, grid1D)
    sample2 = process2.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 427-428

and the third one :

.. GENERATED FROM PYTHON SOURCE LINES 428-431

.. code-block:: Python

    process3 = ot.GaussianProcess(covarianceModel3, grid1D)
    sample3 = process3.getSample(nbTrajectories)








.. GENERATED FROM PYTHON SOURCE LINES 432-433

We draw the trajectories :

.. GENERATED FROM PYTHON SOURCE LINES 433-441

.. code-block:: Python

    graphTraj = sample.drawMarginal(0)
    graphTraj.add(sample2.drawMarginal(0))
    graphTraj.add(sample3.drawMarginal(0))
    graphTraj.setXTitle(r"$x$")
    graphTraj.setYTitle(r"$GP_{\theta}(x)$")
    graphTraj.setColors(["green", "orange", "blue"])
    graphTraj.setLegends([r"$\theta = 0.01$", r"$\theta = 0.1$", r"$\theta = 1.0$"])








.. GENERATED FROM PYTHON SOURCE LINES 442-443

We present each covariance model and the corresponding tracjectory side by side.

.. GENERATED FROM PYTHON SOURCE LINES 443-450

.. code-block:: Python

    fig = pl.figure(figsize=(12, 4))
    ax_pdf = fig.add_subplot(1, 2, 1)
    _ = otv.View(graphModel, figure=fig, axes=[ax_pdf])
    ax_cdf = fig.add_subplot(1, 2, 2)
    _ = otv.View(graphTraj, figure=fig, axes=[ax_cdf])
    _ = fig.suptitle("The Matern model : variation of the correlation length")




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_005.png
   :alt: The Matern model : variation of the correlation length
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_draw_covariance_models_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 451-453

From the previous figure we see that the trajectory of the Gaussian process is smoother with large
correlation length.

.. GENERATED FROM PYTHON SOURCE LINES 455-456

Display figures

.. GENERATED FROM PYTHON SOURCE LINES 456-458

.. code-block:: Python

    plt.show()








.. GENERATED FROM PYTHON SOURCE LINES 459-460

Reset default settings

.. GENERATED FROM PYTHON SOURCE LINES 460-461

.. code-block:: Python

    ot.ResourceMap.Reload()








.. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_draw_covariance_models.py:

.. only:: html

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

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

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

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

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

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

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