:orphan:

.. _Probabilistic modeling:

Probabilistic modeling
======================



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Distributions
-------------



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    <div class="sphx-glr-thumbcontainer" tooltip="Create a deconditioned distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_deconditioned_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_deconditioned_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a deconditioned distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create your own distribution given its quantile function">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_create_your_own_dist_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_create_your_own_dist.py`

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      <div class="sphx-glr-thumbnail-title">Create your own distribution given its quantile function</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a Joint by Conditioning  distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_bayes_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_bayes_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a Joint by Conditioning  distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create an extreme value distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_create_extreme_value_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_create_extreme_value_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create an extreme value distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create and draw scalar distributions">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_create_and_draw_scalar_distributions_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_create_and_draw_scalar_distributions.py`

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      <div class="sphx-glr-thumbnail-title">Create and draw scalar distributions</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Truncate a  distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_truncated_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_truncated_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Truncate a  distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a random mixture">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_create_random_mixture_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_create_random_mixture.py`

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      <div class="sphx-glr-thumbnail-title">Create a random mixture</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Generate random variates by inverting the CDF">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_generate_by_inversion_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_generate_by_inversion.py`

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      <div class="sphx-glr-thumbnail-title">Generate random variates by inverting the CDF</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a deconditioned random vector">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_deconditioned_random_vector_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_deconditioned_random_vector.py`

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      <div class="sphx-glr-thumbnail-title">Create a deconditioned random vector</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Transform a distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_distribution_transformation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_distribution_transformation.py`

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      <div class="sphx-glr-thumbnail-title">Transform a distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Overview of univariate distribution management">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_overview_univariate_distributions_thumb.svg
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  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_overview_univariate_distributions.py`

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      <div class="sphx-glr-thumbnail-title">Overview of univariate distribution management</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Distribution manipulation">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_distribution_manipulation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_distribution_manipulation.py`

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      <div class="sphx-glr-thumbnail-title">Distribution manipulation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Quick start guide to distributions">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_quick_start_guide_distributions_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_quick_start_guide_distributions.py`

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      <div class="sphx-glr-thumbnail-title">Quick start guide to distributions</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a customized distribution or copula">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_python_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_python_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a customized distribution or copula</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a mixture of distributions">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_mixture_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_mixture_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a mixture of distributions</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create and draw multivariate distributions">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_create_draw_multivariate_distributions_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_create_draw_multivariate_distributions.py`

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      <div class="sphx-glr-thumbnail-title">Create and draw multivariate distributions</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a Point Conditional Distribution">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_point_conditional_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_point_conditional_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a Point Conditional Distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Draw minimum volume level sets">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_minimum_volume_level_sets_thumb.svg
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  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_minimum_volume_level_sets.py`

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      <div class="sphx-glr-thumbnail-title">Draw minimum volume level sets</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Compare frequentist and Bayesian estimation">

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  .. image:: /auto_probabilistic_modeling/distributions/images/thumb/sphx_glr_plot_posterior_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_distributions_plot_posterior_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Compare frequentist and Bayesian estimation</div>
    </div>


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    </div>


Copulas
-------



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    <div class="sphx-glr-thumbcontainer" tooltip="Create a copula">

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  .. image:: /auto_probabilistic_modeling/copulas/images/thumb/sphx_glr_plot_create_copula_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_copulas_plot_create_copula.py`

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      <div class="sphx-glr-thumbnail-title">Create a copula</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Extract the copula from a distribution">

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  .. image:: /auto_probabilistic_modeling/copulas/images/thumb/sphx_glr_plot_extract_copula_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_copulas_plot_extract_copula.py`

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      <div class="sphx-glr-thumbnail-title">Extract the copula from a distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Assemble copulas">

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  .. image:: /auto_probabilistic_modeling/copulas/images/thumb/sphx_glr_plot_composed_copula_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_copulas_plot_composed_copula.py`

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      <div class="sphx-glr-thumbnail-title">Assemble copulas</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create the ordinal sum of copulas">

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  .. image:: /auto_probabilistic_modeling/copulas/images/thumb/sphx_glr_plot_ordinal_sum_copula_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_copulas_plot_ordinal_sum_copula.py`

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      <div class="sphx-glr-thumbnail-title">Create the ordinal sum of copulas</div>
    </div>


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    </div>


Order Statistics
----------------



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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Create a maximum entropy order statistics distribution">

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  .. image:: /auto_probabilistic_modeling/order_statistics/images/thumb/sphx_glr_plot_max_entropy_order_statistics_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_order_statistics_plot_max_entropy_order_statistics_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create a maximum entropy order statistics distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create the distribution of the maximum of distributions">

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  .. image:: /auto_probabilistic_modeling/order_statistics/images/thumb/sphx_glr_plot_maximum_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_order_statistics_plot_maximum_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Create the distribution of the maximum of distributions</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Compute the joint distribution of order statistics">

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  .. image:: /auto_probabilistic_modeling/order_statistics/images/thumb/sphx_glr_plot_order_statistics_distribution_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_order_statistics_plot_order_statistics_distribution.py`

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      <div class="sphx-glr-thumbnail-title">Compute the joint distribution of order statistics</div>
    </div>


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    </div>


Random Vectors
--------------



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    <div class="sphx-glr-thumbcontainer" tooltip="Composite random vector">

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  .. image:: /auto_probabilistic_modeling/random_vectors/images/thumb/sphx_glr_plot_composite_random_vector_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_random_vectors_plot_composite_random_vector.py`

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      <div class="sphx-glr-thumbnail-title">Composite random vector</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a random vector">

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  .. image:: /auto_probabilistic_modeling/random_vectors/images/thumb/sphx_glr_plot_random_vector_manipulation_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_random_vectors_plot_random_vector_manipulation.py`

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      <div class="sphx-glr-thumbnail-title">Create a random vector</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Use the Ratio of Uniforms algorithm to sample a distribution">

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  .. image:: /auto_probabilistic_modeling/random_vectors/images/thumb/sphx_glr_plot_ratio_of_uniforms_exploitation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_random_vectors_plot_ratio_of_uniforms_exploitation.py`

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      <div class="sphx-glr-thumbnail-title">Use the Ratio of Uniforms algorithm to sample a distribution</div>
    </div>


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    </div>


Stochastic Processes
--------------------



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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Create a functional basis process">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_functional_basis_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_functional_basis_process.py`

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      <div class="sphx-glr-thumbnail-title">Create a functional basis process</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Add a trend to a process">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_add_trend_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_add_trend.py`

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      <div class="sphx-glr-thumbnail-title">Add a trend to a process</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Create a parametric spectral density function">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_parametric_spectral_density_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_parametric_spectral_density.py`

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      <div class="sphx-glr-thumbnail-title">Create a parametric spectral density function</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Export a field to VTK">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_export_field_vtk_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_export_field_vtk.py`

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      <div class="sphx-glr-thumbnail-title">Export a field to VTK</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a stationary covariance model">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_create_stationary_covmodel_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_create_stationary_covmodel.py`

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      <div class="sphx-glr-thumbnail-title">Create a stationary covariance model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Aggregate processes">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_aggregated_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_aggregated_process.py`

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      <div class="sphx-glr-thumbnail-title">Aggregate processes</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a white noise process">

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  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_white_noise_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_white_noise_process.py`

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      <div class="sphx-glr-thumbnail-title">Create a white noise process</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Manipulate a time series">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_timeseries_manipulation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_timeseries_manipulation.py`

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      <div class="sphx-glr-thumbnail-title">Manipulate a time series</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Use the Box-Cox transformation">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_box_cox_transform_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_box_cox_transform.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Use the Box-Cox transformation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a Gaussian process">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_create_normal_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_create_normal_process.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a Gaussian process</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a random walk process">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_random_walk_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_random_walk_process.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a random walk process</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a spectral model">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_userdefined_spectral_model_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_userdefined_spectral_model.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a spectral model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Draw a field">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_field_manipulation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_field_manipulation.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Draw a field</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a process from random vectors and processes">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_mix_rv_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_mix_rv_process.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a process from random vectors and processes</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A KroneckerCovarianceModel takes a covariance function with 1-d output and makes its output multidimensional.">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_kronecker_covmodel_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_kronecker_covmodel.py`

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      <div class="sphx-glr-thumbnail-title">Sample trajectories from a Gaussian Process with correlated outputs</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a custom covariance model">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_userdefined_covariance_model_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_userdefined_covariance_model.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a custom covariance model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Manipulate stochastic processes">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_process_manipulation_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_process_manipulation.py`

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      <div class="sphx-glr-thumbnail-title">Manipulate stochastic processes</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a discrete Markov chain process">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_discrete_markov_chain_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_discrete_markov_chain_process.py`

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      <div class="sphx-glr-thumbnail-title">Create a discrete Markov chain process</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create and manipulate an ARMA process">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_create_and_manipulate_arma_process_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_create_and_manipulate_arma_process.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create and manipulate an ARMA process</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Trend computation">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_trend_transform_thumb.png
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_trend_transform.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Trend computation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Compare covariance models">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_gaussian_processes_comparison_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_gaussian_processes_comparison.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Compare covariance models</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Create a mesh">

.. only:: html

  .. image:: /auto_probabilistic_modeling/stochastic_processes/images/thumb/sphx_glr_plot_create_mesh_thumb.svg
    :alt:

  :ref:`sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_create_mesh.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Create a mesh</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /auto_probabilistic_modeling/distributions/index.rst
   /auto_probabilistic_modeling/copulas/index.rst
   /auto_probabilistic_modeling/order_statistics/index.rst
   /auto_probabilistic_modeling/random_vectors/index.rst
   /auto_probabilistic_modeling/stochastic_processes/index.rst