..
   Copyright 2021 IRT Saint-Exupéry, https://www.irt-saintexupery.com

   This work is licensed under the Creative Commons Attribution-ShareAlike 4.0
   International License. To view a copy of this license, visit
   http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative
   Commons, PO Box 1866, Mountain View, CA 94042, USA.

.. _gen_distribution_algos:

Probability distributions
=========================

.. warning::
   Some capabilities may require the :ref:`installation <installation>` of |g| with :ref:`all its features <dependencies>`
   and some others may depend on :ref:`plugins <plugins>`.

.. note::
   All the features of the wrapped libraries may not be exposed through |g|.



.. raw:: html

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      cursor: pointer;
    }

    th, td {
      text-align: center;
    }

    tr:hover {
        border: 1px solid black;
    }

    tr:nth-child(even) {
      background-color: #f2f2f2
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    </style>

    <script>
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      table = document.getElementById("myTable");
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            if (x.innerHTML.toLowerCase() < y.innerHTML.toLowerCase()) {
              shouldSwitch = true;
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        if (shouldSwitch) {
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          if (switchcount == 0 && dir == "asc") {
            dir = "desc";
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        }
      }
    }
    </script>



.. _OTDiracDistribution_options:

OTDiracDistribution
-------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.dirac`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    <li>

**variable_value** : *float, optional*

The value of the random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTDistribution_options:

OTDistribution
--------------



Module: :class:`gemseo.uncertainty.distributions.openturns.distribution`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**interfaced_distribution** : *str, optional*

The name of the probability distribution,
typically the name of a class wrapped from an external library,
such as ``"Normal"`` for OpenTURNS or ``"norm"`` for SciPy.


By default it is set to Uniform.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**parameters** : *tuple[Any], optional*

The parameters of the probability distribution.


By default it is set to ().


.. raw:: html

    </li>

    <li>

**standard_parameters** : *StandardParametersType | None, optional*

The parameters of the probability distribution
used for string representation only
(use ``parameters`` for computation).
If ``None``, use ``parameters`` instead.
For instance,
let us consider the interfaced OpenTURNS distribution ``"Dirac"``.
Then,
the string representation of
``OTDistribution("x", "Dirac", (1,), 1, {"loc": 1})``
is ``"Dirac(loc=1)"``
while the string representation of
``OTDistribution("x", "Dirac", (1,))``
is ``"Dirac(1)"``.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTExponentialDistribution_options:

OTExponentialDistribution
-------------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.exponential`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**loc** : *float, optional*

The location of the exponential random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**rate** : *float, optional*

The rate of the exponential random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTJointDistribution_options:

OTJointDistribution
-------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.joint`





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    <dl class="field-list simple">
    <dt class="field-odd">Required parameters</dt>
    <dd class="field-odd"><ul class="simple">

    <li>

**distributions** : *Sequence[OTDistribution]*

The marginal distributions.

.. raw:: html

    </li>

    <li>

**variable** : *str*

The name of the variable, if any;
otherwise,
concatenate the names of the random variables
defined by ``distributions``.

.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**copula** : *ots.Distribution | None, optional*

A copula distribution
defining the dependency structure between random variables;
if ``None``, consider an independent copula.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTNormalDistribution_options:

OTNormalDistribution
--------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.normal`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**mu** : *float, optional*

The mean of the normal random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**sigma** : *float, optional*

The standard deviation
of the normal random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTTriangularDistribution_options:

OTTriangularDistribution
------------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.triangular`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maximum** : *float, optional*

The maximum of the random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the triangular random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**mode** : *float, optional*

The mode of the triangular random variable.


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTUniformDistribution_options:

OTUniformDistribution
---------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.uniform`





.. raw:: html

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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maximum** : *float, optional*

The maximum of the uniform random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the uniform random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _OTWeibullDistribution_options:

OTWeibullDistribution
---------------------



Module: :class:`gemseo.uncertainty.distributions.openturns.weibull`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**location** : *float, optional*

The location parameter :math:`\gamma` of the Weibull distribution.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**lower_bound** : *float | None, optional*

A lower bound to truncate the probability distribution.
If ``None``, no lower truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**scale** : *float, optional*

The scale parameter of the Weibull distribution.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**shape** : *float, optional*

The shape parameter of the Weibull distribution.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1].


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str | None, optional*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If ``None``, no transformation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**upper_bound** : *float | None, optional*

An upper bound to truncate the probability distribution.
If ``None``, no upper truncation.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**use_weibull_min** : *bool, optional*

Whether to use
the Weibull minimum extreme value distribution
(the support of the random variable is :math:`[\gamma,+\infty[`)
or the Weibull maximum extreme value distribution
(the support of the random variable is :math:`]-\infty[,\gamma]`).


By default it is set to True.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPDistribution_options:

SPDistribution
--------------



Module: :class:`gemseo.uncertainty.distributions.scipy.distribution`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**interfaced_distribution** : *str, optional*

The name of the probability distribution,
typically the name of a class wrapped from an external library,
such as ``"Normal"`` for OpenTURNS or ``"norm"`` for SciPy.


By default it is set to uniform.


.. raw:: html

    </li>

    <li>

**parameters** : *Mapping[str, Any], optional*

The parameters of the probability distribution.


By default it is set to {}.


.. raw:: html

    </li>

    <li>

**standard_parameters** : *StandardParametersType | None, optional*

The parameters of the probability distribution
used for string representation only
(use ``parameters`` for computation).
If ``None``, use ``parameters`` instead.
For instance,
let us consider the interfaced SciPy distribution ``"uniform"``.
Then,
the string representation of
``SPDistribution("x", "uniform", parameters, 1, {"min": 1, "max": 3})``
with ``parameters={"loc": 1, "scale": 2}``
is ``"uniform(max=3, min=1)"``
while the string representation of
``SPDistribution("x", "uniform", parameters)``
is ``"uniform(loc=1, scale=2)"``.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPExponentialDistribution_options:

SPExponentialDistribution
-------------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.exponential`





.. raw:: html

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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**loc** : *float, optional*

The location of the exponential random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**rate** : *float, optional*

The rate of the exponential random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPJointDistribution_options:

SPJointDistribution
-------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.joint`





.. raw:: html

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    <dl class="field-list simple">
    <dt class="field-odd">Required parameters</dt>
    <dd class="field-odd"><ul class="simple">

    <li>

**distributions** : *Sequence[SPDistribution]*

The marginal distributions.

.. raw:: html

    </li>

    <li>

**variable** : *str*

The name of the variable, if any;
otherwise,
concatenate the names of the random variables
defined by ``distributions``.

.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**copula** : *None, optional*

A copula distribution
defining the dependency structure between random variables;
if ``None``, consider an independent copula.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPNormalDistribution_options:

SPNormalDistribution
--------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.normal`





.. raw:: html

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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**mu** : *float, optional*

The mean of the normal random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**sigma** : *float, optional*

The standard deviation of the normal random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPTriangularDistribution_options:

SPTriangularDistribution
------------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.triangular`





.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**maximum** : *float, optional*

The maximum of the triangular random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the triangular random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**mode** : *float, optional*

The mode of the triangular random variable.


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPUniformDistribution_options:

SPUniformDistribution
---------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.uniform`





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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**maximum** : *float, optional*

The maximum of the uniform random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the uniform random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _SPWeibullDistribution_options:

SPWeibullDistribution
---------------------



Module: :class:`gemseo.uncertainty.distributions.scipy.weibull`





.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
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    <dl class="field-list simple">
    <dt class="field-odd">Optional parameters</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**dimension** : *int, optional*

The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.BaseJointDistribution` instead.


By default it is set to 1.


.. raw:: html

    </li>

    <li>

**location** : *float, optional*

The location parameter of the Weibull distribution.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**scale** : *float, optional*

The scale parameter of the Weibull distribution.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**shape** : *float, optional*

The shape parameter of the Weibull distribution.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**use_weibull_min** : *bool, optional*

Whether to use
the Weibull minimum extreme value distribution
(the support of the random variable is :math:`[\gamma,+\infty[`)
or the Weibull maximum extreme value distribution
(the support of the random variable is :math:`]-\infty[,\gamma]`).


By default it is set to True.


.. raw:: html

    </li>

    <li>

**variable** : *str, optional*

The name of the random variable.


By default it is set to x.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>