..
   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>`.

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








.. _OTBetaDistribution_options:

OTBetaDistribution
------------------



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




































































.. raw:: html

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


    <li>

**alpha** : *float, optional*

The first shape parameter of the beta random variable.


By default it is set to 2.0.


.. raw:: html

    </li>

    <li>

**beta** : *float, optional*

The second shape parameter of the beta random variable.


By default it is set to 2.0.


.. 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>

**maximum** : *float, optional*

The maximum of the beta random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the beta random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTDiracDistribution_options:

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



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


















































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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_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`






























































.. raw:: html

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


    <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, optional*

The parameters of the probability distribution
used for string representation only
(use ``parameters`` for computation).
If empty, 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 {}.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTExponentialDistribution_options:

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



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
























































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTJointDistribution_options:

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



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






























.. raw:: html

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

    <li>

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

The marginal distributions.

.. raw:: html

    </li>

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



.. raw:: html

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


    <li>

**copula** : *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>



.. _OTLogNormalDistribution_options:

OTLogNormalDistribution
-----------------------



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




































































.. raw:: html

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


    <li>

**location** : *float, optional*

The location of the log-normal 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>

**mu** : *float, optional*

Either the mean of the log-normal random variable
or that of its logarithm when ``set_log`` is ``True``.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**set_log** : *bool, optional*

Whether ``mu`` and ``sigma`` apply
to the logarithm of the log-normal random variable.
Otherwise,
``mu`` and ``sigma`` apply to the log-normal random variable directly.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**sigma** : *float, optional*

Either the standard deviation of the log-normal random variable
or that of its logarithm when ``set_log`` is ``True``.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**threshold** : *float, optional*

A threshold in [0,1]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTNormalDistribution_options:

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



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
























































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTTriangularDistribution_options:

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



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






























































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTUniformDistribution_options:

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



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
























































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _OTWeibullDistribution_options:

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



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




































































.. raw:: html

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


    <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]
(`see OpenTURNS documentation
<http://openturns.github.io/openturns/latest/user_manual/
_generated/openturns.TruncatedDistribution.html>`_).


By default it is set to 0.5.


.. raw:: html

    </li>

    <li>

**transformation** : *str*

A transformation applied
to the random variable,
e.g. :math:`\sin(x)`. If empty, no transformation.


By default it is set to .


.. 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>

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



.. _SPBetaDistribution_options:

SPBetaDistribution
------------------



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












































.. raw:: html

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


    <li>

**alpha** : *float, optional*

The first shape parameter of the beta random variable.


By default it is set to 2.0.


.. raw:: html

    </li>

    <li>

**beta** : *float, optional*

The second shape parameter of the beta random variable.


By default it is set to 2.0.


.. raw:: html

    </li>

    <li>

**maximum** : *float, optional*

The maximum of the beta random variable.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**minimum** : *float, optional*

The minimum of the beta random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

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



.. _SPDistribution_options:

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



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






































.. raw:: html

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


    <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** : *StrKeyMapping | tuple[Any, ...], optional*

The parameters of the probability distribution.


By default it is set to {}.


.. raw:: html

    </li>

    <li>

**standard_parameters** : *StandardParametersType, optional*

The parameters of the probability distribution
used for string representation only
(use ``parameters`` for computation).
If empty, use ``parameters`` instead.
For instance,
let us consider the interfaced SciPy distribution ``"uniform"``.
Then,
the string representation of
``SPDistribution("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("uniform", parameters)``
is ``"uniform(loc=1, scale=2)"``.


By default it is set to {}.


.. raw:: html

    </li>

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



.. _SPExponentialDistribution_options:

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



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
































.. raw:: html

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


    <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>

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



.. _SPJointDistribution_options:

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



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






























.. raw:: html

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

    <li>

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

The marginal distributions.

.. raw:: html

    </li>

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



.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</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>



.. _SPLogNormalDistribution_options:

SPLogNormalDistribution
-----------------------



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












































.. raw:: html

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


    <li>

**location** : *float, optional*

The location of the log-normal random variable.


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**mu** : *float, optional*

Either the mean of the log-normal random variable
or that of its logarithm when ``set_log`` is ``True``.


By default it is set to 1.0.


.. raw:: html

    </li>

    <li>

**set_log** : *bool, optional*

Whether ``mu`` and ``sigma`` apply
to the logarithm of the log-normal random variable.
Otherwise,
``mu`` and ``sigma`` apply to the log-normal random variable directly.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**sigma** : *float, optional*

Either the standard deviation of the log-normal random variable
or that of its logarithm when ``set_log`` is ``True``.


By default it is set to 1.0.


.. raw:: html

    </li>

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



.. _SPNormalDistribution_options:

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



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
































.. raw:: html

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


    <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>

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



.. _SPTriangularDistribution_options:

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



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






































.. raw:: html

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


    <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>

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



.. _SPUniformDistribution_options:

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



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
































.. raw:: html

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


    <li>

**maximum** : *float, optional*

The maximum of the uniform random variable.


By default it is set to 1.0.


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    </li>

    <li>

**minimum** : *float, optional*

The minimum of the uniform random variable.


By default it is set to 0.0.


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    </li>

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



.. _SPWeibullDistribution_options:

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



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












































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


    <li>

**location** : *float, optional*

The location parameter of the Weibull distribution.


By default it is set to 0.0.


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    </li>

    <li>

**scale** : *float, optional*

The scale parameter of the Weibull distribution.


By default it is set to 1.0.


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    </li>

    <li>

**shape** : *float, optional*

The shape parameter of the Weibull distribution.


By default it is set to 1.0.


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    </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.


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    </li>

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