.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_distributions_plot_quick_start_guide_distributions.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_quick_start_guide_distributions.py:


Create univariate and multivariate distributions: a quick start guide to distributions
======================================================================================

Abstract 
---------

In this example, we present classes for univariate and multivariate distributions. We demonstrate the probabilistic programming capabilities of the library. For univariate distributions, we show how to compute the probability density, the cumulated probability density and the quantiles. We also show how to create graphics. The `ComposedDistribution` class, which creates a distribution based on its marginals and its copula, is presented. We show how to truncate any distribution with the `TruncatedDistribution` class.

Univariate distribution
-----------------------

The library is a probabilistic programming library: it is possible to create a random variable and perform operations on this variable *without* generating a sample. 

In the OpenTURNS platform, several *univariate distributions* are implemented. The most commonly used are:

 - `Uniform`,
 - `Normal`,
 - `Beta`, 
 - `LogNormal`, 
 - `Exponential`, 
 - `Weibull`.


.. code-block:: default

    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    ot.Log.Show(ot.Log.NONE)








The uniform distribution
------------------------

Let us create a uniform random variable :math:`\mathcal{U}(2,5)`.


.. code-block:: default

    uniform = ot.Uniform(2,5)








The `drawPDF` method plots the probability density function.


.. code-block:: default

    graph = uniform.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_001.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





The `computePDF` method computes the probability distribution at a specific point.


.. code-block:: default

    uniform.computePDF(3.5)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none


    0.3333333333333333



The `drawCDF` method plots the cumulated distribution function.


.. code-block:: default

    graph = uniform.drawCDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_002.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





The `computeCDF` method computes the value of the cumulated distribution function a given point.


.. code-block:: default

    uniform.computeCDF(3.5)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none


    0.5



The `getSample` method generates a sample.


.. code-block:: default

    sample = uniform.getSample(10)
    sample






.. raw:: html

    <TABLE><TR><TD></TD><TH>X0</TH></TR>
    <TR><TH>0</TH><TD>3.540217</TD></TR>
    <TR><TH>1</TH><TD>4.186363</TD></TR>
    <TR><TH>2</TH><TD>3.796623</TD></TR>
    <TR><TH>3</TH><TD>2.244979</TD></TR>
    <TR><TH>4</TH><TD>4.516745</TD></TR>
    <TR><TH>5</TH><TD>2.706726</TD></TR>
    <TR><TH>6</TH><TD>3.45518</TD></TR>
    <TR><TH>7</TH><TD>4.295657</TD></TR>
    <TR><TH>8</TH><TD>3.640201</TD></TR>
    <TR><TH>9</TH><TD>2.055526</TD></TR>
    </TABLE>
    <br />
    <br />

The most common way to "see" a sample is to plot the empirical histogram. 


.. code-block:: default

    sample = uniform.getSample(1000)
    graph = ot.HistogramFactory().build(sample).drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_003.png
    :alt: X0 PDF
    :class: sphx-glr-single-img





Multivariate distributions with or without independent copula
-------------------------------------------------------------

We can create multivariate distributions by two different methods:

- we can also create a multivariate distribution by combining a list of univariate marginal distribution and a copula,
- some distributions are defined as multivariate distributions: `Normal`, `Dirichlet`, `Student`.

Since the method based on a marginal and a copula is more flexible, we illustrate below this principle.

In the following script, we define a bivariate distribution made of two univariate distributions (Gaussian and uniform) and an independent copula. 

The second input argument of the `ComposedDistribution` class is optional: if it is not specified, the copula is independent by default. 


.. code-block:: default

    normal = ot.Normal()
    uniform = ot.Uniform()
    distribution = ot.ComposedDistribution([normal, uniform])
    distribution






.. raw:: html

    <p>ComposedDistribution(Normal(mu = 0, sigma = 1), Uniform(a = -1, b = 1), IndependentCopula(dimension = 2))</p>
    <br />
    <br />

We can also use the `IndependentCopula` class.


.. code-block:: default

    normal = ot.Normal()
    uniform = ot.Uniform()
    copula = ot.IndependentCopula(2)
    distribution = ot.ComposedDistribution([normal, uniform], copula)
    distribution






.. raw:: html

    <p>ComposedDistribution(Normal(mu = 0, sigma = 1), Uniform(a = -1, b = 1), IndependentCopula(dimension = 2))</p>
    <br />
    <br />

We see that this produces the same result: in the end of this section, we will change the copula and see what happens.

The `getSample` method produces a sample from this distribution.


.. code-block:: default

    distribution.getSample(10)






.. raw:: html

    <TABLE><TR><TD></TD><TH>X0</TH><TH>X1</TH></TR>
    <TR><TH>0</TH><TD>-0.8152716</TD><TD>0.5865111</TD></TR>
    <TR><TH>1</TH><TD>-0.6392132</TD><TD>0.738932</TD></TR>
    <TR><TH>2</TH><TD>1.632357</TD><TD>0.9835865</TD></TR>
    <TR><TH>3</TH><TD>2.147953</TD><TD>-0.2462071</TD></TR>
    <TR><TH>4</TH><TD>-1.546417</TD><TD>-0.1164286</TD></TR>
    <TR><TH>5</TH><TD>1.931777</TD><TD>0.6615492</TD></TR>
    <TR><TH>6</TH><TD>-1.482484</TD><TD>-0.6689347</TD></TR>
    <TR><TH>7</TH><TD>-0.7122513</TD><TD>0.4741733</TD></TR>
    <TR><TH>8</TH><TD>-0.03644661</TD><TD>-0.4998729</TD></TR>
    <TR><TH>9</TH><TD>-0.01658812</TD><TD>-0.2449791</TD></TR>
    </TABLE>
    <br />
    <br />

In order to visualize a bivariate sample, we can use the `Cloud` class.


.. code-block:: default

    sample = distribution.getSample(1000)
    showAxes = True
    graph = ot.Graph("X0~N, X1~U", "X0", "X1", showAxes)
    cloud = ot.Cloud(sample, "blue", "fsquare", "") # Create the cloud
    graph.add(cloud) # Then, add it to the graph
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_004.png
    :alt: X0~N, X1~U
    :class: sphx-glr-single-img





We see that the marginals are Gaussian and uniform and that the copula is independent. 

Define a plot a copula
----------------------

The `NormalCopula` class allows to create a Gaussian copula. Such a copula is defined by its correlation matrix.


.. code-block:: default

    R = ot.CorrelationMatrix(2)
    R[0,1] = 0.6
    copula = ot.NormalCopula(R)
    copula






.. raw:: html

    <p>NormalCopula(R = [[ 1   0.6 ]<br>
     [ 0.6 1   ]])</p>
    <br />
    <br />

We can draw the contours of a copula with the `drawPDF` method.


.. code-block:: default

    graph = copula.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_005.png
    :alt: [X0,X1] iso-PDF
    :class: sphx-glr-single-img





Multivariate distribution with arbitrary copula
-----------------------------------------------

Now that we know that we can define a copula, we create a bivariate distribution with normal and uniform marginals and an arbitrary copula. We select the the Ali-Mikhail-Haq copula as an example of a non trivial dependence. 


.. code-block:: default

    normal = ot.Normal()
    uniform = ot.Uniform()
    theta = 0.9
    copula = ot.AliMikhailHaqCopula(theta)
    distribution = ot.ComposedDistribution([normal, uniform], copula)
    distribution






.. raw:: html

    <p>ComposedDistribution(Normal(mu = 0, sigma = 1), Uniform(a = -1, b = 1), AliMikhailHaqCopula(theta = 0.9))</p>
    <br />
    <br />


.. code-block:: default

    sample = distribution.getSample(1000)
    showAxes = True
    graph = ot.Graph("X0~N, X1~U, Ali-Mikhail-Haq copula", "X0", "X1", showAxes)
    cloud = ot.Cloud(sample, "blue", "fsquare", "") # Create the cloud
    graph.add(cloud) # Then, add it to the graph
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_006.png
    :alt: X0~N, X1~U, Ali-Mikhail-Haq copula
    :class: sphx-glr-single-img





We see that the sample is quite different from the previous sample with independent copula.

Draw several distributions in the same plot
-------------------------------------------

It is sometimes convenient to create a plot presenting the PDF and CDF on the same graphics. This is possible thanks to Matplotlib.


.. code-block:: default

    beta = ot.Beta(5, 7, 9, 10)
    pdfbeta = beta.drawPDF()
    cdfbeta = beta.drawCDF()
    exponential = ot.Exponential(3)
    pdfexp = exponential.drawPDF()
    cdfexp = exponential.drawCDF()









.. code-block:: default

    import openturns.viewer as otv









.. code-block:: default

    import pylab as plt
    fig = plt.figure(figsize=(12, 4))
    ax = fig.add_subplot(2, 2, 1)
    _ = otv.View(pdfbeta, figure=fig, axes=[ax])
    ax = fig.add_subplot(2, 2, 2)
    _ = otv.View(cdfbeta, figure=fig, axes=[ax])
    ax = fig.add_subplot(2, 2, 3)
    _ = otv.View(pdfexp, figure=fig, axes=[ax])
    ax = fig.add_subplot(2, 2, 4)
    _ = otv.View(cdfexp, figure=fig, axes=[ax])




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_007.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





Truncate a distribution
-----------------------

Any distribution can be truncated with the `TruncatedDistribution` class. 

Let :math:`f_X` (resp. :math:`F_X`) the PDF (resp. the CDF) of the real random variable :math:`X`. Let :math:`a` and :math:`b` two reals with :math:`a<b`. Let :math:`Y` be the random variable :math:`max(a, min(b, X))`. Its distribution is the distribution of :math:`X` truncated to the :math:`[a,b]` interval. 

Therefore, the PDF of :math:`Y` is:

.. math::
   f_Y(y) = \frac{f_X(y)}{F_X(b) - F_X(a)}


if :math:`y\in[a,b]` and :math:`f_Y(y)=0` otherwise. 

Consider for example the log-normal variable :math:`X` with mean :math:`\mu=0` and standard deviation :math:`\sigma=1`. 


.. code-block:: default

    X = ot.LogNormal()
    graph = X.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_008.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





We can truncate this distribution to the :math:`[1,2]` interval. We see that the PDF of the distribution becomes discontinuous at the truncation points 1 and 2.


.. code-block:: default

    Y = ot.TruncatedDistribution(X,1.,2.)
    graph = Y.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_009.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





We can also also truncate it with only a lower bound.


.. code-block:: default

    Y = ot.TruncatedDistribution(X,1.,ot.TruncatedDistribution.LOWER)
    graph = Y.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_010.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





We can finally truncate a distribution with an upper bound.


.. code-block:: default

    Y = ot.TruncatedDistribution(X,2.,ot.TruncatedDistribution.UPPER)
    graph = Y.drawPDF()
    view = viewer.View(graph)

    plt.show()



.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_quick_start_guide_distributions_011.png
    :alt: plot quick start guide distributions
    :class: sphx-glr-single-img





In the specific case of the Gaussian distribution, the specialized `TruncatedNormal` distribution can be used instead of the generic `TruncatedDistribution` class.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  1.049 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_quick_start_guide_distributions.py:


.. only :: html

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



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