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

.. only:: html

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

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

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

.. _sphx_glr_auto_examples_plot_low_discrepancy_quadrature.py:


Low-discrepancy quadrature
--------------------------
Monte Carlo
===========
It is often important to quantify statistics of numerical models. Monte Carlo
quadrature is the simplest and most robust method for doing so. For any model
we can compute statistics, such as mean and variance as follows

.. GENERATED FROM PYTHON SOURCE LINES 10-21

.. code-block:: default

    from pyapprox import expdesign
    from pyapprox.benchmarks import setup_benchmark
    benchmark = setup_benchmark("genz", test_name="oscillatory", nvars=2)
    nsamples = 100
    mc_samples = benchmark.variable.rvs(nsamples)
    values = benchmark.fun(mc_samples)
    mean = values.mean(axis=0)
    variance = values.var(axis=0)
    print("mean", mean)
    print("variance", variance)





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

 .. code-block:: none

    mean [-0.49820729]
    variance [0.03984726]




.. GENERATED FROM PYTHON SOURCE LINES 22-32

Although not necessary for this scalar valued function, we in geenral must
use the args axis=0 to mean and variance so that we compute the statistics
for each quantity of interest (column of values)

Sobol Sequences
===============
Large number of samples are needed to compute statistics using Monte Carlo
quadrature. Low-discrepancy sequences can be used to improve accuracy for a
fixed number of samples. We can compute statistics using Sobol sequences as
follows

.. GENERATED FROM PYTHON SOURCE LINES 32-39

.. code-block:: default


    sobol_samples = expdesign.sobol_sequence(
        benchmark.variable.num_vars(), nsamples, variable=benchmark.variable)
    values = benchmark.fun(sobol_samples)
    from pyapprox.variables import print_statistics
    print_statistics(sobol_samples, values)





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

 .. code-block:: none

               z0         z1         y0    
    count  100.000000 100.000000 100.000000
    mean     0.494063   0.497812  -0.464285
    std      0.288799   0.288344   0.198696
    min      0.000000   0.000000  -0.837224
    max      0.992188   0.992188   0.000000




.. GENERATED FROM PYTHON SOURCE LINES 40-48

Here we have used print statistics to compute the sample stats. Note,
the inverse cumulative distribution functions of the univariate variables in
variable are used to ensure that the samples integrate with respect to the
joint distribution of variable. If the variable argument is not provided the
Sobol sequence will be generated on the unit hybercube :math:`[0,1]^D`.

Low-discrepancy sequences are typically more evenly space over the parameter
space. This can be seen by comparing the Monte Carlo and Sobol sequence samples

.. GENERATED FROM PYTHON SOURCE LINES 48-63

.. code-block:: default

    import matplotlib.pyplot as plt
    plt.plot(mc_samples[0, :], mc_samples[1, :], 'ko', label="MC")
    plt.plot(sobol_samples[0, :], sobol_samples[1, :], 'rs', label="Sobol")
    plt.legend()
    plt.show()

    #
    #Halton Sequences
    #================
    #Pyapprox also supports Halton Sequences
    halton_samples = expdesign.halton_sequence(
        benchmark.variable.num_vars(), nsamples, variable=benchmark.variable)
    values = benchmark.fun(halton_samples)
    print_statistics(halton_samples, values)




.. image-sg:: /auto_examples/images/sphx_glr_plot_low_discrepancy_quadrature_001.png
   :alt: plot low discrepancy quadrature
   :srcset: /auto_examples/images/sphx_glr_plot_low_discrepancy_quadrature_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

               z0         z1         y0    
    count  100.000000 100.000000 100.000000
    mean     0.488438   0.484856  -0.454770
    std      0.288831   0.290073   0.199483
    min      0.000000   0.000000  -0.803806
    max      0.984375   0.987654   0.000000




.. GENERATED FROM PYTHON SOURCE LINES 64-71

Latin Hypercube Designs
=======================
Latin Hypercube designs are another popular means to compute low-discrepancy
samples for quadrature. Pyapprox does not support such designs as unlike
Sobol and Halton Sequences they are not nested. For example, when using Latin
Hypercube designs, it is not possible to requrest 100 samples
then request 200 samples while resuing the first 100.


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

   **Total running time of the script:** ( 0 minutes  0.033 seconds)


.. _sphx_glr_download_auto_examples_plot_low_discrepancy_quadrature.py:

.. only:: html

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




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

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

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

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


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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_