.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/variance_reduction/plot_04_latin_hypercube.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_monte_carlo_variance_reduction_plot_04_latin_hypercube.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_monte_carlo_variance_reduction_plot_04_latin_hypercube.py:


McKay-Beckman-Conover Latin hypercube
=====================================

Michael McKay, Richard Beckman and William Conover (1979), at Los Alamos,
needed to run expensive computer codes on few input combinations. A Latin
hypercube design of :math:`n` points stratifies every coordinate into
:math:`n` equal intervals at once: each interval of each axis holds
exactly one point, the axes shuffled independently. The variance of an
additive integrand falls by an order of :math:`n`; interactions are left
alone.

.. GENERATED FROM PYTHON SOURCE LINES 15-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import norm

    from financialkit.monte_carlo import latin_hypercube_estimate, monte_carlo_integrate
    from financialkit.monte_carlo.utils import qmc_engine
    from financialkit.monte_carlo.visualizers import plot_point_set








.. GENERATED FROM PYTHON SOURCE LINES 24-26

One point per row and per column
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    plot_point_set(np.random.default_rng(0).random((10, 2)), ax=axes[0], grid=10, title="10 random points")
    plot_point_set(qmc_engine("lhs", 2, rng=0).random(10), ax=axes[1], grid=10, title="10-point Latin hypercube")




.. image-sg:: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_04_latin_hypercube_001.png
   :alt: 10 random points, 10-point Latin hypercube
   :srcset: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_04_latin_hypercube_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Axes: title={'center': '10-point Latin hypercube'}, xlabel='$u_{1}$', ylabel='$u_{2}$'>



.. GENERATED FROM PYTHON SOURCE LINES 32-36

A basket of five lognormal assets
---------------------------------
The basket's mean value is nearly additive in the inputs; a call on it is
less so. Both gain, the additive one far more.

.. GENERATED FROM PYTHON SOURCE LINES 36-53

.. code-block:: Python


    d, n = 5, 500
    basket = lambda u: np.mean(np.exp(0.2 * norm.ppf(u) - 0.02), axis=1)  # noqa: E731
    call = lambda u: np.maximum(basket(u) - 1.0, 0.0)  # noqa: E731
    labels, ratios = [], []
    for name, f in [("basket mean", basket), ("basket call", call)]:
        crude = monte_carlo_integrate(f, 20 * n, dim=d, rng=1)
        lhs = latin_hypercube_estimate(f, n, d, n_replications=20, rng=2)
        print(f"{name}: crude {crude.estimate:.5f} +/- {crude.std_error:.5f}, LHS {lhs.estimate:.5f} +/- {lhs.std_error:.5f}")
        labels.append(name)
        ratios.append(lhs.variance_ratio(crude))
    axes[2].bar(labels, ratios)
    axes[2].set_yscale("log")
    axes[2].set_ylabel("variance reduction factor")
    axes[2].set_title(f"LHS, n = {n}, d = {d}")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    basket mean: crude 0.99987 +/- 0.00092, LHS 0.99999 +/- 0.00002
    basket call: crude 0.03652 +/- 0.00056, LHS 0.03573 +/- 0.00021





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

   **Total running time of the script:** (0 minutes 0.078 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_variance_reduction_plot_04_latin_hypercube.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/monte_carlo/variance_reduction/plot_04_latin_hypercube.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_latin_hypercube.ipynb <plot_04_latin_hypercube.ipynb>`

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

      :download:`Download Python source code: plot_04_latin_hypercube.py <plot_04_latin_hypercube.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_latin_hypercube.zip <plot_04_latin_hypercube.zip>`


.. only:: html

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

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