.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/quasi_monte_carlo/plot_04_owen_scrambling.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_quasi_monte_carlo_plot_04_owen_scrambling.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_quasi_monte_carlo_plot_04_owen_scrambling.py:


Owen scrambling: randomized QMC
===============================

A deterministic point set gives one number and no error bar. Art Owen
(1995) randomly permuted the digits of a net, nested so that the result
is still a net: every scrambled point is uniform on the cube, so the
estimate is unbiased, independent scramblings give an honest standard
error, and for smooth integrands the root-mean-square error falls like
:math:`N^{-3/2}`, faster than the unscrambled sequence's :math:`N^{-1}`.

.. GENERATED FROM PYTHON SOURCE LINES 14-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.monte_carlo import monte_carlo_integrate, randomized_qmc_estimate, sobol
    from financialkit.monte_carlo.visualizers import plot_error_convergence, plot_point_set








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Scrambling keeps the net structure
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    plot_point_set(sobol(64, 2, scramble=False), ax=axes[0], grid=8, title="unscrambled Sobol")
    plot_point_set(sobol(64, 2, rng=1), ax=axes[1], grid=8, title="Owen-scrambled Sobol")




.. image-sg:: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_04_owen_scrambling_001.png
   :alt: unscrambled Sobol, Owen-scrambled Sobol
   :srcset: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_04_owen_scrambling_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Axes: title={'center': 'Owen-scrambled Sobol'}, xlabel='$u_{1}$', ylabel='$u_{2}$'>



.. GENERATED FROM PYTHON SOURCE LINES 29-31

Error bars and convergence
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python


    d = 3
    f = lambda u: np.prod(np.exp(u) / (np.e - 1), axis=1)  # noqa: E731  integral 1
    ns = 2 ** np.arange(5, 14)
    rqmc = [randomized_qmc_estimate(f, n, d, n_replications=16, rng=n) for n in ns]
    mc = [monte_carlo_integrate(f, n, d, rng=n) for n in ns]
    # Per-replication RMSE = standard error times sqrt(replications).
    plot_error_convergence(ns, [r.std_error * 4 for r in rqmc], ax=axes[2], label="scrambled Sobol (RMSE)", rates=(1.5,))
    plot_error_convergence(ns, [m.std_error for m in mc], ax=axes[2], label="Monte Carlo (SE)", rates=(0.5,))
    res = rqmc[-1]
    print(f"N = {ns[-1]} x 16 scramblings: {res.estimate:.8f} +/- {res.std_error:.1e}, exact 1")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    N = 8192 x 16 scramblings: 0.99999989 +/- 1.5e-07, exact 1





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

   **Total running time of the script:** (0 minutes 0.106 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_quasi_monte_carlo_plot_04_owen_scrambling.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/quasi_monte_carlo/plot_04_owen_scrambling.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_owen_scrambling.ipynb <plot_04_owen_scrambling.ipynb>`

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

      :download:`Download Python source code: plot_04_owen_scrambling.py <plot_04_owen_scrambling.py>`

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

      :download:`Download zipped: plot_04_owen_scrambling.zip <plot_04_owen_scrambling.zip>`


.. only:: html

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

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