.. 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_01_halton.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_01_halton.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_01_halton.py:


Halton sequences
================

Johannes van der Corput (1935) built a one-dimensional sequence that
fills the unit interval evenly by reflecting the binary digits of
:math:`k` about the radix point: 1/2, 1/4, 3/4, 1/8, 5/8, ... John Halton
(1960) took one such radical-inverse sequence per coordinate, in
distinct prime bases, to fill the :math:`d`-dimensional cube. The points
are deterministic, yet integrate far better than random ones in low
dimension, with an error close to :math:`(\log n)^d / n`.

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.monte_carlo import halton, van_der_corput
    from financialkit.monte_carlo.visualizers import plot_error_convergence, plot_point_set








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Van der Corput fills the largest gap
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-35

.. code-block:: Python


    print("base 2:", van_der_corput(8).tolist())
    print("base 3:", np.round(van_der_corput(8, base=3), 4).tolist())

    fig, axes = plt.subplots(1, 4, figsize=(17, 4))
    x = van_der_corput(16)
    axes[0].scatter(x, np.arange(16), c=np.arange(16))
    axes[0].set_xlabel("point")
    axes[0].set_ylabel("index k")
    axes[0].set_title("van der Corput, base 2")




.. image-sg:: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_01_halton_001.png
   :alt: van der Corput, base 2
   :srcset: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_01_halton_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    base 2: [0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625]
    base 3: [0.3333, 0.6667, 0.1111, 0.4444, 0.7778, 0.2222, 0.5556, 0.8889]

    Text(0.5, 1.0, 'van der Corput, base 2')



.. GENERATED FROM PYTHON SOURCE LINES 36-38

Halton points against random points
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python


    plot_point_set(np.random.default_rng(0).random((256, 2)), ax=axes[1], title="256 random points")
    plot_point_set(halton(256, 2), ax=axes[2], title="256 Halton points (bases 2, 3)")





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

 .. code-block:: none


    <Axes: title={'center': '256 Halton points (bases 2, 3)'}, xlabel='$u_{1}$', ylabel='$u_{2}$'>



.. GENERATED FROM PYTHON SOURCE LINES 43-47

Large neighbouring primes correlate
-----------------------------------
Coordinates 15 and 16 (bases 47 and 53) march in step for the first
points; scrambling the digits breaks the pattern.

.. GENERATED FROM PYTHON SOURCE LINES 47-50

.. code-block:: Python


    plot_point_set(halton(256, 16), ax=axes[3], dims=(14, 15), title="dims 15-16 (bases 47, 53)")





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

 .. code-block:: none


    <Axes: title={'center': 'dims 15-16 (bases 47, 53)'}, xlabel='$u_{15}$', ylabel='$u_{16}$'>



.. GENERATED FROM PYTHON SOURCE LINES 51-53

Integration error
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 53-63

.. code-block:: Python


    f = lambda u: np.prod(1 + (u - 0.5), axis=1)  # noqa: E731  integral 1
    ns = 2 ** np.arange(6, 15)
    fig, ax = plt.subplots(figsize=(6, 4))
    mc = [abs(f(np.random.default_rng(n).random((n, 3))).mean() - 1) for n in ns]
    ha = [abs(f(halton(n, 3)).mean() - 1) for n in ns]
    plot_error_convergence(ns, mc, ax=ax, label="random", rates=(0.5,))
    plot_error_convergence(ns, ha, ax=ax, label="Halton", rates=(1.0,))
    ax.set_title("d = 3")
    plt.show()



.. image-sg:: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_01_halton_002.png
   :alt: d = 3
   :srcset: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_01_halton_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_quasi_monte_carlo_plot_01_halton.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_01_halton.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_halton.ipynb <plot_01_halton.ipynb>`

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

      :download:`Download Python source code: plot_01_halton.py <plot_01_halton.py>`

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

      :download:`Download zipped: plot_01_halton.zip <plot_01_halton.zip>`


.. only:: html

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

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