.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/long_memory/plot_01_hurst_exponent.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_time_series_long_memory_plot_01_hurst_exponent.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_time_series_long_memory_plot_01_hurst_exponent.py:


Hurst exponent
==============

To size the Aswan reservoirs, Harold Edwin Hurst (1951) studied the
*range* of cumulative departures of the Nile's annual flow from its mean:
the storage a dam would need to deliver the average flow every year.
Divided by the standard deviation, the range of independent inflows grows
like :math:`n^{1/2}`; Hurst found :math:`n^{0.73}` for the Nile and many
other records. Wet years cluster with wet years. Mandelbrot later linked
the exponent :math:`H` to fractional Brownian motion: :math:`H > 1/2`
means persistence, :math:`H < 1/2` mean reversion, and in markets a
reliable :math:`H \ne 1/2` would make returns predictable.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import hurst_exponent, simulate_fgn
    from financialkit.time_series.visualizers import plot_rescaled_range








.. GENERATED FROM PYTHON SOURCE LINES 24-27

Persistent, independent and anti-persistent noise
-------------------------------------------------
Fractional Gaussian noise with three exponents, cumulated into "flows".

.. GENERATED FROM PYTHON SOURCE LINES 27-37

.. code-block:: Python


    rng = np.random.default_rng(1951)
    series = {H: simulate_fgn(4096, H, rng=rng) for H in (0.3, 0.5, 0.75)}
    fig, ax = plt.subplots(figsize=(8, 3.5))
    for H, x in series.items():
        ax.plot(np.cumsum(x), lw=0.8, label=f"H = {H}")
    ax.set_xlabel("year")
    ax.set_ylabel("cumulative departure")
    ax.legend()




.. image-sg:: /api/gallery/time_series/long_memory/images/sphx_glr_plot_01_hurst_exponent_001.png
   :alt: plot 01 hurst exponent
   :srcset: /api/gallery/time_series/long_memory/images/sphx_glr_plot_01_hurst_exponent_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1169dfa10>



.. GENERATED FROM PYTHON SOURCE LINES 38-44

Rescaled-range analysis
-----------------------
The slope of :math:`\log(R/S)_n` on :math:`\log n` estimates :math:`H`.
The dashed line is Anis and Lloyd's exact expectation for independent
data; its slope exceeds 1/2 in short windows, which is why the corrected
estimate measures the excess over it.

.. GENERATED FROM PYTHON SOURCE LINES 44-53

.. code-block:: Python


    fig, ax = plt.subplots()
    for H, x in series.items():
        res = hurst_exponent(x)
        plot_rescaled_range(res, ax=ax, label=f"true {H}:")
        print(f"true H = {H}: raw slope {res.hurst:.2f}, corrected {res.hurst_corrected:.2f}")
    ax.legend(fontsize=8)
    plt.show()




.. image-sg:: /api/gallery/time_series/long_memory/images/sphx_glr_plot_01_hurst_exponent_002.png
   :alt: plot 01 hurst exponent
   :srcset: /api/gallery/time_series/long_memory/images/sphx_glr_plot_01_hurst_exponent_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    true H = 0.3: raw slope 0.38, corrected 0.34
    true H = 0.5: raw slope 0.52, corrected 0.48
    true H = 0.75: raw slope 0.72, corrected 0.69




.. GENERATED FROM PYTHON SOURCE LINES 54-56

The returns of a random-walk price have :math:`H \approx 1/2` once the
small-sample bias is removed.

.. GENERATED FROM PYTHON SOURCE LINES 56-59

.. code-block:: Python


    returns = 0.01 * np.random.default_rng(0).standard_normal(4096)
    print(f"random-walk returns: corrected H = {hurst_exponent(returns).hurst_corrected:.2f}")




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

 .. code-block:: none

    random-walk returns: corrected H = 0.52





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

   **Total running time of the script:** (0 minutes 0.077 seconds)


.. _sphx_glr_download_api_gallery_time_series_long_memory_plot_01_hurst_exponent.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/time_series/long_memory/plot_01_hurst_exponent.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hurst_exponent.ipynb <plot_01_hurst_exponent.ipynb>`

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

      :download:`Download Python source code: plot_01_hurst_exponent.py <plot_01_hurst_exponent.py>`

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

      :download:`Download zipped: plot_01_hurst_exponent.zip <plot_01_hurst_exponent.zip>`


.. only:: html

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

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