.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/linear_models/plot_01_yule_autoregression.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_linear_models_plot_01_yule_autoregression.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_linear_models_plot_01_yule_autoregression.py:


Yule's autoregression
=====================

Schuster's periodogram treated sunspot numbers as a sum of exact cycles
plus measurement error. Udny Yule (1927) proposed instead a pendulum
"bombarded by peas": a damped oscillation kept alive by random shocks,

.. math:: x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + \varepsilon_t,

with :math:`\phi_1 \approx 1.34` and :math:`\phi_2 \approx -0.65` for
Wolfer's sunspot series. Complex roots of :math:`1 - \phi_1 z - \phi_2 z^2`
give a *pseudo*-cycle whose phase and amplitude wander, which is exactly
what the sunspot record looks like. The coefficients come from the
Yule-Walker equations, a Toeplitz system in the autocorrelations.

.. GENERATED FROM PYTHON SOURCE LINES 19-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import ar_cycle_period, arma_acf, autocorrelation, simulate_arma, yule_walker








.. GENERATED FROM PYTHON SOURCE LINES 25-29

A disturbed pendulum
--------------------
Two hundred "years" of Yule's AR(2) look cyclical, but no two cycles have
the same length or height.

.. GENERATED FROM PYTHON SOURCE LINES 29-42

.. code-block:: Python


    phi = [1.34, -0.65]
    x = simulate_arma(ar=phi, n=200, rng=1927)
    print(f"pseudo-period implied by the roots: {ar_cycle_period(phi):.1f} years")

    fig, ax = plt.subplots(figsize=(8, 3))
    ax.plot(np.arange(1750, 1950), x, color="C0")
    ax.axhline(0, color="k", lw=0.8)
    ax.set_xlabel("year")
    ax.set_ylabel("deviation")
    ax.set_title("Yule's AR(2): a pendulum bombarded by random shocks")
    fig.tight_layout()




.. image-sg:: /api/gallery/time_series/linear_models/images/sphx_glr_plot_01_yule_autoregression_001.png
   :alt: Yule's AR(2): a pendulum bombarded by random shocks
   :srcset: /api/gallery/time_series/linear_models/images/sphx_glr_plot_01_yule_autoregression_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    pseudo-period implied by the roots: 10.7 years




.. GENERATED FROM PYTHON SOURCE LINES 43-48

Estimating it by Yule-Walker
----------------------------
The sample autocorrelations oscillate and decay, as the model's do. Solving
the Yule-Walker equations on a longer simulated record recovers the
coefficients and the period.

.. GENERATED FROM PYTHON SOURCE LINES 48-62

.. code-block:: Python


    long = simulate_arma(ar=phi, n=5000, rng=1)
    fit = yule_walker(long, 2)
    print(f"Yule-Walker estimates: phi = {fit.ar.round(3)}, sigma^2 = {fit.sigma2:.3f}")
    print(f"estimated pseudo-period: {ar_cycle_period(fit.ar):.1f} years")

    fig, ax = plt.subplots()
    k = np.arange(31)
    ax.bar(k, autocorrelation(long, 30), color="0.7", label="sample")
    ax.plot(k, arma_acf(ar=phi, nlags=30), "C3o-", ms=3, label="AR(2) theory")
    ax.set_xlabel("lag (years)")
    ax.set_ylabel("autocorrelation")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/time_series/linear_models/images/sphx_glr_plot_01_yule_autoregression_002.png
   :alt: plot 01 yule autoregression
   :srcset: /api/gallery/time_series/linear_models/images/sphx_glr_plot_01_yule_autoregression_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Yule-Walker estimates: phi = [ 1.343 -0.651], sigma^2 = 1.021
    estimated pseudo-period: 10.7 years





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

   **Total running time of the script:** (0 minutes 0.074 seconds)


.. _sphx_glr_download_api_gallery_time_series_linear_models_plot_01_yule_autoregression.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/linear_models/plot_01_yule_autoregression.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_yule_autoregression.ipynb <plot_01_yule_autoregression.ipynb>`

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

      :download:`Download Python source code: plot_01_yule_autoregression.py <plot_01_yule_autoregression.py>`

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

      :download:`Download zipped: plot_01_yule_autoregression.zip <plot_01_yule_autoregression.zip>`


.. only:: html

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

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