.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_05_damped_pendulum.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_05_damped_pendulum.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_05_damped_pendulum.py:


Damped pendulum
===============

The angle of a pendulum is recorded with a rotary encoder. The oscillation decays
exponentially; the encoder has a small zero offset. Frequency, phase and decay time are
estimated automatically from a periodogram.

.. GENERATED FROM PYTHON SOURCE LINES 11-22

.. code-block:: Python

    import numpy as np

    import fitwise as fw

    rng = np.random.default_rng(5)
    t = np.linspace(0, 20, 400)  # s
    angle = 12 * np.exp(-t / 8) * np.sin(2 * np.pi * 0.62 * t + 0.4) + 0.3 + rng.normal(0, 0.2, t.size)

    result = fw.fit(fw.models.damped_sine + fw.models.constant, t, angle, yerr=0.2)
    print(result)





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

 .. code-block:: none

    Model:      damped_sine(x; amp, freq, phase, tau) + constant(x; c)
    Method:     Trust Region Reflective (scipy least_squares) (auto: nonlinear)
    Converged:  yes (16 evals)
    Data:       400 points, yerr given (absolute)
    Start:      amp, freq, phase, tau: periodogram · c: edge baseline

    Parameter     Value   ± Std.Err    Rel.   Bounds
    amp          12.007       0.045   0.4 %   –
    freq        0.61991     0.00012   0.0 %   –
    phase        0.4023      0.0039   1.0 %   –
    tau           7.986       0.046   0.6 %   –
    c             0.290       0.010   3.5 %   –

    χ²/ndf = 0.938 (395 dof, p = 0.81)    R² = 0.9975    AIC = -20.6
    Correlations > 0.5:  freq–phase (-0.73), amp–tau (-0.72)
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 23-27

.. code-block:: Python

    fig, axes = result.plot()
    axes[0].set_ylabel("angle (°)")
    axes[1].set_xlabel("t (s)")




.. image-sg:: /auto_examples/images/sphx_glr_plot_05_damped_pendulum_001.png
   :alt: plot 05 damped pendulum
   :srcset: /auto_examples/images/sphx_glr_plot_05_damped_pendulum_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 18.503588541666666, 't (s)')



.. GENERATED FROM PYTHON SOURCE LINES 28-31

The quality factor of the pendulum is Q = π f τ. ``as_ufloat()`` returns the parameters as
correlated values of the `uncertainties <https://uncertainties.readthedocs.io>`_ package, so
the error propagation automatically includes the correlation between f and τ:

.. GENERATED FROM PYTHON SOURCE LINES 31-35

.. code-block:: Python

    u = result.params.as_ufloat()
    Q = np.pi * u["freq"] * u["tau"]
    print(f"Q = {Q:.1uS}")





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

 .. code-block:: none

    Q = 15.55(9)




.. GENERATED FROM PYTHON SOURCE LINES 36-37

The 95 % prediction band shows where new readings of the encoder are expected:

.. GENERATED FROM PYTHON SOURCE LINES 37-41

.. code-block:: Python

    fig, axes = result.plot(band="prediction", residuals=False)
    axes[0].set_xlim(0, 5)
    axes[0].set_ylabel("angle (°)")
    axes[0].set_xlabel("t (s)")



.. image-sg:: /auto_examples/images/sphx_glr_plot_05_damped_pendulum_002.png
   :alt: plot 05 damped pendulum
   :srcset: /auto_examples/images/sphx_glr_plot_05_damped_pendulum_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 18.503588541666666, 't (s)')




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

   **Total running time of the script:** (0 minutes 0.508 seconds)


.. _sphx_glr_download_auto_examples_plot_05_damped_pendulum.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_05_damped_pendulum.ipynb <plot_05_damped_pendulum.ipynb>`

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

      :download:`Download Python source code: plot_05_damped_pendulum.py <plot_05_damped_pendulum.py>`

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

      :download:`Download zipped: plot_05_damped_pendulum.zip <plot_05_damped_pendulum.zip>`


.. only:: html

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

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