.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_11_outliers.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_11_outliers.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_11_outliers.py:


Outliers: a robust fit
======================

A temperature sensor cools down in air, and the logger records a few corrupted readings. An
ordinary least-squares fit is pulled towards them; ``loss="huber"`` downweights them. The
model is written as text.

.. GENERATED FROM PYTHON SOURCE LINES 11-24

.. code-block:: Python

    import warnings

    import numpy as np

    import fitwise as fw

    rng = np.random.default_rng(11)
    t = np.linspace(0, 600, 61)  # s
    temperature = 21.5 + 58 * np.exp(-t / 180) + rng.normal(0, 0.2, t.size)  # °C
    temperature[[9, 26, 44]] += [4.0, -3.5, 5.0]  # corrupted readings

    model = "T_room + dT*exp(-x/tau)"








.. GENERATED FROM PYTHON SOURCE LINES 25-26

The ordinary fit warns that χ² is far too large for the given uncertainty of 0.2 °C:

.. GENERATED FROM PYTHON SOURCE LINES 26-30

.. code-block:: Python

    with warnings.catch_warnings(record=True):
        ordinary = fw.fit(model, t, temperature, yerr=0.2)
    print(ordinary)





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

 .. code-block:: none

    Model:      y = T_room + dT*exp(-x/tau)
    Method:     Trust Region Reflective (scipy least_squares) with multi-start (auto: nonlinear, no starting values)
    Converged:  yes (1689 evals)
    Data:       61 points, yerr given (absolute)
    Start:      T_room, dT, tau: multi-start

    Parameter    Value   ± Std.Err    Rel.   Bounds
    T_room      21.752       0.078   0.4 %   –
    dT          58.180       0.097   0.2 %   –
    tau         177.21        0.84   0.5 %   –

    χ²/ndf = 21.8 (58 dof, p = 3e-226)    R² = 0.9965    AIC = 190.9
    Correlations > 0.5:  T_room–tau (-0.87)
    Warnings:   ! χ²/ndf = 21.8 is too large for 58 degrees of freedom (p = 3e-226): the model does not describe the data, or yerr is underestimated




.. GENERATED FROM PYTHON SOURCE LINES 31-32

The robust fit identifies the three readings and is no longer pulled by them:

.. GENERATED FROM PYTHON SOURCE LINES 32-36

.. code-block:: Python

    robust = fw.fit(model, t, temperature, yerr=0.2, loss="huber")
    print(robust)
    print("downweighted readings at t =", t[robust.downweighted], "s")





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

 .. code-block:: none

    Model:      y = T_room + dT*exp(-x/tau)
    Method:     Robust fit (huber loss, scale 1.5) with Trust Region Reflective (auto: nonlinear, no starting values, robust (huber))
    Converged:  yes (1716 evals)
    Data:       61 points, yerr given (absolute)
    Start:      T_room, dT, tau: from ordinary fit (multi-start)

    Parameter    Value   ± Std.Err    Rel.   Bounds
    T_room      21.565       0.081   0.4 %   –
    dT          58.105       0.099   0.2 %   –
    tau         178.88        0.87   0.5 %   –

    χ²/ndf (robust) = 0.999    R² = 0.9965    AIC = 191.5
    Robust:     huber loss; 3 of 61 points downweighted (weight < 0.5)
    Correlations > 0.5:  T_room–tau (-0.88)
    Warnings:   none
    downweighted readings at t = [ 90. 260. 440.] s




.. GENERATED FROM PYTHON SOURCE LINES 37-41

.. code-block:: Python

    fig, axes = robust.plot()
    axes[0].set_ylabel("T (°C)")
    axes[1].set_xlabel("t (s)")




.. image-sg:: /auto_examples/images/sphx_glr_plot_11_outliers_001.png
   :alt: plot 11 outliers
   :srcset: /auto_examples/images/sphx_glr_plot_11_outliers_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 42-43

Cooling time constants in comparison (true value 180 s):

.. GENERATED FROM PYTHON SOURCE LINES 43-46

.. code-block:: Python

    for label, r in [("ordinary", ordinary), ("robust", robust)]:
        tau = r.params["tau"]
        print(f"{label:8s} tau = {tau.value:.1f} ± {tau.stderr:.1f} s")




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

 .. code-block:: none

    ordinary tau = 177.2 ± 0.8 s
    robust   tau = 178.9 ± 0.9 s





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

   **Total running time of the script:** (0 minutes 0.260 seconds)


.. _sphx_glr_download_auto_examples_plot_11_outliers.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_11_outliers.ipynb <plot_11_outliers.ipynb>`

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

      :download:`Download Python source code: plot_11_outliers.py <plot_11_outliers.py>`

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

      :download:`Download zipped: plot_11_outliers.zip <plot_11_outliers.zip>`


.. only:: html

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

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