.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_12_errors_in_x_and_y.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_12_errors_in_x_and_y.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_12_errors_in_x_and_y.py:


Errors in x and y: Ohm's law
============================

A resistor is characterized with a voltmeter and an ammeter. Both readings have
uncertainties, the current even more than the voltage. An ordinary fit treats x as exact and
underestimates the uncertainty of the resistance. With ``xerr``, fitwise uses orthogonal
distance regression, which accounts for the errors in both variables.

.. GENERATED FROM PYTHON SOURCE LINES 12-25

.. code-block:: Python

    import numpy as np

    import fitwise as fw

    rng = np.random.default_rng(12)
    current_true = np.linspace(2, 20, 12)  # mA
    R_true = 0.47  # kΩ, so U = R·I comes out in V
    u_I, u_U = 0.4, 0.05  # instrument uncertainties: mA and V
    current = current_true + rng.normal(0, u_I, current_true.size)
    voltage = R_true * current_true + rng.normal(0, u_U, current_true.size)

    ohm = "R*x"








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Ordinary fit, current treated as exact. The χ² warning says that the voltage errors
alone cannot explain the scatter:

.. GENERATED FROM PYTHON SOURCE LINES 28-34

.. code-block:: Python

    import warnings

    with warnings.catch_warnings(record=True):
        ordinary = fw.fit(ohm, current, voltage, yerr=u_U)
    print(ordinary)





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

 .. code-block:: none

    Model:      y = R*x
    Method:     Linear least squares (direct) (auto: linear in the parameters)
    Converged:  yes (2 evals)
    Data:       12 points, yerr given (absolute)
    Start:      R: not needed (linear model)

    Parameter    Value   ± Std.Err    Rel.   Bounds
    R           0.4688      0.0012   0.2 %   –

    χ²/ndf = 13.4 (11 dof, p = 6.4e-26)    R² = 0.9957    AIC = 32.1
    Warnings:   ! χ²/ndf = 13.4 is too large for 11 degrees of freedom (p = 6.4e-26): the model does not describe the data, or yerr is underestimated




.. GENERATED FROM PYTHON SOURCE LINES 35-36

With ``xerr`` the fit is consistent, and the uncertainty of R is realistic:

.. GENERATED FROM PYTHON SOURCE LINES 36-39

.. code-block:: Python

    odr = fw.fit(ohm, current, voltage, yerr=u_U, xerr=u_I)
    print(odr)





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

 .. code-block:: none

    Model:      y = R*x
    Method:     Orthogonal distance regression (odrpack) (auto: xerr given)
    Converged:  yes (31 evals)
    Data:       12 points, xerr and yerr given (absolute)
    Start:      R: from ordinary fit (linear model)

    Parameter    Value   ± Std.Err    Rel.   Bounds
    R           0.4692      0.0045   1.0 %   –

    χ²/ndf = 0.886 (11 dof, p = 0.55)    R² = 0.9957    AIC = -0.5
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 40-42

The plot shows error bars in both directions. The residuals are normalized by the effective
uncertainty √(yerr² + (R·xerr)²).

.. GENERATED FROM PYTHON SOURCE LINES 42-46

.. code-block:: Python

    fig, axes = odr.plot()
    axes[0].set_ylabel("U (V)")
    axes[1].set_xlabel("I (mA)")




.. image-sg:: /auto_examples/images/sphx_glr_plot_12_errors_in_x_and_y_001.png
   :alt: plot 12 errors in x and y
   :srcset: /auto_examples/images/sphx_glr_plot_12_errors_in_x_and_y_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 18.503588541666666, 'I (mA)')



.. GENERATED FROM PYTHON SOURCE LINES 47-50

.. code-block:: Python

    for label, r in [("ordinary", ordinary), ("ODR", odr)]:
        R = r.params["R"]
        print(f"{label:8s} R = {1000 * R.value:.1f} ± {1000 * R.stderr:.1f} Ω   (true: 470 Ω)")




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

 .. code-block:: none

    ordinary R = 468.8 ± 1.2 Ω   (true: 470 Ω)
    ODR      R = 469.2 ± 4.5 Ω   (true: 470 Ω)





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

   **Total running time of the script:** (0 minutes 0.137 seconds)


.. _sphx_glr_download_auto_examples_plot_12_errors_in_x_and_y.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_12_errors_in_x_and_y.ipynb <plot_12_errors_in_x_and_y.ipynb>`

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

      :download:`Download Python source code: plot_12_errors_in_x_and_y.py <plot_12_errors_in_x_and_y.py>`

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

      :download:`Download zipped: plot_12_errors_in_x_and_y.zip <plot_12_errors_in_x_and_y.zip>`


.. only:: html

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

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