.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/calibration/plot_01_least_squares_calibration.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_analytical_calibration_plot_01_least_squares_calibration.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_analytical_calibration_plot_01_least_squares_calibration.py:


Legendre and Gauss's least squares: fitting a calibration line
=================================================================

The method of least squares picks the straight line that minimizes the
sum of squared residuals :math:`S(m,b)=\sum_i(y_i-mx_i-b)^2`.
:func:`~chemistrykit.analytical.fit_calibration` performs exactly that
fit for a set of calibration standards; here we check that its slope
sits at the bottom of the :math:`S(m)` parabola, and that the residuals
it leaves behind scatter around zero.

.. GENERATED FROM PYTHON SOURCE LINES 14-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import fit_calibration

    rng = np.random.default_rng(42)
    concentration = np.array([0.0, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0])  # ppm
    true_slope, true_intercept, noise_sigma = 12.4, 0.8, 1.5
    signal = true_slope * concentration + true_intercept + rng.normal(scale=noise_sigma, size=concentration.shape)

    cal = fit_calibration(concentration, signal)
    print(f"Least-squares slope:     {cal.slope:.4f} (true {true_slope})")
    print(f"Least-squares intercept: {cal.intercept:.4f} (true {true_intercept})")
    print(f"R^2 = {cal.r_squared:.6f}")





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

 .. code-block:: none

    Least-squares slope:     12.3677 (true 12.4)
    Least-squares intercept: 0.4806 (true 0.8)
    R^2 = 0.999456




.. GENERATED FROM PYTHON SOURCE LINES 30-33

The sum of squared residuals as a function of trial slope (with the
intercept re-optimized for each slope) is a parabola whose minimum is
the least-squares slope -- Legendre's criterion made visible:

.. GENERATED FROM PYTHON SOURCE LINES 33-41

.. code-block:: Python

    trial_slopes = np.linspace(cal.slope - 1.0, cal.slope + 1.0, 401)
    x_mean, y_mean = concentration.mean(), signal.mean()
    S = np.array([np.sum((signal - (m * concentration + (y_mean - m * x_mean))) ** 2) for m in trial_slopes])
    print(f"\nSlope minimizing S on the scan: {trial_slopes[np.argmin(S)]:.4f}")

    residuals = signal - cal.predict_signal(concentration)
    print(f"Sum of residuals (zero for a least-squares line with intercept): {residuals.sum():.2e}")





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

 .. code-block:: none


    Slope minimizing S on the scan: 12.3677
    Sum of residuals (zero for a least-squares line with intercept): -1.99e-13




.. GENERATED FROM PYTHON SOURCE LINES 42-66

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    c_line = np.linspace(0, concentration.max(), 100)
    axes[0].plot(concentration, signal, "o", label="standards")
    axes[0].plot(c_line, cal.predict_signal(c_line), "-", label="least-squares line")
    axes[0].set_xlabel("concentration (ppm)")
    axes[0].set_ylabel("signal")
    axes[0].set_title("Calibration standards and fit")
    axes[0].legend()

    axes[1].plot(trial_slopes, S)
    axes[1].axvline(cal.slope, color="crimson", linestyle="--", label=f"fit slope = {cal.slope:.3f}")
    axes[1].set_xlabel("trial slope m")
    axes[1].set_ylabel(r"$S(m)=\sum_i r_i^2$")
    axes[1].set_title("Sum of squared residuals")
    axes[1].legend()

    axes[2].axhline(0.0, color="gray", linewidth=0.8)
    axes[2].vlines(concentration, 0.0, residuals, color="C0")
    axes[2].plot(concentration, residuals, "o")
    axes[2].set_xlabel("concentration (ppm)")
    axes[2].set_ylabel("residual")
    axes[2].set_title("Residuals about the line")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/analytical/calibration/images/sphx_glr_plot_01_least_squares_calibration_001.png
   :alt: Calibration standards and fit, Sum of squared residuals, Residuals about the line
   :srcset: /api/gallery/analytical/calibration/images/sphx_glr_plot_01_least_squares_calibration_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.667 seconds)


.. _sphx_glr_download_api_gallery_analytical_calibration_plot_01_least_squares_calibration.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_least_squares_calibration.ipynb <plot_01_least_squares_calibration.ipynb>`

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

      :download:`Download Python source code: plot_01_least_squares_calibration.py <plot_01_least_squares_calibration.py>`

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

      :download:`Download zipped: plot_01_least_squares_calibration.zip <plot_01_least_squares_calibration.zip>`


.. only:: html

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

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