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.. "examples/12_sum_model/01_fitting_sum_model.py"
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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_examples_12_sum_model_01_fitting_sum_model.py>`
        to download the full example code.

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

.. _sphx_glr_examples_12_sum_model_01_fitting_sum_model.py:


Fitting a Sum Model
--------------------

In this tutorial, we demonstrate how to fit a sum model consisting of two
covariance models to an empirical variogram.

We will generate synthetic data, compute an empirical variogram, and fit a
sum model combining a Spherical and Gaussian model to it.

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.. code-block:: Python


    import gstools as gs

    x = y = range(100)








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First, we create a synthetic random field based on a known sum model.
This will serve as the ground truth for fitting.

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.. code-block:: Python


    # Define the true sum model
    m0 = gs.Spherical(dim=2, var=2.0, len_scale=5.0)
    m1 = gs.Gaussian(dim=2, var=1.0, len_scale=10.0)
    true_model = m0 + m1

    # Generate synthetic field
    srf = gs.SRF(true_model, seed=20250405)
    field = srf.structured((x, y))








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Next, we calculate the empirical variogram from the synthetic data.

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.. code-block:: Python


    # Compute empirical variogram
    bin_center, gamma = gs.vario_estimate((x, y), field)








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Now we define a sum model to fit to the empirical variogram.
Initially, the parameters of the models are arbitrary.

A sum model can also be created by a list of model classes together with
the common arguments (like dim in this case).

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.. code-block:: Python


    fit_model = gs.SumModel(gs.Spherical, gs.Gaussian, dim=2)








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We fit the sum model to the empirical variogram using GSTools' built-in
fitting capabilities. We deactivate the nugget fitting to not overparameterize
our model.

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.. code-block:: Python


    fit_model.fit_variogram(bin_center, gamma, nugget=False)
    print(f"{true_model=}")
    print(f" {fit_model=}")





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

 .. code-block:: none

    true_model=SumModel(Spherical, Gaussian, dim=2, vars=[2.0, 1.0], len_scales=[5.0, 10.0])
     fit_model=SumModel(Spherical, Gaussian, dim=2, vars=[2.1, 0.923], len_scales=[3.99, 14.2])




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The variance of a sum model is the sum of the sub variances (:any:`SumModel.vars`)
from the contained models. The length scale is a weighted sum of the sub
length scales (:any:`SumModel.len_scales`) where the weights are the ratios
of the sub variances to the total variance of the sum model.

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.. code-block:: Python


    print(f"{true_model.var=:.2}, {true_model.len_scale=:.2}")
    print(f" {fit_model.var=:.2},  {fit_model.len_scale=:.2}")





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

 .. code-block:: none

    true_model.var=3.0, true_model.len_scale=6.7
     fit_model.var=3.0,  fit_model.len_scale=7.1




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After fitting, we can visualize the empirical variogram alongside the
fitted sum model and its components. A Sum Model is subscriptable to access
the individual models its contains.

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.. code-block:: Python


    ax = fit_model.plot(x_max=max(bin_center))
    ax.scatter(bin_center, gamma)
    # Extract individual components
    fit_model[0].plot(x_max=max(bin_center), ax=ax)
    fit_model[1].plot(x_max=max(bin_center), ax=ax)
    # True models
    true_model.plot(x_max=max(bin_center), ax=ax, ls="--", c="C0", label="")
    true_model[0].plot(x_max=max(bin_center), ax=ax, ls="--", c="C1", label="")
    true_model[1].plot(x_max=max(bin_center), ax=ax, ls="--", c="C2", label="")




.. image-sg:: /examples/12_sum_model/images/sphx_glr_01_fitting_sum_model_001.png
   :alt: 01 fitting sum model
   :srcset: /examples/12_sum_model/images/sphx_glr_01_fitting_sum_model_001.png
   :class: sphx-glr-single-img





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As we can see, the fitted sum model closely matches the empirical variogram,
demonstrating its ability to capture multi-scale variability effectively.
The "true" variograms are shown with dashed lines for comparison.


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

   **Total running time of the script:** (0 minutes 0.630 seconds)


.. _sphx_glr_download_examples_12_sum_model_01_fitting_sum_model.py:

.. only:: html

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

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

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

      :download:`Download Python source code: 01_fitting_sum_model.py <01_fitting_sum_model.py>`

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      :download:`Download zipped: 01_fitting_sum_model.zip <01_fitting_sum_model.zip>`


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