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

.. only:: html

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

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

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

.. _sphx_glr_examples_mlearning_regression_model_plot_rbf_regression.py:


Radial basis function (RBF) regression
======================================

An :class:`.RBFRegressor` is an RBF model
based on `SciPy <https://scipy.org/>`__.

.. seealso::
   You can find more information about RBF models on
   `this wikipedia page <https://en.wikipedia.org/wiki/Radial_basis_function_interpolation>`__.

.. GENERATED FROM PYTHON SOURCE LINES 32-47

.. code-block:: Python


    from __future__ import annotations

    from matplotlib import pyplot as plt
    from numpy import array

    from gemseo import configure_logger
    from gemseo import create_design_space
    from gemseo import create_discipline
    from gemseo import sample_disciplines
    from gemseo.mlearning import create_regression_model
    from gemseo.mlearning.regression.algos.rbf_settings import RBF

    configure_logger()





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

 .. code-block:: none


    <RootLogger root (INFO)>



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Problem
-------
In this example,
we represent the function :math:`f(x)=(6x-2)^2\sin(12x-4)` :cite:`forrester2008`
by the :class:`.AnalyticDiscipline`

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.. code-block:: Python

    discipline = create_discipline(
        "AnalyticDiscipline",
        name="f",
        expressions={"y": "(6*x-2)**2*sin(12*x-4)"},
    )







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and seek to approximate it over the input space

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.. code-block:: Python

    input_space = create_design_space()
    input_space.add_variable("x", lower_bound=0.0, upper_bound=1.0)








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To do this,
we create a training dataset with 6 equispaced points:

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.. code-block:: Python

    training_dataset = sample_disciplines(
        [discipline], input_space, "y", algo_name="PYDOE_FULLFACT", n_samples=6
    )





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

 .. code-block:: none

     WARNING - 20:35:12: No coupling in MDA, switching chain_linearize to True.
        INFO - 20:35:12: *** Start Sampling execution ***
        INFO - 20:35:12: Sampling
        INFO - 20:35:12:    Disciplines: f
        INFO - 20:35:12:    MDO formulation: MDF
        INFO - 20:35:12: Running the algorithm PYDOE_FULLFACT:
        INFO - 20:35:12:     17%|█▋        | 1/6 [00:00<00:00, 623.22 it/sec]
        INFO - 20:35:12:     33%|███▎      | 2/6 [00:00<00:00, 1014.34 it/sec]
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        INFO - 20:35:12:    100%|██████████| 6/6 [00:00<00:00, 2041.85 it/sec]
        INFO - 20:35:12: *** End Sampling execution ***




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Basics
------
Training
~~~~~~~~
Then,
we train an RBF regression model from these samples:

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.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset)
    model.learn()








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Prediction
~~~~~~~~~~
Once it is built,
we can predict the output value of :math:`f` at a new input point:

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.. code-block:: Python

    input_value = {"x": array([0.65])}
    output_value = model.predict(input_value)
    output_value





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

 .. code-block:: none


    {'y': array([-2.16802353])}



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as well as its Jacobian value:

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.. code-block:: Python

    jacobian_value = model.predict_jacobian(input_value)
    jacobian_value





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

 .. code-block:: none


    {'y': {'x': array([[-45.81825011]])}}



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Plotting
~~~~~~~~
You can see that the RBF model is pretty good on the right, but bad on the left:

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.. code-block:: Python

    test_dataset = sample_disciplines(
        [discipline], input_space, "y", algo_name="PYDOE_FULLFACT", n_samples=100
    )
    input_data = test_dataset.get_view(variable_names=model.input_names).to_numpy()
    reference_output_data = test_dataset.get_view(variable_names="y").to_numpy().ravel()
    predicted_output_data = model.predict(input_data).ravel()
    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_001.png
   :alt: plot rbf regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

     WARNING - 20:35:12: No coupling in MDA, switching chain_linearize to True.
        INFO - 20:35:12: *** Start Sampling execution ***
        INFO - 20:35:12: Sampling
        INFO - 20:35:12:    Disciplines: f
        INFO - 20:35:12:    MDO formulation: MDF
        INFO - 20:35:12: Running the algorithm PYDOE_FULLFACT:
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        INFO - 20:35:12: *** End Sampling execution ***




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Settings
--------
The :class:`.RBFRegressor` has many options
defined in the :class:`.RBFRegressor_Settings` Pydantic model.

Function
~~~~~~~~
The default RBF is the multiquadratic function :math:`\sqrt{(r/\epsilon)^2 + 1}`
depending on a radius :math:`r` representing a distance between two points
and an adjustable constant :math:`\epsilon`.
The RBF can be changed using the ``function`` option,
which can be either an :class:`.RBF`:

.. GENERATED FROM PYTHON SOURCE LINES 123-128

.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset, function=RBF.GAUSSIAN)
    model.learn()
    predicted_output_data_g = model.predict(input_data).ravel()









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or a Python function:

.. GENERATED FROM PYTHON SOURCE LINES 130-147

.. code-block:: Python

    def rbf(self, r: float) -> float:
        """Evaluate a cubic RBF.

        An RBF must take 2 arguments, namely ``(self, r)``.

        Args:
            r: The radius.

        Returns:
            The RBF value.
        """
        return r**3


    model = create_regression_model("RBFRegressor", training_dataset, function=rbf)
    model.learn()
    predicted_output_data_c = model.predict(input_data).ravel()







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We can see that the predictions are different:

.. GENERATED FROM PYTHON SOURCE LINES 149-157

.. code-block:: Python

    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
    plt.plot(input_data.ravel(), predicted_output_data_g, label="Regression - Gaussian RBF")
    plt.plot(input_data.ravel(), predicted_output_data_c, label="Regression - Cubic RBF")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_002.png
   :alt: plot rbf regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 158-165

Epsilon
~~~~~~~
Some RBFs depend on an ``epsilon`` parameter
whose default value is the average distance between input data.
This is the case of ``"multiquadric"``, ``"gaussian"`` and ``"inverse"`` RBFs.
For example,
we can train a first multiquadric RBF model with an ``epsilon`` set to 0.5

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.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset, epsilon=0.5)
    model.learn()
    predicted_output_data_1 = model.predict(input_data).ravel()







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a second one with an ``epsilon`` set to 1.0:

.. GENERATED FROM PYTHON SOURCE LINES 170-173

.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset, epsilon=1.0)
    model.learn()
    predicted_output_data_2 = model.predict(input_data).ravel()







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and a last one with an ``epsilon`` set to 2.0:

.. GENERATED FROM PYTHON SOURCE LINES 175-178

.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset, epsilon=2.0)
    model.learn()
    predicted_output_data_3 = model.predict(input_data).ravel()







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and see that this parameter represents the regularity of the regression model:

.. GENERATED FROM PYTHON SOURCE LINES 180-189

.. code-block:: Python

    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
    plt.plot(input_data.ravel(), predicted_output_data_1, label="Regression - Epsilon(0.5)")
    plt.plot(input_data.ravel(), predicted_output_data_2, label="Regression - Epsilon(1)")
    plt.plot(input_data.ravel(), predicted_output_data_3, label="Regression - Epsilon(2)")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_003.png
   :alt: plot rbf regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 190-196

Smooth
~~~~~~
By default,
an RBF model interpolates the training points.
This is parametrized by the ``smooth`` option which is set to 0.
We can increase the smoothness of the model by increasing this value:

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.. code-block:: Python

    model = create_regression_model("RBFRegressor", training_dataset, smooth=0.1)
    model.learn()
    predicted_output_data_ = model.predict(input_data).ravel()







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and see that the model is not interpolating:

.. GENERATED FROM PYTHON SOURCE LINES 201-208

.. code-block:: Python

    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
    plt.plot(input_data.ravel(), predicted_output_data_, label="Regression - Smooth")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_004.png
   :alt: plot rbf regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 209-216

Thin plate spline (TPS)
-----------------------
TPS regression is a specific case of RBF regression
where the RBF is the thin plate radial basis function for :math:`r^2\log(r)`.
The :class:`.TPSRegressor` class
deriving from :class:`.RBFRegressor`
implements this case:

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.. code-block:: Python

    model = create_regression_model("TPSRegressor", training_dataset)
    model.learn()
    predicted_output_data_ = model.predict(input_data).ravel()







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We can see that the difference between this model
and the default multiquadric RBF model:

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.. code-block:: Python

    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
    plt.plot(input_data.ravel(), predicted_output_data_, label="Regression - TPS")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_005.png
   :alt: plot rbf regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_rbf_regression_005.png
   :class: sphx-glr-single-img





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The :class:`.TPSRegressor` can be customized with the :class:`.TPSRegressor_Settings`.


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

   **Total running time of the script:** (0 minutes 0.347 seconds)


.. _sphx_glr_download_examples_mlearning_regression_model_plot_rbf_regression.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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