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

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

.. _sphx_glr_examples_mlearning_regression_model_plot_polynomial_regression.py:


Polynomial regression
=====================

A :class:`.PolynomialRegressor` is a polynomial regression model
based on a :class:`.LinearRegressor`.
This design choice was made
because a polynomial regression model is a generalized linear model
whose basis functions are monomials.
Thus,
a :class:`.PolynomialRegressor` benefits
from the same settings as :class:`.LinearRegressor`:
offset can be set to zero and regularization techniques can be used.

.. seealso::
   You will find more information about these settings
   in :ref:`the example about the linear regression model <sphx_glr_examples_mlearning_regression_model_plot_polynomial_regression.py>`.

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.. 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

    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 - 20:35:11: No coupling in MDA, switching chain_linearize to True.
        INFO - 20:35:11: *** Start Sampling execution ***
        INFO - 20:35:11: Sampling
        INFO - 20:35:11:    Disciplines: f
        INFO - 20:35:11:    MDO formulation: MDF
        INFO - 20:35:11: Running the algorithm PYDOE_FULLFACT:
        INFO - 20:35:11:     17%|█▋        | 1/6 [00:00<00:00, 600.73 it/sec]
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        INFO - 20:35:11:    100%|██████████| 6/6 [00:00<00:00, 1973.64 it/sec]
        INFO - 20:35:11: *** End Sampling execution ***




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Basics
------
Training
~~~~~~~~
Then,
we train a polynomial regression model with a degree of 2 (default)
from these samples:

.. GENERATED FROM PYTHON SOURCE LINES 84-87

.. code-block:: Python

    model = create_regression_model("PolynomialRegressor", 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([-0.90980781])}



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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([[20.65451891]])}}



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Plotting
~~~~~~~~
Of course,
you can see that the quadratic model is no good at all here:

.. GENERATED FROM PYTHON SOURCE LINES 106-118

.. 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_polynomial_regression_001.png
   :alt: plot polynomial regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_polynomial_regression_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

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        INFO - 20:35:11: Sampling
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        INFO - 20:35:11:    MDO formulation: MDF
        INFO - 20:35:11: Running the algorithm PYDOE_FULLFACT:
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        INFO - 20:35:11: *** End Sampling execution ***




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Settings
--------
The :class:`.PolynomialRegressor` has many options
defined in the :class:`.PolynomialRegressor_Settings` Pydantic model.
Most of them are presented
in :ref:`the example about the linear regression model <sphx_glr_examples_mlearning_regression_model_plot_polynomial_regression.py>`.

The only one we will look at here is the degree of the polynomial regression model.
This information can be set with the ``degree`` keyword.
For example,
we can use a cubic regression model instead of a quadratic one:

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.. code-block:: Python

    model = create_regression_model("PolynomialRegressor", training_dataset, degree=3)
    model.learn()







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and see that this model seems to be better:

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.. code-block:: Python

    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.plot(input_data.ravel(), predicted_output_data_, label="Regression - Degree(3)")
    plt.grid()
    plt.legend()
    plt.show()



.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_polynomial_regression_002.png
   :alt: plot polynomial regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_polynomial_regression_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.165 seconds)


.. _sphx_glr_download_examples_mlearning_regression_model_plot_polynomial_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_polynomial_regression.ipynb <plot_polynomial_regression.ipynb>`

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

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

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

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


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