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

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

.. _sphx_glr_examples_mlearning_regression_model_plot_pce_regression.py:


PCE regression
==============

We want to approximate a discipline with two inputs and two outputs:

- :math:`y_1=1+2x_1+3x_2`
- :math:`y_2=-1-2x_1-3x_2`

over the unit hypercube :math:`[0,1]\times[0,1]`.

.. GENERATED FROM PYTHON SOURCE LINES 32-48

.. code-block:: Python


    from __future__ import annotations

    from numpy import array

    from gemseo import configure_logger
    from gemseo import create_design_space
    from gemseo import create_discipline
    from gemseo import create_parameter_space
    from gemseo import create_scenario
    from gemseo.mlearning import create_regression_model
    from gemseo.mlearning import import_regression_model

    configure_logger()






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

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 49-53

Create the discipline to learn
------------------------------
We can implement this analytic discipline by means of the
:class:`.AnalyticDiscipline` class.

.. GENERATED FROM PYTHON SOURCE LINES 53-58

.. code-block:: Python

    expressions = {"y_1": "1+2*x_1+3*x_2", "y_2": "-1-2*x_1-3*x_2"}
    discipline = create_discipline(
        "AnalyticDiscipline", name="func", expressions=expressions
    )








.. GENERATED FROM PYTHON SOURCE LINES 59-62

Create the input sampling space
-------------------------------
We create the input sampling space by adding the variables one by one.

.. GENERATED FROM PYTHON SOURCE LINES 62-66

.. code-block:: Python

    design_space = create_design_space()
    design_space.add_variable("x_1", l_b=0.0, u_b=1.0)
    design_space.add_variable("x_2", l_b=0.0, u_b=1.0)








.. GENERATED FROM PYTHON SOURCE LINES 67-72

Create the learning set
-----------------------
We can build a learning set by means of a
:class:`.DOEScenario` with a full factorial design of
experiments. The number of samples can be equal to 9 for example.

.. GENERATED FROM PYTHON SOURCE LINES 72-77

.. code-block:: Python

    scenario = create_scenario(
        [discipline], "DisciplinaryOpt", "y_1", design_space, scenario_type="DOE"
    )
    scenario.execute({"algo": "fullfact", "n_samples": 9})





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

 .. code-block:: none

        INFO - 13:55:47:  
        INFO - 13:55:47: *** Start DOEScenario execution ***
        INFO - 13:55:47: DOEScenario
        INFO - 13:55:47:    Disciplines: func
        INFO - 13:55:47:    MDO formulation: DisciplinaryOpt
        INFO - 13:55:47: Optimization problem:
        INFO - 13:55:47:    minimize y_1(x_1, x_2)
        INFO - 13:55:47:    with respect to x_1, x_2
        INFO - 13:55:47:    over the design space:
        INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
        INFO - 13:55:47:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
        INFO - 13:55:47:       | x_1  |      0      |  None |      1      | float |
        INFO - 13:55:47:       | x_2  |      0      |  None |      1      | float |
        INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
        INFO - 13:55:47: Solving optimization problem with algorithm fullfact:
        INFO - 13:55:47:     11%|█         | 1/9 [00:00<00:00, 353.12 it/sec, obj=1]
        INFO - 13:55:47:     22%|██▏       | 2/9 [00:00<00:00, 559.61 it/sec, obj=2]
        INFO - 13:55:47:     33%|███▎      | 3/9 [00:00<00:00, 713.36 it/sec, obj=3]
        INFO - 13:55:47:     44%|████▍     | 4/9 [00:00<00:00, 828.79 it/sec, obj=2.5]
        INFO - 13:55:47:     56%|█████▌    | 5/9 [00:00<00:00, 918.11 it/sec, obj=3.5]
        INFO - 13:55:47:     67%|██████▋   | 6/9 [00:00<00:00, 990.00 it/sec, obj=4.5]
        INFO - 13:55:47:     78%|███████▊  | 7/9 [00:00<00:00, 1038.30 it/sec, obj=4]
        INFO - 13:55:47:     89%|████████▉ | 8/9 [00:00<00:00, 1085.52 it/sec, obj=5]
        INFO - 13:55:47:    100%|██████████| 9/9 [00:00<00:00, 1127.00 it/sec, obj=6]
        INFO - 13:55:47: Optimization result:
        INFO - 13:55:47:    Optimizer info:
        INFO - 13:55:47:       Status: None
        INFO - 13:55:47:       Message: None
        INFO - 13:55:47:       Number of calls to the objective function by the optimizer: 9
        INFO - 13:55:47:    Solution:
        INFO - 13:55:47:       Objective: 1.0
        INFO - 13:55:47:       Design space:
        INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
        INFO - 13:55:47:          | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
        INFO - 13:55:47:          | x_1  |      0      |   0   |      1      | float |
        INFO - 13:55:47:          | x_2  |      0      |   0   |      1      | float |
        INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
        INFO - 13:55:47: *** End DOEScenario execution (time: 0:00:00.019796) ***

    {'eval_jac': False, 'n_samples': 9, 'algo': 'fullfact'}



.. GENERATED FROM PYTHON SOURCE LINES 78-82

Create the regression model
---------------------------
Then, we build the linear regression model from the database and
displays this model.

.. GENERATED FROM PYTHON SOURCE LINES 82-92

.. code-block:: Python

    prob_space = create_parameter_space()
    prob_space.add_random_variable("x_1", "OTUniformDistribution")
    prob_space.add_random_variable("x_2", "OTUniformDistribution")
    dataset = scenario.to_dataset(opt_naming=False)
    model = create_regression_model(
        "PCERegressor", data=dataset, probability_space=prob_space, transformer=None
    )
    model.learn()
    model






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div style='background-color: #fafae2; padding: 10px;'>PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=Uncertain space:
    +------+-------------------------------+
    | Name |          Distribution         |
    +------+-------------------------------+
    | x_1  | Uniform(lower=0.0, upper=1.0) |
    | x_2  | Uniform(lower=0.0, upper=1.0) |
    +------+-------------------------------+, use_cleaning=False, use_lars=False)<br/><ul><li>based on the OpenTURNS library</li><li>built from 9 learning samples</li></ul></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 93-96

Predict output
--------------
Once it is built, we can use it for prediction.

.. GENERATED FROM PYTHON SOURCE LINES 96-100

.. code-block:: Python

    input_value = {"x_1": array([1.0]), "x_2": array([2.0])}
    output_value = model.predict(input_value)
    output_value





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

 .. code-block:: none


    {'y_1': array([9.])}



.. GENERATED FROM PYTHON SOURCE LINES 101-104

Save the regression model
-------------------------
Lastly, we save the model.

.. GENERATED FROM PYTHON SOURCE LINES 104-106

.. code-block:: Python

    directory = model.to_pickle()





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

 .. code-block:: none

    /home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/gemseo/mlearning/core/ml_algo.py:359: UserWarning: Pandas doesn't allow columns to be created via a new attribute name - see https://pandas.pydata.org/pandas-docs/stable/indexing.html#attribute-access
      self.learning_set.data = {}




.. GENERATED FROM PYTHON SOURCE LINES 107-110

Load the regression model
-------------------------
In an other study, we could load this model.

.. GENERATED FROM PYTHON SOURCE LINES 110-113

.. code-block:: Python

    loaded_model = import_regression_model(directory)
    loaded_model






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div style='background-color: #fafae2; padding: 10px;'>PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=Uncertain space:
    +------+-------------------------------+
    | Name |          Distribution         |
    +------+-------------------------------+
    | x_1  | Uniform(lower=0.0, upper=1.0) |
    | x_2  | Uniform(lower=0.0, upper=1.0) |
    +------+-------------------------------+, use_cleaning=False, use_lars=False)<br/><ul><li>based on the OpenTURNS library</li><li>built from 9 learning samples</li></ul></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 114-117

Use the loaded regression model
-------------------------------
And use it!

.. GENERATED FROM PYTHON SOURCE LINES 117-118

.. code-block:: Python

    loaded_model.predict(input_value)




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

 .. code-block:: none


    {'y_1': array([9.])}




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

   **Total running time of the script:** (0 minutes 0.184 seconds)


.. _sphx_glr_download_examples_mlearning_regression_model_plot_pce_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_pce_regression.ipynb <plot_pce_regression.ipynb>`

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

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


.. only:: html

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

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