.. 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 ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_mlearning_regression_model_plot_pce_regression.py: Polynomial chaos expansion (PCE) ================================ A :class:`.PCERegressor` is a PCE model based on `OpenTURNS `__. .. GENERATED FROM PYTHON SOURCE LINES 28-39 .. code-block:: Python from __future__ import annotations from matplotlib import pyplot as plt from numpy import array from gemseo import create_discipline from gemseo import create_parameter_space from gemseo import sample_disciplines from gemseo.mlearning import create_regression_model .. GENERATED FROM PYTHON SOURCE LINES 40-45 Problem ------- In this example, we represent the function :math:`f(x)=(6x-2)^2\sin(12x-4)` :cite:`forrester2008` by the :class:`.AnalyticDiscipline` .. GENERATED FROM PYTHON SOURCE LINES 45-50 .. code-block:: Python discipline = create_discipline( "AnalyticDiscipline", name="f", expressions={"y": "(6*x-2)**2*sin(12*x-4)"}, ) .. GENERATED FROM PYTHON SOURCE LINES 51-52 and seek to approximate it over the input space .. GENERATED FROM PYTHON SOURCE LINES 52-55 .. code-block:: Python input_space = create_parameter_space() input_space.add_random_variable("x", "OTUniformDistribution") .. GENERATED FROM PYTHON SOURCE LINES 56-58 To do this, we create a training dataset with 6 equispaced points: .. GENERATED FROM PYTHON SOURCE LINES 58-62 .. code-block:: Python training_dataset = sample_disciplines( [discipline], input_space, "y", algo_name="PYDOE_FULLFACT", n_samples=10 ) .. rst-class:: sphx-glr-script-out .. code-block:: none INFO - 16:19:13: *** Start Sampling execution *** INFO - 16:19:13: Sampling INFO - 16:19:13: Disciplines: f INFO - 16:19:13: MDO formulation: MDF INFO - 16:19:13: Running the algorithm PYDOE_FULLFACT: INFO - 16:19:13: 10%|█ | 1/10 [00:00<00:00, 586.78 it/sec] INFO - 16:19:13: 20%|██ | 2/10 [00:00<00:00, 875.55 it/sec] INFO - 16:19:13: 30%|███ | 3/10 [00:00<00:00, 1067.34 it/sec] INFO - 16:19:13: 40%|████ | 4/10 [00:00<00:00, 1213.72 it/sec] INFO - 16:19:13: 50%|█████ | 5/10 [00:00<00:00, 1318.47 it/sec] INFO - 16:19:13: 60%|██████ | 6/10 [00:00<00:00, 1404.11 it/sec] INFO - 16:19:13: 70%|███████ | 7/10 [00:00<00:00, 1473.90 it/sec] INFO - 16:19:13: 80%|████████ | 8/10 [00:00<00:00, 1530.00 it/sec] INFO - 16:19:13: 90%|█████████ | 9/10 [00:00<00:00, 1585.95 it/sec] INFO - 16:19:13: 100%|██████████| 10/10 [00:00<00:00, 1588.21 it/sec] INFO - 16:19:13: *** End Sampling execution *** .. GENERATED FROM PYTHON SOURCE LINES 63-69 Basics ------ Training ~~~~~~~~ Then, we train an PCE regression model from these samples: .. GENERATED FROM PYTHON SOURCE LINES 69-72 .. code-block:: Python model = create_regression_model("PCERegressor", training_dataset) model.learn() .. rst-class:: sphx-glr-script-out .. code-block:: none WARNING - 16:19:13: Remove input data transformation because PCERegressor does not support transformers. .. GENERATED FROM PYTHON SOURCE LINES 73-77 Prediction ~~~~~~~~~~ Once it is built, we can predict the output value of :math:`f` at a new input point: .. GENERATED FROM PYTHON SOURCE LINES 77-81 .. 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([-0.81106394])} .. GENERATED FROM PYTHON SOURCE LINES 82-83 as well as its Jacobian value: .. GENERATED FROM PYTHON SOURCE LINES 83-86 .. code-block:: Python jacobian_value = model.predict_jacobian(input_value) jacobian_value .. rst-class:: sphx-glr-script-out .. code-block:: none {'y': {'x': array([[18.2279622]])}} .. GENERATED FROM PYTHON SOURCE LINES 87-91 Plotting ~~~~~~~~ Of course, you can see that the quadratic model is no good at all here: .. GENERATED FROM PYTHON SOURCE LINES 91-103 .. 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_pce_regression_001.png :alt: plot pce regression :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_pce_regression_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none INFO - 16:19:13: *** Start Sampling execution *** INFO - 16:19:13: Sampling INFO - 16:19:13: Disciplines: f INFO - 16:19:13: MDO formulation: MDF INFO - 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GENERATED FROM PYTHON SOURCE LINES 104-111 Settings -------- The :class:`.PCERegressor` has many options defined in the :class:`.PCERegressor_Settings` Pydantic model. Degree ~~~~~~ .. GENERATED FROM PYTHON SOURCE LINES 111-113 .. code-block:: Python model = create_regression_model("PCERegressor", training_dataset, degree=3) model.learn() .. rst-class:: sphx-glr-script-out .. code-block:: none WARNING - 16:19:13: Remove input data transformation because PCERegressor does not support transformers. .. GENERATED FROM PYTHON SOURCE LINES 114-115 and see that this model seems to be better: .. GENERATED FROM PYTHON SOURCE LINES 115-122 .. 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_pce_regression_002.png :alt: plot pce regression :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_pce_regression_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.260 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 ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_pce_regression.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_pce_regression.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_