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

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

.. _sphx_glr_examples_mlearning_regression_model_plot_gp_regression.py:


Gaussian process (GP) regression
================================

A :class:`.GaussianProcessRegressor` is a GP regression model
based on `scikit-learn <https://scikit-learn.org/>`__.

.. seealso::
   You can find more information about building GP models with scikit-learn on
   `this page <https://scikit-learn.org/stable/modules/gaussian_process.html>`__.

.. GENERATED FROM PYTHON SOURCE LINES 32-48

.. code-block:: Python


    from __future__ import annotations

    from matplotlib import pyplot as plt
    from numpy import array
    from sklearn.gaussian_process.kernels import RBF
    from sklearn.gaussian_process.kernels import Matern

    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)>



.. GENERATED FROM PYTHON SOURCE LINES 49-54

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 54-59

.. code-block:: Python

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







.. GENERATED FROM PYTHON SOURCE LINES 60-61

and seek to approximate it over the input space

.. GENERATED FROM PYTHON SOURCE LINES 61-64

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 65-67

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

.. GENERATED FROM PYTHON SOURCE LINES 67-71

.. 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 - 20:35:07: No coupling in MDA, switching chain_linearize to True.
        INFO - 20:35:07: *** Start Sampling execution ***
        INFO - 20:35:07: Sampling
        INFO - 20:35:07:    Disciplines: f
        INFO - 20:35:07:    MDO formulation: MDF
        INFO - 20:35:07: Running the algorithm PYDOE_FULLFACT:
        INFO - 20:35:07:     17%|█▋        | 1/6 [00:00<00:00, 550.94 it/sec]
        INFO - 20:35:07:     33%|███▎      | 2/6 [00:00<00:00, 899.00 it/sec]
        INFO - 20:35:07:     50%|█████     | 3/6 [00:00<00:00, 1204.68 it/sec]
        INFO - 20:35:07:     67%|██████▋   | 4/6 [00:00<00:00, 1468.21 it/sec]
        INFO - 20:35:07:     83%|████████▎ | 5/6 [00:00<00:00, 1699.34 it/sec]
        INFO - 20:35:07:    100%|██████████| 6/6 [00:00<00:00, 1878.89 it/sec]
        INFO - 20:35:07: *** End Sampling execution ***




.. GENERATED FROM PYTHON SOURCE LINES 72-78

Basics
------
Training
~~~~~~~~
Then,
we train a GP regression model from these samples:

.. GENERATED FROM PYTHON SOURCE LINES 78-81

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 82-86

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

.. GENERATED FROM PYTHON SOURCE LINES 86-90

.. 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([2.20380214])}



.. GENERATED FROM PYTHON SOURCE LINES 91-92

but cannot predict its Jacobian value:

.. GENERATED FROM PYTHON SOURCE LINES 92-97

.. code-block:: Python

    try:
        model.predict_jacobian(input_value)
    except NotImplementedError:
        print("The derivatives are not available for GaussianProcessRegressor.")





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

 .. code-block:: none

    The derivatives are not available for GaussianProcessRegressor.




.. GENERATED FROM PYTHON SOURCE LINES 98-108

Uncertainty
~~~~~~~~~~~
GP models are often valued for their ability to provide model uncertainty.
Indeed,
a GP model is a random process fully characterized
by its mean function
and a covariance structure.
Given an input point :math:`x`,
the prediction is equal to the mean at :math:`x`
and the uncertainty is equal to the standard deviation at :math:`x`:

.. GENERATED FROM PYTHON SOURCE LINES 108-111

.. code-block:: Python

    standard_deviation = model.predict_std(input_value)
    standard_deviation





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

 .. code-block:: none


    array([[0.3140468]])



.. GENERATED FROM PYTHON SOURCE LINES 112-118

Plotting
~~~~~~~~
You can see that the GP model interpolates the training points
but is very bad elsewhere.
This case-dependent problem is due to poor auto-tuning of these length scales.
We will look at how to correct this next.

.. GENERATED FROM PYTHON SOURCE LINES 118-130

.. 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_gp_regression_001.png
   :alt: plot gp regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

     WARNING - 20:35:07: No coupling in MDA, switching chain_linearize to True.
        INFO - 20:35:07: *** Start Sampling execution ***
        INFO - 20:35:07: Sampling
        INFO - 20:35:07:    Disciplines: f
        INFO - 20:35:07:    MDO formulation: MDF
        INFO - 20:35:07: Running the algorithm PYDOE_FULLFACT:
        INFO - 20:35:07:      1%|          | 1/100 [00:00<00:00, 3492.34 it/sec]
        INFO - 20:35:07:      2%|▏         | 2/100 [00:00<00:00, 3633.00 it/sec]
        INFO - 20:35:07:      3%|▎         | 3/100 [00:00<00:00, 3855.06 it/sec]
        INFO - 20:35:07:      4%|▍         | 4/100 [00:00<00:00, 4025.24 it/sec]
        INFO - 20:35:07:      5%|▌         | 5/100 [00:00<00:00, 4095.20 it/sec]
        INFO - 20:35:07:      6%|▌         | 6/100 [00:00<00:00, 4155.52 it/sec]
        INFO - 20:35:07:      7%|▋         | 7/100 [00:00<00:00, 4235.45 it/sec]
        INFO - 20:35:07:      8%|▊         | 8/100 [00:00<00:00, 4301.85 it/sec]
        INFO - 20:35:07:      9%|▉         | 9/100 [00:00<00:00, 4311.68 it/sec]
        INFO - 20:35:07:     10%|█         | 10/100 [00:00<00:00, 4301.41 it/sec]
        INFO - 20:35:07:     11%|█         | 11/100 [00:00<00:00, 4333.37 it/sec]
        INFO - 20:35:07:     12%|█▏        | 12/100 [00:00<00:00, 4367.17 it/sec]
        INFO - 20:35:07:     13%|█▎        | 13/100 [00:00<00:00, 4402.58 it/sec]
        INFO - 20:35:07:     14%|█▍        | 14/100 [00:00<00:00, 4442.45 it/sec]
        INFO - 20:35:07:     15%|█▌        | 15/100 [00:00<00:00, 4423.44 it/sec]
        INFO - 20:35:07:     16%|█▌        | 16/100 [00:00<00:00, 4453.73 it/sec]
        INFO - 20:35:07:     17%|█▋        | 17/100 [00:00<00:00, 4488.43 it/sec]
        INFO - 20:35:07:     18%|█▊        | 18/100 [00:00<00:00, 4515.40 it/sec]
        INFO - 20:35:07:     19%|█▉        | 19/100 [00:00<00:00, 4536.96 it/sec]
        INFO - 20:35:07:     20%|██        | 20/100 [00:00<00:00, 4524.84 it/sec]
        INFO - 20:35:07:     21%|██        | 21/100 [00:00<00:00, 4538.82 it/sec]
        INFO - 20:35:07:     22%|██▏       | 22/100 [00:00<00:00, 4558.35 it/sec]
        INFO - 20:35:07:     23%|██▎       | 23/100 [00:00<00:00, 4579.15 it/sec]
        INFO - 20:35:07:     24%|██▍       | 24/100 [00:00<00:00, 4572.69 it/sec]
        INFO - 20:35:07:     25%|██▌       | 25/100 [00:00<00:00, 4582.34 it/sec]
        INFO - 20:35:07:     26%|██▌       | 26/100 [00:00<00:00, 4597.47 it/sec]
        INFO - 20:35:07:     27%|██▋       | 27/100 [00:00<00:00, 4613.07 it/sec]
        INFO - 20:35:07:     28%|██▊       | 28/100 [00:00<00:00, 4630.02 it/sec]
        INFO - 20:35:07:     29%|██▉       | 29/100 [00:00<00:00, 4632.65 it/sec]
        INFO - 20:35:07:     30%|███       | 30/100 [00:00<00:00, 4642.80 it/sec]
        INFO - 20:35:07:     31%|███       | 31/100 [00:00<00:00, 4657.17 it/sec]
        INFO - 20:35:07:     32%|███▏      | 32/100 [00:00<00:00, 4670.23 it/sec]
        INFO - 20:35:07:     33%|███▎      | 33/100 [00:00<00:00, 4682.25 it/sec]
        INFO - 20:35:07:     34%|███▍      | 34/100 [00:00<00:00, 4681.91 it/sec]
        INFO - 20:35:07:     35%|███▌      | 35/100 [00:00<00:00, 4688.32 it/sec]
        INFO - 20:35:07:     36%|███▌      | 36/100 [00:00<00:00, 4695.85 it/sec]
        INFO - 20:35:07:     37%|███▋      | 37/100 [00:00<00:00, 4696.02 it/sec]
        INFO - 20:35:07:     38%|███▊      | 38/100 [00:00<00:00, 4706.72 it/sec]
        INFO - 20:35:07:     39%|███▉      | 39/100 [00:00<00:00, 4707.55 it/sec]
        INFO - 20:35:07:     40%|████      | 40/100 [00:00<00:00, 4713.50 it/sec]
        INFO - 20:35:07:     41%|████      | 41/100 [00:00<00:00, 4657.06 it/sec]
        INFO - 20:35:07:     42%|████▏     | 42/100 [00:00<00:00, 4659.35 it/sec]
        INFO - 20:35:07:     43%|████▎     | 43/100 [00:00<00:00, 4657.45 it/sec]
        INFO - 20:35:07:     44%|████▍     | 44/100 [00:00<00:00, 4661.87 it/sec]
        INFO - 20:35:07:     45%|████▌     | 45/100 [00:00<00:00, 4666.91 it/sec]
        INFO - 20:35:07:     46%|████▌     | 46/100 [00:00<00:00, 4674.79 it/sec]
        INFO - 20:35:07:     47%|████▋     | 47/100 [00:00<00:00, 4679.92 it/sec]
        INFO - 20:35:07:     48%|████▊     | 48/100 [00:00<00:00, 4678.75 it/sec]
        INFO - 20:35:07:     49%|████▉     | 49/100 [00:00<00:00, 4683.81 it/sec]
        INFO - 20:35:07:     50%|█████     | 50/100 [00:00<00:00, 4691.41 it/sec]
        INFO - 20:35:07:     51%|█████     | 51/100 [00:00<00:00, 4698.52 it/sec]
        INFO - 20:35:07:     52%|█████▏    | 52/100 [00:00<00:00, 4705.89 it/sec]
        INFO - 20:35:07:     53%|█████▎    | 53/100 [00:00<00:00, 4701.34 it/sec]
        INFO - 20:35:07:     54%|█████▍    | 54/100 [00:00<00:00, 4705.94 it/sec]
        INFO - 20:35:07:     55%|█████▌    | 55/100 [00:00<00:00, 4713.18 it/sec]
        INFO - 20:35:07:     56%|█████▌    | 56/100 [00:00<00:00, 4721.32 it/sec]
        INFO - 20:35:07:     57%|█████▋    | 57/100 [00:00<00:00, 4727.42 it/sec]
        INFO - 20:35:07:     58%|█████▊    | 58/100 [00:00<00:00, 4724.60 it/sec]
        INFO - 20:35:07:     59%|█████▉    | 59/100 [00:00<00:00, 4728.46 it/sec]
        INFO - 20:35:07:     60%|██████    | 60/100 [00:00<00:00, 4734.87 it/sec]
        INFO - 20:35:07:     61%|██████    | 61/100 [00:00<00:00, 4741.00 it/sec]
        INFO - 20:35:07:     62%|██████▏   | 62/100 [00:00<00:00, 4747.03 it/sec]
        INFO - 20:35:07:     63%|██████▎   | 63/100 [00:00<00:00, 4746.99 it/sec]
        INFO - 20:35:07:     64%|██████▍   | 64/100 [00:00<00:00, 4750.31 it/sec]
        INFO - 20:35:07:     65%|██████▌   | 65/100 [00:00<00:00, 4755.28 it/sec]
        INFO - 20:35:07:     66%|██████▌   | 66/100 [00:00<00:00, 4760.44 it/sec]
        INFO - 20:35:07:     67%|██████▋   | 67/100 [00:00<00:00, 4765.37 it/sec]
        INFO - 20:35:07:     68%|██████▊   | 68/100 [00:00<00:00, 4764.34 it/sec]
        INFO - 20:35:07:     69%|██████▉   | 69/100 [00:00<00:00, 4767.75 it/sec]
        INFO - 20:35:07:     70%|███████   | 70/100 [00:00<00:00, 4772.45 it/sec]
        INFO - 20:35:07:     71%|███████   | 71/100 [00:00<00:00, 4777.11 it/sec]
        INFO - 20:35:07:     72%|███████▏  | 72/100 [00:00<00:00, 4781.65 it/sec]
        INFO - 20:35:07:     73%|███████▎  | 73/100 [00:00<00:00, 4778.83 it/sec]
        INFO - 20:35:07:     74%|███████▍  | 74/100 [00:00<00:00, 4780.86 it/sec]
        INFO - 20:35:07:     75%|███████▌  | 75/100 [00:00<00:00, 4785.40 it/sec]
        INFO - 20:35:07:     76%|███████▌  | 76/100 [00:00<00:00, 4788.95 it/sec]
        INFO - 20:35:07:     77%|███████▋  | 77/100 [00:00<00:00, 4791.78 it/sec]
        INFO - 20:35:07:     78%|███████▊  | 78/100 [00:00<00:00, 4790.26 it/sec]
        INFO - 20:35:07:     79%|███████▉  | 79/100 [00:00<00:00, 4791.83 it/sec]
        INFO - 20:35:07:     80%|████████  | 80/100 [00:00<00:00, 4795.96 it/sec]
        INFO - 20:35:07:     81%|████████  | 81/100 [00:00<00:00, 4799.72 it/sec]
        INFO - 20:35:07:     82%|████████▏ | 82/100 [00:00<00:00, 4803.53 it/sec]
        INFO - 20:35:07:     83%|████████▎ | 83/100 [00:00<00:00, 4802.48 it/sec]
        INFO - 20:35:07:     84%|████████▍ | 84/100 [00:00<00:00, 4804.08 it/sec]
        INFO - 20:35:07:     85%|████████▌ | 85/100 [00:00<00:00, 4806.03 it/sec]
        INFO - 20:35:07:     86%|████████▌ | 86/100 [00:00<00:00, 4804.09 it/sec]
        INFO - 20:35:07:     87%|████████▋ | 87/100 [00:00<00:00, 4805.36 it/sec]
        INFO - 20:35:07:     88%|████████▊ | 88/100 [00:00<00:00, 4802.85 it/sec]
        INFO - 20:35:07:     89%|████████▉ | 89/100 [00:00<00:00, 4804.90 it/sec]
        INFO - 20:35:07:     90%|█████████ | 90/100 [00:00<00:00, 4807.47 it/sec]
        INFO - 20:35:07:     91%|█████████ | 91/100 [00:00<00:00, 4810.04 it/sec]
        INFO - 20:35:07:     92%|█████████▏| 92/100 [00:00<00:00, 4812.68 it/sec]
        INFO - 20:35:07:     93%|█████████▎| 93/100 [00:00<00:00, 4807.43 it/sec]
        INFO - 20:35:07:     94%|█████████▍| 94/100 [00:00<00:00, 4807.46 it/sec]
        INFO - 20:35:07:     95%|█████████▌| 95/100 [00:00<00:00, 4810.16 it/sec]
        INFO - 20:35:07:     96%|█████████▌| 96/100 [00:00<00:00, 4813.14 it/sec]
        INFO - 20:35:07:     97%|█████████▋| 97/100 [00:00<00:00, 4816.70 it/sec]
        INFO - 20:35:07:     98%|█████████▊| 98/100 [00:00<00:00, 4814.55 it/sec]
        INFO - 20:35:07:     99%|█████████▉| 99/100 [00:00<00:00, 4817.35 it/sec]
        INFO - 20:35:07:    100%|██████████| 100/100 [00:00<00:00, 4820.71 it/sec]
        INFO - 20:35:07: *** End Sampling execution ***




.. GENERATED FROM PYTHON SOURCE LINES 131-148

Settings
--------
The :class:`.GaussianProcessRegressor` has many options
defined in the :class:`.GaussianProcessRegressor_Settings` Pydantic model.
Here are the main ones.

Kernel
~~~~~~
The ``kernel`` option defines the kernel function
parametrizing the Gaussian process regressor
and must be passed as a scikit-learn object.
The default kernel is the Matérn 5/2 covariance function
with input length scales belonging to the interval :math:`[0.01,100]`,
initialized at 1
and optimized by the L-BFGS-B algorithm.
We can replace this kernel by the Matérn 5/2 kernel
with input length scales fixed at 1:

.. GENERATED FROM PYTHON SOURCE LINES 148-155

.. code-block:: Python

    model = create_regression_model(
        "GaussianProcessRegressor",
        training_dataset,
        kernel=Matern(length_scale=1.0, length_scale_bounds="fixed", nu=2.5),
    )
    model.learn()
    predicted_output_data_1 = model.predict(input_data).ravel()







.. GENERATED FROM PYTHON SOURCE LINES 156-158

or a squared exponential covariance kernel
with input length scales fixed at 1:

.. GENERATED FROM PYTHON SOURCE LINES 158-165

.. code-block:: Python

    model = create_regression_model(
        "GaussianProcessRegressor",
        training_dataset,
        kernel=RBF(length_scale=1.0, length_scale_bounds="fixed"),
    )
    model.learn()
    predicted_output_data_2 = model.predict(input_data).ravel()







.. GENERATED FROM PYTHON SOURCE LINES 166-170

These two models are much better than the previous one,
notably the one with the Matérn 5/2 kernel,
which highlights that the concern with the initial model is
the value of the length scales found by numerical optimization:

.. GENERATED FROM PYTHON SOURCE LINES 170-179

.. 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 - Kernel(Matern 2.5)"
    )
    plt.plot(input_data.ravel(), predicted_output_data_2, label="Regression - Kernel(RBF)")
    plt.grid()
    plt.legend()
    plt.show()



.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_002.png
   :alt: plot gp regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 180-183

Bounds
~~~~~~
The ``bounds`` option defines the bounds of the input length scales;

.. GENERATED FROM PYTHON SOURCE LINES 183-187

.. code-block:: Python

    model = create_regression_model(
        "GaussianProcessRegressor", training_dataset, bounds=(1e-1, 1e2)
    )
    model.learn()







.. GENERATED FROM PYTHON SOURCE LINES 188-189

Increasing the lower bounds can facilitate the training as in this example:

.. GENERATED FROM PYTHON SOURCE LINES 189-197

.. 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 - Bounds")
    plt.grid()
    plt.legend()
    plt.show()




.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_003.png
   :alt: plot gp regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 198-209

Alpha
~~~~~
The ``alpha`` parameter (default: 1e-10),
often called *nugget effect*,
is the value added to the diagonal of the training kernel matrix
to avoid overfitting.
When ``alpha`` is equal to zero,
the GP model interpolates the training points
at which the standard deviation is equal to zero.
The larger ``alpha`` is, the less interpolating the GP model is.
For example, we can increase the value to 0.1:

.. GENERATED FROM PYTHON SOURCE LINES 209-214

.. code-block:: Python

    predicted_output_data_1 = predicted_output_data_
    model = create_regression_model(
        "GaussianProcessRegressor", training_dataset, bounds=(1e-1, 1e2), alpha=0.1
    )
    model.learn()







.. GENERATED FROM PYTHON SOURCE LINES 215-216

and see that the model moves away from the training points:

.. GENERATED FROM PYTHON SOURCE LINES 216-223

.. code-block:: Python

    predicted_output_data_2 = model.predict(input_data).ravel()
    plt.plot(input_data.ravel(), reference_output_data, label="Reference")
    plt.plot(input_data.ravel(), predicted_output_data_1, label="Regression - Alpha(1e-10)")
    plt.plot(input_data.ravel(), predicted_output_data_2, label="Regression - Alpha(1e-1)")
    plt.grid()
    plt.legend()
    plt.show()



.. image-sg:: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_004.png
   :alt: plot gp regression
   :srcset: /examples/mlearning/regression_model/images/sphx_glr_plot_gp_regression_004.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.311 seconds)


.. _sphx_glr_download_examples_mlearning_regression_model_plot_gp_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_gp_regression.ipynb <plot_gp_regression.ipynb>`

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

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

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

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


.. only:: html

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

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