.. _example_gaussian_process_plot_gp_regression.py:


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Gaussian Processes regression: basic introductory example
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A simple one-dimensional regression exercise computed in two different ways:

1. A noise-free case with a cubic correlation model
2. A noisy case with a squared Euclidean correlation model

In both cases, the model parameters are estimated using the maximum
likelihood principle.

The figures illustrate the interpolating property of the Gaussian Process
model as well as its probabilistic nature in the form of a pointwise 95%
confidence interval.

Note that the parameter ``nugget`` is applied as a Tikhonov regularization
of the assumed covariance between the training points.  In the special case
of the squared euclidean correlation model, nugget is mathematically equivalent
to a normalized variance:  That is

.. math::
   \mathrm{nugget}_i = \left[\frac{\sigma_i}{y_i}\right]^2




.. rst-class:: horizontal


    *

      .. image:: images/plot_gp_regression_001.png
            :scale: 47

    *

      .. image:: images/plot_gp_regression_002.png
            :scale: 47




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

.. literalinclude:: plot_gp_regression.py
    :lines: 30-

**Total running time of the example:**  1.27 seconds
( 0 minutes  1.27 seconds)