.. _example_gaussian_process_plot_gpr_noisy.py:


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Gaussian process regression (GPR) with noise-level estimation
=============================================================

This example illustrates that GPR with a sum-kernel including a WhiteKernel can
estimate the noise level of data. An illustration of the
log-marginal-likelihood (LML) landscape shows that there exist two local
maxima of LML. The first corresponds to a model with a high noise level and a
large length scale, which explains all variations in the data by noise. The
second one has a smaller noise level and shorter length scale, which explains
most of the variation by the noise-free functional relationship. The second
model has a higher likelihood; however, depending on the initial value for the
hyperparameters, the gradient-based optimization might also converge to the
high-noise solution. It is thus important to repeat the optimization several
times for different initializations.



.. rst-class:: horizontal


    *

      .. image:: images/plot_gpr_noisy_000.png
            :scale: 47

    *

      .. image:: images/plot_gpr_noisy_001.png
            :scale: 47

    *

      .. image:: images/plot_gpr_noisy_002.png
            :scale: 47




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

.. literalinclude:: plot_gpr_noisy.py
    :lines: 18-

**Total running time of the example:**  9.36 seconds
( 0 minutes  9.36 seconds)