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<li><a class="reference internal" href="#">Gaussian process regression (GPR) with noise-level estimation</a><ul>
<li><a class="reference internal" href="#data-generation">Data generation</a></li>
<li><a class="reference internal" href="#optimisation-of-kernel-hyperparameters-in-gpr">Optimisation of kernel hyperparameters in GPR</a></li>
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<section class="sphx-glr-example-title" id="gaussian-process-regression-gpr-with-noise-level-estimation">
<span id="sphx-glr-auto-examples-gaussian-process-plot-gpr-noisy-py"></span><h1>Gaussian process regression (GPR) with noise-level estimation<a class="headerlink" href="#gaussian-process-regression-gpr-with-noise-level-estimation" title="Permalink to this heading">¶</a></h1>
<p>This example shows the ability of the
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel"><code class="xref py py-class docutils literal notranslate"><span class="pre">WhiteKernel</span></code></a> to estimate the noise
level in the data. Moreover, we show the importance of kernel hyperparameters
initialization.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Jan Hendrik Metzen <[email protected]></span>
<span class="c1"># Guillaume Lemaitre <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="data-generation">
<h2>Data generation<a class="headerlink" href="#data-generation" title="Permalink to this heading">¶</a></h2>
<p>We will work in a setting where <code class="docutils literal notranslate"><span class="pre">X</span></code> will contain a single feature. We create a
function that will generate the target to be predicted. We will add an
option to add some noise to the generated target.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">def</span> <span class="nf">target_generator</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">add_noise</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="n">target</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sin.html#numpy.sin" title="numpy.sin" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sin</span></a><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">X</span><span class="p">)</span>
<span class="k">if</span> <span class="n">add_noise</span><span class="p">:</span>
<span class="n">rng</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/legacy.html#numpy.random.RandomState" title="numpy.random.RandomState" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span></a><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">target</span> <span class="o">+=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">return</span> <span class="n">target</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
</pre></div>
</div>
<p>Let’s have a look to the target generator where we will not add any noise to
observe the signal that we would like to predict.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">target_generator</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">add_noise</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Expected signal"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"X"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_noisy_001.png" srcset="../../_images/sphx_glr_plot_gpr_noisy_001.png" alt="plot gpr noisy" class = "sphx-glr-single-img"/><p>The target is transforming the input <code class="docutils literal notranslate"><span class="pre">X</span></code> using a sine function. Now, we will
generate few noisy training samples. To illustrate the noise level, we will
plot the true signal together with the noisy training samples.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">rng</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/legacy.html#numpy.random.RandomState" title="numpy.random.RandomState" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">target_generator</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">add_noise</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Expected signal"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span>
<span class="n">x</span><span class="o">=</span><span class="n">X_train</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">y</span><span class="o">=</span><span class="n">y_train</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"Observations"</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"X"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_noisy_002.png" srcset="../../_images/sphx_glr_plot_gpr_noisy_002.png" alt="plot gpr noisy" class = "sphx-glr-single-img"/></section>
<section id="optimisation-of-kernel-hyperparameters-in-gpr">
<h2>Optimisation of kernel hyperparameters in GPR<a class="headerlink" href="#optimisation-of-kernel-hyperparameters-in-gpr" title="Permalink to this heading">¶</a></h2>
<p>Now, we will create a
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html#sklearn.gaussian_process.GaussianProcessRegressor" title="sklearn.gaussian_process.GaussianProcessRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianProcessRegressor</span></code></a>
using an additive kernel adding a
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBF</span></code></a> and
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel"><code class="xref py py-class docutils literal notranslate"><span class="pre">WhiteKernel</span></code></a> kernels.
The <a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel"><code class="xref py py-class docutils literal notranslate"><span class="pre">WhiteKernel</span></code></a> is a kernel that
will able to estimate the amount of noise present in the data while the
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBF</span></code></a> will serve at fitting the
non-linearity between the data and the target.</p>
<p>However, we will show that the hyperparameter space contains several local
minima. It will highlights the importance of initial hyperparameter values.</p>
<p>We will create a model using a kernel with a high noise level and a large
length scale, which will explain all variations in the data by noise.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.gaussian_process</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html#sklearn.gaussian_process.GaussianProcessRegressor" title="sklearn.gaussian_process.GaussianProcessRegressor" class="sphx-glr-backref-module-sklearn-gaussian_process sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianProcessRegressor</span></a>
<span class="kn">from</span> <span class="nn">sklearn.gaussian_process.kernels</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RBF</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">WhiteKernel</span></a>
<span class="n">kernel</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">*</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RBF</span></a><span class="p">(</span><span class="n">length_scale</span><span class="o">=</span><span class="mf">1e1</span><span class="p">,</span> <span class="n">length_scale_bounds</span><span class="o">=</span><span class="p">(</span><span class="mf">1e-2</span><span class="p">,</span> <span class="mf">1e3</span><span class="p">))</span> <span class="o">+</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">WhiteKernel</span></a><span class="p">(</span>
<span class="n">noise_level</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">noise_level_bounds</span><span class="o">=</span><span class="p">(</span><span class="mf">1e-5</span><span class="p">,</span> <span class="mf">1e1</span><span class="p">)</span>
<span class="p">)</span>
<span class="n">gpr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html#sklearn.gaussian_process.GaussianProcessRegressor" title="sklearn.gaussian_process.GaussianProcessRegressor" class="sphx-glr-backref-module-sklearn-gaussian_process sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianProcessRegressor</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.0</span><span class="p">)</span>
<span class="n">gpr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_mean</span><span class="p">,</span> <span class="n">y_std</span> <span class="o">=</span> <span class="n">gpr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">return_std</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>/home/circleci/project/sklearn/gaussian_process/kernels.py:430: ConvergenceWarning:
The optimal value found for dimension 0 of parameter k1__k2__length_scale is close to the specified upper bound 1000.0. Increasing the bound and calling fit again may find a better value.
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Expected signal"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">X_train</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">y</span><span class="o">=</span><span class="n">y_train</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Observations"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y_mean</span><span class="p">,</span> <span class="n">y_std</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"X"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span>
<span class="sa">f</span><span class="s2">"Initial: </span><span class="si">{</span><span class="n">kernel</span><span class="si">}</span><span class="se">\n</span><span class="s2">Optimum: </span><span class="si">{</span><span class="n">gpr</span><span class="o">.</span><span class="n">kernel_</span><span class="si">}</span><span class="se">\n</span><span class="s2">Log-Marginal-Likelihood: "</span>
<span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">gpr</span><span class="o">.</span><span class="n">log_marginal_likelihood</span><span class="p">(</span><span class="n">gpr</span><span class="o">.</span><span class="n">kernel_</span><span class="o">.</span><span class="n">theta</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_noisy_003.png" srcset="../../_images/sphx_glr_plot_gpr_noisy_003.png" alt="Initial: 1**2 * RBF(length_scale=10) + WhiteKernel(noise_level=1) Optimum: 0.763**2 * RBF(length_scale=1e+03) + WhiteKernel(noise_level=0.525) Log-Marginal-Likelihood: -23.499266455424188" class = "sphx-glr-single-img"/><p>We see that the optimum kernel found still have a high noise level and
an even larger length scale. Furthermore, we observe that the
model does not provide faithful predictions.</p>
<p>Now, we will initialize the
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF"><code class="xref py py-class docutils literal notranslate"><span class="pre">RBF</span></code></a> with a
larger <code class="docutils literal notranslate"><span class="pre">length_scale</span></code> and the
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel"><code class="xref py py-class docutils literal notranslate"><span class="pre">WhiteKernel</span></code></a>
with a smaller noise level lower bound.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">kernel</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">*</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.RBF.html#sklearn.gaussian_process.kernels.RBF" title="sklearn.gaussian_process.kernels.RBF" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RBF</span></a><span class="p">(</span><span class="n">length_scale</span><span class="o">=</span><span class="mf">1e-1</span><span class="p">,</span> <span class="n">length_scale_bounds</span><span class="o">=</span><span class="p">(</span><span class="mf">1e-2</span><span class="p">,</span> <span class="mf">1e3</span><span class="p">))</span> <span class="o">+</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.WhiteKernel.html#sklearn.gaussian_process.kernels.WhiteKernel" title="sklearn.gaussian_process.kernels.WhiteKernel" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">WhiteKernel</span></a><span class="p">(</span>
<span class="n">noise_level</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span> <span class="n">noise_level_bounds</span><span class="o">=</span><span class="p">(</span><span class="mf">1e-10</span><span class="p">,</span> <span class="mf">1e1</span><span class="p">)</span>
<span class="p">)</span>
<span class="n">gpr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html#sklearn.gaussian_process.GaussianProcessRegressor" title="sklearn.gaussian_process.GaussianProcessRegressor" class="sphx-glr-backref-module-sklearn-gaussian_process sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianProcessRegressor</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.0</span><span class="p">)</span>
<span class="n">gpr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_mean</span><span class="p">,</span> <span class="n">y_std</span> <span class="o">=</span> <span class="n">gpr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">return_std</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Expected signal"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">X_train</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">y</span><span class="o">=</span><span class="n">y_train</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Observations"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y_mean</span><span class="p">,</span> <span class="n">y_std</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"X"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span>
<span class="sa">f</span><span class="s2">"Initial: </span><span class="si">{</span><span class="n">kernel</span><span class="si">}</span><span class="se">\n</span><span class="s2">Optimum: </span><span class="si">{</span><span class="n">gpr</span><span class="o">.</span><span class="n">kernel_</span><span class="si">}</span><span class="se">\n</span><span class="s2">Log-Marginal-Likelihood: "</span>
<span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">gpr</span><span class="o">.</span><span class="n">log_marginal_likelihood</span><span class="p">(</span><span class="n">gpr</span><span class="o">.</span><span class="n">kernel_</span><span class="o">.</span><span class="n">theta</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_noisy_004.png" srcset="../../_images/sphx_glr_plot_gpr_noisy_004.png" alt="Initial: 1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01) Optimum: 1.05**2 * RBF(length_scale=0.569) + WhiteKernel(noise_level=0.134) Log-Marginal-Likelihood: -18.429732528984058" class = "sphx-glr-single-img"/><p>First, we see that the model’s predictions are more precise than the
previous model’s: this new model is able to estimate the noise-free
functional relationship.</p>
<p>Looking at the kernel hyperparameters, we see that the best combination found
has a smaller noise level and shorter length scale than the first model.</p>
<p>We can inspect the Log-Marginal-Likelihood (LML) of
<a class="reference internal" href="../../modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html#sklearn.gaussian_process.GaussianProcessRegressor" title="sklearn.gaussian_process.GaussianProcessRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianProcessRegressor</span></code></a>
for different hyperparameters to get a sense of the local minima.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">matplotlib.colors</span> <span class="kn">import</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.colors.LogNorm.html#matplotlib.colors.LogNorm" title="matplotlib.colors.LogNorm" class="sphx-glr-backref-module-matplotlib-colors sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogNorm</span></a>
<span class="n">length_scale</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.logspace.html#numpy.logspace" title="numpy.logspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">noise_level</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.logspace.html#numpy.logspace" title="numpy.logspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">length_scale_grid</span><span class="p">,</span> <span class="n">noise_level_grid</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.meshgrid.html#numpy.meshgrid" title="numpy.meshgrid" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><span class="n">length_scale</span><span class="p">,</span> <span class="n">noise_level</span><span class="p">)</span>
<span class="n">log_marginal_likelihood</span> <span class="o">=</span> <span class="p">[</span>
<span class="n">gpr</span><span class="o">.</span><span class="n">log_marginal_likelihood</span><span class="p">(</span><span class="n">theta</span><span class="o">=</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.log.html#numpy.log" title="numpy.log" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log</span></a><span class="p">([</span><span class="mf">0.36</span><span class="p">,</span> <span class="n">scale</span><span class="p">,</span> <span class="n">noise</span><span class="p">]))</span>
<span class="k">for</span> <span class="n">scale</span><span class="p">,</span> <span class="n">noise</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">length_scale_grid</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">noise_level_grid</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
<span class="p">]</span>
<span class="n">log_marginal_likelihood</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.reshape.html#numpy.reshape" title="numpy.reshape" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">reshape</span></a><span class="p">(</span>
<span class="n">log_marginal_likelihood</span><span class="p">,</span> <span class="n">newshape</span><span class="o">=</span><span class="n">noise_level_grid</span><span class="o">.</span><span class="n">shape</span>
<span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">log_marginal_likelihood</span><span class="p">)</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="mi">50</span>
<span class="n">level</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.around.html#numpy.around" title="numpy.around" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">around</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.logspace.html#numpy.logspace" title="numpy.logspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.log10.html#numpy.log10" title="numpy.log10" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log10</span></a><span class="p">(</span><span class="n">vmin</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.log10.html#numpy.log10" title="numpy.log10" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log10</span></a><span class="p">(</span><span class="n">vmax</span><span class="p">),</span> <span class="n">num</span><span class="o">=</span><span class="mi">50</span><span class="p">),</span> <span class="n">decimals</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour" title="matplotlib.pyplot.contour" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">contour</span></a><span class="p">(</span>
<span class="n">length_scale_grid</span><span class="p">,</span>
<span class="n">noise_level_grid</span><span class="p">,</span>
<span class="o">-</span><span class="n">log_marginal_likelihood</span><span class="p">,</span>
<span class="n">levels</span><span class="o">=</span><span class="n">level</span><span class="p">,</span>
<span class="n">norm</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.colors.LogNorm.html#matplotlib.colors.LogNorm" title="matplotlib.colors.LogNorm" class="sphx-glr-backref-module-matplotlib-colors sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogNorm</span></a><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vmax</span><span class="p">),</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.colorbar.html#matplotlib.pyplot.colorbar" title="matplotlib.pyplot.colorbar" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xscale.html#matplotlib.pyplot.xscale" title="matplotlib.pyplot.xscale" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xscale</span></a><span class="p">(</span><span class="s2">"log"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.yscale.html#matplotlib.pyplot.yscale" title="matplotlib.pyplot.yscale" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">yscale</span></a><span class="p">(</span><span class="s2">"log"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Length-scale"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Noise-level"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Log-marginal-likelihood"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_noisy_005.png" srcset="../../_images/sphx_glr_plot_gpr_noisy_005.png" alt="Log-marginal-likelihood" class = "sphx-glr-single-img"/><p>We see that there are two local minima that correspond to the combination
of hyperparameters previously found. Depending on the initial values for the
hyperparameters, the gradient-based optimization might converge whether or
not to the best model. It is thus important to repeat the optimization
several times for different initializations.</p>
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