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<ul>
<li><a class="reference internal" href="#">Gaussian process regression (GPR) on Mauna Loa CO2 data</a><ul>
<li><a class="reference internal" href="#build-the-dataset">Build the dataset</a></li>
<li><a class="reference internal" href="#design-the-proper-kernel">Design the proper kernel</a></li>
<li><a class="reference internal" href="#model-fitting-and-extrapolation">Model fitting and extrapolation</a></li>
<li><a class="reference internal" href="#interpretation-of-kernel-hyperparameters">Interpretation of kernel hyperparameters</a></li>
</ul>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-gaussian-process-plot-gpr-co2-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
</div>
<section class="sphx-glr-example-title" id="gaussian-process-regression-gpr-on-mauna-loa-co2-data">
<span id="sphx-glr-auto-examples-gaussian-process-plot-gpr-co2-py"></span><h1>Gaussian process regression (GPR) on Mauna Loa CO2 data<a class="headerlink" href="#gaussian-process-regression-gpr-on-mauna-loa-co2-data" title="Permalink to this heading">¶</a></h1>
<p>This example is based on Section 5.4.3 of “Gaussian Processes for Machine
Learning” <a class="reference internal" href="../../modules/gaussian_process.html#rw2006" id="id1"><span>[RW2006]</span></a>. It illustrates an example of complex kernel engineering
and hyperparameter optimization using gradient ascent on the
log-marginal-likelihood. The data consists of the monthly average atmospheric
CO2 concentrations (in parts per million by volume (ppm)) collected at the
Mauna Loa Observatory in Hawaii, between 1958 and 2001. The objective is to
model the CO2 concentration as a function of the time <span class="math notranslate nohighlight">\(t\)</span> and extrapolate
for years after 2001.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</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="build-the-dataset">
<h2>Build the dataset<a class="headerlink" href="#build-the-dataset" title="Permalink to this heading">¶</a></h2>
<p>We will derive a dataset from the Mauna Loa Observatory that collected air
samples. We are interested in estimating the concentration of CO2 and
extrapolate it for further year. First, we load the original dataset available
in OpenML.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a>
<span class="n">co2</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a><span class="p">(</span><span class="n">data_id</span><span class="o">=</span><span class="mi">41187</span><span class="p">,</span> <span class="n">as_frame</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">parser</span><span class="o">=</span><span class="s2">"pandas"</span><span class="p">)</span>
<span class="n">co2</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div class="output_subarea output_html rendered_html output_result">
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>year</th>
<th>month</th>
<th>day</th>
<th>weight</th>
<th>flag</th>
<th>station</th>
<th>co2</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>1958</td>
<td>3</td>
<td>29</td>
<td>4</td>
<td>0</td>
<td>MLO</td>
<td>316.1</td>
</tr>
<tr>
<th>1</th>
<td>1958</td>
<td>4</td>
<td>5</td>
<td>6</td>
<td>0</td>
<td>MLO</td>
<td>317.3</td>
</tr>
<tr>
<th>2</th>
<td>1958</td>
<td>4</td>
<td>12</td>
<td>4</td>
<td>0</td>
<td>MLO</td>
<td>317.6</td>
</tr>
<tr>
<th>3</th>
<td>1958</td>
<td>4</td>
<td>19</td>
<td>6</td>
<td>0</td>
<td>MLO</td>
<td>317.5</td>
</tr>
<tr>
<th>4</th>
<td>1958</td>
<td>4</td>
<td>26</td>
<td>2</td>
<td>0</td>
<td>MLO</td>
<td>316.4</td>
</tr>
</tbody>
</table>
</div>
</div>
<br />
<br /><p>First, we process the original dataframe to create a date index and select
only the CO2 column.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="n">co2_data</span> <span class="o">=</span> <span class="n">co2</span><span class="o">.</span><span class="n">frame</span>
<span class="n">co2_data</span><span class="p">[</span><span class="s2">"date"</span><span class="p">]</span> <span class="o">=</span> <a href="https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.to_datetime.html#pandas.to_datetime" title="pandas.to_datetime" class="sphx-glr-backref-module-pandas sphx-glr-backref-type-py-function"><span class="n">pd</span><span class="o">.</span><span class="n">to_datetime</span></a><span class="p">(</span><span class="n">co2_data</span><span class="p">[[</span><span class="s2">"year"</span><span class="p">,</span> <span class="s2">"month"</span><span class="p">,</span> <span class="s2">"day"</span><span class="p">]])</span>
<span class="n">co2_data</span> <span class="o">=</span> <span class="n">co2_data</span><span class="p">[[</span><span class="s2">"date"</span><span class="p">,</span> <span class="s2">"co2"</span><span class="p">]]</span><span class="o">.</span><span class="n">set_index</span><span class="p">(</span><span class="s2">"date"</span><span class="p">)</span>
<span class="n">co2_data</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div class="output_subarea output_html rendered_html output_result">
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>co2</th>
</tr>
<tr>
<th>date</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<th>1958-03-29</th>
<td>316.1</td>
</tr>
<tr>
<th>1958-04-05</th>
<td>317.3</td>
</tr>
<tr>
<th>1958-04-12</th>
<td>317.6</td>
</tr>
<tr>
<th>1958-04-19</th>
<td>317.5</td>
</tr>
<tr>
<th>1958-04-26</th>
<td>316.4</td>
</tr>
</tbody>
</table>
</div>
</div>
<br />
<br /><div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">co2_data</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">co2_data</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
</pre></div>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>(Timestamp('1958-03-29 00:00:00'), Timestamp('2001-12-29 00:00:00'))
</pre></div>
</div>
<p>We see that we get CO2 concentration for some days from March, 1958 to
December, 2001. We can plot these raw information to have a better
understanding.</p>
<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>
<span class="n">co2_data</span><span class="o">.</span><span class="n">plot</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">"CO$_2$ concentration (ppm)"</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="s2">"Raw air samples measurements from the Mauna Loa Observatory"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_co2_001.png" srcset="../../_images/sphx_glr_plot_gpr_co2_001.png" alt="Raw air samples measurements from the Mauna Loa Observatory" class = "sphx-glr-single-img"/><p>We will preprocess the dataset by taking a monthly average and drop month
for which no measurements were collected. Such a processing will have an
smoothing effect on the data.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">co2_data</span> <span class="o">=</span> <span class="n">co2_data</span><span class="o">.</span><span class="n">resample</span><span class="p">(</span><span class="s2">"M"</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">"index"</span><span class="p">,</span> <span class="n">how</span><span class="o">=</span><span class="s2">"any"</span><span class="p">)</span>
<span class="n">co2_data</span><span class="o">.</span><span class="n">plot</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">"Monthly average of CO$_2$ concentration (ppm)"</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="s2">"Monthly average of air samples measurements</span><span class="se">\n</span><span class="s2">from the Mauna Loa Observatory"</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_co2_002.png" srcset="../../_images/sphx_glr_plot_gpr_co2_002.png" alt="Monthly average of air samples measurements from the Mauna Loa Observatory" class = "sphx-glr-single-img"/><p>The idea in this example will be to predict the CO2 concentration in function
of the date. We are as well interested in extrapolating for upcoming year
after 2001.</p>
<p>As a first step, we will divide the data and the target to estimate. The data
being a date, we will convert it into a numeric.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="p">(</span><span class="n">co2_data</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">year</span> <span class="o">+</span> <span class="n">co2_data</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">month</span> <span class="o">/</span> <span class="mi">12</span><span class="p">)</span><span class="o">.</span><span class="n">to_numpy</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">co2_data</span><span class="p">[</span><span class="s2">"co2"</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
</pre></div>
</div>
</section>
<section id="design-the-proper-kernel">
<h2>Design the proper kernel<a class="headerlink" href="#design-the-proper-kernel" title="Permalink to this heading">¶</a></h2>
<p>To design the kernel to use with our Gaussian process, we can make some
assumption regarding the data at hand. We observe that they have several
characteristics: we see a long term rising trend, a pronounced seasonal
variation and some smaller irregularities. We can use different appropriate
kernel that would capture these features.</p>
<p>First, the long term rising trend could be fitted using a radial basis
function (RBF) kernel with a large length-scale parameter. The RBF kernel
with a large length-scale enforces this component to be smooth. An trending
increase is not enforced as to give a degree of freedom to our model. The
specific length-scale and the amplitude are free hyperparameters.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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="n">long_term_trend_kernel</span> <span class="o">=</span> <span class="mf">50.0</span><span class="o">**</span><span class="mi">2</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">50.0</span><span class="p">)</span>
</pre></div>
</div>
<p>The seasonal variation is explained by the periodic exponential sine squared
kernel with a fixed periodicity of 1 year. The length-scale of this periodic
component, controlling its smoothness, is a free parameter. In order to allow
decaying away from exact periodicity, the product with an RBF kernel is
taken. The length-scale of this RBF component controls the decay time and is
a further free parameter. This type of kernel is also known as locally
periodic kernel.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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.ExpSineSquared.html#sklearn.gaussian_process.kernels.ExpSineSquared" title="sklearn.gaussian_process.kernels.ExpSineSquared" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ExpSineSquared</span></a>
<span class="n">seasonal_kernel</span> <span class="o">=</span> <span class="p">(</span>
<span class="mf">2.0</span><span class="o">**</span><span class="mi">2</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">100.0</span><span class="p">)</span>
<span class="o">*</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.ExpSineSquared.html#sklearn.gaussian_process.kernels.ExpSineSquared" title="sklearn.gaussian_process.kernels.ExpSineSquared" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ExpSineSquared</span></a><span class="p">(</span><span class="n">length_scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">periodicity</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">periodicity_bounds</span><span class="o">=</span><span class="s2">"fixed"</span><span class="p">)</span>
<span class="p">)</span>
</pre></div>
</div>
<p>The small irregularities are to be explained by a rational quadratic kernel
component, whose length-scale and alpha parameter, which quantifies the
diffuseness of the length-scales, are to be determined. A rational quadratic
kernel is equivalent to an RBF kernel with several length-scale and will
better accommodate the different irregularities.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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.RationalQuadratic.html#sklearn.gaussian_process.kernels.RationalQuadratic" title="sklearn.gaussian_process.kernels.RationalQuadratic" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RationalQuadratic</span></a>
<span class="n">irregularities_kernel</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <a href="../../modules/generated/sklearn.gaussian_process.kernels.RationalQuadratic.html#sklearn.gaussian_process.kernels.RationalQuadratic" title="sklearn.gaussian_process.kernels.RationalQuadratic" class="sphx-glr-backref-module-sklearn-gaussian_process-kernels sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RationalQuadratic</span></a><span class="p">(</span><span class="n">length_scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
</pre></div>
</div>
<p>Finally, the noise in the dataset can be accounted with a kernel consisting
of an RBF kernel contribution, which shall explain the correlated noise
components such as local weather phenomena, and a white kernel contribution
for the white noise. The relative amplitudes and the RBF’s length scale are
further free parameters.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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.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">noise_kernel</span> <span class="o">=</span> <span class="mf">0.1</span><span class="o">**</span><span class="mi">2</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">0.1</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">0.1</span><span class="o">**</span><span class="mi">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-5</span><span class="p">,</span> <span class="mf">1e5</span><span class="p">)</span>
<span class="p">)</span>
</pre></div>
</div>
<p>Thus, our final kernel is an addition of all previous kernel.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">co2_kernel</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">long_term_trend_kernel</span> <span class="o">+</span> <span class="n">seasonal_kernel</span> <span class="o">+</span> <span class="n">irregularities_kernel</span> <span class="o">+</span> <span class="n">noise_kernel</span>
<span class="p">)</span>
<span class="n">co2_kernel</span>
</pre></div>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01)
</pre></div>
</div>
</section>
<section id="model-fitting-and-extrapolation">
<h2>Model fitting and extrapolation<a class="headerlink" href="#model-fitting-and-extrapolation" title="Permalink to this heading">¶</a></h2>
<p>Now, we are ready to use a Gaussian process regressor and fit the available
data. To follow the example from the literature, we will subtract the mean
from the target. We could have used <code class="docutils literal notranslate"><span class="pre">normalize_y=True</span></code>. However, doing so
would have also scaled the target (dividing <code class="docutils literal notranslate"><span class="pre">y</span></code> by its standard deviation).
Thus, the hyperparameters of the different kernel would have had different
meaning since they would not have been expressed in ppm.</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="n">y_mean</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="n">gaussian_process</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">co2_kernel</span><span class="p">,</span> <span class="n">normalize_y</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">gaussian_process</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="n">y_mean</span><span class="p">)</span>
</pre></div>
</div>
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See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-29 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-29" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>GaussianProcessRegressor(kernel=50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-116" type="checkbox" checked><label for="sk-estimator-id-116" class="sk-toggleable__label sk-toggleable__label-arrow">GaussianProcessRegressor</label><div class="sk-toggleable__content"><pre>GaussianProcessRegressor(kernel=50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01))</pre></div></div></div></div></div>
</div>
<br />
<br /><p>Now, we will use the Gaussian process to predict on:</p>
<ul class="simple">
<li><p>training data to inspect the goodness of fit;</p></li>
<li><p>future data to see the extrapolation done by the model.</p></li>
</ul>
<p>Thus, we create synthetic data from 1958 to the current month. In addition,
we need to add the subtracted mean computed during training.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">datetime</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">today</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">datetime</span><span class="o">.</span><span class="n">now</span><span class="p">()</span>
<span class="n">current_month</span> <span class="o">=</span> <span class="n">today</span><span class="o">.</span><span class="n">year</span> <span class="o">+</span> <span class="n">today</span><span class="o">.</span><span class="n">month</span> <span class="o">/</span> <span class="mi">12</span>
<span class="n">X_test</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="n">start</span><span class="o">=</span><span class="mi">1958</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="n">current_month</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">1_000</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">mean_y_pred</span><span class="p">,</span> <span class="n">std_y_pred</span> <span class="o">=</span> <span class="n">gaussian_process</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">return_std</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">mean_y_pred</span> <span class="o">+=</span> <span class="n">y_mean</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">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">"dashed"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Measurements"</span><span class="p">)</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_test</span><span class="p">,</span> <span class="n">mean_y_pred</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"tab:blue"</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">"Gaussian process"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.fill_between.html#matplotlib.pyplot.fill_between" title="matplotlib.pyplot.fill_between" 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">fill_between</span></a><span class="p">(</span>
<span class="n">X_test</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span>
<span class="n">mean_y_pred</span> <span class="o">-</span> <span class="n">std_y_pred</span><span class="p">,</span>
<span class="n">mean_y_pred</span> <span class="o">+</span> <span class="n">std_y_pred</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"tab:blue"</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</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">"Year"</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">"Monthly average of CO$_2$ concentration (ppm)"</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="s2">"Monthly average of air samples measurements</span><span class="se">\n</span><span class="s2">from the Mauna Loa Observatory"</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_gpr_co2_003.png" srcset="../../_images/sphx_glr_plot_gpr_co2_003.png" alt="Monthly average of air samples measurements from the Mauna Loa Observatory" class = "sphx-glr-single-img"/><p>Our fitted model is capable to fit previous data properly and extrapolate to
future year with confidence.</p>
</section>
<section id="interpretation-of-kernel-hyperparameters">
<h2>Interpretation of kernel hyperparameters<a class="headerlink" href="#interpretation-of-kernel-hyperparameters" title="Permalink to this heading">¶</a></h2>
<p>Now, we can have a look at the hyperparameters of the kernel.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">gaussian_process</span><span class="o">.</span><span class="n">kernel_</span>
</pre></div>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>44.8**2 * RBF(length_scale=51.6) + 2.64**2 * RBF(length_scale=91.5) * ExpSineSquared(length_scale=1.48, periodicity=1) + 0.536**2 * RationalQuadratic(alpha=2.89, length_scale=0.968) + 0.188**2 * RBF(length_scale=0.122) + WhiteKernel(noise_level=0.0367)
</pre></div>
</div>
<p>Thus, most of the target signal, with the mean subtracted, is explained by a
long-term rising trend for ~45 ppm and a length-scale of ~52 years. The
periodic component has an amplitude of ~2.6ppm, a decay time of ~90 years and
a length-scale of ~1.5. The long decay time indicates that we have a
component very close to a seasonal periodicity. The correlated noise has an
amplitude of ~0.2 ppm with a length scale of ~0.12 years and a white-noise
contribution of ~0.04 ppm. Thus, the overall noise level is very small,
indicating that the data can be very well explained by the model.</p>
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