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<li><a class="reference internal" href="#">Comparison of kernel ridge regression and SVR</a><ul>
<li><a class="reference internal" href="#generate-sample-data">Generate sample data</a></li>
<li><a class="reference internal" href="#construct-the-kernel-based-regression-models">Construct the kernel-based regression models</a></li>
<li><a class="reference internal" href="#compare-times-of-svr-and-kernel-ridge-regression">Compare times of SVR and Kernel Ridge Regression</a></li>
<li><a class="reference internal" href="#look-at-the-results">Look at the results</a></li>
<li><a class="reference internal" href="#visualize-training-and-prediction-times">Visualize training and prediction times</a></li>
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<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#sphx-glr-download-auto-examples-miscellaneous-plot-kernel-ridge-regression-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
</div>
<section class="sphx-glr-example-title" id="comparison-of-kernel-ridge-regression-and-svr">
<span id="sphx-glr-auto-examples-miscellaneous-plot-kernel-ridge-regression-py"></span><h1>Comparison of kernel ridge regression and SVR<a class="headerlink" href="#comparison-of-kernel-ridge-regression-and-svr" title="Link to this heading">¶</a></h1>
<p>Both kernel ridge regression (KRR) and SVR learn a non-linear function by
employing the kernel trick, i.e., they learn a linear function in the space
induced by the respective kernel which corresponds to a non-linear function in
the original space. They differ in the loss functions (ridge versus
epsilon-insensitive loss). In contrast to SVR, fitting a KRR can be done in
closed-form and is typically faster for medium-sized datasets. On the other
hand, the learned model is non-sparse and thus slower than SVR at
prediction-time.</p>
<p>This example illustrates both methods on an artificial dataset, which
consists of a sinusoidal target function and strong noise added to every fifth
datapoint.</p>
<p>Authors: Jan Hendrik Metzen <<a class="reference external" href="mailto:jhm%40informatik.uni-bremen.de">jhm<span>@</span>informatik<span>.</span>uni-bremen<span>.</span>de</a>>
License: BSD 3 clause</p>
<section id="generate-sample-data">
<h2>Generate sample data<a class="headerlink" href="#generate-sample-data" title="Link to this heading">¶</a></h2>
<div class="highlight-Python 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="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">42</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">rng</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</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="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="c1"># Add noise to targets</span>
<span class="n">y</span><span class="p">[::</span><span class="mi">5</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">3</span> <span class="o">*</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">-</span> <span class="n">rng</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">//</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">X_plot</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="mi">100000</span><span class="p">)[:,</span> <span class="kc">None</span><span class="p">]</span>
</pre></div>
</div>
</section>
<section id="construct-the-kernel-based-regression-models">
<h2>Construct the kernel-based regression models<a class="headerlink" href="#construct-the-kernel-based-regression-models" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.kernel_ridge</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge" class="sphx-glr-backref-module-sklearn-kernel_ridge sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelRidge</span></a>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSearchCV</span></a>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">SVR</span></a>
<span class="n">train_size</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">svr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSearchCV</span></a><span class="p">(</span>
<a href="../../modules/generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">SVR</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">),</span>
<span class="n">param_grid</span><span class="o">=</span><span class="p">{</span><span class="s2">"C"</span><span class="p">:</span> <span class="p">[</span><span class="mf">1e0</span><span class="p">,</span> <span class="mf">1e1</span><span class="p">,</span> <span class="mf">1e2</span><span class="p">,</span> <span class="mf">1e3</span><span class="p">],</span> <span class="s2">"gamma"</span><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><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)},</span>
<span class="p">)</span>
<span class="n">kr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSearchCV</span></a><span class="p">(</span>
<a href="../../modules/generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge" class="sphx-glr-backref-module-sklearn-kernel_ridge sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelRidge</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">),</span>
<span class="n">param_grid</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="p">[</span><span class="mf">1e0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">1e-2</span><span class="p">,</span> <span class="mf">1e-3</span><span class="p">],</span> <span class="s2">"gamma"</span><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><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)},</span>
<span class="p">)</span>
</pre></div>
</div>
</section>
<section id="compare-times-of-svr-and-kernel-ridge-regression">
<h2>Compare times of SVR and Kernel Ridge Regression<a class="headerlink" href="#compare-times-of-svr-and-kernel-ridge-regression" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">svr</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">train_size</span><span class="p">],</span> <span class="n">y</span><span class="p">[:</span><span class="n">train_size</span><span class="p">])</span>
<span class="n">svr_fit</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Best SVR with params: </span><span class="si">{</span><span class="n">svr</span><span class="o">.</span><span class="n">best_params_</span><span class="si">}</span><span class="s2"> and R2 score: </span><span class="si">{</span><span class="n">svr</span><span class="o">.</span><span class="n">best_score_</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"SVR complexity and bandwidth selected and model fitted in </span><span class="si">%.3f</span><span class="s2"> s"</span> <span class="o">%</span> <span class="n">svr_fit</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">kr</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">train_size</span><span class="p">],</span> <span class="n">y</span><span class="p">[:</span><span class="n">train_size</span><span class="p">])</span>
<span class="n">kr_fit</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Best KRR with params: </span><span class="si">{</span><span class="n">kr</span><span class="o">.</span><span class="n">best_params_</span><span class="si">}</span><span class="s2"> and R2 score: </span><span class="si">{</span><span class="n">kr</span><span class="o">.</span><span class="n">best_score_</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"KRR complexity and bandwidth selected and model fitted in </span><span class="si">%.3f</span><span class="s2"> s"</span> <span class="o">%</span> <span class="n">kr_fit</span><span class="p">)</span>
<span class="n">sv_ratio</span> <span class="o">=</span> <span class="n">svr</span><span class="o">.</span><span class="n">best_estimator_</span><span class="o">.</span><span class="n">support_</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">train_size</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Support vector ratio: </span><span class="si">%.3f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">sv_ratio</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">y_svr</span> <span class="o">=</span> <span class="n">svr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_plot</span><span class="p">)</span>
<span class="n">svr_predict</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"SVR prediction for </span><span class="si">%d</span><span class="s2"> inputs in </span><span class="si">%.3f</span><span class="s2"> s"</span> <span class="o">%</span> <span class="p">(</span><span class="n">X_plot</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">svr_predict</span><span class="p">))</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">y_kr</span> <span class="o">=</span> <span class="n">kr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_plot</span><span class="p">)</span>
<span class="n">kr_predict</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"KRR prediction for </span><span class="si">%d</span><span class="s2"> inputs in </span><span class="si">%.3f</span><span class="s2"> s"</span> <span class="o">%</span> <span class="p">(</span><span class="n">X_plot</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">kr_predict</span><span class="p">))</span>
</pre></div>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Best SVR with params: {'C': 1.0, 'gamma': 0.09999999999999999} and R2 score: 0.737
SVR complexity and bandwidth selected and model fitted in 0.494 s
Best KRR with params: {'alpha': 0.1, 'gamma': 0.09999999999999999} and R2 score: 0.723
KRR complexity and bandwidth selected and model fitted in 0.192 s
Support vector ratio: 0.340
SVR prediction for 100000 inputs in 0.132 s
KRR prediction for 100000 inputs in 0.092 s
</pre></div>
</div>
</section>
<section id="look-at-the-results">
<h2>Look at the results<a class="headerlink" href="#look-at-the-results" title="Link to this heading">¶</a></h2>
<div class="highlight-Python 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">sv_ind</span> <span class="o">=</span> <span class="n">svr</span><span class="o">.</span><span class="n">best_estimator_</span><span class="o">.</span><span class="n">support_</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="p">[</span><span class="n">sv_ind</span><span class="p">],</span>
<span class="n">y</span><span class="p">[</span><span class="n">sv_ind</span><span class="p">],</span>
<span class="n">c</span><span class="o">=</span><span class="s2">"r"</span><span class="p">,</span>
<span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"SVR support vectors"</span><span class="p">,</span>
<span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">edgecolors</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</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="p">[:</span><span class="mi">100</span><span class="p">],</span> <span class="n">y</span><span class="p">[:</span><span class="mi">100</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"data"</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">edgecolors</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</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_plot</span><span class="p">,</span>
<span class="n">y_svr</span><span class="p">,</span>
<span class="n">c</span><span class="o">=</span><span class="s2">"r"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"SVR (fit: </span><span class="si">%.3f</span><span class="s2">s, predict: </span><span class="si">%.3f</span><span class="s2">s)"</span> <span class="o">%</span> <span class="p">(</span><span class="n">svr_fit</span><span class="p">,</span> <span class="n">svr_predict</span><span class="p">),</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_plot</span><span class="p">,</span> <span class="n">y_kr</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">"g"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"KRR (fit: </span><span class="si">%.3f</span><span class="s2">s, predict: </span><span class="si">%.3f</span><span class="s2">s)"</span> <span class="o">%</span> <span class="p">(</span><span class="n">kr_fit</span><span class="p">,</span> <span class="n">kr_predict</span><span class="p">)</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">"data"</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">"target"</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">"SVR versus Kernel Ridge"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</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>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kernel_ridge_regression_001.png" srcset="../../_images/sphx_glr_plot_kernel_ridge_regression_001.png" alt="SVR versus Kernel Ridge" class = "sphx-glr-single-img"/><p>The previous figure compares the learned model of KRR and SVR when both
complexity/regularization and bandwidth of the RBF kernel are optimized using
grid-search. The learned functions are very similar; however, fitting KRR is
approximately 3-4 times faster than fitting SVR (both with grid-search).</p>
<p>Prediction of 100000 target values could be in theory approximately three
times faster with SVR since it has learned a sparse model using only
approximately 1/3 of the training datapoints as support vectors. However, in
practice, this is not necessarily the case because of implementation details
in the way the kernel function is computed for each model that can make the
KRR model as fast or even faster despite computing more arithmetic
operations.</p>
</section>
<section id="visualize-training-and-prediction-times">
<h2>Visualize training and prediction times<a class="headerlink" href="#visualize-training-and-prediction-times" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" 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">figure</span></a><span class="p">()</span>
<span class="n">sizes</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="mi">1</span><span class="p">,</span> <span class="mf">3.8</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">estimator</span> <span class="ow">in</span> <span class="p">{</span>
<span class="s2">"KRR"</span><span class="p">:</span> <a href="../../modules/generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge" class="sphx-glr-backref-module-sklearn-kernel_ridge sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelRidge</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">10</span><span class="p">),</span>
<span class="s2">"SVR"</span><span class="p">:</span> <a href="../../modules/generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">SVR</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mf">1e2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">10</span><span class="p">),</span>
<span class="p">}</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">train_time</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">test_time</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">train_test_size</span> <span class="ow">in</span> <span class="n">sizes</span><span class="p">:</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">estimator</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">train_test_size</span><span class="p">],</span> <span class="n">y</span><span class="p">[:</span><span class="n">train_test_size</span><span class="p">])</span>
<span class="n">train_time</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span><span class="p">)</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_plot</span><span class="p">[:</span><span class="mi">1000</span><span class="p">])</span>
<span class="n">test_time</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="https://docs.python.org/3/library/time.html#time.time" title="time.time" class="sphx-glr-backref-module-time sphx-glr-backref-type-py-function"><span class="n">time</span><span class="o">.</span><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</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">sizes</span><span class="p">,</span>
<span class="n">train_time</span><span class="p">,</span>
<span class="s2">"o-"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"r"</span> <span class="k">if</span> <span class="n">name</span> <span class="o">==</span> <span class="s2">"SVR"</span> <span class="k">else</span> <span class="s2">"g"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"</span><span class="si">%s</span><span class="s2"> (train)"</span> <span class="o">%</span> <span class="n">name</span><span class="p">,</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">sizes</span><span class="p">,</span>
<span class="n">test_time</span><span class="p">,</span>
<span class="s2">"o--"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"r"</span> <span class="k">if</span> <span class="n">name</span> <span class="o">==</span> <span class="s2">"SVR"</span> <span class="k">else</span> <span class="s2">"g"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"</span><span class="si">%s</span><span class="s2"> (test)"</span> <span class="o">%</span> <span class="n">name</span><span class="p">,</span>
<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">"Train size"</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">"Time (seconds)"</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">"Execution Time"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</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><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kernel_ridge_regression_002.png" srcset="../../_images/sphx_glr_plot_kernel_ridge_regression_002.png" alt="Execution Time" class = "sphx-glr-single-img"/><p>This figure compares the time for fitting and prediction of KRR and SVR for
different sizes of the training set. Fitting KRR is faster than SVR for
medium-sized training sets (less than a few thousand samples); however, for
larger training sets SVR scales better. With regard to prediction time, SVR
should be faster than KRR for all sizes of the training set because of the
learned sparse solution, however this is not necessarily the case in practice
because of implementation details. Note that the degree of sparsity and thus
the prediction time depends on the parameters epsilon and C of the SVR.</p>
</section>
<section id="visualize-the-learning-curves">
<h2>Visualize the learning curves<a class="headerlink" href="#visualize-the-learning-curves" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">LearningCurveDisplay</span>
<span class="n">_</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" 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">subplots</span></a><span class="p">()</span>
<span class="n">svr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.SVR.html#sklearn.svm.SVR" title="sklearn.svm.SVR" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">SVR</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mf">1e1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">kr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.kernel_ridge.KernelRidge.html#sklearn.kernel_ridge.KernelRidge" title="sklearn.kernel_ridge.KernelRidge" class="sphx-glr-backref-module-sklearn-kernel_ridge sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelRidge</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">common_params</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"X"</span><span class="p">:</span> <span class="n">X</span><span class="p">[:</span><span class="mi">100</span><span class="p">],</span>
<span class="s2">"y"</span><span class="p">:</span> <span class="n">y</span><span class="p">[:</span><span class="mi">100</span><span class="p">],</span>
<span class="s2">"train_sizes"</span><span class="p">:</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="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span>
<span class="s2">"scoring"</span><span class="p">:</span> <span class="s2">"neg_mean_squared_error"</span><span class="p">,</span>
<span class="s2">"negate_score"</span><span class="p">:</span> <span class="kc">True</span><span class="p">,</span>
<span class="s2">"score_name"</span><span class="p">:</span> <span class="s2">"Mean Squared Error"</span><span class="p">,</span>
<span class="s2">"score_type"</span><span class="p">:</span> <span class="s2">"test"</span><span class="p">,</span>
<span class="s2">"std_display_style"</span><span class="p">:</span> <span class="kc">None</span><span class="p">,</span>
<span class="s2">"ax"</span><span class="p">:</span> <span class="n">ax</span><span class="p">,</span>
<span class="p">}</span>
<a href="../../modules/generated/sklearn.model_selection.LearningCurveDisplay.html#sklearn.model_selection.LearningCurveDisplay.from_estimator" title="sklearn.model_selection.LearningCurveDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-model_selection-LearningCurveDisplay sphx-glr-backref-type-py-method"><span class="n">LearningCurveDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span><span class="n">svr</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span>
<a href="../../modules/generated/sklearn.model_selection.LearningCurveDisplay.html#sklearn.model_selection.LearningCurveDisplay.from_estimator" title="sklearn.model_selection.LearningCurveDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-model_selection-LearningCurveDisplay sphx-glr-backref-type-py-method"><span class="n">LearningCurveDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span><span class="n">kr</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Learning curves"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">handles</span><span class="o">=</span><span class="n">ax</span><span class="o">.</span><span class="n">get_legend_handles_labels</span><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="n">labels</span><span class="o">=</span><span class="p">[</span><span class="s2">"SVR"</span><span class="p">,</span> <span class="s2">"KRR"</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_kernel_ridge_regression_003.png" srcset="../../_images/sphx_glr_plot_kernel_ridge_regression_003.png" alt="Learning curves" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 8.155 seconds)</p>
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<p><a class="reference download internal" download="" href="../../_downloads/1dcd684ce26b8c407ec2c2d2101c5c73/plot_kernel_ridge_regression.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_kernel_ridge_regression.py</span></code></a></p>
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<p class="rubric">Related examples</p>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="Toy example of 1D regression using linear, polynomial and RBF kernels."><img alt="" src="../../_images/sphx_glr_plot_svm_regression_thumb.png" />
<p><a class="reference internal" href="../svm/plot_svm_regression.html#sphx-glr-auto-examples-svm-plot-svm-regression-py"><span class="std std-ref">Support Vector Regression (SVR) using linear and non-linear kernels</span></a></p>
<div class="sphx-glr-thumbnail-title">Support Vector Regression (SVR) using linear and non-linear kernels</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the effect of the parameters gamma and C of the Radial Basis Function ..."><img alt="" src="../../_images/sphx_glr_plot_rbf_parameters_thumb.png" />
<p><a class="reference internal" href="../svm/plot_rbf_parameters.html#sphx-glr-auto-examples-svm-plot-rbf-parameters-py"><span class="std std-ref">RBF SVM parameters</span></a></p>
<div class="sphx-glr-thumbnail-title">RBF SVM parameters</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the predicted probability of GPC for an RBF kernel with different choi..."><img alt="" src="../../_images/sphx_glr_plot_gpc_thumb.png" />
<p><a class="reference internal" href="../gaussian_process/plot_gpc.html#sphx-glr-auto-examples-gaussian-process-plot-gpc-py"><span class="std std-ref">Probabilistic predictions with Gaussian process classification (GPC)</span></a></p>
<div class="sphx-glr-thumbnail-title">Probabilistic predictions with Gaussian process classification (GPC)</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Demonstrate how model complexity influences both prediction accuracy and computational performa..."><img alt="" src="../../_images/sphx_glr_plot_model_complexity_influence_thumb.png" />
<p><a class="reference internal" href="../applications/plot_model_complexity_influence.html#sphx-glr-auto-examples-applications-plot-model-complexity-influence-py"><span class="std std-ref">Model Complexity Influence</span></a></p>
<div class="sphx-glr-thumbnail-title">Model Complexity Influence</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The dataset used in this example is a preprocessed excerpt of the "Labeled Faces in the Wild", ..."><img alt="" src="../../_images/sphx_glr_plot_face_recognition_thumb.png" />
<p><a class="reference internal" href="../applications/plot_face_recognition.html#sphx-glr-auto-examples-applications-plot-face-recognition-py"><span class="std std-ref">Faces recognition example using eigenfaces and SVMs</span></a></p>
<div class="sphx-glr-thumbnail-title">Faces recognition example using eigenfaces and SVMs</div>
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