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<li><a class="reference internal" href="#">Scaling the regularization parameter for SVCs</a><ul>
<li><a class="reference internal" href="#l1-penalty-case">L1-penalty case</a></li>
<li><a class="reference internal" href="#l2-penalty-case">L2-penalty case</a></li>
</ul>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-svm-plot-svm-scale-c-py"><span class="std std-ref">here</span></a>
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<section class="sphx-glr-example-title" id="scaling-the-regularization-parameter-for-svcs">
<span id="sphx-glr-auto-examples-svm-plot-svm-scale-c-py"></span><h1>Scaling the regularization parameter for SVCs<a class="headerlink" href="#scaling-the-regularization-parameter-for-svcs" title="Permalink to this heading">¶</a></h1>
<p>The following example illustrates the effect of scaling the
regularization parameter when using <a class="reference internal" href="../../modules/svm.html#svm"><span class="std std-ref">Support Vector Machines</span></a> for
<a class="reference internal" href="../../modules/svm.html#svm-classification"><span class="std std-ref">classification</span></a>.
For SVC classification, we are interested in a risk minimization for the
equation:</p>
<div class="math notranslate nohighlight">
\[C \sum_{i=1, n} \mathcal{L} (f(x_i), y_i) + \Omega (w)\]</div>
<p>where</p>
<blockquote>
<div><ul class="simple">
<li><p><span class="math notranslate nohighlight">\(C\)</span> is used to set the amount of regularization</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathcal{L}\)</span> is a <code class="docutils literal notranslate"><span class="pre">loss</span></code> function of our samples
and our model parameters.</p></li>
<li><p><span class="math notranslate nohighlight">\(\Omega\)</span> is a <code class="docutils literal notranslate"><span class="pre">penalty</span></code> function of our model parameters</p></li>
</ul>
</div></blockquote>
<p>If we consider the loss function to be the individual error per
sample, then the data-fit term, or the sum of the error for each sample, will
increase as we add more samples. The penalization term, however, will not
increase.</p>
<p>When using, for example, <a class="reference internal" href="../../modules/cross_validation.html#cross-validation"><span class="std std-ref">cross validation</span></a>, to
set the amount of regularization with <code class="docutils literal notranslate"><span class="pre">C</span></code>, there will be a
different amount of samples between the main problem and the smaller problems
within the folds of the cross validation.</p>
<p>Since our loss function is dependent on the amount of samples, the latter
will influence the selected value of <code class="docutils literal notranslate"><span class="pre">C</span></code>.
The question that arises is “How do we optimally adjust C to
account for the different amount of training samples?”</p>
<p>In the remainder of this example, we will investigate the effect of scaling
the value of the regularization parameter <code class="docutils literal notranslate"><span class="pre">C</span></code> in regards to the number of
samples for both L1 and L2 penalty. We will generate some synthetic datasets
that are appropriate for each type of regularization.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Andreas Mueller <[email protected]></span>
<span class="c1"># Jaques Grobler <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="l1-penalty-case">
<h2>L1-penalty case<a class="headerlink" href="#l1-penalty-case" title="Permalink to this heading">¶</a></h2>
<p>In the L1 case, theory says that prediction consistency (i.e. that under
given hypothesis, the estimator learned predicts as well as a model knowing
the true distribution) is not possible because of the bias of the L1. It
does say, however, that model consistency, in terms of finding the right set
of non-zero parameters as well as their signs, can be achieved by scaling
<code class="docutils literal notranslate"><span class="pre">C</span></code>.</p>
<p>We will demonstrate this effect by using a synthetic dataset. This
dataset will be sparse, meaning that only a few features will be informative
and useful for the model.</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.make_classification.html#sklearn.datasets.make_classification" title="sklearn.datasets.make_classification" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_classification</span></a>
<span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">300</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_classification.html#sklearn.datasets.make_classification" title="sklearn.datasets.make_classification" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_classification</span></a><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="o">=</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_informative</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span>
<span class="p">)</span>
</pre></div>
</div>
<p>Now, we can define a linear SVC with the <code class="docutils literal notranslate"><span class="pre">l1</span></code> penalty.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearSVC</span></a>
<span class="n">model_l1</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearSVC</span></a><span class="p">(</span><span class="n">penalty</span><span class="o">=</span><span class="s2">"l1"</span><span class="p">,</span> <span class="n">loss</span><span class="o">=</span><span class="s2">"squared_hinge"</span><span class="p">,</span> <span class="n">dual</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
</pre></div>
</div>
<p>We will compute the mean test score for different values of <code class="docutils literal notranslate"><span class="pre">C</span></code>.</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="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.validation_curve.html#sklearn.model_selection.validation_curve" title="sklearn.model_selection.validation_curve" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">validation_curve</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.model_selection.ShuffleSplit.html#sklearn.model_selection.ShuffleSplit" title="sklearn.model_selection.ShuffleSplit" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShuffleSplit</span></a>
<span class="n">Cs</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="mf">2.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">train_sizes</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="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">"fraction: </span><span class="si">{</span><span class="n">train_size</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">train_size</span> <span class="ow">in</span> <span class="n">train_sizes</span><span class="p">]</span>
<span class="n">results</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"C"</span><span class="p">:</span> <span class="n">Cs</span><span class="p">}</span>
<span class="k">for</span> <span class="n">label</span><span class="p">,</span> <span class="n">train_size</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">train_sizes</span><span class="p">):</span>
<span class="n">cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.ShuffleSplit.html#sklearn.model_selection.ShuffleSplit" title="sklearn.model_selection.ShuffleSplit" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShuffleSplit</span></a><span class="p">(</span><span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">n_splits</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">train_scores</span><span class="p">,</span> <span class="n">test_scores</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.validation_curve.html#sklearn.model_selection.validation_curve" title="sklearn.model_selection.validation_curve" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">validation_curve</span></a><span class="p">(</span>
<span class="n">model_l1</span><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">param_name</span><span class="o">=</span><span class="s2">"C"</span><span class="p">,</span> <span class="n">param_range</span><span class="o">=</span><span class="n">Cs</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="n">cv</span>
<span class="p">)</span>
<span class="n">results</span><span class="p">[</span><span class="n">label</span><span class="p">]</span> <span class="o">=</span> <span class="n">test_scores</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">results</span> <span class="o">=</span> <a href="https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.html#pandas.DataFrame" title="pandas.DataFrame" class="sphx-glr-backref-module-pandas sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span></a><span class="p">(</span><span class="n">results</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>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</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">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="c1"># plot results without scaling C</span>
<span class="n">results</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">"C"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"CV score"</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"No scaling"</span><span class="p">)</span>
<span class="c1"># plot results by scaling C</span>
<span class="k">for</span> <span class="n">train_size_idx</span><span class="p">,</span> <span class="n">label</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">labels</span><span class="p">):</span>
<span class="n">results_scaled</span> <span class="o">=</span> <span class="n">results</span><span class="p">[[</span><span class="n">label</span><span class="p">]]</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span>
<span class="n">C_scaled</span><span class="o">=</span><span class="n">Cs</span> <span class="o">*</span> <span class="nb">float</span><span class="p">(</span><span class="n">n_samples</span> <span class="o">*</span> <span class="n">train_sizes</span><span class="p">[</span><span class="n">train_size_idx</span><span class="p">])</span>
<span class="p">)</span>
<span class="n">results_scaled</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">"C_scaled"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Scaling C by 1 / n_samples"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Effect of scaling C with L1 penalty"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_scale_c_001.png" srcset="../../_images/sphx_glr_plot_svm_scale_c_001.png" alt="Effect of scaling C with L1 penalty, No scaling, Scaling C by 1 / n_samples" class = "sphx-glr-single-img"/><p>Here, we observe that the cross-validation-error correlates best with the
test-error, when scaling our <code class="docutils literal notranslate"><span class="pre">C</span></code> with the number of samples, <code class="docutils literal notranslate"><span class="pre">n</span></code>.</p>
</section>
<section id="l2-penalty-case">
<h2>L2-penalty case<a class="headerlink" href="#l2-penalty-case" title="Permalink to this heading">¶</a></h2>
<p>We can repeat a similar experiment with the <code class="docutils literal notranslate"><span class="pre">l2</span></code> penalty. In this case, we
don’t need to use a sparse dataset.</p>
<p>In this case, the theory says that in order to achieve prediction
consistency, the penalty parameter should be kept constant as the number of
samples grow.</p>
<p>So we will repeat the same experiment by creating a linear SVC classifier
with the <code class="docutils literal notranslate"><span class="pre">l2</span></code> penalty and check the test score via cross-validation and
plot the results with and without scaling the parameter <code class="docutils literal notranslate"><span class="pre">C</span></code>.</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">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.sign.html#numpy.sign" title="numpy.sign" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sign</span></a><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">n_samples</span><span class="p">))</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">//</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><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">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">//</span> <span class="mi">5</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">model_l2</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearSVC</span></a><span class="p">(</span><span class="n">penalty</span><span class="o">=</span><span class="s2">"l2"</span><span class="p">,</span> <span class="n">loss</span><span class="o">=</span><span class="s2">"squared_hinge"</span><span class="p">,</span> <span class="n">dual</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">Cs</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="mf">4.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">"fraction: </span><span class="si">{</span><span class="n">train_size</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">train_size</span> <span class="ow">in</span> <span class="n">train_sizes</span><span class="p">]</span>
<span class="n">results</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"C"</span><span class="p">:</span> <span class="n">Cs</span><span class="p">}</span>
<span class="k">for</span> <span class="n">label</span><span class="p">,</span> <span class="n">train_size</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">train_sizes</span><span class="p">):</span>
<span class="n">cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.ShuffleSplit.html#sklearn.model_selection.ShuffleSplit" title="sklearn.model_selection.ShuffleSplit" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShuffleSplit</span></a><span class="p">(</span><span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">n_splits</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">train_scores</span><span class="p">,</span> <span class="n">test_scores</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.validation_curve.html#sklearn.model_selection.validation_curve" title="sklearn.model_selection.validation_curve" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">validation_curve</span></a><span class="p">(</span>
<span class="n">model_l2</span><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">param_name</span><span class="o">=</span><span class="s2">"C"</span><span class="p">,</span> <span class="n">param_range</span><span class="o">=</span><span class="n">Cs</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="n">cv</span>
<span class="p">)</span>
<span class="n">results</span><span class="p">[</span><span class="n">label</span><span class="p">]</span> <span class="o">=</span> <span class="n">test_scores</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">results</span> <span class="o">=</span> <a href="https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.html#pandas.DataFrame" title="pandas.DataFrame" class="sphx-glr-backref-module-pandas sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span></a><span class="p">(</span><span class="n">results</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>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</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">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="c1"># plot results without scaling C</span>
<span class="n">results</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">"C"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"CV score"</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"No scaling"</span><span class="p">)</span>
<span class="c1"># plot results by scaling C</span>
<span class="k">for</span> <span class="n">train_size_idx</span><span class="p">,</span> <span class="n">label</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">labels</span><span class="p">):</span>
<span class="n">results_scaled</span> <span class="o">=</span> <span class="n">results</span><span class="p">[[</span><span class="n">label</span><span class="p">]]</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span>
<span class="n">C_scaled</span><span class="o">=</span><span class="n">Cs</span> <span class="o">*</span> <span class="nb">float</span><span class="p">(</span><span class="n">n_samples</span> <span class="o">*</span> <span class="n">train_sizes</span><span class="p">[</span><span class="n">train_size_idx</span><span class="p">])</span>
<span class="p">)</span>
<span class="n">results_scaled</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">"C_scaled"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Scaling C by 1 / n_samples"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Effect of scaling C with L2 penalty"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_scale_c_002.png" srcset="../../_images/sphx_glr_plot_svm_scale_c_002.png" alt="Effect of scaling C with L2 penalty, No scaling, Scaling C by 1 / n_samples" class = "sphx-glr-single-img"/><p>So or the L2 penalty case, the best result comes from the case where <code class="docutils literal notranslate"><span class="pre">C</span></code> is
not scaled.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></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>
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