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<li><a class="reference internal" href="#">Single estimator versus bagging: bias-variance decomposition</a><ul>
<li><a class="reference internal" href="#references">References</a></li>
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<section class="sphx-glr-example-title" id="single-estimator-versus-bagging-bias-variance-decomposition">
<span id="sphx-glr-auto-examples-ensemble-plot-bias-variance-py"></span><h1>Single estimator versus bagging: bias-variance decomposition<a class="headerlink" href="#single-estimator-versus-bagging-bias-variance-decomposition" title="Link to this heading">¶</a></h1>
<p>This example illustrates and compares the bias-variance decomposition of the
expected mean squared error of a single estimator against a bagging ensemble.</p>
<p>In regression, the expected mean squared error of an estimator can be
decomposed in terms of bias, variance and noise. On average over datasets of
the regression problem, the bias term measures the average amount by which the
predictions of the estimator differ from the predictions of the best possible
estimator for the problem (i.e., the Bayes model). The variance term measures
the variability of the predictions of the estimator when fit over different
random instances of the same problem. Each problem instance is noted “LS”, for
“Learning Sample”, in the following. Finally, the noise measures the irreducible part
of the error which is due the variability in the data.</p>
<p>The upper left figure illustrates the predictions (in dark red) of a single
decision tree trained over a random dataset LS (the blue dots) of a toy 1d
regression problem. It also illustrates the predictions (in light red) of other
single decision trees trained over other (and different) randomly drawn
instances LS of the problem. Intuitively, the variance term here corresponds to
the width of the beam of predictions (in light red) of the individual
estimators. The larger the variance, the more sensitive are the predictions for
<code class="docutils literal notranslate"><span class="pre">x</span></code> to small changes in the training set. The bias term corresponds to the
difference between the average prediction of the estimator (in cyan) and the
best possible model (in dark blue). On this problem, we can thus observe that
the bias is quite low (both the cyan and the blue curves are close to each
other) while the variance is large (the red beam is rather wide).</p>
<p>The lower left figure plots the pointwise decomposition of the expected mean
squared error of a single decision tree. It confirms that the bias term (in
blue) is low while the variance is large (in green). It also illustrates the
noise part of the error which, as expected, appears to be constant and around
<code class="docutils literal notranslate"><span class="pre">0.01</span></code>.</p>
<p>The right figures correspond to the same plots but using instead a bagging
ensemble of decision trees. In both figures, we can observe that the bias term
is larger than in the previous case. In the upper right figure, the difference
between the average prediction (in cyan) and the best possible model is larger
(e.g., notice the offset around <code class="docutils literal notranslate"><span class="pre">x=2</span></code>). In the lower right figure, the bias
curve is also slightly higher than in the lower left figure. In terms of
variance however, the beam of predictions is narrower, which suggests that the
variance is lower. Indeed, as the lower right figure confirms, the variance
term (in green) is lower than for single decision trees. Overall, the bias-
variance decomposition is therefore no longer the same. The tradeoff is better
for bagging: averaging several decision trees fit on bootstrap copies of the
dataset slightly increases the bias term but allows for a larger reduction of
the variance, which results in a lower overall mean squared error (compare the
red curves int the lower figures). The script output also confirms this
intuition. The total error of the bagging ensemble is lower than the total
error of a single decision tree, and this difference indeed mainly stems from a
reduced variance.</p>
<p>For further details on bias-variance decomposition, see section 7.3 of <a class="footnote-reference brackets" href="#id2" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>.</p>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading">¶</a></h2>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p>T. Hastie, R. Tibshirani and J. Friedman,
“Elements of Statistical Learning”, Springer, 2009.</p>
</aside>
</aside>
<img src="../../_images/sphx_glr_plot_bias_variance_001.png" srcset="../../_images/sphx_glr_plot_bias_variance_001.png" alt="Tree, Bagging(Tree)" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Tree: 0.0255 (error) = 0.0003 (bias^2) + 0.0152 (var) + 0.0098 (noise)
Bagging(Tree): 0.0196 (error) = 0.0004 (bias^2) + 0.0092 (var) + 0.0098 (noise)
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Gilles Louppe <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.ensemble.BaggingRegressor.html#sklearn.ensemble.BaggingRegressor" title="sklearn.ensemble.BaggingRegressor" class="sphx-glr-backref-module-sklearn-ensemble sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">BaggingRegressor</span></a>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="sklearn.tree.DecisionTreeRegressor" class="sphx-glr-backref-module-sklearn-tree sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DecisionTreeRegressor</span></a>
<span class="c1"># Settings</span>
<span class="n">n_repeat</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># Number of iterations for computing expectations</span>
<span class="n">n_train</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># Size of the training set</span>
<span class="n">n_test</span> <span class="o">=</span> <span class="mi">1000</span> <span class="c1"># Size of the test set</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span> <span class="c1"># Standard deviation of the noise</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Change this for exploring the bias-variance decomposition of other</span>
<span class="c1"># estimators. This should work well for estimators with high variance (e.g.,</span>
<span class="c1"># decision trees or KNN), but poorly for estimators with low variance (e.g.,</span>
<span class="c1"># linear models).</span>
<span class="n">estimators</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">(</span><span class="s2">"Tree"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="sklearn.tree.DecisionTreeRegressor" class="sphx-glr-backref-module-sklearn-tree sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DecisionTreeRegressor</span></a><span class="p">()),</span>
<span class="p">(</span><span class="s2">"Bagging(Tree)"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.ensemble.BaggingRegressor.html#sklearn.ensemble.BaggingRegressor" title="sklearn.ensemble.BaggingRegressor" class="sphx-glr-backref-module-sklearn-ensemble sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">BaggingRegressor</span></a><span class="p">(</span><a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="sklearn.tree.DecisionTreeRegressor" class="sphx-glr-backref-module-sklearn-tree sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DecisionTreeRegressor</span></a><span class="p">())),</span>
<span class="p">]</span>
<span class="n">n_estimators</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">estimators</span><span class="p">)</span>
<span class="c1"># Generate data</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="k">return</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.exp.html#numpy.exp" title="numpy.exp" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.exp.html#numpy.exp" title="numpy.exp" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.rand.html#numpy.random.rand" title="numpy.random.rand" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">-</span> <span class="mi">5</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sort.html#numpy.sort" title="numpy.sort" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">sort</span></a><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n_repeat</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">y</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</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">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</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">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_train</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span>
<span class="n">X_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">y_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_test</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="n">n_repeat</span><span class="p">)</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">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="c1"># Loop over estimators to compare</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">estimators</span><span class="p">):</span>
<span class="c1"># Compute predictions</span>
<span class="n">y_predict</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_test</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><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_train</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">estimator</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="c1"># Bias^2 + Variance + Noise decomposition of the mean squared error</span>
<span class="n">y_error</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">(</span><span class="n">n_test</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">y_error</span> <span class="o">+=</span> <span class="p">(</span><span class="n">y_test</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">y_error</span> <span class="o">/=</span> <span class="n">n_repeat</span> <span class="o">*</span> <span class="n">n_repeat</span>
<span class="n">y_noise</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.var.html#numpy.var" title="numpy.var" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">var</span></a><span class="p">(</span><span class="n">y_test</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">y_bias</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span> <span class="o">-</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_predict</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="o">**</span> <span class="mi">2</span>
<span class="n">y_var</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.var.html#numpy.var" title="numpy.var" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">var</span></a><span class="p">(</span><span class="n">y_predict</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="nb">print</span><span class="p">(</span>
<span class="s2">"</span><span class="si">{0}</span><span class="s2">: </span><span class="si">{1:.4f}</span><span class="s2"> (error) = </span><span class="si">{2:.4f}</span><span class="s2"> (bias^2) "</span>
<span class="s2">" + </span><span class="si">{3:.4f}</span><span class="s2"> (var) + </span><span class="si">{4:.4f}</span><span class="s2"> (noise)"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
<span class="n">name</span><span class="p">,</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_error</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_bias</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_var</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_noise</span><span class="p">)</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="c1"># Plot figures</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_estimators</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.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">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="s2">"b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$f(x)$"</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_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s2">".b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"LS ~ $y = f(x)+noise$"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</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_test</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">"r"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\^y(x)$"</span><span class="p">)</span>
<span class="k">else</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">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">"r"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.05</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> <a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">y_predict</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="s2">"c"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\mathbb</span><span class="si">{E}</span><span class="s2">_</span><span class="si">{LS}</span><span class="s2"> \^y(x)$"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" 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">xlim</span></a><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</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="n">name</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.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="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
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<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">y_bias</span><span class="p">,</span> <span class="s2">"b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$bias^2(x)$"</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">y_var</span><span class="p">,</span> <span class="s2">"g"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$variance(x)$"</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">y_noise</span><span class="p">,</span> <span class="s2">"c"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$noise(x)$"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" 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">xlim</span></a><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
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<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.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="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="matplotlib.pyplot.subplots_adjust" 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_adjust</span></a><span class="p">(</span><span class="n">right</span><span class="o">=</span><span class="mf">0.75</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>
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