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<ul>
<li><a class="reference internal" href="#">Effect of transforming the targets in regression model</a><ul>
<li><a class="reference internal" href="#synthetic-example">Synthetic example</a></li>
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<p class="first admonition-title">Note</p>
<p class="last">Click <a class="reference internal" href="#sphx-glr-download-auto-examples-preprocessing-plot-transformed-target-py"><span class="std std-ref">here</span></a> to download the full example code or run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="effect-of-transforming-the-targets-in-regression-model">
<span id="sphx-glr-auto-examples-preprocessing-plot-transformed-target-py"></span><h1>Effect of transforming the targets in regression model<a class="headerlink" href="#effect-of-transforming-the-targets-in-regression-model" title="Permalink to this headline">¶</a></h1>
<p>In this example, we give an overview of the
<a class="reference internal" href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="sklearn.compose.TransformedTargetRegressor"><code class="xref py py-class docutils literal"><span class="pre">sklearn.compose.TransformedTargetRegressor</span></code></a>. Two examples
illustrate the benefit of transforming the targets before learning a linear
regression model. The first example uses synthetic data while the second
example is based on the Boston housing data set.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="c1"># Author: Guillaume Lemaitre <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span><span class="p">,</span> <span class="n">division</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
</pre></div>
</div>
<div class="section" id="synthetic-example">
<h2>Synthetic example<a class="headerlink" href="#synthetic-example" title="Permalink to this headline">¶</a></h2>
<div class="highlight-python"><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_regression.html#sklearn.datasets.make_regression" title="View documentation for sklearn.datasets.make_regression"><span class="n">make_regression</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.train_test_split.html#sklearn.model_selection.train_test_split" title="View documentation for sklearn.model_selection.train_test_split"><span class="n">train_test_split</span></a>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="View documentation for sklearn.linear_model.RidgeCV"><span class="n">RidgeCV</span></a>
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="View documentation for sklearn.compose.TransformedTargetRegressor"><span class="n">TransformedTargetRegressor</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="View documentation for sklearn.metrics.median_absolute_error"><span class="n">median_absolute_error</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="View documentation for sklearn.metrics.r2_score"><span class="n">r2_score</span></a>
</pre></div>
</div>
<p>A synthetic random regression problem is generated. The targets <code class="docutils literal"><span class="pre">y</span></code> are
modified by: (i) translating all targets such that all entries are
non-negative and (ii) applying an exponential function to obtain non-linear
targets which cannot be fitted using a simple linear model.</p>
<p>Therefore, a logarithmic (<cite>np.log1p</cite>) and an exponential function
(<cite>np.expm1</cite>) will be used to transform the targets before training a linear
regression model and using it for prediction.</p>
<div class="highlight-python"><div class="highlight"><pre><span></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_regression.html#sklearn.datasets.make_regression" title="View documentation for sklearn.datasets.make_regression"><span class="n">make_regression</span></a><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.exp.html#numpy.exp" title="View documentation for numpy.exp"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">((</span><span class="n">y</span> <span class="o">+</span> <span class="nb">abs</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">min</span><span class="p">()))</span> <span class="o">/</span> <span class="mi">200</span><span class="p">)</span>
<span class="n">y_trans</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.log1p.html#numpy.log1p" title="View documentation for numpy.log1p"><span class="n">np</span><span class="o">.</span><span class="n">log1p</span></a><span class="p">(</span><span class="n">y</span><span class="p">)</span>
</pre></div>
</div>
<p>The following illustrate the probability density functions of the target
before and after applying the logarithmic functions.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="View documentation for matplotlib.pyplot.subplots"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Probability'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Target'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Target distribution'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">y_trans</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Probability'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Target'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Transformed target distribution'</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Synthetic data"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">0.035</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">(</span><span class="n">rect</span><span class="o">=</span><span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">])</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="View documentation for sklearn.model_selection.train_test_split"><span class="n">train_test_split</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
<img alt="../../_images/sphx_glr_plot_transformed_target_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_transformed_target_001.png" />
<p>At first, a linear model will be applied on the original targets. Due to the
non-linearity, the model trained will not be precise during the
prediction. Subsequently, a logarithmic function is used to linearize the
targets, allowing better prediction even with a similar linear model as
reported by the median absolute error (MAE).</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="View documentation for matplotlib.pyplot.subplots"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">regr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="View documentation for sklearn.linear_model.RidgeCV"><span class="n">RidgeCV</span></a><span class="p">()</span>
<span class="n">regr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">regr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">],</span> <span class="s1">'--k'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Target predicted'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'True Target'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Ridge regression </span><span class="se">\n</span><span class="s1"> without target transformation'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1750</span><span class="p">,</span> <span class="sa">r</span><span class="s1">'$R^2$=</span><span class="si">%.2f</span><span class="s1">, MAE=</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span>
<a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="View documentation for sklearn.metrics.r2_score"><span class="n">r2_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">),</span> <a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="View documentation for sklearn.metrics.median_absolute_error"><span class="n">median_absolute_error</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)))</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">])</span>
<span class="n">regr_trans</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="View documentation for sklearn.compose.TransformedTargetRegressor"><span class="n">TransformedTargetRegressor</span></a><span class="p">(</span><span class="n">regressor</span><span class="o">=</span><a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="View documentation for sklearn.linear_model.RidgeCV"><span class="n">RidgeCV</span></a><span class="p">(),</span>
<span class="n">func</span><span class="o">=</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.log1p.html#numpy.log1p" title="View documentation for numpy.log1p"><span class="n">np</span><span class="o">.</span><span class="n">log1p</span></a><span class="p">,</span>
<span class="n">inverse_func</span><span class="o">=</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.expm1.html#numpy.expm1" title="View documentation for numpy.expm1"><span class="n">np</span><span class="o">.</span><span class="n">expm1</span></a><span class="p">)</span>
<span class="n">regr_trans</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">regr_trans</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">],</span> <span class="s1">'--k'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Target predicted'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'True Target'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Ridge regression </span><span class="se">\n</span><span class="s1"> with target transformation'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1750</span><span class="p">,</span> <span class="sa">r</span><span class="s1">'$R^2$=</span><span class="si">%.2f</span><span class="s1">, MAE=</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span>
<a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="View documentation for sklearn.metrics.r2_score"><span class="n">r2_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">),</span> <a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="View documentation for sklearn.metrics.median_absolute_error"><span class="n">median_absolute_error</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)))</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">])</span>
<span class="n">f</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Synthetic data"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">0.035</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">(</span><span class="n">rect</span><span class="o">=</span><span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">])</span>
</pre></div>
</div>
<img alt="../../_images/sphx_glr_plot_transformed_target_002.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_transformed_target_002.png" />
</div>
<div class="section" id="real-world-data-set">
<h2>Real-world data set<a class="headerlink" href="#real-world-data-set" title="Permalink to this headline">¶</a></h2>
<p>In a similar manner, the boston housing data set is used to show the impact
of transforming the targets before learning a model. In this example, the
targets to be predicted corresponds to the weighted distances to the five
Boston employment centers.</p>
<div class="highlight-python"><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.load_boston.html#sklearn.datasets.load_boston" title="View documentation for sklearn.datasets.load_boston"><span class="n">load_boston</span></a>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.preprocessing.QuantileTransformer.html#sklearn.preprocessing.QuantileTransformer" title="View documentation for sklearn.preprocessing.QuantileTransformer"><span class="n">QuantileTransformer</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.preprocessing.quantile_transform.html#sklearn.preprocessing.quantile_transform" title="View documentation for sklearn.preprocessing.quantile_transform"><span class="n">quantile_transform</span></a>
<span class="n">dataset</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.load_boston.html#sklearn.datasets.load_boston" title="View documentation for sklearn.datasets.load_boston"><span class="n">load_boston</span></a><span class="p">()</span>
<span class="n">target</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span><span class="n">dataset</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"DIS"</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">dataset</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.logical_not.html#numpy.logical_not" title="View documentation for numpy.logical_not"><span class="n">np</span><span class="o">.</span><span class="n">logical_not</span></a><span class="p">(</span><span class="n">target</span><span class="p">)]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">dataset</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span> <span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
<span class="n">y_trans</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.preprocessing.quantile_transform.html#sklearn.preprocessing.quantile_transform" title="View documentation for sklearn.preprocessing.quantile_transform"><span class="n">quantile_transform</span></a><span class="p">(</span><span class="n">dataset</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span> <span class="n">target</span><span class="p">],</span>
<span class="n">output_distribution</span><span class="o">=</span><span class="s1">'normal'</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
</pre></div>
</div>
<p>A <a class="reference internal" href="../../modules/generated/sklearn.preprocessing.QuantileTransformer.html#sklearn.preprocessing.QuantileTransformer" title="sklearn.preprocessing.QuantileTransformer"><code class="xref py py-class docutils literal"><span class="pre">sklearn.preprocessing.QuantileTransformer</span></code></a> is used such that the
targets follows a normal distribution before applying a
<a class="reference internal" href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="sklearn.linear_model.RidgeCV"><code class="xref py py-class docutils literal"><span class="pre">sklearn.linear_model.RidgeCV</span></code></a> model.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="View documentation for matplotlib.pyplot.subplots"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Probability'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Target'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Target distribution'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">y_trans</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Probability'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Target'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Transformed target distribution'</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Boston housing data: distance to employment centers"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">0.035</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">(</span><span class="n">rect</span><span class="o">=</span><span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">])</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="View documentation for sklearn.model_selection.train_test_split"><span class="n">train_test_split</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<img alt="../../_images/sphx_glr_plot_transformed_target_003.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_transformed_target_003.png" />
<p>The effect of the transformer is weaker than on the synthetic data. However,
the transform induces a decrease of the MAE.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="View documentation for matplotlib.pyplot.subplots"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">regr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="View documentation for sklearn.linear_model.RidgeCV"><span class="n">RidgeCV</span></a><span class="p">()</span>
<span class="n">regr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">regr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="s1">'--k'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Target predicted'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'True Target'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Ridge regression </span><span class="se">\n</span><span class="s1"> without target transformation'</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="sa">r</span><span class="s1">'$R^2$=</span><span class="si">%.2f</span><span class="s1">, MAE=</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span>
<a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="View documentation for sklearn.metrics.r2_score"><span class="n">r2_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">),</span> <a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="View documentation for sklearn.metrics.median_absolute_error"><span class="n">median_absolute_error</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)))</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">regr_trans</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="View documentation for sklearn.compose.TransformedTargetRegressor"><span class="n">TransformedTargetRegressor</span></a><span class="p">(</span>
<span class="n">regressor</span><span class="o">=</span><a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="View documentation for sklearn.linear_model.RidgeCV"><span class="n">RidgeCV</span></a><span class="p">(),</span>
<span class="n">transformer</span><span class="o">=</span><a href="../../modules/generated/sklearn.preprocessing.QuantileTransformer.html#sklearn.preprocessing.QuantileTransformer" title="View documentation for sklearn.preprocessing.QuantileTransformer"><span class="n">QuantileTransformer</span></a><span class="p">(</span><span class="n">output_distribution</span><span class="o">=</span><span class="s1">'normal'</span><span class="p">))</span>
<span class="n">regr_trans</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">regr_trans</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="s1">'--k'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Target predicted'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'True Target'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Ridge regression </span><span class="se">\n</span><span class="s1"> with target transformation'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="sa">r</span><span class="s1">'$R^2$=</span><span class="si">%.2f</span><span class="s1">, MAE=</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span>
<a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="View documentation for sklearn.metrics.r2_score"><span class="n">r2_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">),</span> <a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="View documentation for sklearn.metrics.median_absolute_error"><span class="n">median_absolute_error</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)))</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">f</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"Boston housing data: distance to employment centers"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">0.035</span><span class="p">)</span>
<span class="n">f</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">(</span><span class="n">rect</span><span class="o">=</span><span class="p">[</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">])</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
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