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<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>
<li><a class="reference internal" href="#real-world-data-set">Real-world data set</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-compose-plot-transformed-target-py"><span class="std std-ref">here</span></a>
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<section class="sphx-glr-example-title" id="effect-of-transforming-the-targets-in-regression-model">
<span id="sphx-glr-auto-examples-compose-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 heading">¶</a></h1>
<p>In this example, we give an overview of
<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 notranslate"><span class="pre">TransformedTargetRegressor</span></code></a>. We use two examples
to 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 Ames housing data set.</p>
<div class="highlight-default notranslate"><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="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
</pre></div>
</div>
<section id="synthetic-example">
<h2>Synthetic example<a class="headerlink" href="#synthetic-example" title="Permalink to this heading">¶</a></h2>
<blockquote>
<div><p>A synthetic random regression dataset is generated. The targets <code class="docutils literal notranslate"><span class="pre">y</span></code> are
modified by:</p>
<blockquote>
<div><ol class="arabic simple">
<li><p>translating all targets such that all entries are
non-negative (by adding the absolute value of the lowest <code class="docutils literal notranslate"><span class="pre">y</span></code>) and</p></li>
<li><p>applying an exponential function to obtain non-linear
targets which cannot be fitted using a simple linear model.</p></li>
</ol>
</div></blockquote>
<p>Therefore, a logarithmic (<code class="docutils literal notranslate"><span class="pre">np.log1p</span></code>) and an exponential function
(<code class="docutils literal notranslate"><span class="pre">np.expm1</span></code>) will be used to transform the targets before training a linear
regression model and using it for prediction.</p>
</div></blockquote>
<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">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="sklearn.datasets.make_regression" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_regression</span></a>
<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="sklearn.datasets.make_regression" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_regression</span></a><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">10_000</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://numpy.org/doc/stable/reference/generated/numpy.expm1.html#numpy.expm1" title="numpy.expm1" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">expm1</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://numpy.org/doc/stable/reference/generated/numpy.log1p.html#numpy.log1p" title="numpy.log1p" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><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>Below we plot the probability density functions of the target
before and after applying the logarithmic functions.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="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="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">train_test_split</span></a>
<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/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="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">density</span><span class="o">=</span><span class="kc">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="s2">"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="s2">"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="s2">"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">density</span><span class="o">=</span><span class="kc">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="s2">"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="s2">"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="s2">"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">1.05</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><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="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><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 src="../../_images/sphx_glr_plot_transformed_target_001.png" srcset="../../_images/sphx_glr_plot_transformed_target_001.png" alt="Synthetic data, Target distribution, Transformed target distribution" class = "sphx-glr-single-img"/><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
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 (MedAE).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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="sklearn.metrics.median_absolute_error" class="sphx-glr-backref-module-sklearn-metrics sphx-glr-backref-type-py-function"><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="sklearn.metrics.r2_score" class="sphx-glr-backref-module-sklearn-metrics sphx-glr-backref-type-py-function"><span class="n">r2_score</span></a>
<span class="k">def</span> <span class="nf">compute_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">):</span>
<span class="k">return</span> <span class="p">{</span>
<span class="s2">"R2"</span><span class="p">:</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><a href="../../modules/generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="sklearn.metrics.r2_score" class="sphx-glr-backref-module-sklearn-metrics sphx-glr-backref-type-py-function"><span class="n">r2_score</span></a><span class="p">(</span><span class="n">y_true</span><span class="p">,</span><span class="w"> </span><span class="n">y_pred</span><span class="p">)</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="s2">"MedAE"</span><span class="p">:</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><a href="../../modules/generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="sklearn.metrics.median_absolute_error" class="sphx-glr-backref-module-sklearn-metrics sphx-glr-backref-type-py-function"><span class="n">median_absolute_error</span></a><span class="p">(</span><span class="n">y_true</span><span class="p">,</span><span class="w"> </span><span class="n">y_pred</span><span class="p">)</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="p">}</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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="sklearn.compose.TransformedTargetRegressor" class="sphx-glr-backref-module-sklearn-compose sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">TransformedTargetRegressor</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="sklearn.linear_model.RidgeCV" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RidgeCV</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">PredictionErrorDisplay</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/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="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="kc">True</span><span class="p">)</span>
<span class="n">ridge_cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="sklearn.linear_model.RidgeCV" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RidgeCV</span></a><span class="p">()</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_ridge</span> <span class="o">=</span> <span class="n">ridge_cv</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">ridge_cv_with_trans_target</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="sklearn.compose.TransformedTargetRegressor" class="sphx-glr-backref-module-sklearn-compose sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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="sklearn.linear_model.RidgeCV" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RidgeCV</span></a><span class="p">(),</span> <span class="n">func</span><span class="o">=</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.log1p.html#numpy.log1p" title="numpy.log1p" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><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://numpy.org/doc/stable/reference/generated/numpy.expm1.html#numpy.expm1" title="numpy.expm1" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">expm1</span></a>
<span class="p">)</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_ridge_with_trans_target</span> <span class="o">=</span> <span class="n">ridge_cv_with_trans_target</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"actual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">,</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge_with_trans_target</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"actual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<span class="c1"># Add the score in the legend of each axis</span>
<span class="k">for</span> <span class="n">ax</span><span class="p">,</span> <span class="n">y_pred</span> <span class="ow">in</span> <span class="nb">zip</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="p">[</span><span class="n">y_pred_ridge</span><span class="p">,</span> <span class="n">y_pred_ridge_with_trans_target</span><span class="p">]):</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">score</span> <span class="ow">in</span> <span class="n">compute_score</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="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([],</span> <span class="p">[],</span> <span class="s2">" "</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">name</span><span class="si">}</span><span class="s2">=</span><span class="si">{</span><span class="n">score</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper left"</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="s2">"Ridge regression </span><span class="se">\n</span><span class="s2"> without target transformation"</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="s2">"Ridge regression </span><span class="se">\n</span><span class="s2"> with target transformation"</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">1.05</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_transformed_target_002.png" srcset="../../_images/sphx_glr_plot_transformed_target_002.png" alt="Synthetic data, Ridge regression without target transformation, Ridge regression with target transformation" class = "sphx-glr-single-img"/></section>
<section id="real-world-data-set">
<h2>Real-world data set<a class="headerlink" href="#real-world-data-set" title="Permalink to this heading">¶</a></h2>
<blockquote>
<div><p>In a similar manner, the Ames housing data set is used to show the impact
of transforming the targets before learning a model. In this example, the
target to be predicted is the selling price of each house.</p>
</div></blockquote>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.preprocessing.quantile_transform.html#sklearn.preprocessing.quantile_transform" title="sklearn.preprocessing.quantile_transform" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-function"><span class="n">quantile_transform</span></a>
<span class="n">ames</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s2">"house_prices"</span><span class="p">,</span> <span class="n">as_frame</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">parser</span><span class="o">=</span><span class="s2">"pandas"</span><span class="p">)</span>
<span class="c1"># Keep only numeric columns</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">ames</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">select_dtypes</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.number" title="numpy.number" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">number</span></a><span class="p">)</span>
<span class="c1"># Remove columns with NaN or Inf values</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">"LotFrontage"</span><span class="p">,</span> <span class="s2">"GarageYrBlt"</span><span class="p">,</span> <span class="s2">"MasVnrArea"</span><span class="p">])</span>
<span class="c1"># Let the price be in k$</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">ames</span><span class="o">.</span><span class="n">target</span> <span class="o">/</span> <span class="mi">1000</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="sklearn.preprocessing.quantile_transform" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-function"><span class="n">quantile_transform</span></a><span class="p">(</span>
<span class="n">y</span><span class="o">.</span><span class="n">to_frame</span><span class="p">(),</span> <span class="n">n_quantiles</span><span class="o">=</span><span class="mi">900</span><span class="p">,</span> <span class="n">output_distribution</span><span class="o">=</span><span class="s2">"normal"</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="kc">True</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 notranslate"><span class="pre">QuantileTransformer</span></code></a> is used to normalize
the target 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 notranslate"><span class="pre">RidgeCV</span></code></a> model.</p>
<div class="highlight-default notranslate"><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/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="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">density</span><span class="o">=</span><span class="kc">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="s2">"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="s2">"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="s2">"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">density</span><span class="o">=</span><span class="kc">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="s2">"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="s2">"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="s2">"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">"Ames housing data: selling price"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">1.05</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_transformed_target_003.png" srcset="../../_images/sphx_glr_plot_transformed_target_003.png" alt="Ames housing data: selling price, Target distribution, Transformed target distribution" class = "sphx-glr-single-img"/><div class="highlight-default notranslate"><div class="highlight"><pre><span></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="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><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>
<p>The effect of the transformer is weaker than on the synthetic data. However,
the transformation results in an increase in <span class="math notranslate nohighlight">\(R^2\)</span> and large decrease
of the MedAE. The residual plot (predicted target - true target vs predicted
target) without target transformation takes on a curved, ‘reverse smile’
shape due to residual values that vary depending on the value of predicted
target. With target transformation, the shape is more linear indicating
better model fit.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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="sklearn.preprocessing.QuantileTransformer" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">QuantileTransformer</span></a>
<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/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="mi">2</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="s2">"row"</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mf">6.5</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">ridge_cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="sklearn.linear_model.RidgeCV" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RidgeCV</span></a><span class="p">()</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_ridge</span> <span class="o">=</span> <span class="n">ridge_cv</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">ridge_cv_with_trans_target</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.compose.TransformedTargetRegressor.html#sklearn.compose.TransformedTargetRegressor" title="sklearn.compose.TransformedTargetRegressor" class="sphx-glr-backref-module-sklearn-compose sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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="sklearn.linear_model.RidgeCV" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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="sklearn.preprocessing.QuantileTransformer" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">QuantileTransformer</span></a><span class="p">(</span><span class="n">n_quantiles</span><span class="o">=</span><span class="mi">900</span><span class="p">,</span> <span class="n">output_distribution</span><span class="o">=</span><span class="s2">"normal"</span><span class="p">),</span>
<span class="p">)</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_ridge_with_trans_target</span> <span class="o">=</span> <span class="n">ridge_cv_with_trans_target</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"># plot the actual vs predicted values</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"actual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge_with_trans_target</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"actual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax0</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<span class="c1"># Add the score in the legend of each axis</span>
<span class="k">for</span> <span class="n">ax</span><span class="p">,</span> <span class="n">y_pred</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">([</span><span class="n">ax0</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ax0</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="n">y_pred_ridge</span><span class="p">,</span> <span class="n">y_pred_ridge_with_trans_target</span><span class="p">]):</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">score</span> <span class="ow">in</span> <span class="n">compute_score</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="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([],</span> <span class="p">[],</span> <span class="s2">" "</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">name</span><span class="si">}</span><span class="s2">=</span><span class="si">{</span><span class="n">score</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper left"</span><span class="p">)</span>
<span class="n">ax0</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">"Ridge regression </span><span class="se">\n</span><span class="s2"> without target transformation"</span><span class="p">)</span>
<span class="n">ax0</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">"Ridge regression </span><span class="se">\n</span><span class="s2"> with target transformation"</span><span class="p">)</span>
<span class="c1"># plot the residuals vs the predicted values</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"residual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.metrics.PredictionErrorDisplay.html#sklearn.metrics.PredictionErrorDisplay.from_predictions" title="sklearn.metrics.PredictionErrorDisplay.from_predictions" class="sphx-glr-backref-module-sklearn-metrics-PredictionErrorDisplay sphx-glr-backref-type-py-method"><span class="n">PredictionErrorDisplay</span><span class="o">.</span><span class="n">from_predictions</span></a><span class="p">(</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">y_pred_ridge_with_trans_target</span><span class="p">,</span>
<span class="n">kind</span><span class="o">=</span><span class="s2">"residual_vs_predicted"</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
<span class="n">scatter_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">"alpha"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">)</span>
<span class="n">ax1</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">"Ridge regression </span><span class="se">\n</span><span class="s2"> without target transformation"</span><span class="p">)</span>
<span class="n">ax1</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">"Ridge regression </span><span class="se">\n</span><span class="s2"> with target transformation"</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">"Ames housing data: selling price"</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">1.05</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" 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">tight_layout</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_transformed_target_004.png" srcset="../../_images/sphx_glr_plot_transformed_target_004.png" alt="Ames housing data: selling price, Ridge regression without target transformation, Ridge regression with target transformation, Ridge regression without target transformation, Ridge regression with target transformation" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 1.367 seconds)</p>
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