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<div class="section" id="prediction-intervals-for-gradient-boosting-regression">
<span id="sphx-glr-auto-examples-ensemble-plot-gradient-boosting-quantile-py"></span><h1>Prediction Intervals for Gradient Boosting Regression<a class="headerlink" href="#prediction-intervals-for-gradient-boosting-regression" title="Permalink to this headline">¶</a></h1>
<p>This example shows how quantile regression can be used
to create prediction intervals.</p>
<img alt="../../_images/sphx_glr_plot_gradient_boosting_quantile_001.png" class="align-center" src="../../_images/sphx_glr_plot_gradient_boosting_quantile_001.png" />
<div class="highlight-python"><div class="highlight"><pre><span></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="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="View documentation for sklearn.ensemble.GradientBoostingRegressor"><span class="n">GradientBoostingRegressor</span></a>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.seed.html#numpy.random.seed" title="View documentation for numpy.random.seed"><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">1</span><span class="p">)</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="sd">"""The function to predict."""</span>
<span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.sin.html#numpy.sin" title="View documentation for numpy.sin"><span class="n">np</span><span class="o">.</span><span class="n">sin</span></a><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1">#----------------------------------------------------------------------</span>
<span class="c1"># First the noiseless case</span>
<span class="n">X</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.atleast_2d.html#numpy.atleast_2d" title="View documentation for numpy.atleast_2d"><span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span></a><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.uniform.html#numpy.random.uniform" title="View documentation for numpy.random.uniform"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">T</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="c1"># Observations</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><span class="n">ravel</span><span class="p">()</span>
<span class="n">dy</span> <span class="o">=</span> <span class="mf">1.5</span> <span class="o">+</span> <span class="mf">1.0</span> <span class="o">*</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.wald.html#numpy.random" title="View documentation for numpy.random.random"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span></a><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.normal.html#numpy.random.normal" title="View documentation for numpy.random.normal"><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="mi">0</span><span class="p">,</span> <span class="n">dy</span><span class="p">)</span>
<span class="n">y</span> <span class="o">+=</span> <span class="n">noise</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="c1"># Mesh the input space for evaluations of the real function, the prediction and</span>
<span class="c1"># its MSE</span>
<span class="n">xx</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.atleast_2d.html#numpy.atleast_2d" title="View documentation for numpy.atleast_2d"><span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span></a><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1000</span><span class="p">))</span><span class="o">.</span><span class="n">T</span>
<span class="n">xx</span> <span class="o">=</span> <span class="n">xx</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.95</span>
<span class="n">clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.ensemble.GradientBoostingRegressor.html#sklearn.ensemble.GradientBoostingRegressor" title="View documentation for sklearn.ensemble.GradientBoostingRegressor"><span class="n">GradientBoostingRegressor</span></a><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">'quantile'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">,</span>
<span class="n">n_estimators</span><span class="o">=</span><span class="mi">250</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
<span class="n">learning_rate</span><span class="o">=.</span><span class="mi">1</span><span class="p">,</span> <span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">9</span><span class="p">,</span>
<span class="n">min_samples_split</span><span class="o">=</span><span class="mi">9</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Make the prediction on the meshed x-axis</span>
<span class="n">y_upper</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">xx</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">1.0</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Make the prediction on the meshed x-axis</span>
<span class="n">y_lower</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">xx</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">'ls'</span><span class="p">)</span>
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Make the prediction on the meshed x-axis</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">xx</span><span class="p">)</span>
<span class="c1"># Plot the function, the prediction and the 90% confidence interval based on</span>
<span class="c1"># the MSE</span>
<span class="n">fig</span> <span class="o">=</span> <a href="http://matplotlib.org/api/_as_gen/matplotlib.figure.AxesStack.html#matplotlib.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="s1">'g:'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">u</span><span class="s1">'$f(x) = x\,\sin(x)$'</span><span class="p">)</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</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="s1">'b.'</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">u</span><span class="s1">'Observations'</span><span class="p">)</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="s1">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">u</span><span class="s1">'Prediction'</span><span class="p">)</span>
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