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<li><a class="reference internal" href="#">Theil-Sen Regression</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-linear-model-plot-theilsen-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="theil-sen-regression">
<span id="sphx-glr-auto-examples-linear-model-plot-theilsen-py"></span><h1>Theil-Sen Regression<a class="headerlink" href="#theil-sen-regression" title="Permalink to this headline">¶</a></h1>
<p>Computes a Theil-Sen Regression on a synthetic dataset.</p>
<p>See <a class="reference internal" href="../../modules/linear_model.html#theil-sen-regression"><span class="std std-ref">Theil-Sen estimator: generalized-median-based estimator</span></a> for more information on the regressor.</p>
<p>Compared to the OLS (ordinary least squares) estimator, the Theil-Sen
estimator is robust against outliers. It has a breakdown point of about 29.3%
in case of a simple linear regression which means that it can tolerate
arbitrary corrupted data (outliers) of up to 29.3% in the two-dimensional
case.</p>
<p>The estimation of the model is done by calculating the slopes and intercepts
of a subpopulation of all possible combinations of p subsample points. If an
intercept is fitted, p must be greater than or equal to n_features + 1. The
final slope and intercept is then defined as the spatial median of these
slopes and intercepts.</p>
<p>In certain cases Theil-Sen performs better than <a class="reference internal" href="../../modules/linear_model.html#ransac-regression"><span class="std std-ref">RANSAC</span></a> which is also a robust method. This is illustrated in the
second example below where outliers with respect to the x-axis perturb RANSAC.
Tuning the <code class="docutils literal"><span class="pre">residual_threshold</span></code> parameter of RANSAC remedies this but in
general a priori knowledge about the data and the nature of the outliers is
needed.
Due to the computational complexity of Theil-Sen it is recommended to use it
only for small problems in terms of number of samples and features. For larger
problems the <code class="docutils literal"><span class="pre">max_subpopulation</span></code> parameter restricts the magnitude of all
possible combinations of p subsample points to a randomly chosen subset and
therefore also limits the runtime. Therefore, Theil-Sen is applicable to larger
problems with the drawback of losing some of its mathematical properties since
it then works on a random subset.</p>
<ul class="sphx-glr-horizontal">
<li><img alt="../../_images/sphx_glr_plot_theilsen_001.png" class="sphx-glr-multi-img first" src="../../_images/sphx_glr_plot_theilsen_001.png" />
</li>
<li><img alt="../../_images/sphx_glr_plot_theilsen_002.png" class="sphx-glr-multi-img first" src="../../_images/sphx_glr_plot_theilsen_002.png" />
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<div class="highlight-python"><div class="highlight"><pre><span></span><span class="c1"># Author: Florian Wilhelm -- <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">time</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.linear_model</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="View documentation for sklearn.linear_model.LinearRegression"><span class="n">LinearRegression</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.linear_model.TheilSenRegressor.html#sklearn.linear_model.TheilSenRegressor" title="View documentation for sklearn.linear_model.TheilSenRegressor"><span class="n">TheilSenRegressor</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.RANSACRegressor.html#sklearn.linear_model.RANSACRegressor" title="View documentation for sklearn.linear_model.RANSACRegressor"><span class="n">RANSACRegressor</span></a>
<span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="n">estimators</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">'OLS'</span><span class="p">,</span> <a href="../../modules/generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="View documentation for sklearn.linear_model.LinearRegression"><span class="n">LinearRegression</span></a><span class="p">()),</span>
<span class="p">(</span><span class="s1">'Theil-Sen'</span><span class="p">,</span> <a href="../../modules/generated/sklearn.linear_model.TheilSenRegressor.html#sklearn.linear_model.TheilSenRegressor" title="View documentation for sklearn.linear_model.TheilSenRegressor"><span class="n">TheilSenRegressor</span></a><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)),</span>
<span class="p">(</span><span class="s1">'RANSAC'</span><span class="p">,</span> <a href="../../modules/generated/sklearn.linear_model.RANSACRegressor.html#sklearn.linear_model.RANSACRegressor" title="View documentation for sklearn.linear_model.RANSACRegressor"><span class="n">RANSACRegressor</span></a><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)),</span> <span class="p">]</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'OLS'</span><span class="p">:</span> <span class="s1">'turquoise'</span><span class="p">,</span> <span class="s1">'Theil-Sen'</span><span class="p">:</span> <span class="s1">'gold'</span><span class="p">,</span> <span class="s1">'RANSAC'</span><span class="p">:</span> <span class="s1">'lightgreen'</span><span class="p">}</span>
<span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Outliers only in the y direction</span>
<a href="https://docs.scipy.org/doc/numpy/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">0</span><span class="p">)</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">200</span>
<span class="c1"># Linear model y = 3*x + N(2, 0.1**2)</span>
<span class="n">x</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.randn.html#numpy.random.randn" title="View documentation for numpy.random.randn"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="mf">3.</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">2.</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span> <span class="o">*</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.randn.html#numpy.random.randn" title="View documentation for numpy.random.randn"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">w</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">c</span> <span class="o">+</span> <span class="n">noise</span>
<span class="c1"># 10% outliers</span>
<span class="n">y</span><span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">:]</span> <span class="o">+=</span> <span class="o">-</span><span class="mi">20</span> <span class="o">*</span> <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">:]</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <a href="https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html#numpy.newaxis" title="View documentation for numpy.newaxis"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="View documentation for matplotlib.pyplot.scatter"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</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">color</span><span class="o">=</span><span class="s1">'indigo'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'x'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">)</span>
<span class="n">line_x</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="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">estimator</span> <span class="ow">in</span> <span class="n">estimators</span><span class="p">:</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">elapsed_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">line_x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<a href="https://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">line_x</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">name</span><span class="p">],</span> <span class="n">linewidth</span><span class="o">=</span><span class="n">lw</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> (fit time: </span><span class="si">%.2f</span><span class="s1">s)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">elapsed_time</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axis.html#matplotlib.pyplot.axis" title="View documentation for matplotlib.pyplot.axis"><span class="n">plt</span><span class="o">.</span><span class="n">axis</span></a><span class="p">(</span><span class="s1">'tight'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'upper left'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Corrupt y"</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Outliers in the X direction</span>
<a href="https://docs.scipy.org/doc/numpy/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">0</span><span class="p">)</span>
<span class="c1"># Linear model y = 3*x + N(2, 0.1**2)</span>
<span class="n">x</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.randn.html#numpy.random.randn" title="View documentation for numpy.random.randn"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span> <span class="o">*</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.randn.html#numpy.random.randn" title="View documentation for numpy.random.randn"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">noise</span>
<span class="c1"># 10% outliers</span>
<span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">:]</span> <span class="o">=</span> <span class="mf">9.9</span>
<span class="n">y</span><span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">:]</span> <span class="o">+=</span> <span class="mi">22</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <a href="https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html#numpy.newaxis" title="View documentation for numpy.newaxis"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span>
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<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="View documentation for matplotlib.pyplot.scatter"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</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">color</span><span class="o">=</span><span class="s1">'indigo'</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'x'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">)</span>
<span class="n">line_x</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="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">estimator</span> <span class="ow">in</span> <span class="n">estimators</span><span class="p">:</span>
<span class="n">t0</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">elapsed_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">line_x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<a href="https://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">line_x</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">name</span><span class="p">],</span> <span class="n">linewidth</span><span class="o">=</span><span class="n">lw</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> (fit time: </span><span class="si">%.2f</span><span class="s1">s)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">elapsed_time</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axis.html#matplotlib.pyplot.axis" title="View documentation for matplotlib.pyplot.axis"><span class="n">plt</span><span class="o">.</span><span class="n">axis</span></a><span class="p">(</span><span class="s1">'tight'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'upper left'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Corrupt x"</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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