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<li><a class="reference internal" href="#">Plot Ridge coefficients as a function of the regularization</a></li>
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<p class="first admonition-title">Note</p>
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<div class="sphx-glr-example-title section" id="plot-ridge-coefficients-as-a-function-of-the-regularization">
<span id="sphx-glr-auto-examples-linear-model-plot-ridge-path-py"></span><h1>Plot Ridge coefficients as a function of the regularization<a class="headerlink" href="#plot-ridge-coefficients-as-a-function-of-the-regularization" title="Permalink to this headline">¶</a></h1>
<p>Shows the effect of collinearity in the coefficients of an estimator.</p>
<p><a class="reference internal" href="../../modules/generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal"><span class="pre">Ridge</span></code></a> Regression is the estimator used in this example.
Each color represents a different feature of the
coefficient vector, and this is displayed as a function of the
regularization parameter.</p>
<p>This example also shows the usefulness of applying Ridge regression
to highly ill-conditioned matrices. For such matrices, a slight
change in the target variable can cause huge variances in the
calculated weights. In such cases, it is useful to set a certain
regularization (alpha) to reduce this variation (noise).</p>
<p>When alpha is very large, the regularization effect dominates the
squared loss function and the coefficients tend to zero.
At the end of the path, as alpha tends toward zero
and the solution tends towards the ordinary least squares, coefficients
exhibit big oscillations. In practise it is necessary to tune alpha
in such a way that a balance is maintained between both.</p>
<img alt="../../_images/sphx_glr_plot_ridge_path_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_ridge_path_001.png" />
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="c1"># Author: Fabian Pedregosa -- <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</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</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="c1"># X is the 10x10 Hilbert matrix</span>
<span class="n">X</span> <span class="o">=</span> <span class="mf">1.</span> <span class="o">/</span> <span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arange.html#numpy.arange" title="View documentation for numpy.arange"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span> <span class="o">+</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arange.html#numpy.arange" title="View documentation for numpy.arange"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</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>
<span class="n">y</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.ones.html#numpy.ones" title="View documentation for numpy.ones"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Compute paths</span>
<span class="n">n_alphas</span> <span class="o">=</span> <span class="mi">200</span>
<span class="n">alphas</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.logspace.html#numpy.logspace" title="View documentation for numpy.logspace"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_alphas</span><span class="p">)</span>
<span class="n">coefs</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">alphas</span><span class="p">:</span>
<span class="n">ridge</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="View documentation for sklearn.linear_model.Ridge"><span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span></a><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="n">a</span><span class="p">,</span> <span class="n">fit_intercept</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
<span class="n">ridge</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">coefs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ridge</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Display results</span>
<span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.gca.html#matplotlib.pyplot.gca" title="View documentation for matplotlib.pyplot.gca"><span class="n">plt</span><span class="o">.</span><span class="n">gca</span></a><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">alphas</span><span class="p">,</span> <span class="n">coefs</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">ax</span><span class="o">.</span><span class="n">get_xlim</span><span class="p">()[::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="c1"># reverse axis</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s1">'alpha'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s1">'weights'</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="s1">'Ridge coefficients as a function of the regularization'</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.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>
</pre></div>
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