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<li><a class="reference internal" href="#">Plot Ridge coefficients as a function of the L2 regularization</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-ridge-coeffs-py"><span class="std std-ref">here</span></a> to download the full example code</p>
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<div class="sphx-glr-example-title section" id="plot-ridge-coefficients-as-a-function-of-the-l2-regularization">
<span id="sphx-glr-auto-examples-linear-model-plot-ridge-coeffs-py"></span><h1>Plot Ridge coefficients as a function of the L2 regularization<a class="headerlink" href="#plot-ridge-coefficients-as-a-function-of-the-l2-regularization" title="Permalink to this headline">¶</a></h1>
<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 in the left plot represents one different dimension of the
coefficient vector, and this is displayed as a function of the
regularization parameter. The right plot shows how exact the solution
is. This example illustrates how a well defined solution is
found by Ridge regression and how regularization affects the
coefficients and their values. The plot on the right shows how
the difference of the coefficients from the estimator changes
as a function of regularization.</p>
<p>In this example the dependent variable Y is set as a function
of the input features: y = X*w + c. The coefficient vector w is
randomly sampled from a normal distribution, whereas the bias term c is
set to a constant.</p>
<p>As alpha tends toward zero the coefficients found by Ridge
regression stabilize towards the randomly sampled vector w.
For big alpha (strong regularisation) the coefficients
are smaller (eventually converging at 0) leading to a
simpler and biased solution.
These dependencies can be observed on the left plot.</p>
<p>The right plot shows the mean squared error between the
coefficients found by the model and the chosen vector w.
Less regularised models retrieve the exact
coefficients (error is equal to 0), stronger regularised
models increase the error.</p>
<p>Please note that in this example the data is non-noisy, hence
it is possible to extract the exact coefficients.</p>
<img alt="../../_images/sphx_glr_plot_ridge_coeffs_001.png" class="sphx-glr-single-img" src="../../_images/sphx_glr_plot_ridge_coeffs_001.png" />
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="c1"># Author: Kornel Kielczewski -- <[email protected]></span>
<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</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">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="k">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.linear_model</span> <span class="k">import</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">Ridge</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.metrics.mean_squared_error.html#sklearn.metrics.mean_squared_error" title="View documentation for sklearn.metrics.mean_squared_error"><span class="n">mean_squared_error</span></a>
<span class="n">clf</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">Ridge</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">w</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">10</span><span class="p">,</span> <span class="n">n_features</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">coef</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="mf">3.5</span><span class="p">)</span>
<span class="n">coefs</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">errors</span> <span class="o">=</span> <span class="p">[]</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">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
<span class="c1"># Train the model with different regularisation strengths</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">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="n">a</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="n">coefs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
<span class="n">errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><a href="../../modules/generated/sklearn.metrics.mean_squared_error.html#sklearn.metrics.mean_squared_error" title="View documentation for sklearn.metrics.mean_squared_error"><span class="n">mean_squared_error</span></a><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
<span class="c1"># Display results</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.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><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">121</span><span class="p">)</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>
<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.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">122</span><span class="p">)</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">errors</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>
<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">'error'</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">'Coefficient error 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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