M-Estimators for Robust Linear Modeling
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.. _robust_models_1_notebook:

`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/robust_models_1.ipynb>`_

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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span>
   <span class="kn">from</span> <span class="nn">statsmodels.compat</span> <span class="kn">import</span> <span class="n">lmap</span>
   <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
   <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</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">import</span> <span class="nn">statsmodels.api</span> <span class="kn">as</span> <span class="nn">sm</span>
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   <ul>
   <li>An M-estimator minimizes the function </li>
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   <p>$$Q(e_i, \rho) = \sum_i~\rho \left (\frac{e_i}{s}\right )$$</p>
   <p>where $\rho$ is a symmetric function of the residuals </p>
   <ul>
   <li>The effect of $\rho$ is to reduce the influence of outliers</li>
   <li>$s$ is an estimate of scale. </li>
   <li>The robust estimates $\hat{\beta}$ are computed by the iteratively re-weighted least squares algorithm</li>
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   <li>We have several choices available for the weighting functions to be used</li>
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   <div class="highlight"><pre><span class="n">norms</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">norms</span>
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   <div class="highlight"><pre><span class="k">def</span> <span class="nf">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">weights_func</span><span class="p">,</span> <span class="n">xlabels</span><span class="p">,</span> <span class="n">xticks</span><span class="p">):</span>
       <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
       <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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="n">support</span><span class="p">,</span> <span class="n">weights_func</span><span class="p">(</span><span class="n">support</span><span class="p">))</span>
       <span class="n">ax</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">xticks</span><span class="p">)</span>
       <span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">xlabels</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
       <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-.</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">)</span>
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   <h3 id="Andrew's-Wave">Andrew's Wave<a class="anchor-link" href="#Andrew's-Wave">&#182;</a></h3>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">AndrewWave</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.AndrewWave method
       Andrew&apos;s wave weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = sin(z/a)/(z/a)     for \|z\| &lt;= a*pi
       
           weights(z) = 0                  for \|z\| &gt; a*pi
   
   
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   <div class="highlight"><pre><span class="n">a</span> <span class="o">=</span> <span class="mf">1.339</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
   <span class="n">andrew</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">AndrewWave</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">andrew</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;$-\pi*a$&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;$\pi*a$&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">]);</span>
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   <h3 id="Hampel's-17A">Hampel's 17A<a class="anchor-link" href="#Hampel's-17A">&#182;</a></h3>
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   <div class="prompt input_prompt">In&nbsp;[6]:</div>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">Hampel</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   <pre>
   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.Hampel method
       Hampel weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = 1                            for \|z\| &lt;= a
       
           weights(z) = a/\|z\|                        for a &lt; \|z\| &lt;= b
       
           weights(z) = a*(c - \|z\|)/(\|z\|*(c-b))      for b &lt; \|z\| &lt;= c
       
           weights(z) = 0                            for \|z\| &gt; c
   
   
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   <div class="highlight"><pre><span class="n">c</span> <span class="o">=</span> <span class="mi">8</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
   <span class="n">hampel</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">Hampel</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="mf">2.</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mf">4.</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">c</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">hampel</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;3*c&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3*c&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">]);</span>
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   <h3 id="Huber's-t">Huber's t<a class="anchor-link" href="#Huber's-t">&#182;</a></h3>
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   <div class="prompt input_prompt">In&nbsp;[8]:</div>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">HuberT</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   <pre>
   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.HuberT method
       Huber&apos;s t weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = 1          for \|z\| &lt;= t
       
           weights(z) = t/\|z\|      for \|z\| &gt; t
   
   
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   <div class="highlight"><pre><span class="n">t</span> <span class="o">=</span> <span class="mf">1.345</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">t</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">t</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
   <span class="n">huber</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">HuberT</span><span class="p">(</span><span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">huber</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;-3*t&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3*t&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">t</span><span class="p">]);</span>
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   <h3 id="Least-Squares">Least Squares<a class="anchor-link" href="#Least-Squares">&#182;</a></h3>
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   <div class="prompt input_prompt">In&nbsp;[10]:</div>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">LeastSquares</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   <pre>
   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.LeastSquares method
       The least squares estimator weighting function for the IRLS algorithm.
       
       The psi function scaled by the input z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = np.ones(z.shape)
   
   
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   <div class="highlight"><pre><span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><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="mi">1000</span><span class="p">)</span>
   <span class="n">lst_sq</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">LeastSquares</span><span class="p">()</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">lst_sq</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;-3&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]);</span>
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   <h3 id="Ramsay's-Ea">Ramsay's Ea<a class="anchor-link" href="#Ramsay's-Ea">&#182;</a></h3>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">RamsayE</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   <pre>
   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.RamsayE method
       Ramsay&apos;s Ea weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = exp(-a*\|z\|)
   
   
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   <div class="highlight"><pre><span class="n">a</span> <span class="o">=</span> <span class="o">.</span><span class="mi">3</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
   <span class="n">ramsay</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">RamsayE</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">ramsay</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;-3*a&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3*a&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="p">]);</span>
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   <h3 id="Trimmed-Mean">Trimmed Mean<a class="anchor-link" href="#Trimmed-Mean">&#182;</a></h3>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">TrimmedMean</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.TrimmedMean method
       Least trimmed mean weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           weights(z) = 1             for \|z\| &lt;= c
       
           weights(z) = 0             for \|z\| &gt; c
   
   
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   <div class="highlight"><pre><span class="n">c</span> <span class="o">=</span> <span class="mi">2</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
   <span class="n">trimmed</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">TrimmedMean</span><span class="p">(</span><span class="n">c</span><span class="o">=</span><span class="n">c</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">trimmed</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;-3*c&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3*c&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">]);</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
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   <h3 id="Tukey's-Biweight">Tukey's Biweight<a class="anchor-link" href="#Tukey's-Biweight">&#182;</a></h3>
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   <div class="highlight"><pre><span class="n">help</span><span class="p">(</span><span class="n">norms</span><span class="o">.</span><span class="n">TukeyBiweight</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
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   <pre>
   Help on method weights in module statsmodels.robust.norms:
   
   weights(self, z) unbound statsmodels.robust.norms.TukeyBiweight method
       Tukey&apos;s biweight weighting function for the IRLS algorithm
       
       The psi function scaled by z
       
       Parameters
       ----------
       z : array-like
           1d array
       
       Returns
       -------
       weights : array
           psi(z) = (1 - (z/c)**2)**2          for \|z\| &lt;= R
       
           psi(z) = 0                          for \|z\| &gt; R
   
   
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   <div class="highlight"><pre><span class="n">c</span> <span class="o">=</span> <span class="mf">4.685</span>
   <span class="n">support</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
   <span class="n">tukey</span> <span class="o">=</span> <span class="n">norms</span><span class="o">.</span><span class="n">TukeyBiweight</span><span class="p">(</span><span class="n">c</span><span class="o">=</span><span class="n">c</span><span class="p">)</span>
   <span class="n">plot_weights</span><span class="p">(</span><span class="n">support</span><span class="p">,</span> <span class="n">tukey</span><span class="o">.</span><span class="n">weights</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;-3*c&#39;</span><span class="p">,</span> <span class="s1">&#39;0&#39;</span><span class="p">,</span> <span class="s1">&#39;3*c&#39;</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">c</span><span class="p">]);</span>
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   "
   >
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   <h3 id="Scale-Estimators">Scale Estimators<a class="anchor-link" href="#Scale-Estimators">&#182;</a></h3>
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   <ul>
   <li>Robust estimates of the location</li>
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   <div class="prompt input_prompt">In&nbsp;[18]:</div>
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   <div class="highlight"><pre><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">500</span><span class="p">])</span>
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   <ul>
   <li>The mean is not a robust estimator of location</li>
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   <div class="prompt input_prompt">In&nbsp;[19]:</div>
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   <div class="highlight"><pre><span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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   <pre>
   102.0
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   <ul>
   <li>The median, on the other hand, is a robust estimator with a breakdown point of 50%</li>
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   <div class="prompt input_prompt">In&nbsp;[20]:</div>
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
   </pre></div>
   
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   <pre>
   3.0
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   <ul>
   <li>Analagously for the scale</li>
   <li>The standard deviation is not robust</li>
   </ul>
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   <div class="highlight"><pre><span class="n">x</span><span class="o">.</span><span class="n">std</span><span class="p">()</span>
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   <pre>
   199.00251254695254
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   <p>Median Absolute Deviation</p>
   <p>$$ median_i |X_i - median_j(X_j)|) $$</p>
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   <p>Standardized Median Absolute Deviation is a consistent estimator for $\hat{\sigma}$</p>
   <p>$$\hat{\sigma}=K \cdot MAD$$</p>
   <p>where $K$ depends on the distribution. For the normal distribution for example,</p>
   <p>$$K = \Phi^{-1}(.75)$$</p>
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   <div class="highlight"><pre><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="o">.</span><span class="mi">75</span><span class="p">)</span>
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   0.67448975019608171
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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   [  1   2   3   4 500]
   
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   <div class="highlight"><pre><span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">stand_mad</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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   /build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/robust/scale.py:49: FutureWarning: stand_mad is deprecated and will be removed in 0.7.0. Use mad instead.
     &quot;instead.&quot;, FutureWarning)
   
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   1.482602218505602
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mf">5.</span><span class="p">])</span><span class="o">.</span><span class="n">std</span><span class="p">()</span>
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   1.4142135623730951
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   <ul>
   <li>The default for Robust Linear Models is MAD</li>
   <li>another popular choice is Huber&#39;s proposal 2</li>
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">12345</span><span class="p">)</span>
   <span class="n">fat_tails</span> <span class="o">=</span> <span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">40</span><span class="p">)</span>
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   <div class="highlight"><pre><span class="n">kde</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">nonparametric</span><span class="o">.</span><span class="n">KDE</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">)</span>
   <span class="n">kde</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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="n">kde</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde</span><span class="o">.</span><span class="n">density</span><span class="p">);</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-123-605f8dff1422&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>kde <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>nonparametric<span class="ansiblue">.</span>KDE<span class="ansiblue">(</span>fat_tails<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> kde<span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> fig <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>figure<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">12</span><span class="ansiblue">,</span><span class="ansicyan">8</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> ax <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">111</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> ax<span class="ansiblue">.</span>plot<span class="ansiblue">(</span>kde<span class="ansiblue">.</span>support<span class="ansiblue">,</span> kde<span class="ansiblue">.</span>density<span class="ansiblue">)</span><span class="ansiblue">;</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;module&apos; object has no attribute &apos;KDE&apos;</pre>
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">fat_tails</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">fat_tails</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
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   0.0688231044811 1.34716332297
   
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">))</span>
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   (0.068823104481087499, 1.3471633229698652)
   
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">,</span> <span class="n">f0</span><span class="o">=</span><span class="mi">6</span><span class="p">))</span>
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   (6, 0.039009187170278181, 1.0564230978488927)
   
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   <div class="highlight"><pre><span class="n">huber</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">Huber</span><span class="p">()</span>
   <span class="n">loc</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="n">huber</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">loc</span><span class="p">,</span> <span class="n">scale</span><span class="p">)</span>
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   0.0404898433327 1.15571400476
   
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   <div class="highlight"><pre><span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">stand_mad</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">)</span>
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   1.1153350011654151
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   <div class="highlight"><pre><span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">stand_mad</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">stats</span><span class="o">.</span><span class="n">t</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="o">.</span><span class="mi">75</span><span class="p">))</span>
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   1.0483916565928972
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   <div class="highlight"><pre><span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">mad</span><span class="p">(</span><span class="n">fat_tails</span><span class="p">)</span>
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   1.1153350011654151
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   <h3 id="Duncan's-Occupational-Prestige-data---M-estimation-for-outliers">Duncan's Occupational Prestige data - M-estimation for outliers<a class="anchor-link" href="#Duncan's-Occupational-Prestige-data---M-estimation-for-outliers">&#182;</a></h3>
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   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">statsmodels.graphics.api</span> <span class="kn">import</span> <span class="n">abline_plot</span>
   <span class="kn">from</span> <span class="nn">statsmodels.formula.api</span> <span class="kn">import</span> <span class="n">ols</span><span class="p">,</span> <span class="n">rlm</span>
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   <div class="highlight"><pre><span class="n">prestige</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">&quot;Duncan&quot;</span><span class="p">,</span> <span class="s2">&quot;car&quot;</span><span class="p">,</span> <span class="n">cache</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">URLError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-132-d85a43c9ce16&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>prestige <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>datasets<span class="ansiblue">.</span>get_rdataset<span class="ansiblue">(</span><span class="ansiblue">&quot;Duncan&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;car&quot;</span><span class="ansiblue">,</span> cache<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue">.</span>data<span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">get_rdataset</span><span class="ansiblue">(dataname, package, cache)</span>
   <span class="ansigreen">    284</span>                      &quot;master/doc/&quot;+package+&quot;/rst/&quot;)
   <span class="ansigreen">    285</span>     cache <span class="ansiblue">=</span> _get_cache<span class="ansiblue">(</span>cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 286</span><span class="ansired">     </span>data<span class="ansiblue">,</span> from_cache <span class="ansiblue">=</span> _get_data<span class="ansiblue">(</span>data_base_url<span class="ansiblue">,</span> dataname<span class="ansiblue">,</span> cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    287</span>     data <span class="ansiblue">=</span> read_csv<span class="ansiblue">(</span>data<span class="ansiblue">,</span> index_col<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    288</span>     data <span class="ansiblue">=</span> _maybe_reset_index<span class="ansiblue">(</span>data<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">_get_data</span><span class="ansiblue">(base_url, dataname, cache, extension)</span>
   <span class="ansigreen">    215</span>     url <span class="ansiblue">=</span> base_url <span class="ansiblue">+</span> <span class="ansiblue">(</span>dataname <span class="ansiblue">+</span> <span class="ansiblue">&quot;.%s&quot;</span><span class="ansiblue">)</span> <span class="ansiblue">%</span> extension<span class="ansiblue"></span>
   <span class="ansigreen">    216</span>     <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 217</span><span class="ansired">         </span>data<span class="ansiblue">,</span> from_cache <span class="ansiblue">=</span> _urlopen_cached<span class="ansiblue">(</span>url<span class="ansiblue">,</span> cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    218</span>     <span class="ansigreen">except</span> HTTPError <span class="ansigreen">as</span> err<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    219</span>         <span class="ansigreen">if</span> <span class="ansiblue">&apos;404&apos;</span> <span class="ansigreen">in</span> str<span class="ansiblue">(</span>err<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">_urlopen_cached</span><span class="ansiblue">(url, cache)</span>
   <span class="ansigreen">    206</span>     <span class="ansired"># not using the cache or didn&apos;t find it in cache</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">    207</span>     <span class="ansigreen">if</span> <span class="ansigreen">not</span> from_cache<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 208</span><span class="ansired">         </span>data <span class="ansiblue">=</span> urlopen<span class="ansiblue">(</span>url<span class="ansiblue">)</span><span class="ansiblue">.</span>read<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    209</span>         <span class="ansigreen">if</span> cache <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span>  <span class="ansired"># then put it in the cache</span><span class="ansiblue"></span>
   <span class="ansigreen">    210</span>             _cache_it<span class="ansiblue">(</span>data<span class="ansiblue">,</span> cache_path<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">urlopen</span><span class="ansiblue">(url, data, timeout, cafile, capath, cadefault, context)</span>
   <span class="ansigreen">    152</span>     <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    153</span>         opener <span class="ansiblue">=</span> _opener<span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 154</span><span class="ansired">     </span><span class="ansigreen">return</span> opener<span class="ansiblue">.</span>open<span class="ansiblue">(</span>url<span class="ansiblue">,</span> data<span class="ansiblue">,</span> timeout<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    155</span> <span class="ansiblue"></span>
   <span class="ansigreen">    156</span> <span class="ansigreen">def</span> install_opener<span class="ansiblue">(</span>opener<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">open</span><span class="ansiblue">(self, fullurl, data, timeout)</span>
   <span class="ansigreen">    427</span>             req <span class="ansiblue">=</span> meth<span class="ansiblue">(</span>req<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    428</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 429</span><span class="ansired">         </span>response <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_open<span class="ansiblue">(</span>req<span class="ansiblue">,</span> data<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    430</span> <span class="ansiblue"></span>
   <span class="ansigreen">    431</span>         <span class="ansired"># post-process response</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">_open</span><span class="ansiblue">(self, req, data)</span>
   <span class="ansigreen">    445</span>         protocol <span class="ansiblue">=</span> req<span class="ansiblue">.</span>get_type<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    446</span>         result = self._call_chain(self.handle_open, protocol, protocol +
   <span class="ansigreen">--&gt; 447</span><span class="ansired">                                   &apos;_open&apos;, req)
   </span><span class="ansigreen">    448</span>         <span class="ansigreen">if</span> result<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    449</span>             <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">_call_chain</span><span class="ansiblue">(self, chain, kind, meth_name, *args)</span>
   <span class="ansigreen">    405</span>             func <span class="ansiblue">=</span> getattr<span class="ansiblue">(</span>handler<span class="ansiblue">,</span> meth_name<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    406</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 407</span><span class="ansired">             </span>result <span class="ansiblue">=</span> func<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    408</span>             <span class="ansigreen">if</span> result <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    409</span>                 <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">https_open</span><span class="ansiblue">(self, req)</span>
   <span class="ansigreen">   1239</span>         <span class="ansigreen">def</span> https_open<span class="ansiblue">(</span>self<span class="ansiblue">,</span> req<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1240</span>             return self.do_open(httplib.HTTPSConnection, req,
   <span class="ansigreen">-&gt; 1241</span><span class="ansired">                 context=self._context)
   </span><span class="ansigreen">   1242</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1243</span>         https_request <span class="ansiblue">=</span> AbstractHTTPHandler<span class="ansiblue">.</span>do_request_<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">do_open</span><span class="ansiblue">(self, http_class, req, **http_conn_args)</span>
   <span class="ansigreen">   1196</span>         <span class="ansigreen">except</span> socket<span class="ansiblue">.</span>error<span class="ansiblue">,</span> err<span class="ansiblue">:</span> <span class="ansired"># XXX what error?</span><span class="ansiblue"></span>
   <span class="ansigreen">   1197</span>             h<span class="ansiblue">.</span>close<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1198</span><span class="ansired">             </span><span class="ansigreen">raise</span> URLError<span class="ansiblue">(</span>err<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1199</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1200</span>             <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansired">URLError</span>: &lt;urlopen error [Errno -2] Name or service not known&gt;</pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[37]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">prestige</span><span class="o">.</span><span class="n">head</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-133-d3ced65c396c&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">print</span><span class="ansiblue">(</span>prestige<span class="ansiblue">.</span>head<span class="ansiblue">(</span><span class="ansicyan">10</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;prestige&apos; is not defined</pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[38]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">))</span>
   <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s1">&#39;Income&#39;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s1">&#39;Prestige&#39;</span><span class="p">)</span>
   <span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">prestige</span><span class="o">.</span><span class="n">income</span><span class="p">,</span> <span class="n">prestige</span><span class="o">.</span><span class="n">prestige</span><span class="p">)</span>
   <span class="n">xy_outlier</span> <span class="o">=</span> <span class="n">prestige</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="s1">&#39;minister&#39;</span><span class="p">][[</span><span class="s1">&#39;income&#39;</span><span class="p">,</span><span class="s1">&#39;prestige&#39;</span><span class="p">]]</span>
   <span class="n">ax1</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="s1">&#39;Minister&#39;</span><span class="p">,</span> <span class="n">xy_outlier</span><span class="p">,</span> <span class="n">xy_outlier</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
   <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s1">&#39;Education&#39;</span><span class="p">,</span>
                              <span class="n">ylabel</span><span class="o">=</span><span class="s1">&#39;Prestige&#39;</span><span class="p">)</span>
   <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">prestige</span><span class="o">.</span><span class="n">education</span><span class="p">,</span> <span class="n">prestige</span><span class="o">.</span><span class="n">prestige</span><span class="p">);</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-134-ab347cc850d5&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> fig <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>figure<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">12</span><span class="ansiblue">,</span><span class="ansicyan">12</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> ax1 <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">211</span><span class="ansiblue">,</span> xlabel<span class="ansiblue">=</span><span class="ansiblue">&apos;Income&apos;</span><span class="ansiblue">,</span> ylabel<span class="ansiblue">=</span><span class="ansiblue">&apos;Prestige&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 3</span><span class="ansired"> </span>ax1<span class="ansiblue">.</span>scatter<span class="ansiblue">(</span>prestige<span class="ansiblue">.</span>income<span class="ansiblue">,</span> prestige<span class="ansiblue">.</span>prestige<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> xy_outlier <span class="ansiblue">=</span> prestige<span class="ansiblue">.</span>ix<span class="ansiblue">[</span><span class="ansiblue">&apos;minister&apos;</span><span class="ansiblue">]</span><span class="ansiblue">[</span><span class="ansiblue">[</span><span class="ansiblue">&apos;income&apos;</span><span class="ansiblue">,</span><span class="ansiblue">&apos;prestige&apos;</span><span class="ansiblue">]</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> ax1<span class="ansiblue">.</span>annotate<span class="ansiblue">(</span><span class="ansiblue">&apos;Minister&apos;</span><span class="ansiblue">,</span> xy_outlier<span class="ansiblue">,</span> xy_outlier<span class="ansiblue">+</span><span class="ansicyan">1</span><span class="ansiblue">,</span> fontsize<span class="ansiblue">=</span><span class="ansicyan">16</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;prestige&apos; is not defined</pre>
   </div>
   </div>
   
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   >
   </div>
   
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[39]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">ols_model</span> <span class="o">=</span> <span class="n">ols</span><span class="p">(</span><span class="s1">&#39;prestige ~ income + education&#39;</span><span class="p">,</span> <span class="n">prestige</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="k">print</span><span class="p">(</span><span class="n">ols_model</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
   </pre></div>
   
   </div>
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   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-135-94340cd202f8&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>ols_model <span class="ansiblue">=</span> ols<span class="ansiblue">(</span><span class="ansiblue">&apos;prestige ~ income + education&apos;</span><span class="ansiblue">,</span> prestige<span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>ols_model<span class="ansiblue">.</span>summary<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;prestige&apos; is not defined</pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[40]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">infl</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">get_influence</span><span class="p">()</span>
   <span class="n">student</span> <span class="o">=</span> <span class="n">infl</span><span class="o">.</span><span class="n">summary_frame</span><span class="p">()[</span><span class="s1">&#39;student_resid&#39;</span><span class="p">]</span>
   <span class="k">print</span><span class="p">(</span><span class="n">student</span><span class="p">)</span>
   </pre></div>
   
   </div>
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   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-136-762835c5010b&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>infl <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>get_influence<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> student <span class="ansiblue">=</span> infl<span class="ansiblue">.</span>summary_frame<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">[</span><span class="ansiblue">&apos;student_resid&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>student<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;get_influence&apos;</pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[41]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">student</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">student</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">])</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-137-4de70e5d7415&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">print</span><span class="ansiblue">(</span>student<span class="ansiblue">.</span>ix<span class="ansiblue">[</span>np<span class="ansiblue">.</span>abs<span class="ansiblue">(</span>student<span class="ansiblue">)</span> <span class="ansiblue">&gt;</span> <span class="ansicyan">2</span><span class="ansiblue">]</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;student&apos; is not defined</pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[42]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">infl</span><span class="o">.</span><span class="n">summary_frame</span><span class="p">()</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="s1">&#39;minister&#39;</span><span class="p">])</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">KeyError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-138-1b2a9fb1d022&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">print</span><span class="ansiblue">(</span>infl<span class="ansiblue">.</span>summary_frame<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>ix<span class="ansiblue">[</span><span class="ansiblue">&apos;minister&apos;</span><span class="ansiblue">]</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/indexing.pyc</span> in <span class="ansicyan">__getitem__</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">     68</span>             <span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_tuple<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     69</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 70</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_axis<span class="ansiblue">(</span>key<span class="ansiblue">,</span> axis<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     71</span> <span class="ansiblue"></span>
   <span class="ansigreen">     72</span>     <span class="ansigreen">def</span> _get_label<span class="ansiblue">(</span>self<span class="ansiblue">,</span> label<span class="ansiblue">,</span> axis<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/indexing.pyc</span> in <span class="ansicyan">_getitem_axis</span><span class="ansiblue">(self, key, axis)</span>
   <span class="ansigreen">    965</span>                     <span class="ansigreen">return</span> self<span class="ansiblue">.</span>_get_loc<span class="ansiblue">(</span>key<span class="ansiblue">,</span> axis<span class="ansiblue">=</span>axis<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    966</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 967</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_get_label<span class="ansiblue">(</span>key<span class="ansiblue">,</span> axis<span class="ansiblue">=</span>axis<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    968</span> <span class="ansiblue"></span>
   <span class="ansigreen">    969</span>     <span class="ansigreen">def</span> _getitem_iterable<span class="ansiblue">(</span>self<span class="ansiblue">,</span> key<span class="ansiblue">,</span> axis<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/indexing.pyc</span> in <span class="ansicyan">_get_label</span><span class="ansiblue">(self, label, axis)</span>
   <span class="ansigreen">     84</span>             <span class="ansigreen">raise</span> IndexingError<span class="ansiblue">(</span><span class="ansiblue">&apos;no slices here, handle elsewhere&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     85</span> <span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 86</span><span class="ansired">         </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>obj<span class="ansiblue">.</span>_xs<span class="ansiblue">(</span>label<span class="ansiblue">,</span> axis<span class="ansiblue">=</span>axis<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     87</span> <span class="ansiblue"></span>
   <span class="ansigreen">     88</span>     <span class="ansigreen">def</span> _get_loc<span class="ansiblue">(</span>self<span class="ansiblue">,</span> key<span class="ansiblue">,</span> axis<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/generic.pyc</span> in <span class="ansicyan">xs</span><span class="ansiblue">(self, key, axis, level, copy, drop_level)</span>
   <span class="ansigreen">   1484</span>                                                       drop_level=drop_level)
   <span class="ansigreen">   1485</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1486</span><span class="ansired">             </span>loc <span class="ansiblue">=</span> self<span class="ansiblue">.</span>index<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1487</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1488</span>             <span class="ansigreen">if</span> isinstance<span class="ansiblue">(</span>loc<span class="ansiblue">,</span> np<span class="ansiblue">.</span>ndarray<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/index.pyc</span> in <span class="ansicyan">get_loc</span><span class="ansiblue">(self, key, method, tolerance)</span>
   <span class="ansigreen">   1757</span>                                  &apos;backfill or nearest lookups&apos;)
   <span class="ansigreen">   1758</span>             key <span class="ansiblue">=</span> _values_from_object<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1759</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_engine<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1760</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1761</span>         indexer = self.get_indexer([key], method=method,
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3979)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3908)</span><span class="ansiblue">()</span>
   
   <span class="ansired">KeyError</span>: &apos;minister&apos;</pre>
   </div>
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   </div>
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   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[43]:</div>
   <div class="inner_cell">
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   <div class="highlight"><pre><span class="n">sidak</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">outlier_test</span><span class="p">(</span><span class="s1">&#39;sidak&#39;</span><span class="p">)</span>
   <span class="n">sidak</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="s1">&#39;unadj_p&#39;</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">sidak</span><span class="p">)</span>
   </pre></div>
   
   </div>
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   <div class="output_wrapper">
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-139-88381d4a1ea2&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>sidak <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>outlier_test<span class="ansiblue">(</span><span class="ansiblue">&apos;sidak&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> sidak<span class="ansiblue">.</span>sort<span class="ansiblue">(</span><span class="ansiblue">&apos;unadj_p&apos;</span><span class="ansiblue">,</span> inplace<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>sidak<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;outlier_test&apos;</pre>
   </div>
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   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[44]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">fdr</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">outlier_test</span><span class="p">(</span><span class="s1">&#39;fdr_bh&#39;</span><span class="p">)</span>
   <span class="n">fdr</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="s1">&#39;unadj_p&#39;</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">fdr</span><span class="p">)</span>
   </pre></div>
   
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   <div class="output_wrapper">
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-140-69e9cb113222&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>fdr <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>outlier_test<span class="ansiblue">(</span><span class="ansiblue">&apos;fdr_bh&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> fdr<span class="ansiblue">.</span>sort<span class="ansiblue">(</span><span class="ansiblue">&apos;unadj_p&apos;</span><span class="ansiblue">,</span> inplace<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>fdr<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;outlier_test&apos;</pre>
   </div>
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   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[45]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">rlm_model</span> <span class="o">=</span> <span class="n">rlm</span><span class="p">(</span><span class="s1">&#39;prestige ~ income + education&#39;</span><span class="p">,</span> <span class="n">prestige</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="k">print</span><span class="p">(</span><span class="n">rlm_model</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
   </pre></div>
   
   </div>
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   <div class="output_wrapper">
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   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-141-f16d19a78b97&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>rlm_model <span class="ansiblue">=</span> rlm<span class="ansiblue">(</span><span class="ansiblue">&apos;prestige ~ income + education&apos;</span><span class="ansiblue">,</span> prestige<span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>rlm_model<span class="ansiblue">.</span>summary<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;prestige&apos; is not defined</pre>
   </div>
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   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[46]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">rlm_model</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
   </pre></div>
   
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   <div class="output_wrapper">
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   <div class="output_area"><div class="prompt"></div>
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-142-2a0e93853fea&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">print</span><span class="ansiblue">(</span>rlm_model<span class="ansiblue">.</span>weights<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;rlm_model&apos; is not defined</pre>
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   <div class="text_cell_render border-box-sizing rendered_html">
   <h3 id="Hertzprung-Russell-data-for-Star-Cluster-CYG-0B1---Leverage-Points">Hertzprung Russell data for Star Cluster CYG 0B1 - Leverage Points<a class="anchor-link" href="#Hertzprung-Russell-data-for-Star-Cluster-CYG-0B1---Leverage-Points">&#182;</a></h3>
   </div>
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   <div class="cell border-box-sizing text_cell rendered">
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   <ul>
   <li>Data is on the luminosity and temperature of 47 stars in the direction of Cygnus.</li>
   </ul>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[47]:</div>
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   <div class="highlight"><pre><span class="n">dta</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">&quot;starsCYG&quot;</span><span class="p">,</span> <span class="s2">&quot;robustbase&quot;</span><span class="p">,</span> <span class="n">cache</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">data</span>
   </pre></div>
   
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   <div class="output_wrapper">
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">URLError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-143-4880a5abdb55&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>dta <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>datasets<span class="ansiblue">.</span>get_rdataset<span class="ansiblue">(</span><span class="ansiblue">&quot;starsCYG&quot;</span><span class="ansiblue">,</span> <span class="ansiblue">&quot;robustbase&quot;</span><span class="ansiblue">,</span> cache<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue">.</span>data<span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">get_rdataset</span><span class="ansiblue">(dataname, package, cache)</span>
   <span class="ansigreen">    284</span>                      &quot;master/doc/&quot;+package+&quot;/rst/&quot;)
   <span class="ansigreen">    285</span>     cache <span class="ansiblue">=</span> _get_cache<span class="ansiblue">(</span>cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 286</span><span class="ansired">     </span>data<span class="ansiblue">,</span> from_cache <span class="ansiblue">=</span> _get_data<span class="ansiblue">(</span>data_base_url<span class="ansiblue">,</span> dataname<span class="ansiblue">,</span> cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    287</span>     data <span class="ansiblue">=</span> read_csv<span class="ansiblue">(</span>data<span class="ansiblue">,</span> index_col<span class="ansiblue">=</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    288</span>     data <span class="ansiblue">=</span> _maybe_reset_index<span class="ansiblue">(</span>data<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">_get_data</span><span class="ansiblue">(base_url, dataname, cache, extension)</span>
   <span class="ansigreen">    215</span>     url <span class="ansiblue">=</span> base_url <span class="ansiblue">+</span> <span class="ansiblue">(</span>dataname <span class="ansiblue">+</span> <span class="ansiblue">&quot;.%s&quot;</span><span class="ansiblue">)</span> <span class="ansiblue">%</span> extension<span class="ansiblue"></span>
   <span class="ansigreen">    216</span>     <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 217</span><span class="ansired">         </span>data<span class="ansiblue">,</span> from_cache <span class="ansiblue">=</span> _urlopen_cached<span class="ansiblue">(</span>url<span class="ansiblue">,</span> cache<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    218</span>     <span class="ansigreen">except</span> HTTPError <span class="ansigreen">as</span> err<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    219</span>         <span class="ansigreen">if</span> <span class="ansiblue">&apos;404&apos;</span> <span class="ansigreen">in</span> str<span class="ansiblue">(</span>err<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/datasets/utils.pyc</span> in <span class="ansicyan">_urlopen_cached</span><span class="ansiblue">(url, cache)</span>
   <span class="ansigreen">    206</span>     <span class="ansired"># not using the cache or didn&apos;t find it in cache</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">    207</span>     <span class="ansigreen">if</span> <span class="ansigreen">not</span> from_cache<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 208</span><span class="ansired">         </span>data <span class="ansiblue">=</span> urlopen<span class="ansiblue">(</span>url<span class="ansiblue">)</span><span class="ansiblue">.</span>read<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    209</span>         <span class="ansigreen">if</span> cache <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span>  <span class="ansired"># then put it in the cache</span><span class="ansiblue"></span>
   <span class="ansigreen">    210</span>             _cache_it<span class="ansiblue">(</span>data<span class="ansiblue">,</span> cache_path<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">urlopen</span><span class="ansiblue">(url, data, timeout, cafile, capath, cadefault, context)</span>
   <span class="ansigreen">    152</span>     <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    153</span>         opener <span class="ansiblue">=</span> _opener<span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 154</span><span class="ansired">     </span><span class="ansigreen">return</span> opener<span class="ansiblue">.</span>open<span class="ansiblue">(</span>url<span class="ansiblue">,</span> data<span class="ansiblue">,</span> timeout<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    155</span> <span class="ansiblue"></span>
   <span class="ansigreen">    156</span> <span class="ansigreen">def</span> install_opener<span class="ansiblue">(</span>opener<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">open</span><span class="ansiblue">(self, fullurl, data, timeout)</span>
   <span class="ansigreen">    427</span>             req <span class="ansiblue">=</span> meth<span class="ansiblue">(</span>req<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    428</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 429</span><span class="ansired">         </span>response <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_open<span class="ansiblue">(</span>req<span class="ansiblue">,</span> data<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    430</span> <span class="ansiblue"></span>
   <span class="ansigreen">    431</span>         <span class="ansired"># post-process response</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">_open</span><span class="ansiblue">(self, req, data)</span>
   <span class="ansigreen">    445</span>         protocol <span class="ansiblue">=</span> req<span class="ansiblue">.</span>get_type<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    446</span>         result = self._call_chain(self.handle_open, protocol, protocol +
   <span class="ansigreen">--&gt; 447</span><span class="ansired">                                   &apos;_open&apos;, req)
   </span><span class="ansigreen">    448</span>         <span class="ansigreen">if</span> result<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    449</span>             <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">_call_chain</span><span class="ansiblue">(self, chain, kind, meth_name, *args)</span>
   <span class="ansigreen">    405</span>             func <span class="ansiblue">=</span> getattr<span class="ansiblue">(</span>handler<span class="ansiblue">,</span> meth_name<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    406</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 407</span><span class="ansired">             </span>result <span class="ansiblue">=</span> func<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    408</span>             <span class="ansigreen">if</span> result <span class="ansigreen">is</span> <span class="ansigreen">not</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    409</span>                 <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">https_open</span><span class="ansiblue">(self, req)</span>
   <span class="ansigreen">   1239</span>         <span class="ansigreen">def</span> https_open<span class="ansiblue">(</span>self<span class="ansiblue">,</span> req<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1240</span>             return self.do_open(httplib.HTTPSConnection, req,
   <span class="ansigreen">-&gt; 1241</span><span class="ansired">                 context=self._context)
   </span><span class="ansigreen">   1242</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1243</span>         https_request <span class="ansiblue">=</span> AbstractHTTPHandler<span class="ansiblue">.</span>do_request_<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/urllib2.pyc</span> in <span class="ansicyan">do_open</span><span class="ansiblue">(self, http_class, req, **http_conn_args)</span>
   <span class="ansigreen">   1196</span>         <span class="ansigreen">except</span> socket<span class="ansiblue">.</span>error<span class="ansiblue">,</span> err<span class="ansiblue">:</span> <span class="ansired"># XXX what error?</span><span class="ansiblue"></span>
   <span class="ansigreen">   1197</span>             h<span class="ansiblue">.</span>close<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1198</span><span class="ansired">             </span><span class="ansigreen">raise</span> URLError<span class="ansiblue">(</span>err<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1199</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1200</span>             <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansired">URLError</span>: &lt;urlopen error [Errno -2] Name or service not known&gt;</pre>
   </div>
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   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[48]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">matplotlib.patches</span> <span class="kn">import</span> <span class="n">Ellipse</span>
   <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s1">&#39;log(Temp)&#39;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s1">&#39;log(Light)&#39;</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s1">&#39;Hertzsprung-Russell Diagram of Star Cluster CYG OB1&#39;</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="n">dta</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
   <span class="c1"># highlight outliers</span>
   <span class="n">e</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">((</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="o">.</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">25</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">add_patch</span><span class="p">(</span><span class="n">e</span><span class="p">);</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="s1">&#39;Red giants&#39;</span><span class="p">,</span> <span class="n">xy</span><span class="o">=</span><span class="p">(</span><span class="mf">3.6</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">xytext</span><span class="o">=</span><span class="p">(</span><span class="mf">3.8</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span>
               <span class="n">arrowprops</span><span class="o">=</span><span class="nb">dict</span><span class="p">(</span><span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">shrink</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span>
               <span class="n">horizontalalignment</span><span class="o">=</span><span class="s1">&#39;left&#39;</span><span class="p">,</span> <span class="n">verticalalignment</span><span class="o">=</span><span class="s1">&#39;bottom&#39;</span><span class="p">,</span>
               <span class="n">clip_on</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="c1"># clip to the axes bounding box</span>
               <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span>
        <span class="p">)</span>
   <span class="c1"># annotate these with their index</span>
   <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">row</span> <span class="ow">in</span> <span class="n">dta</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="n">dta</span><span class="p">[</span><span class="s1">&#39;log.Te&#39;</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mf">3.8</span><span class="p">]</span><span class="o">.</span><span class="n">iterrows</span><span class="p">():</span>
       <span class="n">ax</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">row</span><span class="p">,</span> <span class="n">row</span> <span class="o">+</span> <span class="o">.</span><span class="mo">01</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
   <span class="n">xlim</span><span class="p">,</span> <span class="n">ylim</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">get_xlim</span><span class="p">(),</span> <span class="n">ax</span><span class="o">.</span><span class="n">get_ylim</span><span class="p">()</span>
   </pre></div>
   
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   <div class="output_area"><div class="prompt"></div>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">TypeError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-144-d061dc5cc6d1&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      2</span> fig <span class="ansiblue">=</span> plt<span class="ansiblue">.</span>figure<span class="ansiblue">(</span>figsize<span class="ansiblue">=</span><span class="ansiblue">(</span><span class="ansicyan">12</span><span class="ansiblue">,</span><span class="ansicyan">8</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> ax <span class="ansiblue">=</span> fig<span class="ansiblue">.</span>add_subplot<span class="ansiblue">(</span><span class="ansicyan">111</span><span class="ansiblue">,</span> xlabel<span class="ansiblue">=</span><span class="ansiblue">&apos;log(Temp)&apos;</span><span class="ansiblue">,</span> ylabel<span class="ansiblue">=</span><span class="ansiblue">&apos;log(Light)&apos;</span><span class="ansiblue">,</span> title<span class="ansiblue">=</span><span class="ansiblue">&apos;Hertzsprung-Russell Diagram of Star Cluster CYG OB1&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 4</span><span class="ansired"> </span>ax<span class="ansiblue">.</span>scatter<span class="ansiblue">(</span><span class="ansiblue">*</span>dta<span class="ansiblue">.</span>values<span class="ansiblue">.</span>T<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> <span class="ansired"># highlight outliers</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">      6</span> e <span class="ansiblue">=</span> Ellipse<span class="ansiblue">(</span><span class="ansiblue">(</span><span class="ansicyan">3.5</span><span class="ansiblue">,</span> <span class="ansicyan">6</span><span class="ansiblue">)</span><span class="ansiblue">,</span> <span class="ansicyan">.2</span><span class="ansiblue">,</span> <span class="ansicyan">1</span><span class="ansiblue">,</span> alpha<span class="ansiblue">=</span><span class="ansicyan">.25</span><span class="ansiblue">,</span> color<span class="ansiblue">=</span><span class="ansiblue">&apos;r&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/__init__.pyc</span> in <span class="ansicyan">inner</span><span class="ansiblue">(ax, *args, **kwargs)</span>
   <span class="ansigreen">   1812</span>                     warnings.warn(msg % (label_namer, func.__name__),
   <span class="ansigreen">   1813</span>                                   RuntimeWarning, stacklevel=2)
   <span class="ansigreen">-&gt; 1814</span><span class="ansired">             </span><span class="ansigreen">return</span> func<span class="ansiblue">(</span>ax<span class="ansiblue">,</span> <span class="ansiblue">*</span>args<span class="ansiblue">,</span> <span class="ansiblue">**</span>kwargs<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1815</span>         pre_doc <span class="ansiblue">=</span> inner<span class="ansiblue">.</span>__doc__<span class="ansiblue"></span>
   <span class="ansigreen">   1816</span>         <span class="ansigreen">if</span> pre_doc <span class="ansigreen">is</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansired">TypeError</span>: scatter() takes at most 14 arguments (24 given)</pre>
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   "
   >
   </div>
   
   </div>
   
   </div>
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   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[49]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
   <span class="n">Image</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s1">&#39;star_diagram.png&#39;</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   <div class="output_wrapper">
   <div class="output">
   
   
   <div class="output_area"><div class="prompt output_prompt">Out[49]:</div>
   
   
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   <div class="prompt input_prompt">In&nbsp;[50]:</div>
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   <div class="highlight"><pre><span class="n">y</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;log.light&#39;</span><span class="p">]</span>
   <span class="n">X</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">dta</span><span class="p">[</span><span class="s1">&#39;log.Te&#39;</span><span class="p">],</span> <span class="n">prepend</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="n">ols_model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">abline_plot</span><span class="p">(</span><span class="n">model_results</span><span class="o">=</span><span class="n">ols_model</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
   </pre></div>
   
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   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">KeyError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-146-6c7349bbf59f&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>y <span class="ansiblue">=</span> dta<span class="ansiblue">[</span><span class="ansiblue">&apos;log.light&apos;</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> X <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>add_constant<span class="ansiblue">(</span>dta<span class="ansiblue">[</span><span class="ansiblue">&apos;log.Te&apos;</span><span class="ansiblue">]</span><span class="ansiblue">,</span> prepend<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> ols_model <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>OLS<span class="ansiblue">(</span>y<span class="ansiblue">,</span> X<span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> abline_plot<span class="ansiblue">(</span>model_results<span class="ansiblue">=</span>ols_model<span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">__getitem__</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1967</span>             <span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_multilevel<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1968</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1969</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_column<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1970</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1971</span>     <span class="ansigreen">def</span> _getitem_column<span class="ansiblue">(</span>self<span class="ansiblue">,</span> key<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">_getitem_column</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1974</span>         <span class="ansired"># get column</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">   1975</span>         <span class="ansigreen">if</span> self<span class="ansiblue">.</span>columns<span class="ansiblue">.</span>is_unique<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1976</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_get_item_cache<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1977</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1978</span>         <span class="ansired"># duplicate columns &amp; possible reduce dimensionality</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/generic.pyc</span> in <span class="ansicyan">_get_item_cache</span><span class="ansiblue">(self, item)</span>
   <span class="ansigreen">   1089</span>         res <span class="ansiblue">=</span> cache<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1090</span>         <span class="ansigreen">if</span> res <span class="ansigreen">is</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1091</span><span class="ansired">             </span>values <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_data<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1092</span>             res <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_box_item_values<span class="ansiblue">(</span>item<span class="ansiblue">,</span> values<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1093</span>             cache<span class="ansiblue">[</span>item<span class="ansiblue">]</span> <span class="ansiblue">=</span> res<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/internals.pyc</span> in <span class="ansicyan">get</span><span class="ansiblue">(self, item, fastpath)</span>
   <span class="ansigreen">   3209</span> <span class="ansiblue"></span>
   <span class="ansigreen">   3210</span>             <span class="ansigreen">if</span> <span class="ansigreen">not</span> isnull<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 3211</span><span class="ansired">                 </span>loc <span class="ansiblue">=</span> self<span class="ansiblue">.</span>items<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   3212</span>             <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   3213</span>                 indexer <span class="ansiblue">=</span> np<span class="ansiblue">.</span>arange<span class="ansiblue">(</span>len<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">[</span>isnull<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/index.pyc</span> in <span class="ansicyan">get_loc</span><span class="ansiblue">(self, key, method, tolerance)</span>
   <span class="ansigreen">   1757</span>                                  &apos;backfill or nearest lookups&apos;)
   <span class="ansigreen">   1758</span>             key <span class="ansiblue">=</span> _values_from_object<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1759</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_engine<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1760</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1761</span>         indexer = self.get_indexer([key], method=method,
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3979)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3843)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12265)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12216)</span><span class="ansiblue">()</span>
   
   <span class="ansired">KeyError</span>: &apos;log.light&apos;</pre>
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   <div class="highlight"><pre><span class="n">rlm_mod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">RLM</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">sm</span><span class="o">.</span><span class="n">robust</span><span class="o">.</span><span class="n">norms</span><span class="o">.</span><span class="n">TrimmedMean</span><span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="p">))</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">abline_plot</span><span class="p">(</span><span class="n">model_results</span><span class="o">=</span><span class="n">rlm_mod</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">ValueError</span>                                Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-147-9210cc7d7045&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> rlm_mod <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>RLM<span class="ansiblue">(</span>y<span class="ansiblue">,</span> X<span class="ansiblue">,</span> sm<span class="ansiblue">.</span>robust<span class="ansiblue">.</span>norms<span class="ansiblue">.</span>TrimmedMean<span class="ansiblue">(</span><span class="ansicyan">.5</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>abline_plot<span class="ansiblue">(</span>model_results<span class="ansiblue">=</span>rlm_mod<span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax<span class="ansiblue">,</span> color<span class="ansiblue">=</span><span class="ansiblue">&apos;red&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/build/statsmodels-ungkPp/statsmodels-0.6.1/debian/python-statsmodels/usr/lib/python2.7/dist-packages/statsmodels/graphics/regressionplots.pyc</span> in <span class="ansicyan">abline_plot</span><span class="ansiblue">(intercept, slope, horiz, vert, model_results, ax, **kwargs)</span>
   <span class="ansigreen">    654</span> <span class="ansiblue"></span>
   <span class="ansigreen">    655</span>     <span class="ansigreen">if</span> model_results<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 656</span><span class="ansired">         </span>intercept<span class="ansiblue">,</span> slope <span class="ansiblue">=</span> model_results<span class="ansiblue">.</span>params<span class="ansiblue"></span>
   <span class="ansigreen">    657</span>         <span class="ansigreen">if</span> x <span class="ansigreen">is</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    658</span>             x = [model_results.model.exog[:, 1].min(),
   
   <span class="ansired">ValueError</span>: too many values to unpack</pre>
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   <ul>
   <li>Why? Because M-estimators are not robust to leverage points.</li>
   </ul>
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   <div class="highlight"><pre><span class="n">infl</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">get_influence</span><span class="p">()</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-148-5ff696623192&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>infl <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>get_influence<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;get_influence&apos;</pre>
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   <div class="highlight"><pre><span class="n">h_bar</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">ols_model</span><span class="o">.</span><span class="n">df_model</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span><span class="o">/</span><span class="n">ols_model</span><span class="o">.</span><span class="n">nobs</span>
   <span class="n">hat_diag</span> <span class="o">=</span> <span class="n">infl</span><span class="o">.</span><span class="n">summary_frame</span><span class="p">()[</span><span class="s1">&#39;hat_diag&#39;</span><span class="p">]</span>
   <span class="n">hat_diag</span><span class="o">.</span><span class="n">ix</span><span class="p">[</span><span class="n">hat_diag</span> <span class="o">&gt;</span> <span class="n">h_bar</span><span class="p">]</span>
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   Series([], Name: hat_diag, dtype: float64)
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   <div class="highlight"><pre><span class="n">sidak2</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">outlier_test</span><span class="p">(</span><span class="s1">&#39;sidak&#39;</span><span class="p">)</span>
   <span class="n">sidak2</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="s1">&#39;unadj_p&#39;</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">sidak2</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-150-a6c575f5e012&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>sidak2 <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>outlier_test<span class="ansiblue">(</span><span class="ansiblue">&apos;sidak&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> sidak2<span class="ansiblue">.</span>sort<span class="ansiblue">(</span><span class="ansiblue">&apos;unadj_p&apos;</span><span class="ansiblue">,</span> inplace<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>sidak2<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;outlier_test&apos;</pre>
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   <div class="highlight"><pre><span class="n">fdr2</span> <span class="o">=</span> <span class="n">ols_model</span><span class="o">.</span><span class="n">outlier_test</span><span class="p">(</span><span class="s1">&#39;fdr_bh&#39;</span><span class="p">)</span>
   <span class="n">fdr2</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="s1">&#39;unadj_p&#39;</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">fdr2</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">AttributeError</span>                            Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-151-cf26bbbd14f6&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>fdr2 <span class="ansiblue">=</span> ols_model<span class="ansiblue">.</span>outlier_test<span class="ansiblue">(</span><span class="ansiblue">&apos;fdr_bh&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> fdr2<span class="ansiblue">.</span>sort<span class="ansiblue">(</span><span class="ansiblue">&apos;unadj_p&apos;</span><span class="ansiblue">,</span> inplace<span class="ansiblue">=</span>True<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> <span class="ansigreen">print</span><span class="ansiblue">(</span>fdr2<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">AttributeError</span>: &apos;OLS&apos; object has no attribute &apos;outlier_test&apos;</pre>
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   <li>Let&#39;s delete that line</li>
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   <div class="highlight"><pre><span class="k">del</span> <span class="n">ax</span><span class="o">.</span><span class="n">lines</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">IndexError</span>                                Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-152-346dae874d03&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">del</span> ax<span class="ansiblue">.</span>lines<span class="ansiblue">[</span><span class="ansiblue">-</span><span class="ansicyan">1</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansired">IndexError</span>: list assignment index out of range</pre>
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   <div class="highlight"><pre><span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
   <span class="n">weights</span><span class="p">[</span><span class="n">X</span><span class="p">[</span><span class="n">X</span><span class="p">[</span><span class="s1">&#39;log.Te&#39;</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mf">3.8</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">values</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
   <span class="n">wls_model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">WLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">weights</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   <span class="n">abline_plot</span><span class="p">(</span><span class="n">model_results</span><span class="o">=</span><span class="n">wls_model</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;green&#39;</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">KeyError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-153-e3ec53a40864&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> weights <span class="ansiblue">=</span> np<span class="ansiblue">.</span>ones<span class="ansiblue">(</span>len<span class="ansiblue">(</span>X<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>weights<span class="ansiblue">[</span>X<span class="ansiblue">[</span>X<span class="ansiblue">[</span><span class="ansiblue">&apos;log.Te&apos;</span><span class="ansiblue">]</span> <span class="ansiblue">&lt;</span> <span class="ansicyan">3.8</span><span class="ansiblue">]</span><span class="ansiblue">.</span>index<span class="ansiblue">.</span>values <span class="ansiblue">-</span> <span class="ansicyan">1</span><span class="ansiblue">]</span> <span class="ansiblue">=</span> <span class="ansicyan">0</span><span class="ansiblue"></span>
   <span class="ansigreen">      3</span> wls_model <span class="ansiblue">=</span> sm<span class="ansiblue">.</span>WLS<span class="ansiblue">(</span>y<span class="ansiblue">,</span> X<span class="ansiblue">,</span> weights<span class="ansiblue">=</span>weights<span class="ansiblue">)</span><span class="ansiblue">.</span>fit<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> abline_plot<span class="ansiblue">(</span>model_results<span class="ansiblue">=</span>wls_model<span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax<span class="ansiblue">,</span> color<span class="ansiblue">=</span><span class="ansiblue">&apos;green&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">__getitem__</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1967</span>             <span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_multilevel<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1968</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1969</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_column<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1970</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1971</span>     <span class="ansigreen">def</span> _getitem_column<span class="ansiblue">(</span>self<span class="ansiblue">,</span> key<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">_getitem_column</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1974</span>         <span class="ansired"># get column</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">   1975</span>         <span class="ansigreen">if</span> self<span class="ansiblue">.</span>columns<span class="ansiblue">.</span>is_unique<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1976</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_get_item_cache<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1977</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1978</span>         <span class="ansired"># duplicate columns &amp; possible reduce dimensionality</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/generic.pyc</span> in <span class="ansicyan">_get_item_cache</span><span class="ansiblue">(self, item)</span>
   <span class="ansigreen">   1089</span>         res <span class="ansiblue">=</span> cache<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1090</span>         <span class="ansigreen">if</span> res <span class="ansigreen">is</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1091</span><span class="ansired">             </span>values <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_data<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1092</span>             res <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_box_item_values<span class="ansiblue">(</span>item<span class="ansiblue">,</span> values<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1093</span>             cache<span class="ansiblue">[</span>item<span class="ansiblue">]</span> <span class="ansiblue">=</span> res<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/internals.pyc</span> in <span class="ansicyan">get</span><span class="ansiblue">(self, item, fastpath)</span>
   <span class="ansigreen">   3209</span> <span class="ansiblue"></span>
   <span class="ansigreen">   3210</span>             <span class="ansigreen">if</span> <span class="ansigreen">not</span> isnull<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 3211</span><span class="ansired">                 </span>loc <span class="ansiblue">=</span> self<span class="ansiblue">.</span>items<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   3212</span>             <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   3213</span>                 indexer <span class="ansiblue">=</span> np<span class="ansiblue">.</span>arange<span class="ansiblue">(</span>len<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">[</span>isnull<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/index.pyc</span> in <span class="ansicyan">get_loc</span><span class="ansiblue">(self, key, method, tolerance)</span>
   <span class="ansigreen">   1757</span>                                  &apos;backfill or nearest lookups&apos;)
   <span class="ansigreen">   1758</span>             key <span class="ansiblue">=</span> _values_from_object<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1759</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_engine<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1760</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1761</span>         indexer = self.get_indexer([key], method=method,
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3979)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3843)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12265)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12216)</span><span class="ansiblue">()</span>
   
   <span class="ansired">KeyError</span>: &apos;log.Te&apos;</pre>
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   <ul>
   <li>MM estimators are good for this type of problem, unfortunately, we don&#39;t yet have these yet. </li>
   <li>It&#39;s being worked on, but it gives a good excuse to look at the R cell magics in the notebook.</li>
   </ul>
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   <div class="prompt input_prompt">In&nbsp;[58]:</div>
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   <div class="highlight"><pre><span class="n">yy</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">values</span><span class="p">[:,</span><span class="bp">None</span><span class="p">]</span>
   <span class="n">xx</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="s1">&#39;log.Te&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">[:,</span><span class="bp">None</span><span class="p">]</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">KeyError</span>                                  Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-154-a5672cb240e0&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">      1</span> yy <span class="ansiblue">=</span> y<span class="ansiblue">.</span>values<span class="ansiblue">[</span><span class="ansiblue">:</span><span class="ansiblue">,</span>None<span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">----&gt; 2</span><span class="ansired"> </span>xx <span class="ansiblue">=</span> X<span class="ansiblue">[</span><span class="ansiblue">&apos;log.Te&apos;</span><span class="ansiblue">]</span><span class="ansiblue">.</span>values<span class="ansiblue">[</span><span class="ansiblue">:</span><span class="ansiblue">,</span>None<span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">__getitem__</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1967</span>             <span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_multilevel<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1968</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1969</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_getitem_column<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1970</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1971</span>     <span class="ansigreen">def</span> _getitem_column<span class="ansiblue">(</span>self<span class="ansiblue">,</span> key<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/frame.pyc</span> in <span class="ansicyan">_getitem_column</span><span class="ansiblue">(self, key)</span>
   <span class="ansigreen">   1974</span>         <span class="ansired"># get column</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">   1975</span>         <span class="ansigreen">if</span> self<span class="ansiblue">.</span>columns<span class="ansiblue">.</span>is_unique<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1976</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_get_item_cache<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1977</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1978</span>         <span class="ansired"># duplicate columns &amp; possible reduce dimensionality</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/generic.pyc</span> in <span class="ansicyan">_get_item_cache</span><span class="ansiblue">(self, item)</span>
   <span class="ansigreen">   1089</span>         res <span class="ansiblue">=</span> cache<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1090</span>         <span class="ansigreen">if</span> res <span class="ansigreen">is</span> None<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1091</span><span class="ansired">             </span>values <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_data<span class="ansiblue">.</span>get<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1092</span>             res <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_box_item_values<span class="ansiblue">(</span>item<span class="ansiblue">,</span> values<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1093</span>             cache<span class="ansiblue">[</span>item<span class="ansiblue">]</span> <span class="ansiblue">=</span> res<span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/internals.pyc</span> in <span class="ansicyan">get</span><span class="ansiblue">(self, item, fastpath)</span>
   <span class="ansigreen">   3209</span> <span class="ansiblue"></span>
   <span class="ansigreen">   3210</span>             <span class="ansigreen">if</span> <span class="ansigreen">not</span> isnull<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 3211</span><span class="ansired">                 </span>loc <span class="ansiblue">=</span> self<span class="ansiblue">.</span>items<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>item<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   3212</span>             <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   3213</span>                 indexer <span class="ansiblue">=</span> np<span class="ansiblue">.</span>arange<span class="ansiblue">(</span>len<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue">[</span>isnull<span class="ansiblue">(</span>self<span class="ansiblue">.</span>items<span class="ansiblue">)</span><span class="ansiblue">]</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/core/index.pyc</span> in <span class="ansicyan">get_loc</span><span class="ansiblue">(self, key, method, tolerance)</span>
   <span class="ansigreen">   1757</span>                                  &apos;backfill or nearest lookups&apos;)
   <span class="ansigreen">   1758</span>             key <span class="ansiblue">=</span> _values_from_object<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1759</span><span class="ansired">             </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>_engine<span class="ansiblue">.</span>get_loc<span class="ansiblue">(</span>key<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1760</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1761</span>         indexer = self.get_indexer([key], method=method,
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3979)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/index.so</span> in <span class="ansicyan">pandas.index.IndexEngine.get_loc (pandas/index.c:3843)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12265)</span><span class="ansiblue">()</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/pandas/hashtable.so</span> in <span class="ansicyan">pandas.hashtable.PyObjectHashTable.get_item (pandas/hashtable.c:12216)</span><span class="ansiblue">()</span>
   
   <span class="ansired">KeyError</span>: &apos;log.Te&apos;</pre>
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   <div class="prompt input_prompt">In&nbsp;[59]:</div>
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   <div class="highlight"><pre><span class="o">%</span><span class="k">load_ext</span> <span class="n">rmagic</span>
   
   <span class="o">%</span><span class="k">R</span> <span class="n">library</span><span class="p">(</span><span class="n">robustbase</span><span class="p">)</span>
   <span class="o">%</span><span class="k">Rpush</span> <span class="n">yy</span> <span class="n">xx</span>
   <span class="o">%</span><span class="k">R</span> <span class="n">mod</span> <span class="o">&lt;-</span> <span class="n">lmrob</span><span class="p">(</span><span class="n">yy</span> <span class="o">~</span> <span class="n">xx</span><span class="p">);</span>
   <span class="o">%</span><span class="k">R</span> <span class="n">params</span> <span class="o">&lt;-</span> <span class="n">mod</span><span class="err">$</span><span class="n">coefficients</span><span class="p">;</span>
   <span class="o">%</span><span class="k">Rpull</span> <span class="n">params</span>
   </pre></div>
   
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">ImportError</span>                               Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-155-53d6967d0401&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>get_ipython<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>magic<span class="ansiblue">(</span><span class="ansiblue">u&apos;load_ext rmagic&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      2</span> <span class="ansiblue"></span>
   <span class="ansigreen">      3</span> get_ipython<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>magic<span class="ansiblue">(</span><span class="ansiblue">u&apos;R library(robustbase)&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      4</span> get_ipython<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>magic<span class="ansiblue">(</span><span class="ansiblue">u&apos;Rpush yy xx&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">      5</span> get_ipython<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>magic<span class="ansiblue">(</span><span class="ansiblue">u&apos;R mod &lt;- lmrob(yy ~ xx);&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/core/interactiveshell.pyc</span> in <span class="ansicyan">magic</span><span class="ansiblue">(self, arg_s)</span>
   <span class="ansigreen">   2203</span>         magic_name<span class="ansiblue">,</span> _<span class="ansiblue">,</span> magic_arg_s <span class="ansiblue">=</span> arg_s<span class="ansiblue">.</span>partition<span class="ansiblue">(</span><span class="ansiblue">&apos; &apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   2204</span>         magic_name <span class="ansiblue">=</span> magic_name<span class="ansiblue">.</span>lstrip<span class="ansiblue">(</span>prefilter<span class="ansiblue">.</span>ESC_MAGIC<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2205</span><span class="ansired">         </span><span class="ansigreen">return</span> self<span class="ansiblue">.</span>run_line_magic<span class="ansiblue">(</span>magic_name<span class="ansiblue">,</span> magic_arg_s<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   2206</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2207</span>     <span class="ansired">#-------------------------------------------------------------------------</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/core/interactiveshell.pyc</span> in <span class="ansicyan">run_line_magic</span><span class="ansiblue">(self, magic_name, line)</span>
   <span class="ansigreen">   2124</span>                 kwargs<span class="ansiblue">[</span><span class="ansiblue">&apos;local_ns&apos;</span><span class="ansiblue">]</span> <span class="ansiblue">=</span> sys<span class="ansiblue">.</span>_getframe<span class="ansiblue">(</span>stack_depth<span class="ansiblue">)</span><span class="ansiblue">.</span>f_locals<span class="ansiblue"></span>
   <span class="ansigreen">   2125</span>             <span class="ansigreen">with</span> self<span class="ansiblue">.</span>builtin_trap<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2126</span><span class="ansired">                 </span>result <span class="ansiblue">=</span> fn<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">,</span><span class="ansiblue">**</span>kwargs<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   2127</span>             <span class="ansigreen">return</span> result<span class="ansiblue"></span>
   <span class="ansigreen">   2128</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">&lt;decorator-gen-64&gt;</span> in <span class="ansicyan">load_ext</span><span class="ansiblue">(self, module_str)</span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/core/magic.pyc</span> in <span class="ansicyan">&lt;lambda&gt;</span><span class="ansiblue">(f, *a, **k)</span>
   <span class="ansigreen">    191</span>     <span class="ansired"># but it&apos;s overkill for just that one bit of state.</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">    192</span>     <span class="ansigreen">def</span> magic_deco<span class="ansiblue">(</span>arg<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 193</span><span class="ansired">         </span>call <span class="ansiblue">=</span> <span class="ansigreen">lambda</span> f<span class="ansiblue">,</span> <span class="ansiblue">*</span>a<span class="ansiblue">,</span> <span class="ansiblue">**</span>k<span class="ansiblue">:</span> f<span class="ansiblue">(</span><span class="ansiblue">*</span>a<span class="ansiblue">,</span> <span class="ansiblue">**</span>k<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    194</span> <span class="ansiblue"></span>
   <span class="ansigreen">    195</span>         <span class="ansigreen">if</span> callable<span class="ansiblue">(</span>arg<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/core/magics/extension.pyc</span> in <span class="ansicyan">load_ext</span><span class="ansiblue">(self, module_str)</span>
   <span class="ansigreen">     61</span>         <span class="ansigreen">if</span> <span class="ansigreen">not</span> module_str<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">     62</span>             <span class="ansigreen">raise</span> UsageError<span class="ansiblue">(</span><span class="ansiblue">&apos;Missing module name.&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 63</span><span class="ansired">         </span>res <span class="ansiblue">=</span> self<span class="ansiblue">.</span>shell<span class="ansiblue">.</span>extension_manager<span class="ansiblue">.</span>load_extension<span class="ansiblue">(</span>module_str<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     64</span> <span class="ansiblue"></span>
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   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/core/extensions.pyc</span> in <span class="ansicyan">load_extension</span><span class="ansiblue">(self, module_str)</span>
   <span class="ansigreen">     96</span>             <span class="ansigreen">if</span> module_str <span class="ansigreen">not</span> <span class="ansigreen">in</span> sys<span class="ansiblue">.</span>modules<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">     97</span>                 <span class="ansigreen">with</span> prepended_to_syspath<span class="ansiblue">(</span>self<span class="ansiblue">.</span>ipython_extension_dir<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 98</span><span class="ansired">                     </span>__import__<span class="ansiblue">(</span>module_str<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     99</span>             mod <span class="ansiblue">=</span> sys<span class="ansiblue">.</span>modules<span class="ansiblue">[</span>module_str<span class="ansiblue">]</span><span class="ansiblue"></span>
   <span class="ansigreen">    100</span>             <span class="ansigreen">if</span> self<span class="ansiblue">.</span>_call_load_ipython_extension<span class="ansiblue">(</span>mod<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/IPython/extensions/rmagic.py</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">     54</span> <span class="ansigreen">import</span> numpy <span class="ansigreen">as</span> np<span class="ansiblue"></span>
   <span class="ansigreen">     55</span> <span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 56</span><span class="ansired"> </span><span class="ansigreen">import</span> rpy2<span class="ansiblue">.</span>rinterface <span class="ansigreen">as</span> ri<span class="ansiblue"></span>
   <span class="ansigreen">     57</span> <span class="ansigreen">import</span> rpy2<span class="ansiblue">.</span>robjects <span class="ansigreen">as</span> ro<span class="ansiblue"></span>
   <span class="ansigreen">     58</span> <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansired">ImportError</span>: No module named rpy2.rinterface</pre>
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   <div class="highlight"><pre><span class="o">%</span><span class="k">R</span> <span class="k">print</span><span class="p">(</span><span class="n">mod</span><span class="p">)</span>
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   ERROR: Line magic function &#96;%R&#96; not found.
   
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   <div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">params</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-157-73ac4b936803&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span><span class="ansigreen">print</span><span class="ansiblue">(</span>params<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;params&apos; is not defined</pre>
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   <div class="highlight"><pre><span class="n">abline_plot</span><span class="p">(</span><span class="n">intercept</span><span class="o">=</span><span class="n">params</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">slope</span><span class="o">=</span><span class="n">params</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;green&#39;</span><span class="p">)</span>
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   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">NameError</span>                                 Traceback (most recent call last)
   <span class="ansigreen">&lt;ipython-input-158-e1a9f35b3320&gt;</span> in <span class="ansicyan">&lt;module&gt;</span><span class="ansiblue">()</span>
   <span class="ansigreen">----&gt; 1</span><span class="ansired"> </span>abline_plot<span class="ansiblue">(</span>intercept<span class="ansiblue">=</span>params<span class="ansiblue">[</span><span class="ansicyan">0</span><span class="ansiblue">]</span><span class="ansiblue">,</span> slope<span class="ansiblue">=</span>params<span class="ansiblue">[</span><span class="ansicyan">1</span><span class="ansiblue">]</span><span class="ansiblue">,</span> ax<span class="ansiblue">=</span>ax<span class="ansiblue">,</span> color<span class="ansiblue">=</span><span class="ansiblue">&apos;green&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">NameError</span>: name &apos;params&apos; is not defined</pre>
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   <h3 id="Exercise:-Breakdown-points-of-M-estimator">Exercise: Breakdown points of M-estimator<a class="anchor-link" href="#Exercise:-Breakdown-points-of-M-estimator">&#182;</a></h3>
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">12345</span><span class="p">)</span>
   <span class="n">nobs</span> <span class="o">=</span> <span class="mi">200</span>
   <span class="n">beta_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">])</span>
   <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">20</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">nobs</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">beta_true</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
   <span class="c1"># stack a constant in front</span>
   <span class="n">X</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">prepend</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="c1"># np.c_[np.ones(nobs), X]</span>
   <span class="n">mc_iter</span> <span class="o">=</span> <span class="mi">500</span>
   <span class="n">contaminate</span> <span class="o">=</span> <span class="o">.</span><span class="mi">25</span> <span class="c1"># percentage of response variables to contaminate</span>
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   <div class="highlight"><pre><span class="n">all_betas</span> <span class="o">=</span> <span class="p">[]</span>
   <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">mc_iter</span><span class="p">):</span>
       <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">beta_true</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
       <span class="n">random_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nobs</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="n">contaminate</span> <span class="o">*</span> <span class="n">nobs</span><span class="p">))</span>
       <span class="n">y</span><span class="p">[</span><span class="n">random_idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">750</span><span class="p">,</span> <span class="mi">750</span><span class="p">)</span>
       <span class="n">beta_hat</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">RLM</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span><span class="o">.</span><span class="n">params</span>
       <span class="n">all_betas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">beta_hat</span><span class="p">)</span>
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   <div class="highlight"><pre><span class="n">all_betas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">all_betas</span><span class="p">)</span>
   <span class="n">se_loss</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span> <span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
   <span class="n">se_beta</span> <span class="o">=</span> <span class="n">lmap</span><span class="p">(</span><span class="n">se_loss</span><span class="p">,</span> <span class="n">all_betas</span> <span class="o">-</span> <span class="n">beta_true</span><span class="p">)</span>
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   <h4 id="Squared-error-loss">Squared error loss<a class="anchor-link" href="#Squared-error-loss">&#182;</a></h4>
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   <div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">se_beta</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
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   0.44502948730686182
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   <div class="highlight"><pre><span class="n">all_betas</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
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   array([ 2.99711706,  0.99898147,  2.49909344,  2.99712918, -3.99626521])
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   <div class="highlight"><pre><span class="n">beta_true</span>
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   array([ 3. ,  1. ,  2.5,  3. , -4. ])
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   <div class="highlight"><pre><span class="n">se_loss</span><span class="p">(</span><span class="n">all_betas</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">-</span> <span class="n">beta_true</span><span class="p">)</span>
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   3.2360913286754188e-05
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