Robust Linear Models
====================


.. _robust_models_0_notebook:

`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/robust_models_0.ipynb>`_

.. raw:: html

   
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[1]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <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">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
   <span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="kn">as</span> <span class="nn">sm</span>
   <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
   <span class="kn">from</span> <span class="nn">statsmodels.sandbox.regression.predstd</span> <span class="kn">import</span> <span class="n">wls_prediction_std</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <h2 id="estimation">Estimation</h2>
   <p>Load data:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[2]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">data</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">stackloss</span><span class="o">.</span><span class="n">load</span><span class="p">()</span>
   <span class="n">data</span><span class="o">.</span><span class="n">exog</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">data</span><span class="o">.</span><span class="n">exog</span><span class="p">)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Huber&#39;s T norm with the (default) median absolute deviation scaling</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[3]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">huber_t</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">data</span><span class="o">.</span><span class="n">endog</span><span class="p">,</span> <span class="n">data</span><span class="o">.</span><span class="n">exog</span><span class="p">,</span> <span class="n">M</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><span class="o">.</span><span class="n">HuberT</span><span class="p">())</span>
   <span class="n">hub_results</span> <span class="o">=</span> <span class="n">huber_t</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">hub_results</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">hub_results</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">hub_results</span><span class="o">.</span><span class="n">summary</span><span class="p">(</span><span class="n">yname</span><span class="o">=</span><span class="s1">&#39;y&#39;</span><span class="p">,</span>
               <span class="n">xname</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;var_</span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">i</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="nb">len</span><span class="p">(</span><span class="n">hub_results</span><span class="o">.</span><span class="n">params</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_stream output_stdout output_text">
   <pre>
   [-41.02649835   0.82938433   0.92606597  -0.12784672]
   [ 9.79189854  0.11100521  0.30293016  0.12864961]
                       Robust linear Model Regression Results                    
   ==============================================================================
   Dep. Variable:                      y   No. Observations:                   21
   Model:                            RLM   Df Residuals:                       17
   Method:                          IRLS   Df Model:                            3
   Norm:                          HuberT                                         
   Scale Est.:                       mad                                         
   Cov Type:                          H1                                         
   Date:                Wed, 27 Apr 2016                                         
   Time:                        23:32:50                                         
   No. Iterations:                    19                                         
   ==============================================================================
                    coef    std err          z      P&gt;|z|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   var_0        -41.0265      9.792     -4.190      0.000       -60.218   -21.835
   var_1          0.8294      0.111      7.472      0.000         0.612     1.047
   var_2          0.9261      0.303      3.057      0.002         0.332     1.520
   var_3         -0.1278      0.129     -0.994      0.320        -0.380     0.124
   ==============================================================================
   
   If the model instance has been used for another fit with different fit
   parameters, then the fit options might not be the correct ones anymore .
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Huber&#39;s T norm with &#39;H2&#39; covariance matrix</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[4]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">hub_results2</span> <span class="o">=</span> <span class="n">huber_t</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">cov</span><span class="o">=</span><span class="s2">&quot;H2&quot;</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">hub_results2</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">hub_results2</span><span class="o">.</span><span class="n">bse</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_stream output_stdout output_text">
   <pre>
   [-41.02649835   0.82938433   0.92606597  -0.12784672]
   [ 9.08950419  0.11945975  0.32235497  0.11796313]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Andrew&#39;s Wave norm with Huber&#39;s Proposal 2 scaling and &#39;H3&#39; covariance matrix</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[5]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">andrew_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">data</span><span class="o">.</span><span class="n">endog</span><span class="p">,</span> <span class="n">data</span><span class="o">.</span><span class="n">exog</span><span class="p">,</span> <span class="n">M</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><span class="o">.</span><span class="n">AndrewWave</span><span class="p">())</span>
   <span class="n">andrew_results</span> <span class="o">=</span> <span class="n">andrew_mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">scale_est</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">HuberScale</span><span class="p">(),</span> <span class="n">cov</span><span class="o">=</span><span class="s2">&quot;H3&quot;</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="s1">&#39;Parameters: &#39;</span><span class="p">,</span> <span class="n">andrew_results</span><span class="o">.</span><span class="n">params</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_stream output_stdout output_text">
   <pre>
   Parameters:  [-40.8817957    0.79276138   1.04857556  -0.13360865]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>See <code>help(sm.RLM.fit)</code> for more options and <code>module sm.robust.scale</code> for scale options</p>
   <h2 id="comparing-ols-and-rlm">Comparing OLS and RLM</h2>
   <p>Artificial data with outliers:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[6]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">nsample</span> <span class="o">=</span> <span class="mi">50</span>
   <span class="n">x1</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="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="n">nsample</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">column_stack</span><span class="p">((</span><span class="n">x1</span><span class="p">,</span> <span class="p">(</span><span class="n">x1</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</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">X</span><span class="p">)</span>
   <span class="n">sig</span> <span class="o">=</span> <span class="mf">0.3</span>   <span class="c1"># smaller error variance makes OLS&lt;-&gt;RLM contrast bigger</span>
   <span class="n">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">]</span>
   <span class="n">y_true2</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</span><span class="p">)</span>
   <span class="n">y2</span> <span class="o">=</span> <span class="n">y_true2</span> <span class="o">+</span> <span class="n">sig</span><span class="o">*</span><span class="mf">1.</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="n">nsample</span><span class="p">)</span>
   <span class="n">y2</span><span class="p">[[</span><span class="mi">39</span><span class="p">,</span><span class="mi">41</span><span class="p">,</span><span class="mi">43</span><span class="p">,</span><span class="mi">45</span><span class="p">,</span><span class="mi">48</span><span class="p">]]</span> <span class="o">-=</span> <span class="mi">5</span>   <span class="c1"># add some outliers (10% of nsample)</span>
   </pre></div>
   
   </div>
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <h3 id="example-1-quadratic-function-with-linear-truth">Example 1: quadratic function with linear truth</h3>
   <p>Note that the quadratic term in OLS regression will capture outlier effects. </p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[7]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">res</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">y2</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="k">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">predict</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_stream output_stdout output_text">
   <pre>
   [ 5.02448278  0.52714658 -0.01436688]
   [ 0.47195451  0.07286341  0.00644729]
   [  4.66531069   4.93671942   5.20334118   5.46517597   5.7222238
      5.97448466   6.22195856   6.46464549   6.70254545   6.93565844
      7.16398447   7.38752353   7.60627563   7.82024076   8.02941892
      8.23381012   8.43341434   8.62823161   8.8182619    9.00350523
      9.18396159   9.35963099   9.53051342   9.69660888   9.85791738
     10.0144389   10.16617347  10.31312106  10.45528169  10.59265535
     10.72524205  10.85304178  10.97605454  11.09428034  11.20771917
     11.31637103  11.42023593  11.51931386  11.61360482  11.70310881
     11.78782584  11.86775591  11.942899    12.01325513  12.0788243
     12.13960649  12.19560172  12.24680998  12.29323128  12.33486561]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Estimate RLM:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[8]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">resrlm</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">y2</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="k">print</span><span class="p">(</span><span class="n">resrlm</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">resrlm</span><span class="o">.</span><span class="n">bse</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_stream output_stdout output_text">
   <pre>
   [  4.92561014e+00   5.16693155e-01  -4.23172786e-03]
   [ 0.15687747  0.02421976  0.00214308]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Draw a plot to compare OLS estimates to the robust estimates:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[9]:</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">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">x1</span><span class="p">,</span> <span class="n">y2</span><span class="p">,</span> <span class="s1">&#39;o&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;data&quot;</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">x1</span><span class="p">,</span> <span class="n">y_true2</span><span class="p">,</span> <span class="s1">&#39;b-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True&quot;</span><span class="p">)</span>
   <span class="n">prstd</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="n">iv_u</span> <span class="o">=</span> <span class="n">wls_prediction_std</span><span class="p">(</span><span class="n">res</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">x1</span><span class="p">,</span> <span class="n">res</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s1">&#39;r-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;OLS&quot;</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">x1</span><span class="p">,</span> <span class="n">iv_u</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</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">x1</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</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">x1</span><span class="p">,</span> <span class="n">resrlm</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s1">&#39;g.-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RLM&quot;</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</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[9]:</div>
   
   
   <div class="output_text output_subarea output_pyout">
   <pre>
   &lt;matplotlib.legend.Legend at 0x7fb77363ef10&gt;
   </pre>
   </div>
   
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_stream output_stdout output_text">
   <pre>
   Error in callback &lt;function post_execute at 0x7fb786441500&gt; (for post_execute):
   
   </pre>
   </div>
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">RuntimeError</span>                              Traceback (most recent call last)
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc</span> in <span class="ansicyan">post_execute</span><span class="ansiblue">()</span>
   <span class="ansigreen">    145</span>             <span class="ansigreen">def</span> post_execute<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    146</span>                 <span class="ansigreen">if</span> matplotlib<span class="ansiblue">.</span>is_interactive<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 147</span><span class="ansired">                     </span>draw_all<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    148</span> <span class="ansiblue"></span>
   <span class="ansigreen">    149</span>             <span class="ansired"># IPython &gt;= 2</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc</span> in <span class="ansicyan">draw_all</span><span class="ansiblue">(cls, force)</span>
   <span class="ansigreen">    148</span>         <span class="ansigreen">for</span> f_mgr <span class="ansigreen">in</span> cls<span class="ansiblue">.</span>get_all_fig_managers<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    149</span>             <span class="ansigreen">if</span> force <span class="ansigreen">or</span> f_mgr<span class="ansiblue">.</span>canvas<span class="ansiblue">.</span>figure<span class="ansiblue">.</span>stale<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 150</span><span class="ansired">                 </span>f_mgr<span class="ansiblue">.</span>canvas<span class="ansiblue">.</span>draw_idle<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    151</span> <span class="ansiblue"></span>
   <span class="ansigreen">    152</span> atexit<span class="ansiblue">.</span>register<span class="ansiblue">(</span>Gcf<span class="ansiblue">.</span>destroy_all<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc</span> in <span class="ansicyan">draw_idle</span><span class="ansiblue">(self, *args, **kwargs)</span>
   <span class="ansigreen">   2024</span>         <span class="ansigreen">if</span> <span class="ansigreen">not</span> self<span class="ansiblue">.</span>_is_idle_drawing<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   2025</span>             <span class="ansigreen">with</span> self<span class="ansiblue">.</span>_idle_draw_cntx<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2026</span><span class="ansired">                 </span>self<span class="ansiblue">.</span>draw<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">   2027</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2028</span>     <span class="ansigreen">def</span> draw_cursor<span class="ansiblue">(</span>self<span class="ansiblue">,</span> event<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self)</span>
   <span class="ansigreen">    472</span> <span class="ansiblue"></span>
   <span class="ansigreen">    473</span>         <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 474</span><span class="ansired">             </span>self<span class="ansiblue">.</span>figure<span class="ansiblue">.</span>draw<span class="ansiblue">(</span>self<span class="ansiblue">.</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    475</span>         <span class="ansigreen">finally</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    476</span>             RendererAgg<span class="ansiblue">.</span>lock<span class="ansiblue">.</span>release<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer)</span>
   <span class="ansigreen">   1157</span>         dsu<span class="ansiblue">.</span>sort<span class="ansiblue">(</span>key<span class="ansiblue">=</span>itemgetter<span class="ansiblue">(</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1158</span>         <span class="ansigreen">for</span> zorder<span class="ansiblue">,</span> a<span class="ansiblue">,</span> func<span class="ansiblue">,</span> args <span class="ansigreen">in</span> dsu<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1159</span><span class="ansired">             </span>func<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1160</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1161</span>         renderer<span class="ansiblue">.</span>close_group<span class="ansiblue">(</span><span class="ansiblue">&apos;figure&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer, inframe)</span>
   <span class="ansigreen">   2322</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2323</span>         <span class="ansigreen">for</span> zorder<span class="ansiblue">,</span> a <span class="ansigreen">in</span> dsu<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2324</span><span class="ansired">             </span>a<span class="ansiblue">.</span>draw<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   2325</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2326</span>         renderer<span class="ansiblue">.</span>close_group<span class="ansiblue">(</span><span class="ansiblue">&apos;axes&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer, *args, **kwargs)</span>
   <span class="ansigreen">   1106</span>         ticks_to_draw <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_update_ticks<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1107</span>         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
   <span class="ansigreen">-&gt; 1108</span><span class="ansired">                                                                 renderer)
   </span><span class="ansigreen">   1109</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1110</span>         <span class="ansigreen">for</span> tick <span class="ansigreen">in</span> ticks_to_draw<span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc</span> in <span class="ansicyan">_get_tick_bboxes</span><span class="ansiblue">(self, ticks, renderer)</span>
   <span class="ansigreen">   1056</span>         <span class="ansigreen">for</span> tick <span class="ansigreen">in</span> ticks<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1057</span>             <span class="ansigreen">if</span> tick<span class="ansiblue">.</span>label1On <span class="ansigreen">and</span> tick<span class="ansiblue">.</span>label1<span class="ansiblue">.</span>get_visible<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1058</span><span class="ansired">                 </span>extent <span class="ansiblue">=</span> tick<span class="ansiblue">.</span>label1<span class="ansiblue">.</span>get_window_extent<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1059</span>                 ticklabelBoxes<span class="ansiblue">.</span>append<span class="ansiblue">(</span>extent<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1060</span>             <span class="ansigreen">if</span> tick<span class="ansiblue">.</span>label2On <span class="ansigreen">and</span> tick<span class="ansiblue">.</span>label2<span class="ansiblue">.</span>get_visible<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/text.pyc</span> in <span class="ansicyan">get_window_extent</span><span class="ansiblue">(self, renderer, dpi)</span>
   <span class="ansigreen">    959</span>             <span class="ansigreen">raise</span> RuntimeError<span class="ansiblue">(</span><span class="ansiblue">&apos;Cannot get window extent w/o renderer&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    960</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 961</span><span class="ansired">         </span>bbox<span class="ansiblue">,</span> info<span class="ansiblue">,</span> descent <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_get_layout<span class="ansiblue">(</span>self<span class="ansiblue">.</span>_renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    962</span>         x<span class="ansiblue">,</span> y <span class="ansiblue">=</span> self<span class="ansiblue">.</span>get_unitless_position<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    963</span>         x<span class="ansiblue">,</span> y <span class="ansiblue">=</span> self<span class="ansiblue">.</span>get_transform<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>transform_point<span class="ansiblue">(</span><span class="ansiblue">(</span>x<span class="ansiblue">,</span> y<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/text.pyc</span> in <span class="ansicyan">_get_layout</span><span class="ansiblue">(self, renderer)</span>
   <span class="ansigreen">    350</span>         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent(&apos;lp&apos;,
   <span class="ansigreen">    351</span>                                                          self<span class="ansiblue">.</span>_fontproperties<span class="ansiblue">,</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 352</span><span class="ansired">                                                          ismath=False)
   </span><span class="ansigreen">    353</span>         offsety <span class="ansiblue">=</span> <span class="ansiblue">(</span>lp_h <span class="ansiblue">-</span> lp_bl<span class="ansiblue">)</span> <span class="ansiblue">*</span> self<span class="ansiblue">.</span>_linespacing<span class="ansiblue"></span>
   <span class="ansigreen">    354</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc</span> in <span class="ansicyan">get_text_width_height_descent</span><span class="ansiblue">(self, s, prop, ismath)</span>
   <span class="ansigreen">    227</span>             fontsize <span class="ansiblue">=</span> prop<span class="ansiblue">.</span>get_size_in_points<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    228</span>             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
   <span class="ansigreen">--&gt; 229</span><span class="ansired">                                                                renderer=self)
   </span><span class="ansigreen">    230</span>             <span class="ansigreen">return</span> w<span class="ansiblue">,</span> h<span class="ansiblue">,</span> d<span class="ansiblue"></span>
   <span class="ansigreen">    231</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc</span> in <span class="ansicyan">get_text_width_height_descent</span><span class="ansiblue">(self, tex, fontsize, renderer)</span>
   <span class="ansigreen">    673</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    674</span>             <span class="ansired"># use dviread. It sometimes returns a wrong descent.</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 675</span><span class="ansired">             </span>dvifile <span class="ansiblue">=</span> self<span class="ansiblue">.</span>make_dvi<span class="ansiblue">(</span>tex<span class="ansiblue">,</span> fontsize<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    676</span>             dvi <span class="ansiblue">=</span> dviread<span class="ansiblue">.</span>Dvi<span class="ansiblue">(</span>dvifile<span class="ansiblue">,</span> <span class="ansicyan">72</span> <span class="ansiblue">*</span> dpi_fraction<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    677</span>             <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc</span> in <span class="ansicyan">make_dvi</span><span class="ansiblue">(self, tex, fontsize)</span>
   <span class="ansigreen">    420</span>                      <span class="ansiblue">&apos;string:\n%s\nHere is the full report generated by &apos;</span><span class="ansiblue"></span>
   <span class="ansigreen">    421</span>                      <span class="ansiblue">&apos;LaTeX: \n\n&apos;</span> <span class="ansiblue">%</span> repr<span class="ansiblue">(</span>tex<span class="ansiblue">.</span>encode<span class="ansiblue">(</span><span class="ansiblue">&apos;unicode_escape&apos;</span><span class="ansiblue">)</span><span class="ansiblue">)</span> <span class="ansiblue">+</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 422</span><span class="ansired">                      report))
   </span><span class="ansigreen">    423</span>             <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    424</span>                 mpl<span class="ansiblue">.</span>verbose<span class="ansiblue">.</span>report<span class="ansiblue">(</span>report<span class="ansiblue">,</span> <span class="ansiblue">&apos;debug&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">RuntimeError</span>: LaTeX was not able to process the following string:
   &apos;lp&apos;
   Here is the full report generated by LaTeX: 
   
   </pre>
   </div>
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   
   
   <div class="output_text output_subarea ">
   <pre>
   &lt;matplotlib.figure.Figure at 0x7fb773b8ba50&gt;
   </pre>
   </div>
   
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <h3 id="example-2-linear-function-with-linear-truth">Example 2: linear function with linear truth</h3>
   <p>Fit a new OLS model using only the linear term and the constant:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[10]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">X2</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]]</span> 
   <span class="n">res2</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">y2</span><span class="p">,</span> <span class="n">X2</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">res2</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">res2</span><span class="o">.</span><span class="n">bse</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_stream output_stdout output_text">
   <pre>
   [ 5.60355615  0.38347775]
   [ 0.40991973  0.03532034]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Estimate RLM:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[11]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">resrlm2</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">y2</span><span class="p">,</span> <span class="n">X2</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">resrlm2</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="n">resrlm2</span><span class="o">.</span><span class="n">bse</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_stream output_stdout output_text">
   <pre>
   [ 5.07805872  0.47912024]
   [ 0.12256837  0.01056099]
   
   </pre>
   </div>
   </div>
   
   </div>
   </div>
   
   </div>
   <div class="cell border-box-sizing text_cell rendered">
   <div class="prompt input_prompt">
   </div>
   <div class="inner_cell">
   <div class="text_cell_render border-box-sizing rendered_html">
   <p>Draw a plot to compare OLS estimates to the robust estimates:</p>
   </div>
   </div>
   </div>
   <div class="cell border-box-sizing code_cell rendered">
   <div class="input">
   <div class="prompt input_prompt">In&nbsp;[12]:</div>
   <div class="inner_cell">
       <div class="input_area">
   <div class="highlight"><pre><span class="n">prstd</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="n">iv_u</span> <span class="o">=</span> <span class="n">wls_prediction_std</span><span class="p">(</span><span class="n">res2</span><span class="p">)</span>
   
   <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">6</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">x1</span><span class="p">,</span> <span class="n">y2</span><span class="p">,</span> <span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;data&quot;</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">x1</span><span class="p">,</span> <span class="n">y_true2</span><span class="p">,</span> <span class="s1">&#39;b-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True&quot;</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">x1</span><span class="p">,</span> <span class="n">res2</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s1">&#39;r-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;OLS&quot;</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">x1</span><span class="p">,</span> <span class="n">iv_u</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</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">x1</span><span class="p">,</span> <span class="n">iv_l</span><span class="p">,</span> <span class="s1">&#39;r--&#39;</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">x1</span><span class="p">,</span> <span class="n">resrlm2</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s1">&#39;g.-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RLM&quot;</span><span class="p">)</span>
   <span class="n">legend</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</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_stream output_stdout output_text">
   <pre>
   Error in callback &lt;function post_execute at 0x7fb786441500&gt; (for post_execute):
   
   </pre>
   </div>
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   <div class="output_subarea output_text output_pyerr">
   <pre>
   <span class="ansired">---------------------------------------------------------------------------</span>
   <span class="ansired">RuntimeError</span>                              Traceback (most recent call last)
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/pyplot.pyc</span> in <span class="ansicyan">post_execute</span><span class="ansiblue">()</span>
   <span class="ansigreen">    145</span>             <span class="ansigreen">def</span> post_execute<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    146</span>                 <span class="ansigreen">if</span> matplotlib<span class="ansiblue">.</span>is_interactive<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 147</span><span class="ansired">                     </span>draw_all<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    148</span> <span class="ansiblue"></span>
   <span class="ansigreen">    149</span>             <span class="ansired"># IPython &gt;= 2</span><span class="ansiblue"></span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc</span> in <span class="ansicyan">draw_all</span><span class="ansiblue">(cls, force)</span>
   <span class="ansigreen">    148</span>         <span class="ansigreen">for</span> f_mgr <span class="ansigreen">in</span> cls<span class="ansiblue">.</span>get_all_fig_managers<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    149</span>             <span class="ansigreen">if</span> force <span class="ansigreen">or</span> f_mgr<span class="ansiblue">.</span>canvas<span class="ansiblue">.</span>figure<span class="ansiblue">.</span>stale<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 150</span><span class="ansired">                 </span>f_mgr<span class="ansiblue">.</span>canvas<span class="ansiblue">.</span>draw_idle<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    151</span> <span class="ansiblue"></span>
   <span class="ansigreen">    152</span> atexit<span class="ansiblue">.</span>register<span class="ansiblue">(</span>Gcf<span class="ansiblue">.</span>destroy_all<span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backend_bases.pyc</span> in <span class="ansicyan">draw_idle</span><span class="ansiblue">(self, *args, **kwargs)</span>
   <span class="ansigreen">   2024</span>         <span class="ansigreen">if</span> <span class="ansigreen">not</span> self<span class="ansiblue">.</span>_is_idle_drawing<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   2025</span>             <span class="ansigreen">with</span> self<span class="ansiblue">.</span>_idle_draw_cntx<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2026</span><span class="ansired">                 </span>self<span class="ansiblue">.</span>draw<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">   2027</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2028</span>     <span class="ansigreen">def</span> draw_cursor<span class="ansiblue">(</span>self<span class="ansiblue">,</span> event<span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self)</span>
   <span class="ansigreen">    472</span> <span class="ansiblue"></span>
   <span class="ansigreen">    473</span>         <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 474</span><span class="ansired">             </span>self<span class="ansiblue">.</span>figure<span class="ansiblue">.</span>draw<span class="ansiblue">(</span>self<span class="ansiblue">.</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    475</span>         <span class="ansigreen">finally</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    476</span>             RendererAgg<span class="ansiblue">.</span>lock<span class="ansiblue">.</span>release<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/figure.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer)</span>
   <span class="ansigreen">   1157</span>         dsu<span class="ansiblue">.</span>sort<span class="ansiblue">(</span>key<span class="ansiblue">=</span>itemgetter<span class="ansiblue">(</span><span class="ansicyan">0</span><span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1158</span>         <span class="ansigreen">for</span> zorder<span class="ansiblue">,</span> a<span class="ansiblue">,</span> func<span class="ansiblue">,</span> args <span class="ansigreen">in</span> dsu<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1159</span><span class="ansired">             </span>func<span class="ansiblue">(</span><span class="ansiblue">*</span>args<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1160</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1161</span>         renderer<span class="ansiblue">.</span>close_group<span class="ansiblue">(</span><span class="ansiblue">&apos;figure&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axes/_base.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer, inframe)</span>
   <span class="ansigreen">   2322</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2323</span>         <span class="ansigreen">for</span> zorder<span class="ansiblue">,</span> a <span class="ansigreen">in</span> dsu<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 2324</span><span class="ansired">             </span>a<span class="ansiblue">.</span>draw<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   2325</span> <span class="ansiblue"></span>
   <span class="ansigreen">   2326</span>         renderer<span class="ansiblue">.</span>close_group<span class="ansiblue">(</span><span class="ansiblue">&apos;axes&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/artist.pyc</span> in <span class="ansicyan">draw_wrapper</span><span class="ansiblue">(artist, renderer, *args, **kwargs)</span>
   <span class="ansigreen">     59</span>     <span class="ansigreen">def</span> draw_wrapper<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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="ansiblue"></span>
   <span class="ansigreen">     60</span>         before<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">---&gt; 61</span><span class="ansired">         </span>draw<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<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">     62</span>         after<span class="ansiblue">(</span>artist<span class="ansiblue">,</span> renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">     63</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc</span> in <span class="ansicyan">draw</span><span class="ansiblue">(self, renderer, *args, **kwargs)</span>
   <span class="ansigreen">   1106</span>         ticks_to_draw <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_update_ticks<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1107</span>         ticklabelBoxes, ticklabelBoxes2 = self._get_tick_bboxes(ticks_to_draw,
   <span class="ansigreen">-&gt; 1108</span><span class="ansired">                                                                 renderer)
   </span><span class="ansigreen">   1109</span> <span class="ansiblue"></span>
   <span class="ansigreen">   1110</span>         <span class="ansigreen">for</span> tick <span class="ansigreen">in</span> ticks_to_draw<span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/axis.pyc</span> in <span class="ansicyan">_get_tick_bboxes</span><span class="ansiblue">(self, ticks, renderer)</span>
   <span class="ansigreen">   1056</span>         <span class="ansigreen">for</span> tick <span class="ansigreen">in</span> ticks<span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">   1057</span>             <span class="ansigreen">if</span> tick<span class="ansiblue">.</span>label1On <span class="ansigreen">and</span> tick<span class="ansiblue">.</span>label1<span class="ansiblue">.</span>get_visible<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">-&gt; 1058</span><span class="ansired">                 </span>extent <span class="ansiblue">=</span> tick<span class="ansiblue">.</span>label1<span class="ansiblue">.</span>get_window_extent<span class="ansiblue">(</span>renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1059</span>                 ticklabelBoxes<span class="ansiblue">.</span>append<span class="ansiblue">(</span>extent<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">   1060</span>             <span class="ansigreen">if</span> tick<span class="ansiblue">.</span>label2On <span class="ansigreen">and</span> tick<span class="ansiblue">.</span>label2<span class="ansiblue">.</span>get_visible<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/text.pyc</span> in <span class="ansicyan">get_window_extent</span><span class="ansiblue">(self, renderer, dpi)</span>
   <span class="ansigreen">    959</span>             <span class="ansigreen">raise</span> RuntimeError<span class="ansiblue">(</span><span class="ansiblue">&apos;Cannot get window extent w/o renderer&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    960</span> <span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 961</span><span class="ansired">         </span>bbox<span class="ansiblue">,</span> info<span class="ansiblue">,</span> descent <span class="ansiblue">=</span> self<span class="ansiblue">.</span>_get_layout<span class="ansiblue">(</span>self<span class="ansiblue">.</span>_renderer<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    962</span>         x<span class="ansiblue">,</span> y <span class="ansiblue">=</span> self<span class="ansiblue">.</span>get_unitless_position<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    963</span>         x<span class="ansiblue">,</span> y <span class="ansiblue">=</span> self<span class="ansiblue">.</span>get_transform<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue">.</span>transform_point<span class="ansiblue">(</span><span class="ansiblue">(</span>x<span class="ansiblue">,</span> y<span class="ansiblue">)</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/text.pyc</span> in <span class="ansicyan">_get_layout</span><span class="ansiblue">(self, renderer)</span>
   <span class="ansigreen">    350</span>         tmp, lp_h, lp_bl = renderer.get_text_width_height_descent(&apos;lp&apos;,
   <span class="ansigreen">    351</span>                                                          self<span class="ansiblue">.</span>_fontproperties<span class="ansiblue">,</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 352</span><span class="ansired">                                                          ismath=False)
   </span><span class="ansigreen">    353</span>         offsety <span class="ansiblue">=</span> <span class="ansiblue">(</span>lp_h <span class="ansiblue">-</span> lp_bl<span class="ansiblue">)</span> <span class="ansiblue">*</span> self<span class="ansiblue">.</span>_linespacing<span class="ansiblue"></span>
   <span class="ansigreen">    354</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/backends/backend_agg.pyc</span> in <span class="ansicyan">get_text_width_height_descent</span><span class="ansiblue">(self, s, prop, ismath)</span>
   <span class="ansigreen">    227</span>             fontsize <span class="ansiblue">=</span> prop<span class="ansiblue">.</span>get_size_in_points<span class="ansiblue">(</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    228</span>             w, h, d = texmanager.get_text_width_height_descent(s, fontsize,
   <span class="ansigreen">--&gt; 229</span><span class="ansired">                                                                renderer=self)
   </span><span class="ansigreen">    230</span>             <span class="ansigreen">return</span> w<span class="ansiblue">,</span> h<span class="ansiblue">,</span> d<span class="ansiblue"></span>
   <span class="ansigreen">    231</span> <span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc</span> in <span class="ansicyan">get_text_width_height_descent</span><span class="ansiblue">(self, tex, fontsize, renderer)</span>
   <span class="ansigreen">    673</span>         <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    674</span>             <span class="ansired"># use dviread. It sometimes returns a wrong descent.</span><span class="ansiblue"></span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 675</span><span class="ansired">             </span>dvifile <span class="ansiblue">=</span> self<span class="ansiblue">.</span>make_dvi<span class="ansiblue">(</span>tex<span class="ansiblue">,</span> fontsize<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    676</span>             dvi <span class="ansiblue">=</span> dviread<span class="ansiblue">.</span>Dvi<span class="ansiblue">(</span>dvifile<span class="ansiblue">,</span> <span class="ansicyan">72</span> <span class="ansiblue">*</span> dpi_fraction<span class="ansiblue">)</span><span class="ansiblue"></span>
   <span class="ansigreen">    677</span>             <span class="ansigreen">try</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   
   <span class="ansigreen">/usr/lib/python2.7/dist-packages/matplotlib/texmanager.pyc</span> in <span class="ansicyan">make_dvi</span><span class="ansiblue">(self, tex, fontsize)</span>
   <span class="ansigreen">    420</span>                      <span class="ansiblue">&apos;string:\n%s\nHere is the full report generated by &apos;</span><span class="ansiblue"></span>
   <span class="ansigreen">    421</span>                      <span class="ansiblue">&apos;LaTeX: \n\n&apos;</span> <span class="ansiblue">%</span> repr<span class="ansiblue">(</span>tex<span class="ansiblue">.</span>encode<span class="ansiblue">(</span><span class="ansiblue">&apos;unicode_escape&apos;</span><span class="ansiblue">)</span><span class="ansiblue">)</span> <span class="ansiblue">+</span><span class="ansiblue"></span>
   <span class="ansigreen">--&gt; 422</span><span class="ansired">                      report))
   </span><span class="ansigreen">    423</span>             <span class="ansigreen">else</span><span class="ansiblue">:</span><span class="ansiblue"></span>
   <span class="ansigreen">    424</span>                 mpl<span class="ansiblue">.</span>verbose<span class="ansiblue">.</span>report<span class="ansiblue">(</span>report<span class="ansiblue">,</span> <span class="ansiblue">&apos;debug&apos;</span><span class="ansiblue">)</span><span class="ansiblue"></span>
   
   <span class="ansired">RuntimeError</span>: LaTeX was not able to process the following string:
   &apos;lp&apos;
   Here is the full report generated by LaTeX: 
   
   </pre>
   </div>
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   
   
   <div class="output_text output_subarea ">
   <pre>
   &lt;matplotlib.figure.Figure at 0x7fb773ba9d10&gt;
   </pre>
   </div>
   
   </div>
   
   </div>
   </div>
   
   </div>

   <script src="https://c328740.ssl.cf1.rackcdn.com/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"type="text/javascript"></script>
   <script type="text/javascript">
   init_mathjax = function() {
       if (window.MathJax) {
           // MathJax loaded
           MathJax.Hub.Config({
               tex2jax: {
               // I'm not sure about the \( and \[ below. It messes with the
               // prompt, and I think it's an issue with the template. -SS
                   inlineMath: [ ['$','$'], ["\\(","\\)"] ],
                   displayMath: [ ['$$','$$'], ["\\[","\\]"] ]
               },
               displayAlign: 'left', // Change this to 'center' to center equations.
               "HTML-CSS": {
                   styles: {'.MathJax_Display': {"margin": 0}}
               }
           });
           MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
       }
   }
   init_mathjax();

   // since we have to load this in a ..raw:: directive we will add the css
   // after the fact
   function loadcssfile(filename){
       var fileref=document.createElement("link")
       fileref.setAttribute("rel", "stylesheet")
       fileref.setAttribute("type", "text/css")
       fileref.setAttribute("href", filename)

       document.getElementsByTagName("head")[0].appendChild(fileref)
   }
   // loadcssfile({{pathto("_static/nbviewer.pygments.css", 1) }})
   // loadcssfile({{pathto("_static/nbviewer.min.css", 1) }})
   loadcssfile("../../../_static/nbviewer.pygments.css")
   loadcssfile("../../../_static/ipython.min.css")
   </script>