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


.. _robust_models_0_notebook:

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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">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>
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   <h2 id="estimation">Estimation</h2>
   <p>Load data:</p>
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   <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>
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   <p>Huber&#39;s T norm with the (default) median absolute deviation scaling</p>
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   <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="s">&#39;y&#39;</span><span class="p">,</span>
               <span class="n">xname</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;var_</span><span class="si">%d</span><span class="s">&#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>
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   [-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:                Thu, 21 May 2015                                         
   Time:                        05:57:49                                         
   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 .
   
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   <p>Huber&#39;s T norm with &#39;H2&#39; covariance matrix</p>
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   <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="s">&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>
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   [-41.02649835   0.82938433   0.92606597  -0.12784672]
   [ 9.08950419  0.11945975  0.32235497  0.11796313]
   
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   <p>Andrew&#39;s Wave norm with Huber&#39;s Proposal 2 scaling and &#39;H3&#39; covariance matrix</p>
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   <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="s">&quot;H3&quot;</span><span class="p">)</span>
   <span class="k">print</span><span class="p">(</span><span class="s">&#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>
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   Parameters:  [-40.8817957    0.79276138   1.04857556  -0.13360865]
   
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   <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>
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   <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="c"># 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="c"># add some outliers (10% of nsample)</span>
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   <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>
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   <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>
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   [ 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]
   
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   <p>Estimate RLM:</p>
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   <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>
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   [  4.92561014e+00   5.16693155e-01  -4.23172786e-03]
   [ 0.15687747  0.02421976  0.00214308]
   
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   <p>Draw a plot to compare OLS estimates to the robust estimates:</p>
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   <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="s">&#39;o&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#39;b-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#39;r-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#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="s">&#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="s">&#39;g.-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&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 0x7fb59733a390&gt;
   </pre>
   </div>
   
   </div>
   
   <div class="output_area"><div class="prompt"></div>
   
   
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   >
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   <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>
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   In&nbsp;[10]:
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   <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>
   
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   <pre>
   [ 5.60355615  0.38347775]
   [ 0.40991973  0.03532034]
   
   </pre>
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   <p>Estimate RLM:</p>
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   In&nbsp;[11]:
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   <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>
   
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   [ 5.07805872  0.47912024]
   [ 0.12256837  0.01056099]
   
   </pre>
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   <p>Draw a plot to compare OLS estimates to the robust estimates:</p>
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   In&nbsp;[12]:
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   <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="s">&#39;o&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#39;b-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#39;r-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&#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="s">&#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="s">&#39;g.-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&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="s">&quot;best&quot;</span><span class="p">)</span>
   </pre></div>
   
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