Robust Linear Models
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.. _robust_models_0_notebook:
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">statsmodels.sandbox.regression.predstd</span> <span class="k">import</span> <span class="n">wls_prediction_std</span>
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<h2 id="Estimation">Estimation<a class="anchor-link" href="#Estimation">¶</a></h2><p>Load data:</p>
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<div class=" highlight hl-ipython3"><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's T norm with the (default) median absolute deviation scaling</p>
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<div class=" highlight hl-ipython3"><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="nb">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="nb">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="nb">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">'y'</span><span class="p">,</span>
<span class="n">xname</span><span class="o">=</span><span class="p">[</span><span class="s">'var_%d'</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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<p>Huber's T norm with 'H2' covariance matrix</p>
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<div class=" highlight hl-ipython3"><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">"H2"</span><span class="p">)</span>
<span class="nb">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="nb">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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<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: Mon, 20 Jul 2015
Time: 17:44:35
No. Iterations: 19
==============================================================================
coef std err z P>|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>Andrew's Wave norm with Huber's Proposal 2 scaling and 'H3' covariance matrix</p>
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<div class=" highlight hl-ipython3"><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">"H3"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s">'Parameters: '</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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<pre>[-41.02649835 0.82938433 0.92606597 -0.12784672]
[ 9.08950419 0.11945975 0.32235497 0.11796313]
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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<a class="anchor-link" href="#Comparing-OLS-and-RLM">¶</a></h2><p>Artificial data with outliers:</p>
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<div class=" highlight hl-ipython3"><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<->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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<pre>Parameters: [-40.8817957 0.79276138 1.04857556 -0.13360865]
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<h3 id="Example-1:-quadratic-function-with-linear-truth">Example 1: quadratic function with linear truth<a class="anchor-link" href="#Example-1:-quadratic-function-with-linear-truth">¶</a></h3><p>Note that the quadratic term in OLS regression will capture outlier effects.</p>
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<div class=" highlight hl-ipython3"><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="nb">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="nb">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="nb">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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<p>Estimate RLM:</p>
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<div class=" highlight hl-ipython3"><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="nb">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="nb">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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<pre>[ 5.05563298 0.52746225 -0.01292805]
[ 0.47889156 0.07393439 0.00654205]
[ 4.73243167 4.99833616 5.25993311 5.51722249 5.77020432
6.0188786 6.26324532 6.50330448 6.73905608 6.97050013
7.19763663 7.42046557 7.63898695 7.85320077 8.06310704
8.26870576 8.46999692 8.66698052 8.85965656 9.04802505
9.23208599 9.41183936 9.58728518 9.75842345 9.92525416
10.08777731 10.24599291 10.39990095 10.54950143 10.69479436
10.83577974 10.97245755 11.10482781 11.23289052 11.35664567
11.47609326 11.59123329 11.70206578 11.8085907 11.91080807
12.00871788 12.10232014 12.19161484 12.27660198 12.35728157
12.4336536 12.50571808 12.573475 12.63692436 12.69606617]
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<p>Draw a plot to compare OLS estimates to the robust estimates:</p>
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<div class=" highlight hl-ipython3"><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">'o'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s">"data"</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">'b-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"True"</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">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"OLS"</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">'r--'</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">'r--'</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">'g.-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"RLM"</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">"best"</span><span class="p">)</span>
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<pre>[ 4.97550374e+00 5.13254535e-01 -2.01895684e-03]
[ 0.10367794 0.01600647 0.00141633]
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<h3 id="Example-2:-linear-function-with-linear-truth">Example 2: linear function with linear truth<a class="anchor-link" href="#Example-2:-linear-function-with-linear-truth">¶</a></h3><p>Fit a new OLS model using only the linear term and the constant:</p>
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<div class=" highlight hl-ipython3"><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="nb">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="nb">print</span><span class="p">(</span><span class="n">res2</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
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<p>Estimate RLM:</p>
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<div class=" highlight hl-ipython3"><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="nb">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="nb">print</span><span class="p">(</span><span class="n">resrlm2</span><span class="o">.</span><span class="n">bse</span><span class="p">)</span>
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<pre>[ 5.57671265 0.39818173]
[ 0.41167908 0.03547193]
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<p>Draw a plot to compare OLS estimates to the robust estimates:</p>
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<div class=" highlight hl-ipython3"><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">'o'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"data"</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">'b-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"True"</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">'r-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"OLS"</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">'r--'</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">'r--'</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">'g.-'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">"RLM"</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">"best"</span><span class="p">)</span>
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<pre>[ 5.03066706 0.49664337]
[ 0.08236985 0.00709732]
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