Prediction (out of sample)
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.. _predict_notebook:
`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/predict.ipynb>`_
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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>
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<h2 id="Artificial-data">Artificial data<a class="anchor-link" href="#Artificial-data">¶</a></h2>
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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">sig</span> <span class="o">=</span> <span class="mf">0.25</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="n">np</span><span class="o">.</span><span class="n">sin</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">beta</span> <span class="o">=</span> <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.02</span><span class="p">]</span>
<span class="n">y_true</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">y</span> <span class="o">=</span> <span class="n">y_true</span> <span class="o">+</span> <span class="n">sig</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>
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<h2 id="Estimation">Estimation<a class="anchor-link" href="#Estimation">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">olsmod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
<span class="n">olsres</span> <span class="o">=</span> <span class="n">olsmod</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">olsres</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<h2 id="In-sample-prediction">In-sample prediction<a class="anchor-link" href="#In-sample-prediction">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">ypred</span> <span class="o">=</span> <span class="n">olsres</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">ypred</span><span class="p">)</span>
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<pre> OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.981
Model: OLS Adj. R-squared: 0.980
Method: Least Squares F-statistic: 793.2
Date: Mon, 20 Jul 2015 Prob (F-statistic): 1.34e-39
Time: 17:44:19 Log-Likelihood: -3.9531
No. Observations: 50 AIC: 15.91
Df Residuals: 46 BIC: 23.55
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
const 4.9076 0.093 52.736 0.000 4.720 5.095
x1 0.5215 0.014 36.334 0.000 0.493 0.550
x2 0.5079 0.056 9.002 0.000 0.394 0.621
x3 -0.0217 0.001 -17.249 0.000 -0.024 -0.019
==============================================================================
Omnibus: 0.097 Durbin-Watson: 2.249
Prob(Omnibus): 0.953 Jarque-Bera (JB): 0.210
Skew: 0.096 Prob(JB): 0.900
Kurtosis: 2.747 Cond. No. 221.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
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<h2 id="Create-a-new-sample-of-explanatory-variables-Xnew,-predict-and-plot">Create a new sample of explanatory variables Xnew, predict and plot<a class="anchor-link" href="#Create-a-new-sample-of-explanatory-variables-Xnew,-predict-and-plot">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">x1n</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="mf">20.5</span><span class="p">,</span><span class="mi">25</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">Xnew</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">x1n</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x1n</span><span class="p">),</span> <span class="p">(</span><span class="n">x1n</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">Xnew</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">Xnew</span><span class="p">)</span>
<span class="n">ynewpred</span> <span class="o">=</span> <span class="n">olsres</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">Xnew</span><span class="p">)</span> <span class="c"># predict out of sample</span>
<span class="nb">print</span><span class="p">(</span><span class="n">ynewpred</span><span class="p">)</span>
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<pre>[ 4.36419585 4.86372083 5.32288364 5.71400582 6.01939792
6.23426583 6.36749844 6.4402072 6.48225762 6.52736237
6.60754246 6.74786624 6.96233076 7.25156207 7.60271237
7.9915709 8.38654164 8.7538345 9.06301692 9.29201382
9.43073454 9.48273062 9.4646124 9.40331987 9.33169661
9.28309362 9.28588985 9.35882964 9.50794309 9.72555529
9.99154729 10.27666165 10.54730926 10.77108802 10.92210636
10.98523716 10.95860313 10.85388518 10.69440335 10.51128533
10.33835231 10.20656224 10.13892353 10.14671482 10.22763195
10.36616652 10.53615388 10.70507189 10.83938423 10.91005034]
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<h2 id="Plot-comparison">Plot comparison<a class="anchor-link" href="#Plot-comparison">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</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">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</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_true</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">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">x1</span><span class="p">,</span> <span class="n">x1n</span><span class="p">)),</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">ypred</span><span class="p">,</span> <span class="n">ynewpred</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 prediction"</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>[ 10.88191737 10.71896653 10.44111456 10.09374917 9.73661657
9.4291935 9.21612524 9.11629492 9.11820024 9.18276965]
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<h2 id="Predicting-with-Formulas">Predicting with Formulas<a class="anchor-link" href="#Predicting-with-Formulas">¶</a></h2>
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<p>Using formulas can make both estimation and prediction a lot easier</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">from</span> <span class="nn">statsmodels.formula.api</span> <span class="k">import</span> <span class="n">ols</span>
<span class="n">data</span> <span class="o">=</span> <span class="p">{</span><span class="s">"x1"</span> <span class="p">:</span> <span class="n">x1</span><span class="p">,</span> <span class="s">"y"</span> <span class="p">:</span> <span class="n">y</span><span class="p">}</span>
<span class="n">res</span> <span class="o">=</span> <span class="n">ols</span><span class="p">(</span><span class="s">"y ~ x1 + np.sin(x1) + I((x1-5)**2)"</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">data</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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<p>We use the <code>I</code> to indicate use of the Identity transform. Ie., we don't want any expansion magic from using <code>**2</code></p>
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<div class=" highlight hl-ipython3"><pre><span class="n">res</span><span class="o">.</span><span class="n">params</span>
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<p>Now we only have to pass the single variable and we get the transformed right-hand side variables automatically</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">res</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">exog</span><span class="o">=</span><span class="nb">dict</span><span class="p">(</span><span class="n">x1</span><span class="o">=</span><span class="n">x1n</span><span class="p">))</span>
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