mhjensen
393 lines
26 KiB
HTML
393 lines
26 KiB
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<!-- navigation toc: --> <li><a href="._Regression-bs001.html#___sec0" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">General linear models</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, follows</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">Simple regression model</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">Simple regression model, now using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">Simple linear regression model using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">Simple linear regression model</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">Less noise</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">How to study our fits</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Minimizing the cost function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">Relative error</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">The richness of <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">Functions in <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Other functions in <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The mean absolute error and other functions in <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Cubic polynomial in <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Polynomial Regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">The singular value decompostion</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Code examples for Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">From standard regression to Ridge regressions</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">A second-order polynomial with Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Fitting vs. predicting when data is in the model class</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Fitting versus predicting when data is not in the model class</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">The code</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Generating test data</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Lasso regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Logistic regression</a></li>
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<!-- !split -->
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<h2 id="___sec38" class="anchor">A second-order polynomial with Ridge and Lasso </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> r2_score
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">4155</span>)
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n_samples <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples,<span style="color: #666666">1</span>)
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y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">*</span>x <span style="color: #666666">+</span> <span style="color: #666666">0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples,<span style="color: #666666">1</span>)
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<span style="color: #408080; font-style: italic"># Centering x and y.</span>
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x_ <span style="color: #666666">=</span> x <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(x)
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y_ <span style="color: #666666">=</span> y <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y) <span style="color: #408080; font-style: italic"># beta_0 = mean(y)</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n_samples,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">**2</span>]
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X_ <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[x_, x_<span style="color: #666666">**2</span>]
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<span style="color: #408080; font-style: italic">### 1.</span>
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lmb_values <span style="color: #666666">=</span> [<span style="color: #666666">1e-4</span>, <span style="color: #666666">1e-3</span>, <span style="color: #666666">1e-2</span>, <span style="color: #666666">10</span>, <span style="color: #666666">1e2</span>, <span style="color: #666666">1e4</span>]
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num_values <span style="color: #666666">=</span> <span style="color: #008000">len</span>(lmb_values)
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<span style="color: #408080; font-style: italic">## Ridge-regression of centered and not centered data</span>
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beta_ridge <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">3</span>,num_values))
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beta_ridge_centered <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">3</span>,num_values))
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I3 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">3</span>)
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I2 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">2</span>)
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<span style="color: #008000; font-weight: bold">for</span> i,lmb <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmb_values):
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beta_ridge[:,i] <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv( X<span style="color: #666666">.</span>T @ X <span style="color: #666666">+</span> lmb<span style="color: #666666">*</span>I3) @ X<span style="color: #666666">.</span>T @ y)<span style="color: #666666">.</span>flatten()
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beta_ridge_centered[<span style="color: #666666">1</span>:,i] <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv( X_<span style="color: #666666">.</span>T @ X_ <span style="color: #666666">+</span> lmb<span style="color: #666666">*</span>I2) @ X_<span style="color: #666666">.</span>T @ y_)<span style="color: #666666">.</span>flatten()
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<span style="color: #408080; font-style: italic"># sett beta_0 = np.mean(y)</span>
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beta_ridge_centered[<span style="color: #666666">0</span>,:] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(y)
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<span style="color: #408080; font-style: italic">## OLS (ordinary least squares) solution </span>
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beta_ls <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv( X<span style="color: #666666">.</span>T @ X ) @ X<span style="color: #666666">.</span>T @ y
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<span style="color: #408080; font-style: italic">## Evaluate the models</span>
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pred_ls <span style="color: #666666">=</span> X @ beta_ls
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pred_ridge <span style="color: #666666">=</span> X @ beta_ridge
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pred_ridge_centered <span style="color: #666666">=</span> X_ @ beta_ridge_centered[<span style="color: #666666">1</span>:] <span style="color: #666666">+</span> beta_ridge_centered[<span style="color: #666666">0</span>,:]
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<span style="color: #408080; font-style: italic">## Plot the results</span>
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<span style="color: #408080; font-style: italic"># Sorting</span>
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sort_ind <span style="color: #666666">=</span> np<span style="color: #666666">.</span>argsort(x[:,<span style="color: #666666">0</span>])
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x_plot <span style="color: #666666">=</span> x[sort_ind,<span style="color: #666666">0</span>]
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x_centered_plot <span style="color: #666666">=</span> x_[sort_ind,<span style="color: #666666">0</span>]
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pred_ls_plot <span style="color: #666666">=</span> pred_ls[sort_ind,<span style="color: #666666">0</span>]
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pred_ridge_plot <span style="color: #666666">=</span> pred_ridge[sort_ind,:]
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pred_ridge_centered_plot <span style="color: #666666">=</span> pred_ridge_centered[sort_ind,:]
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<span style="color: #408080; font-style: italic"># Plott not centered</span>
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plt<span style="color: #666666">.</span>plot(x_plot,pred_ls_plot,label<span style="color: #666666">=</span><span style="color: #BA2121">'ls'</span>)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(num_values):
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plt<span style="color: #666666">.</span>plot(x_plot,pred_ridge_plot[:,i],label<span style="color: #666666">=</span><span style="color: #BA2121">'ridge, lmb=</span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>lmb_values[i])
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plt<span style="color: #666666">.</span>plot(x,y,<span style="color: #BA2121">'ro'</span>)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'linear regression on un-centered data'</span>)
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plt<span style="color: #666666">.</span>legend()
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<span style="color: #408080; font-style: italic"># Plott centered</span>
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plt<span style="color: #666666">.</span>figure()
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(num_values):
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plt<span style="color: #666666">.</span>plot(x_centered_plot,pred_ridge_centered_plot[:,i],label<span style="color: #666666">=</span><span style="color: #BA2121">'ridge, lmb=</span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>lmb_values[i])
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plt<span style="color: #666666">.</span>plot(x_,y,<span style="color: #BA2121">'ro'</span>)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'linear regression on centered data'</span>)
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plt<span style="color: #666666">.</span>legend()
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<span style="color: #408080; font-style: italic"># 2.</span>
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pred_ridge_scikit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n_samples,num_values))
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<span style="color: #008000; font-weight: bold">for</span> i,lmb <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmb_values):
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pred_ridge_scikit[:,i] <span style="color: #666666">=</span> (Ridge(alpha<span style="color: #666666">=</span>lmb,fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>)<span style="color: #666666">.</span>fit(X,y)<span style="color: #666666">.</span>predict(X))<span style="color: #666666">.</span>flatten() <span style="color: #408080; font-style: italic"># fit_intercept=False fordi bias er allerede i X</span>
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>plot(x_plot,pred_ls_plot,label<span style="color: #666666">=</span><span style="color: #BA2121">'ls'</span>)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(num_values):
|
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plt<span style="color: #666666">.</span>plot(x_plot,pred_ridge_scikit[sort_ind,i],label<span style="color: #666666">=</span><span style="color: #BA2121">'scikit-ridge, lmb=</span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>lmb_values[i])
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plt<span style="color: #666666">.</span>plot(x,y,<span style="color: #BA2121">'ro'</span>)
|
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plt<span style="color: #666666">.</span>legend()
|
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'linear regression using scikit'</span>)
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plt<span style="color: #666666">.</span>show()
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<span style="color: #408080; font-style: italic">### R2-score of the results</span>
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(num_values):
|
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'lambda = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>lmb_values[i])
|
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'r2 for scikit: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>r2_score(y,pred_ridge_scikit[:,i]))
|
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'r2 for own code, not centered: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>r2_score(y,pred_ridge[:,i]))
|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'r2 for own, centered: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>r2_score(y,pred_ridge_centered[:,i]))
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</pre></div>
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