mhjensen
317 lines
17 KiB
HTML
317 lines
17 KiB
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<ul class="dropdown-menu">
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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%;">From standard regression to Ridge regressions</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" 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-bs039.html#___sec38" 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-bs040.html#___sec39" style="font-size: 80%;">An example code without the model assessment part</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Generating test data</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">How can we effectively evaluate the various models?</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Code examples for Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">A second-order polynomial with Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Log-likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="#___sec48" style="font-size: 80%;">Computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Predicted Residual Error Sum of Squares</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Bootstrap</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="___sec48" class="anchor">Computationally expensive </h2>
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<p>
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The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
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<ul>
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<li> The validation estimate of the test error rate can be highly variable, depending on precisely which observations are included in the training set and which observations are included in the validation set.</li>
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<li> In the validation approach, only a subset of the observations, those that are included in the training set rather than in the validation set are used to fit the model. Since statistical methods tend to perform worse when trained on fewer observations, this suggests that the validation set error rate may tend to overestimate the test error rate for the model fit on the entire data set.</li>
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