306 lines
16 KiB
HTML
306 lines
16 KiB
HTML
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<a class="navbar-brand" href="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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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="#___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="._Regression-bs039.html#___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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</ul>
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</li>
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</ul>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0024"></a>
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<!-- !split -->
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<h2 id="___sec23" class="anchor">How to study our fits </h2>
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<p>
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We need more rigorous criteria in defining whether we have succeeded or
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not in modeling our training data. You will be surprised to see that
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many scientists seldomly venture beyond this 'by the eye' approach. A
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standard approach for the <em>cost</em> function is the so-called \( \chi^2 \)
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function
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$$ \chi^2 = \frac{1}{n}
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\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
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$$
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<p>
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where \( \sigma_i^2 \) is the variance (to be defined later) of the entry
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\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves
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however the aim of scaling the equations and make the cost function
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dimensionless.
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<p>
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<p>
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