Morten Hjorth-Jensen
416 lines
24 KiB
HTML
416 lines
24 KiB
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<!-- navigation toc: --> <li><a href="._week38-bs001.html#plans-for-week-38" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#thursday-september-23" style="font-size: 80%;">Thursday September 23</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#ridge-and-lasso-regression-reminder" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" 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="._week38-bs006.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#to-think-about-first-part" style="font-size: 80%;">To think about, first part</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#more-thinking" style="font-size: 80%;">More thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#still-thinking" style="font-size: 80%;">Still thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;">What does centering (subtracting the mean values) mean mathematically?</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#further-manipulations" style="font-size: 80%;">Further Manipulations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;">Linear Regression code, Intercept handling first</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#code-examples" style="font-size: 80%;">Code Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#taking-out-the-mean" style="font-size: 80%;">Taking out the mean</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#more-complicated-example-the-ising-model" style="font-size: 80%;">More complicated Example: The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#reformulating-the-problem-to-suit-regression" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#linear-regression" style="font-size: 80%;">Linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#singular-value-decomposition" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#the-one-dimensional-ising-model" style="font-size: 80%;">The one-dimensional Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#ridge-regression" style="font-size: 80%;">Ridge regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#lasso-regression" style="font-size: 80%;">LASSO regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#performance-as-function-of-the-regularization-parameter" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
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<!-- navigation toc: --> <li><a href="#finding-the-optimal-value-of-lambda" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#logistic-regression" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#classification-problems" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#optimization-and-deep-learning" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#basics" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#linear-classifier" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#some-selected-properties" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#simple-example" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs033.html#plotting-the-mean-value-for-each-group" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#the-logistic-function" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#two-parameters" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#maximum-likelihood" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#the-cost-function-rewritten" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#minimizing-the-cross-entropy" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#a-more-compact-expression" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#extending-to-more-predictors" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#including-more-classes" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs043.html#more-classes" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs044.html#friday-september-24" style="font-size: 80%;">Friday September 24</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs045.html#wisconsin-cancer-data" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs046.html#using-the-correlation-matrix" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs047.html#discussing-the-correlation-data" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs048.html#other-measures-in-classification-studies-cancer-data-again" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs049.html#friday-september-25" style="font-size: 80%;">Friday September 25</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs050.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs051.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs052.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs053.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs054.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs055.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs056.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs057.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs058.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs059.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs060.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs061.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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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="part0025"></a>
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<!-- !split -->
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<h2 id="finding-the-optimal-value-of-lambda" class="anchor">Finding the optimal value of \( \lambda \) </h2>
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<p>
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To determine which value of \( \lambda \) is best we plot the accuracy of
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the models when predicting the training and the testing set. We expect
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the accuracy of the training set to be quite good, but if the accuracy
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of the testing set is much lower this tells us that we might be
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subject to an overfit model. The ideal scenario is an accuracy on the
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testing set that is close to the accuracy of the training set.
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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>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
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colors <span style="color: #666666">=</span> {
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<span style="color: #BA2121">"ols_sk"</span>: <span style="color: #BA2121">"r"</span>,
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<span style="color: #BA2121">"ridge_sk"</span>: <span style="color: #BA2121">"y"</span>,
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<span style="color: #BA2121">"lasso_sk"</span>: <span style="color: #BA2121">"c"</span>
|
|
}
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
|
|
plt<span style="color: #666666">.</span>semilogx(
|
|
lambdas,
|
|
train_errors[key],
|
|
colors[key],
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">"Train </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
|
linewidth<span style="color: #666666">=4.0</span>
|
|
)
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> test_errors:
|
|
plt<span style="color: #666666">.</span>semilogx(
|
|
lambdas,
|
|
test_errors[key],
|
|
colors[key] <span style="color: #666666">+</span> <span style="color: #BA2121">"--"</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">"Test </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
|
linewidth<span style="color: #666666">=4.0</span>
|
|
)
|
|
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>, fontsize<span style="color: #666666">=18</span>)
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$\lambda$"</span>, fontsize<span style="color: #666666">=18</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$R^2$"</span>, fontsize<span style="color: #666666">=18</span>)
|
|
plt<span style="color: #666666">.</span>tick_params(labelsize<span style="color: #666666">=18</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
|
|
achieves a very good accuracy on the test set. This by far surpasses the
|
|
other models for all values of \( \lambda \).
|
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<p>
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