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<!-- navigation toc: --> <li><a href="._week38-bs001.html#___sec0" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#___sec7" style="font-size: 80%;">Bias-Variance tradeoff with Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs009.html#___sec8" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs010.html#___sec9" style="font-size: 80%;">Cross-validation with Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">The Ising model</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Linear regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs014.html#___sec13" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#___sec16" style="font-size: 80%;">LASSO regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs018.html#___sec17" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs020.html#___sec19" style="font-size: 80%;">Friday September 18: Intro to Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#___sec21" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#___sec24" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#___sec26" style="font-size: 80%;">Simple example</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs028.html#___sec27" style="font-size: 80%;">Plotting the mean value for each group</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">The logistic function</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs031.html#___sec30" style="font-size: 80%;">Two parameters</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs032.html#___sec31" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#___sec34" style="font-size: 80%;">A more compact expression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Using the correlation matrix</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs041.html#___sec40" style="font-size: 80%;">Discussing the correlation data</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs042.html#___sec41" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<h2 id="___sec18" class="anchor">Finding the optimal value of \( \lambda \) </h2>
<p>
To determine which value of \( \lambda \) is best we plot the accuracy of
the models when predicting the training and the testing set. We expect
the accuracy of the training set to be quite good, but if the accuracy
of the testing set is much lower this tells us that we might be
subject to an overfit model. The ideal scenario is an accuracy on the
testing set that is close to the accuracy of the training set.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<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>))
colors <span style="color: #666666">=</span> {
<span style="color: #BA2121">&quot;ols_sk&quot;</span>: <span style="color: #BA2121">&quot;r&quot;</span>,
<span style="color: #BA2121">&quot;ridge_sk&quot;</span>: <span style="color: #BA2121">&quot;y&quot;</span>,
<span style="color: #BA2121">&quot;lasso_sk&quot;</span>: <span style="color: #BA2121">&quot;c&quot;</span>
}
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
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lambdas,
train_errors[key],
colors[key],
label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Train </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">&quot;</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">&quot;--&quot;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Test </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">&quot;</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">&quot;best&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&quot;$\lambda$&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&quot;$R^2$&quot;</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 \( \lambda \).
<p>
<p>
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