changed layout

doc/LectureNotes/_build/html/week37.html

@@ -1623,7 +1623,7 @@ theorem.</p>
 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Bootstrap Statistics :
 original           bias      std. error
- 100.106  15.0037        100.104        0.149019
+ 100.135  15.1329        100.136        0.151103
 </pre></div>
 </div>
 </div>
@@ -1848,9 +1848,7 @@ Error: 0.10398646080125035
 Bias^2: 0.1007711427354898
 Var: 0.0032153180657605116
 0.10398646080125035 &gt;= 0.1007711427354898 + 0.0032153180657605116 = 0.10398646080125032
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 3
+Polynomial degree: 3
 Error: 0.06547790180152355
 Bias^2: 0.06208238634231949
 Var: 0.0033955154592040936
@@ -1889,9 +1887,7 @@ Error: 0.02660572763718093
 Bias^2: 0.010018312644137363
 Var: 0.016587414993043573
 0.02660572763718093 &gt;= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 10
+Polynomial degree: 10
 Error: 0.021592704588025025
 Bias^2: 0.010516485576645508
 Var: 0.011076219011379514
@@ -1901,7 +1897,9 @@ Error: 0.07160048164233104
 Bias^2: 0.014436800088904942
 Var: 0.05716368155342608
 0.07160048164233104 &gt;= 0.014436800088904942 + 0.05716368155342608 = 0.07160048164233102
-Polynomial degree: 12
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 12
 Error: 0.11547777218872497
 Bias^2: 0.01628578269596628
 Var: 0.09919198949275869
@@ -1913,7 +1911,7 @@ Var: 0.20867052175034223
 0.22842468702219465 &gt;= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
 </pre></div>
 </div>
-<img alt="_images/week37_139_5.png" src="_images/week37_139_5.png" />
+<img alt="_images/week37_139_4.png" src="_images/week37_139_4.png" />
 </div>
 </div>
 </div>
@@ -1946,31 +1944,24 @@ flexible statistical methods have higher variance.</p>
 </div>
 <div class="section" id="another-example-from-scikit-learn-s-repository">
 <h2>Another Example from Scikit-Learn’s Repository<a class="headerlink" href="#another-example-from-scikit-learn-s-repository" title="Permalink to this headline">¶</a></h2>
+<p>This example demonstrates the problems of underfitting and overfitting and
+how we can use linear regression with polynomial features to approximate
+nonlinear functions. The plot shows the function that we want to approximate,
+which is a part of the cosine function. In addition, the samples from the
+real function and the approximations of different models are displayed. The
+models have polynomial features of different degrees. We can see that a
+linear function (polynomial with degree 1) is not sufficient to fit the
+training samples. This is called <strong>underfitting</strong>. A polynomial of degree 4
+approximates the true function almost perfectly. However, for higher degrees
+the model will <strong>overfit</strong> the training data, i.e. it learns the noise of the
+training data.
+We evaluate quantitatively overfitting and underfitting by using
+cross-validation. We calculate the mean squared error (MSE) on the validation
+set, the higher, the less likely the model generalizes correctly from the
+training data.</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">============================</span>
-<span class="sd">Underfitting vs. Overfitting</span>
-<span class="sd">============================</span>
-
-<span class="sd">This example demonstrates the problems of underfitting and overfitting and</span>
-<span class="sd">how we can use linear regression with polynomial features to approximate</span>
-<span class="sd">nonlinear functions. The plot shows the function that we want to approximate,</span>
-<span class="sd">which is a part of the cosine function. In addition, the samples from the</span>
-<span class="sd">real function and the approximations of different models are displayed. The</span>
-<span class="sd">models have polynomial features of different degrees. We can see that a</span>
-<span class="sd">linear function (polynomial with degree 1) is not sufficient to fit the</span>
-<span class="sd">training samples. This is called **underfitting**. A polynomial of degree 4</span>
-<span class="sd">approximates the true function almost perfectly. However, for higher degrees</span>
-<span class="sd">the model will **overfit** the training data, i.e. it learns the noise of the</span>
-<span class="sd">training data.</span>
-<span class="sd">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
-<span class="sd">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
-<span class="sd">set, the higher, the less likely the model generalizes correctly from the</span>
-<span class="sd">training data.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#print(__doc__)</span>
 
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
 <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
@@ -2023,28 +2014,7 @@ flexible statistical methods have higher variance.</p>
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>============================
-Underfitting vs. Overfitting
-============================
-
-This example demonstrates the problems of underfitting and overfitting and
-how we can use linear regression with polynomial features to approximate
-nonlinear functions. The plot shows the function that we want to approximate,
-which is a part of the cosine function. In addition, the samples from the
-real function and the approximations of different models are displayed. The
-models have polynomial features of different degrees. We can see that a
-linear function (polynomial with degree 1) is not sufficient to fit the
-training samples. This is called **underfitting**. A polynomial of degree 4
-approximates the true function almost perfectly. However, for higher degrees
-the model will **overfit** the training data, i.e. it learns the noise of the
-training data.
-We evaluate quantitatively **overfitting** / **underfitting** by using
-cross-validation. We calculate the mean squared error (MSE) on the validation
-set, the higher, the less likely the model generalizes correctly from the
-training data.
-</pre></div>
-</div>
-<img alt="_images/week37_142_1.png" src="_images/week37_142_1.png" />
+<img alt="_images/week37_142_0.png" src="_images/week37_142_0.png" />
 </div>
 </div>
 </div>
@@ -2372,9 +2342,9 @@ Mean squared error on training data: 0.00063866
 Mean squared error on test data: 3099.60342978
 </pre></div>
 </div>
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_96719/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label=&#39;Training Error&#39;)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_96719/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
 </pre></div>
 </div>
@@ -2459,7 +2429,7 @@ Mean squared error on test data: 3099.60342978
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_96719/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label=&#39;Test Error&#39;)
 </pre></div>
 </div>

doc/LectureNotes/_build/jupyter_execute/week37.py

@@ -975,33 +975,27 @@ plt.show()
 # You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest.
 
 # ## Another Example from Scikit-Learn's Repository
+# 
+# This example demonstrates the problems of underfitting and overfitting and
+# how we can use linear regression with polynomial features to approximate
+# nonlinear functions. The plot shows the function that we want to approximate,
+# which is a part of the cosine function. In addition, the samples from the
+# real function and the approximations of different models are displayed. The
+# models have polynomial features of different degrees. We can see that a
+# linear function (polynomial with degree 1) is not sufficient to fit the
+# training samples. This is called **underfitting**. A polynomial of degree 4
+# approximates the true function almost perfectly. However, for higher degrees
+# the model will **overfit** the training data, i.e. it learns the noise of the
+# training data.
+# We evaluate quantitatively overfitting and underfitting by using
+# cross-validation. We calculate the mean squared error (MSE) on the validation
+# set, the higher, the less likely the model generalizes correctly from the
+# training data.
 
 # In[5]:
 
 
-"""
-============================
-Underfitting vs. Overfitting
-============================
-
-This example demonstrates the problems of underfitting and overfitting and
-how we can use linear regression with polynomial features to approximate
-nonlinear functions. The plot shows the function that we want to approximate,
-which is a part of the cosine function. In addition, the samples from the
-real function and the approximations of different models are displayed. The
-models have polynomial features of different degrees. We can see that a
-linear function (polynomial with degree 1) is not sufficient to fit the
-training samples. This is called **underfitting**. A polynomial of degree 4
-approximates the true function almost perfectly. However, for higher degrees
-the model will **overfit** the training data, i.e. it learns the noise of the
-training data.
-We evaluate quantitatively **overfitting** / **underfitting** by using
-cross-validation. We calculate the mean squared error (MSE) on the validation
-set, the higher, the less likely the model generalizes correctly from the
-training data.
-"""
-
-print(__doc__)
+#print(__doc__)
 
 import numpy as np
 import matplotlib.pyplot as plt