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Morten Hjorth-Jensen
2024-10-19 09:58:39 +02:00
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commit 9a699212fa
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@@ -338,6 +338,11 @@ const thebe_selector_output = ".output, .cell_output"
Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40
</a>
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<li class="toctree-l1">
<a class="reference internal" href="week43.html">
Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
@@ -874,10 +879,10 @@ number <span class="math notranslate nohighlight">\(i\)</span> is left out. Usin
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.15199 sec
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.132756 sec
Jackknife Statistics :
original bias std. error
100.028 100.018 0.150301
99.9573 99.9473 0.149003
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@@ -1096,7 +1101,7 @@ theorem.</p>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Bootstrap Statistics :
original bias std. error
99.9349 14.9929 99.9366 0.148477
100.299 15.0473 100.299 0.149379
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@@ -1308,9 +1313,7 @@ Error: 0.10398646080125035
Bias^2: 0.1007711427354898
Var: 0.0032153180657605116
0.10398646080125035 &gt;= 0.1007711427354898 + 0.0032153180657605116 = 0.10398646080125032
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<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
@@ -1335,14 +1338,14 @@ Error: 0.02760977349102253
Bias^2: 0.022999498260366312
Var: 0.004610275230656212
0.02760977349102253 &gt;= 0.022999498260366312 + 0.004610275230656212 = 0.027609773491022525
Polynomial degree: 8
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 8
Error: 0.017355848195593347
Bias^2: 0.010331721306655127
Var: 0.007024126888938232
0.017355848195593347 &gt;= 0.010331721306655127 + 0.007024126888938232 = 0.01735584819559336
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 9
Polynomial degree: 9
Error: 0.02660572763718093
Bias^2: 0.010018312644137363
Var: 0.016587414993043573
@@ -1371,7 +1374,7 @@ Var: 0.20867052175034223
0.22842468702219465 &gt;= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
</pre></div>
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<img alt="_images/chapter3_66_4.png" src="_images/chapter3_66_4.png" />
<img alt="_images/chapter3_66_3.png" src="_images/chapter3_66_3.png" />
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<p>The bias-variance tradeoff summarizes the fundamental tension in
@@ -1592,12 +1595,12 @@ Mean squared error on test data: 123711.53703498
Degree of polynomial: 3
Mean squared error on training data: 9011.85263220
Mean squared error on test data: 10913.84780262
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 4
Degree of polynomial: 4
Mean squared error on training data: 303.47610036
Mean squared error on test data: 426.30787294
Degree of polynomial: 5
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 5
Mean squared error on training data: 3.80354994
Mean squared error on test data: 5.98822371
Degree of polynomial: 6
@@ -1606,12 +1609,12 @@ Mean squared error on test data: 8.14812206
Degree of polynomial: 7
Mean squared error on training data: 0.47075725
Mean squared error on test data: 2.00607783
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 8
Degree of polynomial: 8
Mean squared error on training data: 0.04912436
Mean squared error on test data: 0.21596432
Degree of polynomial: 9
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 9
Mean squared error on training data: 0.02522069
Mean squared error on test data: 0.08576932
Degree of polynomial: 10
@@ -1620,12 +1623,12 @@ Mean squared error on test data: 1.20015436
Degree of polynomial: 11
Mean squared error on training data: 0.01640891
Mean squared error on test data: 1.35533773
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 12
Degree of polynomial: 12
Mean squared error on training data: 0.00813803
Mean squared error on test data: 0.17446471
Degree of polynomial: 13
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 13
Mean squared error on training data: 0.00759119
Mean squared error on test data: 1.08131003
Degree of polynomial: 14
@@ -1634,12 +1637,12 @@ Mean squared error on test data: 0.81333808
Degree of polynomial: 15
Mean squared error on training data: 0.00410478
Mean squared error on test data: 92.09163947
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 16
Degree of polynomial: 16
Mean squared error on training data: 0.00315593
Mean squared error on test data: 234.38827994
Degree of polynomial: 17
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 17
Mean squared error on training data: 0.00242999
Mean squared error on test data: 1271.34367970
Degree of polynomial: 18
@@ -1648,12 +1651,12 @@ Mean squared error on test data: 108.21093775
Degree of polynomial: 19
Mean squared error on training data: 0.00156374
Mean squared error on test data: 1385.79778008
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 20
Degree of polynomial: 20
Mean squared error on training data: 0.00137814
Mean squared error on test data: 1944.86062977
Degree of polynomial: 21
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 21
Mean squared error on training data: 0.00118584
Mean squared error on test data: 14716.58827236
Degree of polynomial: 22
@@ -1662,12 +1665,12 @@ Mean squared error on test data: 877.21517262
Degree of polynomial: 23
Mean squared error on training data: 0.00085892
Mean squared error on test data: 5567.04664255
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 24
Degree of polynomial: 24
Mean squared error on training data: 0.00084707
Mean squared error on test data: 1325.26124692
Degree of polynomial: 25
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 25
Mean squared error on training data: 0.00079125
Mean squared error on test data: 129012.83870189
Degree of polynomial: 26
@@ -1676,19 +1679,19 @@ Mean squared error on test data: 18388.59354079
Degree of polynomial: 27
Mean squared error on training data: 0.00069123
Mean squared error on test data: 2351.97979891
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 28
Degree of polynomial: 28
Mean squared error on training data: 0.00062592
Mean squared error on test data: 3983.63037846
Degree of polynomial: 29
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 29
Mean squared error on training data: 0.00060704
Mean squared error on test data: 3262.26814548
</pre></div>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_64203/626635268.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_87367/626635268.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_64203/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87367/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
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@@ -1922,7 +1925,7 @@ cross-validation (LOOCV).</p>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_64203/3817475779.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_87367/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(estimated_mse_sklearn), label=&#39;Test Error&#39;)
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@@ -2811,7 +2814,7 @@ linear system as an equation would reduce this down to
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87367/4162706317.py:7: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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@@ -2955,7 +2958,7 @@ with the form utilized in linear regression, viz.</p>
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cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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@@ -2995,7 +2998,7 @@ cost function is given by</p>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87367/438060758.py:10: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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@@ -3030,7 +3033,7 @@ cost function is given by</p>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87367/3544313922.py:9: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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@@ -3083,43 +3086,43 @@ constant as opposed to ridge and OLS. We get a sparse solution with
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model = cd_fast.enet_coordinate_descent(
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