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Morten Hjorth-Jensen
2024-10-15 10:49:25 +02:00
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@@ -333,6 +333,11 @@ const thebe_selector_output = ".output, .cell_output"
Week 42 Constructing a Neural Network code with examples
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<li class="toctree-l1">
<a class="reference internal" href="additionweek42.html">
Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40
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</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
@@ -869,10 +874,10 @@ 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: 0.136066 sec
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.15199 sec
Jackknife Statistics :
original bias std. error
99.8762 99.8662 0.150735
100.028 100.018 0.150301
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@@ -1091,7 +1096,7 @@ 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
99.9623 14.9594 99.9606 0.14934
99.9349 14.9929 99.9366 0.148477
</pre></div>
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@@ -1303,7 +1308,9 @@ Error: 0.10398646080125035
Bias^2: 0.1007711427354898
Var: 0.0032153180657605116
0.10398646080125035 &gt;= 0.1007711427354898 + 0.0032153180657605116 = 0.10398646080125032
Polynomial degree: 3
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 3
Error: 0.06547790180152355
Bias^2: 0.06208238634231949
Var: 0.0033955154592040936
@@ -1313,9 +1320,7 @@ Error: 0.06844519414009445
Bias^2: 0.06453579006728324
Var: 0.003909404072811226
0.06844519414009445 &gt;= 0.06453579006728324 + 0.003909404072811226 = 0.06844519414009446
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 5
Polynomial degree: 5
Error: 0.05227921801205686
Bias^2: 0.0481872773043029
Var: 0.004091940707753939
@@ -1335,14 +1340,14 @@ Error: 0.017355848195593347
Bias^2: 0.010331721306655127
Var: 0.007024126888938232
0.017355848195593347 &gt;= 0.010331721306655127 + 0.007024126888938232 = 0.01735584819559336
Polynomial degree: 9
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 9
Error: 0.02660572763718093
Bias^2: 0.010018312644137363
Var: 0.016587414993043573
0.02660572763718093 &gt;= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936
</pre></div>
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<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
@@ -1357,14 +1362,16 @@ Error: 0.11547777218872497
Bias^2: 0.01628578269596628
Var: 0.09919198949275869
0.11547777218872497 &gt;= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497
Polynomial degree: 13
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 13
Error: 0.22842468702219465
Bias^2: 0.01975416527185249
Var: 0.20867052175034223
0.22842468702219465 &gt;= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
</pre></div>
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<img alt="_images/chapter3_66_3.png" src="_images/chapter3_66_3.png" />
<img alt="_images/chapter3_66_4.png" src="_images/chapter3_66_4.png" />
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<p>The bias-variance tradeoff summarizes the fundamental tension in
@@ -1585,12 +1592,12 @@ 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
Degree of polynomial: 4
Mean squared error on training data: 303.47610036
Mean squared error on test data: 426.30787294
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 5
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 4
Mean squared error on training data: 303.47610036
Mean squared error on test data: 426.30787294
Degree of polynomial: 5
Mean squared error on training data: 3.80354994
Mean squared error on test data: 5.98822371
Degree of polynomial: 6
@@ -1599,12 +1606,12 @@ 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
Degree of polynomial: 8
Mean squared error on training data: 0.04912436
Mean squared error on test data: 0.21596432
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 9
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 8
Mean squared error on training data: 0.04912436
Mean squared error on test data: 0.21596432
Degree of polynomial: 9
Mean squared error on training data: 0.02522069
Mean squared error on test data: 0.08576932
Degree of polynomial: 10
@@ -1613,12 +1620,12 @@ 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
Degree of polynomial: 12
Mean squared error on training data: 0.00813803
Mean squared error on test data: 0.17446471
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 13
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 12
Mean squared error on training data: 0.00813803
Mean squared error on test data: 0.17446471
Degree of polynomial: 13
Mean squared error on training data: 0.00759119
Mean squared error on test data: 1.08131003
Degree of polynomial: 14
@@ -1627,12 +1634,12 @@ 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
Degree of polynomial: 16
Mean squared error on training data: 0.00315593
Mean squared error on test data: 234.38827994
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 17
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 16
Mean squared error on training data: 0.00315593
Mean squared error on test data: 234.38827994
Degree of polynomial: 17
Mean squared error on training data: 0.00242999
Mean squared error on test data: 1271.34367970
Degree of polynomial: 18
@@ -1641,12 +1648,12 @@ 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
Degree of polynomial: 20
Mean squared error on training data: 0.00137814
Mean squared error on test data: 1944.86062977
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 21
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 20
Mean squared error on training data: 0.00137814
Mean squared error on test data: 1944.86062977
Degree of polynomial: 21
Mean squared error on training data: 0.00118584
Mean squared error on test data: 14716.58827236
Degree of polynomial: 22
@@ -1655,12 +1662,12 @@ 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
Degree of polynomial: 24
Mean squared error on training data: 0.00084707
Mean squared error on test data: 1325.26124692
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 25
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 24
Mean squared error on training data: 0.00084707
Mean squared error on test data: 1325.26124692
Degree of polynomial: 25
Mean squared error on training data: 0.00079125
Mean squared error on test data: 129012.83870189
Degree of polynomial: 26
@@ -1669,19 +1676,19 @@ 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
Degree of polynomial: 28
Mean squared error on training data: 0.00062592
Mean squared error on test data: 3983.63037846
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 29
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 28
Mean squared error on training data: 0.00062592
Mean squared error on test data: 3983.63037846
Degree of polynomial: 29
Mean squared error on training data: 0.00060704
Mean squared error on test data: 3262.26814548
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59625/626635268.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_64203/626635268.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_59625/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_64203/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
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@@ -1915,7 +1922,7 @@ 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_59625/3817475779.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_64203/3817475779.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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@@ -2804,7 +2811,7 @@ 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_64203/4162706317.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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@@ -2948,7 +2955,7 @@ 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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@@ -2988,7 +2995,7 @@ 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_64203/438060758.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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@@ -3023,7 +3030,7 @@ 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_64203/3544313922.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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@@ -3076,43 +3083,43 @@ 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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