update
This commit is contained in:
Morten Hjorth-Jensen
@@ -323,6 +323,16 @@ const thebe_selector_output = ".output, .cell_output"
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Week 41 Neural networks and constructing a neural network code
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="exercisesweek42.html">
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Exercises week 42
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="week42.html">
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Week 42 Constructing a Neural Network code with examples
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</a>
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</li>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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@@ -1305,10 +1315,10 @@ covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.009134699065945493
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4.0244965108017645
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[[0.85613835 2.50655379]
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[2.50655379 8.3404509 ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.04718566894028431
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4.11080997912276
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[[ 1.10517643 3.48455788]
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[ 3.48455788 12.00216162]]
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</pre></div>
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</div>
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@@ -1345,10 +1355,10 @@ a more brute force way. Here we scale the mean values for each column of the des
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07971187802560528
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1.800782161095708
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[[1. 0.59411814]
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[0.59411814 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07836997022107646
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1.1378267322316808
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[[1. 0.63980097]
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[0.63980097 1. ]]
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</pre></div>
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</div>
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</div>
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@@ -1378,30 +1388,30 @@ this matrix we easily see that it is a positive definite matrix.</p>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 0.81395716 1.89155934]
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[-1.34726166 -4.13453411]
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[-0.46229544 -2.34061974]
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[ 0.24429334 1.4051634 ]
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[ 0.41971814 1.6405671 ]
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[ 2.02456235 5.03973227]
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[-1.97311824 -4.72521196]
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[ 0.10738656 0.24578123]
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[-0.52702419 -2.34023682]
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[ 0.69978197 3.31779928]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1.34931214 3.06139439]
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[-0.44476964 -2.60794187]
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[ 0.02225493 0.16388664]
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[-1.91193672 -3.82324216]
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[-0.2044881 -1.56027537]
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[-1.15572395 -3.25982474]
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[ 0.94217756 1.49888671]
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[ 0.28472162 2.92474572]
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[ 2.38943 7.14118216]
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[-1.27097785 -3.5388115 ]]
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0 1
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0 0.813957 1.891559
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1 -1.347262 -4.134534
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2 -0.462295 -2.340620
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3 0.244293 1.405163
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4 0.419718 1.640567
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5 2.024562 5.039732
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6 -1.973118 -4.725212
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7 0.107387 0.245781
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8 -0.527024 -2.340237
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9 0.699782 3.317799
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0 1.349312 3.061394
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1 -0.444770 -2.607942
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2 0.022255 0.163887
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3 -1.911937 -3.823242
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4 -0.204488 -1.560275
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5 -1.155724 -3.259825
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6 0.942178 1.498887
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7 0.284722 2.924746
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8 2.389430 7.141182
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9 -1.270978 -3.538811
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0 1
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0 1.000000 0.969413
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1 0.969413 1.000000
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0 1.000000 0.950873
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1 0.950873 1.000000
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</pre></div>
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</div>
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</div>
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@@ -1458,37 +1468,37 @@ this matrix we easily see that it is a positive definite matrix.</p>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1 2 3 4 5 6 7 \
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0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
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1 0.0 0.092746 0.090302 0.090561 0.088058 0.085463 0.081530 0.078898
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2 0.0 0.090302 0.088694 0.089052 0.086797 0.084423 0.080286 0.077782
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3 0.0 0.090561 0.089052 0.095106 0.092533 0.089858 0.089404 0.086489
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4 0.0 0.088058 0.086797 0.092533 0.090114 0.087585 0.086913 0.084127
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5 0.0 0.085463 0.084423 0.089858 0.087585 0.085197 0.084340 0.081681
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6 0.0 0.081530 0.080286 0.089404 0.086913 0.084340 0.086471 0.083596
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7 0.0 0.078898 0.077782 0.086489 0.084127 0.081681 0.083596 0.080849
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8 0.0 0.076334 0.075334 0.083645 0.081405 0.079080 0.080793 0.078170
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9 0.0 0.073841 0.072946 0.080875 0.078753 0.076543 0.078067 0.075562
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10 0.0 0.072973 0.071789 0.082361 0.079982 0.077541 0.081296 0.078544
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11 0.0 0.070485 0.069391 0.079518 0.077254 0.074928 0.078454 0.075823
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12 0.0 0.068088 0.067077 0.076777 0.074622 0.072404 0.075712 0.073198
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13 0.0 0.065778 0.064845 0.074134 0.072083 0.069970 0.073071 0.070667
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14 0.0 0.063554 0.062693 0.071588 0.069637 0.067623 0.070527 0.068229
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1 0.0 0.074334 0.080585 0.077061 0.078751 0.080220 0.070657 0.071406
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2 0.0 0.080585 0.088425 0.082009 0.084289 0.086338 0.074009 0.075052
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3 0.0 0.077061 0.082009 0.085147 0.086339 0.087297 0.081324 0.081796
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4 0.0 0.078751 0.084289 0.086339 0.087789 0.089007 0.081926 0.082537
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5 0.0 0.080220 0.086338 0.087297 0.089007 0.090492 0.082307 0.083061
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6 0.0 0.070657 0.074009 0.081324 0.081926 0.082307 0.079874 0.080032
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7 0.0 0.071406 0.075052 0.081796 0.082537 0.083061 0.080032 0.080271
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8 0.0 0.072148 0.076101 0.082240 0.083128 0.083801 0.080150 0.080474
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9 0.0 0.072902 0.077180 0.082670 0.083714 0.084548 0.080237 0.080651
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10 0.0 0.063646 0.065859 0.075320 0.075498 0.075472 0.075478 0.075409
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11 0.0 0.064071 0.066452 0.075576 0.075838 0.075897 0.075542 0.075525
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12 0.0 0.064514 0.067074 0.075838 0.076189 0.076340 0.075602 0.075639
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13 0.0 0.064980 0.067731 0.076108 0.076555 0.076803 0.075658 0.075753
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14 0.0 0.065472 0.068429 0.076389 0.076938 0.077292 0.075711 0.075868
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8 9 10 11 12 13 14
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0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
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1 0.076334 0.073841 0.072973 0.070485 0.068088 0.065778 0.063554
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2 0.075334 0.072946 0.071789 0.069391 0.067077 0.064845 0.062693
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3 0.083645 0.080875 0.082361 0.079518 0.076777 0.074134 0.071588
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4 0.081405 0.078753 0.079982 0.077254 0.074622 0.072083 0.069637
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5 0.079080 0.076543 0.077541 0.074928 0.072404 0.069970 0.067623
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6 0.080793 0.078067 0.081296 0.078454 0.075712 0.073071 0.070527
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7 0.078170 0.075562 0.078544 0.075823 0.073198 0.070667 0.068229
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8 0.075609 0.073115 0.075864 0.073260 0.070747 0.068324 0.065989
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9 0.073115 0.070730 0.073260 0.070769 0.068364 0.066044 0.063809
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10 0.075864 0.073260 0.077608 0.074866 0.072223 0.069676 0.067224
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11 0.073260 0.070769 0.074866 0.072241 0.069711 0.067273 0.064924
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12 0.070747 0.068364 0.072223 0.069711 0.067288 0.064954 0.062705
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13 0.068324 0.066044 0.069676 0.067273 0.064954 0.062719 0.060565
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14 0.065989 0.063809 0.067224 0.064924 0.062705 0.060565 0.058503
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1 0.072148 0.072902 0.063646 0.064071 0.064514 0.064980 0.065472
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2 0.076101 0.077180 0.065859 0.066452 0.067074 0.067731 0.068429
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3 0.082240 0.082670 0.075320 0.075576 0.075838 0.076108 0.076389
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4 0.083128 0.083714 0.075498 0.075838 0.076189 0.076555 0.076938
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5 0.083801 0.084548 0.075472 0.075897 0.076340 0.076803 0.077292
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6 0.080150 0.080237 0.075478 0.075542 0.075602 0.075658 0.075711
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7 0.080474 0.080651 0.075409 0.075525 0.075639 0.075753 0.075868
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8 0.080766 0.081038 0.075293 0.075463 0.075634 0.075809 0.075988
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9 0.081038 0.081411 0.075136 0.075363 0.075595 0.075834 0.076082
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10 0.075293 0.075136 0.072406 0.072329 0.072240 0.072140 0.072028
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11 0.075463 0.075363 0.072329 0.072286 0.072234 0.072173 0.072101
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12 0.075634 0.075595 0.072240 0.072234 0.072220 0.072199 0.072171
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13 0.075809 0.075834 0.072140 0.072173 0.072199 0.072221 0.072238
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14 0.075988 0.076082 0.072028 0.072101 0.072171 0.072238 0.072303
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</pre></div>
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</div>
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@@ -1969,11 +1979,13 @@ We select values of the hyperparameter <span class="math notranslate nohighlight
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[2. 2.]
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Training MSE for OLS
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training MSE for OLS
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3.0
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</pre></div>
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</div>
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<img alt="_images/chapter2_252_1.png" src="_images/chapter2_252_1.png" />
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<img alt="_images/chapter2_252_2.png" src="_images/chapter2_252_2.png" />
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<p>We see here that we reach a plateau for the Ridge results. Writing out the coefficients <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>, we observe that they are getting smaller and smaller and our error stabilizes since the predicted values of <span class="math notranslate nohighlight">\(\tilde{\boldsymbol{y}}\)</span> approach zero.</p>
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