last update, perhaps
This commit is contained in:
Morten Hjorth-Jensen
@@ -1315,10 +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>0.04718566894028431
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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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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.014394967608286841
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4.011594819155615
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[[ 1.24508783 3.8595836 ]
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[ 3.8595836 12.92663007]]
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</pre></div>
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@@ -1355,10 +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>0.07836997022107646
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1.1378267322316808
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[[1. 0.63980097]
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[0.63980097 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07630326327869198
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1.6893421391051477
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[[1. 0.6373454]
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[0.6373454 1. ]]
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</pre></div>
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@@ -1388,30 +1388,30 @@ 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>[[ 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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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 0.20396326 0.93053605]
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[ 0.51936974 2.03440431]
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[-0.53851084 -1.2527027 ]
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[-0.50429483 0.72966563]
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[ 0.71314288 1.60421343]
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[-0.29995377 -2.31356849]
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[-0.14484831 -2.19932442]
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[-0.00570826 -0.32643011]
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[-0.22821607 0.56775345]
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[ 0.28505619 0.22545286]]
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0 1
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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 0.950873
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1 0.950873 1.000000
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0 0.203963 0.930536
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1 0.519370 2.034404
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2 -0.538511 -1.252703
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3 -0.504295 0.729666
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4 0.713143 1.604213
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5 -0.299954 -2.313568
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6 -0.144848 -2.199324
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7 -0.005708 -0.326430
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8 -0.228216 0.567753
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9 0.285056 0.225453
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0 1
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0 1.0000 0.6373
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1 0.6373 1.0000
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</pre></div>
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@@ -1468,37 +1468,40 @@ 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 0.074334 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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1 0.0 0.072835 0.076616 0.071253 0.074193 0.077337 0.063283 0.065305
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2 0.0 0.076616 0.082149 0.073794 0.077643 0.081811 0.064373 0.066929
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3 0.0 0.071253 0.073794 0.075423 0.077534 0.079745 0.070414 0.071978
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4 0.0 0.074193 0.077643 0.077534 0.080199 0.083034 0.071575 0.073483
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5 0.0 0.077337 0.081811 0.079745 0.083034 0.086576 0.072733 0.075028
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6 0.0 0.063283 0.064373 0.070414 0.071575 0.072733 0.068008 0.068984
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7 0.0 0.065305 0.066929 0.071978 0.073483 0.075028 0.068984 0.070181
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8 0.0 0.067523 0.069754 0.073667 0.075563 0.077550 0.070010 0.071459
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9 0.0 0.069964 0.072888 0.075496 0.077839 0.080330 0.071092 0.072826
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10 0.0 0.055866 0.055925 0.064324 0.064790 0.065183 0.063660 0.064182
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11 0.0 0.057298 0.057682 0.065508 0.066192 0.066830 0.064471 0.065139
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12 0.0 0.058866 0.059624 0.066786 0.067719 0.068637 0.065330 0.066162
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13 0.0 0.060590 0.061775 0.068171 0.069388 0.070625 0.066241 0.067260
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14 0.0 0.062489 0.064164 0.069675 0.071217 0.072820 0.067209 0.068440
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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 0.072148 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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1 0.067523 0.069964 0.055866 0.057298 0.058866 0.060590 0.062489
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2 0.069754 0.072888 0.055925 0.057682 0.059624 0.061775 0.064164
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3 0.073667 0.075496 0.064324 0.065508 0.066786 0.068171 0.069675
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4 0.075563 0.077839 0.064790 0.066192 0.067719 0.069388 0.071217
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5 0.077550 0.080330 0.065183 0.066830 0.068637 0.070625 0.072820
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6 0.070010 0.071092 0.063660 0.064471 0.065330 0.066241 0.067209
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7 0.071459 0.072826 0.064182 0.065139 0.066162 0.067260 0.068440
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8 0.073021 0.074713 0.064703 0.065823 0.067032 0.068341 0.069761
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9 0.074713 0.076776 0.065219 0.066524 0.067943 0.069492 0.071186
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10 0.064703 0.065219 0.060678 0.061187 0.061707 0.062241 0.062786
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11 0.065823 0.066524 0.061187 0.061793 0.062424 0.063082 0.063768
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12 0.067032 0.067943 0.061707 0.062424 0.063180 0.063978 0.064823
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13 0.068341 0.069492 0.062241 0.063082 0.063978 0.064935 0.065960
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14 0.069761 0.071186 0.062786 0.063768 0.064823 0.065960 0.067192
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</pre></div>
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@@ -1979,13 +1982,11 @@ We select values of the hyperparameter <span class="math notranslate nohighlight
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[2. 2.]
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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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Training MSE for OLS
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3.0
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</pre></div>
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<img alt="_images/chapter2_252_2.png" src="_images/chapter2_252_2.png" />
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<img alt="_images/chapter2_252_1.png" src="_images/chapter2_252_1.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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