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This commit is contained in:
Morten Hjorth-Jensen
@@ -1204,10 +1204,10 @@ 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>0.04279651270127165
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4.090014704675496
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[[0.77618224 2.31644071]
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[2.31644071 8.00233155]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.024023115996453476
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4.231224729143838
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[[ 0.86171505 2.958476 ]
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[ 2.958476 11.14451625]]
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</pre></div>
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@@ -1244,10 +1244,10 @@ a more brute force way. Here we scale the mean values for each column of the des
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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>0.08322642264994606
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1.7484669886413582
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[[1. 0.6960326]
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[0.6960326 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07897647347778382
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1.7546383166870465
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[[1. 0.61406871]
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[0.61406871 1. ]]
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</pre></div>
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</div>
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@@ -1277,30 +1277,30 @@ 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>[[-0.84886051 -1.52171908]
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[-0.91616374 -3.74126425]
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[ 1.4399427 4.17240522]
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[-0.993865 -2.87927126]
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[ 0.51782322 0.94077605]
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[ 0.80279641 3.93037171]
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[ 0.53952479 0.67970864]
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[ 1.40829683 5.9212905 ]
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[ 0.03326365 -0.11962537]
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[-1.98275836 -7.38267217]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1.4224801 5.74297504]
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[-0.01053024 -1.16804444]
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[ 0.01257962 -1.29082851]
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[ 0.56500639 2.63754524]
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[-0.33680101 -1.01023891]
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[-0.38906684 -0.89335182]
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[ 0.56437897 2.06942103]
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[-1.51131471 -4.53479276]
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[ 0.55518724 0.75623206]
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[-0.87191952 -2.30891693]]
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0 1
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0 -0.848861 -1.521719
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1 -0.916164 -3.741264
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2 1.439943 4.172405
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3 -0.993865 -2.879271
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4 0.517823 0.940776
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5 0.802796 3.930372
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6 0.539525 0.679709
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7 1.408297 5.921291
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8 0.033264 -0.119625
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9 -1.982758 -7.382672
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0 1
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0 1.00000 0.97202
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1 0.97202 1.00000
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0 1.422480 5.742975
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1 -0.010530 -1.168044
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2 0.012580 -1.290829
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3 0.565006 2.637545
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4 -0.336801 -1.010239
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5 -0.389067 -0.893352
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6 0.564379 2.069421
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7 -1.511315 -4.534793
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8 0.555187 0.756232
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9 -0.871920 -2.308917
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0 1
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0 1.000000 0.954988
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1 0.954988 1.000000
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</pre></div>
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</div>
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</div>
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@@ -1357,37 +1357,37 @@ 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 0.088476 0.091031 0.086154 0.087387 0.088765 0.076329 0.077009
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2 0.0 0.091031 0.094722 0.088119 0.090039 0.092126 0.077467 0.078609
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3 0.0 0.086154 0.088119 0.089551 0.090358 0.091266 0.082786 0.083087
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4 0.0 0.087387 0.090039 0.090358 0.091620 0.092998 0.083026 0.083648
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5 0.0 0.088765 0.092126 0.091266 0.092998 0.094867 0.083320 0.084277
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6 0.0 0.076329 0.077467 0.082786 0.083026 0.083320 0.078845 0.078710
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7 0.0 0.077009 0.078609 0.083087 0.083648 0.084277 0.078710 0.078812
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8 0.0 0.077822 0.079901 0.083495 0.084391 0.085368 0.078651 0.078999
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9 0.0 0.078776 0.081357 0.084018 0.085264 0.086608 0.078672 0.079278
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10 0.0 0.067141 0.067560 0.075017 0.074773 0.074552 0.073013 0.072523
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11 0.0 0.067449 0.068197 0.074992 0.074986 0.075012 0.072651 0.072342
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12 0.0 0.067864 0.068953 0.075061 0.075302 0.075586 0.072361 0.072240
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13 0.0 0.068391 0.069838 0.075228 0.075728 0.076282 0.072147 0.072222
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14 0.0 0.069036 0.070860 0.075498 0.076270 0.077108 0.072011 0.072290
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1 0.0 0.082939 0.082189 0.090270 0.084728 0.079489 0.086174 0.080200
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2 0.0 0.082189 0.084434 0.087894 0.084147 0.080455 0.082852 0.078167
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3 0.0 0.090270 0.087894 0.103822 0.096894 0.090337 0.102449 0.095171
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4 0.0 0.084728 0.084147 0.096894 0.091477 0.086249 0.095313 0.089306
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5 0.0 0.079489 0.080455 0.090337 0.086249 0.082204 0.088560 0.083681
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6 0.0 0.086174 0.082852 0.102449 0.095313 0.088560 0.103257 0.095902
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7 0.0 0.080200 0.078167 0.095171 0.089306 0.083681 0.095902 0.089683
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8 0.0 0.074838 0.073925 0.088606 0.083849 0.079214 0.089246 0.084019
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9 0.0 0.070034 0.070105 0.082683 0.078899 0.075138 0.083217 0.078860
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10 0.0 0.080407 0.076760 0.097602 0.090730 0.084223 0.099777 0.092769
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11 0.0 0.074763 0.072145 0.090755 0.084979 0.079447 0.092879 0.086877
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12 0.0 0.069748 0.068019 0.084645 0.079819 0.075136 0.086705 0.081576
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13 0.0 0.065282 0.064329 0.079179 0.075181 0.071243 0.081163 0.076795
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14 0.0 0.061297 0.061028 0.074276 0.071003 0.067720 0.076173 0.072471
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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 0.077822 0.078776 0.067141 0.067449 0.067864 0.068391 0.069036
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2 0.079901 0.081357 0.067560 0.068197 0.068953 0.069838 0.070860
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3 0.083495 0.084018 0.075017 0.074992 0.075061 0.075228 0.075498
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4 0.084391 0.085264 0.074773 0.074986 0.075302 0.075728 0.076270
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5 0.085368 0.086608 0.074552 0.075012 0.075586 0.076282 0.077108
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6 0.078651 0.078672 0.073013 0.072651 0.072361 0.072147 0.072011
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7 0.078999 0.079278 0.072523 0.072342 0.072240 0.072222 0.072290
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8 0.079443 0.079991 0.072084 0.072091 0.072184 0.072370 0.072652
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9 0.079991 0.080821 0.071698 0.071901 0.072198 0.072598 0.073105
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10 0.072084 0.071698 0.068724 0.068098 0.067527 0.067011 0.066553
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11 0.072091 0.071901 0.068098 0.067615 0.067191 0.066828 0.066528
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12 0.072184 0.072198 0.067527 0.067191 0.066919 0.066714 0.066580
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13 0.072370 0.072598 0.067011 0.066828 0.066714 0.066675 0.066714
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14 0.072652 0.073105 0.066553 0.066528 0.066580 0.066714 0.066934
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1 0.074838 0.070034 0.080407 0.074763 0.069748 0.065282 0.061297
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2 0.073925 0.070105 0.076760 0.072145 0.068019 0.064329 0.061028
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3 0.088606 0.082683 0.097602 0.090755 0.084645 0.079179 0.074276
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4 0.083849 0.078899 0.090730 0.084979 0.079819 0.075181 0.071003
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5 0.079214 0.075138 0.084223 0.079447 0.075136 0.071243 0.067720
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6 0.089246 0.083217 0.099777 0.092879 0.086705 0.081163 0.076173
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7 0.084019 0.078860 0.092769 0.086877 0.081576 0.076795 0.072471
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8 0.079225 0.074833 0.086409 0.081396 0.076862 0.072751 0.069015
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9 0.074833 0.071120 0.080630 0.076389 0.072529 0.069010 0.065796
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10 0.086409 0.080630 0.097396 0.090830 0.084936 0.079631 0.074839
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11 0.081396 0.076389 0.090830 0.085165 0.080056 0.075433 0.071239
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12 0.076862 0.072529 0.084936 0.080056 0.075629 0.071604 0.067934
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13 0.072751 0.069010 0.079631 0.075433 0.071604 0.068104 0.064896
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14 0.069015 0.065796 0.074839 0.071239 0.067934 0.064896 0.062097
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</pre></div>
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</div>
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</div>
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@@ -1611,7 +1611,7 @@ We have already analyzed the OLS solutions in terms of the eigenvectors (the col
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\[
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\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
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\]</div>
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<p>with the vectors <span class="math notranslate nohighlight">\(\boldsymbol{u}_j\)</span> being the columns of <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> from the SVD of the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
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<p>with the vectors <span class="math notranslate nohighlight">\(\boldsymbol{u}_j\)</span> being the columns of <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> from the SVD of the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>. Note that the sums goes to <span class="math notranslate nohighlight">\(p-1\)</span> since, by definition, <span class="math notranslate nohighlight">\(\sigma_j=0\)</span> for <span class="math notranslate nohighlight">\(j > p-1\)</span>.</p>
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<p>Since <span class="math notranslate nohighlight">\(\lambda \geq 0\)</span>, it means that compared to OLS, we have</p>
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<div class="math notranslate nohighlight">
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\[
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