small typos
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>-0.08343519179767796
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3.6669838269597004
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[[ 1.09677394 3.55026099]
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[ 3.55026099 12.59182949]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.030407120354722424
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3.7843182645426894
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[[ 1.22134069 3.4744219 ]
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[ 3.4744219 10.84203538]]
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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 class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07844310540708652
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1.689230294669661
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[[1. 0.69023787]
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[0.69023787 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.09023660662586945
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1.5216048821598704
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[[1. 0.62896882]
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[0.62896882 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>[[ 1.79876149 5.76881337]
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[-0.03184647 -0.35207264]
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[ 0.02207532 -0.60802823]
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[ 0.19096968 -0.0202458 ]
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[ 0.18826299 -1.20664007]
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[-1.23792491 -3.7873493 ]
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[-0.07212695 0.3644017 ]
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[-0.5554964 -1.02354476]
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[-0.09252364 1.11231934]
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[-0.2101511 -0.2476536 ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-1.30936179 -2.92208477]
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[ 0.61757228 1.64238022]
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[ 0.23816792 1.87666423]
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[-0.45402701 -2.54845458]
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[ 0.26848435 0.35074881]
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[-0.92143477 -1.96473528]
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[ 0.10516924 -0.36535019]
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[ 0.70721787 3.41211579]
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[ 0.75819326 1.03069324]
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[-0.00998135 -0.51197747]]
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0 1
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0 1.798761 5.768813
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1 -0.031846 -0.352073
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2 0.022075 -0.608028
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3 0.190970 -0.020246
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4 0.188263 -1.206640
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5 -1.237925 -3.787349
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6 -0.072127 0.364402
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7 -0.555496 -1.023545
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8 -0.092524 1.112319
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9 -0.210151 -0.247654
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0 -1.309362 -2.922085
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1 0.617572 1.642380
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2 0.238168 1.876664
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3 -0.454027 -2.548455
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4 0.268484 0.350749
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5 -0.921435 -1.964735
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6 0.105169 -0.365350
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7 0.707218 3.412116
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8 0.758193 1.030693
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9 -0.009981 -0.511977
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0 1
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0 1.000000 0.930683
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1 0.930683 1.000000
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0 1.000000 0.889686
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1 0.889686 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 0.093979 0.084151 0.094362 0.088519 0.082921 0.085936 0.081262
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2 0.0 0.084151 0.076518 0.083152 0.078646 0.074346 0.075262 0.071569
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3 0.0 0.094362 0.083152 0.100670 0.093844 0.087235 0.094848 0.089505
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4 0.0 0.088519 0.078646 0.093844 0.087891 0.082122 0.088261 0.083573
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5 0.0 0.082921 0.074346 0.087235 0.082122 0.077164 0.081848 0.077785
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6 0.0 0.085936 0.075262 0.094848 0.088261 0.081848 0.091311 0.086178
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7 0.0 0.081262 0.071569 0.089505 0.083573 0.077785 0.086178 0.081547
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8 0.0 0.076905 0.068136 0.084476 0.079158 0.073959 0.081321 0.077158
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9 0.0 0.072828 0.064937 0.079719 0.074980 0.070339 0.076697 0.072974
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10 0.0 0.077359 0.067693 0.087224 0.081211 0.075335 0.085249 0.080555
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11 0.0 0.073369 0.064483 0.082716 0.077228 0.071852 0.080922 0.076637
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12 0.0 0.069667 0.061505 0.078508 0.073506 0.068593 0.076866 0.072958
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13 0.0 0.066228 0.058740 0.074569 0.070018 0.065537 0.073052 0.069494
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14 0.0 0.063024 0.056169 0.070871 0.066741 0.062665 0.069454 0.066220
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1 0.0 0.092621 0.084912 0.094477 0.085405 0.077338 0.086802 0.078432
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2 0.0 0.084912 0.079260 0.088946 0.081311 0.074401 0.083242 0.075899
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3 0.0 0.094477 0.088946 0.102401 0.093930 0.086203 0.097710 0.089206
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4 0.0 0.085405 0.081311 0.093930 0.086841 0.080288 0.090630 0.083300
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5 0.0 0.077338 0.074401 0.086203 0.080288 0.074743 0.084032 0.077724
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6 0.0 0.086802 0.083242 0.097710 0.090630 0.084032 0.095617 0.088029
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7 0.0 0.078432 0.075899 0.089206 0.083300 0.077724 0.088029 0.081520
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8 0.0 0.071086 0.069386 0.081637 0.076726 0.072021 0.081198 0.075618
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9 0.0 0.064634 0.063612 0.074905 0.070835 0.066873 0.075058 0.070276
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10 0.0 0.078704 0.076525 0.090849 0.085027 0.079495 0.090503 0.083913
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11 0.0 0.071326 0.069907 0.083000 0.078157 0.073489 0.083251 0.077607
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12 0.0 0.064859 0.064062 0.076055 0.072041 0.068110 0.076783 0.071949
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13 0.0 0.059180 0.058890 0.069903 0.066590 0.063287 0.071009 0.066870
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14 0.0 0.054182 0.054305 0.064442 0.061725 0.058957 0.065846 0.062305
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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 0.076905 0.072828 0.077359 0.073369 0.069667 0.066228 0.063024
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2 0.068136 0.064937 0.067693 0.064483 0.061505 0.058740 0.056169
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3 0.084476 0.079719 0.087224 0.082716 0.078508 0.074569 0.070871
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4 0.079158 0.074980 0.081211 0.077228 0.073506 0.070018 0.066741
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5 0.073959 0.070339 0.075335 0.071852 0.068593 0.065537 0.062665
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6 0.081321 0.076697 0.085249 0.080922 0.076866 0.073052 0.069454
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7 0.077158 0.072974 0.080555 0.076637 0.072958 0.069494 0.066220
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8 0.073206 0.069436 0.076096 0.072558 0.069231 0.066093 0.063125
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9 0.069436 0.066056 0.071836 0.068654 0.065657 0.062827 0.060146
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10 0.076096 0.071836 0.080502 0.076532 0.072798 0.069276 0.065942
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11 0.072558 0.068654 0.076532 0.072898 0.069476 0.066241 0.063173
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12 0.069231 0.065657 0.072798 0.069476 0.066340 0.063371 0.060550
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13 0.066093 0.062827 0.069276 0.066241 0.063371 0.060649 0.058058
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14 0.063125 0.060146 0.065942 0.063173 0.060550 0.058058 0.055684
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1 0.071086 0.064634 0.078704 0.071326 0.064859 0.059180 0.054182
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2 0.069386 0.063612 0.076525 0.069907 0.064062 0.058890 0.054305
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3 0.081637 0.074905 0.090849 0.083000 0.076055 0.069903 0.064442
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4 0.076726 0.070835 0.085027 0.078157 0.072041 0.066590 0.061725
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5 0.072021 0.066873 0.079495 0.073489 0.068110 0.063287 0.058957
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6 0.081198 0.075058 0.090503 0.083251 0.076783 0.071009 0.065846
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7 0.075618 0.070276 0.083913 0.077607 0.071949 0.066870 0.062305
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8 0.070518 0.065869 0.077921 0.072435 0.067484 0.063014 0.058974
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9 0.065869 0.061821 0.072483 0.067710 0.063375 0.059437 0.055859
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10 0.077921 0.072483 0.086809 0.080332 0.074513 0.069281 0.064573
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11 0.072435 0.067710 0.080332 0.074708 0.069626 0.065032 0.060875
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12 0.067484 0.063375 0.074513 0.069626 0.065183 0.061144 0.057469
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13 0.063014 0.059437 0.069281 0.065032 0.061144 0.057588 0.054335
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14 0.058974 0.055859 0.064573 0.060875 0.057469 0.054335 0.051451
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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>. 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>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.</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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@@ -1631,7 +1631,7 @@ eigenvalues ordered in a descending way, that is <span class="math notranslate n
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<p>In this case the standard OLS results in</p>
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<div class="math notranslate nohighlight">
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\[
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\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\]</div>
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<p>and</p>
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<div class="math notranslate nohighlight">
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