more typos
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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>0.024023115996453476
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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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<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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</pre></div>
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</div>
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</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>
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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>0.07897647347778382
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1.7546383166870465
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[[1. 0.61406871]
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[0.61406871 1. ]]
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<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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</pre></div>
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</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>[[ 1.4224801 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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<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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0 1
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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 1.798761 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
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0 1.000000 0.954988
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1 0.954988 1.000000
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0 1.000000 0.930683
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1 0.930683 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 0.082939 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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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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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 0.074838 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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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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</pre></div>
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</div>
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</div>
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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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\[
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\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
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\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\]</div>
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<p>and</p>
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<div class="math notranslate nohighlight">
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