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This commit is contained in:
Morten Hjorth-Jensen
2022-09-07 15:43:17 +02:00
parent 350ab4336b
commit 3be32782b7
9 changed files with 1176 additions and 1176 deletions
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doc/BookChapters/chapter2.do.txt
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@@ -1288,7 +1288,7 @@ For the sake of simplicity, let us assume that the design matrix is orthonormal,
In this case the standard OLS results in
!bt
\[
\bm{\beta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_j\bm{u}_j^T\bm{y},
\bm{\beta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{n-1}\bm{u}_i\bm{u}_i^T\bm{y},
\]
!et
doc/LectureNotes/_build/.doctrees/chapter2.doctree
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doc/LectureNotes/_build/.doctrees/environment.pickle
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doc/LectureNotes/_build/html/_sources/chapter2.ipynb
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doc/LectureNotes/_build/html/chapter2.html
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@@ -1204,10 +1204,10 @@ covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.024023115996453476
4.231224729143838
[[ 0.86171505 2.958476 ]
[ 2.958476 11.14451625]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.08343519179767796
3.6669838269597004
[[ 1.09677394 3.55026099]
[ 3.55026099 12.59182949]]
</pre></div>
</div>
</div>
@@ -1244,10 +1244,10 @@ a more brute force way. Here we scale the mean values for each column of the des
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07897647347778382
1.7546383166870465
[[1. 0.61406871]
[0.61406871 1. ]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07844310540708652
1.689230294669661
[[1. 0.69023787]
[0.69023787 1. ]]
</pre></div>
</div>
</div>
@@ -1277,30 +1277,30 @@ this matrix we easily see that it is a positive definite matrix.</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1.4224801 5.74297504]
[-0.01053024 -1.16804444]
[ 0.01257962 -1.29082851]
[ 0.56500639 2.63754524]
[-0.33680101 -1.01023891]
[-0.38906684 -0.89335182]
[ 0.56437897 2.06942103]
[-1.51131471 -4.53479276]
[ 0.55518724 0.75623206]
[-0.87191952 -2.30891693]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1.79876149 5.76881337]
[-0.03184647 -0.35207264]
[ 0.02207532 -0.60802823]
[ 0.19096968 -0.0202458 ]
[ 0.18826299 -1.20664007]
[-1.23792491 -3.7873493 ]
[-0.07212695 0.3644017 ]
[-0.5554964 -1.02354476]
[-0.09252364 1.11231934]
[-0.2101511 -0.2476536 ]]
0 1
0 1.422480 5.742975
1 -0.010530 -1.168044
2 0.012580 -1.290829
3 0.565006 2.637545
4 -0.336801 -1.010239
5 -0.389067 -0.893352
6 0.564379 2.069421
7 -1.511315 -4.534793
8 0.555187 0.756232
9 -0.871920 -2.308917
0 1.798761 5.768813
1 -0.031846 -0.352073
2 0.022075 -0.608028
3 0.190970 -0.020246
4 0.188263 -1.206640
5 -1.237925 -3.787349
6 -0.072127 0.364402
7 -0.555496 -1.023545
8 -0.092524 1.112319
9 -0.210151 -0.247654
0 1
0 1.000000 0.954988
1 0.954988 1.000000
0 1.000000 0.930683
1 0.930683 1.000000
</pre></div>
</div>
</div>
@@ -1357,37 +1357,37 @@ this matrix we easily see that it is a positive definite matrix.</p>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1 2 3 4 5 6 7 \
0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.0 0.082939 0.082189 0.090270 0.084728 0.079489 0.086174 0.080200
2 0.0 0.082189 0.084434 0.087894 0.084147 0.080455 0.082852 0.078167
3 0.0 0.090270 0.087894 0.103822 0.096894 0.090337 0.102449 0.095171
4 0.0 0.084728 0.084147 0.096894 0.091477 0.086249 0.095313 0.089306
5 0.0 0.079489 0.080455 0.090337 0.086249 0.082204 0.088560 0.083681
6 0.0 0.086174 0.082852 0.102449 0.095313 0.088560 0.103257 0.095902
7 0.0 0.080200 0.078167 0.095171 0.089306 0.083681 0.095902 0.089683
8 0.0 0.074838 0.073925 0.088606 0.083849 0.079214 0.089246 0.084019
9 0.0 0.070034 0.070105 0.082683 0.078899 0.075138 0.083217 0.078860
10 0.0 0.080407 0.076760 0.097602 0.090730 0.084223 0.099777 0.092769
11 0.0 0.074763 0.072145 0.090755 0.084979 0.079447 0.092879 0.086877
12 0.0 0.069748 0.068019 0.084645 0.079819 0.075136 0.086705 0.081576
13 0.0 0.065282 0.064329 0.079179 0.075181 0.071243 0.081163 0.076795
14 0.0 0.061297 0.061028 0.074276 0.071003 0.067720 0.076173 0.072471
1 0.0 0.093979 0.084151 0.094362 0.088519 0.082921 0.085936 0.081262
2 0.0 0.084151 0.076518 0.083152 0.078646 0.074346 0.075262 0.071569
3 0.0 0.094362 0.083152 0.100670 0.093844 0.087235 0.094848 0.089505
4 0.0 0.088519 0.078646 0.093844 0.087891 0.082122 0.088261 0.083573
5 0.0 0.082921 0.074346 0.087235 0.082122 0.077164 0.081848 0.077785
6 0.0 0.085936 0.075262 0.094848 0.088261 0.081848 0.091311 0.086178
7 0.0 0.081262 0.071569 0.089505 0.083573 0.077785 0.086178 0.081547
8 0.0 0.076905 0.068136 0.084476 0.079158 0.073959 0.081321 0.077158
9 0.0 0.072828 0.064937 0.079719 0.074980 0.070339 0.076697 0.072974
10 0.0 0.077359 0.067693 0.087224 0.081211 0.075335 0.085249 0.080555
11 0.0 0.073369 0.064483 0.082716 0.077228 0.071852 0.080922 0.076637
12 0.0 0.069667 0.061505 0.078508 0.073506 0.068593 0.076866 0.072958
13 0.0 0.066228 0.058740 0.074569 0.070018 0.065537 0.073052 0.069494
14 0.0 0.063024 0.056169 0.070871 0.066741 0.062665 0.069454 0.066220
8 9 10 11 12 13 14
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.074838 0.070034 0.080407 0.074763 0.069748 0.065282 0.061297
2 0.073925 0.070105 0.076760 0.072145 0.068019 0.064329 0.061028
3 0.088606 0.082683 0.097602 0.090755 0.084645 0.079179 0.074276
4 0.083849 0.078899 0.090730 0.084979 0.079819 0.075181 0.071003
5 0.079214 0.075138 0.084223 0.079447 0.075136 0.071243 0.067720
6 0.089246 0.083217 0.099777 0.092879 0.086705 0.081163 0.076173
7 0.084019 0.078860 0.092769 0.086877 0.081576 0.076795 0.072471
8 0.079225 0.074833 0.086409 0.081396 0.076862 0.072751 0.069015
9 0.074833 0.071120 0.080630 0.076389 0.072529 0.069010 0.065796
10 0.086409 0.080630 0.097396 0.090830 0.084936 0.079631 0.074839
11 0.081396 0.076389 0.090830 0.085165 0.080056 0.075433 0.071239
12 0.076862 0.072529 0.084936 0.080056 0.075629 0.071604 0.067934
13 0.072751 0.069010 0.079631 0.075433 0.071604 0.068104 0.064896
14 0.069015 0.065796 0.074839 0.071239 0.067934 0.064896 0.062097
1 0.076905 0.072828 0.077359 0.073369 0.069667 0.066228 0.063024
2 0.068136 0.064937 0.067693 0.064483 0.061505 0.058740 0.056169
3 0.084476 0.079719 0.087224 0.082716 0.078508 0.074569 0.070871
4 0.079158 0.074980 0.081211 0.077228 0.073506 0.070018 0.066741
5 0.073959 0.070339 0.075335 0.071852 0.068593 0.065537 0.062665
6 0.081321 0.076697 0.085249 0.080922 0.076866 0.073052 0.069454
7 0.077158 0.072974 0.080555 0.076637 0.072958 0.069494 0.066220
8 0.073206 0.069436 0.076096 0.072558 0.069231 0.066093 0.063125
9 0.069436 0.066056 0.071836 0.068654 0.065657 0.062827 0.060146
10 0.076096 0.071836 0.080502 0.076532 0.072798 0.069276 0.065942
11 0.072558 0.068654 0.076532 0.072898 0.069476 0.066241 0.063173
12 0.069231 0.065657 0.072798 0.069476 0.066340 0.063371 0.060550
13 0.066093 0.062827 0.069276 0.066241 0.063371 0.060649 0.058058
14 0.063125 0.060146 0.065942 0.063173 0.060550 0.058058 0.055684
</pre></div>
</div>
</div>
@@ -1631,7 +1631,7 @@ eigenvalues ordered in a descending way, that is <span class="math notranslate n
<p>In this case the standard OLS results in</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
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doc/LectureNotes/_build/jupyter_execute/chapter2.py
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@@ -1164,7 +1164,7 @@ print(covariance_matrix)
# In this case the standard OLS results in
# $$
# \boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
# \boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
# $$
# and
doc/LectureNotes/chapter2.ipynb
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