lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

update book

Browse Source
This commit is contained in:
Morten Hjorth-Jensen
2022-08-23 11:19:05 +02:00
parent 602b1c6ee0
commit 398fba5b97
233 changed files with 5964 additions and 6430 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/LectureNotes/_build/html/chapter2.html
+222 -129
View File
@@ -7,8 +7,8 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>4. Ridge and Lasso Regression &#8212; Applied Data Analysis and Machine Learning</title>
<link href="_static/css/theme.css" rel="stylesheet" />
<link href="_static/css/index.c5995385ac14fb8791e8eb36b4908be2.css" rel="stylesheet" />
<link href="_static/css/theme.css" rel="stylesheet">
<link href="_static/css/index.ff1ffe594081f20da1ef19478df9384b.css" rel="stylesheet">
<link rel="stylesheet"
@@ -31,7 +31,7 @@
<link rel="stylesheet" type="text/css" href="_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-variables.06eb56fa6e07937060861dad626602ad.css" />
<link rel="preload" as="script" href="_static/js/index.1c5a1a01449ed65a7b51.js">
<link rel="preload" as="script" href="_static/js/index.be7d3bbb2ef33a8344ce.js">
<script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
<script src="_static/jquery.js"></script>
@@ -41,22 +41,24 @@
<script src="_static/clipboard.min.js"></script>
<script src="_static/copybutton.js"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
<script src="_static/sphinx-book-theme.12a9622fbb08dcb3a2a40b2c02b83a57.js"></script>
<script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<script async="async" src="https://unpkg.com/thebe@0.5.1/lib/index.js"></script>
<script>
const thebe_selector = ".thebe"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output"
</script>
<script src="_static/sphinx-book-theme.d59cb220de22ca1c485ebbdc042f0030.js"></script>
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
const thebe_selector = ".thebe,.cell"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output, .cell_output"
</script>
<script async="async" src="_static/sphinx-thebe.js"></script>
<script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="next" title="5. Resampling Methods" href="chapter3.html" />
<link rel="prev" title="3. Linear Regression" href="chapter1.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="en" />
<meta name="docsearch:language" content="None">
<!-- Google Analytics -->
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
@@ -91,11 +93,11 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="intro.html">
Applied Data Analysis and Machine Learning, FYS-STK3155/4155 at the University of Oslo, Norway
Applied Data Analysis and Machine Learning
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
About the course
</span>
@@ -117,7 +119,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Review of Statistics with Resampling Techniques and Linear Algebra
</span>
@@ -134,7 +136,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
From Regression to Support Vector Machines
</span>
@@ -171,7 +173,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Decision Trees, Ensemble Methods and Boosting
</span>
@@ -188,7 +190,7 @@
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Dimensionality Reduction
</span>
@@ -199,8 +201,13 @@
11. Basic ideas of the Principal Component Analysis (PCA)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="clustering.html">
12. Clustering and Unsupervised Learning
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Deep Learning Methods
</span>
@@ -208,17 +215,27 @@
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter9.html">
12. Neural networks
13. Neural networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter10.html">
13. Building a Feed Forward Neural Network
14. Building a Feed Forward Neural Network
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter11.html">
14. Solving Differential Equations with Deep Learning
15. Solving Differential Equations with Deep Learning
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter12.html">
16. Convolutional Neural Networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter13.html">
17. Recurrent neural networks: Overarching view
</a>
</li>
</ul>
@@ -267,7 +284,7 @@
data-placement="left">.ipynb</button></a>
<!-- Download PDF via print -->
<button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
onClick="window.print()" data-toggle="tooltip" data-placement="left">.pdf</button>
onclick="printPdf(this)" data-toggle="tooltip" data-placement="left">.pdf</button>
</div>
</div>
@@ -285,7 +302,7 @@
</div>
<!-- Table of contents -->
<div class="d-none d-md-block col-md-2 bd-toc show">
<div class="d-none d-md-block col-md-2 bd-toc show noprint">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
@@ -370,7 +387,95 @@
</div>
<div id="main-content" class="row">
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
<!-- Table of contents that is only displayed when printing the page -->
<div id="jb-print-docs-body" class="onlyprint">
<h1>Ridge and Lasso Regression</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#mathematical-interpretation-of-ordinary-least-squares">
4.1. Mathematical Interpretation of Ordinary Least Squares
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-singular-value-decomposition">
4.2. The singular value decomposition
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#basic-math-of-the-svd">
4.3. Basic math of the SVD
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#codes-for-the-svd">
4.4. Codes for the SVD
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#code-for-svd-and-inversion-of-matrices">
4.5. Code for SVD and Inversion of Matrices
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#mathematics-of-the-svd-and-implications">
4.6. Mathematics of the SVD and implications
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#further-properties-important-for-our-analyses-later">
4.7. Further properties (important for our analyses later)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#meet-the-covariance-matrix">
4.8. Meet the Covariance Matrix
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#linking-with-the-svd">
4.9. Linking with the SVD
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
4.10. Ridge and Lasso Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
4.11. Linking the regression analysis with a statistical interpretation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#deriving-ols-from-a-probability-distribution">
4.12. Deriving OLS from a probability distribution
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#bayes-theorem-and-ridge-and-lasso-regression">
4.13. Bayes’ Theorem and Ridge and Lasso Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#linking-bayes-theorem-with-ridge-and-lasso-regression">
4.14. Linking Bayes’ Theorem with Ridge and Lasso Regression
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div>
<div class="tex2jax_ignore mathjax_ignore section" id="ridge-and-lasso-regression">
@@ -721,13 +826,13 @@ The simple answer is to use the linear algebra function for the pseudoinverse, t
[2 4 5]
[3 5 6]]
test U
[[ 2.22044605e-16 -1.11362930e-15 -8.52945798e-16]
[-1.11362930e-15 0.00000000e+00 -1.37835429e-16]
[-8.52945798e-16 -1.37835429e-16 -1.11022302e-16]]
[[ 4.44089210e-16 -4.69484813e-16 -6.67314874e-16]
[-4.69484813e-16 -4.44089210e-16 -1.54041041e-16]
[-6.67314874e-16 -1.54041041e-16 1.11022302e-16]]
test VT
[[ 1.11022302e-16 -1.84228957e-16 2.68545647e-16]
[-1.84228957e-16 0.00000000e+00 -6.33166055e-17]
[ 2.68545647e-16 -6.33166055e-17 -1.11022302e-16]]
[[ 2.22044605e-16 3.78156479e-17 1.85278920e-16]
[ 3.78156479e-17 0.00000000e+00 -6.33166055e-17]
[ 1.85278920e-16 -6.33166055e-17 -1.11022302e-16]]
[[0. 0. 0.]
[0. 0. 0.]
[0. 0. 0.]]
@@ -1109,10 +1214,10 @@ 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>0.0934597075922044
4.185278747229417
[[0.84292394 2.47432993]
[2.47432993 8.32458459]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.13876586436927824
3.722047011333792
[[ 1.233528 3.58428804]
[ 3.58428804 11.47942814]]
</pre></div>
</div>
</div>
@@ -1149,10 +1254,10 @@ 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>0.08673755293381497
1.3925515884752442
[[1. 0.62364974]
[0.62364974 1. ]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.08464758160254343
1.8503720991789538
[[1. 0.65626043]
[0.65626043 1. ]]
</pre></div>
</div>
</div>
@@ -1182,30 +1287,30 @@ 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>[[ 0.77034458 1.03056589]
[-0.46986815 -1.26822717]
[ 0.82650876 -0.37655936]
[ 1.1323603 2.97788031]
[-0.07989327 0.08630331]
[-1.8608479 -5.59839245]
[-0.3528556 0.06021285]
[ 0.25286618 1.40116777]
[-1.1224392 -2.14042769]
[ 0.90382431 3.82747653]]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-1.41876267 -4.93248252]
[ 1.83687444 5.28097861]
[ 0.37429133 0.59766 ]
[ 0.59159438 1.71869727]
[-0.80315282 -0.89348922]
[-0.38748219 -2.12288563]
[-2.08917679 -5.64933923]
[ 0.27803645 0.89944994]
[ 1.23703839 3.2321528 ]
[ 0.38073947 1.86925797]]
0 1
0 0.770345 1.030566
1 -0.469868 -1.268227
2 0.826509 -0.376559
3 1.132360 2.977880
4 -0.079893 0.086303
5 -1.860848 -5.598392
6 -0.352856 0.060213
7 0.252866 1.401168
8 -1.122439 -2.140428
9 0.903824 3.827477
0 -1.418763 -4.932483
1 1.836874 5.280979
2 0.374291 0.597660
3 0.591594 1.718697
4 -0.803153 -0.893489
5 -0.387482 -2.122886
6 -2.089177 -5.649339
7 0.278036 0.899450
8 1.237038 3.232153
9 0.380739 1.869258
0 1
0 1.000000 0.900449
1 0.900449 1.000000
0 1.000000 0.977418
1 0.977418 1.000000
</pre></div>
</div>
</div>
@@ -1262,37 +1367,37 @@ 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 0.088611 0.086567 0.090229 0.089425 0.088631 0.082174 0.081773
2 0.0 0.086567 0.085416 0.088176 0.087834 0.087501 0.080517 0.080406
3 0.0 0.090229 0.088176 0.097360 0.096623 0.095871 0.091891 0.091630
4 0.0 0.089425 0.087834 0.096623 0.096173 0.095702 0.091426 0.091363
5 0.0 0.088631 0.087501 0.095871 0.095702 0.095510 0.090929 0.091060
6 0.0 0.082174 0.080517 0.091891 0.091426 0.090929 0.088853 0.088816
7 0.0 0.081773 0.080406 0.091630 0.091363 0.091060 0.088816 0.088926
8 0.0 0.081431 0.080347 0.091414 0.091340 0.091224 0.088809 0.089059
9 0.0 0.081140 0.080334 0.091236 0.091349 0.091416 0.088825 0.089212
10 0.0 0.073695 0.072492 0.084408 0.084224 0.084000 0.083053 0.083220
11 0.0 0.073531 0.072530 0.084400 0.084364 0.084282 0.083227 0.083506
12 0.0 0.073423 0.072617 0.084444 0.084549 0.084604 0.083441 0.083829
13 0.0 0.073368 0.072750 0.084536 0.084777 0.084965 0.083692 0.084184
14 0.0 0.073362 0.072928 0.084670 0.085044 0.085361 0.083977 0.084570
1 0.0 0.081253 0.083015 0.080297 0.079157 0.078040 0.071112 0.069847
2 0.0 0.083015 0.086807 0.084909 0.084904 0.084702 0.076955 0.076338
3 0.0 0.080297 0.084909 0.084414 0.084862 0.085018 0.077793 0.077402
4 0.0 0.079157 0.084904 0.084862 0.086076 0.086872 0.079281 0.079383
5 0.0 0.078040 0.084702 0.085018 0.086872 0.088212 0.080319 0.080847
6 0.0 0.071112 0.076955 0.077793 0.079281 0.080319 0.073719 0.074034
7 0.0 0.069847 0.076338 0.077402 0.079383 0.080847 0.074034 0.074693
8 0.0 0.068735 0.075760 0.077003 0.079405 0.081233 0.074237 0.075195
9 0.0 0.067769 0.075241 0.076628 0.079389 0.081533 0.074375 0.075594
10 0.0 0.062136 0.068318 0.069888 0.071921 0.073449 0.067622 0.068376
11 0.0 0.061149 0.067723 0.069408 0.071767 0.073583 0.067612 0.068608
12 0.0 0.060309 0.067213 0.068991 0.071632 0.073699 0.067599 0.068806
13 0.0 0.059601 0.066789 0.068642 0.071529 0.073816 0.067597 0.068992
14 0.0 0.059013 0.066449 0.068363 0.071467 0.073949 0.067620 0.069181
8 9 10 11 12 13 14
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.081431 0.081140 0.073695 0.073531 0.073423 0.073368 0.073362
2 0.080347 0.080334 0.072492 0.072530 0.072617 0.072750 0.072928
3 0.091414 0.091236 0.084408 0.084400 0.084444 0.084536 0.084670
4 0.091340 0.091349 0.084224 0.084364 0.084549 0.084777 0.085044
5 0.091224 0.091416 0.084000 0.084282 0.084604 0.084965 0.085361
6 0.088809 0.088825 0.083053 0.083227 0.083441 0.083692 0.083977
7 0.089059 0.089212 0.083220 0.083506 0.083829 0.084184 0.084570
8 0.089329 0.089614 0.083404 0.083799 0.084226 0.084683 0.085167
9 0.089614 0.090028 0.083600 0.084101 0.084629 0.085184 0.085764
10 0.083404 0.083600 0.078667 0.078992 0.079347 0.079729 0.080137
11 0.083799 0.084101 0.078992 0.079406 0.079847 0.080312 0.080801
12 0.084226 0.084629 0.079347 0.079847 0.080370 0.080916 0.081483
13 0.084683 0.085184 0.079729 0.080312 0.080916 0.081540 0.082183
14 0.085167 0.085764 0.080137 0.080801 0.081483 0.082183 0.082900
1 0.068735 0.067769 0.062136 0.061149 0.060309 0.059601 0.059013
2 0.075760 0.075241 0.068318 0.067723 0.067213 0.066789 0.066449
3 0.077003 0.076628 0.069888 0.069408 0.068991 0.068642 0.068363
4 0.079405 0.079389 0.071921 0.071767 0.071632 0.071529 0.071467
5 0.081233 0.081533 0.073449 0.073583 0.073699 0.073816 0.073949
6 0.074237 0.074375 0.067622 0.067612 0.067599 0.067597 0.067620
7 0.075195 0.075594 0.068376 0.068608 0.068806 0.068992 0.069181
8 0.075959 0.076589 0.068965 0.069409 0.069796 0.070150 0.070491
9 0.076589 0.077425 0.069441 0.070074 0.070630 0.071136 0.071614
10 0.068965 0.069441 0.063052 0.063364 0.063631 0.063874 0.064110
11 0.069409 0.070074 0.063364 0.063851 0.064274 0.064658 0.065020
12 0.069796 0.070630 0.063631 0.064274 0.064838 0.065348 0.065826
13 0.070150 0.071136 0.063874 0.064658 0.065348 0.065974 0.066558
14 0.070491 0.071614 0.064110 0.065020 0.065826 0.066558 0.067240
</pre></div>
</div>
</div>
@@ -2149,9 +2254,9 @@ set of <span class="math notranslate nohighlight">\(\lambda\)</span> values.</p>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 2.03099776 -0.17917768 5.18029127]
Training MSE for OLS
0.009163470508352211
0.009163470508352218
Test MSE OLS
0.008675369724976501
0.008675369724975977
</pre></div>
</div>
<img alt="_images/chapter2_249_1.png" src="_images/chapter2_249_1.png" />
@@ -2428,7 +2533,7 @@ The following code example solves the simpler problem we discussed above, where
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">ModuleNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="o">/</span><span class="n">var</span><span class="o">/</span><span class="n">folders</span><span class="o">/</span><span class="n">jy</span><span class="o">/</span><span class="n">g42mrgv128v34gnnhxwk9nrc0000gp</span><span class="o">/</span><span class="n">T</span><span class="o">/</span><span class="n">ipykernel_42449</span><span class="o">/</span><span class="mf">3530606977.</span><span class="n">py</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="o">&lt;</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">12</span><span class="o">-</span><span class="n">d670a873ab0c</span><span class="o">&gt;</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="ne">----&gt; </span><span class="mi">1</span> <span class="kn">from</span> <span class="nn">cvxopt</span> <span class="kn">import</span> <span class="n">matrix</span><span class="p">,</span> <span class="n">spdiag</span><span class="p">,</span> <span class="n">mul</span><span class="p">,</span> <span class="n">div</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">setseed</span>
<span class="g g-Whitespace"> </span><span class="mi">2</span> <span class="kn">from</span> <span class="nn">cvxopt</span> <span class="kn">import</span> <span class="n">blas</span><span class="p">,</span> <span class="n">lapack</span><span class="p">,</span> <span class="n">solvers</span><span class="p">,</span> <span class="n">sparse</span><span class="p">,</span> <span class="n">spmatrix</span>
<span class="g g-Whitespace"> </span><span class="mi">3</span> <span class="kn">import</span> <span class="nn">math</span>
@@ -3049,54 +3154,42 @@ decreasing <span class="math notranslate nohighlight">\(\lambda\)</span> and shr
</div>
<div class='prev-next-bottom'>
<div id="prev">
<a class="left-prev" href="chapter1.html" title="previous page">
<i class="prevnext-label fas fa-angle-left"></i>
<div class="prevnext-info">
<p class="prevnext-label">previous</p>
<p class="prevnext-title"><span class="section-number">3. </span>Linear Regression</p>
</div>
</a>
<!-- Previous / next buttons -->
<div class='prev-next-area'>
<a class='left-prev' id="prev-link" href="chapter1.html" title="previous page">
<i class="fas fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title"><span class="section-number">3. </span>Linear Regression</p>
</div>
</a>
<a class='right-next' id="next-link" href="chapter3.html" title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title"><span class="section-number">5. </span>Resampling Methods</p>
</div>
<div id="next">
<a class="right-next" href="chapter3.html" title="next page">
<div class="prevnext-info">
<p class="prevnext-label">next</p>
<p class="prevnext-title"><span class="section-number">5. </span>Resampling Methods</p>
</div>
<i class="prevnext-label fas fa-angle-right"></i>
</a>
</div>
<i class="fas fa-angle-right"></i>
</a>
</div>
</div>
</div>
<footer class="footer">
<div class="container">
<p>
By Morten Hjorth-Jensen<br/>
&copy; Copyright 2021.<br/>
</p>
</div>
</footer>
<p>
By Morten Hjorth-Jensen<br/>
&copy; Copyright 2021.<br/>
</p>
</footer>
</main>
</div>
</div>
<script src="_static/js/index.1c5a1a01449ed65a7b51.js"></script>
<script src="_static/js/index.be7d3bbb2ef33a8344ce.js"></script>
</body>
</html>