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Morten Hjorth-Jensen
2022-10-04 17:50:07 +02:00
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+202 -71
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@@ -7,8 +7,8 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>3. Linear Regression &#8212; Applied Data Analysis and Machine Learning</title>
<link href="_static/css/theme.css" rel="stylesheet" />
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@@ -31,31 +31,37 @@
<link rel="stylesheet" type="text/css" href="_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
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<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>
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@@ -94,7 +100,7 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
About the course
</span>
@@ -116,7 +122,7 @@ const thebe_selector_output = ".output, .cell_output"
</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>
@@ -133,7 +139,7 @@ const thebe_selector_output = ".output, .cell_output"
</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>
@@ -170,7 +176,7 @@ const thebe_selector_output = ".output, .cell_output"
</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>
@@ -187,7 +193,7 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
</ul>
<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Dimensionality Reduction
</span>
@@ -204,7 +210,7 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
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<p class="caption" role="heading">
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Deep Learning Methods
</span>
@@ -281,7 +287,7 @@ const thebe_selector_output = ".output, .cell_output"
data-placement="left">.ipynb</button></a>
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@@ -299,7 +305,7 @@ const thebe_selector_output = ".output, .cell_output"
</div>
<!-- Table of contents -->
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<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
@@ -427,7 +433,138 @@ const thebe_selector_output = ".output, .cell_output"
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<!-- Table of contents that is only displayed when printing the page -->
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<h1>Linear Regression</h1>
<!-- Table of contents -->
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<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="#introduction">
3.1. Introduction
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-machine-learning">
3.2. What is Machine Learning?
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-frequentist-approach-to-data-analysis">
3.2.1. A Frequentist approach to data analysis
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-a-good-model">
3.2.2. What is a good model?
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-linear-regression-model-using-scikit-learn">
3.3. Simple linear regression model using
<strong>
scikit-learn
</strong>
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#organizing-our-data">
3.3.1. Organizing our data
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#linear-regression-basic-elements">
3.4. Linear Regression, basic elements
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-chi-2-function">
3.4.1. The
<span class="math notranslate nohighlight">
\(\chi^2\)
</span>
function
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#fitting-an-equation-of-state-for-dense-nuclear-matter">
3.4.2. Fitting an Equation of State for Dense Nuclear Matter
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#splitting-our-data-in-training-and-test-data">
3.5. Splitting our Data in Training and Test data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-boston-housing-data-example">
3.6. The Boston housing data example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#housing-data-the-code">
3.7. Housing data, the code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reducing-the-number-of-degrees-of-freedom-overarching-view">
3.8. Reducing the number of degrees of freedom, overarching view
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#testing-the-means-squared-error-as-function-of-complexity">
3.9. Testing the Means Squared Error as function of Complexity
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercises">
3.10. Exercises
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-1-setting-up-various-python-environments">
3.11. Exercise 1: Setting up various Python environments
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-2-making-your-own-data-and-exploring-scikit-learn">
3.12. Exercise 2: making your own data and exploring scikit-learn
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-3-normalizing-our-data">
3.13. Exercise 3: Normalizing our data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-4-adding-ridge-regression">
3.14. Exercise 4: Adding Ridge Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-5-analytical-exercises">
3.15. Exercise 5: Analytical exercises
</a>
</li>
</ul>
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</div>
</div>
</div>
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
@@ -820,13 +957,13 @@ example of the functionality of <strong>Scikit-Learn</strong>.</p>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The intercept alpha:
[1.99274513]
[2.02408959]
Coefficient beta :
[[5.03570747]]
Mean squared error: 0.27
Variance score: 0.88
[[4.92811987]]
Mean squared error: 0.25
Variance score: 0.89
Mean squared log error: 0.01
Mean absolute error: 0.42
Mean absolute error: 0.39
</pre></div>
</div>
<img alt="_images/chapter1_19_1.png" src="_images/chapter1_19_1.png" />
@@ -926,7 +1063,7 @@ a linear <span class="math notranslate nohighlight">\(x\)</span>-dependence we s
</div>
<div class="cell_output docutils container">
<img alt="_images/chapter1_33_0.png" src="_images/chapter1_33_0.png" />
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.005000000000000011
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.005
</pre></div>
</div>
</div>
@@ -1111,26 +1248,32 @@ covert them into the <strong>pandas</strong> DataFrame structure.</p>
<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">ValueError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<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">8</span><span class="o">-</span><span class="mi">30</span><span class="n">b53504a633</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="nn">Input In [8],</span> in <span class="ni">&lt;cell line: 2&gt;</span><span class="nt">()</span>
<span class="g g-Whitespace"> </span><span class="mi">1</span> <span class="c1"># Read the experimental data with Pandas</span>
<span class="ne">----&gt; </span><span class="mi">2</span> <span class="n">Masses</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_fwf</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">usecols</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">11</span><span class="p">),</span>
<span class="g g-Whitespace"> </span><span class="mi">3</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">,</span> <span class="s1">&#39;Z&#39;</span><span class="p">,</span> <span class="s1">&#39;A&#39;</span><span class="p">,</span> <span class="s1">&#39;Element&#39;</span><span class="p">,</span> <span class="s1">&#39;Ebinding&#39;</span><span class="p">),</span>
<span class="g g-Whitespace"> </span><span class="mi">4</span> <span class="n">widths</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span>
<span class="g g-Whitespace"> </span><span class="mi">5</span> <span class="n">header</span><span class="o">=</span><span class="mi">39</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">6</span> <span class="n">index_col</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">8</span> <span class="c1"># Extrapolated values are indicated by &#39;#&#39; in place of the decimal place, so</span>
<span class="g g-Whitespace"> </span><span class="mi">9</span> <span class="c1"># the Ebinding column won&#39;t be numeric. Coerce to float and drop these entries.</span>
<span class="g g-Whitespace"> </span><span class="mi">10</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/util/_decorators.py</span> in <span class="ni">wrapper</span><span class="nt">(*args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">309</span> <span class="n">stacklevel</span><span class="o">=</span><span class="n">stacklevel</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">310</span> <span class="p">)</span>
<span class="ne">--&gt; </span><span class="mi">311</span> <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">312</span>
<span class="g g-Whitespace"> </span><span class="mi">313</span> <span class="k">return</span> <span class="n">wrapper</span>
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/util/_decorators.py:311,</span> in <span class="ni">deprecate_nonkeyword_arguments.&lt;locals&gt;.decorate.&lt;locals&gt;.wrapper</span><span class="nt">(*args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">305</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">num_allow_args</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">306</span> <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span>
<span class="g g-Whitespace"> </span><span class="mi">307</span> <span class="n">msg</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">arguments</span><span class="o">=</span><span class="n">arguments</span><span class="p">),</span>
<span class="g g-Whitespace"> </span><span class="mi">308</span> <span class="ne">FutureWarning</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">309</span> <span class="n">stacklevel</span><span class="o">=</span><span class="n">stacklevel</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">310</span> <span class="p">)</span>
<span class="ne">--&gt; </span><span class="mi">311</span> <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/io/parsers/readers.py</span> in <span class="ni">read_fwf</span><span class="nt">(filepath_or_buffer, colspecs, widths, infer_nrows, **kwds)</span>
<span class="g g-Whitespace"> </span><span class="mi">869</span> <span class="n">len_index</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">index_col</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">870</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">names</span><span class="p">)</span> <span class="o">+</span> <span class="n">len_index</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">colspecs</span><span class="p">):</span>
<span class="ne">--&gt; </span><span class="mi">871</span> <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Length of colspecs must match length of names&quot;</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">872</span>
<span class="g g-Whitespace"> </span><span class="mi">873</span> <span class="n">kwds</span><span class="p">[</span><span class="s2">&quot;colspecs&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">colspecs</span>
<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/io/parsers/readers.py:871,</span> in <span class="ni">read_fwf</span><span class="nt">(filepath_or_buffer, colspecs, widths, infer_nrows, **kwds)</span>
<span class="g g-Whitespace"> </span><span class="mi">869</span> <span class="n">len_index</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">index_col</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">870</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">names</span><span class="p">)</span> <span class="o">+</span> <span class="n">len_index</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">colspecs</span><span class="p">):</span>
<span class="ne">--&gt; </span><span class="mi">871</span> <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Length of colspecs must match length of names&quot;</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">873</span> <span class="n">kwds</span><span class="p">[</span><span class="s2">&quot;colspecs&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">colspecs</span>
<span class="g g-Whitespace"> </span><span class="mi">874</span> <span class="n">kwds</span><span class="p">[</span><span class="s2">&quot;infer_nrows&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">infer_nrows</span>
<span class="ne">ValueError</span>: Length of colspecs must match length of names
</pre></div>
@@ -2991,54 +3134,42 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
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