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Morten Hjorth-Jensen
2024-08-25 21:36:53 +02:00
parent 053f351848
commit 757cad43cd
16 changed files with 1952 additions and 3003 deletions
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"cells": [
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"id": "a50ea987",
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@@ -14,7 +14,7 @@
},
{
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@@ -27,7 +27,7 @@
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@@ -39,7 +39,7 @@
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@@ -108,7 +108,7 @@
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"metadata": {
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@@ -122,20 +122,21 @@
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"id": "278aa34f",
"id": "81c11edc",
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"x = np.random.rand(100,1)\n",
"y = 2.0+5*x*x+0.1*np.random.randn(100,1)"
]
},
{
"cell_type": "markdown",
"id": "1ff20e45",
"id": "970bd604",
"metadata": {
"editable": true
},
@@ -149,7 +150,7 @@
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@@ -162,7 +163,7 @@
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@@ -173,7 +174,7 @@
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@@ -185,7 +186,7 @@
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@@ -195,7 +196,7 @@
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@@ -207,7 +208,7 @@
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@@ -218,7 +219,7 @@
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@@ -237,7 +238,7 @@
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@@ -253,7 +254,7 @@
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@@ -263,7 +264,7 @@
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@@ -274,7 +275,7 @@
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@@ -286,7 +287,7 @@
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doc/LectureNotes/_build/html/_sources/week35.ipynb
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@@ -480,22 +480,12 @@ license.</p>
The following simple Python instructions define our <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> values (with 100 data points).</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
</div>
<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">NameError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">1</span>
<span class="ne">----&gt; </span><span class="mi">1</span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">2</span> <span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="ne">NameError</span>: name &#39;np&#39; is not defined
</pre></div>
</div>
</div>
</div>
<ol class="simple">
<li><p>Write your own code (following the examples under the <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html">regression notes</a>) for computing the parametrization of the data set fitting a second-order polynomial.</p></li>
doc/LectureNotes/_build/html/searchindex.js
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doc/LectureNotes/_build/html/week35.html
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@@ -351,8 +351,8 @@ const thebe_selector_output = ".output, .cell_output"
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
<a class="reference internal nav-link" href="#for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
@@ -536,18 +536,8 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-boston-housing-data-example">
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">
Housing data, the code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#material-for-lecture-thursday-august-31">
Material for lecture Thursday, August 31
<a class="reference internal nav-link" href="#material-for-lecture-monday-august-26">
Material for lecture Monday, August 26
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
@@ -746,8 +736,8 @@ const thebe_selector_output = ".output, .cell_output"
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
<a class="reference internal nav-link" href="#for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
@@ -931,18 +921,8 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-boston-housing-data-example">
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">
Housing data, the code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#material-for-lecture-thursday-august-31">
Material for lecture Thursday, August 31
<a class="reference internal nav-link" href="#material-for-lecture-monday-august-26">
Material for lecture Monday, August 26
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
@@ -1121,14 +1101,14 @@ const thebe_selector_output = ".output, .cell_output"
doconce format html week35.do.txt --no_mako -->
<!-- dom:TITLE: Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression --><div class="tex2jax_ignore mathjax_ignore section" id="week-35-from-ordinary-linear-regression-to-ridge-and-lasso-regression">
<h1>Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression<a class="headerlink" href="#week-35-from-ordinary-linear-regression-to-ridge-and-lasso-regression" title="Permalink to this headline"></a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</p>
<p>Date: <strong>August 26-30</strong></p>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo</p>
<p>Date: <strong>August 26-30, 2024</strong></p>
<div class="section" id="plans-for-week-35">
<h2>Plans for week 35<a class="headerlink" href="#plans-for-week-35" title="Permalink to this headline"></a></h2>
<p>The main topics are:</p>
<ol class="simple">
<li><p>Brief repetition from last week</p></li>
<li><p>Derivation of the equations for ordinary least squares</p></li>
<li><p>Discussions of the equations for ordinary least squares</p></li>
<li><p>Discussion on how to prepare data and examples of applications of linear regression</p></li>
<li><p>Material for the lecture on Monday: Mathematical interpretations of linear regression</p></li>
<li><p>Monday: Ridge and Lasso regression and Singular Value Decomposition</p></li>
@@ -1136,13 +1116,15 @@ doconce format html week35.do.txt --no_mako -->
<div class="section" id="reading-recommendations">
<h3>Reading recommendations:<a class="headerlink" href="#reading-recommendations" title="Permalink to this headline"></a></h3>
<ol class="simple">
<li><p>See lecture notes for week 35 at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/web/course.html">https://compphysics.github.io/MachineLearning/doc/web/course.html</a></p></li>
<li><p>These lecture notes</p></li>
<li><p>Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)</p></li>
<li><p>Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.</p></li>
<li><p>For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL”<a class="reference external" href="https://mml-book.github.io/">https://mml-book.github.io/</a>” (section 5.5 on derivatives) is very useful for exercise 1 this coming week.</p></li>
</ol>
</div>
</div>
<div class="section" id="why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
<h2>Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week<a class="headerlink" href="#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" title="Permalink to this headline"></a></h2>
<div class="section" id="for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week">
<h2>For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week<a class="headerlink" href="#for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" title="Permalink to this headline"></a></h2>
<p>We need first a reminder from last week about linear regression.</p>
<p>Fitting a continuous function with linear parameterization in terms of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.</p>
<ul class="simple">
@@ -1610,7 +1592,7 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9952638231265687
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9966579875073338
</pre></div>
</div>
</div>
@@ -1627,7 +1609,7 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.011761161707539526
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.007652435292578527
</pre></div>
</div>
</div>
@@ -1642,23 +1624,31 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.00035753 0.04937621 0.02268114 0.03112297 0.01856502 0.05322899
0.01397927 0.03999935 0.02508836 0.01834042 0.0652633 0.00887114
0.01956921 0.01987248 0.06294313 0.01152476 0.00328431 0.03564082
0.02337045 0.01743586 0.01278035 0.02400968 0.08388138 0.03270341
0.00335956 0.01031903 0.09575469 0.00956774 0.00900879 0.01867985
0.01191227 0.02090296 0.04045671 0.03582958 0.06692165 0.06615863
0.06594872 0.03756493 0.00488992 0.01405237 0.00117071 0.0017567
0.05006306 0.02545639 0.03456485 0.00373984 0.02576476 0.03530741
0.00092902 0.0275742 0.05948007 0.01321652 0.18500705 0.00382166
0.00327313 0.01853877 0.01771317 0.05293662 0.07199977 0.00836148
0.01541649 0.00343257 0.00626797 0.05350297 0.01548272 0.05235058
0.04310698 0.00225189 0.02356396 0.01690512 0.03467756 0.00064364
0.02593596 0.00019607 0.0029508 0.0180194 0.06825695 0.01659559
0.01971341 0.02012338 0.02241311 0.00135736 0.0095653 0.05695438
0.00395659 0.07068033 0.02699873 0.00919237 0.02493299 0.00803115
0.0293055 0.02178063 0.00594353 0.04081883 0.01325225 0.0386443
0.01889695 0.02810253 0.0181166 0.01135924]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[3.25042813e-02 2.38298328e-02 2.64093905e-02 1.31145723e-02
7.85640706e-02 2.08298871e-02 2.08187418e-02 6.69029803e-04
2.18905573e-02 2.14957387e-02 2.32863661e-02 4.66181587e-03
8.93710544e-03 2.89564974e-03 2.51647799e-03 3.61336683e-02
1.11766707e-02 6.39623570e-02 7.54190070e-03 2.85193598e-03
4.90018622e-02 1.46957262e-02 4.86592590e-02 9.69161166e-05
1.11309850e-02 1.47605918e-02 1.80419748e-03 3.18023984e-02
2.92063704e-02 3.53149684e-02 1.18573920e-02 3.32775737e-03
1.71196799e-02 1.96079426e-02 1.69232893e-02 1.89273223e-02
3.13021533e-02 2.96574402e-03 4.06270382e-02 4.92293994e-02
1.07282462e-02 1.29064873e-02 8.37618253e-03 4.02472584e-02
2.44637348e-02 1.00149403e-02 2.21129518e-02 2.30674822e-03
4.87391702e-02 2.53371547e-02 8.26280707e-03 1.80203363e-02
2.38115471e-02 7.62038116e-02 4.34816464e-02 3.92213792e-02
6.68818255e-03 8.88623007e-03 8.28473779e-02 4.53859920e-02
8.26121775e-02 4.67869899e-03 1.62034461e-02 2.78672237e-02
8.98401894e-03 3.17303133e-02 3.51729923e-02 2.90612093e-03
5.70221196e-03 3.49065324e-02 3.22091393e-02 2.15706394e-02
1.40920261e-02 3.82602592e-03 4.59101492e-02 3.65313891e-02
1.52725477e-02 3.05989176e-02 1.28347954e-03 3.00674454e-02
8.92998014e-02 1.63727258e-02 3.77117804e-02 1.17886788e-02
8.76071777e-02 4.11958697e-02 4.99322932e-03 7.57875488e-02
1.16954396e-04 1.90884438e-02 1.41441328e-02 5.38923192e-02
9.21237677e-03 1.36654107e-02 9.03904224e-04 9.90027459e-03
3.01828101e-02 4.71038982e-02 3.87200322e-02 1.62397973e-02]
</pre></div>
</div>
</div>
@@ -1727,15 +1717,15 @@ but now splitting the data into a training set and a test set.</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 2.00507876 0.39026883 3.20764972 2.73806921 -1.39609089]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 1.86544433 1.28573357 0.76075617 5.54611024 -2.45878472]
Training R2
0.9972493421341901
0.9958543328528107
Training MSE
0.007021475481484069
0.011031108072207247
Test R2
0.9976639055733267
0.9958824304846606
Test MSE
0.00575158411598985
0.006546368424400406
</pre></div>
</div>
</div>
@@ -2729,270 +2719,8 @@ back the intercept, we will get an MSE which then contains the
intercept. This becomes more important when we discuss Ridge and Lasso
regression next week.</p>
</div>
<div class="section" id="the-boston-housing-data-example">
<h2>The Boston housing data example<a class="headerlink" href="#the-boston-housing-data-example" title="Permalink to this headline"></a></h2>
<p>The Boston housing<br />
data set was originally a part of UCI Machine Learning Repository
and has been removed now. The data set is now included in <strong>Scikit-Learn</strong>’s
library. There are 506 samples and 13 feature (predictor) variables
in this data set. The objective is to predict the value of prices of
the house using the features (predictors) listed here.</p>
<p>The features/predictors are</p>
<ol class="simple">
<li><p>CRIM: Per capita crime rate by town</p></li>
<li><p>ZN: Proportion of residential land zoned for lots over 25000 square feet</p></li>
<li><p>INDUS: Proportion of non-retail business acres per town</p></li>
<li><p>CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)</p></li>
<li><p>NOX: Nitric oxide concentration (parts per 10 million)</p></li>
<li><p>RM: Average number of rooms per dwelling</p></li>
<li><p>AGE: Proportion of owner-occupied units built prior to 1940</p></li>
<li><p>DIS: Weighted distances to five Boston employment centers</p></li>
<li><p>RAD: Index of accessibility to radial highways</p></li>
<li><p>TAX: Full-value property tax rate per USD10000</p></li>
<li><p>B: <span class="math notranslate nohighlight">\(1000(Bk - 0.63)^2\)</span>, where <span class="math notranslate nohighlight">\(Bk\)</span> is the proportion of [people of African American descent] by town</p></li>
<li><p>LSTAT: Percentage of lower status of the population</p></li>
<li><p>MEDV: Median value of owner-occupied homes in USD 1000s</p></li>
</ol>
</div>
<div class="section" id="housing-data-the-code">
<h2>Housing data, the code<a class="headerlink" href="#housing-data-the-code" title="Permalink to this headline"></a></h2>
<p>We start by importing the libraries</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
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<p>and load the Boston Housing DataSet from <strong>Scikit-Learn</strong></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_boston</span>
<span class="n">boston_dataset</span> <span class="o">=</span> <span class="n">load_boston</span><span class="p">()</span>
<span class="c1"># boston_dataset is a dictionary</span>
<span class="c1"># let&#39;s check what it contains</span>
<span class="n">boston_dataset</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span>
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<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span>---------------------------------------------------------------------------
ImportError Traceback (most recent call last)
Cell In[16], line 1
----&gt; 1 from sklearn.datasets import load_boston
3 boston_dataset = load_boston()
5 # boston_dataset is a dictionary
6 # let&#39;s check what it contains
File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/datasets/__init__.py:157, in __getattr__(name)
108 if name == &quot;load_boston&quot;:
109 msg = textwrap.dedent(&quot;&quot;&quot;
110 `load_boston` has been removed from scikit-learn since version 1.2.
111
(...)
155 &lt;https://www.researchgate.net/publication/4974606_Hedonic_housing_prices_and_the_demand_for_clean_air&gt;
156 &quot;&quot;&quot;)
--&gt; 157 raise ImportError(msg)
158 try:
159 return globals()[name]
ImportError:
`load_boston` has been removed from scikit-learn since version 1.2.
The Boston housing prices dataset has an ethical problem: as
investigated in [1], the authors of this dataset engineered a
non-invertible variable &quot;B&quot; assuming that racial self-segregation had a
positive impact on house prices [2]. Furthermore the goal of the
research that led to the creation of this dataset was to study the
impact of air quality but it did not give adequate demonstration of the
validity of this assumption.
The scikit-learn maintainers therefore strongly discourage the use of
this dataset unless the purpose of the code is to study and educate
about ethical issues in data science and machine learning.
In this special case, you can fetch the dataset from the original
source::
import pandas as pd
import numpy as np
data_url = &quot;http://lib.stat.cmu.edu/datasets/boston&quot;
raw_df = pd.read_csv(data_url, sep=&quot;\s+&quot;, skiprows=22, header=None)
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]
Alternative datasets include the California housing dataset and the
Ames housing dataset. You can load the datasets as follows::
from sklearn.datasets import fetch_california_housing
housing = fetch_california_housing()
for the California housing dataset and::
from sklearn.datasets import fetch_openml
housing = fetch_openml(name=&quot;house_prices&quot;, as_frame=True)
for the Ames housing dataset.
[1] M Carlisle.
&quot;Racist data destruction?&quot;
&lt;https://medium.com/@docintangible/racist-data-destruction-113e3eff54a8&gt;
[2] Harrison Jr, David, and Daniel L. Rubinfeld.
&quot;Hedonic housing prices and the demand for clean air.&quot;
Journal of environmental economics and management 5.1 (1978): 81-102.
&lt;https://www.researchgate.net/publication/4974606_Hedonic_housing_prices_and_the_demand_for_clean_air&gt;
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<p>Then we invoke Pandas</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">boston</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">boston_dataset</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">boston_dataset</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">boston</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">boston_dataset</span><span class="o">.</span><span class="n">target</span>
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<p>and preprocess the data</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># check for missing values in all the columns</span>
<span class="n">boston</span><span class="o">.</span><span class="n">isnull</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
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<p>We can then visualize the data</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># set the size of the figure</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">:(</span><span class="mf">11.7</span><span class="p">,</span><span class="mf">8.27</span><span class="p">)})</span>
<span class="c1"># plot a histogram showing the distribution of the target values</span>
<span class="n">sns</span><span class="o">.</span><span class="n">distplot</span><span class="p">(</span><span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">],</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>It is now useful to look at the correlation matrix</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># compute the pair wise correlation for all columns </span>
<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">boston</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># use the heatmap function from seaborn to plot the correlation matrix</span>
<span class="c1"># annot = True to print the values inside the square</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">correlation_matrix</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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<p>From the above coorelation plot we can see that <strong>MEDV</strong> is strongly correlated to <strong>LSTAT</strong> and <strong>RM</strong>. We see also that <strong>RAD</strong> and <strong>TAX</strong> are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">,</span> <span class="s1">&#39;RM&#39;</span><span class="p">]</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">features</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">features</span><span class="p">)</span> <span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="n">col</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">target</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;MEDV&#39;</span><span class="p">)</span>
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<p>Now we start training our model</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">boston</span><span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">],</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;RM&#39;</span><span class="p">]],</span> <span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">,</span><span class="s1">&#39;RM&#39;</span><span class="p">])</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span>
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<p>We split the data into training and test sets</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># splits the training and test data set in 80% : 20%</span>
<span class="c1"># assign random_state to any value.This ensures consistency.</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">test_size</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Y_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
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<p>Then we use the linear regression functionality from <strong>Scikit-Learn</strong></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span><span class="p">,</span> <span class="n">r2_score</span>
<span class="n">lin_model</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">lin_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="c1"># model evaluation for training set</span>
<span class="n">y_train_predict</span> <span class="o">=</span> <span class="n">lin_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">rmse</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">y_train_predict</span><span class="p">)))</span>
<span class="n">r2</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">y_train_predict</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The model performance for training set&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;--------------------------------------&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;RMSE is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;R2 score is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">r2</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># model evaluation for testing set</span>
<span class="n">y_test_predict</span> <span class="o">=</span> <span class="n">lin_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># root mean square error of the model</span>
<span class="n">rmse</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)))</span>
<span class="c1"># r-squared score of the model</span>
<span class="n">r2</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The model performance for testing set&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;--------------------------------------&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;RMSE is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;R2 score is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">r2</span><span class="p">))</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plotting the y_test vs y_pred</span>
<span class="c1"># ideally should have been a straight line</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="material-for-lecture-thursday-august-31">
<h2>Material for lecture Thursday, August 31<a class="headerlink" href="#material-for-lecture-thursday-august-31" title="Permalink to this headline"></a></h2>
<div class="section" id="material-for-lecture-monday-august-26">
<h2>Material for lecture Monday, August 26<a class="headerlink" href="#material-for-lecture-monday-august-26" title="Permalink to this headline"></a></h2>
</div>
<div class="section" id="mathematical-interpretation-of-ordinary-least-squares">
<h2>Mathematical Interpretation of Ordinary Least Squares<a class="headerlink" href="#mathematical-interpretation-of-ordinary-least-squares" title="Permalink to this headline"></a></h2>
@@ -3268,6 +2996,25 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1. -1.]
[ 1. -1.]]
test U
[[0. 0.]
[0. 0.]]
test VT
[[0. 0.]
[0. 0.]]
[[-0.70710678 -0.70710678]
[-0.70710678 0.70710678]]
[2.00000000e+00 3.35470445e-17]
[[-0.70710678 0.70710678]
[ 0.70710678 0.70710678]]
[[-3.33066907e-16 4.44089210e-16]
[ 0.00000000e+00 2.22044605e-16]]
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<p>The matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> has columns that are linearly dependent. The first
column is the row-wise sum of the other two columns. The rank of a
@@ -3623,6 +3370,14 @@ covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.10790125813226321
4.340071371496255
[[ 1.04193203 3.08165104]
[ 3.08165104 10.18383522]]
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<div class="section" id="correlation-matrix">
@@ -3658,6 +3413,14 @@ a more brute force way. Here we scale the mean values for each column of the des
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.08881497884574564
1.7086067479626619
[[1. 0.66080313]
[0.66080313 1. ]]
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<p>We see that the matrix elements along the diagonal are one as they
should be and that the matrix is symmetric. Furthermore, diagonalizing
@@ -3686,6 +3449,34 @@ this matrix we easily see that it is a positive definite matrix.</p>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-0.40620066 -2.01265755]
[ 0.01458611 0.37737221]
[-1.0895387 -3.65442354]
[ 0.2338675 1.12044974]
[ 0.4676059 1.54393936]
[-0.65891389 -3.16304863]
[-0.1715252 0.39197698]
[ 0.71142161 2.95511792]
[ 0.39214397 0.13069442]
[ 0.50655336 2.3105791 ]]
0 1
0 -0.406201 -2.012658
1 0.014586 0.377372
2 -1.089539 -3.654424
3 0.233868 1.120450
4 0.467606 1.543939
5 -0.658914 -3.163049
6 -0.171525 0.391977
7 0.711422 2.955118
8 0.392144 0.130694
9 0.506553 2.310579
0 1
0 1.000000 0.952387
1 0.952387 1.000000
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</div>
</div>
<p>We expand this model to the Franke function discussed above.</p>
</div>
@@ -3739,6 +3530,43 @@ this matrix we easily see that it is a positive definite matrix.</p>
</pre></div>
</div>
</div>
<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 0.078974 0.081276 0.075889 0.077168 0.078540 0.065410 0.066474
2 0.0 0.081276 0.084076 0.078336 0.079946 0.081655 0.067707 0.069028
3 0.0 0.075889 0.078336 0.077460 0.078986 0.080616 0.069452 0.070737
4 0.0 0.077168 0.079946 0.078986 0.080764 0.082653 0.070986 0.072476
5 0.0 0.078540 0.081655 0.080616 0.082653 0.084809 0.072621 0.074323
6 0.0 0.065410 0.067707 0.069452 0.070986 0.072621 0.064074 0.065378
7 0.0 0.066474 0.069028 0.070737 0.072476 0.074323 0.065378 0.066854
8 0.0 0.067637 0.070457 0.072132 0.074084 0.076150 0.066787 0.068441
9 0.0 0.068906 0.072000 0.073644 0.075816 0.078110 0.068307 0.070146
10 0.0 0.055734 0.057835 0.060872 0.062337 0.063894 0.057393 0.058645
11 0.0 0.056683 0.058996 0.062016 0.063653 0.065390 0.058552 0.059951
12 0.0 0.057722 0.060254 0.063260 0.065077 0.066999 0.059807 0.061359
13 0.0 0.058854 0.061614 0.064609 0.066612 0.068727 0.061163 0.062874
14 0.0 0.060083 0.063080 0.066066 0.068264 0.070582 0.062624 0.064501
8 9 10 11 12 13 14
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.067637 0.068906 0.055734 0.056683 0.057722 0.058854 0.060083
2 0.070457 0.072000 0.057835 0.058996 0.060254 0.061614 0.063080
3 0.072132 0.073644 0.060872 0.062016 0.063260 0.064609 0.066066
4 0.074084 0.075816 0.062337 0.063653 0.065077 0.066612 0.068264
5 0.076150 0.078110 0.063894 0.065390 0.066999 0.068727 0.070582
6 0.066787 0.068307 0.057393 0.058552 0.059807 0.061163 0.062624
7 0.068441 0.070146 0.058645 0.059951 0.061359 0.062874 0.064501
8 0.070213 0.072111 0.059993 0.061452 0.063019 0.064699 0.066500
9 0.072111 0.074210 0.061443 0.063061 0.064793 0.066647 0.068629
10 0.059993 0.061443 0.052305 0.053417 0.054617 0.055910 0.057300
11 0.061452 0.063061 0.053417 0.054655 0.055987 0.057418 0.058952
12 0.063019 0.064793 0.054617 0.055987 0.057457 0.059031 0.060716
13 0.064699 0.066647 0.055910 0.057418 0.059031 0.060756 0.062599
14 0.066500 0.068629 0.057300 0.058952 0.060716 0.062599 0.064606
</pre></div>
</div>
</div>
</div>
<p>We note here that the covariance is zero for the first rows and
columns since all matrix elements in the design matrix were set to one
doc/LectureNotes/_build/jupyter_execute/exercisesweek34.ipynb
+21 -32
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@@ -2,7 +2,7 @@
"cells": [
{
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"id": "72dc0997",
"id": "a50ea987",
"metadata": {
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@@ -14,7 +14,7 @@
},
{
"cell_type": "markdown",
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"id": "291e6015",
"metadata": {
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@@ -27,7 +27,7 @@
},
{
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"id": "1ad21876",
"id": "141ff111",
"metadata": {
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@@ -39,7 +39,7 @@
},
{
"cell_type": "markdown",
"id": "4eaa91e2",
"id": "32a1fc9e",
"metadata": {
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@@ -108,7 +108,7 @@
},
{
"cell_type": "markdown",
"id": "360adb56",
"id": "f103f4c1",
"metadata": {
"editable": true
},
@@ -122,32 +122,21 @@
{
"cell_type": "code",
"execution_count": 1,
"id": "278aa34f",
"id": "81c11edc",
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [
{
"ename": "NameError",
"evalue": "name 'np' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[1], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m x \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mrandom\u001b[38;5;241m.\u001b[39mrand(\u001b[38;5;241m100\u001b[39m,\u001b[38;5;241m1\u001b[39m)\n\u001b[1;32m 2\u001b[0m y \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m2.0\u001b[39m\u001b[38;5;241m+\u001b[39m\u001b[38;5;241m5\u001b[39m\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m+\u001b[39m\u001b[38;5;241m0.1\u001b[39m\u001b[38;5;241m*\u001b[39mnp\u001b[38;5;241m.\u001b[39mrandom\u001b[38;5;241m.\u001b[39mrandn(\u001b[38;5;241m100\u001b[39m,\u001b[38;5;241m1\u001b[39m)\n",
"\u001b[0;31mNameError\u001b[0m: name 'np' is not defined"
]
}
],
"outputs": [],
"source": [
"import numpy as np\n",
"x = np.random.rand(100,1)\n",
"y = 2.0+5*x*x+0.1*np.random.randn(100,1)"
]
},
{
"cell_type": "markdown",
"id": "1ff20e45",
"id": "970bd604",
"metadata": {
"editable": true
},
@@ -161,7 +150,7 @@
},
{
"cell_type": "markdown",
"id": "e8809de1",
"id": "bda3ed04",
"metadata": {
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@@ -174,7 +163,7 @@
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"id": "ca995691",
"metadata": {
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@@ -185,7 +174,7 @@
},
{
"cell_type": "markdown",
"id": "85a5881c",
"id": "27cbfeed",
"metadata": {
"editable": true
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@@ -197,7 +186,7 @@
},
{
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"id": "4fb35d0d",
"id": "9e063f6d",
"metadata": {
"editable": true
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@@ -207,7 +196,7 @@
},
{
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"id": "608f9780",
"id": "e582c2bb",
"metadata": {
"editable": true
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@@ -219,7 +208,7 @@
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"id": "014bbc09",
"id": "59e0ed7a",
"metadata": {
"editable": true
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@@ -230,7 +219,7 @@
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"id": "051e71aa",
"metadata": {
"editable": true
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@@ -249,7 +238,7 @@
{
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"execution_count": 2,
"id": "85f75a99",
"id": "c82d8b64",
"metadata": {
"collapsed": false,
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@@ -265,7 +254,7 @@
},
{
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"metadata": {
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@@ -275,7 +264,7 @@
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@@ -286,7 +275,7 @@
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"id": "1b0cea0a",
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@@ -298,7 +287,7 @@
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doc/LectureNotes/_build/jupyter_execute/exercisesweek34.py
+1
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@@ -83,6 +83,7 @@
# In[1]:
import numpy as np
x = np.random.rand(100,1)
y = 2.0+5*x*x+0.1*np.random.randn(100,1)
doc/LectureNotes/_build/jupyter_execute/week35.ipynb
+574 -824
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doc/LectureNotes/_build/jupyter_execute/week35.py
+15 -210
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@@ -6,9 +6,9 @@
# <!-- dom:TITLE: Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression -->
# # Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo
#
# Date: **August 26-30**
# Date: **August 26-30, 2024**
# ## Plans for week 35
#
@@ -16,7 +16,7 @@
#
# 1. Brief repetition from last week
#
# 2. Derivation of the equations for ordinary least squares
# 2. Discussions of the equations for ordinary least squares
#
# 3. Discussion on how to prepare data and examples of applications of linear regression
#
@@ -26,11 +26,15 @@
# ### Reading recommendations:
#
# 1. See lecture notes for week 35 at <https://compphysics.github.io/MachineLearning/doc/web/course.html>
# 1. These lecture notes
#
# 2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)
#
# 3. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.
#
# 4. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL"https://mml-book.github.io/" (section 5.5 on derivatives) is very useful for exercise 1 this coming week.
# ## Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
# ## For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
#
# We need first a reminder from last week about linear regression.
#
@@ -1264,206 +1268,7 @@ plt.show()
# intercept. This becomes more important when we discuss Ridge and Lasso
# regression next week.
# ## The Boston housing data example
#
# The Boston housing
# data set was originally a part of UCI Machine Learning Repository
# and has been removed now. The data set is now included in **Scikit-Learn**'s
# library. There are 506 samples and 13 feature (predictor) variables
# in this data set. The objective is to predict the value of prices of
# the house using the features (predictors) listed here.
#
# The features/predictors are
# 1. CRIM: Per capita crime rate by town
#
# 2. ZN: Proportion of residential land zoned for lots over 25000 square feet
#
# 3. INDUS: Proportion of non-retail business acres per town
#
# 4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
#
# 5. NOX: Nitric oxide concentration (parts per 10 million)
#
# 6. RM: Average number of rooms per dwelling
#
# 7. AGE: Proportion of owner-occupied units built prior to 1940
#
# 8. DIS: Weighted distances to five Boston employment centers
#
# 9. RAD: Index of accessibility to radial highways
#
# 10. TAX: Full-value property tax rate per USD10000
#
# 11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town
#
# 12. LSTAT: Percentage of lower status of the population
#
# 13. MEDV: Median value of owner-occupied homes in USD 1000s
# ## Housing data, the code
# We start by importing the libraries
# In[15]:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
# and load the Boston Housing DataSet from **Scikit-Learn**
# In[16]:
from sklearn.datasets import load_boston
boston_dataset = load_boston()
# boston_dataset is a dictionary
# let's check what it contains
boston_dataset.keys()
# Then we invoke Pandas
# In[17]:
boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)
boston.head()
boston['MEDV'] = boston_dataset.target
# and preprocess the data
# In[18]:
# check for missing values in all the columns
boston.isnull().sum()
# We can then visualize the data
# In[19]:
# set the size of the figure
sns.set(rc={'figure.figsize':(11.7,8.27)})
# plot a histogram showing the distribution of the target values
sns.distplot(boston['MEDV'], bins=30)
plt.show()
# It is now useful to look at the correlation matrix
# In[20]:
# compute the pair wise correlation for all columns
correlation_matrix = boston.corr().round(2)
# use the heatmap function from seaborn to plot the correlation matrix
# annot = True to print the values inside the square
sns.heatmap(data=correlation_matrix, annot=True)
# From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity
# In[21]:
plt.figure(figsize=(20, 5))
features = ['LSTAT', 'RM']
target = boston['MEDV']
for i, col in enumerate(features):
plt.subplot(1, len(features) , i+1)
x = boston[col]
y = target
plt.scatter(x, y, marker='o')
plt.title(col)
plt.xlabel(col)
plt.ylabel('MEDV')
# Now we start training our model
# In[22]:
X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])
Y = boston['MEDV']
# We split the data into training and test sets
# In[23]:
from sklearn.model_selection import train_test_split
# splits the training and test data set in 80% : 20%
# assign random_state to any value.This ensures consistency.
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)
print(X_train.shape)
print(X_test.shape)
print(Y_train.shape)
print(Y_test.shape)
# Then we use the linear regression functionality from **Scikit-Learn**
# In[24]:
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
lin_model = LinearRegression()
lin_model.fit(X_train, Y_train)
# model evaluation for training set
y_train_predict = lin_model.predict(X_train)
rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))
r2 = r2_score(Y_train, y_train_predict)
print("The model performance for training set")
print("--------------------------------------")
print('RMSE is {}'.format(rmse))
print('R2 score is {}'.format(r2))
print("\n")
# model evaluation for testing set
y_test_predict = lin_model.predict(X_test)
# root mean square error of the model
rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))
# r-squared score of the model
r2 = r2_score(Y_test, y_test_predict)
print("The model performance for testing set")
print("--------------------------------------")
print('RMSE is {}'.format(rmse))
print('R2 score is {}'.format(r2))
# In[25]:
# plotting the y_test vs y_pred
# ideally should have been a straight line
plt.scatter(Y_test, y_test_predict)
plt.show()
# ## Material for lecture Thursday, August 31
# ## Material for lecture Monday, August 26
# ## Mathematical Interpretation of Ordinary Least Squares
#
@@ -1738,7 +1543,7 @@ plt.show()
# ## Codes for the SVD
# In[26]:
# In[15]:
import numpy as np
@@ -2155,7 +1960,7 @@ print(C-X)
# function **np.mean(x)**. We can also extract the eigenvalues of the
# covariance matrix through the **np.linalg.eig()** function.
# In[27]:
# In[16]:
# Importing various packages
@@ -2178,7 +1983,7 @@ print(C)
# code which sets up the correlations matrix for the previous example in
# a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\times 2$ correlation matrix (since we have only two vectors).
# In[28]:
# In[17]:
import numpy as np
@@ -2214,7 +2019,7 @@ print(C)
#
# We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code
# In[29]:
# In[18]:
import numpy as np
@@ -2237,7 +2042,7 @@ print(correlation_matrix)
# ## Correlation Matrix with Pandas and the Franke function
# In[30]:
# In[19]:
# Common imports
doc/LectureNotes/exercisesweek34.ipynb
+20 -19
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@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
"id": "72dc0997",
"id": "a50ea987",
"metadata": {
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@@ -14,7 +14,7 @@
},
{
"cell_type": "markdown",
"id": "8c591125",
"id": "291e6015",
"metadata": {
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@@ -27,7 +27,7 @@
},
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"cell_type": "markdown",
"id": "1ad21876",
"id": "141ff111",
"metadata": {
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@@ -39,7 +39,7 @@
},
{
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@@ -108,7 +108,7 @@
},
{
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"id": "360adb56",
"id": "f103f4c1",
"metadata": {
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@@ -122,20 +122,21 @@
{
"cell_type": "code",
"execution_count": 1,
"id": "278aa34f",
"id": "81c11edc",
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"x = np.random.rand(100,1)\n",
"y = 2.0+5*x*x+0.1*np.random.randn(100,1)"
]
},
{
"cell_type": "markdown",
"id": "1ff20e45",
"id": "970bd604",
"metadata": {
"editable": true
},
@@ -149,7 +150,7 @@
},
{
"cell_type": "markdown",
"id": "e8809de1",
"id": "bda3ed04",
"metadata": {
"editable": true
},
@@ -162,7 +163,7 @@
},
{
"cell_type": "markdown",
"id": "8c9e5277",
"id": "ca995691",
"metadata": {
"editable": true
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@@ -173,7 +174,7 @@
},
{
"cell_type": "markdown",
"id": "85a5881c",
"id": "27cbfeed",
"metadata": {
"editable": true
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@@ -185,7 +186,7 @@
},
{
"cell_type": "markdown",
"id": "4fb35d0d",
"id": "9e063f6d",
"metadata": {
"editable": true
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@@ -195,7 +196,7 @@
},
{
"cell_type": "markdown",
"id": "608f9780",
"id": "e582c2bb",
"metadata": {
"editable": true
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@@ -207,7 +208,7 @@
},
{
"cell_type": "markdown",
"id": "014bbc09",
"id": "59e0ed7a",
"metadata": {
"editable": true
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@@ -218,7 +219,7 @@
},
{
"cell_type": "markdown",
"id": "f28863ad",
"id": "051e71aa",
"metadata": {
"editable": true
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@@ -237,7 +238,7 @@
{
"cell_type": "code",
"execution_count": 2,
"id": "85f75a99",
"id": "c82d8b64",
"metadata": {
"collapsed": false,
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@@ -253,7 +254,7 @@
},
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"id": "bee8d814",
"id": "3e810bba",
"metadata": {
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@@ -263,7 +264,7 @@
},
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"cell_type": "markdown",
"id": "7c94f8bd",
"id": "c1198fce",
"metadata": {
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@@ -274,7 +275,7 @@
},
{
"cell_type": "markdown",
"id": "c0e789dd",
"id": "1b0cea0a",
"metadata": {
"editable": true
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@@ -286,7 +287,7 @@
},
{
"cell_type": "markdown",
"id": "5ee1d70e",
"id": "5194ba08",
"metadata": {
"editable": true
},
doc/LectureNotes/week35.ipynb
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doc/src/week34/exercisesweek34.do.txt
+1
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@@ -78,6 +78,7 @@ We recommend using _Anaconda_ if you are not too familiar with setting paths in
We will generate our own dataset for a function $y(x)$ where $x \in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\cal {N}(0,1)$.
The following simple Python instructions define our $x$ and $y$ values (with 100 data points).
!bc pycod
import numpy as np
x = np.random.rand(100,1)
y = 2.0+5*x*x+0.1*np.random.randn(100,1)
!ec
doc/src/week34/exercisesweek34.ipynb
+303
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@@ -0,0 +1,303 @@
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"source": [
"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
"doconce format html exercisesweek34.do.txt -->\n",
"<!-- dom:TITLE: Exercises week 34 -->"
]
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{
"cell_type": "markdown",
"id": "291e6015",
"metadata": {
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"source": [
"# Exercises week 34\n",
"**FYS-STK3155/4155**\n",
"\n",
"Date: **August 19-23, 2024**"
]
},
{
"cell_type": "markdown",
"id": "141ff111",
"metadata": {
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"source": [
"## Exercises\n",
"\n",
"Here are three possible exercises for week 34"
]
},
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"id": "32a1fc9e",
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"source": [
"## Exercise 1: Setting up various Python environments\n",
"\n",
"The first exercise here is of a mere technical art. We want you to have \n",
"* git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo [GitHub facilities](https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html). \n",
"\n",
"* Install various Python packages\n",
"\n",
"We will make extensive use of Python as programming language and its\n",
"myriad of available libraries. You will find\n",
"IPython/Jupyter notebooks invaluable in your work. You can run **R**\n",
"codes in the Jupyter/IPython notebooks, with the immediate benefit of\n",
"visualizing your data. You can also use compiled languages like C++,\n",
"Rust, Fortran etc if you prefer. The focus in these lectures will be\n",
"on Python.\n",
"\n",
"If you have Python installed (we recommend Python3) and you feel\n",
"pretty familiar with installing different packages, we recommend that\n",
"you install the following Python packages via **pip** as \n",
"\n",
"1. pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow \n",
"\n",
"For **Tensorflow**, we recommend following the instructions in the text of \n",
"[Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly](http://shop.oreilly.com/product/0636920052289.do)\n",
"\n",
"We will come back to **tensorflow** later. \n",
"\n",
"For Python3, replace **pip** with **pip3**.\n",
"\n",
"For OSX users we recommend, after having installed Xcode, to\n",
"install **brew**. Brew allows for a seamless installation of additional\n",
"software via for example \n",
"\n",
"1. brew install python3\n",
"\n",
"For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n",
"you can use **pip** as well and simply install Python as \n",
"\n",
"1. sudo apt-get install python3 (or python for Python2.7)\n",
"\n",
"If you don't want to perform these operations separately and venture\n",
"into the hassle of exploring how to set up dependencies and paths, we\n",
"recommend two widely used distrubutions which set up all relevant\n",
"dependencies for Python, namely \n",
"\n",
"* [Anaconda](https://docs.anaconda.com/), \n",
"\n",
"which is an open source\n",
"distribution of the Python and R programming languages for large-scale\n",
"data processing, predictive analytics, and scientific computing, that\n",
"aims to simplify package management and deployment. Package versions\n",
"are managed by the package management system **conda**. \n",
"\n",
"* [Enthought canopy](https://www.enthought.com/product/canopy/) \n",
"\n",
"is a Python\n",
"distribution for scientific and analytic computing distribution and\n",
"analysis environment, available for free and under a commercial\n",
"license.\n",
"\n",
"We recommend using **Anaconda** if you are not too familiar with setting paths in a terminal environment."
]
},
{
"cell_type": "markdown",
"id": "f103f4c1",
"metadata": {
"editable": true
},
"source": [
"## Exercise 2: making your own data and exploring scikit-learn\n",
"\n",
"We will generate our own dataset for a function $y(x)$ where $x \\in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\\cal {N}(0,1)$.\n",
"The following simple Python instructions define our $x$ and $y$ values (with 100 data points)."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "81c11edc",
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"x = np.random.rand(100,1)\n",
"y = 2.0+5*x*x+0.1*np.random.randn(100,1)"
]
},
{
"cell_type": "markdown",
"id": "970bd604",
"metadata": {
"editable": true
},
"source": [
"1. Write your own code (following the examples under the [regression notes](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html)) for computing the parametrization of the data set fitting a second-order polynomial. \n",
"\n",
"2. Use thereafter **scikit-learn** (see again the examples in the regression slides) and compare with your own code. \n",
"\n",
"3. Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as"
]
},
{
"cell_type": "markdown",
"id": "bda3ed04",
"metadata": {
"editable": true
},
"source": [
"$$\n",
"MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "ca995691",
"metadata": {
"editable": true
},
"source": [
"and the $R^2$ score function.\n",
"If $\\tilde{\\boldsymbol{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
]
},
{
"cell_type": "markdown",
"id": "27cbfeed",
"metadata": {
"editable": true
},
"source": [
"$$\n",
"R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "9e063f6d",
"metadata": {
"editable": true
},
"source": [
"where we have defined the mean value of $\\boldsymbol{y}$ as"
]
},
{
"cell_type": "markdown",
"id": "e582c2bb",
"metadata": {
"editable": true
},
"source": [
"$$\n",
"\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "59e0ed7a",
"metadata": {
"editable": true
},
"source": [
"You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. \n",
"Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits."
]
},
{
"cell_type": "markdown",
"id": "051e71aa",
"metadata": {
"editable": true
},
"source": [
"## Exercise 3: Split data in test and training data\n",
"\n",
"In this exercise we want you to to compute the MSE for the training\n",
"data and the test data as function of the complexity of a polynomial,\n",
"that is the degree of a given polynomial.\n",
"\n",
"The aim is to reproduce Figure 2.11 of [Hastie et al](https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf).\n",
"\n",
"Our data is defined by $x\\in [-3,3]$ with a total of for example $n=100$ data points. You should try to vary the number of data points $n$ in your analysis."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "c82d8b64",
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"np.random.seed()\n",
"n = 100\n",
"# Make data set.\n",
"x = np.linspace(-3, 3, n).reshape(-1, 1)\n",
"y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)"
]
},
{
"cell_type": "markdown",
"id": "3e810bba",
"metadata": {
"editable": true
},
"source": [
"where $y$ is the function we want to fit with a given polynomial."
]
},
{
"cell_type": "markdown",
"id": "c1198fce",
"metadata": {
"editable": true
},
"source": [
"**a)**\n",
"Write a first code which sets up a design matrix $X$ defined by a fifth-order polynomial and split your data set in training and test data."
]
},
{
"cell_type": "markdown",
"id": "1b0cea0a",
"metadata": {
"editable": true
},
"source": [
"**b)**\n",
"Write thereafter (using either **scikit-learn** or your matrix inversion code using for example **numpy**)\n",
"and perform an ordinary least squares fitting and compute the mean squared error for the training data and the test data. These calculations should apply to a model given by a fifth-order polynomial."
]
},
{
"cell_type": "markdown",
"id": "5194ba08",
"metadata": {
"editable": true
},
"source": [
"**c)**\n",
"Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE for the training and test data and plot both test and training data MSE as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?"
]
}
],
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