Morten Hjorth-Jensen
681 lines
34 KiB
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681 lines
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
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<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter5.html">8. Support Vector Machines, overarching aims</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Decision Trees, Ensemble Methods and Boosting</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter6.html">9. Decision trees, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter7.html">10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="clustering.html">12. Clustering and Unsupervised Learning</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Statistical interpretation of Linear Regression and Resampling techniques</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Statistical interpretations and Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Logistic Regression and Optimization</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 9 (midnight), 2023</a></li>
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<h1>Exercises week 34</h1>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-setting-up-various-python-environments">Exercise 1: Setting up various Python environments</a></li>
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<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">Exercise 2: making your own data and exploring scikit-learn</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-split-data-in-test-and-training-data">Exercise 3: Split data in test and training data</a></li>
|
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<!-- dom:TITLE: Exercises week 34 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-34">
|
||
<h1>Exercises week 34<a class="headerlink" href="#exercises-week-34" title="Permalink to this heading">#</a></h1>
|
||
<p><strong>FYS-STK3155/4155</strong></p>
|
||
<p>Date: <strong>August 21-25, 2023</strong></p>
|
||
<section id="exercises">
|
||
<h2>Exercises<a class="headerlink" href="#exercises" title="Permalink to this heading">#</a></h2>
|
||
<p>Here are three possible exercises for week 34</p>
|
||
</section>
|
||
<section id="exercise-1-setting-up-various-python-environments">
|
||
<h2>Exercise 1: Setting up various Python environments<a class="headerlink" href="#exercise-1-setting-up-various-python-environments" title="Permalink to this heading">#</a></h2>
|
||
<p>The first exercise here is of a mere technical art. We want you to have</p>
|
||
<ul class="simple">
|
||
<li><p>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 <a class="reference external" href="https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html">GitHub facilities</a>.</p></li>
|
||
<li><p>Install various Python packages</p></li>
|
||
</ul>
|
||
<p>We will make extensive use of Python as programming language and its
|
||
myriad of available libraries. You will find
|
||
IPython/Jupyter notebooks invaluable in your work. You can run <strong>R</strong>
|
||
codes in the Jupyter/IPython notebooks, with the immediate benefit of
|
||
visualizing your data. You can also use compiled languages like C++,
|
||
Rust, Fortran etc if you prefer. The focus in these lectures will be
|
||
on Python.</p>
|
||
<p>If you have Python installed (we recommend Python3) and you feel
|
||
pretty familiar with installing different packages, we recommend that
|
||
you install the following Python packages via <strong>pip</strong> as</p>
|
||
<ol class="arabic simple">
|
||
<li><p>pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow</p></li>
|
||
</ol>
|
||
<p>For <strong>Tensorflow</strong>, we recommend following the instructions in the text of
|
||
<a class="reference external" href="http://shop.oreilly.com/product/0636920052289.do">Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O’Reilly</a></p>
|
||
<p>We will come back to <strong>tensorflow</strong> later.</p>
|
||
<p>For Python3, replace <strong>pip</strong> with <strong>pip3</strong>.</p>
|
||
<p>For OSX users we recommend, after having installed Xcode, to
|
||
install <strong>brew</strong>. Brew allows for a seamless installation of additional
|
||
software via for example</p>
|
||
<ol class="arabic simple">
|
||
<li><p>brew install python3</p></li>
|
||
</ol>
|
||
<p>For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
|
||
you can use <strong>pip</strong> as well and simply install Python as</p>
|
||
<ol class="arabic simple">
|
||
<li><p>sudo apt-get install python3 (or python for Python2.7)</p></li>
|
||
</ol>
|
||
<p>If you don’t want to perform these operations separately and venture
|
||
into the hassle of exploring how to set up dependencies and paths, we
|
||
recommend two widely used distrubutions which set up all relevant
|
||
dependencies for Python, namely</p>
|
||
<ul class="simple">
|
||
<li><p><a class="reference external" href="https://docs.anaconda.com/">Anaconda</a>,</p></li>
|
||
</ul>
|
||
<p>which is an open source
|
||
distribution of the Python and R programming languages for large-scale
|
||
data processing, predictive analytics, and scientific computing, that
|
||
aims to simplify package management and deployment. Package versions
|
||
are managed by the package management system <strong>conda</strong>.</p>
|
||
<ul class="simple">
|
||
<li><p><a class="reference external" href="https://www.enthought.com/product/canopy/">Enthought canopy</a></p></li>
|
||
</ul>
|
||
<p>is a Python
|
||
distribution for scientific and analytic computing distribution and
|
||
analysis environment, available for free and under a commercial
|
||
license.</p>
|
||
<p>We recommend using <strong>Anaconda</strong> if you are not too familiar with setting paths in a terminal environment.</p>
|
||
</section>
|
||
<section id="exercise-2-making-your-own-data-and-exploring-scikit-learn">
|
||
<h2>Exercise 2: making your own data and exploring scikit-learn<a class="headerlink" href="#exercise-2-making-your-own-data-and-exploring-scikit-learn" title="Permalink to this heading">#</a></h2>
|
||
<p>We will generate our own dataset for a function <span class="math notranslate nohighlight">\(y(x)\)</span> where <span class="math notranslate nohighlight">\(x \in [0,1]\)</span> and defined by random numbers computed with the uniform distribution. The function <span class="math notranslate nohighlight">\(y\)</span> is a quadratic polynomial in <span class="math notranslate nohighlight">\(x\)</span> with added stochastic noise according to the normal distribution <span class="math notranslate nohighlight">\(\cal {N}(0,1)\)</span>.
|
||
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>
|
||
<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">----> </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 'np' is not defined
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<ol class="arabic 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>
|
||
<li><p>Use thereafter <strong>scikit-learn</strong> (see again the examples in the regression slides) and compare with your own code.</p></li>
|
||
<li><p>Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</p></li>
|
||
</ol>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
|
||
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
||
\]</div>
|
||
<p>and the <span class="math notranslate nohighlight">\(R^2\)</span> score function.
|
||
If <span class="math notranslate nohighlight">\(\tilde{\boldsymbol{y}}_i\)</span> is the predicted value of the <span class="math notranslate nohighlight">\(i-th\)</span> sample and <span class="math notranslate nohighlight">\(y_i\)</span> is the corresponding true value, then the score <span class="math notranslate nohighlight">\(R^2\)</span> is defined as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
||
\]</div>
|
||
<p>where we have defined the mean value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
||
\]</div>
|
||
<p>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.
|
||
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.</p>
|
||
</section>
|
||
<section id="exercise-3-split-data-in-test-and-training-data">
|
||
<h2>Exercise 3: Split data in test and training data<a class="headerlink" href="#exercise-3-split-data-in-test-and-training-data" title="Permalink to this heading">#</a></h2>
|
||
<p>In this exercise we want you to to compute the MSE for the training
|
||
data and the test data as function of the complexity of a polynomial,
|
||
that is the degree of a given polynomial.</p>
|
||
<p>The aim is to reproduce Figure 2.11 of <a class="reference external" href="https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf">Hastie et al</a>.</p>
|
||
<p>Our data is defined by <span class="math notranslate nohighlight">\(x\in [-3,3]\)</span> with a total of for example <span class="math notranslate nohighlight">\(n=100\)</span> data points. You should try to vary the number of data points <span class="math notranslate nohighlight">\(n\)</span> in your analysis.</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">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">()</span>
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="c1"># Make data set.</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</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="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>where <span class="math notranslate nohighlight">\(y\)</span> is the function we want to fit with a given polynomial.</p>
|
||
<p><strong>a)</strong>
|
||
Write a first code which sets up a design matrix <span class="math notranslate nohighlight">\(X\)</span> defined by a fifth-order polynomial and split your data set in training and test data.</p>
|
||
<p><strong>b)</strong>
|
||
Write thereafter (using either <strong>scikit-learn</strong> or your matrix inversion code using for example <strong>numpy</strong>)
|
||
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.</p>
|
||
<p><strong>c)</strong>
|
||
Add now a model which allows you to make polynomials up to degree <span class="math notranslate nohighlight">\(15\)</span>. 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)?</p>
|
||
</section>
|
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</section>
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<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">Exercise 2: making your own data and exploring scikit-learn</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3-split-data-in-test-and-training-data">Exercise 3: Split data in test and training data</a></li>
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