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<h1>Project 1 on Machine Learning, deadline October 6 (midnight), 2025</h1>
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<h2> Contents </h2>
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<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools">Preamble: Note on writing reports, using reference material, AI and other tools</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-analysis-and-resampling-methods">Regression analysis and resampling methods</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-a-ordinary-least-square-ols-for-the-runge-function">Part a : Ordinary Least Square (OLS) for the Runge function</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-b-adding-ridge-regression-for-the-runge-function">Part b: Adding Ridge regression for the Runge function</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-c-writing-your-own-gradient-descent-code">Part c: Writing your own gradient descent code</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate">Part d: Including momentum and more advanced ways to update the learning the rate</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-e-writing-our-own-code-for-lasso-regression">Part e: Writing our own code for Lasso regression</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-f-stochastic-gradient-descent">Part f: Stochastic gradient descent</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-g-bias-variance-trade-off-and-resampling-techniques">Part g: Bias-variance trade-off and resampling techniques</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-h-cross-validation-as-resampling-techniques-adding-more-complexity">Part h): Cross-validation as resampling techniques, adding more complexity</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-numerical-projects">Introduction to numerical projects</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#format-for-electronic-delivery-of-report-and-programs">Format for electronic delivery of report and programs</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#software-and-needed-installations">Software and needed installations</a></li>
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
doconce format html Project1.do.txt -->
<!-- dom:TITLE: Project 1 on Machine Learning, deadline October 6 (midnight), 2025 --><section class="tex2jax_ignore mathjax_ignore" id="project-1-on-machine-learning-deadline-october-6-midnight-2025">
<h1>Project 1 on Machine Learning, deadline October 6 (midnight), 2025<a class="headerlink" href="#project-1-on-machine-learning-deadline-october-6-midnight-2025" title="Link to this heading">#</a></h1>
<p><strong>Data Analysis and Machine Learning FYS-STK3155/FYS4155</strong>, University of Oslo, Norway</p>
<p>Date: <strong>September 2</strong></p>
<section id="preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools">
<h2>Preamble: Note on writing reports, using reference material, AI and other tools<a class="headerlink" href="#preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools" title="Link to this heading">#</a></h2>
<p>We want you to answer the three different projects by handing in
reports written like a standard scientific/technical report. The
links at
<a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects">CompPhysics/MachineLearning</a>
contain more information. There you can find examples of previous
reports, the projects themselves, how we grade reports etc. How to
write reports will also be discussed during the various lab
sessions. Please do ask us if you are in doubt.</p>
<p>When using codes and material from other sources, you should refer to
these in the bibliography of your report, indicating wherefrom you for
example got the code, whether this is from the lecture notes,
softwares like Scikit-Learn, TensorFlow, PyTorch or other sources. These sources
should always be cited correctly. How to cite some
of the libraries is often indicated from their corresponding GitHub
sites or websites, see for example how to cite Scikit-Learn at
<a class="reference external" href="https://scikit-learn.org/dev/about.html">https://scikit-learn.org/dev/about.html</a>.</p>
<p>We enocurage you to use tools like
<a class="reference external" href="https://openai.com/chatgpt/">ChatGPT</a> or similar in writing the report. If you use for example ChatGPT,
please do cite it properly and include (if possible) your questions and answers as an addition to the report. This can
be uploaded to for example your website, GitHub/GitLab or similar as supplemental material.</p>
<p>If you would like to study other data sets, feel free to propose other
sets. What we have proposed here are mere suggestions from our
side. If you opt for another data set, consider using a set which has
been studied in the scientific literature. This makes it easier for
you to compare and analyze your results. Comparing with existing
results from the scientific literature is also an essential element of
the scientific discussion. The University of California at Irvine
with its Machine Learning repository at
<a class="reference external" href="https://archive.ics.uci.edu/ml/index.php">https://archive.ics.uci.edu/ml/index.php</a> is an excellent site to
look up for examples and
inspiration. <a class="reference external" href="https://www.kaggle.com/">Kaggle.com</a> is an equally
interesting site. Feel free to explore these sites. When selecting
other data sets, make sure these are sets used for regression problems
(not classification).</p>
</section>
<section id="regression-analysis-and-resampling-methods">
<h2>Regression analysis and resampling methods<a class="headerlink" href="#regression-analysis-and-resampling-methods" title="Link to this heading">#</a></h2>
<p>The main aim of this project is to study in more detail various
regression methods, including Ordinary Least Squares (OLS) reegression, Ridge regression and LASSO regression.
In addition to the scientific part, in this course we want also to
give you an experience in writing scientific reports.</p>
<p>We will study how to fit polynomials to specific
one-dimensional functions (feel free to replace the suggested function with more complicated ones).</p>
<p>We will use Runge’s function (see <a class="reference external" href="https://en.wikipedia.org/wiki/Runge%27s_phenomenon">https://en.wikipedia.org/wiki/Runge’s_phenomenon</a> for a discussion). The one-dimensional function we will study is</p>
<div class="math notranslate nohighlight">
\[
f(x) = \frac{1}{1+25x^2}.
\]</div>
<p>Our first step will be to perform an OLS regression analysis of this
function, trying out a polynomial fit with an <span class="math notranslate nohighlight">\(x\)</span> dependence of the
form <span class="math notranslate nohighlight">\([x,x^2,\dots]\)</span>. You can use a uniform distribution to set up the
arrays of values for <span class="math notranslate nohighlight">\(x \in [-1,1]\)</span>, or alternatively use a fixed step size.
Thereafter we will repeat many of the same steps when using the Ridge and Lasso regression methods,
introducing thereby a dependence on the hyperparameter (penalty) <span class="math notranslate nohighlight">\(\lambda\)</span>.</p>
<p>We will also include bootstrap as a resampling technique in order to
study the so-called <strong>bias-variance tradeoff</strong>. After that we will
include the so-called cross-validation technique.</p>
<section id="part-a-ordinary-least-square-ols-for-the-runge-function">
<h3>Part a : Ordinary Least Square (OLS) for the Runge function<a class="headerlink" href="#part-a-ordinary-least-square-ols-for-the-runge-function" title="Link to this heading">#</a></h3>
<p>We will generate our own dataset for abovementioned function
<span class="math notranslate nohighlight">\(\mathrm{Runge}(x)\)</span> function with <span class="math notranslate nohighlight">\(x\in [-1,1]\)</span>. You should explore also the addition
of an added stochastic noise to this function using the normal
distribution <span class="math notranslate nohighlight">\(N(0,1)\)</span>.</p>
<p><em>Write your own code</em> (using for example the pseudoinverse function <strong>pinv</strong> from <strong>Numpy</strong> ) and perform a standard <strong>ordinary least square regression</strong>
analysis using polynomials in <span class="math notranslate nohighlight">\(x\)</span> up to order <span class="math notranslate nohighlight">\(15\)</span> or higher. Explore the dependence on the number of data points and the polynomial degree.</p>
<p>Evaluate the mean Squared error (MSE)</p>
<div class="math notranslate nohighlight">
\[
MSE(\boldsymbol{y},\tilde{\boldsymbol{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
\]</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>Plot the resulting scores (MSE and R<span class="math notranslate nohighlight">\(^2\)</span>) as functions of the polynomial degree (here up to polymial degree 15).
Plot also the parameters <span class="math notranslate nohighlight">\(\theta\)</span> as you increase the order of the polynomial. Comment your results.</p>
<p>Your code has to include a scaling/centering of the data (for example by
subtracting the mean value), and
a split of the data in training and test data. For the scaling you can
either write your own code or use for example the function for
splitting training data provided by the library <strong>Scikit-Learn</strong> (make
sure you have installed it). This function is called
<span class="math notranslate nohighlight">\(train\_test\_split\)</span>. <strong>You should present a critical discussion of why and how you have scaled or not scaled the data</strong>.</p>
<p>It is normal in essentially all Machine Learning studies to split the
data in a training set and a test set (eventually also an additional
validation set). There
is no explicit recipe for how much data should be included as training
data and say test data. An accepted rule of thumb is to use
approximately <span class="math notranslate nohighlight">\(2/3\)</span> to <span class="math notranslate nohighlight">\(4/5\)</span> of the data as training data.</p>
<p>You can easily reuse the solutions to your exercises from week 35.
See also the lecture slides from week 35 and week 36.</p>
<p>On scaling, we recommend reading the following section from the scikit-learn software description, see <a class="reference external" href="https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section">https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section</a>.</p>
</section>
<section id="part-b-adding-ridge-regression-for-the-runge-function">
<h3>Part b: Adding Ridge regression for the Runge function<a class="headerlink" href="#part-b-adding-ridge-regression-for-the-runge-function" title="Link to this heading">#</a></h3>
<p>Write your own code for the Ridge method as done in the previous
exercise. The lecture notes from week 35 and 36 contain more information. Furthermore, the results from the exercise set from week 36 is something you can reuse here.</p>
<p>Perform the same analysis as you did in the previous exercise but now for different values of <span class="math notranslate nohighlight">\(\lambda\)</span>. Compare and
analyze your results with those obtained in part a) with the OLS method. Study the
dependence on <span class="math notranslate nohighlight">\(\lambda\)</span>.</p>
</section>
<section id="part-c-writing-your-own-gradient-descent-code">
<h3>Part c: Writing your own gradient descent code<a class="headerlink" href="#part-c-writing-your-own-gradient-descent-code" title="Link to this heading">#</a></h3>
<p>Replace now the analytical expressions for the optimal parameters
<span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> with your own gradient descent code. In this exercise we
focus only on the simplest gradient descent approach with a fixed
learning rate (see the exercises from week 37 and the lecture notes
from week 36).</p>
<p>Study and compare your results from parts a) and b) with your gradient
descent approch. Discuss in particular the role of the learning rate.</p>
</section>
<section id="part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate">
<h3>Part d: Including momentum and more advanced ways to update the learning the rate<a class="headerlink" href="#part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate" title="Link to this heading">#</a></h3>
<p>We keep our focus on OLS and Ridge regression and update our code for
the gradient descent method by including <strong>momentum</strong>, <strong>ADAgrad</strong>,
<strong>RMSprop</strong> and <strong>ADAM</strong> as methods fro iteratively updating your learning
rate. Discuss the results and compare the different methods applied to
the one-dimensional Runge function. The lecture notes from week 37 contain several examples on how to implement these methods.</p>
</section>
<section id="part-e-writing-our-own-code-for-lasso-regression">
<h3>Part e: Writing our own code for Lasso regression<a class="headerlink" href="#part-e-writing-our-own-code-for-lasso-regression" title="Link to this heading">#</a></h3>
<p>LASSO regression (see lecture slides from week 36 and week 37)
represents our first encounter with a machine learning method which
cannot be solved through analytical expressions (as in OLS and Ridge regression). Use the gradient
descent methods you developed in parts c) and d) to solve the LASSO
optimization problem. You can compare your results with
the functionalities of <strong>Scikit-Learn</strong>.</p>
<p>Discuss (critically) your results for the Runge function from OLS,
Ridge and LASSO regression using the various gradient descent
approaches.</p>
</section>
<section id="part-f-stochastic-gradient-descent">
<h3>Part f: Stochastic gradient descent<a class="headerlink" href="#part-f-stochastic-gradient-descent" title="Link to this heading">#</a></h3>
<p>Our last gradient step is to include stochastic gradient descent using
the same methods to update the learning rates as in parts c-e).
Compare and discuss your results with and without stochastic gradient
and give a critical assessment of the various methods.</p>
</section>
<section id="part-g-bias-variance-trade-off-and-resampling-techniques">
<h3>Part g: Bias-variance trade-off and resampling techniques<a class="headerlink" href="#part-g-bias-variance-trade-off-and-resampling-techniques" title="Link to this heading">#</a></h3>
<p>Our aim here is to study the bias-variance trade-off by implementing
the <strong>bootstrap</strong> resampling technique. <strong>We will only use the simpler
ordinary least squares here</strong>.</p>
<p>With a code which does OLS and includes resampling techniques,
we will now discuss the bias-variance trade-off in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks and basically all Machine Learning algorithms.</p>
<p>Before you perform an analysis of the bias-variance trade-off on your
test data, make first a figure similar to Fig. 2.11 of Hastie,
Tibshirani, and Friedman. Figure 2.11 of this reference displays only
the test and training MSEs. The test MSE can be used to indicate
possible regions of low/high bias and variance. You will most likely
not get an equally smooth curve! You may also need to increase the
polynomial order and play around with the number of data points as
well (see also the exercise set from week 35).</p>
<p>With this result we move on to the bias-variance trade-off analysis.</p>
<p>Consider a
dataset <span class="math notranslate nohighlight">\(\mathcal{L}\)</span> consisting of the data
<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</span>.</p>
<p>We assume that the true data is generated from a noisy model</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}.
\]</div>
<p>Here <span class="math notranslate nohighlight">\(\epsilon\)</span> is normally distributed with mean zero and standard
deviation <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
<p>In our derivation of the ordinary least squares method we defined then
an approximation to the function <span class="math notranslate nohighlight">\(f\)</span> in terms of the parameters
<span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which embody our model,
that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\theta}\)</span>.</p>
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> are in turn found by optimizing the mean
squared error via the so-called cost function</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{X},\boldsymbol{\theta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
\]</div>
<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
<p>Show that you can rewrite this in terms of a term which contains the
variance of the model itself (the so-called variance term), a term
which measures the deviation from the true data and the mean value of
the model (the bias term) and finally the variance of the noise.</p>
<p>That is, show that</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2,
\]</div>
<p>with (we approximate <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\approx \boldsymbol{y}\)</span>)</p>
<div class="math notranslate nohighlight">
\[
\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\]</div>
<p><strong>Important note</strong>: Since the function <span class="math notranslate nohighlight">\(f(x)\)</span> is unknown, in order to be able to evalute the bias, we replace <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> in the expression for the bias with <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>.</p>
<p>The answer to this exercise should be included in the theory part of
the report. This exercise is also part of the weekly exercises of
week 38. Explain what the terms mean and discuss their
interpretations.</p>
<p>Perform then a bias-variance analysis of the Runge function by
studying the MSE value as function of the complexity of your model.</p>
<p>Discuss the bias and variance trade-off as function
of your model complexity (the degree of the polynomial) and the number
of data points, and possibly also your training and test data using the <strong>bootstrap</strong> resampling method.
You can follow the code example in the jupyter-book at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff</a>.</p>
</section>
<section id="part-h-cross-validation-as-resampling-techniques-adding-more-complexity">
<h3>Part h): Cross-validation as resampling techniques, adding more complexity<a class="headerlink" href="#part-h-cross-validation-as-resampling-techniques-adding-more-complexity" title="Link to this heading">#</a></h3>
<p>The aim here is to implement another widely popular
resampling technique, the so-called cross-validation method.</p>
<p>Implement the <span class="math notranslate nohighlight">\(k\)</span>-fold cross-validation algorithm (feel free to use
the functionality of <strong>Scikit-Learn</strong> or write your own code) and
evaluate again the MSE function resulting from the test folds.</p>
<p>Compare the MSE you get from your cross-validation code with the one
you got from your <strong>bootstrap</strong> code from the previous exercise. Comment and interpret your results.</p>
<p>In addition to using the ordinary least squares method, you should
include both Ridge and Lasso regression in the final analysis.</p>
</section>
</section>
<section id="background-literature">
<h2>Background literature<a class="headerlink" href="#background-literature" title="Link to this heading">#</a></h2>
<ol class="arabic simple">
<li><p>For a discussion and derivation of the variances and mean squared errors using linear regression, see the <a class="reference external" href="https://arxiv.org/abs/1509.09169">Lecture notes on ridge regression by Wessel N. van Wieringen</a></p></li>
<li><p>The textbook of <a class="reference external" href="https://www.springer.com/gp/book/9780387848570">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>, chapters 3 and 7 are the most relevant ones for the analysis of parts g) and h).</p></li>
</ol>
</section>
<section id="introduction-to-numerical-projects">
<h2>Introduction to numerical projects<a class="headerlink" href="#introduction-to-numerical-projects" title="Link to this heading">#</a></h2>
<p>Here follows a brief recipe and recommendation on how to answer the various questions when preparing your answers.</p>
<ul class="simple">
<li><p>Give a short description of the nature of the problem and the eventual numerical methods you have used.</p></li>
<li><p>Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.</p></li>
<li><p>Include the source code of your program. Comment your program properly. You should have the code at your GitHub/GitLab link. You can also place the code in an appendix of your report.</p></li>
<li><p>If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.</p></li>
<li><p>Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.</p></li>
<li><p>Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.</p></li>
<li><p>Try to give an interpretation of you results in your answers to the problems.</p></li>
<li><p>Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you’ve made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.</p></li>
<li><p>Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don’t properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.</p></li>
</ul>
</section>
<section id="format-for-electronic-delivery-of-report-and-programs">
<h2>Format for electronic delivery of report and programs<a class="headerlink" href="#format-for-electronic-delivery-of-report-and-programs" title="Link to this heading">#</a></h2>
<p>The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008, Julia or Python. The following prescription should be followed when preparing the report:</p>
<ul class="simple">
<li><p>Use Canvas to hand in your projects, log in at <a class="reference external" href="https://www.uio.no/english/services/it/education/canvas/">https://www.uio.no/english/services/it/education/canvas/</a> with your normal UiO username and password.</p></li>
<li><p>Upload <strong>only</strong> the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.</p></li>
<li><p>In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.</p></li>
</ul>
<p>Finally,
we encourage you to collaborate. Optimal working groups consist of
2-3 students. You can then hand in a common report.</p>
</section>
<section id="software-and-needed-installations">
<h2>Software and needed installations<a class="headerlink" href="#software-and-needed-installations" title="Link to this heading">#</a></h2>
<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 tensorflow sympy pandas pillow</p></li>
</ol>
<p>For Python3, replace <strong>pip</strong> with <strong>pip3</strong>.</p>
<p>See below for a discussion of <strong>tensorflow</strong> and <strong>scikit-learn</strong>.</p>
<p>For OSX users we recommend also, 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>etc etc.</p>
<p>If you don’t want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely</p>
<ol class="arabic simple">
<li><p><a class="reference external" href="https://docs.anaconda.com/">Anaconda</a> Anaconda 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></li>
<li><p><a class="reference external" href="https://www.enthought.com/product/canopy/">Enthought canopy</a> is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.</p></li>
</ol>
<p>Popular software packages written in Python for ML are</p>
<ul class="simple">
<li><p><a class="reference external" href="http://scikit-learn.org/stable/">Scikit-learn</a>,</p></li>
<li><p><a class="reference external" href="https://www.tensorflow.org/">Tensorflow</a>,</p></li>
<li><p><a class="reference external" href="http://pytorch.org/">PyTorch</a> and</p></li>
<li><p><a class="reference external" href="https://keras.io/">Keras</a>.</p></li>
</ul>
<p>These are all freely available at their respective GitHub sites. They
encompass communities of developers in the thousands or more. And the number
of code developers and contributors keeps increasing.</p>
</section>
</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools">Preamble: Note on writing reports, using reference material, AI and other tools</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-analysis-and-resampling-methods">Regression analysis and resampling methods</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-a-ordinary-least-square-ols-for-the-runge-function">Part a : Ordinary Least Square (OLS) for the Runge function</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-b-adding-ridge-regression-for-the-runge-function">Part b: Adding Ridge regression for the Runge function</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-c-writing-your-own-gradient-descent-code">Part c: Writing your own gradient descent code</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate">Part d: Including momentum and more advanced ways to update the learning the rate</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-e-writing-our-own-code-for-lasso-regression">Part e: Writing our own code for Lasso regression</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-f-stochastic-gradient-descent">Part f: Stochastic gradient descent</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-g-bias-variance-trade-off-and-resampling-techniques">Part g: Bias-variance trade-off and resampling techniques</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#part-h-cross-validation-as-resampling-techniques-adding-more-complexity">Part h): Cross-validation as resampling techniques, adding more complexity</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#background-literature">Background literature</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-numerical-projects">Introduction to numerical projects</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#format-for-electronic-delivery-of-report-and-programs">Format for electronic delivery of report and programs</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#software-and-needed-installations">Software and needed installations</a></li>
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