mhjensen
288 lines
11 KiB
HTML
288 lines
11 KiB
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<a class="navbar-brand" href="hw1-bs.html">Homework 1</a>
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<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Exercise 1</a></li>
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<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">Exercise 2</a></li>
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<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Exercise 3, variance of the parameters \( \beta \) in linear regression</a></li>
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Homework 1</h1></center> <!-- document title -->
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<!-- author(s): <a href="http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html" target="_self">Data Analysis and Machine Learning FYS-STK3155/FYS4155</a> -->
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<center>
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<b><a href="http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html" target="_self">Data Analysis and Machine Learning FYS-STK3155/FYS4155</a></b>
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<!-- institution -->
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<center><b>Department of Physics, University of Oslo, Norway</b></center>
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<br>
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<center><h4>Aug 27, 2018</h4></center> <!-- date -->
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<br>
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<h2 id="___sec0" class="anchor">Exercise 1 </h2>
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<p>
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The first exercise here is of a mere technical art. We want you have installed
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<ul>
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<li> 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 href="https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html" target="_self">GitHub facilities</a>.</li>
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<li> Install various Python packages</li>
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</ul>
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We will make extensive use of Python as programming language and its
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myriad of available libraries. You will find
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IPython/Jupyter notebooks invaluable in your work. You can run <b>R</b>
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codes in the Jupyter/IPython notebooks, with the immediate benefit of
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visualizing your data. You can also use compiled languages like C++,
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Rust, Fortran etc if you prefer. The focus in these lectures will be
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on Python, but we will provide many code examples for those of you who
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prefer R or compiled languages. You can integrate C++ codes and R in for example
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a Jupyter notebook.
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<p>
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If you have Python installed (we recommend Python3) and you feel
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pretty familiar with installing different packages, we recommend that
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you install the following Python packages via <b>pip</b> as
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<ol>
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<li> pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow</li>
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</ol>
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For <b>Tensorflow</b>, we recommend following the instructions in the text of
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<a href="http://shop.oreilly.com/product/0636920052289.do" target="_self">Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly</a>
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<p>
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We will come back to <b>tensorflow</b> later.
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<p>
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For Python3, replace <b>pip</b> with <b>pip3</b>.
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<p>
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For OSX users we recommend, after having installed Xcode, to
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install <b>brew</b>. Brew allows for a seamless installation of additional
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software via for example
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<ol>
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<li> brew install python3</li>
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</ol>
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For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
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you can use <b>pip</b> as well and simply install Python as
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<ol>
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<li> sudo apt-get install python3 (or python for pyhton2.7)</li>
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</ol>
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If you don't want to perform these operations separately and venture
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into the hassle of exploring how to set up dependencies and paths, we
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recommend two widely used distrubutions which set up all relevant
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dependencies for Python, namely
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<ul>
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<li> <a href="https://docs.anaconda.com/" target="_self">Anaconda</a>,</li>
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</ul>
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which is an open source
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distribution of the Python and R programming languages for large-scale
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data processing, predictive analytics, and scientific computing, that
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aims to simplify package management and deployment. Package versions
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are managed by the package management system <b>conda</b>.
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<ul>
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<li> <a href="https://www.enthought.com/product/canopy/" target="_self">Enthought canopy</a></li>
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</ul>
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is a Python
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distribution for scientific and analytic computing distribution and
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analysis environment, available for free and under a commercial
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license.
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<p>
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We recommend using <b>Anaconda</b>.
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<h2 id="___sec1" class="anchor">Exercise 2 </h2>
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<p>
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We will generate our own dataset for function \( y(x) \) where \( x \in [0,2] \) 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 \( \cal {N}(0,1) \).
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The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points).
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
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y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
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</pre></div>
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<ol>
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<li> Write your own code (following the examples under the <a href="https://compphysics.github.io/MachineLearning/doc/pub/Regression/html/Regression-bs.html" target="_self">regression slides</a> for computing the parametrization of the data set fitting a second-order polynomial.</li>
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<li> Use thereafter <b>scikit-learn</b> (see again the examples in the regression slides) and compare with your own code.</li>
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<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as</li>
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</ol>
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$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
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\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
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$$
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and the \( R^2 \) score function.
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If \( \tilde{\hat{y}}_i \) 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
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$$
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R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
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$$
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where we have defined the mean value of \( \hat{y} \) as
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$$
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\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
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$$
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<p>
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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.
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<h2 id="___sec2" class="anchor">Exercise 3, variance of the parameters \( \beta \) in linear regression </h2>
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<p>
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Show that the variance of the parameters \( \beta \) in the linear regression method (chapter 3, equation (3.8) of <a href="https://www.springer.com/gp/book/9780387848570" target="_self">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>) is given as
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$$
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mathrm{Var}(\hat{\beta}) = \left(\hat{X}^T\hat{X}\right)^{-1}\sigma^2,
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$$
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with
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$$
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\sigma^2 = \frac{1}{N-p-1}\sum_{i=1}{N} (y_i-\tilde{y}_i)^2,
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$$
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where we have assumed that we fit a function of degree \( p-1 \) (for example a polynomial in \( x \)).
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<p>
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