Homework 1 Fall Semester 2019

Data Analysis and Machine Learning FYS-STK3155/FYS4155

Department of Physics, University of Oslo, Norway

Aug 20, 2019

Exercise 1

The first exercise here is of a mere technical art. We want you to have

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.
Install various Python packages

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 R 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.

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 pip as

pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow

For Tensorflow, we recommend following the instructions in the text of Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly

We will come back to tensorflow later.

For Python3, replace pip with pip3.

For OSX users we recommend, after having installed Xcode, to install brew. Brew allows for a seamless installation of additional software via for example

brew install python3

For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, you can use pip as well and simply install Python as

sudo apt-get install python3 (or python for pyhton2.7)

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

Anaconda,

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 conda.

Enthought canopy

is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.

We recommend using Anaconda.

Exercise 2

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).

x = np.random.rand(100,1)
y = 5*x*x+0.1*np.random.randn(100,1)

Write your own code (following the examples under the regression slides) for computing the parametrization of the data set fitting a second-order polynomial.
Use thereafter scikit-learn (see again the examples in the regression slides) and compare with your own code.
Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as

$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, $$ and the $ R^2 $ score function. 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 $$ R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, $$ where we have defined the mean value of $ \hat{y} $ as $$ \bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. $$ 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.

Exercise 3, mean values and variances in linear regression

This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer).

The assumption we have made is that there exists a function $ f(\boldsymbol{x}) $ and a normal distributed error $ \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) $ which describes our data $$ \boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} $$

We then approximate this function with our model from the solution of the linear regression equations (ordinary least squares OLS), that is our function $ f $ is approximated by $ \boldsymbol{\tilde{y}} $ where we minimized $ (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 $, with $$ \boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. $$ The matrix $ \boldsymbol{X} $ is the so-called design matrix.

Show that the expectation value of $ \boldsymbol{y} $ for a given element $ i $ $$ \begin{align*} \mathbb{E}(y_i) & =\mathbf{X}_{i, \ast} \, \beta, \end{align*} $$ and that its variance is $$ \begin{align*} \mbox{Var}(y_i) & = \sigma^2. \end{align*} $$ Hence, $ y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) $, that is $ \boldsymbol{y} $ follows a normal distribution with mean value $ \boldsymbol{X}\boldsymbol{\beta} $ and variance $ \sigma^2 $.

With the OLS expressions for the parameters $ \boldsymbol{\beta} $ show that $$ \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}. $$ This means that the estimator of the regression parameters is unbiased.

Show finally that the variance of $ \boldsymbol{\beta} $ is $$ \begin{eqnarray*} \mbox{Var}(\boldsymbol{\beta}) & = & \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. \end{eqnarray*} $$