+ + +
The main aim of this project is to study in more detail various +regression methods, including the Ordinary Least Squares (OLS) method. +In addition to the scientific part, in this course we want also to +give you an experience in writing scientific reports. +
+ +We will study how to fit polynomials to specific +one-dimensional functions (feel free to replace the suggested function with more complicated ones). +
+ +We will use Runge's function (see https://en.wikipedia.org/wiki/Runge%27s_phenomenon for a discussion). The one-dimensional function we will study is
+$$ +f(x) = \frac{1}{1+25x^2}. +$$ + +Our first step will be to perform an OLS regression analysis of this +function, trying out a polynomial fit with an \( x \) dependence of the +form \( [x,x^2,\dots] \). We can use a uniform distribution to set up the +arrays of values for \( x \in [-1,1] \), or alternatively use a fixed step size. +Thereafter we will repeat much of the +same procedure using the Ridge and Lasso regression methods, +introducing thus a dependence on the hyperparameter (penalty) \( \lambda \). +
+ +We will also include bootstrap as a resampling technique in order to +study the so-called bias-variance tradeoff. After that we will +include the cross-validation technique. +
+We will generate our own dataset for a function +\( \mathrm{Runge}(x) \) function with \( x\in [-1,1] \). You should explore also the addition +of an added stochastic noise to this function using the normal +distribution \( N(0,1) \). +
+ +Write your own code (using for example the pseudoinverse function pinv from Numpy ) and perform a standard ordinary least square regression +analysis using polynomials in \( x \) up to order \( 15 \). Explore the dependence on the number of data points and the polynomial degree. +
+ +Evaluate the mean Squared error (MSE)
+ +$$ MSE(\boldsymbol{y},\tilde{\boldsymbol{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +$$ + +and the \( R^2 \) score function. 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 +
+ +$$ +R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +$$ + +where we have defined the mean value of \( \boldsymbol{y} \) as
+ +$$ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +$$ + +Plot the resulting scores (MSE and R$^2$) as functions of the polynomial degree (here up to polymial degree 20). +Plot also the parameters \( \theta \) as you increase the order of the polynomial. Comment your results. +
+ +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 this exercise you can +either write your own code or use for example the function for +splitting training data provided by the library Scikit-Learn (make +sure you have installed it). This function is called +\( train\_test\_split \). You should present a critical discussion of why and how you have scaled or not scaled the data. +
+ +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 \( 2/3 \) to \( 4/5 \) of the data as training data. +
+ +You can easily reuse the solutions to your exercises from week 35. +See also the lecture slides from week 35 and week 36. +
+ +On scaling, we recommend reading the following section from the scikit-learn software description, see https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section.
+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 exercise from week 36 is something you can reuse here. +
+ +Perform the same analysis as you did in the previous exercise but now for different values of \( \lambda \). Compare and +analyze your results with those obtained in part a) with the ordinary least squares method. Study the +dependence on \( \lambda \). +
+Replace now the analytical expressions for the optimal parameters +\( \boldsymbol{\theta} \) 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). +
+ +Study and compare your results from parts a) and b) with your gradient +descent approch. Discuss in particular the role of the learning rate. +
+We keep our focus on OLS and Ridge regression and update our code for +the gradient descent method by including momentum, ADAgrad, +RMSprop and ADAM as methods fro iteratively updating your learning +rate. Discuss the results and compare the different methods applied to +the one-dimensional Runge function. +
+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. Use the gradient +descent methods you developed in parts c) and d) to solve the LASSO +optimization problem. You can compare your results using +the functionalities of Scikit-Learn. +
+ +Discuss (critically) your results for the Runge function from OLS, +Ridge and LASSO regression using the various gradient descent +approaches. +
+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. +
+Our aim here is to study the bias-variance trade-off by implementing the bootstrap resampling technique. +We will only use the simpler ordinary least squares here. +
+ +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. +
+ +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! +
+ +With this result we move on to the bias-variance trade-off analysis.
+ +Consider a +dataset \( \mathcal{L} \) consisting of the data +\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +
+ +As in part d), we assume that the true data is generated from a noisy model
+ +$$ +\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}. +$$ + +Here \( \epsilon \) is normally distributed with mean zero and standard +deviation \( \sigma^2 \). +
+ +In our derivation of the ordinary least squares method we defined then +an approximation to the function \( f \) in terms of the parameters +\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model, +that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \). +
+ +The parameters \( \boldsymbol{\beta} \) are in turn found by optimizing the mean +squared error via the so-called cost function +
+ +$$ +C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. +$$ + +Here the expected value \( \mathbb{E} \) is the sample value.
+ +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. +That is, show that +
+$$ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2, +$$ + +with
+$$ +\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right], +$$ + +and
+$$ +\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. +$$ + +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. +
+ +Perform then a bias-variance analysis of the Runge function by +studying the MSE value as function of the complexity of your model. +
+ +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 bootstrap resampling method. +You can follow the code example in the jupyter-book at https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff. +
+The aim here is to implement another widely popular +resampling technique, the so-called cross-validation method. +
+ +Implement the \( k \)-fold cross-validation algorithm (feel free to use the functionality of Scikit-Learn or write your own code) and evaluate again the MSE function resulting +from the test folds. +
+ +Compare the MSE you get from your cross-validation code with the one +you got from your bootstrap code. Comment your results. Try \( 5-10 \) +folds. +
+ +In addition to using the ordinary least squares method, you should include both Ridge and Lasso regression in the analysis.
+Here follows a brief recipe and recommendation on how to answer the various questions when preparing your answers.
+ +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:
+ +Finally, +we encourage you to collaborate. Optimal working groups consist of +2-3 students. You can then hand in a common report. +
+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 +
+For Python3, replace pip with pip3.
+ +See below for a discussion of tensorflow and scikit-learn.
+ +For OSX users we recommend also, after having installed Xcode, to install brew. Brew allows +for a seamless installation of additional software via for example +
+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 +
+etc etc.
+ +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
+Popular software packages written in Python for ML are
+ +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. +
+ ++ +
+ +
- - -
We want you to answer the three different projects by handing in -reports written like a standard scientific/technical report. The link -at -https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ProjectWriting/projectwriting.ipynb -gives some guidance. See also the grading suggestion at -https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md. -
- -Furthermore, at -https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/ReportExample/ -you can find examples of previous reports. How to write reports will -also be discussed during the various lab sessions. Please do ask us if you are in doubt. -
- -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 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 https://scikit-learn.org/dev/about.html. -
- -We enocurage you to use tools like -ChatGPT 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. -
-The main aim of this project is to study in more detail various -regression methods, including the Ordinary Least Squares (OLS) method. -In addition to the scientific part, in this course we want also to -give you an experience in writing scientific reports. -
- -A small recommendation when developing the codes here. Instead of -jumping on to the two-dimensional function described below, we -recommend to do the code development and testing with a simpler -one-dimensional function, similar to those discussed in the exercises -of weeks 35 and 36. A simple test, as discussed during the lectures the first -three weeks is to set the design matrix equal to the identity -matrix. Then your model should give a mean square error which is exactly equal to zero. -When you are sure that your codes function well, you can then replace -the one-dimensional test function with the two-dimensional Franke function -discussed here. -
- -The Franke function serves as a stepping stone towards the analysis of -real topographic data. The latter is the last part of this project. -
-We will first study how to fit polynomials to a specific -two-dimensional function called Franke's -function. This -is a function which has been widely used when testing various -interpolation and fitting algorithms. Furthermore, after having -established the model and the method, we will employ resamling -techniques such as cross-validation and/or bootstrap in order to perform a -proper assessment of our models. We will also study in detail the -so-called Bias-Variance trade off. -
- -The Franke function, which is a weighted sum of four exponentials reads as follows
-$$ -\begin{align*} -f(x,y) &= \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49}- \frac{(9y+1)}{10}\right)} \\ -&+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }. -\end{align*} -$$ - -The function will be defined for \( x,y\in [0,1] \). In a sense, our data are thus scaled to a particular domain for the input values.
- -Our first step will -be to perform an OLS regression analysis of this function, trying out -a polynomial fit with an \( x \) and a \( y \) dependence of the form \( [x, y, -x^2, y^2, xy, \dots] \). We will also include bootstrap first as a -resampling technique. After that we will include the cross-validation -technique. -
- -We can -use a uniform distribution to set up the arrays of values for \( x \) and -\( y \), or as in the example below just a set of fixed values for \( x \) and -\( y \) with a given step size. We will fit a function (for example a -polynomial) of \( x \) and \( y \). Thereafter we will repeat much of the -same procedure using the Ridge and Lasso regression methods, -introducing thus a dependence on the bias (penalty) \( \lambda \). -
- -Finally we are going to use (real) digital terrain data and try to -reproduce these data using the same methods. We will also try to go -beyond the second-order polynomials metioned above and explore -which polynomial fits the data best. -
- -The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)
- - -from mpl_toolkits.mplot3d import Axes3D
-import matplotlib.pyplot as plt
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import numpy as np
-from random import random, seed
-
-fig = plt.figure()
-ax = fig.add_subplot(projection = '3d')
-# Make data.
-x = np.arange(0, 1, 0.05)
-y = np.arange(0, 1, 0.05)
-x, y = np.meshgrid(x,y)
-
-
-def FrankeFunction(x,y):
- term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
- term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
- term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
- term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
- return term1 + term2 + term3 + term4
-
-
-z = FrankeFunction(x, y)
-
-# Plot the surface.
-surf = ax.plot_surface(x, y, z, cmap=cm.coolwarm,
- linewidth=0, antialiased=False)
-
-# Customize the z axis.
-ax.set_zlim(-0.10, 1.40)
-ax.zaxis.set_major_locator(LinearLocator(10))
-ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
-
-# Add a color bar which maps values to colors.
-fig.colorbar(surf, shrink=0.5, aspect=5)
-
-plt.show()
-
-If you wish to compare your results with other on the Franke function or other popular functions tested with linear regression, see the list in Figure 1 of the article by Cook et al at https://arxiv.org/abs/2401.11694.
-We will generate our own dataset for a function -\( \mathrm{FrankeFunction}(x,y) \) with \( x,y \in [0,1] \). The function -\( f(x,y) \) is the Franke function. You should explore also the addition -of an added stochastic noise to this function using the normal -distribution \( N(0,1) \). -
- -Write your own code (using either a matrix inversion or a singular -value decomposition from e.g., numpy ) and perform a standard ordinary least square regression -analysis using polynomials in \( x \) and \( y \) up to fifth order. -
- -Evaluate the mean Squared error (MSE)
- -$$ MSE(\boldsymbol{y},\tilde{\boldsymbol{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -$$ - -and the \( R^2 \) score function. 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 -
- -$$ -R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -$$ - -where we have defined the mean value of \( \boldsymbol{y} \) as
- -$$ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -$$ - -Plot the resulting scores (MSE and R$^2$) as functions of the polynomial degree (here up to polymial degree five). -Plot also the parameters \( \beta \) as you increase the order of the polynomial. Comment your results. -
- -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 this exercise you can -either write your own code or use for example the function for -splitting training data provided by the library Scikit-Learn (make -sure you have installed it). This function is called -\( train\_test\_split \). You should present a critical discussion of why and how you have scaled or not scaled the data. -
- -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 \( 2/3 \) to \( 4/5 \) of the data as training data. -
- -You can easily reuse the solutions to your exercises from week 35 and week 36. -See also the lecture slides from week 35 and week 36. -
- -On scaling, we recommend reading the following section from the scikit-learn software description, see https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section.
-Write your own code for the Ridge method, either using matrix -inversion or the singular value decomposition as done in the previous -exercise. The lecture notes from week 35 and 36 contain more information. Furthermore, the numerical exercise from week 36 is something you can reuse here. -
- -Perform the same analysis as you did in the previous exercise but now for different values of \( \lambda \). Compare and -analyze your results with those obtained in part a) with the ordinary least squares method. Study the -dependence on \( \lambda \). -
-This exercise is essentially a repeat of the previous two ones, but now -with Lasso regression. Write either your own code (difficult and optional) or, in this case, -you can also use the functionalities of Scikit-Learn (recommended). Keep in mind that the library Scikit-Learn excludes the intercept by default. -Give a -critical discussion of the three methods and a judgement of which -model fits the data best. -
-This exercise deals with various mean values and 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 exercise is also part of the weekly exercises for week 37.
- -The assumption we have made is -that there exists a continuous function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim N(0, \sigma^2) \) -which describes our data -
-$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ - -We then approximate this function \( f(\boldsymbol{x}) \) with our model \( \boldsymbol{\tilde{y}} \) 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 or feature matrix.
- -Show that the expectation value of \( \boldsymbol{y} \) for a given element \( i \)
-$$ -\mathbb{E}(y_i) =\sum_{j}x_{ij} \beta_j=\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, -$$ - -and that -its variance is -
-$$ -\mbox{Var}(y_i) = \sigma^2. -$$ - -Hence, \( y_i \sim N( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), 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 optimal parameters \( \boldsymbol{\hat{\beta}} \) show that
-$$ -\mathbb{E}(\boldsymbol{\hat{\beta}}) = \boldsymbol{\beta}. -$$ - -Show finally that the variance of \( \boldsymbol{\beta} \) is
-$$ -\mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. -$$ - -We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). -A given parameter \( \beta_j \) is given by the diagonal matrix element of the above matrix. -
-Our aim here is to study the bias-variance trade-off by implementing the bootstrap resampling technique. -We will only use the simpler ordinary least squares here. -
- -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. -
- -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! -
- -With this result we move on to the bias-variance trade-off analysis.
- -Consider a -dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). -
- -As in part d), we assume that the true data is generated from a noisy model
- -$$ -\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}. -$$ - -Here \( \epsilon \) is normally distributed with mean zero and standard -deviation \( \sigma^2 \). -
- -In our derivation of the ordinary least squares method we defined then -an approximation to the function \( f \) in terms of the parameters -\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model, -that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \). -
- -The parameters \( \boldsymbol{\beta} \) are in turn found by optimizing the mean -squared error via the so-called cost function -
- -$$ -C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. -$$ - -Here the expected value \( \mathbb{E} \) is the sample value.
- -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. -That is, show that -
-$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2, -$$ - -with
-$$ -\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right], -$$ - -and
-$$ -\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. -$$ - -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. -
- -Perform then a bias-variance analysis of the Franke function by -studying the MSE value as function of the complexity of your model. -
- -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 bootstrap resampling method. -You can follow the code example in the jupyter-book at https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff. -
-The aim here is to implement another widely popular -resampling technique, the so-called cross-validation method. -
- -Implement the \( k \)-fold cross-validation algorithm (write your own -code or use the functionality of Scikit-Learn) and evaluate again the MSE function resulting -from the test folds. -
- -Compare the MSE you get from your cross-validation code with the one -you got from your bootstrap code. Comment your results. Try \( 5-10 \) -folds. -
- -In addition to using the ordinary least squares method, you should include both Ridge and Lasso regression.
-With our codes functioning and having been tested properly on a -simpler function we are now ready to look at real data. We will -essentially repeat in this exercise what was done in exercises a-f. However, we -need first to download the data and prepare properly the inputs to our -codes. We are going to download digital terrain data from the website -https://earthexplorer.usgs.gov/, -
- -Or, if you prefer, we have placed selected datafiles at https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects/2023/Project1/DataFiles
- -In order to obtain data for a specific region, you need to register as -a user (free) at this website and then decide upon which area you want -to fetch the digital terrain data from. In order to be able to read -the data properly, you need to specify that the format should be SRTM -Arc-Second Global and download the data as a GeoTIF file. The -files are then stored in tif format which can be imported into a -Python program using -
- - - -scipy.misc.imread
-
-Here is a simple part of a Python code which reads and plots the data -from such files -
- - - -import numpy as np
-from imageio import imread
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-
-# Load the terrain
-terrain1 = imread('SRTM_data_Norway_1.tif')
-# Show the terrain
-plt.figure()
-plt.title('Terrain over Norway 1')
-plt.imshow(terrain1, cmap='gray')
-plt.xlabel('X')
-plt.ylabel('Y')
-plt.show()
-
-If you should have problems in downloading the digital terrain data, -we provide two examples under the data folder of project 1. One is -from a region close to Stavanger in Norway and the other Møsvatn -Austfjell, again in Norway. -Feel free to produce your own terrain data. -
- -Alternatively, if you would like to use another data set, feel free to do so. This could be data close to your reseach area or simply a data set you found interesting. See for example kaggle.com for examples.
- -Our final part deals with the parameterization of your digital terrain -data (or your own data). We will apply all three methods for linear regression, the same type (or higher order) of polynomial -approximation and cross-validation as resampling technique to evaluate which -model fits the data best. -
- -At the end, you should present a critical evaluation of your results -and discuss the applicability of these regression methods to the type -of data presented here (either the terrain data we propose or other data sets). -
-Here follows a brief recipe and recommendation on how to answer the various questions when preparing your answers.
- -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:
- -Finally, -we encourage you to collaborate. Optimal working groups consist of -2-3 students. You can then hand in a common report. -
-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 -
-For Python3, replace pip with pip3.
- -See below for a discussion of tensorflow and scikit-learn.
- -For OSX users we recommend also, after having installed Xcode, to install brew. Brew allows -for a seamless installation of additional software via for example -
-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 -
-etc etc.
- -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
-Popular software packages written in Python for ML are
- -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. -
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