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-The main aim of this project is to study in more detail various +
The main aim of this project is to study in more detail various regression methods, including the Ordinary Least Squares (OLS) method, The total score is 100 points. Each subtask has its own final score. +
--We will first study how to fit polynomials to a specific +
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 @@ -190,9 +175,9 @@ 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 +
The Franke function, which is a weighted sum of four exponentials reads as follows
$$ \begin{align*} f(x,y) &= \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49}- \frac{(9y+1)}{10}\right)} \\ @@ -200,8 +185,7 @@ f(x,y) &= \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)} \end{align*} $$ --The function will be defined for \( x,y\in [0,1] \). Our first step will +
The function will be defined for \( x,y\in [0,1] \). Our first step will be to perform an OLS regression analysis of this function, trying out a polynomial fit with an \( x \) and \( y \) dependence of the form \( [x, y, x^2, y^2, xy, \dots] \). We will also include bootstrap first as @@ -214,19 +198,23 @@ 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 +
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) -
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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
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import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
@@ -265,220 +253,245 @@ ax.zaxis..colorbar(surf, shrink=0.5, aspect=5)
plt.show()
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+-We will generate our own dataset for a function +
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 +
Write your own code (using either a matrix inversion or a singular value decomposition from e.g., numpy ) or use your code from homeworks 1 and 2 and perform a standard least square regression -analysis using polynomials in \( x \) and \( y \) up to fifth order. Find the -confidence intervals of the parameters (estimators) \( \beta \) by computing their -variances, evaluate the Mean Squared error (MSE) +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 +
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 +
where we have defined the mean value of \( \boldsymbol{y} \) as
$$ \bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. $$ --Your code has to include a scaling of the data (for example by +
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 +
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. - +
You can easily reuse the solutions to your exercises from week 35 and week 36.
-Our aim here is to study the bias-variance trade-off by implementing the bootstrap resampling technique. +
Our aim here is to study the bias-variance trade-off by implementing the bootstrap resampling technique.
--With a code which does OLS and includes resampling techniques, +
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. +tasks and basically all Machine Learning algorithms. +
--Before you perform an analysis of the bias-variance trade-off on your test data, make +
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. +
With this result we move on to the bias-variance trade-off analysis.
--Consider a +
Consider a dataset \( \mathcal{L} \) consisting of the data \( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +
--Let us assume that the true data is generated from a noisy model +
Let us 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 +
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 +
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 means +
The parameters \( \boldsymbol{\beta} \) are in turn found by optimizing the means 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. +Here the expected value \( \mathbb{E} \) is the sample value.
--Show that you can rewrite this as +
Show that you can rewrite this as
$$ \mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. $$ --Explain what the terms mean, which one is the bias and which one is +
Explain what the terms mean, which one is the bias and which one is the variance and discuss their interpretations. +
--Perform then a bias-variance analysis of the Franke function by +
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 +
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. +
--Note also that when you calculate the bias, in all applications you don't know the function values \( f_i \). You would hence replace them with the actual data points \( y_i \). - +
Note also that when you calculate the bias, in all applications you don't know the function values \( f_i \). You would hence replace them with the actual data points \( y_i \).
-The aim here is to write your own code for another widely popular +
The aim here is to write your own code for another widely popular resampling technique, the so-called cross-validation method. Again, before you start with cross-validation approach, you should scale your data if you think this is needed. +
--Implement the \( k \)-fold cross-validation algorithm (write your own +
Implement the \( k \)-fold cross-validation algorithm (write your own code) and evaluate again the MSE function resulting from the test folds. You can compare your own code with that from -Scikit-Learn if needed. +Scikit-Learn if needed. +
--Compare the MSE you get from your cross-validation code with the one +
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. You can also compare your own cross-validation code with the one provided by Scikit-Learn. - +
-Write your own code for the Ridge method, either using matrix +
Write your own code for the Ridge method, either using matrix inversion or the singular value decomposition as done in the previous exercise. Perform the same bootstrap analysis as in the Exercise 2 (for the same polynomials) and the cross-validation in exercise 3 but now for different values of \( \lambda \). Compare and analyze your results with those obtained in exercises 1-3. Study the dependence on \( \lambda \). +
--Study also the bias-variance trade-off as function of various values of -the parameter \( \lambda \). For the bias-variance trade-off, use the bootstrap resampling method. Comment your results. - +
Study also the bias-variance trade-off as function of various values of +the parameter \( \lambda \). For the bias-variance trade-off, use the bootstrap resampling method. Comment your results. +
-This exercise is essentially a repeat of the previous two ones, but now +
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). Give a critical discussion of the three methods and a judgement of which -model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the bootstrap resampling technique and an analysis of the mean squared error using cross-validation. - +model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the bootstrap resampling technique and an analysis of the mean squared error using cross-validation. +
-With our codes functioning and having been tested properly on a +
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 1-5. 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/2021/Project1/DataFiles +
Or, if you prefer, we have placed selected datafiles at https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects/2021/Project1/DataFiles
--In order to obtain data for a specific region, you need to register as +
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 +
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scipy.misc.imread
--Here is a simple part of a Python code which reads and plots the data +
scipy.misc.imread
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+Here is a simple part of a Python code which reads and plots the data from such files +
--
import numpy as np
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+ import numpy as np
from imageio import imread
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
@@ -493,39 +506,49 @@ plt.imshow(terrain1, cmap.xlabel('X')
plt.ylabel('Y')
plt.show()
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+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. +
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 +
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 +
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. Note that you can answer question by question and there is no need to structure your report as a scientific report with abstract, introduction, theory, results and discussions, conclusions etc. But you have the following elements in mind when you answer the various questions. +
Here follows a brief recipe and recommendation on how to answer the various questions when preparing your answers. Note that you can answer question by question and there is no need to structure your report as a scientific report with abstract, introduction, theory, results and discussions, conclusions etc. But you have the following elements in mind when you answer the various questions.
-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: +
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. - +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, +
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.
-For Python3, replace pip with pip3. +See below for a discussion of tensorflow and scikit-learn.
--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 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.
-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 - +
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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