From a2a1bb885f70afe650238343b2254bc831c629c5 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Tue, 11 Sep 2018 21:56:08 +0200 Subject: [PATCH] First iteration of project 1 --- .../2018/Project1/html/._Project1-bs000.html | 459 ++++++++++++++++++ .../2018/Project1/html/Project1-bs.html | 263 +++++++++- doc/Projects/2018/Project1/html/Project1.html | 246 +++++++++- .../Project1/ipynb/ipynb-Project1-src.tar.gz | Bin 210 -> 210 bytes doc/Projects/2018/Project1/pdf/Project1.p.tex | 179 ++++++- doc/Projects/2018/Project1/pdf/Project1.pdf | Bin 177657 -> 250188 bytes doc/Projects/2018/Project1/pdf/Project1.tex | 175 ++++++- .../Projects/2018/Project1/Project1.do.txt | 2 +- 8 files changed, 1272 insertions(+), 52 deletions(-) create mode 100644 doc/Projects/2018/Project1/html/._Project1-bs000.html diff --git a/doc/Projects/2018/Project1/html/._Project1-bs000.html b/doc/Projects/2018/Project1/html/._Project1-bs000.html new file mode 100644 index 000000000..07aa51264 --- /dev/null +++ b/doc/Projects/2018/Project1/html/._Project1-bs000.html @@ -0,0 +1,459 @@ + + + + + + + +Project 1 on Machine Learning, deadline October 1 + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +

 

 

 

+ + + + + + +
+

Project 1 on Machine Learning, deadline October 1

+ +

+ + +

+Data Analysis and Machine Learning FYS-STK3155/FYS4155 +
+ +

+ + +

Department of Physics, University of Oslo, Norway
+
+

+

Sep 11, 2018

+
+

+

+ +

Regression analysis and resampling methods

+ +

+The main aim of this project is to study in more detail various +regression methods, including the Ordinary Least Squares (OLS) method, +Ridge regression and finally Lasso regression. +The methods are in turn combined with resampling techniques. + +

+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 etsablished the model and the +method, we will employ resamling techniques such as the cross-validation and +the bootstrap methods, in order to perform a proper assessment of our models. + +

+The Franke function, which is a weighted sum of four exponentials reads as follows +$$ +f(x,y) = \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49} - \frac{(9y+1)}{10}\right)}+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }. +$$ + +

+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 cross-validation and +bootstrap as resampling techniques. As in homeworks 1 and 2, 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 fix values for \( x \) and \( y \) with a given step size. +In this case we will have two predictors and need to fit a +function (for example a polynomial) of \( x \) and \( y \). Thereafter we will +repeat much of the same procedure using the the Ridge and +Lasso regression methods, introducing thus a dependence on the bias +(penalty) \( \lambda \). + +

+Thereafter 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 fucntion 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.gca(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()
+
+ +

Part a): Ordinary Least Square on the Franke function with resampling

+ +

+We will thus again generate our own dataset for a function \( \mathrm{FrankeFunction}(x,y) \) where +\( x,y \in [0,1] \) could be defined by random numbers computed with the uniform +distribution. The function \( f(x,y) \) is the Franke function. You should explore also the addition +an added stochastic noise to this function using the normal distribution \( \cal{N}(0,1) \). + +

+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 \( \beta \) by computing their variances, evaluate the Mean Squared error (MSE) +$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +$$ + +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 +$$ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +$$ + +

+Perform a resampling of the data where you split the data in training data and test data. Implement the \( k \)-fold cross-validation algorithm and the bootstrap algorithm +and evaluate again the MSE and the \( R^2 \) functions resulting from the test data. + +

Part b): Ridge Regression on the Franke function with resampling

+ +

+Write your own code for the Ridge method, either using matrix inversion or the singular value decomposition as done in the previous exercise or howework 2 (see also chapter 3.4 of Hastie et al., equations (3.43) and (3.44)). Perform the same analysis as in the previous exercise (for the same polynomials and include resampling techniques) but now for different values of \( \lambda \). Compare and analyze your results with those obtained in part a). Study the dependence on \( \lambda \) while also varying eventually the strength of the noise in your expression for \( \mathrm{FrankeFunction}(x,y) \). + +

Part c): Lasso Regression on the Franke function with resampling

+ +

+This part is essentially a repeat of the previous two ones, but now with Lasso regression. Write either your own code or, in this case, you can also use the functionalities of scikit-learn. Give a critical discussion of the three methods and a judgement of which model fits the data best. + +

Part d): Introducing real data

+ +

+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 part e) what was done in parts a-c). 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/, + +

+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 scipy.misc 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. + +

Part e) OLS, Ridge and Lasso regression with resampling

+ +

+Our final part deals with the parameterization of your digital terrain data. We will apply all three methods for linear regression as in parts a-c), the same type (or higher order) of polynomial approximation and the same resampling techniques to evaluate which model fits the data best. + +

+At the end, you should pesent a critical evaluation of your results and discuss the applicability of these regression methods to the type of data presented here. + +

Background literature

+ +
    +
  1. For a discussion and derivation of the variances and mean squared errors using linear regression, see the Lecture notes on ridge regression by Wessel N. van Wieringen
  2. +
  3. The textbook of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer, chapters 3 and 7 are the most relevant ones for the analysis here.
  4. +
+ +

Introduction to numerical projects

+ +

+Here follows a brief recipe and recommendation on how to write a report for each +project. + +

+ +

Format for electronic delivery of report and programs

+ +

+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 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. + +

Software and needed installations

+ +

+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 + +

    +
  1. pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow
  2. +
+ +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 + +

    +
  1. brew install python3
  2. +
+ +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 + +
    +
  1. sudo apt-get install python3 (or python for python2.7)
  2. +
+ +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 + +

    +
  1. Anaconda 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 conda
  2. +
  3. Enthought canopy is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.
  4. +
+ +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. + +

+ +

+ +

+ + +
+ + + + + + + +
+ © 1999-2018, "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html". Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/Projects/2018/Project1/html/Project1-bs.html b/doc/Projects/2018/Project1/html/Project1-bs.html index 7084beb83..07aa51264 100644 --- a/doc/Projects/2018/Project1/html/Project1-bs.html +++ b/doc/Projects/2018/Project1/html/Project1-bs.html @@ -40,20 +40,57 @@ Automatically generated HTML file from DocOnce source + + + + + + + -

Regression analysis and classification

+

Regression analysis and resampling methods

-

Introduction

+

+The main aim of this project is to study in more detail various +regression methods, including the Ordinary Least Squares (OLS) method, +Ridge regression and finally Lasso regression. +The methods are in turn combined with resampling techniques. -

Part a): The data

+

+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 etsablished the model and the +method, we will employ resamling techniques such as the cross-validation and +the bootstrap methods, in order to perform a proper assessment of our models. -

Background literature

+

+The Franke function, which is a weighted sum of four exponentials reads as follows +$$ +f(x,y) = \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49} - \frac{(9y+1)}{10}\right)}+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }. +$$ -

Introduction to numerical projects

+

+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 cross-validation and +bootstrap as resampling techniques. As in homeworks 1 and 2, 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 fix values for \( x \) and \( y \) with a given step size. +In this case we will have two predictors and need to fit a +function (for example a polynomial) of \( x \) and \( y \). Thereafter we will +repeat much of the same procedure using the the Ridge and +Lasso regression methods, introducing thus a dependence on the bias +(penalty) \( \lambda \). + +

+Thereafter 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 fucntion 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.gca(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()
+
+ +

Part a): Ordinary Least Square on the Franke function with resampling

+ +

+We will thus again generate our own dataset for a function \( \mathrm{FrankeFunction}(x,y) \) where +\( x,y \in [0,1] \) could be defined by random numbers computed with the uniform +distribution. The function \( f(x,y) \) is the Franke function. You should explore also the addition +an added stochastic noise to this function using the normal distribution \( \cal{N}(0,1) \). + +

+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 \( \beta \) by computing their variances, evaluate the Mean Squared error (MSE) +$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +$$ + +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 +$$ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +$$ + +

+Perform a resampling of the data where you split the data in training data and test data. Implement the \( k \)-fold cross-validation algorithm and the bootstrap algorithm +and evaluate again the MSE and the \( R^2 \) functions resulting from the test data. + +

Part b): Ridge Regression on the Franke function with resampling

+ +

+Write your own code for the Ridge method, either using matrix inversion or the singular value decomposition as done in the previous exercise or howework 2 (see also chapter 3.4 of Hastie et al., equations (3.43) and (3.44)). Perform the same analysis as in the previous exercise (for the same polynomials and include resampling techniques) but now for different values of \( \lambda \). Compare and analyze your results with those obtained in part a). Study the dependence on \( \lambda \) while also varying eventually the strength of the noise in your expression for \( \mathrm{FrankeFunction}(x,y) \). + +

Part c): Lasso Regression on the Franke function with resampling

+ +

+This part is essentially a repeat of the previous two ones, but now with Lasso regression. Write either your own code or, in this case, you can also use the functionalities of scikit-learn. Give a critical discussion of the three methods and a judgement of which model fits the data best. + +

Part d): Introducing real data

+ +

+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 part e) what was done in parts a-c). 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/, + +

+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 scipy.misc 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. + +

Part e) OLS, Ridge and Lasso regression with resampling

+ +

+Our final part deals with the parameterization of your digital terrain data. We will apply all three methods for linear regression as in parts a-c), the same type (or higher order) of polynomial approximation and the same resampling techniques to evaluate which model fits the data best. + +

+At the end, you should pesent a critical evaluation of your results and discuss the applicability of these regression methods to the type of data presented here. + +

Background literature

+ +
    +
  1. For a discussion and derivation of the variances and mean squared errors using linear regression, see the Lecture notes on ridge regression by Wessel N. van Wieringen
  2. +
  3. The textbook of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer, chapters 3 and 7 are the most relevant ones for the analysis here.
  4. +
+ +

Introduction to numerical projects

Here follows a brief recipe and recommendation on how to write a report for each @@ -142,7 +357,7 @@ project.

  • 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.
  • -

    Format for electronic delivery of report and programs

    +

    Format for electronic delivery of report and programs

    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 or Python. The following prescription should be followed when preparing the report: @@ -156,10 +371,10 @@ The preferred format for the report is a PDF file. You can also use DOC or posts Finally, -we encourage you to work two and two together. Optimal working groups consist of +we encourage you to collaborate. Optimal working groups consist of 2-3 students. You can then hand in a common report. -

    Software and needed installations

    +

    Software and needed installations

    If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, @@ -212,6 +427,8 @@ 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. +

    +

    -

    Format for electronic delivery of report and programs

    +

    Format for electronic delivery of report and programs

    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 or Python. The following prescription should be followed when preparing the report: @@ -116,10 +328,10 @@ The preferred format for the report is a PDF file. You can also use DOC or posts Finally, -we encourage you to work two and two together. Optimal working groups consist of +we encourage you to collaborate. Optimal working groups consist of 2-3 students. You can then hand in a common report. -

    Software and needed installations

    +

    Software and needed installations

    If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, @@ -172,6 +384,8 @@ 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. +

    + diff --git a/doc/Projects/2018/Project1/ipynb/ipynb-Project1-src.tar.gz b/doc/Projects/2018/Project1/ipynb/ipynb-Project1-src.tar.gz index cb896340ec3eea0b1feed83a9d6ddfb63e34cf02..ee46cfcadbba39582d2f3ccd018b7707347b1423 100644 GIT binary patch literal 210 zcmb2|=3r=-oe|Bz{Pz68EG9#NV}S)yz6C)K-VH z+HN%$eEGlpr$m}WT3q^h{qD%nx>GOGrUWfp_jE?@%y~h!b6*RumA*LHGk1!ao9Uvq zq%5~ZdF7X`+KJC@3*NPs`(|tW^R|s=pY_{54}UU=|Iymdk3xwIJ!3J&xF*j z=TeN9}x`LE3Tpc?;g+To9@_cJpfga5}EnM*Tw JF=#L_004czU&H_a literal 210 zcmb2|=3p=k&WL7UetZ647L%dCvBL9qM{N#H+VQ$d?@DufQ-{95-5b4EkEjXW*Ru(+QFYJLzE5-Zjro`I*7{%jZ*{#t*Np$$ow$kB{{&B8Q=NHs_Z>EUyOiBr551om z=Df9feMr`7|0J~=C7E7Z?z@YxmAK!TDK$OzLz=nGkq?q*?L-fU?`LK}2LF#q>hEpd J$)Lf&002RiVG95N diff --git a/doc/Projects/2018/Project1/pdf/Project1.p.tex b/doc/Projects/2018/Project1/pdf/Project1.p.tex index dcd8e3f9b..06eb6c25b 100644 --- a/doc/Projects/2018/Project1/pdf/Project1.p.tex +++ b/doc/Projects/2018/Project1/pdf/Project1.p.tex @@ -45,6 +45,12 @@ final, % draft: marks overfull hboxes, figures with paths \usepackage[pdftex]{graphicx} +\usepackage{ptex2tex} +% #ifdef MINTED +\usepackage{minted} +\usemintedstyle{default} +% #endif + \usepackage[T1]{fontenc} %\usepackage[latin1]{inputenc} \usepackage{ucs} @@ -149,22 +155,180 @@ Project 1 on Machine Learning, deadline October 1 % --- begin date --- \begin{center} -May 2018 +Sep 11, 2018 \end{center} % --- end date --- \vspace{1cm} -\subsection{Regression analysis and classification} +\subsection{Regression analysis and resampling methods} + +The main aim of this project is to study in more detail various +regression methods, including the Ordinary Least Squares (OLS) method, +Ridge regression and finally Lasso regression. +The methods are in turn combined with resampling techniques. + +We will first study how +to fit polynomials to a specific two-dimensional function called +\href{{http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf}}{Franke's +function}. This +is a function which has been widely used when testing various interpolation and fitting +algorithms. Furthermore, after having etsablished the model and the +method, we will employ resamling techniques such as the cross-validation and +the bootstrap methods, in order to perform a proper assessment of our models. + + +The Franke function, which is a weighted sum of four exponentials reads as follows +\[ +f(x,y) = \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49} - \frac{(9y+1)}{10}\right)}+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }. +\] + +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 cross-validation and +bootstrap as resampling techniques. As in homeworks 1 and 2, 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 fix values for $x$ and $y$ with a given step size. +In this case we will have two predictors and need to fit a +function (for example a polynomial) of $x$ and $y$. Thereafter we will +repeat much of the same procedure using the the Ridge and +Lasso regression methods, introducing thus a dependence on the bias +(penalty) $\lambda$. + +Thereafter 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 fucntion for the Franke function is included here (it performs also a three-dimensional plot of it) +\bpycod +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.gca(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() + +\epycod + + +\paragraph{Part a): Ordinary Least Square on the Franke function with resampling.} +We will thus again generate our own dataset for a function $\mathrm{FrankeFunction}(x,y)$ where +$x,y \in [0,1]$ could be defined by random numbers computed with the uniform +distribution. The function $f(x,y)$ is the Franke function. You should explore also the addition +an added stochastic noise to this function using the normal distribution $\cal{N}(0,1)$. + +Write your own code (using either a matrix inversion or a singular value decomposition from e.g., \textbf{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 $\beta$ by computing their variances, evaluate the Mean Squared error (MSE) +\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\] +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 +\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\] + +Perform a resampling of the data where you split the data in training data and test data. Implement the $k$-fold cross-validation algorithm and the bootstrap algorithm +and evaluate again the MSE and the $R^2$ functions resulting from the test data. + + + +\paragraph{Part b): Ridge Regression on the Franke function with resampling.} +Write your own code for the Ridge method, either using matrix inversion or the singular value decomposition as done in the previous exercise or howework 2 (see also chapter 3.4 of Hastie \emph{et al.}, equations (3.43) and (3.44)). Perform the same analysis as in the previous exercise (for the same polynomials and include resampling techniques) but now for different values of $\lambda$. Compare and analyze your results with those obtained in part a). Study the dependence on $\lambda$ while also varying eventually the strength of the noise in your expression for $\mathrm{FrankeFunction}(x,y)$. + +\paragraph{Part c): Lasso Regression on the Franke function with resampling.} +This part is essentially a repeat of the previous two ones, but now with Lasso regression. Write either your own code or, in this case, you can also use the functionalities of \textbf{scikit-learn}. Give a critical discussion of the three methods and a judgement of which model fits the data best. + +\paragraph{Part d): Introducing real data.} +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 part e) what was done in parts a-c). 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 \href{{https://earthexplorer.usgs.gov/}}{\nolinkurl{https://earthexplorer.usgs.gov/}}, + +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 \textbf{SRTM Arc-Second Global} and download the data as a \textbf{GeoTIF} file. +The files are then stored in \emph{tif} format which can be imported into a Python program using +\bpycod +scipy.misc.imread +\epycod + +Here is a simple part of a Python code which reads and plots the data from such files +\bpycod +import numpy as np +from scipy.misc 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() +\epycod + +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. + + +\paragraph{Part e) OLS, Ridge and Lasso regression with resampling.} +Our final part deals with the parameterization of your digital terrain data. We will apply all three methods for linear regression as in parts a-c), the same type (or higher order) of polynomial approximation and the same resampling techniques to evaluate which model fits the data best. + +At the end, you should pesent a critical evaluation of your results and discuss the applicability of these regression methods to the type of data presented here. + + + -\paragraph{Introduction.} -\paragraph{Part a): The data.} \subsection{Background literature} +\begin{enumerate} +\item For a discussion and derivation of the variances and mean squared errors using linear regression, see the \href{{https://arxiv.org/abs/1509.09169}}{Lecture notes on ridge regression by Wessel N. van Wieringen} +\item The textbook of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}, chapters 3 and 7 are the most relevant ones for the analysis here. +\end{enumerate} - +\noindent \subsection{Introduction to numerical projects} Here follows a brief recipe and recommendation on how to write a report for each @@ -209,7 +373,7 @@ The preferred format for the report is a PDF file. You can also use DOC or posts \noindent Finally, -we encourage you to work two and two together. Optimal working groups consist of +we encourage you to collaborate. Optimal working groups consist of 2-3 students. You can then hand in a common report. @@ -269,6 +433,9 @@ encompass communities of developers in the thousands or more. 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