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starting to edit project 1

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2019-08-26 17:16:25 +02:00
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doc/pub/Regression/ipynb/Regression.ipynb
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doc/src/Projects/2019/Project1/Project1.do.txt
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TITLE: Project 1 on Machine Learning, deadline October 1
TITLE: Project 1 on Machine Learning, deadline September 30, 2019
AUTHOR: "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html" {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo, Norway
DATE: today
@@ -10,87 +10,6 @@ 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":"http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf". 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/or
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
!bt
\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*}
!et
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)
!bc pycod
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()
!ec
=== Part a): Ordinary Least Square on the Franke function with resampling ===
@@ -125,51 +44,17 @@ 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 ===
=== Part b): Ridge Regression 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 ===
=== Part c): Lasso Regression 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 URL:"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
!bc pycod
scipy.misc.imread
!ec
Here is a simple part of a Python code which reads and plots the data from such files
!bc pycod
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()
!ec
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.
doc/src/Projects/2019/Project1/make.sh
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@@ -48,7 +48,7 @@ mv -f $name.pdf ${name}.pdf
cp $name.tex ${name}.tex
# Publish
dest=../../../../Projects/2018
dest=../../../../Projects/2019
if [ ! -d $dest/$name ]; then
mkdir $dest/$name
mkdir $dest/$name/pdf