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 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 \( x,y\in [0,1] \). 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 (or bootstrap) as resampling technique. 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 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 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()
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 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. Here you can write your own function or use the function for splitting training data provided by Scikit-Learn. This function is called \( train\_test\_split \).
It is normal in essentially all Machine Learning studies to split the data in a training set and a test set (sometimes 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.
Implement the \( k \)-fold cross-validation algorithm (write your own code) and evaluate again the MSE function resulting from the test data. You can compare your own code with that from Scikit-Learn if needed. You can alternatively write your own bootstrap code.
With a code which does OLS and includes resampling techniques, we will now discuss the bias-variance tradeoff 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.
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 $$ \boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}. $$
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 means squared error via the so-called cost function $$ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. $$
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 the variance and discuss their interpretations.
Discuss the bias and variance tradeoff 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.
Try to make a figure similar to Fig. 2.11 of Hastie, Tibshirani, and Friedman, see the references below. You should include an analysis of the bias and variance for the test results. Figure 2.11 displays only the test and training MSEs while indicating regions of low/high bias and variance. You will most likely not get an equally smooth curve! 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 \( y_i \).
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 exercises (for the same polynomials and include resampling techniques) but now for different values of \( \lambda \). Compare and analyze your results with those obtained in parts a-c). Study the dependence on \( \lambda \).
Study also the bias-variance tradeoff as function of various values of the parameter \( \lambda \). Comment your results.
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 (recommended). Give a critical discussion of the three methods and a judgement of which model fits the data best.
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 g) what was done in parts a-e). 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 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.
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.
Here follows a brief recipe and recommendation on how to write a report for each project.
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:
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
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
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