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"plt.legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercises and Projects\n",
"\n",
"\n",
"\n",
"The main aim of this project is to study in more detail various\n",
"regression methods, including the Ordinary Least Squares (OLS) method,\n",
"The total score is **100** points. Each subtask has its own final score.\n",
"\n",
"\n",
"We will first study how to fit polynomials to a specific\n",
"two-dimensional function called [Franke's\n",
"function](http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf). This\n",
"is a function which has been widely used when testing various\n",
"interpolation and fitting algorithms. Furthermore, after having\n",
"established the model and the method, we will employ resamling\n",
"techniques such as cross-validation and/or bootstrap in order to perform a\n",
"proper assessment of our models. We will also study in detail the\n",
"so-called Bias-Variance trade off.\n",
"\n",
"\n",
"The Franke function, which is a weighted sum of four exponentials reads as follows"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"f(x,y) &= \\frac{3}{4}\\exp{\\left(-\\frac{(9x-2)^2}{4} - \\frac{(9y-2)^2}{4}\\right)}+\\frac{3}{4}\\exp{\\left(-\\frac{(9x+1)^2}{49}- \\frac{(9y+1)}{10}\\right)} \\\\\n",
"&+\\frac{1}{2}\\exp{\\left(-\\frac{(9x-7)^2}{4} - \\frac{(9y-3)^2}{4}\\right)} -\\frac{1}{5}\\exp{\\left(-(9x-4)^2 - (9y-7)^2\\right) }.\n",
"\\end{align*}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function will be defined for $x,y\\in [0,1]$. Our first step will\n",
"be to perform an OLS regression analysis of this function, trying out\n",
"a polynomial fit with an $x$ and $y$ dependence of the form $[x, y,\n",
"x^2, y^2, xy, \\dots]$. We will also include bootstrap first as\n",
"a resampling technique. After that we will include the cross-validation technique. As in homeworks 1 and 2, we can use a uniform\n",
"distribution to set up the arrays of values for $x$ and $y$, or as in\n",
"the example below just a set of fixed \n",
"values for $x$ and $y$ with a given step\n",
"size. We will fit a\n",
"function (for example a polynomial) of $x$ and $y$. Thereafter we\n",
"will repeat much of the same procedure using the Ridge and Lasso\n",
"regression methods, introducing thus a dependence on the bias\n",
"(penalty) $\\lambda$.\n",
"\n",
"Finally we are going to use (real) digital terrain data and try to\n",
"reproduce these data using the same methods. We will also try to go\n",
"beyond the second-order polynomials metioned above and explore \n",
"which polynomial fits the data best.\n",
"\n",
"\n",
"The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false,
"editable": true
},
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"source": [
"from mpl_toolkits.mplot3d import Axes3D\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib import cm\n",
"from matplotlib.ticker import LinearLocator, FormatStrFormatter\n",
"import numpy as np\n",
"from random import random, seed\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"\n",
"# Make data.\n",
"x = np.arange(0, 1, 0.05)\n",
"y = np.arange(0, 1, 0.05)\n",
"x, y = np.meshgrid(x,y)\n",
"\n",
"\n",
"def FrankeFunction(x,y):\n",
" term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n",
" term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n",
" term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n",
" term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n",
" return term1 + term2 + term3 + term4\n",
"\n",
"\n",
"z = FrankeFunction(x, y)\n",
"\n",
"# Plot the surface.\n",
"surf = ax.plot_surface(x, y, z, cmap=cm.coolwarm,\n",
" linewidth=0, antialiased=False)\n",
"\n",
"# Customize the z axis.\n",
"ax.set_zlim(-0.10, 1.40)\n",
"ax.zaxis.set_major_locator(LinearLocator(10))\n",
"ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))\n",
"\n",
"# Add a color bar which maps values to colors.\n",
"fig.colorbar(surf, shrink=0.5, aspect=5)\n",
"\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise: Ordinary Least Square (OLS) on the Franke function\n",
"\n",
"We will generate our own dataset for a function\n",
"$\\mathrm{FrankeFunction}(x,y)$ with $x,y \\in [0,1]$. The function\n",
"$f(x,y)$ is the Franke function. You should explore also the addition\n",
"of an added stochastic noise to this function using the normal\n",
"distribution $N(0,1)$.\n",
"\n",
"*Write your own code* (using either a matrix inversion or a singular\n",
"value decomposition from e.g., **numpy** ) or use your code from\n",
"homeworks 1 and 2 and perform a standard least square regression\n",
"analysis using polynomials in $x$ and $y$ up to fifth order. Find the\n",
"[confidence intervals](https://en.wikipedia.org/wiki/Confidence_interval) of the parameters (estimators) $\\beta$ by computing their\n",
"variances, evaluate the Mean Squared error (MSE)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and the $R^2$ score function. If $\\tilde{\\hat{y}}_i$ is the predicted\n",
"value of the $i-th$ sample and $y_i$ is the corresponding true value,\n",
"then the score $R^2$ is defined as"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where we have defined the mean value of $\\hat{y}$ as"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Your code has to include a scaling of the data (for example by\n",
"subtracting the mean value), and\n",
"a split of the data in training and test data. For this exercise you can\n",
"either write your own code or use for example the function for\n",
"splitting training data provided by the library **Scikit-Learn** (make\n",
"sure you have installed it). This function is called\n",
"$train\\_test\\_split$. **You should present a critical discussion of why and how you have scaled or not scaled the data**.\n",
"\n",
"It is normal in essentially all Machine Learning studies to split the\n",
"data in a training set and a test set (eventually also an additional\n",
"validation set). There\n",
"is no explicit recipe for how much data should be included as training\n",
"data and say test data. An accepted rule of thumb is to use\n",
"approximately $2/3$ to $4/5$ of the data as training data.\n",
"\n",
"\n",
"You can easily reuse the solutions to your exercises from week 35 and week 36.\n",
"\n",
"\n",
"\n",
"### Exercise: Bias-variance trade-off and resampling techniques\n",
"\n",
"Our aim here is to study the bias-variance trade-off by implementing the **bootstrap** resampling technique.\n",
"\n",
"With a code which does OLS and includes resampling techniques, \n",
"we will now discuss the bias-variance trade-off in the context of\n",
"continuous predictions such as regression. However, many of the\n",
"intuitions and ideas discussed here also carry over to classification\n",
"tasks and basically all Machine Learning algorithms. \n",
"\n",
"Before you perform an analysis of the bias-variance trade-off on your test data, make\n",
"first a figure similar to Fig. 2.11 of Hastie, Tibshirani, and\n",
"Friedman. Figure 2.11 of this reference displays only the test and training MSEs. The test MSE can be used to \n",
"indicate possible regions of low/high bias and variance. You will most likely not get an\n",
"equally smooth curve!\n",
"\n",
"With this result we move on to the bias-variance trade-off analysis.\n",
"\n",
"Consider a\n",
"dataset $\\mathcal{L}$ consisting of the data\n",
"$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$.\n",
"\n",
"Let us assume that the true data is generated from a noisy model"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here $\\epsilon$ is normally distributed with mean zero and standard\n",
"deviation $\\sigma^2$.\n",
"\n",
"In our derivation of the ordinary least squares method we defined then\n",
"an approximation to the function $f$ in terms of the parameters\n",
"$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n",
"that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$.\n",
"\n",
"The parameters $\\boldsymbol{\\beta}$ are in turn found by optimizing the means\n",
"squared error via the so-called cost function"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here the expected value $\\mathbb{E}$ is the sample value. \n",
"\n",
"Show that you can rewrite this as"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Explain what the terms mean, which one is the bias and which one is\n",
"the variance and discuss their interpretations.\n",
"\n",
"Perform then a bias-variance analysis of the Franke function by\n",
"studying the MSE value as function of the complexity of your model.\n",
"\n",
"Discuss the bias and variance trade-off as function\n",
"of your model complexity (the degree of the polynomial) and the number\n",
"of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n",
"\n",
"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 $y_i$.\n",
"\n",
"\n",
"### Exercise: Cross-validation as resampling techniques, adding more complexity\n",
"\n",
"The aim here is to write your own code for another widely popular\n",
"resampling technique, the so-called cross-validation method. Again,\n",
"before you start with cross-validation approach, you should scale your\n",
"data.\n",
"\n",
"Implement the $k$-fold cross-validation algorithm (write your own\n",
"code) and evaluate again the MSE function resulting\n",
"from the test folds. You can compare your own code with that from\n",
"**Scikit-Learn** if needed. \n",
"\n",
"Compare the MSE you get from your cross-validation code with the one\n",
"you got from your **bootstrap** code. Comment your results. Try $5-10$\n",
"folds. You can also compare your own cross-validation code with the\n",
"one provided by **Scikit-Learn**.\n",
"\n",
"\n",
"### Exercise: Ridge Regression on the Franke function with resampling\n",
"\n",
"Write your own code for the Ridge method, either using matrix\n",
"inversion or the singular value decomposition as done in the previous\n",
"exercise. Perform the same bootstrap analysis as in the\n",
"Exercise 2 (for the same polynomials) and the cross-validation in exercise 3 but now for different values of $\\lambda$. Compare and\n",
"analyze your results with those obtained in exercises 1-3. Study the\n",
"dependence on $\\lambda$.\n",
"\n",
"Study also the bias-variance trade-off as function of various values of\n",
"the parameter $\\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results. \n",
"\n",
"### Exercise: Lasso Regression on the Franke function with resampling\n",
"\n",
"This exercise is essentially a repeat of the previous two ones, but now\n",
"with Lasso regression. Write either your own code (difficult and optional) or, in this case,\n",
"you can also use the functionalities of **Scikit-Learn** (recommended). \n",
"Give a\n",
"critical discussion of the three methods and a judgement of which\n",
"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. \n",
"\n",
"### Exercise: Analysis of real data\n",
"\n",
"With our codes functioning and having been tested properly on a\n",
"simpler function we are now ready to look at real data. We will\n",
"essentially repeat in this exercise what was done in exercises 1-5. However, we\n",
"need first to download the data and prepare properly the inputs to our\n",
"codes. We are going to download digital terrain data from the website\n",
"<https://earthexplorer.usgs.gov/>,\n",
"\n",
"Or, if you prefer, we have placed selected datafiles at <https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects/2021/Project1/DataFiles>\n",
"\n",
"In order to obtain data for a specific region, you need to register as\n",
"a user (free) at this website and then decide upon which area you want\n",
"to fetch the digital terrain data from. In order to be able to read\n",
"the data properly, you need to specify that the format should be **SRTM\n",
"Arc-Second Global** and download the data as a **GeoTIF** file. The\n",
"files are then stored in *tif* format which can be imported into a\n",
"Python program using"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"scipy.misc.imread"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is a simple part of a Python code which reads and plots the data\n",
"from such files"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false,
"editable": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"from imageio import imread\n",
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"from matplotlib import cm\n",
"\n",
"# Load the terrain\n",
"terrain1 = imread('SRTM_data_Norway_1.tif')\n",
"# Show the terrain\n",
"plt.figure()\n",
"plt.title('Terrain over Norway 1')\n",
"plt.imshow(terrain1, cmap='gray')\n",
"plt.xlabel('X')\n",
"plt.ylabel('Y')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If you should have problems in downloading the digital terrain data,\n",
"we provide two examples under the data folder of project 1. One is\n",
"from a region close to Stavanger in Norway and the other Møsvatn\n",
"Austfjell, again in Norway.\n",
"Feel free to produce your own terrain data.\n",
"\n",
"\n",
"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](https://www.kaggle.com/datasets) for examples.\n",
"\n",
"\n",
"Our final part deals with the parameterization of your digital terrain\n",
"data (or your own data). We will apply all three methods for linear regression, the same type (or higher order) of polynomial\n",
"approximation and cross-validation as resampling technique to evaluate which\n",
"model fits the data best.\n",
"\n",
"At the end, you should present a critical evaluation of your results\n",
"and discuss the applicability of these regression methods to the type\n",
"of data presented here (either the terrain data we propose or other data sets)."
]
}
],
"metadata": {},