From 1f9e4412cb7943e5a2aca21d217ffd776df56f48 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 14 Sep 2021 22:56:12 +0200 Subject: [PATCH] updated book --- doc/BookChapters/chapter3.do.txt | 323 ++++++++++ .../_build/.doctrees/chapter3.doctree | Bin 337013 -> 442382 bytes .../_build/.doctrees/environment.pickle | Bin 172629 -> 177132 bytes .../_build/.doctrees/schedule.doctree | Bin 45950 -> 47398 bytes .../_build/html/_images/chapter3_38_4.png | Bin 0 -> 16322 bytes .../_build/html/_images/chapter3_41_10.png | Bin 0 -> 16300 bytes .../_build/html/_images/chapter3_54_0.png | Bin 0 -> 32458 bytes .../_build/html/_sources/chapter3.ipynb | 427 +++++++++++++ .../_build/html/_sources/schedule.md | 9 +- doc/LectureNotes/_build/html/chapter3.html | 417 +++++++++++-- doc/LectureNotes/_build/html/schedule.html | 7 +- doc/LectureNotes/_build/html/searchindex.js | 2 +- .../_build/jupyter_execute/chapter3.ipynb | 573 ++++++++++++++++-- .../_build/jupyter_execute/chapter3.py | 309 +++++++++- .../_build/jupyter_execute/chapter3_38_4.png | Bin 0 -> 16322 bytes .../_build/jupyter_execute/chapter3_41_10.png | Bin 0 -> 16300 bytes .../_build/jupyter_execute/chapter3_54_0.png | Bin 0 -> 32458 bytes doc/LectureNotes/chapter3.ipynb | 427 +++++++++++++ 18 files changed, 2373 insertions(+), 121 deletions(-) create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_38_4.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_41_10.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_54_0.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_38_4.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_41_10.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_54_0.png diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index 567f2e082..e29143ddf 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -1014,3 +1014,326 @@ plt.show() !ec +===== Exercises and Projects ===== + + + +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 +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 +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 +!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 bootstrap first as +a resampling technique. After that we will include the cross-validation 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 code 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 + + +=== Exercise: Ordinary Least Square (OLS) on the Franke 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 +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":"https://en.wikipedia.org/wiki/Confidence_interval" of the parameters (estimators) $\beta$ by computing their +variances, evaluate the Mean Squared error (MSE) + +!bt +\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\] +!et + +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 + +!bt +\[ +R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +\] +!et + +where we have defined the mean value of $\hat{y}$ as + +!bt +\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\] +!et + +Your code has to include a scaling 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 +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. + + + +=== Exercise: Bias-variance trade-off and resampling techniques === + +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, +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. + +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. + +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 + +!bt +\[ +\bm{y}=f(\boldsymbol{x}) + \bm{\epsilon}. +\] +!et + +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 +$\bm{\beta}$ and the design matrix $\bm{X}$ which embody our model, +that is $\bm{\tilde{y}}=\bm{X}\bm{\beta}$. + +The parameters $\bm{\beta}$ are in turn found by optimizing the means +squared error via the so-called cost function + +!bt +\[ +C(\bm{X},\bm{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]. +\] +!et +Here the expected value $\mathbb{E}$ is the sample value. + +Show that you can rewrite this as +!bt +\[ +\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\sigma^2. +\] +!et + +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 +studying the MSE value as function of the complexity of your model. + +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 $y_i$. + + +=== Exercise: Cross-validation as resampling techniques, adding more complexity === + + +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. + +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. + +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_. + + +=== Exercise: 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. 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. + +=== Exercise: Lasso Regression on the Franke function with resampling === + +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. + +=== Exercise: Analysis of 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 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 +URL:"https://earthexplorer.usgs.gov/", + +Or, if you prefer, we have placed selected datafiles at URL:"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 +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. +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":"https://www.kaggle.com/datasets" for examples. + + +Our final part deals with the parameterization of your digital terrain +data (or your own data). 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--git a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb index 95758bd41..b7775f687 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb @@ -1282,6 +1282,433 @@ "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 + }, + "outputs": [], + "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", + ",\n", + "\n", + "Or, if you prefer, we have placed selected datafiles at \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": {}, diff --git a/doc/LectureNotes/_build/html/_sources/schedule.md b/doc/LectureNotes/_build/html/_sources/schedule.md index 9a54ff154..03fa6f0c1 100644 --- a/doc/LectureNotes/_build/html/_sources/schedule.md +++ b/doc/LectureNotes/_build/html/_sources/schedule.md @@ -52,8 +52,13 @@ For the reading assignments we use the following abbreviations: - Lab Wednesday: - Lecture Thursday: Resampling methods, cross-validation and Bootstrap - Lecture Friday: More on Resampling methods and summary of linear regression -- Reading recommendations: See lecture notes for week 37 at https://compphysics.github.io/MachineLearning/doc/web/course.html. - - Chapter +- Reading recommendations: +- Recommended Reading: + - Lectures on Resampling methods for week 37 at https://compphysics.github.io/MachineLearning/doc/web/course.html. + - Bishop 1.3 (cross-validation) and 3.2 (bias-variance tradeoff) + - Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). This chapter is better than Bishop's on these topics. Goodfellow et al discuss some of these topics in sections 5.2-5.5. + + ### Week 38 September 20-24 - Lab Wednesday: - Lecture Thursday: Classification problems and Logistic Regression, from binary cases to several categories diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 2db258533..7352494cb 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -321,6 +321,43 @@ 5.5. Cross-validation +
  • + + 5.6. Exercises and Projects + + +
    • @@ -592,10 +629,10 @@ number \(i\) is left out. Usin
    -
    Runtime: 0.1375 sec
+
    Runtime: 0.137546 sec
 Jackknife Statistics :
 original           bias      std. error
- 100.029        100.019        0.150581
+ 99.9031        99.8931        0.149233
    @@ -841,14 +878,14 @@ Error: 0.32149601703519126 Bias^2: 0.3123314713548606 Var: 0.009164545680330616 0.32149601703519126 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912 -
    -
    -
    Polynomial degree: 1
+Polynomial degree: 1
 Error: 0.08426840630693411
 Bias^2: 0.07968918676726028
 Var: 0.004579219539673833
 0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411
-Polynomial degree: 2
+
    +
    +
    Polynomial degree: 2
 Error: 0.10398646080125035
 Bias^2: 0.10077114273548986
 Var: 0.0032153180657605086
@@ -868,21 +905,20 @@ Error: 0.05227921801205707
 Bias^2: 0.048187277304303125
 Var: 0.004091940707753964
 0.05227921801205707 >= 0.048187277304303125 + 0.004091940707753964 = 0.05227921801205709
-
    -
    -
    Polynomial degree: 6
+Polynomial degree: 6
 Error: 0.03781367141738898
 Bias^2: 0.03365768507152761
 Var: 0.004155986345861379
 0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899
-Polynomial degree: 7
+Polynomial degree:
+
    +
    +
     7
 Error: 0.027609773491022498
 Bias^2: 0.02299949826036597
 Var: 0.004610275230656537
 0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505
-
    -
    -
    Polynomial degree: 8
+Polynomial degree: 8
 Error: 0.017355848195591973
 Bias^2: 0.010331721306655588
 Var: 0.007024126888936384
@@ -892,7 +928,10 @@ Error: 0.026605727637189085
 Bias^2: 0.010018312644140933
 Var: 0.016587414993048166
 0.026605727637189085 >= 0.010018312644140933 + 0.016587414993048166 = 0.0266057276371891
-Polynomial degree: 10
+Polynomial degree:
+
    +
    +
     10
 Error: 0.021592704588043153
 Bias^2: 0.010516485576652981
 Var: 0.011076219011390184
@@ -902,23 +941,19 @@ Error: 0.07160048164228314
 Bias^2: 0.01443680008897583
 Var: 0.0571636815533073
 0.07160048164228314 >= 0.01443680008897583 + 0.0571636815533073 = 0.07160048164228312
-
    -
    -
    Polynomial degree: 12
+Polynomial degree: 12
 Error: 0.1154777721897675
 Bias^2: 0.01628578269590588
 Var: 0.09919198949386163
 0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751
-
    -
    -
    Polynomial degree: 13
+Polynomial degree: 13
 Error: 0.22842468702166951
 Bias^2: 0.01975416527163567
 Var: 0.20867052175003387
 0.22842468702166951 >= 0.01975416527163567 + 0.20867052175003387 = 0.22842468702166954
    -_images/chapter3_38_6.png +_images/chapter3_38_4.png

    The bias-variance tradeoff summarizes the fundamental tension in @@ -1139,11 +1174,11 @@ Mean squared error on test data: 123711.53703498 Degree of polynomial: 3 Mean squared error on training data: 9011.85263220 Mean squared error on test data: 10913.84780262 -Degree of polynomial: 4 -Mean squared error on training data: 303.47610036 -

    Mean squared error on test data: 426.30787294
+
    Degree of polynomial:   4
+Mean squared error on training data: 303.47610036
+Mean squared error on test data: 426.30787294
 Degree of polynomial:   5
 Mean squared error on training data: 3.80354994
 Mean squared error on test data: 5.98822371
@@ -1175,64 +1210,62 @@ Mean squared error on test data: 0.17446471
 Degree of polynomial:  13
 Mean squared error on training data: 0.00759119
 Mean squared error on test data: 1.08131003
-
    
-
    
-
    Degree of polynomial:  14
+Degree of polynomial:  14
 Mean squared error on training data: 0.00472199
 Mean squared error on test data: 0.81333793
-Degree of polynomial:  15
+
    
+
    
+
    Degree of polynomial:  15
 Mean squared error on training data: 0.00410478
 Mean squared error on test data: 92.09145189
 Degree of polynomial:  16
 Mean squared error on training data: 0.00315593
 Mean squared error on test data: 234.39716546
-
    
-
    
-
    Degree of polynomial:  17
+Degree of polynomial:  17
 Mean squared error on training data: 0.00242998
 Mean squared error on test data: 1271.05295709
-Degree of polynomial:  18
+
    
+
    
+
    Degree of polynomial:  18
 Mean squared error on training data: 0.00228740
 Mean squared error on test data: 108.42208194
 Degree of polynomial:  19
 Mean squared error on training data: 0.00156372
 Mean squared error on test data: 1388.41078073
-
    
-
    
-
    Degree of polynomial:  20
+Degree of polynomial:  20
 Mean squared error on training data: 0.00137982
 Mean squared error on test data: 1761.43341615
-Degree of polynomial:  21
+
    
+
    
+
    Degree of polynomial:  21
 Mean squared error on training data: 0.00118170
 Mean squared error on test data: 15061.31603087
 Degree of polynomial:  22
 Mean squared error on training data: 0.00092354
 Mean squared error on test data: 890.63488525
-
    
-
    
-
    Degree of polynomial:  23
+Degree of polynomial:  23
 Mean squared error on training data: 0.00085887
 Mean squared error on test data: 5483.16796929
-Degree of polynomial:  24
+
    
+
    
+
    Degree of polynomial:  24
 Mean squared error on training data: 0.00084589
 Mean squared error on test data: 1695.57143061
 Degree of polynomial:  25
 Mean squared error on training data: 0.00078806
 Mean squared error on test data: 131343.30655001
-
    
-
    
-
    Degree of polynomial:  26
+Degree of polynomial:  26
 Mean squared error on training data: 0.00076916
 Mean squared error on test data: 17709.14370264
-Degree of polynomial:  27
+
    
+
    
+
    Degree of polynomial:  27
 Mean squared error on training data: 0.00068970
 Mean squared error on test data: 2975.38903780
 Degree of polynomial:  28
 Mean squared error on training data: 0.00062588
 Mean squared error on test data: 3848.64522721
-
    
-
    
-
    Degree of polynomial:  29
+Degree of polynomial:  29
 Mean squared error on training data: 0.00060728
 Mean squared error on test data: 2988.64001211
 
    
@@ -1243,7 +1276,7 @@ Mean squared error on test data: 2988.64001211
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
    -_images/chapter3_41_11.png +_images/chapter3_41_10.png @@ -1525,6 +1558,294 @@ cross-validation (LOOCV).

    +
    +

    5.6. Exercises and Projects

    +

    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 +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

    +
    +\[\begin{split} +\begin{align*} +f(x,y) &= \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49}- \frac{(9y+1)}{10}\right)} \\ +&+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }. +\end{align*} +\end{split}\]
    +

    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 +a resampling technique. After that we will include the cross-validation 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 code 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()
+
    +
    +
    +
    +_images/chapter3_54_0.png +
    +
    +
    +

    5.6.1. Exercise: Ordinary Least Square (OLS) on the Franke 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 +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)

    +
    +\[ +MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\]
    +

    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

    +
    +\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\]
    +

    Your code has to include a scaling 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 +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.

    +
    +
    +

    5.6.2. Exercise: Bias-variance trade-off and resampling techniques

    +

    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, +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.

    +

    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.

    +

    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

    +
    +\[ +C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. +\]
    +

    Here the expected value \(\mathbb{E}\) is the sample value.

    +

    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.

    +

    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 +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 \(y_i\).

    +
    +
    +

    5.6.3. Exercise: Cross-validation as resampling techniques, adding more complexity

    +

    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.

    +

    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.

    +

    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.

    +
    +
    +

    5.6.4. Exercise: 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. 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.

    +
    +
    +

    5.6.5. Exercise: Lasso Regression on the Franke function with resampling

    +

    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.

    +
    +
    +

    5.6.6. Exercise: Analysis of 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 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

    +

    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
+
    +
    +
    +
    +
    ---------------------------------------------------------------------------
+NameError                                 Traceback (most recent call last)
+<ipython-input-10-d985fb40c43d> in <module>
+----> 1 scipy.misc.imread
+
+NameError: name 'scipy' is not defined
+
    +
    +
    +
    +

    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.

    +

    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 +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 +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).

    +
    +