v
This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer).
+This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). The exercise is also part of the weekly exercises for week 37.
The assumption we have made is that there exists a continuous function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim N(0, \sigma^2) \) @@ -430,7 +428,7 @@ $$ \mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. $$ -
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). . +
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). A given parameter \( \beta_j \) is given by the diagonal matrix element of the above matrix.
The answer to this exercise should be included in the theory part of the report. +
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations.
diff --git a/doc/Projects/2023/Project1/html/Project1-bs.html b/doc/Projects/2023/Project1/html/Project1-bs.html index 3a8cec6ec..96ecf6e87 100644 --- a/doc/Projects/2023/Project1/html/Project1-bs.html +++ b/doc/Projects/2023/Project1/html/Project1-bs.html @@ -381,11 +381,9 @@ Give a critical discussion of the three methods and a judgement of which model fits the data best. - -v
This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer).
+This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). The exercise is also part of the weekly exercises for week 37.
The assumption we have made is that there exists a continuous function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim N(0, \sigma^2) \) @@ -430,7 +428,7 @@ $$ \mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. $$ -
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). . +
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). A given parameter \( \beta_j \) is given by the diagonal matrix element of the above matrix.
The answer to this exercise should be included in the theory part of the report. +
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations.
diff --git a/doc/Projects/2023/Project1/html/Project1.html b/doc/Projects/2023/Project1/html/Project1.html index ef7099584..5a28b77ea 100644 --- a/doc/Projects/2023/Project1/html/Project1.html +++ b/doc/Projects/2023/Project1/html/Project1.html @@ -417,11 +417,9 @@ Give a critical discussion of the three methods and a judgement of which model fits the data best. - -v
This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer).
+This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). The exercise is also part of the weekly exercises for week 37.
The assumption we have made is that there exists a continuous function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim N(0, \sigma^2) \) @@ -466,7 +464,7 @@ $$ \mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. $$ -
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). . +
We can use the last expression when we define a so-called confidence interval for the parameters \( \beta \). A given parameter \( \beta_j \) is given by the diagonal matrix element of the above matrix.
The answer to this exercise should be included in the theory part of the report. +
The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations.
diff --git a/doc/Projects/2023/Project1/ipynb/Project1.ipynb b/doc/Projects/2023/Project1/ipynb/Project1.ipynb index c867ffa98..c7f114b4a 100644 --- a/doc/Projects/2023/Project1/ipynb/Project1.ipynb +++ b/doc/Projects/2023/Project1/ipynb/Project1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "1ceb1efb", + "id": "24b1316d", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "1f73d83b", + "id": "cf021550", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "df2b7d33", + "id": "30acee10", "metadata": { "editable": true }, @@ -63,7 +63,7 @@ }, { "cell_type": "markdown", - "id": "83787bae", + "id": "589ea9f7", "metadata": { "editable": true }, @@ -85,7 +85,7 @@ }, { "cell_type": "markdown", - "id": "d7ee2bfd", + "id": "6280c2c1", "metadata": { "editable": true }, @@ -100,7 +100,7 @@ }, { "cell_type": "markdown", - "id": "958a455e", + "id": "8ab92f24", "metadata": { "editable": true }, @@ -133,7 +133,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "3deb0a35", + "id": "80b126e0", "metadata": { "collapsed": false, "editable": true @@ -185,7 +185,7 @@ }, { "cell_type": "markdown", - "id": "742ae4b5", + "id": "bbd0b625", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "e49ae3a8", + "id": "0754efce", "metadata": { "editable": true }, @@ -220,7 +220,7 @@ }, { "cell_type": "markdown", - "id": "091e8f0d", + "id": "b87c82f6", "metadata": { "editable": true }, @@ -232,7 +232,7 @@ }, { "cell_type": "markdown", - "id": "3fad86e6", + "id": "8832d308", "metadata": { "editable": true }, @@ -244,7 +244,7 @@ }, { "cell_type": "markdown", - "id": "d281bfb0", + "id": "55129a1a", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "af6fb771", + "id": "44243865", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "0d89eb4e", + "id": "ff13dcbf", "metadata": { "editable": true }, @@ -295,7 +295,7 @@ }, { "cell_type": "markdown", - "id": "e1c30216", + "id": "c2078519", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "b121745d", + "id": "f718aec9", "metadata": { "editable": true }, @@ -325,21 +325,19 @@ "you can also use the functionalities of **Scikit-Learn** (recommended). Keep in mind that the library **Scikit-Learn** excludes the intercept by default. \n", "Give a\n", "critical discussion of the three methods and a judgement of which\n", - "model fits the data best.\n", - "\n", - "v" + "model fits the data best." ] }, { "cell_type": "markdown", - "id": "56d96043", + "id": "d8b10104", "metadata": { "editable": true }, "source": [ "### Part d): Paper and pencil part\n", "\n", - "This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570)).\n", + "This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570)). The exercise is also part of the weekly exercises for week 37.\n", "\n", "The assumption we have made is \n", "that there exists a continuous function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0, \\sigma^2)$\n", @@ -348,7 +346,7 @@ }, { "cell_type": "markdown", - "id": "e64c004b", + "id": "c5267aae", "metadata": { "editable": true }, @@ -360,7 +358,7 @@ }, { "cell_type": "markdown", - "id": "6f17d2da", + "id": "fd9640e3", "metadata": { "editable": true }, @@ -371,7 +369,7 @@ }, { "cell_type": "markdown", - "id": "fc4fe2fb", + "id": "630acebb", "metadata": { "editable": true }, @@ -383,7 +381,7 @@ }, { "cell_type": "markdown", - "id": "93babb22", + "id": "254ecf9c", "metadata": { "editable": true }, @@ -395,7 +393,7 @@ }, { "cell_type": "markdown", - "id": "4b4061ba", + "id": "6b4ac35f", "metadata": { "editable": true }, @@ -407,7 +405,7 @@ }, { "cell_type": "markdown", - "id": "1ca5e57a", + "id": "404fb6a5", "metadata": { "editable": true }, @@ -418,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "da25c4ba", + "id": "8d8f00ea", "metadata": { "editable": true }, @@ -430,7 +428,7 @@ }, { "cell_type": "markdown", - "id": "10ce6cfd", + "id": "60f92a7f", "metadata": { "editable": true }, @@ -443,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "8d3f716a", + "id": "6eab356b", "metadata": { "editable": true }, @@ -455,7 +453,7 @@ }, { "cell_type": "markdown", - "id": "e30af070", + "id": "d41efd26", "metadata": { "editable": true }, @@ -465,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "1bc3edc7", + "id": "f2ffda3b", "metadata": { "editable": true }, @@ -477,18 +475,18 @@ }, { "cell_type": "markdown", - "id": "050e1f04", + "id": "78ec374f", "metadata": { "editable": true }, "source": [ - "We can use the last expression when we define a so-called confidence interval for the parameters $\\beta$. .\n", + "We can use the last expression when we define a so-called confidence interval for the parameters $\\beta$. \n", "A given parameter $\\beta_j$ is given by the diagonal matrix element of the above matrix." ] }, { "cell_type": "markdown", - "id": "52486468", + "id": "74bec52c", "metadata": { "editable": true }, @@ -521,7 +519,7 @@ }, { "cell_type": "markdown", - "id": "008e92a6", + "id": "afdd7a0b", "metadata": { "editable": true }, @@ -533,7 +531,7 @@ }, { "cell_type": "markdown", - "id": "123f65c5", + "id": "87974f8e", "metadata": { "editable": true }, @@ -552,7 +550,7 @@ }, { "cell_type": "markdown", - "id": "a4398be8", + "id": "335ac425", "metadata": { "editable": true }, @@ -564,7 +562,7 @@ }, { "cell_type": "markdown", - "id": "3e5bee01", + "id": "16b4fea1", "metadata": { "editable": true }, @@ -578,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "e35fdf55", + "id": "c0f81bfe", "metadata": { "editable": true }, @@ -590,7 +588,7 @@ }, { "cell_type": "markdown", - "id": "ac0994d8", + "id": "ac7c8a94", "metadata": { "editable": true }, @@ -600,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "65ba61ff", + "id": "4cbbc7f8", "metadata": { "editable": true }, @@ -612,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "dd4e4536", + "id": "f52e099f", "metadata": { "editable": true }, @@ -622,7 +620,7 @@ }, { "cell_type": "markdown", - "id": "7730a37d", + "id": "0838f22d", "metadata": { "editable": true }, @@ -634,12 +632,12 @@ }, { "cell_type": "markdown", - "id": "8cefeb53", + "id": "9e5f062d", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report.\n", + "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of the Franke function by\n", @@ -653,7 +651,7 @@ }, { "cell_type": "markdown", - "id": "8ec208c4", + "id": "76ec4768", "metadata": { "editable": true }, @@ -678,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "b142cd65", + "id": "c7b05bee", "metadata": { "editable": true }, @@ -706,7 +704,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "fe878605", + "id": "a562d2f5", "metadata": { "collapsed": false, "editable": true @@ -718,7 +716,7 @@ }, { "cell_type": "markdown", - "id": "10b84fbb", + "id": "9e5a57d7", "metadata": { "editable": true }, @@ -730,7 +728,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "e845fd10", + "id": "b97b3537", "metadata": { "collapsed": false, "editable": true @@ -756,7 +754,7 @@ }, { "cell_type": "markdown", - "id": "0745b3d5", + "id": "db97fd61", "metadata": { "editable": true }, @@ -781,7 +779,7 @@ }, { "cell_type": "markdown", - "id": "2ce7a62f", + "id": "a8a6b74a", "metadata": { "editable": true }, @@ -795,7 +793,7 @@ }, { "cell_type": "markdown", - "id": "34b72e9a", + "id": "4fc8cfed", "metadata": { "editable": true }, @@ -825,7 +823,7 @@ }, { "cell_type": "markdown", - "id": "d0f3a4fc", + "id": "eff5bb90", "metadata": { "editable": true }, @@ -847,7 +845,7 @@ }, { "cell_type": "markdown", - "id": "436bcd57", + "id": "ca7c2f59", "metadata": { "editable": true }, diff --git a/doc/Projects/2023/Project1/ipynb/ipynb-Project1-src.tar.gz b/doc/Projects/2023/Project1/ipynb/ipynb-Project1-src.tar.gz index ca520b06a..8eef8c769 100644 Binary files a/doc/Projects/2023/Project1/ipynb/ipynb-Project1-src.tar.gz and b/doc/Projects/2023/Project1/ipynb/ipynb-Project1-src.tar.gz differ diff --git a/doc/Projects/2023/Project1/pdf/Project1.p.tex b/doc/Projects/2023/Project1/pdf/Project1.p.tex index da544d658..06ea38047 100644 --- a/doc/Projects/2023/Project1/pdf/Project1.p.tex +++ b/doc/Projects/2023/Project1/pdf/Project1.p.tex @@ -362,10 +362,8 @@ Give a critical discussion of the three methods and a judgement of which model fits the data best. -v - \paragraph{Part d): Paper and pencil part.} -This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). +This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). The exercise is also part of the weekly exercises for week 37. The assumption we have made is that there exists a continuous function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim N(0, \sigma^2)$ @@ -402,7 +400,7 @@ Show finally that the variance of $\bm{\beta}$ is \mbox{Var}(\bm{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. \] -We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. . +We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. A given parameter $\beta_j$ is given by the diagonal matrix element of the above matrix. \paragraph{Part e): Bias-variance trade-off and resampling techniques.} @@ -463,7 +461,7 @@ and \[ \mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2. \] -The answer to this exercise should be included in the theory part of the report. +The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations. Perform then a bias-variance analysis of the Franke function by diff --git a/doc/Projects/2023/Project1/pdf/Project1.pdf b/doc/Projects/2023/Project1/pdf/Project1.pdf index c0705095b..3cbf8be5e 100644 Binary files a/doc/Projects/2023/Project1/pdf/Project1.pdf and b/doc/Projects/2023/Project1/pdf/Project1.pdf differ diff --git a/doc/Projects/2023/Project1/pdf/Project1.tex b/doc/Projects/2023/Project1/pdf/Project1.tex index b495b5aeb..11c692a26 100644 --- a/doc/Projects/2023/Project1/pdf/Project1.tex +++ b/doc/Projects/2023/Project1/pdf/Project1.tex @@ -332,10 +332,8 @@ Give a critical discussion of the three methods and a judgement of which model fits the data best. -v - \paragraph{Part d): Paper and pencil part.} -This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). +This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). The exercise is also part of the weekly exercises for week 37. The assumption we have made is that there exists a continuous function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim N(0, \sigma^2)$ @@ -372,7 +370,7 @@ Show finally that the variance of $\bm{\beta}$ is \mbox{Var}(\bm{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. \] -We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. . +We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. A given parameter $\beta_j$ is given by the diagonal matrix element of the above matrix. \paragraph{Part e): Bias-variance trade-off and resampling techniques.} @@ -433,7 +431,7 @@ and \[ \mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2. \] -The answer to this exercise should be included in the theory part of the report. +The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations. Perform then a bias-variance analysis of the Franke function by diff --git a/doc/src/Projects/2023/Project1/Project1.do.txt b/doc/src/Projects/2023/Project1/Project1.do.txt index cc9a9b3af..80add4cbe 100644 --- a/doc/src/Projects/2023/Project1/Project1.do.txt +++ b/doc/src/Projects/2023/Project1/Project1.do.txt @@ -207,12 +207,12 @@ Give a critical discussion of the three methods and a judgement of which model fits the data best. -v + === Part d): Paper and pencil part === -This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of "Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer":"https://www.springer.com/gp/book/9780387848570"). +This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of "Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer":"https://www.springer.com/gp/book/9780387848570"). The exercise is also part of the weekly exercises for week 37. The assumption we have made is that there exists a continuous function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim N(0, \sigma^2)$ @@ -263,7 +263,7 @@ Show finally that the variance of $\bm{\beta}$ is !et -We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. . +We can use the last expression when we define a so-called confidence interval for the parameters $\beta$. A given parameter $\beta_j$ is given by the diagonal matrix element of the above matrix. @@ -339,7 +339,7 @@ and \mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2. \] !et -The answer to this exercise should be included in the theory part of the report. +The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. Explain what the terms mean and discuss their interpretations. Perform then a bias-variance analysis of the Franke function by