updating jupyter-book
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Morten Hjorth-Jensen
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{
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"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
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"doconce format html exercisesweek37.do.txt -->\n",
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"<!-- dom:TITLE: Exercises week 37 -->"
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"metadata": {
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"source": [
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"# Exercises week 37\n",
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"**September 9-13, 2024**\n",
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"\n",
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"Date: **Deadline is Friday September 13 at midnight**"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0c0df373",
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"metadata": {
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"editable": true
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"source": [
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"## Overarching aims of the exercises this week\n",
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"\n",
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"This exercise deals with various mean values and variances in linear\n",
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"regression method (here it may be useful to look up chapter 3,\n",
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"equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome\n",
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"H. Friedman, The Elements of Statistical Learning,\n",
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"Springer](https://www.springer.com/gp/book/9780387848570)). The\n",
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"exercise is also a part of project 1 and can be reused in the theory\n",
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"part of the project.\n",
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"\n",
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"For more discussions on Ridge regression and calculation of\n",
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"expectation values, [Wessel van\n",
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"Wieringen's](https://arxiv.org/abs/1509.09169) article is highly\n",
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"recommended.\n",
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"\n",
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"The assumption we have made is that there exists a continuous function\n",
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"$f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0,\n",
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"\\sigma^2)$ which describes our data"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a1ac8666",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ee6ce4be",
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"metadata": {
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"editable": true
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},
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"source": [
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"We then approximate this function $f(\\boldsymbol{x})$ with our model $\\boldsymbol{\\tilde{y}}$ from the solution of the linear regression equations (ordinary least squares OLS), that is our\n",
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"function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we minimized $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, with"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2d50b2e4",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3645cccd",
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"metadata": {
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"editable": true
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},
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"source": [
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"The matrix $\\boldsymbol{X}$ is the so-called design or feature matrix."
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]
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},
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{
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"cell_type": "markdown",
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"id": "6a378434",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Exercise 1: Expectation values for ordinary least squares expressions\n",
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"\n",
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"Show that the expectation value of $\\boldsymbol{y}$ for a given element $i$"
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]
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},
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{
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"metadata": {
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},
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"source": [
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"$$\n",
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"\\mathbb{E}(y_i) =\\sum_{j}x_{ij} \\beta_j=\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta},\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a6435e39",
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"source": [
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"and that\n",
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"its variance is"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\mbox{Var}(y_i) = \\sigma^2.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b5aef4d3",
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"metadata": {
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"editable": true
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},
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"source": [
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"Hence, $y_i \\sim N( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
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"mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$.\n",
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"\n",
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"With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ show that"
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]
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},
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{
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||||
"cell_type": "markdown",
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"metadata": {
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"editable": true
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||||
},
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"source": [
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"$$\n",
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"\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\boldsymbol{\\beta}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4da3a821",
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"metadata": {
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"editable": true
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},
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"source": [
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"Show finally that the variance of $\\boldsymbol{\\boldsymbol{\\beta}}$ is"
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]
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||||
},
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{
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||||
"cell_type": "markdown",
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||||
"id": "0ba77c7b",
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||||
"metadata": {
|
||||
"editable": true
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||||
},
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||||
"source": [
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||||
"$$\n",
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"\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) = \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}.\n",
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||||
"$$"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "5c66de05",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
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||||
"source": [
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||||
"We can use the last expression when we define a [so-called confidence interval](https://en.wikipedia.org/wiki/Confidence_interval) for the parameters $\\beta$. \n",
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||||
"A given parameter $\\beta_j$ is given by the diagonal matrix element of the above matrix."
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||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "43b92138",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
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"source": [
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||||
"## Exercise 2: Expectation values for Ridge regression\n",
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"\n",
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||||
"Show that"
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||||
]
|
||||
},
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||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "afea98ca",
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||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
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||||
"\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "13d7fd96",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"We see clearly that\n",
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||||
"$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\mathbb{E} \\big[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}\\big ]$ for any $\\lambda > 0$.\n",
|
||||
"\n",
|
||||
"Show also that the variance is"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "77d1d47e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T}\\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
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"id": "2fa6bdfd",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of the Ridge parameters $\\boldsymbol{\\beta}$ goes to zero."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
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"nbformat_minor": 5
|
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}
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<h1 class="site-logo" id="site-title">Applied Data Analysis and Machine Learning</h1>
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading">
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About the course
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Teaching schedule with links to material
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Teachers and Grading
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Textbooks
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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||||
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|
||||
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|
||||
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|
||||
<li class="toctree-l1">
|
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8. Support Vector Machines, overarching aims
|
||||
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|
||||
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|
||||
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|
||||
<p aria-level="2" class="caption" role="heading">
|
||||
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Decision Trees, Ensemble Methods and Boosting
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
|
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|
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|
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Dimensionality Reduction
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12. Clustering and Unsupervised Learning
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|
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Deep Learning Methods
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14. Building a Feed Forward Neural Network
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Week 36: Linear Regression and Statistical interpretations
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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<!-- dom:TITLE: Exercises week 37 --><div class="tex2jax_ignore mathjax_ignore section" id="exercises-week-37">
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<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Permalink to this headline">¶</a></h1>
|
||||
<p><strong>September 9-13, 2024</strong></p>
|
||||
<p>Date: <strong>Deadline is Friday September 13 at midnight</strong></p>
|
||||
<div class="section" id="overarching-aims-of-the-exercises-this-week">
|
||||
<h2>Overarching aims of the exercises this week<a class="headerlink" href="#overarching-aims-of-the-exercises-this-week" title="Permalink to this headline">¶</a></h2>
|
||||
<p>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 <a class="reference external" href="https://www.springer.com/gp/book/9780387848570">Trevor Hastie, Robert Tibshirani, Jerome
|
||||
H. Friedman, The Elements of Statistical Learning,
|
||||
Springer</a>). The
|
||||
exercise is also a part of project 1 and can be reused in the theory
|
||||
part of the project.</p>
|
||||
<p>For more discussions on Ridge regression and calculation of
|
||||
expectation values, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van
|
||||
Wieringen’s</a> article is highly
|
||||
recommended.</p>
|
||||
<p>The assumption we have made is that there exists a continuous function
|
||||
<span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim N(0,
|
||||
\sigma^2)\)</span> which describes our data</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
|
||||
\]</div>
|
||||
<p>We then approximate this function <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> with our model <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> from the solution of the linear regression equations (ordinary least squares OLS), that is our
|
||||
function <span class="math notranslate nohighlight">\(f\)</span> is approximated by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> where we minimized <span class="math notranslate nohighlight">\((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\)</span>, with</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
|
||||
\]</div>
|
||||
<p>The matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is the so-called design or feature matrix.</p>
|
||||
</div>
|
||||
<div class="section" id="exercise-1-expectation-values-for-ordinary-least-squares-expressions">
|
||||
<h2>Exercise 1: Expectation values for ordinary least squares expressions<a class="headerlink" href="#exercise-1-expectation-values-for-ordinary-least-squares-expressions" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Show that the expectation value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> for a given element <span class="math notranslate nohighlight">\(i\)</span></p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathbb{E}(y_i) =\sum_{j}x_{ij} \beta_j=\mathbf{X}_{i, \ast} \, \boldsymbol{\beta},
|
||||
\]</div>
|
||||
<p>and that
|
||||
its variance is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mbox{Var}(y_i) = \sigma^2.
|
||||
\]</div>
|
||||
<p>Hence, <span class="math notranslate nohighlight">\(y_i \sim N( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> follows a normal distribution with
|
||||
mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
|
||||
<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> show that</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathbb{E}(\boldsymbol{\hat{\beta}}) = \boldsymbol{\beta}.
|
||||
\]</div>
|
||||
<p>Show finally that the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\boldsymbol{\beta}}\)</span> is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mbox{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}.
|
||||
\]</div>
|
||||
<p>We can use the last expression when we define a <a class="reference external" href="https://en.wikipedia.org/wiki/Confidence_interval">so-called confidence interval</a> for the parameters <span class="math notranslate nohighlight">\(\beta\)</span>.
|
||||
A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span> is given by the diagonal matrix element of the above matrix.</p>
|
||||
</div>
|
||||
<div class="section" id="exercise-2-expectation-values-for-ridge-regression">
|
||||
<h2>Exercise 2: Expectation values for Ridge regression<a class="headerlink" href="#exercise-2-expectation-values-for-ridge-regression" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Show that</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
|
||||
\]</div>
|
||||
<p>We see clearly that
|
||||
<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\big ]\)</span> for any <span class="math notranslate nohighlight">\(\lambda > 0\)</span>.</p>
|
||||
<p>Show also that the variance is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
|
||||
\]</div>
|
||||
<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of the Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
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|
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<a class="reference internal" href="exercisesweek37.html">
|
||||
Exercises week 37
|
||||
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|
||||
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|
||||
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|
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|
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Week 37: Statistical interpretations and Resampling Methods
|
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||||
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Week 36: Linear Regression and Statistical interpretations
|
||||
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|
||||
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|
||||
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|
||||
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||||
Exercises week 37
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
Week 37: Statistical interpretations and Resampling Methods
|
||||
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||||
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||||
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Week 36: Linear Regression and Statistical interpretations
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||||
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||||
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|
||||
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|
||||
<a class="reference internal" href="exercisesweek37.html">
|
||||
Exercises week 37
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
Week 37: Statistical interpretations and Resampling Methods
|
||||
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||||
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||||
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "2a3463de",
|
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"editable": true
|
||||
},
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"source": [
|
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"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
|
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"doconce format html exercisesweek37.do.txt -->\n",
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"<!-- dom:TITLE: Exercises week 37 -->"
|
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]
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|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 37\n",
|
||||
"**September 9-13, 2024**\n",
|
||||
"\n",
|
||||
"Date: **Deadline is Friday September 13 at midnight**"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0c0df373",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Overarching aims of the exercises this week\n",
|
||||
"\n",
|
||||
"This exercise deals with various mean values and variances in linear\n",
|
||||
"regression method (here it may be useful to look up chapter 3,\n",
|
||||
"equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome\n",
|
||||
"H. Friedman, The Elements of Statistical Learning,\n",
|
||||
"Springer](https://www.springer.com/gp/book/9780387848570)). The\n",
|
||||
"exercise is also a part of project 1 and can be reused in the theory\n",
|
||||
"part of the project.\n",
|
||||
"\n",
|
||||
"For more discussions on Ridge regression and calculation of\n",
|
||||
"expectation values, [Wessel van\n",
|
||||
"Wieringen's](https://arxiv.org/abs/1509.09169) article is highly\n",
|
||||
"recommended.\n",
|
||||
"\n",
|
||||
"The assumption we have made is that there exists a continuous function\n",
|
||||
"$f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0,\n",
|
||||
"\\sigma^2)$ which describes our data"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a1ac8666",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ee6ce4be",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"We then approximate this function $f(\\boldsymbol{x})$ with our model $\\boldsymbol{\\tilde{y}}$ from the solution of the linear regression equations (ordinary least squares OLS), that is our\n",
|
||||
"function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we minimized $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, with"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "2d50b2e4",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "3645cccd",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"The matrix $\\boldsymbol{X}$ is the so-called design or feature matrix."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "6a378434",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 1: Expectation values for ordinary least squares expressions\n",
|
||||
"\n",
|
||||
"Show that the expectation value of $\\boldsymbol{y}$ for a given element $i$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ce9b87a9",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E}(y_i) =\\sum_{j}x_{ij} \\beta_j=\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a6435e39",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"and that\n",
|
||||
"its variance is"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "75d89e64",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mbox{Var}(y_i) = \\sigma^2.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b5aef4d3",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Hence, $y_i \\sim N( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
|
||||
"mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$.\n",
|
||||
"\n",
|
||||
"With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ show that"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "17012dce",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\boldsymbol{\\beta}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4da3a821",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Show finally that the variance of $\\boldsymbol{\\boldsymbol{\\beta}}$ is"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0ba77c7b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) = \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "5c66de05",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"We can use the last expression when we define a [so-called confidence interval](https://en.wikipedia.org/wiki/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": "43b92138",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 2: Expectation values for Ridge regression\n",
|
||||
"\n",
|
||||
"Show that"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "afea98ca",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "13d7fd96",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"We see clearly that\n",
|
||||
"$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\mathbb{E} \\big[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}\\big ]$ for any $\\lambda > 0$.\n",
|
||||
"\n",
|
||||
"Show also that the variance is"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "77d1d47e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T}\\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "2fa6bdfd",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of the Ridge parameters $\\boldsymbol{\\beta}$ goes to zero."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
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||||
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|
||||
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|
||||
- file: exercisesweek37.ipynb
|
||||
- file: week37.ipynb
|
||||
- caption: Projects
|
||||
numbered: false
|
||||
chapters:
|
||||
|
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Block a user