typo in exercise 2

doc/LectureNotes/_build/html/_sources/exercisesweek37.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "1ff4a4e9",
+   "id": "8b21a389",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0569f7a9",
+   "id": "10a3b32a",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6f34e507",
+   "id": "940daa26",
    "metadata": {
     "editable": true
    },
@@ -54,7 +54,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ff574a4",
+   "id": "d51b0da4",
    "metadata": {
     "editable": true
    },
@@ -66,7 +66,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "096a3165",
+   "id": "40a14559",
    "metadata": {
     "editable": true
    },
@@ -77,7 +77,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc9fb258",
+   "id": "e4afb2ea",
    "metadata": {
     "editable": true
    },
@@ -89,7 +89,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "daf4eb5d",
+   "id": "b0153681",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b491679",
+   "id": "17b98331",
    "metadata": {
     "editable": true
    },
@@ -111,7 +111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "93a49739",
+   "id": "0e35faa1",
    "metadata": {
     "editable": true
    },
@@ -123,7 +123,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3fa80db",
+   "id": "c1c7837e",
    "metadata": {
     "editable": true
    },
@@ -134,7 +134,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e6e2defd",
+   "id": "0ab57c17",
    "metadata": {
     "editable": true
    },
@@ -146,7 +146,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa19afa",
+   "id": "345dbeca",
    "metadata": {
     "editable": true
    },
@@ -159,7 +159,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5fa82a47",
+   "id": "fea2ddcf",
    "metadata": {
     "editable": true
    },
@@ -171,7 +171,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "33099abb",
+   "id": "683a72cf",
    "metadata": {
     "editable": true
    },
@@ -181,7 +181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d03f001",
+   "id": "0dfd010a",
    "metadata": {
     "editable": true
    },
@@ -193,7 +193,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94bd151d",
+   "id": "d4e4aa80",
    "metadata": {
     "editable": true
    },
@@ -204,7 +204,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0b84941",
+   "id": "acb128a7",
    "metadata": {
     "editable": true
    },
@@ -216,33 +216,32 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60370a63",
+   "id": "83af53d9",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
-    "\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af251217",
+   "id": "1bbd0f50",
    "metadata": {
     "editable": true
    },
    "source": [
     "We see clearly that\n",
-    "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\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": "02e978da",
+   "id": "617deac1",
    "metadata": {
     "editable": true
    },
@@ -254,12 +253,12 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3656a896",
+   "id": "bee378b0",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero."
+    "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."
    ]
   }
  ],

doc/LectureNotes/_build/html/exercisesweek37.html

@@ -497,17 +497,16 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
 <p>Show that</p>
 <div class="math notranslate nohighlight">
 \[
-\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
-\boldsymbol{\beta}^{\mathrm{OLS}}.
+\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
 \]</div>
 <p>We see clearly that
-<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>.</p>
+<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">\(\lambda &gt; 0\)</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 Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
+<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>
 </div>
 </div>
 

doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "1ff4a4e9",
+   "id": "8b21a389",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0569f7a9",
+   "id": "10a3b32a",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6f34e507",
+   "id": "940daa26",
    "metadata": {
     "editable": true
    },
@@ -54,7 +54,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ff574a4",
+   "id": "d51b0da4",
    "metadata": {
     "editable": true
    },
@@ -66,7 +66,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "096a3165",
+   "id": "40a14559",
    "metadata": {
     "editable": true
    },
@@ -77,7 +77,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc9fb258",
+   "id": "e4afb2ea",
    "metadata": {
     "editable": true
    },
@@ -89,7 +89,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "daf4eb5d",
+   "id": "b0153681",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b491679",
+   "id": "17b98331",
    "metadata": {
     "editable": true
    },
@@ -111,7 +111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "93a49739",
+   "id": "0e35faa1",
    "metadata": {
     "editable": true
    },
@@ -123,7 +123,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3fa80db",
+   "id": "c1c7837e",
    "metadata": {
     "editable": true
    },
@@ -134,7 +134,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e6e2defd",
+   "id": "0ab57c17",
    "metadata": {
     "editable": true
    },
@@ -146,7 +146,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa19afa",
+   "id": "345dbeca",
    "metadata": {
     "editable": true
    },
@@ -159,7 +159,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5fa82a47",
+   "id": "fea2ddcf",
    "metadata": {
     "editable": true
    },
@@ -171,7 +171,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "33099abb",
+   "id": "683a72cf",
    "metadata": {
     "editable": true
    },
@@ -181,7 +181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d03f001",
+   "id": "0dfd010a",
    "metadata": {
     "editable": true
    },
@@ -193,7 +193,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94bd151d",
+   "id": "d4e4aa80",
    "metadata": {
     "editable": true
    },
@@ -204,7 +204,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0b84941",
+   "id": "acb128a7",
    "metadata": {
     "editable": true
    },
@@ -216,33 +216,32 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60370a63",
+   "id": "83af53d9",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
-    "\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af251217",
+   "id": "1bbd0f50",
    "metadata": {
     "editable": true
    },
    "source": [
     "We see clearly that\n",
-    "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\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": "02e978da",
+   "id": "617deac1",
    "metadata": {
     "editable": true
    },
@@ -254,12 +253,12 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3656a896",
+   "id": "bee378b0",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero."
+    "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."
    ]
   }
  ],

doc/LectureNotes/exercisesweek37.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "1ff4a4e9",
+   "id": "8b21a389",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0569f7a9",
+   "id": "10a3b32a",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6f34e507",
+   "id": "940daa26",
    "metadata": {
     "editable": true
    },
@@ -54,7 +54,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ff574a4",
+   "id": "d51b0da4",
    "metadata": {
     "editable": true
    },
@@ -66,7 +66,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "096a3165",
+   "id": "40a14559",
    "metadata": {
     "editable": true
    },
@@ -77,7 +77,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc9fb258",
+   "id": "e4afb2ea",
    "metadata": {
     "editable": true
    },
@@ -89,7 +89,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "daf4eb5d",
+   "id": "b0153681",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b491679",
+   "id": "17b98331",
    "metadata": {
     "editable": true
    },
@@ -111,7 +111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "93a49739",
+   "id": "0e35faa1",
    "metadata": {
     "editable": true
    },
@@ -123,7 +123,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3fa80db",
+   "id": "c1c7837e",
    "metadata": {
     "editable": true
    },
@@ -134,7 +134,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e6e2defd",
+   "id": "0ab57c17",
    "metadata": {
     "editable": true
    },
@@ -146,7 +146,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa19afa",
+   "id": "345dbeca",
    "metadata": {
     "editable": true
    },
@@ -159,7 +159,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5fa82a47",
+   "id": "fea2ddcf",
    "metadata": {
     "editable": true
    },
@@ -171,7 +171,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "33099abb",
+   "id": "683a72cf",
    "metadata": {
     "editable": true
    },
@@ -181,7 +181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d03f001",
+   "id": "0dfd010a",
    "metadata": {
     "editable": true
    },
@@ -193,7 +193,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94bd151d",
+   "id": "d4e4aa80",
    "metadata": {
     "editable": true
    },
@@ -204,7 +204,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0b84941",
+   "id": "acb128a7",
    "metadata": {
     "editable": true
    },
@@ -216,33 +216,32 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60370a63",
+   "id": "83af53d9",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
-    "\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af251217",
+   "id": "1bbd0f50",
    "metadata": {
     "editable": true
    },
    "source": [
     "We see clearly that\n",
-    "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\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": "02e978da",
+   "id": "617deac1",
    "metadata": {
     "editable": true
    },
@@ -254,12 +253,12 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3656a896",
+   "id": "bee378b0",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero."
+    "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."
    ]
   }
  ],

doc/src/week37/exercisesweek37.do.txt

@@ -80,12 +80,11 @@ A given parameter $\beta_j$ is given by the diagonal matrix element of the above
 Show that
 !bt
 \[
-\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
-\bm{\beta}}.
+\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\bm{\beta}.
 \]
 !et
 We see clearly that
-$\mathbb{E} \big[ \bm{\beta}^{\mathrm{Ridge}} \big] \not= \bm{\beta}^{\mathrm{OLS}}$ for any $\lambda > 0$.
+$\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\bm{\beta}}^{\mathrm{OLS}}\big ]$ for any $\lambda > 0$.
 
 
 Show also that the variance is
@@ -95,7 +94,7 @@ Show also that the variance is
 \mbox{Var}[\hat{\bm{\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},
 \]
 !et
-and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\bm{\beta}$ goes to zero.
+and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of the Ridge parameters $\bm{\beta}$ goes to zero.