From 7fb0038ca121abb787346644b7d3828f5c7d072e Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Mon, 2 Sep 2024 16:32:42 +0200
Subject: [PATCH] added link to video

---
 doc/pub/week36/html/._week36-bs001.html      |    6 +-
 doc/pub/week36/html/week36-reveal.html       |    8 +-
 doc/pub/week36/html/week36-solarized.html    |    6 +-
 doc/pub/week36/html/week36.html              |    6 +-
 doc/pub/week36/ipynb/ipynb-week36-src.tar.gz |  Bin 192 -> 191 bytes
 doc/pub/week36/ipynb/week36.ipynb            | 2115 ++++++++++--------
 doc/src/week36/week36.do.txt                 |    4 +-
 7 files changed, 1220 insertions(+), 925 deletions(-)

diff --git a/doc/pub/week36/html/._week36-bs001.html b/doc/pub/week36/html/._week36-bs001.html
index c92892f0c..12ef2a484 100644
--- a/doc/pub/week36/html/._week36-bs001.html
+++ b/doc/pub/week36/html/._week36-bs001.html
@@ -271,9 +271,9 @@ MathJax.Hub.Config({
   <li> Technicalities related to scaling and other issues with data handling</li>
   <li> Linear Regression, ordinary least squares, Ridge and Lasso  and links with Statistics</li>
   <li> Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see <a href="https://www.deeplearningbook.org/" target="_self"><tt>https://www.deeplearningbook.org/</tt></a></li>
-  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.
-<!-- * <a href="https://youtu.be/Kc20CFK0z7Y" target="_self">Video of lecture</a> -->
-<!-- * <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf" target="_self">Whiteboard notes</a> --></li>
+  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.</li>
+  <li> <a href="https://youtu.be/oHjqjUB36KE" target="_self">Video of lecture</a></li>  
+  <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf" target="_self">Whiteboard notes</a></li>
 </ul>
 <li> Material for the active learning sessions on Tuesday and Wednesday</li>
 <ul>
diff --git a/doc/pub/week36/html/week36-reveal.html b/doc/pub/week36/html/week36-reveal.html
index 1a806e011..5b120b111 100644
--- a/doc/pub/week36/html/week36-reveal.html
+++ b/doc/pub/week36/html/week36-reveal.html
@@ -207,9 +207,11 @@ MathJax.Hub.Config({
 
 <p><li> Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see <a href="https://www.deeplearningbook.org/" target="_blank"><tt>https://www.deeplearningbook.org/</tt></a></li>
 
-<p><li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.
-<!-- * <a href="https://youtu.be/Kc20CFK0z7Y" target="_blank">Video of lecture</a> -->
-<!-- * <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf" target="_blank">Whiteboard notes</a> --></li>
+<p><li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.</li>
+
+<p><li> <a href="https://youtu.be/oHjqjUB36KE" target="_blank">Video of lecture</a></li>
+
+<p><li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf" target="_blank">Whiteboard notes</a></li>
 </ul>
 <p>
 <p><li> Material for the active learning sessions on Tuesday and Wednesday</li>
diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html
index 69b16a4d3..61248b081 100644
--- a/doc/pub/week36/html/week36-solarized.html
+++ b/doc/pub/week36/html/week36-solarized.html
@@ -246,9 +246,9 @@ MathJax.Hub.Config({
   <li> Technicalities related to scaling and other issues with data handling</li>
   <li> Linear Regression, ordinary least squares, Ridge and Lasso  and links with Statistics</li>
   <li> Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see <a href="https://www.deeplearningbook.org/" target="_blank"><tt>https://www.deeplearningbook.org/</tt></a></li>
-  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.
-<!-- * <a href="https://youtu.be/Kc20CFK0z7Y" target="_blank">Video of lecture</a> -->
-<!-- * <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf" target="_blank">Whiteboard notes</a> --></li>
+  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.</li>
+  <li> <a href="https://youtu.be/oHjqjUB36KE" target="_blank">Video of lecture</a></li>  
+  <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf" target="_blank">Whiteboard notes</a></li>
 </ul>
 <li> Material for the active learning sessions on Tuesday and Wednesday</li>
 <ul>
diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html
index d5b40cacc..7a8cc45e0 100644
--- a/doc/pub/week36/html/week36.html
+++ b/doc/pub/week36/html/week36.html
@@ -323,9 +323,9 @@ MathJax.Hub.Config({
   <li> Technicalities related to scaling and other issues with data handling</li>
   <li> Linear Regression, ordinary least squares, Ridge and Lasso  and links with Statistics</li>
   <li> Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see <a href="https://www.deeplearningbook.org/" target="_blank"><tt>https://www.deeplearningbook.org/</tt></a></li>
-  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.
-<!-- * <a href="https://youtu.be/Kc20CFK0z7Y" target="_blank">Video of lecture</a> -->
-<!-- * <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf" target="_blank">Whiteboard notes</a> --></li>
+  <li> Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.</li>
+  <li> <a href="https://youtu.be/oHjqjUB36KE" target="_blank">Video of lecture</a></li>  
+  <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf" target="_blank">Whiteboard notes</a></li>
 </ul>
 <li> Material for the active learning sessions on Tuesday and Wednesday</li>
 <ul>
diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz
index 1e75c1d223128d2e7226e62da3a712242dfe157a..76339cb8f845efff5b060c8190c47ce2aa68fe7d 100644
GIT binary patch
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z`v5&DUWN$$UH*iGknFS7dYdKw?xWcdLIh(l<Wlf4$zrNUlyaa<2%a)QmjVb)IpYzK
z`A&LiotL&hr7q8?9qM;;{n+vNVbAmmJo8T+O66d)>wIZ6XyY+Arh*${m5M|)oz9^!
t=!O<pytc|BNZo-f3gwm2@)9@JoUE)}694*{AP9osYYz*{*<t_)000M@UWEVv

literal 192
zcmV;x06+g9iwFRASJh?!1MSbv3c@f92k@Qu6nTQtZEZaZ?%+WX@dY}TxjNU*wnO*!
z?gR9sco`z}cli?%LUPE~n_U*Uy9*XW2;r2%m@CevB*|P);+R29IcEt=8RtOalu;G{
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uqa9md@Y+d(AXE>cD5R5GiA&fTeKMl4QTXd;JkRsIuRQ<;uf+%e2mk=0DqGY5

diff --git a/doc/pub/week36/ipynb/week36.ipynb b/doc/pub/week36/ipynb/week36.ipynb
index 95789382c..9d7c9b89a 100644
--- a/doc/pub/week36/ipynb/week36.ipynb
+++ b/doc/pub/week36/ipynb/week36.ipynb
@@ -2,8 +2,10 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "66ea8ed4",
-   "metadata": {},
+   "id": "03b68af2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
     "doconce format html week36.do.txt --no_mako -->\n",
@@ -12,8 +14,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "79ce282c",
-   "metadata": {},
+   "id": "91ddfc70",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "# Week 36: Linear Rgeression and Statistical interpretations\n",
     "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
@@ -23,8 +27,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ed85c38c",
-   "metadata": {},
+   "id": "c25f7c42",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Plans for week 36\n",
     "\n",
@@ -37,8 +43,10 @@
     "  * Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see <https://www.deeplearningbook.org/>\n",
     "\n",
     "  * Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.\n",
-    "<!-- * [Video of lecture](https://youtu.be/Kc20CFK0z7Y) -->\n",
-    "<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf) -->\n",
+    "\n",
+    "  * [Video of lecture](https://youtu.be/oHjqjUB36KE)  \n",
+    "\n",
+    "  * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf)\n",
     "\n",
     "* Material for the active learning sessions on Tuesday and Wednesday\n",
     "\n",
@@ -51,16 +59,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "faf98f14",
-   "metadata": {},
+   "id": "f5230b8b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Material for lecture Monday September 2"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "203d6afa",
-   "metadata": {},
+   "id": "fbd9ed4d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Important technicalities: More on Rescaling data\n",
     "\n",
@@ -114,8 +126,11 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "e3f345b9",
-   "metadata": {},
+   "id": "a5ec261b",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "\"\"\"\n",
@@ -138,8 +153,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d382904b",
-   "metadata": {},
+   "id": "5cc0694a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us try to understand what this may imply mathematically when we\n",
     "subtract the mean values, also known as *zero centering*. For\n",
@@ -150,8 +167,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a2d4217",
-   "metadata": {},
+   "id": "e3b8bc0b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
@@ -160,8 +179,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1963de4b",
-   "metadata": {},
+   "id": "fd6de781",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Recall also that we use the squared value. This expression can lead to an\n",
     "increased penalty for higher differences between predicted and\n",
@@ -175,8 +196,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "09a545ff",
-   "metadata": {},
+   "id": "bfd7a040",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
@@ -185,16 +208,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6384894b",
-   "metadata": {},
+   "id": "41e1b93b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "for all $j$. For $\\beta_0$ we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b72ca042",
-   "metadata": {},
+   "id": "24749eb9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
@@ -203,16 +230,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b193f08c",
-   "metadata": {},
+   "id": "b922308a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Multiplying away the constant $2/n$, we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d05df72f",
-   "metadata": {},
+   "id": "70be741b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
@@ -221,8 +252,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "708ab439",
-   "metadata": {},
+   "id": "b295f0ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
     "Our result for $\\beta_0$ simplifies then to"
@@ -230,8 +263,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bfc8322f",
-   "metadata": {},
+   "id": "bb1e5813",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
@@ -240,16 +275,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea8a6fa5",
-   "metadata": {},
+   "id": "578817a0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We obtain then"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "db3ef9e5",
-   "metadata": {},
+   "id": "78bb7489",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
@@ -258,16 +297,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0fcdc688",
-   "metadata": {},
+   "id": "45cdf117",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we define"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3e7f32ff",
-   "metadata": {},
+   "id": "d671bea1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n",
@@ -276,16 +319,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5803cedf",
-   "metadata": {},
+   "id": "f1e037f7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the mean value of the outputs as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "884ca2e2",
-   "metadata": {},
+   "id": "5592094a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
@@ -294,16 +341,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "633c4189",
-   "metadata": {},
+   "id": "3784e759",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1c60b743",
-   "metadata": {},
+   "id": "238e4a7c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
@@ -312,16 +363,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb3025b6",
-   "metadata": {},
+   "id": "6d9e90e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "01d272fc",
-   "metadata": {},
+   "id": "9ecc0ad6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
@@ -330,16 +385,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "794acece",
-   "metadata": {},
+   "id": "5bb35680",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can rewrite the latter equation as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "997ab0ab",
-   "metadata": {},
+   "id": "748c35c7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
@@ -348,16 +407,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "952c8b2c",
-   "metadata": {},
+   "id": "06776f59",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "08835cf9",
-   "metadata": {},
+   "id": "2d6d58ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
@@ -366,8 +429,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "416bf355",
-   "metadata": {},
+   "id": "cc9ff189",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
     "\n",
@@ -376,8 +441,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c37bccef",
-   "metadata": {},
+   "id": "cc5996b6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
@@ -386,16 +453,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "461d332f",
-   "metadata": {},
+   "id": "beac8256",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "572cd133",
-   "metadata": {},
+   "id": "70ffda09",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
@@ -404,8 +475,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d73a2b5d",
-   "metadata": {},
+   "id": "8d6e0895",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
     "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
@@ -415,8 +488,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3da49213",
-   "metadata": {},
+   "id": "b314aa68",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
@@ -425,8 +500,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e099555",
-   "metadata": {},
+   "id": "9c4a4032",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
     "\n",
@@ -436,42 +513,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
-   "id": "fc066ea4",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "True beta: [2, 0.5, 3.7]\n",
-      "Fitted beta: [2.08376632 0.19569961 3.97898392]\n",
-      "Sklearn fitted beta: [2.08376632 0.19569961 3.97898392]\n",
-      "MSE with intercept column\n",
-      "0.00411363461744314\n",
-      "MSE with intercept column from SKL\n",
-      "0.004113634617443147\n",
-      "Manual intercept: 2.083766322923899\n",
-      "Fitted beta (wiothout intercept): [0.19569961 3.97898392]\n",
-      "Sklearn intercept: 2.0837663229239043\n",
-      "Sklearn fitted beta (without intercept): [0.19569961 3.97898392]\n",
-      "MSE with Manual intercept\n",
-      "0.00411363461744314\n",
-      "MSE with Sklearn intercept\n",
-      "0.004113634617443131\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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fVEhJYSGEEEKY0J9zR5Owbid3MqyJaDgNH7+ifV+pojUVVQghhChAIs4cxHPZVtySrdjtGUDPtn0tHZLJSY+FEEIIYQI6rZazY4fglgy3PaD1rPWWDskspLAwE5VKxYYNG3J83d/fnzlz5pgtnmfxxRdf8M477xieK4rCkCFDcHd3R6VScerUKZo3b87IkSOf+T2WLl1KiRIlnj9Y8UQDBgzg1VdfzZe2YmJi8PT05ObNm/nSnhBFxW//6UjlSB0ZalCPGoa7t5+lQzILKSzyQUxMDO+88w7lypXDzs4OHx8f2rZtyz///GPp0PLNnTt3+Pbbb/nkk08M20JDQ1m6dCmbNm0iOjqaoKAg1q1bx6RJkwz75LVg6tmzJxcvXsxTbM9bzJjb7t27UalUPHjw4Lna8ff3R6VSGT3Kli2bP0HmgZeXF3379mXcuHFmf28hCqq9a78n8K+rAIS3Kk/T7sMtHJH5yByLfNCtWzc0Gg3Lli2jQoUK3Llzh7///pu4uDhLh2YkIyMDW1vbZzr2559/pmHDhvj7+5OQkABAREQEvr6+NGrUyLCfu7v7c8Xo4OCAg4PDc7XxrJ4nP5YyceJEBg8ebHiuVqstEsfAgQMJCQnhm2++oWTJkhaJQYiCIu7OdXSzf8BWB5cC1PSYucnSIZlVwe6xUBTISM79Q5OSt/2f9MjlLc0fPHjA/v37mTZtGi1atKB8+fKEhITwySef0KFDhxyPmzhxIt7e3pw6dSrb1+Pj4xkyZAheXl64urrSsmVLozujRkRE0KVLF7y9vXF2dubFF1/kr7/+MmrD39+fr776igEDBuDm5sbgwYMNQw3btm2jevXqODs7065dO6Kjo5/4OVevXk3nzp0NzwcOHMj777/PtWvXUKlU+Pv7A8a9B82bN+fq1auMGjXK8Bf10zw+FDJ+/Hhq167N8uXL8ff3x83NjV69epGYmAhkdunv2bOHb7/91vAeUVFRAISHh9O+fXucnZ3x9vamb9++xMbGGtpu3rw5w4cPZ/To0ZQqVYrWrVsDcPbsWTp06ICrqysuLi40adKEiIgIw3FLliyhevXq2NvbU61aNebNm2d4LSoqCpVKxerVq2nUqBH29vbUqFHDcKfbqKgoWrRoAUDJkiVRqVQMGDDgqXnJiYuLCz4+PoaHp6cnOp2Ot956i4CAABwcHKhatSrffvvtE9tZu3YtNWvWxMHBAQ8PD1q1akVycnKuPjNAzZo18fHxYf364jGGLMSTHFn0AVZ6hQdOUGP6giK3ZPfTFOxPq0mByaVztasVUCI/3/vTW2Dr9NTdnJ2dcXZ2ZsOGDTRo0AA7O7sn7q8oCiNHjmTDhg3s37+fypUrZ7tPhw4dcHd3Z8uWLbi5ufHTTz/x8ssvc/HiRdzd3UlKSqJ9+/Z89dVX2Nvbs2zZMjp16sSFCxcoV66coa1vvvmGL774gs8//xyA/fv3k5KSwowZM1i+fDlWVla8+eabjBkzhl9++SXbmO/fv09YWBj16tUzbJszZw6VKlViwYIFHD16NNu/lNetW0dwcDBDhgwx+qs6ryIiItiwYQObNm3i/v379OjRg6lTp/L111/z7bffcvHiRYKCgpg4cSIAnp6eREdH06xZMwYPHsysWbNITU3lo48+okePHuzcudPQ9rJlyxg6dCgHDhxAURRu3rxJ06ZNad68OTt37sTV1ZUDBw6g1WoBWLhwIePGjeP777+nTp06nDx5ksGDB+Pk5ET//v0N7X744YfMmTOHwMBAZs2aRefOnYmMjMTPz4/ff/+dbt26ceHCBVxdXQ09NJMnT2by5MlPzMXWrVtp0qTJE/fR6/WULVuW3377jVKlSnHw4EGGDBmCr68vPXr0yLJ/dHQ0vXv3Zvr06XTt2pXExET27duH8t/iOrefOSQkhH379jFo0KAnxidEUXZi23LaaXaS8LKaw5VH07Bmo6cfVMQU7MKiELC2tmbp0qUMHjyYH3/8kbp169KsWTN69epFrVq1jPbVarX069ePY8eOceDAgRzHw3ft2sWZM2eIiYkxFCozZsxgw4YNrF27liFDhhAcHExwcLDhmK+++or169fzxx9/MHz4/8byWrZsyZgxYwzP9+/fj0aj4ccff6RixYoADB8+3HBSzs7Vq1dRFIXSpf9X5Lm5ueHi4oJarcbHxyfb49zd3VGr1Ya/qp+VXq9n6dKluLi4ANC3b1/+/vtvvv76a9zc3LC1tcXR0dHoPebPn0/dunWNTtSLFy/Gz8+PixcvUqVKFQAqVarE9OnTDft8+umnuLm5sXr1amxsbAAM+wJMmjSJmTNn8tprrwEQEBBAeHg4P/30k9FJdvjw4XTr1s0QS2hoKD///DNjx441DBd5eXkZ9c68++67Rid+vV5PUlISzs7OWFlldi6WKVPGKDcfffSRoWiEzOLkgw8+YMKECYZtAQEBHDx4kN9++y3HwkKr1fLaa69Rvnx5ILMHIq+fuUyZMpw8eTJL+0IUF9HXLlHhn48BOOvXm9b9P3/KEUVTwS4sbBwzew5yQa/Xk5CYiKuLi+GX8HO/dy5169aNDh06sG/fPv755x9CQ0OZPn06ixYtMurmHjVqFHZ2dhw6dIhSpUrl2N7x48dJSkrCw8PDaHtqaqqhSz45OZkJEyawadMmbt26hVarJTU1lWvXrhkd82gvw0OOjo6GogLA19eXmJiYHONJTU0FwN7ePuckmJC/v7+hqICnxwuZOdy1axfOzs5ZXouIiDAUC4/n59SpUzRp0sRQVDzq7t27XL9+nbfeesuoB0ar1eLm5ma0b8OGDQ3/b21tTb169Th37twTY3Z3dzeao6LX60lISMDV1TXH7/SHH35o9B17+L368ccfWbRoEVevXiU1NZWMjAxq166dbRvBwcG8/PLL1KxZk7Zt29KmTRu6d+9OyZIl8/SZHRwcSElJeeJnFKKoSk9N4fjgV7HxtaV8xYq8MGCmpUOymIJdWKhUuRqOAECvBxtd5v75UVjkkb29Pa1bt6Z169Z8+eWXvP3224wbN87ol37r1q1ZtWoV27Zt44033sixLb1ej6+vr2Fc/lEP/8L98MMP2bZtGzNmzKBSpUo4ODjQvXt3MjIyjPZ3csqav8dPmiqVytDtnZ2HJ6v79+9nKXbMIbt49Xr9E4/R6/V06tSJadOmZXnN19fX8P+P5+dJE0cfvufChQupX7++0Wu5mTT5tDkmzzIUUqpUKSpVMl7F77fffmPUqFHMnDmThg0b4uLiwjfffMPhw4ezbVOtVrNjxw4OHjzI9u3b+e677/jss884fPgwjo6ZBXZuPnNcXByenp5PjF+IomrdiHbUvqon7ZY9KQM+w9bOMn+IFQQFu7AoxAIDA7OsW9G5c2c6depEnz59UKvV9OrVK9tj69aty+3bt7G2tjZMinzcvn37GDBgAF27dgUgKSnJMGkxv1WsWBFXV1fCw8OznMSextbWFp1OZ5K4nvQedevW5ffff8ff3x/rPEycqlWrFsuWLUOj0WQpaLy9vSlTpgxXrlx5YmEIcOjQIZo2bQpk/nV//PhxwxDVwytPHo/5WYZCsrNv3z4aNWrEe++9Z9j26OTT7KhUKho3bkzjxo358ssvKV++POvXr2f06NG5/sxhYWE0b978qfEJUdRsWzyRmvvuAnCpYyA9mnezcESWVbCvCikE7t27R8uWLVmxYgX//vsvkZGRrFmzhunTp9OlS5cs+3ft2pXly5czcOBA1q5dm22brVq1omHDhrz66qts27aNqKgoDh48yOeff86xY8eAzLkB69at49SpU5w+fZo+ffo89a/4Z2VlZUWrVq3Yv39/no/19/dn79693Lx50+iKjPzk7+/P4cOHiYqKIjY2Fr1ez7Bhw4iLi6N3794cOXKEK1eusH37dgYNGvTEQmf48OEkJCTQq1cvjh07xqVLl1i+fDkXLlwAMq9SmTJlimHS6JkzZ1iyZAmzZs0yaueHH35g/fr1nD9/nmHDhnH//n3DpMby5cujUqnYtGkTd+/eJSkpCcgcCqlUqZLRo0KFCkbPc3MpbqVKlTh27Bjbtm3j4sWLfPHFFxw9ejTH/Q8fPszkyZM5duwY165dY926ddy9e5fq1avn+jOnpKRw/Phx2rRp89T4hChKbkaG4/TjKtQKnKtqQ7dJv1o6JIuTwuI5OTs7U79+fWbPnk3Tpk0JCgriiy++YPDgwXz//ffZHtO9e3eWLVtG3759WbduXZbXVSoVW7ZsoWnTpgwaNIgqVarQq1cvoqKi8Pb2BmD27NmULFmSRo0a0alTJ9q2bUvdunVN9jmHDBnC6tWr81y8TJw4kaioKCpWrGiybvIxY8agVqsJDAzE09OTa9euUbp0aQ4cOIBOp6Nt27YEBQUxYsQI3NzcnjgHx8PDg507d5KUlESzZs144YUXWLhwoaH34u2332bRokUsXbqUmjVr0qxZM5YuXUpAQIBRO1OnTmXatGkEBwezb98+Nm7caBhSKlOmDBMmTODjjz/G29vbaLJtfnj33Xd57bXX6NmzJ/Xr1+fevXtGvRePc3V1Ze/evbRv354qVarw+eefM3PmTF555ZVcf+aNGzdSrly5p16xIkRRotNqOTyyFx4JcM8V6s9ZXewuLc2OSnnS4LoJJCQk4ObmRnx8PK6urkavpaWlERkZSUBAQJ4nCuZmopt4doqi0KBBAz744APDGg+S56yioqIICAjg5MmTOU6WzK3C9J0OCQlh5MiR9OnTJ8d9nufn25Q0Gg1btmyhffv22U7aFfmjKOb5t0+6UXN9ODoV3PqoD20GfGHpkEya5yedvx9VsH9biQJDpVKxYMECw3oOQjwUExND9+7d6d27t6VDEcJsrl8+g8udU2it4EwTzwJRVBQUUliIXAsODqZv3+e75e8rr7xiWFTs8cfTrogQBZOXlxdjx47N1cqqQhQFGelppK0eyCvlYonp4spr34ZaOqQCRQaDhFktWrTIsC7G4573PiMFgb+//xMv3RVCFH5HFo/kJe0lHuBM0LCV2Dnkft2j4kAKC2FWublcUgghCqo/vx2Fy6+7OdfInrQ206hTtuLTDypmZChECCGEyIWIMwfx/L9QvONUnLlRljpt3rR0SAWSFBZCCCHEU+i0Ws6OHYJbMtz2gFbfbrB0SAWWDIUIIYQQj9HpFY5ExhGTmIaXiz1R375B7UgdGWqw/s9w3L39LB1igSWFhRBCCPGI0LBoJvwZTnR8GgBN0v5mzN9XATjXujy9XhtmyfAKPBkKMbHmzZszcuTIZzp2586dVKtWzWRLdZvb7t27UalUPHjwIMd9VCpVlnusFAebNm2iTp06RebfWojCKjQsmqErThiKClddHIOObMVWBxf8rXAdsNDCERZ8UljkgwEDBqBSqbI8Ll++zLp165g0aZJhX39/f+bMmZOrdseOHctnn31W4FddzE/R0dGGpaRzY+nSpYY7vhYW2X0HOnbsiEqlYuXKlZYJSgiBTq8w4c9wHr1gfJTVr6Q76XngBLOC3mHS1ovo9HJJ+ZMUnzOWibVr147o6GijR0BAAO7u7ri4uOS5vYMHD3Lp0iVef/11E0RbcPn4+GBnZ2f299XpdBbvLRg4cCDfffedRWMQojg7Ehln6KkAaGt1hIEO+3i54R0WN2/HNeuKRMencSQyzoJRFnxSWOQTOzs7fHx8jB5qtdpoKKR58+ZcvXqVUaNGGXo1crJ69WratGljdE+F8ePHU7t2bRYvXky5cuVwdnZm6NCh6HQ6pk+fjo+PD15eXnz99ddGbc2aNYuaNWvi5OSEn58f7733nuGOmvC/v/q3bdtG9erVcXZ2NhRKD2U3pPPqq68yYMAAw/MVK1ZQr149XFxc8PHxoU+fPsTExOQpj48OhURFRaFSqVi3bh0tWrTA0dGR4OBg/vnnHyBzaGXgwIHEx8cb8jl+/HgAMjIyGDt2LGXKlMHJyYn69euze/fuLJ9506ZNBAYGYmdnx9WrV0lPT2fs2LH4+flhZ2dH5cqV+fnnnw3HhYeH0759e5ydnfH29qZv375Gd21t3rw5w4cPZ/jw4ZQoUQIPDw8+//xzw6JZT/oOdO7c2XAnViGE+cUk/q+oqKiPYJpN5rDHz/qO7LBple1+IqtCUVikaFJyfKTr0nO9b5o2LVf7msq6desoW7YsEydONPRq5GTv3r3Uq1cvy/aIiAi2bt1KaGgoq1atYvHixXTo0IEbN26wZ88epk2bxueff86hQ4cMx1hZWTF37lzCwsJYtmwZO3fuZOzYsUbtpqSkMGPGDJYvX87evXu5du0aY8aMydPny8jIYNKkSZw+fZoNGzYQGRlpVHg8q88++4wxY8Zw6tQpqlSpQu/evdFqtTRq1Ig5c+bg6upqyOfDmAcOHMiBAwdYvXo1//77L6+//jrt2rXj0qVLRp95ypQpLFq0iLNnz+Ll5UW/fv1YvXo1c+fO5dy5c/z44484OzsDmcM0zZo1o3bt2hw7dozQ0FDu3LlDjx49jOJdtmwZ1tbWHD58mLlz5zJ79mwWLVoEPPk7UL58eby8vNi3b99z50wIkXdeLpl/yJXQxfLpwfnsP+PEUU0AM7WvZ7ufyF6huCqk/sr6Ob7WpEwT5rWaZ3jeYk0L0nTZV5P1vOuxpN0Sw/N2v7fjfvr9LPud6X8mzzFu2rTJcAKCzHtirFmzxmgfd3d31Gq14S/6J4mKiqJ06dJZtuv1ehYvXoyLiwuBgYG0aNGCCxcusGXLFqysrKhatSrTpk1j9+7dNGjQAMCopyEgIIBJkyYxdOhQ5s37X940Gg0//vgjFStmriI3fPhwJk6cmKccDBo0yPD/FSpUYO7cuYSEhJCUlGSUm7waM2YMHTp0AGDChAnUqFGDy5cvU61aNdzc3FCpVEb5jIiIYNWqVdy4ccOQwzFjxhAaGsqSJUsM9yTRaDTMmzeP4OBgAC5evMhvv/3Gjh07aNWqleFzPDR//nzq1q1rdE+TxYsX4+fnx8WLF6lSpQoAfn5+zJ49G5VKRdWqVTlz5gyzZ89m8ODBT/0OlClThqioqGfOlRDi2YUEuOPtbM3YQzMoHQvxKdZ8GtAHjU3mqVIF+LjZExJQ+G8/YEqForAoDFq0aMH8+fMNz52cnJ6rvdTU1GxvLe3v7280Z8Pb2xu1Wm00wdPb29toCGLXrl1MnjyZ8PBwEhIS0Gq1pKWlkZycbIjT0dHRUFQA+Pr65nkY4+TJk4wfP55Tp04RFxdnmLNw7do1AgMD89TWo2rVqmUUF2TeUbNatWrZ7n/ixAkURTGc6B9KT0/Hw8PD8NzW1tao7VOnTqFWq2nWrFm27R4/fpxdu3ZlWyRFREQY3q9BgwZGQxwNGzZk5syZ6HQ61Gr1Ez+rg4MDKSmm6zUTQuRMbaViWNTXBF7WorWC/2vcmEs2NYDMogJgXKdA1FZyw70nKRSFxeE+h3N8TW1l/It61+u7cryKwkplvD20W/7dkc7JyYlKlSrlW3ulSpXi/v2svSk2NjZGz1UqVbbbHp7Ur169Svv27Xn33XeZNGkS7u7u7N+/n7feeguNRvPEdh+9mZaVlVWWm2s9enxycjJt2rShTZs2rFixAk9PT65du0bbtm3JyMjI46fP+TM/PGE/aaKlXq9HrVZz/PjxLCfyR4sCBwcHowLAwcHhiXHo9Xo6derEtGnTsrz2sOB5XnFxcXh6euZLW0KIvNkwYxgv7L8LwF+NyrLFqavhNR83e8Z1CqRdUP78rBdlhaKwcLTJ/Z3jHG0cc315Zl7azS+2trbodLqn7lenTh3Cw8Of+/2OHTuGVqtl5syZhrz89ttveW7H09PTaD6ATqcjLCyMFi1aAHD+/HliY2OZOnUqfn5+hvc2tezyWadOHXQ6HTExMTRp0iTXbdWsWRO9Xs+ePXsMQyGPqlu3Lr///jv+/v5YW+f8o/Po/JaHzytXrmwocnL6DqSlpREREUGdOnVyHbMQIn8c3rYCv+U7sQLCajnwwYLtNHhk5c2QAHfpqcilQjF5syjx9/dn79693Lx50+hqgse1bduW/fv3P/f7VaxYEa1Wy3fffceVK1dYvnw5P/74Y57badmyJZs3b2bz5s1cvHiRYcOGGS10Va5cOWxtbQ3v88cffxit32Eq/v7+JCUl8ffffxMbG0tKSgpVqlThjTfeoF+/fqxbt47IyEiOHj3KtGnT2LJlyxPb6t+/P4MGDTJMPt29e7ehEBs2bBhxcXH07t3bcPXG9u3bGTRokFGhcP36dUaPHs2FCxdYtWoV3333HSNGjDB6n+y+A4cOHcLOzo6GDRuaIFNCiJzE3b1F8qSvcUyHq6VVvLLwL9RWKhpW9KBL7TI0rOghRUUeSGFhZhMnTiQqKoqKFSs+scv7zTffJDw8nAsXLjzX+9WuXZtZs2Yxbdo0goKC+OWXX5gyZUqe2xk0aBD9+/dnwIABdOzYkYCAAENvBWT2aCxdupQ1a9YQGBjI1KlTmTFjxnPFnhuNGjXi3XffpWfPnnh6ejJ9+nQAlixZQr9+/fjPf/5D1apV6dy5M4cPHzb0puRk/vz5dO/enffee49q1aoxePBgkpOTAShdujQHDhxAp9PRtm1bgoKCGDFiBG5ubka9ZP369SM1NZWQkBCGDRvG+++/z5AhQwyv5/QdWLVqFW+88QaOjubvSROiuNLrdFxd8haOdRKJLgUBM+fh7CaTM5+HSnl84NzEEhIScHNzIz4+HldXV6PX0tLSiIyMJCAgINuJi0+i1+tJSEjA1dW1yKxUOXbsWOLj4/npp58sHYpBUcxzfmrevDm1a9fO9eqqD929e5dq1apx7NgxAgICgKKX6+f5+TYljUbDli1baN++fZa5RiL/FNQ8/7P4QxpeW0C6YsOVjr9S/cWXLR3SczFlnp90/n5U4f9tVYR99tlnlC9fPldzMkThFhkZybx58wxFhRDC9DbMHEbpC0sBOF17XKEvKgqKQjF5s7hyc3Pj008/tXQYwgxCQkIICQmxdBhCFBsH/1hIuWU7uWtVijPdq9Gx6/uWDqnIkMJCiHz06LLhQoiC6XpEGJopsyiZAZFlrWgxYqmlQypSZChECCFEsZGemsKpYT3xug9xrlB19kKcXNwsHVaRIoWFEEKIYmP90JepFKUn3RoyRg+mYs1Glg6pyJHCQgghRLGwdmI/gg89AOBi12Ba9Bpt2YCKKCkshBBCFHmX//0H571HAPi3nis9Jq22cERFlxQWQgghirT4e3dwXN+fZo2iCWtkT+efdlg6pCJNCgshhBBFVkZ6GtcW9KK0coc4tTdtvgnFwSnnxZ3E85PLTYUQQhRZ699qhrM+lvIV7cjotRw3D29Lh1TkSY+FmahUKjZs2JDj6/7+/nleBtrcvvjiC9555x3Dc0VRGDJkCO7u7qhUKk6dOkXz5s0ZOXLkM7/H0qVLKVGixPMHK0xi06ZN1KlT54m3rReioFjzeU9qHUugwglbdpV8nYAa9S0dUrEghUU+iImJ4Z133qFcuXLY2dnh4+ND27Zt+eeffywdWr65c+cO3377LZ988olhW2hoKEuXLmXTpk1ER0cTFBTEunXrjO5qmteCqWfPnly8eDFPsT1vMWNuu3fvRqVSGd0d9lmOf9Jj6dKl+RrzQx07dkSlUrFy5UqTtC9Eftm5aiZVNvwLwOkGJegy+gcLR1R8yFBIPujWrRsajYZly5ZRoUIF7ty5w99//01cXJylQzOSkZGBra3tMx37888/07BhQ/z9/UlISAAgIiICX19fGjX633Xg7u7Pd1dABwcHHBwcnquNZ/U8+TGnRo0aER0dbXg+YsQIEhISWLJkiWGbm9v/FvzR6XSoVKp8u5HZwIED+e6773jzzTfzpT0h8tul0wewm7UIWy1cDlDTdf7flg6pWCkUPRb6lJScH+npud83LS1X++bFgwcP2L9/P9OmTaNFixaUL1+ekJAQPvnkEzp06JDjcRMnTsTb25tTp05l+3p8fDxDhgzBy8sLV1dXWrZsyenTpw2vR0RE0KVLF7y9vXF2dubFF1/kr7/+MmrD39+fr776igEDBuDm5sbgwYMNQw3btm2jevXqODs7065dO6MTVXZWr15N586dDc8HDhzI+++/z7Vr11CpVPj7+wPGvQfNmzfn6tWrjBo1yvCX9NM8PhQyfvx4ateuzfLly/H398fNzY1evXqRmJgIwIABA9izZw/ffvut4T2ioqIACA8Pp3379jg7O+Pt7U3fvn2JjY01tN28eXOGDx/O6NGjKVWqFK1btwbg7NmzdOjQAVdXV1xcXGjSpAkRERGG45YsWUL16tWxt7enWrVqzJs3z/BaVFQUKpWK1atX06hRI+zt7alRo4Zhqe+oqCjD7eZLliyJSqViwIABT83Lo2xtbfHx8TE8HBwcDD1lPj4+hIaG4uvry6ZNmwgMDMTOzo6rV69m27Pz6quvGr1/RkYGY8eOpUyZMjg5OVG/fv0sy5R37tyZI0eOcOXKlTzFLYQ5JCfGc+k/g3FPhJiSUPv71dg5OFo6rGKlUPRYXKj7Qo6vOTVrSrlHbit+uUlTlNTUbPd1fPFFyi//v//t+3IrdPfvZ9mv+vlzuY7N2dkZZ2dnNmzYQIMGDbCzs3vi/oqiMHLkSDZs2MD+/fupXLlytvt06NABd3d3tmzZgpubGz/99BMvv/wyFy9exN3dnaSkJNq3b89XX32Fvb09y5Yto1OnTly4cIFy5coZ2vrmm2/44osv+PzzzwHYv38/KSkpzJgxg+XLl2NlZcWbb77JmDFj+OWXX7KN+f79+4SFhVGvXj3Dtjlz5lCpUiUWLFjA0aNHUavVWY5bt24dwcHBDBkyhMGDB+cqn9mJiIhgw4YNbNq0ifv379OjRw+mTp3K119/zbfffsvFixcJCgpi4sSJAHh6ehIdHU2zZs0YPHgws2bNIjU1lY8++ogePXqwc+dOQ9vLli1j6NChHDhwAEVRuHnzJk2bNqV58+bs3LkTV1dXDhw4gFarBWDhwoWMGzeO77//njp16nDy5EkGDx6Mk5MT/fv3N7T74YcfMmfOHAIDA5k1axadO3cmMjISPz8/fv/9d7p168aFCxdwdXU19NBMnjyZyZMnPzEXW7dupUmTJk/NWUpKClOmTGHRokV4eHjg5eWVq1wPHDiQqKgoVq9eTenSpVm/fj3t2rXjzJkzhu9q+fLl8fLyYt++fVSoUCFX7QphLpuHvEzNGwqptmD72Rj8KgZZOqRip1AUFgWZtbU1S5cuZfDgwfz444/UrVuXZs2a0atXL2rVqmW0r1arpV+/fhw7dowDBw5QtmzZbNvctWsXZ86cISYmxlCozJgxgw0bNrB27VqGDBlCcHAwwcHBhmO++uor1q9fzx9//MHw4cMN21u2bMmYMWMMz/fv349Go+HHH3+kYsWKAAwfPtxwUs7O1atXURSF0qVLG7a5ubnh4uKCWq3Gx8cn2+Pc3d1Rq9W4uLjkuE9u6PV6li5diouLCwB9+/bl77//5uuvv8bNzQ1bW1scHR2N3mP+/PnUrVvX6ES9ePFi/Pz8uHjxIlWqVAGgUqVKTJ8+3bDPp59+ipubG6tXr8bGxgbAsC/ApEmTmDlzJq+99hoAAQEBhIeH89NPPxkVFsOHD6dbt26GWEJDQ/n5558ZO3asYbjIy8vLqHfm3XffpUePHkafOykpCWdnZ8MwRpkyZXKVM41Gw7x584y+I08TERHBqlWruHHjhuHfesyYMYSGhrJkyRKjXJYpU8bQMyREQXFk/Vy8XGNIt3Hiap8mdO34lqVDKpYKRWFR9cTxnF987C/lSvv25jyW/Nj2Sn//lf1+edStWzc6dOjAvn37+OeffwgNDWX69OksWrTIqJt51KhR2NnZcejQIUqVKpVje8ePHycpKQkPDw+j7ampqYYu+eTkZCZMmMCmTZu4desWWq2W1NRUrl27ZnTMo70MDzk6OhqKCgBfX19iYmJyjCf1vz1A9vb2OSfBhPz9/Q1FBTw9XsjM4a5du3B2ds7yWkREhKFYeDw/p06dokmTJoai4lF3797l+vXrvPXWW0Y9MFqt1mhOA0DDhg0N/29tbU29evU4d+7JPWHu7u5Gc1T0ej0JCQm4urrmeX6Era1tlsL2aU6cOIGiKEaFFEB6enqW76KDgwMpeRw2FMKULhzbSe1TE7D11bJjSHu6vv+jpUMqtgpFYWHlmPvxMStHx1z/Es5Lu09jb29P69atad26NV9++SVvv/0248aNMyosWrduzapVq9i2bRtvvPFGjm3p9Xp8fX2zvQX3w79wP/zwQ7Zt28aMGTOoVKkSDg4OdO/enYyMDKP9nZycsrTx+ElTpVKhKEqO8Twsgu7fv5/lBGMO2cX7tMsd9Xo9nTp1Ytq0aVle8/X1Nfz/4/l50sTRh++5cOFC6tc3vmwtu6Ggxz1tjkl+DoU4ODhkeT8rK6ss/84ajcbw/3q9HrVazfHjx7N8nscLtLi4ODw9PZ8ahxDmcOHUHvS/v42tnZaTjo15+T25AsSSCkVhURgFBgZmWbeic+fOdOrUiT59+qBWq+nVq1e2x9atW5fbt29jbW1tmBT5uH379jFgwAC6du0KQFJSksm6pitWrIirqyvh4eFUqlQpT8fa2tqi0+lMEteT3qNu3br8/vvv+Pv7Y22d+695rVq1WLZsGRqNJktB4+3tTZkyZbhy5coTC0OAQ4cO0bRpUyCzR+P48eOGIaqHV548HnN+DoVk5+Hck4d0Oh1hYWGGyaR16tRBp9MRExPzxOIlLS2NiIgI6tSp88yxCJFfkuLjiBw9FNcEO/a0KMsLo3/BKheFvjCdQnFVSEF27949WrZsyYoVK/j333+JjIxkzZo1TJ8+nS5dumTZv2vXrixfvpyBAweydu3abNts1aoVDRs25NVXX2Xbtm1ERUVx8OBBPv/8c44dOwZkzg1Yt24dp06d4vTp0/Tp08dkixZZWVnRqlUr9u/fn+dj/f392bt3Lzdv3jS6IiM/+fv7c/jwYaKiooiNjUWv1zNs2DDi4uLo3bu34QqG7du3M2jQoCcWOsOHDychIYFevXpx7NgxLl26xPLly7lw4QKQeZXKlClTDJNGz5w5w5IlS5g1a5ZROz/88APr16/n/PnzDBs2jPv37zNo0CAgc/KjSqVi06ZN3L17l6SkJCBzKKRSpUpGjwoVKhg9f55LcVu2bMnmzZvZvHkz58+f57333jNaS6NKlSq88cYb9OvXj3Xr1hEZGcnRo0eZNm0aW7ZsMex36NAh7OzsjIZ7hLCUrYNbUf6Wgp0GHFp/grNrSUuHVOxJYfGcnJ2dqV+/PrNnz6Zp06YEBQXxxRdfMHjwYL7//vtsj+nevTvLli2jb9++rFu3LsvrKpWKLVu20LRpUwYNGkSVKlXo1asXUVFReHtnLkc7e/ZsSpYsSaNGjejUqRNt27albt26JvucQ4YMYfXq1XkuXiZOnEhUVBQVK1Y0Wdf5mDFjUKvVBAYG4unpybVr1yhdujQHDhxAp9PRtm1bgoKCGDFiBG5ubk8cKvPw8GDnzp0kJSXRrFkzXnjhBRYuXGjovXj77bdZtGgRS5cupWbNmjRr1oylS5cSEBBg1M7UqVOZNm0awcHB7Nu3j40bNxqGlMqUKcOECRP4+OOP8fb2Nppsa0qDBg2if//+9OvXj2bNmhEQEGDorXhoyZIl9OvXj//85z9UrVqVzp07c/jwYfz8/Az7rFq1ijfeeAPHfBxKFOJZ/PqfjgT9m4oeuNGvJSFtZG2VgkClPGlw3QQSEhJwc3MjPj4eV1fjG8GkpaURGRlJQEBAnicKPs9EN/F0iqLQoEEDPvjgA8MaD5LnrKKioggICODkyZPUrl37udoqiN/pu3fvUq1aNY4dO5almHqa5/n5NiWNRsOWLVto3759tpN2Rf7I7zxvWzSO0rN+w1oPp5p70/vH3c8fZBFgyu/zk87fjyoYv61EgadSqViwYIFhPQdRPEVGRjJv3rw8FxVC5Kezh7fjNi+zqDhf2YYe3+fPFX4if0hhIXItODiYvn37Plcbr7zyimFRsccfT7siQlheSEgIPXv2tHQYohhLS03m4uRRuKVAdClosmAj6jxM0BamJ/8awqwWLVpkWBfjcc97n5GCwN/f/4mX7gohnp2i13Pmx0G8XPMWu/GizAdfUcpXes8KGikshFk9z+WSQoji7fCvU2kQH4pOrSLgP9Op2TTrlXfC8gpkYSF/8QlR9MjPtXgem+d9hPrIGvTl4WiV0TSQoqLAytMci/HjxxvuIPnw8Tz3gHjcw9X+Hl89UghR+D1cAlyuvBB5dWrfRjwX/kH5Q3asiQmkfu/PLR2SeII891jUqFHD6PbcuVnKONfBWFvj6OjI3bt3sbGxydMldnq9noyMDNLS0grMpXlFkeTZfIpKrhVFISUlhZiYGEqUKJGvvzNE0Rd35zqxn31MmVS44QVtJq5FVYh/HoqDPBcW1tbW+dpL8SiVSoWvry+RkZFcvXo1T8cqikJqamq290gQ+UfybD5FLdclSpQw2e8OUTTptFp2v9OR6jGQ4AjeU6ZT0lPmaRV0eS4sLl26ROnSpbGzs6N+/fpMnjyZChUq5Lh/eno66enphucJCQlA5iIej94A6SGVSoW/vz8ajSZPY7JarZaDBw/SqFGjPN0bQuSN5Nl8ikquVSoV1tbWqNXqArkOysPfQ9n9PhL551nyvHZkO2qfz0Cngti3u9AmpJ38Oz2FKb/PuW0zTytvbt26lZSUFKpUqcKdO3f46quvOH/+PGfPns3xrpfjx49nwoQJWbavXLlSlgQWQgiRrbtHfqXhupNYKXComTfu7UdZOqRiLyUlhT59+jx15c3nWtI7OTmZihUrMnbsWEaPHp3tPtn1WPj5+REbG/vEwPJKo9GwY8cOWrduLZPDTEjybD6Sa/OQPJtHXvJ8K+oc4dNew++ADecD7ei44h9ZBCuXTPl9TkhIoFSpUk8tLJ7rX8rJyYmaNWty6dKlHPexs7PDzs4uy3YbGxuT/BCbql1hTPJsPpJr85A8m8fT8pyc+AD9qr68UvYu2zsE0OqTrdg/x119iytTfJ9z295zTa1NT0/n3Llz+Pr6Pk8zQgghBDqtltPf9yZAH0UsJag9aj0lPOT8UtjkqbAYM2YMe/bsITIyksOHD9O9e3cSEhLo37+/qeITQghRTPz23sto/ojkQrIDse0X4VVGlusujPI0FHLjxg169+5NbGwsnp6eNGjQgEOHDlG+fHlTxSeEEKIY2DBjKLX2xmCFilPUp2dIa0uHJJ5RngqL1atXmyoOIYQQxdThrcvwW74bKyAs2IGe32y0dEjiOcjyZUIIISwm+up5UiZNxTEdrpZW8cqCv55+kCjQpLAQQghhEZqMdI681x2fOLjvDAEz5+Hs5m7psMRzksJCCCGERfz+QRuqROjIUEPKiP5UrdPc0iGJfCCFhRBCCLM7uX0FLUuGcb2MwoXOgbTq+7GlQxL5RJYyE0IIYVZXzx2nyoH/4GSnxblrY9oM/9nSIYl8JD0WQgghzObmlTDOfvcmTqo0ztoG8+KQeZYOSeQzKSyEEEKYlE6vcCQyDq0mg39HvEnATthwxRfft1dhY5v1lg+icJOhECGEECYTGhbNhD/DiUtK5Yuo6QRG6Um3hgcN3sTdq4ylwxMmID0WQgghTCI0LJqhK04QHZ/GG7ELeOF4EgB/NqzIhGt1CA2LtnCEwhSksBBCCJHvdHqFCX+GowCtUzbR+WDmXbAPBTvxs8dQACb8GY5Or1gwSmEKUlgIIYTId0ci44iOT+PF9IMM3rcbOw1cLm/FNxU+BUABouPTOBIZZ9E4Rf6TwkIIIUS+i0lMw5s4+iauxSUVrntDypuj0VnZZtlPFC1SWAghhMh3riSzzHYancveJqIFTKs/FHvXUln283Kxt0B0wpTkqhAhhBD56v7dm7hu6Es1q+vcVkoytcRH3LUqBegM+6gAHzd7QgLk3iBFjRQWQggh8k1qcgL7+rfFOVFLxEsuvMdH3FQ8seV/kzRV//3vuE6BqK1U2TckCi0ZChFCCJEvdFotm/s1pfIVHR73VVyqOIhRb7yKj5vxcIePmz3z36xLuyBfC0UqTEl6LIQQQuSLNW81IfhsOnoVRLzxEq+9PRGA1oE+HLocQ+y5Qyzu/yINKnlJT0URJj0WQgghntvq99sQfPgBAGEdq/DaJwsNr6mtVIa5FCEB7lJUFHFSWAghhHgua77oTfCO6wCcauZFz282WjgiYUlSWAghhHhmJ3asxHf7KQDO1HGixw9/WzYgYXEyx0IIIcQzuXhiN9X2jySuhZ6D18vRdckB1NZyWinupMdCCCFEnkWEHcLzjzdxVKVz360mry4+KLdAF4AUFkIIIfIo/OgObgweyNkYFZesKxMwbB22drKCpsgkhYUQQohcu3bpNLdHfYDXfdCccsZ1wK84u5a0dFiiAJHCQgghRK7cv3uT8Hd74xsLD5ygxJTJeJetaOmwRAEjhYUQQoinSk1OYG//tpS/qZBiB5pPhlGnWVdLhyUKICkshBBCPJFOq2Vz/2ZUuaIjwxpihr1G0+7DLR2WKKCksBBCCPFEaz/uQI2wtP8u1d2YV4Z8bemQRAEmFxwLIYTI0aGVX9Hd6RBba3mSXL4GPT9ZZOmQRAEnPRZCCCGydWzTAhpc/Aa1Ckq17yFLdYtckR4LIYQQWfz57UjYuYWMWnDSpzv1+31l6ZBEISGFhRBCCCN/LZ9K2UXbsNeoWV+qCq+PX4DKSjq4Re7IN0UIIYTBoa3LcJ21DHsNXClnRYdZ27BSqy0dlihEpLAQQggBwLmjf6EdNxWXVLjhreKFRRtwdnO3dFiikJHCQgghBNcjwoge9T4eCRBTEgK+X4hPucqWDksUQlJYCCFEMZeSlMC/Q3sYlup2nfo1lWo2tnRYopCSwkIIIYoxrSaDC/N64lv9AXGuD5fqfs3SYYlCTK4KEUKIYkrR6zkxbwAhKQdJL2HDhRmTqdVU7v8hno/0WAghRDH169DmlLzxNzpFRXjjOVJUiHwhhYUQQhRDqz9oS/Ceu8TvcGdvhQ+o0+ZNS4ckiggpLIQQophZM643wduvAXApxJMW/SdaOCJRlEhhIYQQxcgfc0ZSbc0pAM7UdqLHvJ2WDUgUOVJYCCFEMfHX8qn4/bwNaz2cr2pD1/87gNpa5vCL/CWFhRBCFANHtq8wWqr75aV/YWNrZ+mwRBEkpaoQQhRxd29FUXLPeK54qInXqHlh0QZcS3pZOixRRElhIYQQRVj8/ViSFnWhss1dVI1Lo+qxRJbqFiYlQyFCCFFE3b97k52ftSZAH0UsJXAcsJ6KNepbOixRxElhIYQQRVBqcgJ7B7Sl2s40/rzkRXy3XykdUM3SYYliQAoLIYQoYnRaLZv7N6NKhI4MNVi91J2KNRtYOixRTEhhIYQQRcyawU2pEZaGXgWX+zSkw9Aplg5JFCMyeVMIIQoRnV7hSGQcMYlpeLnYExLgjtpKZXh91Yh21P7nPgBhHSrT87PFlgpVFFNSWAghRCERGhbNhD/DiY5PM2zzdbNnXKdA2gX5smZcH2ptuwrA6Sae9Jrxh6VCFcWYFBZCCFEIhIZFM3TFCZTHtt+OT2PoihN888I9XK8fAuw4U9uR1+fLUt3CMqSwEEKIAk6nV5jwZ3iWogJAAWqqrvBK2CScyqWz0bM2XcZvlaW6hcXIN08IIQq4I5FxRsMfj6qbcYSZDotxUqVzyjqYdhO2YmfvaOYIhfgfKSyEEKKAi0nMvqiorAln9P7fiLJ14kbjEsS8+hO1pagQFiaXmwohRAHn5WKfZZu39iYfH16MRwKotSq+0L5NaS+5/4ewPCkshBCigAsJcMfXzZ6HF5W66B8w7uQcSsdCvBPMaNSb9FJBhAS4WzROIUAKCyGEKPDUVirGdQoEwEafysSwKQTcVEixhXkvteOM7QuM6xRotJ6FEJYicyyEEKIQaBfky5yu/ugm9qTalcylupc1acAln47M/+86FkIUBFJYCCFEIZCWkoT71qGoYnVo1HDstfr0HjCD2Y+tvCmEpUlhIYQQBVxy4gOivutME05zqbkT58v04q0PZlk6LCGyJYWFEEIUYDeuhHF6bl86lLhCsmKP5tWf6dTwFUuHJUSOZPKmEEIUUFfCj3BuwOuU35bGrlgPbnb5lUApKkQBJ4WFEEIUQBdO7ubq4P6UjYFkB1A1HUGVus0tHZYQTyVDIUIIUcD8e3Az9/8zBp/78MAZGD+G5h3fsnRYQuSKFBZCCFGAHP97DamffIlXAtxzBfsp46j3ci9LhyVErklhIYQQBcTJvRvJ+OhLPJIgpiSUnDGdWo07WTosIfJE5lgIIUQBcCXsMH5/D+deOS23PcDn+/lSVIhC6bkKiylTpqBSqRg5cmQ+hSOEEMXPxRN78FjblVKqBCrXdqLCkpVUfaG5pcMS4pk881DI0aNHWbBgAbVq1crPeIQQoljZuuAztJvX4h+czAXbavgM24xbyVKWDkuIZ/ZMPRZJSUm88cYbLFy4kJIlS+Z3TEIIUSz8OXc0vnPXUemCFRuv+lPmg1ApKkSh90w9FsOGDaNDhw60atWKr7766on7pqenk56ebniekJAAgEajQaPRPMvbZ+thW/nZpshK8mw+kmvzsFSe/5z5HhWX78dWB5f9rWg+9Q/sHJyL7L+3fJ/Nw5R5zm2bKkVRlLw0vHr1ar7++muOHj2Kvb09zZs3p3bt2syZMyfb/cePH8+ECROybF+5ciWOjo55eWshhCgS7u5ZRP3Qy1jr4UJFNdq+H2Pj4GLpsIR4opSUFPr06UN8fDyurq457penwuL69evUq1eP7du3ExwcDPDUwiK7Hgs/Pz9iY2OfGFheaTQaduzYQevWrbGxscm3doUxybP5SK7Nw9x5XvdlL4I2hGOlwLlqNry8ZCeOzm4mf19Lk++zeZgyzwkJCZQqVeqphUWehkKOHz9OTEwML7zwgmGbTqdj7969fP/996Snp6NWq42OsbOzw87OLktbNjY2JvlymapdYUzybD6Sa/MwR553Lv6Mypszi4qzQfZ0Wn4AO4fi1XMr32fzMEWec9tengqLl19+mTNnzhhtGzhwINWqVeOjjz7KUlQIIYTIdOiXCbS89j1Hmzvx770yvLZkPza2Wf/oEqKwy1Nh4eLiQlBQkNE2JycnPDw8smwXQggBOq2WnT++T+vYlQBoa3Sjx+BvUVnJ+oSiaJIlvYUQwkR0Wi1r3mpCwJkHXG5lx93qA2nQf4oUFaJIe+7CYvfu3fkQhhBCFC2ajHTW929M8MlkAI5l1KXXwGkWjkoI05OyWQgh8ll6agob+zSg5n+LitPtK9BrTqiFoxLCPKSwEEKIfJSSFM/m3g2pEZaGXgVnXqtBr1mbLR2WEGYjhYUQQuSThPsxbO/1EtXPZ6C1gnO969Fj8lpLhyWEWUlhIYQQ+SAtJYnzP/TAMVmDRg2X+zel+5fLLR2WEGYnV4UIIcRzSk58QNR3nQnRnuZOQ3uO+Q2g61CZqCmKJykshBDiOVyPCOPY9Dfo6htFkuLAvc5L6dCgnaXDEsJiZChECCGe0ZXwI1wY+DrV9mSw+Zont7qsJlCKClHMSWEhhBDP4MLJ3Vwd3J8yMZDgCM6vjKRK3eaWDksIi5OhECGEyKN/D27m/n/G4HMfHjiDMv5DmnUcZOmwhCgQpMdCCFFs6fQKRyLjADgSGYdOrzz1mON//0bCyDF43Yd7rmAzbRyNpKgQwkB6LIQQxVJoWDQT/gwnLimV6SEwaNlR3J0dGNcpkHZBvtkec+7oTtI/GodHEsSUhJIzZ1CrUQczRy5EwSY9FkKIYic0LJqhK04QHZ9mtP12fBpDV5wgNCw6yzERZw7hvbk/iUFp3PYAn3nzpagQIhtSWAghihWdXmHCn+FkN+jxcNuEP8ONhkXOH9tJqd9fw50EKlYsSbWVG6hap7kZohWi8JHCQghRrByJjMvSU/EoBYiOTzPMvdi64DOiPn4PRZvKeevqeA7fjm/5qmaKVojCR+ZYCCGKlZjEnIuKx/f789uR+C3chp1WxV+Xy/LKwlCcXEqYNkAhCjkpLIQQxYqXi32u9nuw4QvqrPkHGx1c9rfi5e+kqBAiN2QoRAhRrIQEuOPrZo8qh9dVwJD4hdT9LbOouFDJmmardlHSs4w5wxSi0JLCQghRrKitVIzrFAiQpbhQAUNiv6PL7gtY6+FcNVva/noA15JeZo9TiMJKCgshRLHTLsiX+W/WxcfNeFhkoM0Wmp2+ipUCZ4Ps6bDqHxycXC0UpRCFk8yxEEIUS+2CfGkd6MOhyzHEnjvE7HIHaR+9gsst7DgeE8BrC/ZiY2tn6TCFKHSkx0IIUWyprVS84OdKyvFltI/+AYC7lXrSY/E/UlQI8Yykx0IIUWzptFo2vNOCwOPxHH3ZGW2dN2nQfwoqK/mbS4hnJT89Qohi6f7dm2zs/gLBR+Kx1cFlpQYNB06TokKI5yQ/QUKIYufsoVCOdW9N9fMZ6FTwT8vSdJ+12dJhCVEkyFCIEKJY2b5kIq7fr6JsMiQ6wJ23OuLh95KlwxKiyJAeCyFEsbH5+5H4zFiFWzJElwKbWV/R7p3Jlg5LiCJFeiyEEEWeVpPBsYXDeOXub2yp6I1Wb0ejnzbgVaYiGo3G0uEJUaRIYSGEKNKuXTjB7d9G0ED3L6jAtcMrNBw4XS4nFcJEpLAQQhRZ+zcuQD9lNknuehLr2XGp0XSathtg6bCEKNKksBBCFEnrpw6h/Mp9OGSAld6KC82+p97LPSwdlhBFnhQWQogiRafV8tt7Lam19y5WQGRZK6rNXUKFwBBLhyZEsSCFhRCiyIiNjmTfO12ofTFzQuaZ2o50WLgTJxc3C0cmRPEhl5sKIYqE29cucWhgB6pd1KC1gtMdK9Fj9XEpKoQwMykshBCFXvihUGwXt6R84H3iXODGyNfoNeNPS4clRLEkQyFCiEItdP4YXr69GBuVjvslK+D6y1LKVwm2dFhCFFtSWAghCqXkxHi2vN2SymdTuNzOhhSfJlR/9/9wdJahDyEsSYZChBCFTsSZg+zp2pCg0ynYaOF0Ri3qjl4vRYUQBYAUFkKIQmXP2u+IfustAm4opNrCxYFN6TV3h9zuXIgCQoZChBCFxu+T+lPxtyPYaeBuCVB/OoqunYdYOiwhxCOksBBCFHh6nY5147tRY80FAK6Us6Lmd8spV7WuhSMTQjxOCgshRIGWnPiAiz++wauq/ewo70NiKVc6LfgbBydXS4cmhMiGFBZCiALr5J4NOO78nDqq62SorHF7ewivvP4fS4clhHgCKSyEEAXSph8+xHvhJi5V0FKyZgniOv5MoxdbWTosIcRTSGEhhChQdFota8Z0JGjbVdQKPLhnQ3KvlVSr/qKlQxNC5IIUFkKIAiMpPo7Qt14mOCwNgHPVbGmxcAslPctYODIhRG7Jhd9CiALhwqm97O/6EjXC0tADp1v60mXtcSkqhChkpLAQQljchRO7uTn8HcrfUkixg8uDW9Jr3k7U1tKpKkRhI4WFEMKijm1eSLmNr+NcN5HbHqCZ+jFd/vODpcMSQjwj+XNACGERmox0ts8eRIfUTaAC+9LBVPhsGaW8ZehDiMJMCgshhNlFXz3P0Xe7UzpaR1RbO24HdOfFt+fK0IcQRYAMhQghzOrI9hWc792VypE6bLRw3PkVGrw7T4oKIYoI+UkWQpjNH3NGUHrJdnzS4b4zpIwcQLc3P7J0WEKIfCSFhRDC5HRaLWtGvkLNv29gpcA1XxXlZ3xPoxdaWjo0IUQ+k8JCCGFSaanJbBzThuC/4wA4W8OONou241rSy8KRCSFMQeZYCCFMJvbWVa7ObMGr7mFcL61wurUfXX89JkWFEEWY9FgIIUxiz5q5VD4zm6pWccSrnfD+4mvatOhm6bCEECYmhYUQIt/99tnrVN0QxslaVmRU98O6z2qCKwVZOiwhhBlIYSGEyDfpqSlsGNyCWscSANDFO+A27C9KevhYODIhhLnIHAshRL6IOneM7V1fNBQVpxuW5JUNx6WoEKKYkR4LIcRzWzdlMKXX7KdSCqRbw+XuL9Br/ApLhyWEsAApLIQQzyw+7i4nF7yF/6pLOGTAbQ+I7PMG1dsNRadXUFupLB2iEMLMpLAQQjyT07vW4LtnLM2J4+/6JTmd5MlMv5EkXS8BCw/h62bPuE6BtAvytXSoQggzksJCCJEnt6LOcXBMH8qVjsHLPYmrlOaHUkM44VHFaL/b8WkMXXGC+W/WleJCiGJEJm8KIXLtz7mjiXz9NWqEpaH5x5X9Ht3oZzODE0qVLPsq//3vhD/D0emVLK8LIYomKSyEEE8VczOCNT3rUmneVtwTIdYNEga/hrr9N1xNzPk4BYiOT+NIZJzZYhVCWJYMhQghnmjzvI9xW7qRoMyrSDlT25GWs9dSyjeAjadu5qqNmMQ0E0YohChIpLAQQmQrNTmRHTPfpPLKiwDcc4UH/TrQY/gMwz5eLva5aiu3+wkhCj8pLIQQWZw/sgOnrR/QWbnF5srepNo703jGSnzLVzPaLyTAHV83e27Hp5HdLAoV4ONmT0iAu1niFkJYnsyxEEIYPLgXzeoBDSm5vhd+yi1icKf0iAl0X3MiS1EBoLZSMa5TIJBZRDzq4fNxnQJlPQshihEpLIQQAPz1f5M51aklwYcecOzfUhwt8Qp2I45Sp1XvJx7XLsiX+W/WxcfNeLjDx81eLjUVohjK01DI/PnzmT9/PlFRUQDUqFGDL7/8kldeecUUsQkhzCDhfgxbRnWh5uEHWCkQ7wQZTZvz4sifct1GuyBfWgf6cCQyjpjENLxcMoc/pKdCiOInT4VF2bJlmTp1KpUqVQJg2bJldOnShZMnT1KjRg2TBCiEMJ1dq2ehfL+Q4NjM5+eq2VJ3ys80qF4vz22prVQ0rOiRzxEKIQqbPBUWnTp1Mnr+9ddfM3/+fA4dOiSFhRCFiCYjnQ3jX6fahktY6yHBEW50DaHbF8ssHZoQopB75qtCdDoda9asITk5mYYNG+a4X3p6Ounp6YbnCQmZF8NrNBo0Gs2zvn0WD9vKzzZFVpJn8zFVrqPCj6L6YxhtraI44+zDXS9rqn41j841GhTLf1f5TpuH5Nk8TJnn3LapUhQlT2vtnjlzhoYNG5KWloazszMrV66kffv2Oe4/fvx4JkyYkGX7ypUrcXR0zMtbCyGegyYjlaQ9C+jlfgp7Kx0PFGf+dOqKc8WWWFmrLR2eEKKAS0lJoU+fPsTHx+Pq6prjfnkuLDIyMrh27RoPHjzg999/Z9GiRezZs4fAwMBs98+ux8LPz4/Y2NgnBpZXGo2GHTt20Lp1a2xsbPKtXWFM8mw++ZnrQ5uXkjp7Nn53FG6+lIJH5er49pmHh49fPkVbeMl32jwkz+ZhyjwnJCRQqlSppxYWeR4KsbW1NUzerFevHkePHuXbb7/lp5+yn0FuZ2eHnZ1dlu02NjYm+XKZql1hTPJsPs+T6/TUFNb/pxPV99yilA5S7CDOtwEvf7gKlZVcbf4o+U6bh+TZPEyR59y299wrbyqKYtQjIYQoGI5sX0Hc1MkE38rslLzsb0WFCbPoUb+thSMTQhRleSosPv30U1555RX8/PxITExk9erV7N69m9DQUFPFJ4TII71Ox5rPXqPqpouU10KqLVxqV5Xuk9eitpZV/IUQppWn3zJ37tyhb9++REdH4+bmRq1atQgNDaV169amik8IkQc3r5zjwerBVM+IQK0rwZVyVpT5YjI9m3SxdGhCiGIiT4XFzz//bKo4hBDPQavJYOsPI3n5/lrKqNJJcbFj+6DGdBw5D2sbW0uHJ4QoRqRfVIgCSKdXOBIZB8CRyDgaVPLKcXnsfw/8yfUJH1Pupp5bHSC9RE1K9FrIqxWqmzNkIYQApLAQosAJDYtmwp/hxCWlMj0EBi07iruzA+M6BRrd0Eun1bL20+5UDr1AhQxIt4bj1o15/aO1WKllXQohhGXI9WZCFCChYdEMXXGC6Pg0o+2349MYuuIEoWHRAJw9vJ0tHYOp9ccFHDLgmq+K9Bmf0HPyeikqhBAWJT0WQhQQOr3ChD/DyW7FOgVQAeM3hpGwegQV/jxDpXTIUMO5ZqXpOvNP7BxkJVshhOVJYSFEAXEkMi5LT8WjSvGAiWk/o7sWhVO6Hde9VTj/ZxS9Og82Y5RCCPFkUlgIUUDEJGZfVCh6HV30OxnvsJaSqiSSy6v50/1Funy9Bgen/FsWXwgh8oMUFkIUEF4u9lm2Jd29wpQzP+NxX4tjyyTOqspzrdlMerWStWOEEAWTFBZCFBAhAe74utlzOz4NBeh172cqbT2PawroVCrm3a/PWt/R7G3ZytKhCiFEjqSwEKKAUFupGNcpkFULp9Ht/A4qX9UDEO0By19ozW7HtszvEpzjehZCCFEQSGEhRAERee44KZ8P4uPzGQBoreB0HWe+8RuNo5sX8x9bx0IIIQoiKSyEsLDY29eIWDuOunc3cj7NE7DiXFUbdL2G4+LozbzqDZ648qYQQhQkUlgIYSE3I8PZN/FtGvhEUt8uBVRg84Intwe8xmu9RqPRaNiyZQshAe5SVAghCg0pLIQws/t3b7JtXD8qHrxFcBqEBTmTXrsc2hZf0qpxB0uHJ4QQz0UKCyHMJDU5gT/G98Vv10WCkzK33fYATb3mVBn7EyorWWFfCFH4SWEhhIkpej3rpryFx+ZD1Mq8YSlxrnDj5UBe/XK5LMUthChSpLAQwoTC9m3Ebs8k7M48wDvOhkQHuPKSH+3GL6Oxh1zhIYQoeqSwEMIE/v5lOvqwdbS2OweAQ0UH9jtXotmXiwgpX83C0QkhhOlIYSFEPjq0dRnR82ZS7ZKGq3560hqqOeX9GpW7j6eXd1lLhyeEECYnhYUQ+eDs4e2cm/ER1cLSqKaAHkh2tie693YaVK9n6fCEEMJspLAQ4jlcu3CCf74aSrWTCdTQZm67VEFNybeH0u21YZYNTgghLEAKCyGeQUpSPKd/n0b8/l+pddQWgKtlVNC7O53fnmjh6IQQwnKksBAiD5IT49n7f1/y4t0NNOQBGb4QWrEM+uZN6TTqe9TW8iMlhCje5LegELmgyUhnw9cD8d52EhcFXNs84Ja1N7fqjabDl29LQSGEEP8lvw2FeAKdVsvmuaNw3PAXQTGZ2xIcYbt7H9q8N5vSdvaWDVAIIQoYKSyEyMH2pZPQrlhF5RsKAKm2cLG+Jy3HL6F+mYoWjk4IIQomKSyEeMzVc8c5v2I05X7P7KLIUMP52i68+Nn39AoMsXB0QghRsElhIcR/RZ47Tuy2b6h7P5TyNgpb/H1IcXKgyuhJ9GzcydLhCSFEoSCFhSj2roQf4ejXwwgIT8K/XQxqO4WTTi9RdcY4KgZJD4UQQuSFFBai2Lpz4zK7xg+kypFYamVkbtt7pyw135tNnXotLRucEEIUUlJYiGIn4X4Mm8f3pcK+awSnZG676QUpr7Wh6/CZcumoEEI8B/kNKooNnVbLkY3fkz7zJ2rHZW67WwJi2tWly6eLsbG1s2h8QghRFEhhIYo8nVbLmd1rcDs4hYb6q2wq64ldug3Xmlek4/hfcHJxs3SIQghRZEhhIYqsiDMHOfLTl/icvIlX/QcEuKaQgCOOL3ei6nejaejtZ+kQhRCiyJHCQhQpmox0ts4bi+7vXVSK0FBbn7k9MtyVhK6vEdj9S1p6eFs2SCGEKMKksBBFwtWLp/ln+nDK/htL5YT/bb/hrSKuXgVeGj6dMgGBlgtQCCGKCSksRKGVlppC+J7fsD61gsCUY5Q+64NHgopkO7hS3Qmvrm/SuudIS4cphBDFihQWotA5uHkJ11f9iE9EAi+2isZRrYAV3HvBkTslKtNqxCzqeZYBQKdXOBIZR0xiGl4u9oQEuKO2Uln4EwghRNElhYUoFGKjI/l77hjcjpyj/E2Fkv/dvu+uJ651X8Gv5RC6VgoyOiY0LJoJf4YTHZ9m2ObrZs+4ToG0C/I1Y/RCCFF8SGEhCixFr+fQlqXc+L/vqXA+1bA6pk4FVwLU6Jo2ot37M3Bwcs1ybGhYNENXnEB5bPvt+DSGrjjB/DfrSnEhhBAmIIWFKHDu343mwl9L8Lz8Gz6JtyjxrweQuZjVrWBvar39GZ1fbJ3j8Tq9woQ/w7MUFQAKoAIm/BlO60AfGRYRQoh8JoWFKBA0GelsW/g5GTu2Y6tPo2Pd2wCkO9iwoYETTi+2oO3gr3K1OuaRyDij4Y/HKUB0fBpHIuNoWNEjvz6CEEIIpLAQFnbu6F+cXvQVpU/foeKDzG1aKytO1SpHWo3eVG/zFj3zuO5ETGLORcWz7CeEECL3pLAQZqfVZLB13liUbX9RIVJH8H/HLFJtIaKaA+5detG85+hnvhmYl4t9vu4nhBAi96SwEGZz7eK/3Ny9iEq3/sDuqgq/K5kn9qtlVMS/WI2XR8ykrm/Ac79PSIA7vm723I5Py3aehQrwccu89FQIIUT+ksJCmNT9u7f469tROB0Kw75CEi1LPwCgUmk3jtZzwa/Xu7TrOChf31NtpWJcp0CGrjiBCoyKi4dTNcd1CpSJm0IIYQJSWAiT2L3mO2J+X0aFc8kEpWduu6534N8KldHWfpOgFr2oYGe6oYh2Qb7Mf7NulnUsfGQdCyGEMCkpLES+SYyPY8uk/rgfjaDsHYWHUy7jXOF6rVIEDhxLrcadzBZPuyBfWgf6yMqbQghhRlJYiOei6PVcOPoXif8spsb9nbj9W5Kyd6zQWsHlijZYtWxOu3en0tjB0SLxqa1UckmpEEKYkRQW4plEhB3k1KKJeJ2+RfUmd6lmpwEVqKpbc6qKLyHvTKRrzUaWDlMIIYSZSWEhcu3s4W2E/fEztqfD0VzREazP3H78Rgm8XngR10aDaFPvZVRWVpYNVAghhMVIYSFydPv6Za6f2E7s6S04bYnA8wHUeuT16z4q7terRNP3p+NbvpqlwhRCCFGASGEhDE7t28jFLf+H1cUIrJ3TeTXgFj5Aik7F5URf9Cq45aUipowjZbr2pc3rIywdshBCiAJGCotiSqfV8u++jVzc9gvWF6/gfSMdjwSo+d/Xoz0VtP5WXLGpxD3fEO4NcyK4XX8eaEvhfO4QNtUboNMrcoWFEEIII1JYFBM6rZbzx/4i+eZ5VNcOUObBCVK32FIr8X/7aK3glreK++XccKj9ImlvTaCKa0kg8zbkPX4NJy4pkukhMGjZUdydHWRNCCGEEEaksCiidFotx/5eTdSO37C5fBWfGxnYaOGFLrdQqwAVHC/rTUK8NQ/KueFYO4QGr39AzTIVs7QVGhbN0BUnUAA79f+2345PY+iKE8x/s64UF0IIIQApLIoMRa/n2qV/ObJqKtYnw/G5oaFEsvFkS60VHNBUwabCSzhXaU6L0U1x/m+PRE50eoUJf4Zne88Nhcwlsif8GU7rQB8ZFhFCCFE0CgudXuFIZBwARyLjaFDJq8Cf5B7G/KwrQuq0Wo5sXcrVXesp66lQLf0s5XlA2MVSVLhgC4BGDTd9VMSXL4lz3YY0en0ENb398hTnkcg4oyWxH6cA0fFpHImMk4WohBBCFP7CIjQsmgl/hhOXlFpoxv4fxvzoCdv3Kfew0Go0HNqymBu7NmAXcYPSN7WUSIESwINWiZQqlUi6YoN1WXdOO9jh8kJjGr3+AbU8yzxXrDGJORcVz7KfEEKIoq1QFxaFcez/0Zgf9XjMep2OqHPHiDnzN7ePbab0zjg8UuHRPoF0a7jpa0WqewjhbQdToXZT2jo45Wu8Xi65u1FYbvcTQghRtBXawqIwjv0/KWaVkkGTtD1cnvkNG+7ew750Cu1Kx1IBiFDbkZHqQbpNZiGR4O9ByReb0aj7+9Qu6WXSmEMC3PF1s+d2fFr2cZN5x9CQAHeTxiGEEKJwKLSFRWEc+8+MORVXUvDmLsEJBymXfB2/mDj8onU4PfJxrqAmxdeOyw5BJPmHkFjNgcbd3qe2m3lP4GorFeM6BTJ0xQkeL88ePh/XKbDAFG9CCCEsq9AWFgVx7D8jPY1LJ3dz68IJEm9cIj0mGqv7D7BJTMU+SUt6CT3hgfdwVKWjVyBsT2lsdP87PtUWbvhacd+vFKWbv4JNj1HUsrUzW/w5aRfky/w36xrmsjzk85R5IUIIIYqfQltYmHvs//a1S1w5s5+4K2GkRF9Ffy8Wq/hEcIDaFa0ooY2lpP4+2t99KavLvo2bGgVHVToACThxrSxordTc8HTjdIla7LZ/GY2VA6sGNygwvSwPtQvypXWgD4cuxxB77hCL+79YKK6+EUIIYV6FtrDIr7H/jLRUroQf4ua5o8Rfu0hGzC1UVloqlC2JfdodnDPuErNRj2sKlCTz8aibPgpVykdnPrGCB64K1loVSc4qUp3UZLjYoS/hitrDE2f/qnSNrsS5ZGfSsIM6WWP2LcDzFdRWKkIC3NlyjjxfHiuEEKJ4KLSFRW7G/j9q6s6Jv1YTE3EabXI83l6lIDEa25Q73N4egdsDBddksNZDaTIfADe9FV50OWxo76ZN5iuptpDgDMnOVqQ526BxdUDl483JRp/j7FmOkr7+NPrUBxtb2xzjVv33qhAVGBVEMl9BCCFEUVBoCwv439j/+I1h9E5dSdqKk3wTr8E5WY9rMjhtyNzPmcxioUGLaMOx+xJ9cU/MPIHrgQQnSHJWkeJsRVopJ/4JeA1rtzI4eJTFvmk6JSvVoXq5yvkW8+PrWMh8BSGEEEVBoS4s4H9j//cnvsW5646UemD81366DcQ7Q5KrNcdcW6Fx8kHlWpr7JW6TWsKb0tVeoGLNxjg4uZo95udZeVMIIYQoiAp9YQGZwyIXy/Umou5hbjmXwN63HB4ValA+qDFVyldDbV3wPqbaSlXgJmgKIYQQz6vgnXGf0Yt9v+Kuxxbat2+PjY2NpcMRQgghiiWrvOw8ZcoUXnzxRVxcXPDy8uLVV1/lwoULpopNCCGEEIVMngqLPXv2MGzYMA4dOsSOHTvQarW0adOG5ORkU8UnhBBCiEIkT0MhoaGhRs+XLFmCl5cXx48fp2nTpvkamBBCCCEKn+eaYxEfHw+Au3vOCzqlp6eTnp5ueJ6QkACARqNBo9E8z9sbedhWfrYpspI8m4/k2jwkz+YheTYPU+Y5t22qFEXJbuHKp1IUhS5dunD//n327duX437jx49nwoQJWbavXLkSR0fHZ3lrIYQQQphZSkoKffr0IT4+HlfXnJdoeObCYtiwYWzevJn9+/dTtmzZHPfLrsfCz8+P2NjYJwaWVxqNhh07dtC6dWu5KsSEJM/mI7k2D8mzeUiezcOUeU5ISKBUqVJPLSyeaSjk/fff548//mDv3r1PLCoA7OzssLPLeodOGxsbk3y5TNWuMCZ5Nh/JtXlIns1D8mwepshzbtvLU2GhKArvv/8+69evZ/fu3QQEBDxTcEIIIYQomvJUWAwbNoyVK1eyceNGXFxcuH37NgBubm44ODiYJEAhhBBCFB55Wsdi/vz5xMfH07x5c3x9fQ2PX3/91VTxCSGEEKIQyfNQiBBCCCFETvLUYyGEEEII8SRSWAghhBAi35j97qYPh1MersCZXzQaDSkpKSQkJMilTCYkeTYfybV5SJ7NQ/JsHqbM88Pz9tOmRZi9sEhMTATAz8/P3G8thBBCiOeUmJiIm5tbjq8/88qbz0qv13Pr1i1cXFxQqVT51u7DFT2vX7+eryt6CmOSZ/ORXJuH5Nk8JM/mYco8K4pCYmIipUuXxsoq55kUZu+xsLKyeupqnc/D1dVVvrRmIHk2H8m1eUiezUPybB6myvOTeioeksmbQgghhMg3UlgIIYQQIt8UmcLCzs6OcePGZXvDM5F/JM/mI7k2D8mzeUiezaMg5NnskzeFEEIIUXQVmR4LIYQQQlieFBZCCCGEyDdSWAghhBAi30hhIYQQQoh8U6gKi3nz5hEQEIC9vT0vvPAC+/bte+L+e/bs4YUXXsDe3p4KFSrw448/minSwi0veV63bh2tW7fG09MTV1dXGjZsyLZt28wYbeGV1+/zQwcOHMDa2pratWubNsAiJK+5Tk9P57PPPqN8+fLY2dlRsWJFFi9ebKZoC6+85vmXX34hODgYR0dHfH19GThwIPfu3TNTtIXT3r176dSpE6VLl0alUrFhw4anHmP2c6FSSKxevVqxsbFRFi5cqISHhysjRoxQnJyclKtXr2a7/5UrVxRHR0dlxIgRSnh4uLJw4ULFxsZGWbt2rZkjL1zymucRI0Yo06ZNU44cOaJcvHhR+eSTTxQbGxvlxIkTZo68cMlrnh968OCBUqFCBaVNmzZKcHCweYIt5J4l1507d1bq16+v7NixQ4mMjFQOHz6sHDhwwIxRFz55zfO+ffsUKysr5dtvv1WuXLmi7Nu3T6lRo4by6quvmjnywmXLli3KZ599pvz+++8KoKxfv/6J+1viXFhoCouQkBDl3XffNdpWrVo15eOPP852/7FjxyrVqlUz2vbOO+8oDRo0MFmMRUFe85ydwMBAZcKECfkdWpHyrHnu2bOn8vnnnyvjxo2TwiKX8prrrVu3Km5ubsq9e/fMEV6Rkdc8f/PNN0qFChWMts2dO1cpW7asyWIsanJTWFjiXFgohkIyMjI4fvw4bdq0Mdrepk0bDh48mO0x//zzT5b927Zty7Fjx9BoNCaLtTB7ljw/Tq/Xk5iYiLu7uylCLBKeNc9LliwhIiKCcePGmTrEIuNZcv3HH39Qr149pk+fTpkyZahSpQpjxowhNTXVHCEXSs+S50aNGnHjxg22bNmCoijcuXOHtWvX0qFDB3OEXGxY4lxo9puQPYvY2Fh0Oh3e3t5G2729vbl9+3a2x9y+fTvb/bVaLbGxsfj6+pos3sLqWfL8uJkzZ5KcnEyPHj1MEWKR8Cx5vnTpEh9//DH79u3D2rpQ/NgWCM+S6ytXrrB//37s7e1Zv349sbGxvPfee8TFxck8ixw8S54bNWrEL7/8Qs+ePUlLS0Or1dK5c2e+++47c4RcbFjiXFgoeiweevw264qiPPHW69ntn912YSyveX5o1apVjB8/nl9//RUvLy9ThVdk5DbPOp2OPn36MGHCBKpUqWKu8IqUvHyn9Xo9KpWKX375hZCQENq3b8+sWbNYunSp9Fo8RV7yHB4ezgcffMCXX37J8ePHCQ0NJTIyknfffdccoRYr5j4XFoo/fUqVKoVarc5S+cbExGSpxB7y8fHJdn9ra2s8PDxMFmth9ix5fujXX3/lrbfeYs2aNbRq1cqUYRZ6ec1zYmIix44d4+TJkwwfPhzIPPkpioK1tTXbt2+nZcuWZom9sHmW77Svry9lypQxuj109erVURSFGzduULlyZZPGXBg9S56nTJlC48aN+fDDDwGoVasWTk5ONGnShK+++kp6lfOJJc6FhaLHwtbWlhdeeIEdO3YYbd+xYweNGjXK9piGDRtm2X/79u3Uq1cPGxsbk8VamD1LniGzp2LAgAGsXLlSxkdzIa95dnV15cyZM5w6dcrwePfdd6latSqnTp2ifv365gq90HmW73Tjxo25desWSUlJhm0XL17EysqKsmXLmjTewupZ8pySkoKVlfEpSK1WA//7i1o8P4ucC002LTSfPbyU6eeff1bCw8OVkSNHKk5OTkpUVJSiKIry8ccfK3379jXs//ASm1GjRinh4eHKzz//LJeb5kJe87xy5UrF2tpa+eGHH5To6GjD48GDB5b6CIVCXvP8OLkqJPfymuvExESlbNmySvfu3ZWzZ88qe/bsUSpXrqy8/fbblvoIhUJe87xkyRLF2tpamTdvnhIREaHs379fqVevnhISEmKpj1AoJCYmKidPnlROnjypAMqsWbOUkydPGi7rLQjnwkJTWCiKovzwww9K+fLlFVtbW6Vu3brKnj17DK/1799fadasmdH+u3fvVurUqaPY2toq/v7+yvz5880cceGUlzw3a9ZMAbI8+vfvb/7AC5m8fp8fJYVF3uQ11+fOnVNatWqlODg4KGXLllVGjx6tpKSkmDnqwieveZ47d64SGBioODg4KL6+vsobb7yh3Lhxw8xRFy67du164u/cgnAulNumCyGEECLfFIo5FkIIIYQoHKSwEEIIIUS+kcJCCCGEEPlGCgshhBBC5BspLIQQQgiRb6SwEEIIIUS+kcJCCCGEEPlGCgshhBBC5BspLIQQQgiRb6SwEEIIIUS+kcJCCCGEEPlGCgshhBBC5Jv/B0LFhVYonimoAAAAAElFTkSuQmCC",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "execution_count": 2,
+   "id": "7851af68",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
    "source": [
     "%matplotlib inline\n",
     "\n",
@@ -565,8 +613,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8b52bb8a",
-   "metadata": {},
+   "id": "96995260",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The intercept is the value of our output/target variable\n",
     "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
@@ -584,8 +634,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68ac9f59",
-   "metadata": {},
+   "id": "e02411ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
@@ -594,16 +646,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6d9edc5",
-   "metadata": {},
+   "id": "dd2082ec",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "but when we take out the intercept, this equation becomes"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "95b2e91e",
-   "metadata": {},
+   "id": "701e4d68",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
@@ -612,16 +668,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e156d9fe",
-   "metadata": {},
+   "id": "9d89244c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For Lasso regression we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "85a5a3f6",
-   "metadata": {},
+   "id": "596c0314",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
@@ -630,8 +690,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "229077e1",
-   "metadata": {},
+   "id": "daf1b75f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It means that, when scaling the design matrix and the outputs/targets,\n",
     "by subtracting the mean values, we have an optimization problem which\n",
@@ -645,115 +707,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
-   "id": "7ed86f8e",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "Beta values for own Ridge implementation\n",
-      "[ 1.03032441e+00  6.28336218e-02 -6.24175744e-01  5.21169159e-02\n",
-      "  2.80847477e-01  2.12552073e-01  8.13220608e-02 -1.69634577e-02\n",
-      " -6.50846111e-02 -7.38962192e-02 -5.94226022e-02 -3.50227564e-02\n",
-      " -9.80609616e-03  1.08299273e-02  2.41882037e-02  2.93492130e-02\n",
-      "  2.64742912e-02  1.63249532e-02 -5.01831050e-05 -2.15098090e-02]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 1.03032441e+00  6.28336218e-02 -6.24175744e-01  5.21169159e-02\n",
-      "  2.80847477e-01  2.12552073e-01  8.13220608e-02 -1.69634577e-02\n",
-      " -6.50846112e-02 -7.38962192e-02 -5.94226022e-02 -3.50227564e-02\n",
-      " -9.80609615e-03  1.08299273e-02  2.41882037e-02  2.93492130e-02\n",
-      "  2.64742912e-02  1.63249532e-02 -5.01831190e-05 -2.15098090e-02]\n",
-      "MSE values for own Ridge implementation\n",
-      "4.3632959111950474e-07\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "4.363295916366933e-07\n",
-      "Beta values for own Ridge implementation\n",
-      "[ 1.03630548 -0.01963611 -0.37900111 -0.07062318  0.12182967  0.16343471\n",
-      "  0.13003291  0.07490892  0.02365049 -0.01449782 -0.03814292 -0.04909093\n",
-      " -0.05009826 -0.04389027 -0.03279636 -0.01866537 -0.00289724  0.01348565\n",
-      "  0.02976145  0.04543942]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 1.03630548 -0.01963611 -0.37900111 -0.07062318  0.12182967  0.16343471\n",
-      "  0.13003291  0.07490892  0.02365049 -0.01449782 -0.03814292 -0.04909093\n",
-      " -0.05009826 -0.04389027 -0.03279636 -0.01866537 -0.00289724  0.01348565\n",
-      "  0.02976145  0.04543942]\n",
-      "MSE values for own Ridge implementation\n",
-      "5.194042826649355e-06\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "5.194042826815211e-06\n",
-      "Beta values for own Ridge implementation\n",
-      "[ 1.04220758 -0.10931453 -0.17641709 -0.06020587  0.02208512  0.05789007\n",
-      "  0.06491736  0.05785343  0.04537385  0.03196357  0.01969145  0.00934499\n",
-      "  0.00107405 -0.00526348 -0.00992331 -0.01318643 -0.01531845 -0.01655318\n",
-      " -0.01708852 -0.01708781]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 1.04220758 -0.10931453 -0.17641709 -0.06020587  0.02208512  0.05789007\n",
-      "  0.06491736  0.05785343  0.04537385  0.03196357  0.01969145  0.00934499\n",
-      "  0.00107405 -0.00526348 -0.00992331 -0.01318643 -0.01531845 -0.01655318\n",
-      " -0.01708852 -0.01708781]\n",
-      "MSE values for own Ridge implementation\n",
-      "2.0940821989652176e-05\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "2.0940821989627646e-05\n",
-      "Beta values for own Ridge implementation\n",
-      "[ 1.01219292 -0.06043581 -0.10391807 -0.05651951 -0.01898855  0.00312361\n",
-      "  0.01463049  0.01975848  0.02123176  0.02068067  0.01905883  0.01691985\n",
-      "  0.01458337  0.01223198  0.00996754  0.00784393  0.00588657  0.00410387\n",
-      "  0.00249435  0.00105081]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 1.01219292 -0.06043581 -0.10391807 -0.05651951 -0.01898855  0.00312361\n",
-      "  0.01463049  0.01975848  0.02123176  0.02068067  0.01905883  0.01691985\n",
-      "  0.01458337  0.01223198  0.00996754  0.00784393  0.00588657  0.00410387\n",
-      "  0.00249435  0.00105081]\n",
-      "MSE values for own Ridge implementation\n",
-      "0.00031535148309577417\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "0.0003153514830958095\n",
-      "Beta values for own Ridge implementation\n",
-      "[ 8.38916861e-01  1.31276579e-01  8.97497404e-03 -1.72271878e-02\n",
-      " -2.11744554e-02 -1.91492986e-02 -1.57201944e-02 -1.23002365e-02\n",
-      " -9.30466214e-03 -6.81048318e-03 -4.78184120e-03 -3.15130074e-03\n",
-      " -1.84923989e-03 -8.13661243e-04  7.46984697e-06  6.56636616e-04\n",
-      "  1.16805821e-03  1.56912044e-03  1.88168312e-03  2.12318726e-03]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 8.38916861e-01  1.31276579e-01  8.97497404e-03 -1.72271878e-02\n",
-      " -2.11744554e-02 -1.91492986e-02 -1.57201944e-02 -1.23002365e-02\n",
-      " -9.30466214e-03 -6.81048318e-03 -4.78184120e-03 -3.15130074e-03\n",
-      " -1.84923989e-03 -8.13661243e-04  7.46984697e-06  6.56636616e-04\n",
-      "  1.16805821e-03  1.56912044e-03  1.88168312e-03  2.12318726e-03]\n",
-      "MSE values for own Ridge implementation\n",
-      "0.01507238889517717\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "0.01507238889517706\n",
-      "Beta values for own Ridge implementation\n",
-      "[0.37396662 0.14174745 0.0764924  0.04892055 0.03447512 0.02586427\n",
-      " 0.02024962 0.01633913 0.01347916 0.0113104  0.0096208  0.00827728\n",
-      " 0.00719176 0.00630331 0.00556826 0.0049544  0.00443743 0.0039987\n",
-      " 0.0036237  0.003301  ]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[0.37396662 0.14174745 0.0764924  0.04892055 0.03447512 0.02586427\n",
-      " 0.02024962 0.01633913 0.01347916 0.0113104  0.0096208  0.00827728\n",
-      " 0.00719176 0.00630331 0.00556826 0.0049544  0.00443743 0.0039987\n",
-      " 0.0036237  0.003301  ]\n",
-      "MSE values for own Ridge implementation\n",
-      "0.2640931530791004\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "0.26409315307910025\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "execution_count": 3,
+   "id": "94807025",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
    "source": [
     "import numpy as np\n",
     "import pandas as pd\n",
@@ -824,8 +784,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "30c962cb",
-   "metadata": {},
+   "id": "1c95fe0b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
     "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
@@ -835,137 +797,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
-   "id": "a0ae3089",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "Beta values for own Ridge implementation\n",
-      "[ 3.43579948e-02 -5.43330971e-01 -3.10141413e-03  2.47116868e-01\n",
-      "  2.18613217e-01  1.02054837e-01 -4.25617658e-04 -5.90475506e-02\n",
-      " -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02\n",
-      "  1.11482289e-02  2.88529063e-02  3.67047975e-02  3.38135733e-02\n",
-      "  2.02198703e-02 -3.46383926e-03 -3.63025821e-02]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[ 3.43579948e-02 -5.43330971e-01 -3.10141413e-03  2.47116868e-01\n",
-      "  2.18613217e-01  1.02054837e-01 -4.25617655e-04 -5.90475506e-02\n",
-      " -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02\n",
-      "  1.11482289e-02  2.88529063e-02  3.67047975e-02  3.38135733e-02\n",
-      "  2.02198702e-02 -3.46383925e-03 -3.63025821e-02]\n",
-      "Intercept from own implementation:\n",
-      "1.0330308045187757\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "1.0330308045183194\n",
-      "MSE values for own Ridge implementation\n",
-      "3.139255958997547e-06\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "3.1392559585020426e-06\n",
-      "Beta values for own Ridge implementation\n",
-      "[-0.05807125 -0.29822833 -0.08551306  0.08156108  0.13679863  0.12333649\n",
-      "  0.08251519  0.03815288  0.00111756 -0.02498832 -0.04010697 -0.04566964\n",
-      " -0.04355837 -0.03562355 -0.02348765 -0.00848904  0.00831018  0.0260906\n",
-      "  0.04423486]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[-0.05807125 -0.29822833 -0.08551306  0.08156108  0.13679863  0.12333649\n",
-      "  0.08251519  0.03815288  0.00111756 -0.02498832 -0.04010697 -0.04566964\n",
-      " -0.04355837 -0.03562355 -0.02348765 -0.00848904  0.00831018  0.0260906\n",
-      "  0.04423486]\n",
-      "Intercept from own implementation:\n",
-      "1.0411487294305088\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "1.0411487294305226\n",
-      "MSE values for own Ridge implementation\n",
-      "1.9601304850035702e-05\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "1.9601304850073734e-05\n",
-      "Beta values for own Ridge implementation\n",
-      "[-0.1416398  -0.14021063 -0.05383795  0.01367553  0.04784395  0.05796251\n",
-      "  0.05447415  0.044613    0.03267527  0.02098261  0.01066519  0.00217499\n",
-      " -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081  -0.01416528\n",
-      " -0.01290947]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[-0.1416398  -0.14021063 -0.05383795  0.01367553  0.04784395  0.05796251\n",
-      "  0.05447415  0.044613    0.03267527  0.02098261  0.01066519  0.00217499\n",
-      " -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081  -0.01416528\n",
-      " -0.01290947]\n",
-      "Intercept from own implementation:\n",
-      "1.049556996627824\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "1.0495569966278269\n",
-      "MSE values for own Ridge implementation\n",
-      "5.4959161509357395e-05\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "5.4959161509366685e-05\n",
-      "Beta values for own Ridge implementation\n",
-      "[-0.13535942 -0.08593216 -0.03568439 -0.0036367   0.01397146  0.02229529\n",
-      "  0.02503753  0.0245528   0.02228115  0.01908936  0.01549377  0.01179792\n",
-      "  0.00817631  0.00472512  0.00149311 -0.00149956 -0.00424967 -0.00676387\n",
-      " -0.00905423]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[-0.13535942 -0.08593216 -0.03568439 -0.0036367   0.01397146  0.02229529\n",
-      "  0.02503753  0.0245528   0.02228115  0.01908936  0.01549377  0.01179792\n",
-      "  0.00817631  0.00472512  0.00149311 -0.00149956 -0.00424967 -0.00676387\n",
-      " -0.00905423]\n",
-      "Intercept from own implementation:\n",
-      "1.039967668952797\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "1.0399676689527975\n",
-      "MSE values for own Ridge implementation\n",
-      "7.571105947979344e-05\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "7.571105947979412e-05\n",
-      "Beta values for own Ridge implementation\n",
-      "[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706  -0.00517114\n",
-      " -0.00174276  0.00068734  0.00243186  0.00369758  0.00462287  0.0053018\n",
-      "  0.00579953  0.006162    0.00642221  0.00660427  0.00672607  0.0068011\n",
-      "  0.00683964]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706  -0.00517114\n",
-      " -0.00174276  0.00068734  0.00243186  0.00369758  0.00462287  0.0053018\n",
-      "  0.00579953  0.006162    0.00642221  0.00660427  0.00672607  0.0068011\n",
-      "  0.00683964]\n",
-      "Intercept from own implementation:\n",
-      "0.999955585168597\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "0.999955585168597\n",
-      "MSE values for own Ridge implementation\n",
-      "0.0007698473260556343\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "0.0007698473260556325\n",
-      "Beta values for own Ridge implementation\n",
-      "[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335\n",
-      " -0.00323332 -0.00274989 -0.0023548  -0.00202756 -0.00175331 -0.00152117\n",
-      " -0.001323   -0.0011526  -0.00100519 -0.00087697 -0.00076495 -0.00066668\n",
-      " -0.00058016]\n",
-      "Beta values for Scikit-Learn Ridge implementation\n",
-      "[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335\n",
-      " -0.00323332 -0.00274989 -0.0023548  -0.00202756 -0.00175331 -0.00152117\n",
-      " -0.001323   -0.0011526  -0.00100519 -0.00087697 -0.00076495 -0.00066668\n",
-      " -0.00058016]\n",
-      "Intercept from own implementation:\n",
-      "0.9637117593816477\n",
-      "Intercept from Scikit-Learn Ridge implementation\n",
-      "0.9637117593816477\n",
-      "MSE values for own Ridge implementation\n",
-      "0.0023813163025848865\n",
-      "MSE values for Scikit-Learn Ridge implementation\n",
-      "0.002381316302584885\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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Gtenp6UhJSTF6aLXaYvfbrFkz9O3bFyEhITh69Cji4+MxdOhQPPbYY0bb7dq1K77//nu0adMGtra2huZq48aNjzRvUmlee+01yGQyLF++vNjXQ0NDcfXqVUyYMAGXLl3C999/X+Q055gxY7B3714sWbIEV65cwVdffYV9+/YZjVJNnz4d69evx8yZM3HhwgUkJCRgy5Yt+PDDD03OPGnSJMTGxuKdd95BXFwcrly5gj179mDMmDEAgB9//BFffPEF4uLi8Oeff2L9+vXQ6/Vo3rw5gMI5qk6cOIHr168jLS3toaN4RETlpdcLRN7+BgAwwCPkIdVVg40UPRJbW1vDXEQPsra2hq+vLz799FN07twZrVu3xrRp0xASEoKlS5ca1U6fPh2urq5Gjw8++KDE/a5ZswY+Pj548cUX4efnByEE9u7da3R6rlu3bigoKDBqmrp06YKCgoIi10dVFKVSiXfffReLFi0yTK3wX+7u7oiIiMAPP/yAtm3bYuXKlZg3b55RzTPPPIOVK1diyZIlaNu2Lfbv34/x48cbnbILDAzEjz/+iMjISHTo0AEdO3bEkiVL4OHhYXLmNm3aIDo6GleuXEGnTp3Qrl07TJs2Da6urgAAe3t77NixA8899xxatmyJlStXYtOmTWjVqhUA4P3334dcLoeXlxcaNGiApKQkkzMQEZXFlz8cRp7tZSDXGguCX5E6DgBAJspy0QaVi1arhZ2dHTQaTZFmIycnB4mJifD09DT5mhaqe0JCQvD777/jyJEjUkcpF/6+E1FFmL1pH+aenIAm5p2QsGhVpe2ntM/vB/EaKaJqaPHixejevTusrKywb98+rFu3rsTThUREdcX0V3viw6AeSNNUn28Os5EiqoZ+++03LFq0COnp6WjSpAm++OILjBw5UupYRESSMzOTwcmhgr899QjYSBFVQ1u3bpU6AhFRtaHXC0xauwPhL/dCPVsLqeMY4cXmREREVK19vT8Wi2+8jAYfNUNWTp7UcYywkSIiIqJqbfGvhTOZNxHdYaku+X6mUmAjRURERNXWn//cwx/qLQCAic9LP5P5g9hIERERUbU1acP3gCIbKk0rjAzsKHWcIthIERERUbWk1wvs/qtwvqheLiEwM5M9ZI2qx0aKiIiIqqXvfjmFHPt4IF+Fj4cGSx2nWGykiEwQFRUFmUyGe/fulVizdu1a2NvbV1kmIqLaavupwwCAxlkvo2nDehKnKR4bKTLJ8OHDIZPJEBoaWuS10aNHQyaTYfjw4YZlqampGDVqFNzd3aFSqeDi4oLAwEDExsYaaho3bgyZTFbksWDBghJzXLt2Da+++ioaNmwItVqNRo0aoW/fvrh8+bKhRiaTYdeuXWV+b//NYWFhgRYtWuDjjz/Gf++i5O/vj+TkZNjZ2ZV5u5WhuOP138d//wxM1bhxY3z22WcVlpWIqLx+CH8PB1+6glWvzZA6Sok4ISeZzM3NDZs3b8ann34KC4vCidFycnKwadMmuLu7G9UOHDgQeXl5WLduHZo0aYJ//vkHv/zyC+7cuWNUN3v2bISEGN/J28bGptj95+bmonv37mjRogV27NgBV1dX/PXXX9i7dy80Gs0jvbf7OXJycnDw4EG8/fbbsLW1xahRowAU3pTYxcXlkfZREZKTkw0/b9myBdOnT8elS5cMy+7/uRAR1XTPt3tc6gil4ogUmczb2xvu7u7YsWOHYdmOHTvg5uaGdu3aGZbdu3cPR48excKFC9GtWzd4eHjg6aefRnh4OHr37m20TRsbG7i4uBg9rKyKvwXAxYsXce3aNSxfvhwdO3aEh4cHnnnmGcydOxcdOnR4pPd2P0fjxo0xcuRItGnTBgcOHDC8XtypvbVr18Ld3R2Wlpbo378/bt++XWS7c+bMgZOTE2xsbDBy5EhMnjwZTz31lFHNmjVr0LJlS6jVarRo0aLUe+v99zjZ2dlBJpMZLTt8+DB8fHygVqvRpEkTzJo1C/n5+Yb1Z86caRglbNiwIcaOHQsA6Nq1K/7880+MHz/eMLpFRCSFSzfSpI5QJmykqqHM3MwSHzn5OWWuzc7Lfmhteb3xxhtYs2aN4fm3336LESNGGNVYW1vD2toau3btgk6nK/e+HtSgQQOYmZlh+/btKCgoqLDt/pcQAlFRUUhISIBCUfLkbydOnMCIESMwevRoxMXFoVu3bpgzZ45RzcaNGzF37lwsXLgQp0+fhru7O1asWGFU8/XXX2Pq1KmYO3cuEhISMG/ePEybNg3r1q0zOfvPP/+MoUOHYuzYsbh48SK++uorrF27FnPnzgUAbN++HZ9++im++uorXLlyBbt27cKTTz4JoLAhbtSoEWbPno3k5GSjkS8ioqqyJToOLb52hcd7r0KvFw9fQUqCKo1GoxEAhEajKfJadna2uHjxosjOzi7yGmaixEevjb2Mai3nWpZY22VNF6Nax0WORWpMNWzYMNG3b19x69YtoVKpRGJiorh+/bpQq9Xi1q1bom/fvmLYsGGG+u3btwsHBwehVquFv7+/CA8PF/Hx8Ubb9PDwEEqlUlhZWRk9Dh06VGKOpUuXCktLS2FjYyO6desmZs+eLa5evWp8HAGxc+fOMr+3/+ZQKBQCgFCr1eLYsWOGmkOHDgkA4u7du0IIIV599VXRo0cPo+0EBQUJOzs7w3NfX1/xzjvvGNU888wzom3btobnbm5u4vvvvzeq+eijj4Sfn99Dc69Zs8Zof506dRLz5s0zqvnuu++Eq6urEEKITz75RDzxxBMiNze32O15eHiITz/99KH7NUVpv+9ERA9q/cFogZkQbuODJNl/aZ/fD+KIFJWLo6MjevfujXXr1mHNmjXo3bs3HB0di9QNHDgQN2/exJ49exAYGIioqCh4e3tj7dq1RnUTJ05EXFyc0cPX17fE/b/zzjtISUnBhg0b4Ofnh23btqFVq1aIjIx8pPd1P0d0dDS6deuGqVOnwt/fv8T6hIQE+Pn5GS178PmlS5fw9NNPGy377/Nbt27hxo0bePPNNw2jeNbW1pgzZw6uXr1q8ns4ffo0Zs+ebbStkJAQJCcnIysrC4MGDUJ2djaaNGmCkJAQ7Ny50+i0HxGRlFLvZuK8fAMAYMwzIQ+plh4vNq+GMsIzSnxNbiY3ep76fmqJtWYy4z75+rjrj5TrQSNGjMC7774LAFi2bFmJdWq1Gt27d0f37t0xffp0jBw5EjNmzDD6ZpmjoyMef9y0CwptbGzw0ksv4aWXXsKcOXMQGBiIOXPmoHv37uV6P//N8fjjjyMiIgKPP/44OnbsiBdeeKHYeiHKNuT84LVG/11Pr9cDKDy992DzKJcb/3mXhV6vx6xZszBgwIAir6nVari5ueHSpUuIjIzEwYMHMXr0aHz88ceIjo4u9TQmEVFVmLJxG6DSwjy9Ccb36yZ1nIdiI1UNWSmLv8i6KmvLokePHsjNzQUABAYGlnk9Ly8vk6YlKAuZTIYWLVogJiamwrbp4OCAMWPG4P3338eZM2eKvfDay8sLx48fN1r24PPmzZvjt99+Q3Dwv5PJnTp1yvCzs7MzHnvsMVy7dg1Dhgx55Nze3t64dOlSqY2phYWFoQl955130KJFC5w7dw7e3t5QKpWVdu0ZEdHDbPljFeAAPG8fAnN59T9xxkaKyk0ulyMhIcHw84Nu376NQYMGYcSIEWjTpg1sbGxw6tQpLFq0CH379jWqTU9PR0pKitEyS0tL2NraFtluXFwcZsyYgeDgYHh5eUGpVCI6OhrffvstJk2aZFSbmJiIuLg4o2WPP/44rK2ty/Qe33nnHSxcuBARERF4+eWXi7w+duxY+Pv7Y9GiRejXrx8OHDiA/fv3G9WMGTMGISEhaN++Pfz9/bFlyxacPXsWTZo0MdTMnDkTY8eOha2tLXr27AmdTodTp07h7t27mDBhQpmy3jd9+nS8+OKLcHNzw6BBg2BmZoazZ8/i3LlzmDNnDtauXYuCggL4+vrC0tIS3333HSwsLODh4QGgcB6pw4cP45VXXoFKpSr2lC0RUWXYHXMBGQ6xQIE5FgUPlzpO2VT6FVt1WHkvNq/O7l9sXpL/Xmyek5MjJk+eLLy9vYWdnZ2wtLQUzZs3Fx9++KHIysoyrOPh4SEAFHmMGjWq2H3cunVLjB07VrRu3VpYW1sLGxsb8eSTT4rFixeLgoICQ11x2wRQ4kXsJV1kHRISIlq1aiUKCgqKXGwuhBCrV68WjRo1EhYWFqJPnz5i8eLFRhd/CyHE7NmzhaOjo7C2thYjRowQY8eOFR07djSq2bhxo3jqqaeEUqkUDg4OonPnzmLHjh0lHuv7HrzYXAgh9u/fL/z9/YWFhYWwtbUVTz/9tFi1apUQQoidO3cKX19fYWtrK6ysrETHjh3FwYMHDevGxsaKNm3aCJVKJSrqr4ia+vtORFWr3eQwgZkQrmH9Jc1hysXmMiHKeJEHmUyr1cLOzg4ajabIyEpOTg4SExPh6ekJtVotUUKSSvfu3eHi4oLvvvtO6ihVgr/vRFQWicl3MXHDBjzXsh1Gv/isZDlK+/x+EE/tEVWyrKwsrFy5EoGBgZDL5di0aRMOHjz4yN8wJCKqbTxdHbB94hipY5iEjRRRJZPJZNi7dy/mzJkDnU6H5s2bIyIiosRvAhIRUc3BRoqokllYWODgwYNSxyAiqrb2nbyEQd8PwytNQ/HNu8OljmOS6v+9QiIiIqrVpu/6Bpn2J/Dj1R0PL65m2EhJjNf6U13A33MiKok2U4fTBWsBACHe1X8m8wexkZLI/Rmks7KyJE5CVPnuT9xanpnaiah2m7FpN4RFGswyG2JqUE+p45iM10hJRC6Xw97eHqmphbd4sbS0LHbmbKKaTq/X49atW7C0tIS5Of/KISJj684XzmT+jMWbUCtr3t8RNS9xLeLi4gIAhmaKqLYyMzODu7s7/7FAREZ+jbuKuw6/AEKG+YPflDpOubCRkpBMJoOrqyucnJyQl5cndRyiSqNUKmFmxisJiMjYhxGrAXOgviYAz7TykDpOubCRqgbkcjmvHSEiojqnh1cnXP7tf3iz7Sipo5QbbxFTiUyZYp6IiIiqB1M+vznWTkRERFRObKSIiIioSsVeTMJzs2bj5KW/pI7yyNhIERERUZWavHU1DmEGAlcMlzrKI2MjRURERFUmJzcfRzNXAwCGeo2UOM2jYyNFREREVWbe1v3QW/8NWXZ9zBnSX+o4j4yNFBEREVWZVadXAQC8zYbB1kolcZpHx0aKiIiIqsSpy3/jH9ufAACz+9W8GxQXh40UERERVYlJm78FzPSwvdsJvZ5uIXWcCsGZzYmIiKhK5BbkAvmWeK35W1JHqTCc2bwScWZzIiIiY3/d0sLWUlWtr4+qUTObL1++HJ6enlCr1fDx8cGRI0dKrY+OjoaPjw/UajWaNGmClStXFqmJiIiAl5cXVCoVvLy8sHPnTqPX58+fjw4dOsDGxgZOTk7o168fLl26ZFQjhMDMmTPRsGFDWFhYoGvXrrhw4cKjv2EiIqI6rFED22rdRJlK0kZqy5YtCAsLw9SpU3HmzBl06tQJPXv2RFJSUrH1iYmJ6NWrFzp16oQzZ85gypQpGDt2LCIiIgw1sbGxCAoKQnBwMOLj4xEcHIzBgwfjxIkThpro6Gi88847OH78OCIjI5Gfn4+AgABkZmYaahYtWoQlS5Zg6dKlOHnyJFxcXNC9e3ekp6dX3gEhIiKqhS7dSMOGX05LHaNSSHpqz9fXF97e3lixYoVhWcuWLdGvXz/Mnz+/SP2kSZOwZ88eJCQkGJaFhoYiPj4esbGxAICgoCBotVrs27fPUNOjRw84ODhg06ZNxea4desWnJycEB0djc6dO0MIgYYNGyIsLAyTJk0CAOh0Ojg7O2PhwoUYNapsd6nmqT0iIiKg59yF2J8/Gc0y3sDlj7+VOs5D1YhTe7m5uTh9+jQCAgKMlgcEBCAmJqbYdWJjY4vUBwYG4tSpU8jLyyu1pqRtAoBGowEA1KtXD0DhyFdKSorRdlQqFbp06VLqdnQ6HbRardGDiIioLssv0OPgna8BAM+4PStxmoonWSOVlpaGgoICODs7Gy13dnZGSkpKseukpKQUW5+fn4+0tLRSa0raphACEyZMwLPPPovWrVsbtnF/vbJuByi89srOzs7wcHNzK7GWiIioLvh8dxTyba8COhssDA6SOk6Fk/xic5lMZvRcCFFk2cPqH1xuyjbfffddnD17ttjTfqZmCw8Ph0ajMTxu3LhRYi0REVFd8PnRwpnMvQqGwMnBSuI0FU+yeaQcHR0hl8uLjPCkpqYWGQm6z8XFpdh6c3Nz1K9fv9Sa4rY5ZswY7NmzB4cPH0ajRo2M9gMUjky5urqWKRtQePpPpao930QgIiJ6FAlJt3DDqvCb8x/2rD1zR/2XZCNSSqUSPj4+iIyMNFoeGRkJf3//Ytfx8/MrUn/gwAG0b98eCoWi1Jr/blMIgXfffRc7duzAr7/+Ck9PT6N6T09PuLi4GG0nNzcX0dHRJWYjIiIiYx9sXA+Y58Lyng9e7dpO6jiVQtKZzSdMmIDg4GC0b98efn5+WLVqFZKSkhAaGgqg8FTZ33//jfXr1wMo/Ibe0qVLMWHCBISEhCA2NharV682Oi03btw4dO7cGQsXLkTfvn2xe/duHDx4EEePHjXUvPPOO/j++++xe/du2NjYGEaw7OzsYGFhAZlMhrCwMMybNw/NmjVDs2bNMG/ePFhaWuK1116rwiNERERUcx37Zy/gAPT3qJ2jUQAAIbFly5YJDw8PoVQqhbe3t4iOjja8NmzYMNGlSxej+qioKNGuXTuhVCpF48aNxYoVK4psc9u2baJ58+ZCoVCIFi1aiIiICKPXART7WLNmjaFGr9eLGTNmCBcXF6FSqUTnzp3FuXPnTHpvGo1GABAajcak9YiIiGqDbF2e+HD9HvF3mlbqKCYx5fObt4ipRJxHioiIqOapEfNIERERUe10LyMHObn5UseoEmykiIiIqEK9uXwVrD70wNDPVkkdpdKxkSIiIqIKo9cL/JTyNfRWN6HLy5U6TqVjI0VEREQV5tsDJ6CzOw/kqbH49aFSx6l0bKSIiIiowiz6pfB0XpOcwfBwtpc2TBVgI0VEREQVIilVgyvKLQCA97vV4rmj/oONFBEREVWISd99DyizoNS0xKiedeNOIGykiIiIqELs/utrAEAv57dgZiaTOE3VYCNFREREFWLVi1/DKysUi4YESx2lynBm80rEmc2JiIhqHs5sTkRERFQF2EgRERHRI3l7xQY8MXEENv76P6mjVDk2UkRERPRIvrv8Ja5Yr8HG2INSR6lybKSIiIio3LYejkem/W9AgQIfDxkudZwqx0aKiIiIym3O3sIpDx7L6ItWjZ0kTlP12EgRERFRuaRpsnBOtgEA8K5f3ZjJ/EFspIiIiKhcwjdsA9QamKd74v0Bz0sdRxJspIiIiKhctlwpPK3XzW4kzOV1s6Wom++aiIiIHkl+gR4d6gXCXNsUi157Q+o4kuHM5pWIM5sTEVFtp9eLWndfPc5sTkRERFWitjVRpmIjRURERCZZsvMQJn4bgaycPKmjSI6NFBEREZnko+jZWHzjZfRfvFjqKJJjI0VERERl9vOpy7jnEAXozTB74FCp40iOjRQRERGV2bSd3wAAGmh7wLelm8RppMdGioiIiMokIzsXp/LXAgBGtqubM5k/iI0UERERlcn073dDWN6CWaYrpr/SW+o41QIbKSIiIiqTdecKZzL3U4+AWmkucZrqgY0UERERPZQ2U4dcZABChnkvvyl1nGqD7SQRERE9lK2VCumfxSAq/ho6t/GUOk61wREpIiIiKrOubZtIHaFaYSNFREREpfo17ioSk+9KHaNaYiNFREREpXpl3Vg0WdYQY1dtljpKtcNrpIiIiKhEsReTcMtuHyAT6OPjI3WcaocjUkRERFSi8K3fAjIB+7vd0N2nmdRxqh02UkRERFSs3LwCHM1cDQAY2jJE4jTVExspIiIiKta8rftRYP0XZNn18NGQ/lLHqZbYSBEREVGxvjpVOJN5O7NhsLdWS5ymemIjRUREREX8+c89pFj9DACY1Zen9UrCRoqIiIiK8HC2R/yIRIx0XIsXfVtKHafakgkhhNQhaiutVgs7OztoNBrY2tpKHYeIiIjKwJTPb45IERERkZGsnDypI9QYbKSIiIjISPMpQ1Av7HmsP3hK6ijVHmc2JyIiIoPzif/gL+udgDwfliql1HGqPY5IERERkcEH368D5PmwuueLlzu1kTpOtcdGioiIiAAAer3AwTuFc0e97MkpD8qCjRQREREBAD7bHYU82z8AnQ0WDA2SOk6NwEaKiIiIAABfHC0cjWqZ/xpc6llLnKZmYCNFREREuHQjDX9aRgAApgTytF5ZsZEiIiIiODtYY5TrN2ieMRJDn/eROk6NwZnNKxFnNiciIqp5OLM5ERERURVgI0VERFTHdZ4xHX3mf4KrN+9IHaXG4czmREREdVhi8l0cKfgYyM1B37PPoGnDjlJHqlE4IkVERFSHfbBhI6DIgUrTGiMCfKWOU+OwkSIiIqqj9HqBH5NXAQBedH0LZmYyiRPVPGykiIiI6qg1kb8hx+4ckKfGx0OHSh2nRmIjRUREVEctOlg4k3mTnEHwdHWQOE3NxEaKiIioDvrrlhaXlZsAAOO7cCbz8pK8kVq+fDk8PT2hVqvh4+ODI0eOlFofHR0NHx8fqNVqNGnSBCtXrixSExERAS8vL6hUKnh5eWHnzp1Grx8+fBh9+vRBw4YNIZPJsGvXriLbGD58OGQymdGjY0d+k4GIiGqH5DtauGX3gfpeW4zu/azUcWosSRupLVu2ICwsDFOnTsWZM2fQqVMn9OzZE0lJScXWJyYmolevXujUqRPOnDmDKVOmYOzYsYiIiDDUxMbGIigoCMHBwYiPj0dwcDAGDx6MEydOGGoyMzPRtm1bLF26tNR8PXr0QHJysuGxd+/einnjREREEuvQvBGSlmzG3YWneJH5I5D0FjG+vr7w9vbGihUrDMtatmyJfv36Yf78+UXqJ02ahD179iAhIcGwLDQ0FPHx8YiNjQUABAUFQavVYt++fYaaHj16wMHBAZs2bSqyTZlMhp07d6Jfv35Gy4cPH4579+4VO1pVVrxFDBERUc1TI24Rk5ubi9OnTyMgIMBoeUBAAGJiYopdJzY2tkh9YGAgTp06hby8vFJrStpmaaKiouDk5IQnnngCISEhSE1NLbVep9NBq9UaPYiIiKqb91Zvx48nEh5eSA8lWSOVlpaGgoICODs7Gy13dnZGSkpKseukpKQUW5+fn4+0tLRSa0raZkl69uyJjRs34tdff8Unn3yCkydP4rnnnoNOpytxnfnz58POzs7wcHNzM2mfRERElS3lTgaWXB2BPvu9sC7ypNRxajzJbxEjkxmflxVCFFn2sPoHl5u6zeIEBQUZfm7dujXat28PDw8P/PTTTxgwYECx64SHh2PChAmG51qtls0UERFVK+EbtgKqdCi0j2PIcz5Sx6nxJGukHB0dIZfLi4wUpaamFhlRus/FxaXYenNzc9SvX7/UmpK2WVaurq7w8PDAlStXSqxRqVRQqVSPtB8iIqLKtC1xFWAPPF9vJMzlkn95v8aT7AgqlUr4+PggMjLSaHlkZCT8/f2LXcfPz69I/YEDB9C+fXsoFIpSa0raZlndvn0bN27cgKur6yNth4iISCoRR88h0/4EUGCOj18bLnWcWkHSU3sTJkxAcHAw2rdvDz8/P6xatQpJSUkIDQ0FUHiq7O+//8b69esBFH5Db+nSpZgwYQJCQkIQGxuL1atXG30bb9y4cejcuTMWLlyIvn37Yvfu3Th48CCOHj1qqMnIyMAff/xheJ6YmIi4uDjUq1cP7u7uyMjIwMyZMzFw4EC4urri+vXrmDJlChwdHdG/f/8qOjpEREQVa/aPXwMWwGMZfdHa89HO1ND/ExJbtmyZ8PDwEEqlUnh7e4vo6GjDa8OGDRNdunQxqo+KihLt2rUTSqVSNG7cWKxYsaLINrdt2yaaN28uFAqFaNGihYiIiDB6/dChQwJAkcewYcOEEEJkZWWJgIAA0aBBA6FQKIS7u7sYNmyYSEpKMum9aTQaAUBoNBqT1iMiIqpotzVZQjbZXmAmxJzN+6WOU62Z8vkt6TxStR3nkSIioupiwy+n8XpkAMzybZCz8BqvjyqFKZ/fkn9rj4iIiCrf0Od98KLv3zh8nk1UReKRJCIiqiPsrdV4qaOX1DFqFTZSREREtdyvcVeRX6CXOkatxEaKiIioFtNm6vDCZl9YfPAEouKvSR2n1mEjRUREVIt9uHEnhMVtCDMdOrZ0lzpOrcNGioiIqBb77uIqAMAzViOgVvI7ZhWNjRQREVEt9cuZP3DP4RAgZJg3aITUcWolNlJERES11NSIbwAADTQ98EwrD4nT1E5spIiIiGqhjOxc/Ja3BgDwRtsQidPUXmykiIiIaqFFEQcgLFNhlumCGa+8KHWcWotXnREREdVCM1/rjQY/RiPx1j+wVCukjlNrsZEiIiKqhczMZBjzUmepY9R6PLVHRERUy+j1QuoIdQYbKSIiolokJzcflu+3QdvJY5GYfFfqOLUeGykiIqJa5KMte6GzO49zYjMa2FtJHafWYyNFRERUi3zzv68BAD7mw2BtoZQ4Te3HRoqIiKiWOHnpL6Ta7gUAzOnPuaOqAhspIiKiWmLS5m8BMz3s7nZBYPsnpI5TJ7CRIiIiqgVy8wpwOKPwljCvteBoVFVhI0VERFQLLNh+AAXWNyDLccC8oQOljlNncEJOIiKiWuD5Nl746cIHsLawhr21Wuo4dYZMCMFZuyqJVquFnZ0dNBoNbG1tpY5DREREZWDK5zdP7RERERGVExspIiKiGiy/QI9Wk97GnM37kZtXIHWcOoeNFBERUQ22ZOevuGi5EtPiX4E2Syd1nDqHjRQREVENtjSmcCbz1vohcLSzlDhN3WNSI7Vo0SJkZ2cbnh8+fBg63b/db3p6OkaPHl1x6YiIiKhECUm3cMN6JwBgWu+3JE5TN5nUSIWHhyM9Pd3w/MUXX8Tff/9teJ6VlYWvvvqq4tIRERFRiSZuXAfI82B1rwMGd24rdZw6yaRG6sGZEjhzAhERkTT0eoEDaYWn9QZ4cCZzqfAaKSIiohroyx8OI8/2MpBrjQXBr0gdp87izOZEREQ1UJZOB5WmFTzN/dGwvo3Uceoskxupb775BtbW1gCA/Px8rF27Fo6OjgBgdP0UERERVZ7wwQGY9PI53EnPfngxVRqTbhHTuHFjyGSyh9YlJiY+UqjagreIISIiqnlM+fw2aUTq+vXrj5KLiIiIHpFeL/Det9sQPrA3nByspI5T5/FicyIiohrk6/2x+OzvILjOfxxZOXlSx6nzTGqkTpw4gX379hktW79+PTw9PeHk5IS33nrLaIJOIiIiqliLfy2c8qCJvgcs1QqJ05BJjdTMmTNx9uxZw/Nz587hzTffxAsvvIDJkyfjhx9+wPz58ys8JBEREQF//nMPf6i3AAAmPs+ZzKsDkxqpuLg4PP/884bnmzdvhq+vL77++mtMmDABX3zxBbZu3VrhIYmIiAiYtOF7QJENlaYVRgZ2lDoOwcRG6u7du3B2djY8j46ORo8ePQzPO3TogBs3blRcOiIiIgJQeJH57r9WAQB6uYTAzOzh36KnymdSI+Xs7GyY2iA3Nxf/+9//4OfnZ3g9PT0dCgXP1xIREVW07345hRz7eCBfhY+HBksdh/6fSY1Ujx49MHnyZBw5cgTh4eGwtLREp06dDK+fPXsWTZs2rfCQREREdd2u/x0DhAyNs15G04b1pI5D/8+keaTmzJmDAQMGoEuXLrC2tsbatWuhVCoNr3/77bcICAio8JBERER13c5JYTh8ti/yCgqkjkL/YdLM5vdpNBpYW1tDLpcbLb9z5w5sbGx4eu//cWZzIiKimqfSZjYfMWJEmeq+/fZbUzZLREREpbhwPRWtGjtJHYOKYVIjtXbtWnh4eKBdu3Yox0AWERERmWhLdBxe+dUHj6UPQNLirfy2XjVjUiMVGhqKzZs349q1axgxYgSGDh2KevV4wRsREVFlmbP3a8BSDzOYsYmqhkz61t7y5cuRnJyMSZMm4YcffoCbmxsGDx6Mn3/+mSNUREREFSz1bibOyzcAAMY8w5nMqyOTb1qsUqnw6quvIjIyEhcvXkSrVq0wevRoeHh4ICMjozIyEhER1UlTNm4DVFqYpzfB+H7dpI5DxTC5kfovmUwGmUwGIQT0en1FZSIiIiIAW/8ovEHx8/YhMJc/0kc2VRKT/1R0Oh02bdqE7t27o3nz5jh37hyWLl2KpKQkWFtbV0ZGIiKiOmd3zAWkO8QABeZY9NpwqeNQCUy62Hz06NHYvHkz3N3d8cYbb2Dz5s2oX79+ZWUjIiKqs2b/uBpQAa7pfdCmiYvUcagEJk3IaWZmBnd3d7Rr1w4yWcnfHNixY0eFhKvpOCEnERGVV1KqBpO++x6dW7TB272fkTpOnVJpE3K+/vrrpTZQREREVDHcneyw6b23pY5BD2HyhJxEREREVIhfASAiIqpG9p28BKvx7THs82+kjkJlwEaKiIioGpm+6xtk2Z/G/uu7pY5CZcBGioiIqJrQZupwumAtAGCkd4i0YahM2EgRERFVEzM27YawSINZZkNMC+oldRwqAzZSRERE1cT684UzmftbjIBaadL3wUgibKSIiIiqgV/jruKOw0FAyDB/0JtSx6EykryRWr58OTw9PaFWq+Hj44MjR46UWh8dHQ0fHx+o1Wo0adIEK1euLFITEREBLy8vqFQqeHl5YefOnUavHz58GH369EHDhg0hk8mwa9euItsQQmDmzJlo2LAhLCws0LVrV1y4cOGR3isREVFJPoxYDQCorwnAs60bSxuGykzSRmrLli0ICwvD1KlTcebMGXTq1Ak9e/ZEUlJSsfWJiYno1asXOnXqhDNnzmDKlCkYO3YsIiIiDDWxsbEICgpCcHAw4uPjERwcjMGDB+PEiROGmszMTLRt2xZLly4tMduiRYuwZMkSLF26FCdPnoSLiwu6d++O9PT0ijsARERE/69f2+fgdO9FvNl2lNRRyAQm3SKmovn6+sLb2xsrVqwwLGvZsiX69euH+fPnF6mfNGkS9uzZg4SEBMOy0NBQxMfHIzY2FgAQFBQErVaLffv2GWp69OgBBwcHbNq0qcg2ZTIZdu7ciX79+hmWCSHQsGFDhIWFYdKkSQAKb9bs7OyMhQsXYtSosv2S8xYxRERENY8pn9+SjUjl5ubi9OnTCAgIMFoeEBCAmJiYYteJjY0tUh8YGIhTp04hLy+v1JqStlmcxMREpKSkGG1HpVKhS5cupW5Hp9NBq9UaPYiIiKj2kqyRSktLQ0FBAZydnY2WOzs7IyUlpdh1UlJSiq3Pz89HWlpaqTUlbbOk/dxfz5TtzJ8/H3Z2doaHm5tbmfdJRER107ELf6LzjOk4duFPqaNQOUh+sfmDN0EWQpR6Y+Ti6h9cbuo2KypbeHg4NBqN4XHjxg2T90lERHVL+NbVOGL2Efp8zW/q1USSTVLh6OgIuVxeZIQnNTW1yEjQfS4uLsXWm5ubo379+qXWlLTNkvYDFI5Mubq6lnk7KpUKKpWqzPshIqK6LSc3H8eyvgWsgeBWnMm8JpJsREqpVMLHxweRkZFGyyMjI+Hv71/sOn5+fkXqDxw4gPbt20OhUJRaU9I2i+Pp6QkXFxej7eTm5iI6Otqk7RAREZVm3tb90Fv/DVl2fXz0Wj+p41A5SDpt6oQJExAcHIz27dvDz88Pq1atQlJSEkJDQwEUnir7+++/sX79egCF39BbunQpJkyYgJCQEMTGxmL16tVG38YbN24cOnfujIULF6Jv377YvXs3Dh48iKNHjxpqMjIy8McffxieJyYmIi4uDvXq1YO7uztkMhnCwsIwb948NGvWDM2aNcO8efNgaWmJ1157rYqODhER1XarTq8C7AFvs2GwteIZjRpJSGzZsmXCw8NDKJVK4e3tLaKjow2vDRs2THTp0sWoPioqSrRr104olUrRuHFjsWLFiiLb3LZtm2jevLlQKBSiRYsWIiIiwuj1Q4cOCQBFHsOGDTPU6PV6MWPGDOHi4iJUKpXo3LmzOHfunEnvTaPRCABCo9GYtB4REdV+Jy/9JTDdTGAmxA/HL0odh/7DlM9vSeeRqu04jxQREZXkhdlz8IuYBtu7naD57LDUceg/asQ8UkRERHVZgSgAcq3x6hO8yLwm44hUJeKIFBERlSblTgYsVQpeH1XNmPL5LenF5kRERHWZSz1rqSPQI+KpPSIioiqUkHQLq38+Ab2eJ4RqAzZSREREVWj8+tUYebwjmn0wXOooVAHYSBEREVWR/AI9frn7DQCgi0dXacNQhWAjRUREVEU+3x2FfNurgM4WC4YOljoOVQA2UkRERFXk86OrAABeBa/BycFK4jRUEdhIERERVYFLN9Jww2onAGBKIOeOqi3YSBEREVWBDzauB8xzYXnPB0Oe85Y6DlUQNlJERERV4GjKfgBAP3eORtUmnJCTiIioCiR/vA8Lth/AiBeekToKVSA2UkRERFVAqZBj+qs9pY5BFYyn9oiIiCrRvYwcZOXkSR2DKgkbKSIioko0csXXsJnhhlc+WS51FKoEbKSIiIgqiV4v8FPy19Bb/gO94L31aiM2UkRERJXk2wMnkGN3DsizwMfBQ6SOQ5WAjRQREVElWfTL1wCAJjmD4OFsL20YqhRspIiIiCrBX7e0uKLcDAB4v9tbEqehysJGioiIqBJ88N33gDILSk1LjOrpL3UcqiRspIiIiCrBrhuFNyju6RQCMzOZxGmosrCRIiIiqgTrB6zFk9nvYuGQYKmjUCWSCcHvY1YWrVYLOzs7aDQa2NraSh2HiIiIysCUz2+OSBERERGVExspIiKiCvT2ig1o+v7rWBd5UuooVAXYSBEREVWQqPhrWJX4Aa7ZfIctvx2SOg5VATZSREREFeDstRR0Xx8AvVUyVJrWWPVWqNSRqAqwkSIiInpESaka+H7ZA/m2V2Ge7omYt39Gowb8klFdwEaKiIjoEdzRZqPNvJeQYx8Psyxn7B9yAN7NGkodi6oIGykiIqJH8Oyc8dA4HAZ0tvi+13483+5xqSNRFWIjRURE9AhWD58Ci3tP4YuOPyCoy1NSx6EqZi51ACIioprMz8sd2sWnYS7n2ERdxD91IiIiE/WZ/wk+WLPD8JxNVN3FP3kiIiITDPv8G/yY+z4+vj4IEUfPSR2HJMZGioiIqIw+WLMD6++MAgD4FkzEwGeflDgRSY3XSBEREZXBkp2H8PG1VwFzPZ7IeBMxC+dLHYmqATZSRERED7Hhl9N479RLgDIXrvf6I37hSpiZyaSORdUAT+0RERGV4uSlv/D6gZ6AMgP2d7vh4kffQ63kOAQVYiNFRERUinaPN8STsldgcc8b56bugr21WupIVI2wpSYiIiqFudwMZ+Z9jtR7mXCpZy11HKpmOCJFRET0gNS7meg2axYysnMBAGZmMjZRVCw2UkRERP+RkZ0Lr9kDEYWZaPXhMKnjUDXHRoqIiOj/5Rfo0erDYbht/zOQa4lp3cdJHYmqOTZSREREAPR6Ae+pY5FkuxkoUGBOmx0Y2aOj1LGommMjRUREBOD5j2bjnMUyQMgwptF6TA0KlDoS1QBspIiIqM575ZPliMJMAECQzVJ88dYr0gaiGoONFBER1Xkdm3oBOht0wyxsfm+01HGoBuE8UkREVOeF9esK3yfOw7eFm9RRqIbhiBQREdVJ3+w/jp3Hzhue+3m58/55ZDKOSBERUZ2zO+YC3oruBQDYqo/Cy53aSJyIaiqOSBERUZ1y9Px1DNgVAKG+C6uc5ujcuqnUkagGYyNFRER1xoXrqXhuTXforW5CpWmFuIk/wcnBSupYVIOxkSIiojrhr1tadPi8B/Js/4A83QNHQ39G04b1pI5FNRwbKSIiqvXuZeTgybl9kW1/BrKsBtj3aiTaP/GY1LGoFmAjRUREtV6WLq/wB50Nvgvcj+4+zaQNRLUGv7VHRES1XsP6Nkicsw8/n/4dQV2ekjoO1SIckSIiolrr893Rhp/trdVsoqjCsZEiIqJaqfe8jxEW1xVPT/1A6ihUi7GRIiKiWmfEl2uwN6+wgXK0dJQ4DdVmbKSIiKhWCV+3C2vSRgIAOuROxF6OSFElkryRWr58OTw9PaFWq+Hj44MjR46UWh8dHQ0fHx+o1Wo0adIEK1euLFITEREBLy8vqFQqeHl5YefOnSbvd/jw4ZDJZEaPjh07PtqbJSKiSvXZrigs+OMVwEyPZhlv4PhHC6WORLWcpI3Uli1bEBYWhqlTp+LMmTPo1KkTevbsiaSkpGLrExMT0atXL3Tq1AlnzpzBlClTMHbsWERERBhqYmNjERQUhODgYMTHxyM4OBiDBw/GiRMnTN5vjx49kJycbHjs3bu3cg4EERE9sk1RZzD+t5cAcx1c7vXF2bmreBNiqnQyIYSQaue+vr7w9vbGihUrDMtatmyJfv36Yf78+UXqJ02ahD179iAhIcGwLDQ0FPHx8YiNjQUABAUFQavVYt++fYaaHj16wMHBAZs2bSrzfocPH4579+5h165d5X5/Wq0WdnZ20Gg0sLW1Lfd2iIjo4YZ9/g3W3wuB3d0uuDZ7H+rZWkgdiWooUz6/JRuRys3NxenTpxEQEGC0PCAgADExMcWuExsbW6Q+MDAQp06dQl5eXqk197dpyn6joqLg5OSEJ554AiEhIUhNTS31Pel0Omi1WqMHERFVjXXjRuLDJntwdspuNlFUZSRrpNLS0lBQUABnZ2ej5c7OzkhJSSl2nZSUlGLr8/PzkZaWVmrN/W2Wdb89e/bExo0b8euvv+KTTz7ByZMn8dxzz0Gn05X4nubPnw87OzvDw83N7SFHgYiIHkVi8l1cvXnH8Pyj4D5wd7KTMBHVNZLPbC6TGZ+/FkIUWfaw+geXl2WbD6sJCgoy/Ny6dWu0b98eHh4e+OmnnzBgwIBis4WHh2PChAmG51qtls0UEVElSdNkoe3CF5FrpkXM2z/Du1lDqSNRHSRZI+Xo6Ai5XF5k9Ck1NbXIaNF9Li4uxdabm5ujfv36pdbc32Z59gsArq6u8PDwwJUrV0qsUalUUKlUJb5OREQVIysnD14zX0a6QwxkOfb4M/UOGymShGSn9pRKJXx8fBAZGWm0PDIyEv7+/sWu4+fnV6T+wIEDaN++PRQKRak197dZnv0CwO3bt3Hjxg24urqW7Q0SEVGlyC/Qw2vqcNyy3wfkWWBFp5/Q/5nWUseiukpIaPPmzUKhUIjVq1eLixcvirCwMGFlZSWuX78uhBBi8uTJIjg42FB/7do1YWlpKcaPHy8uXrwoVq9eLRQKhdi+fbuh5tixY0Iul4sFCxaIhIQEsWDBAmFubi6OHz9e5v2mp6eL9957T8TExIjExERx6NAh4efnJx577DGh1WrL/P40Go0AIDQazaMeKiIiEkIUFOhFm0ljBGZCYJq5mPX9XqkjUS1kyue3pI2UEEIsW7ZMeHh4CKVSKby9vUV0dLThtWHDhokuXboY1UdFRYl27doJpVIpGjduLFasWFFkm9u2bRPNmzcXCoVCtGjRQkRERJi036ysLBEQECAaNGggFAqFcHd3F8OGDRNJSUkmvTc2UkREFeu5WbMLm6iZEG+v2CB1HKqlTPn8lnQeqdqO80gREVWcv25p0XjRkyiwTsIAi88R8cFYqSNRLVUj5pEiIiIyRaMGtogdeQyDrJayiaJqgyNSlYgjUkREjy4pVcO5oahKcUSKiIhqhdU/n0DjTz0x/putUkchKhYbKSIiqpb2HL+IkEO9INR3seHCGuj1PIFC1Q8bKSIiqnaOXfgT/SMCICzuwOqeL85N2wYzs5LvekEkFTZSRERUrSQk3UK3bwOgt/4bSk1LnHnvJ7jUs5Y6FlGx2EgREVG1cfN2Otp/2hN5tpchz3DH0VEH0KxRfaljEZWIjRQREVUbb3/9NbLsT0OW7YgfBh9Ah+aNpI5EVCo2UkREVG1ETAyDf8EUrH1hH3p2aC51HKKHMpc6ABER1W16vUB+gR5KhRzmcjMcmz1X6khEZcYRKSIiktSzM6bCc9Jg3MvIkToKkcnYSBERkWT6LliCWPP5uGm3Ax/vPCB1HCKTsZEiIiJJvLVsPfbo3gMABMjnYW7wSxInIjIdGykiIqpy0777AV+njgAA+OgmYN+UyRInIiofXmxORERV6ss9hzHn0mBAUYAm6a/j+MKPOWs51VgckSIioipzR5uNsKNBgCIHzvf64Nycb2Au50cR1Vz87SUioipTz9YCX3beBpd7L+HirC2wVCukjkT0SGRCCN5Ou5JotVrY2dlBo9HA1tZW6jhERERUBqZ8fnNEioiIKtWf/9yDY1ggth6OlzoKUYVjI0VERJUmTZOFNvP74LbDAQTvegW5eQVSRyKqUGykiIioUmTl5KHVzMHQOhwFcuywsf8WKBVyqWMRVSg2UkREVOHyC/R48sM3kWr/E5CnxrJnfsTLndpIHYuowrGRIiKiCqXXCzz94Xu4ZvMdoJdjRsvtGP3is1LHIqoUbKSIiKhCvbl0Dc6oPwMAjHJeg5lDeksbiKgSsZEiIqIKNXfIIDjcfR791J9i5ehgqeMQVSreIoaIiCpUw/o2SPn4Z15YTnUCR6SIiOiRLdp+ED3mLIBeXzjHM5soqivYSBER0SNZF3kSk870w88F4Qhd8Z3UcYiqFBspIiIqt72//Y43fukJKDPhcPd5LB4eJHUkoirFRoqIiMrlRMINvLStO4TFbVjd64Dz03bC1koldSyiKsVGioiITHbpRho6fx2AAuu/oNS0wOkJe9Gwvo3UsYiqHBspIiIySVZOHnyW9Eau3e+QZ7jhcMgBNHdzlDoWkSTYSBERkUks1Qr0dR8BWZYT9gw6AN+WblJHIpKMTAghpA5RW2m1WtjZ2UGj0cDW1lbqOEREFeqvW1o0asC/26j2MeXzmyNSRET0UHq9QO95H+PSjTTDMjZRRGykiIioDDrPnIa9eR+g7addkZGdK3UcomqDjRQREZVqwKLPcUw+FwAw0G0MrC2UEiciqj7YSBERUYneXrEBO7PDAAAvmM3BxvGjpA1EVM2wkSIiomLN3PgTVqYMBwA8lTMOP0+dIm0gomrIXOoARERU/Xy1NwazEgYBigJ4aofi5KIlMDOTSR2LqNphI0VEREU0a+gM8xwX1Mv0wvn538JczhMYRMVhI0VEREU891RTnLaNQcP6trBUK6SOQ1Rt8Z8YREQEADif+A8WbIs0PG/TxAWOdpYSJiKq/jgiRURE+POfe+jwRSBybM/jdvoWfDxioNSRiGoEjkgREdVxd7TZaDP/JeTYx8MsxxE9vZ+SOhJRjcFGioioDsvJzUerGa9A63AE0Nni+1778dxTTaWORVRjsJEiIqqj8gv0aD1lJFLs9wB5anzR8QcEdXlK6lhENQobKSKiOkivF+g4/QNctVkH6OWY0nwLxrzUWepYRDUOGykiojpILwQycjUAgJFOqzE3+CWJExHVTPzWHhFRHWQuN8PFhauw/KfX8W6fTlLHIaqxOCJFRFSHrIs8iZzcfACAmZmMTRTRI+KIFBFRDabXC2izdNBk5sDD2d6wPCr+Gi79nYKMnBykZ2cjU5eD67eTsV37Hibv7YGEj7bA3lotXXCiWoKNFBFRBYu7mozkO1pos7KRnp2DjJwcaP+/mdHr9UaTXb69YgPOp1xCTn4OdAXZ0BXkIFefg1yRDb0oQPKnuwy1rSeNxqWCfdCb5UDIcyDMswFzneH17PA8qJWFf60PWz8VSbabi4YzB2SQG+qI6NHw/yQiqlVS72bibkY2tFk5uJeRDW12DjKyCxsZlUKBt3r6GWrHrtqMG3dTkJ2Xg5z8HGTnFzYyuoIcWCtscXLux4baJyaOwN/5Z1Egy0GBLAd6eTbE/zc0ZrkOKPj4L0Nt5y8HId3hWPEBc63xMf5tpLZf3og0+/2AsvhyvV7AzEwGALiXdwv5dtdLfO/aTJ2hQXJUueKmtinkQg25UMNcWMAcajSxegqRH33ERoqogvD/JCKqdHq9QE5uvtHNbz/bFYV/NPdwO0OLu1laZORmIie/sKFxs2uEre+/Y6ht+v7r0OT/g3zkIF+W/W8zY5YNu7wWuP3Zv/eHe2xua+TbXC82h1LTAm/1TDA8X/X7XOjszhc+kQFQ/P8DgDyjEYB/G6nk/ARk2Z8u4f0Z39TXQmaPjBx7yAosYKZXw0yvhlxYwFyooYS1Ue3zjfrg4q3HoZKrYWFuAbW5GhYKNSwVFrBQqqEXAmYobKRWBs1Gyt33YGOhhq2lBWwt1bC3soCdlRr1bCyMmqPT85YAWFJsXiKqOGykiKhEer3AvYwc3LytRfIdLZLvanBLq0Vauha3M7Rwr++ED1/pYahtOjEYWXoNdEILnUyDfLkWBeZaCKUG9bXPIe2znw3bHv9bX0ClLbpTc+D83x0B/NtIXZf9Cr3D38VmzNTYGz030///dT/5SqBADVmBGmYFFpALNeyEp1HtkxY9cEv7JJRmaqjMLKCSq6EyL2xo6jk7GNXO67YQdzIyYK1Ww1pd2MjYWKhha6GGvbWFUe0/n/5Y2mE1svm90WWufdG3ZZlriahqsJEiqoXuN0B/pWmQfEcLawsVnmnlAQDIysnD61+uxL1sDbQ6LTLytMgq0CJbr0WO0KKFtR9OzFkIoHDm6/qfWpa4H8c/ehgaKTMzGa6r9gCq9GJrdTLjpsk+ywf52dlQCVuozWxhYWYNpVwNlZkaTRo2Map9p9li5ObnwVKlhpVSDRsLC0ND42xva1T7z+wzsFQpoFTIH3qc/nvq7mE4WSURFYeNFFE1lZWTh21H4pBqGAHS4G6WFpocLbQ6Ldq7tcXy0CEAgJQ7GfD86Jn/jABpAXm+YVtumiAkLSm88NhcboaIrLGFLyhR5Nqca/esDD8rFXIg1xpQZAI6W8jzbWFeYAuF3hYqmS1a2LY3WneI02KYm8lRz8oW9a1t4WxnByd7W7g62OIxRzuj2ruf/VrmY/HFW6+UuZbfRCOiqsRGiqgC6PUCaZos3LxTeAos9Z4W/2g0SEvXorWbO15/obDhSEy+i16fTkZmvgY5onAEKM+s8KFXaNFavIazC5YCAJLvpGN4zNPF71AJ/HPlFQCFjZS9tRo59meL1gkZoLOFwuzfbkmpkKNJejAUMhWsFbawUdnCTmULe0tb1Leyg1dHN6NNJL+XDEc7S5jLHz7t3Iawt8pwtIiIag/JG6nly5fj448/RnJyMlq1aoXPPvsMnTqVPEFcdHQ0JkyYgAsXLqBhw4b44IMPEBoaalQTERGBadOm4erVq2jatCnmzp2L/v37m7RfIQRmzZqFVatW4e7du/D19cWyZcvQqlWrij0AVGnyC/TQ64XhFE9Gdi5OXf4LGTk6ZObokKXLRaZOh+zcXGTpdPBp6mm4BiUpVYP312+ALl+HdF0GtLlaZOZpkZlf2Py84PYSvp9Q+Ht39Px1dNrWFDDTF5vD6/dQQyOlFwK/W60qMbNWe8fw82OOtpBnuMG8wA7K/x8BspDZwcrcFlYKW/g94WOoVSvNsbBVJOrb2MDZ3hYuDrZo5GhXYgN0dfH6Mh9Hl3rWDy8iIqqjJG2ktmzZgrCwMCxfvhzPPPMMvvrqK/Ts2RMXL16Eu7t7kfrExET06tULISEh2LBhA44dO4bRo0ejQYMGGDiw8OvEsbGxCAoKwkcffYT+/ftj586dGDx4MI4ePQpfX98y73fRokVYsmQJ1q5diyeeeAJz5sxB9+7dcenSJdjY2FTdQarm9HqBLF0etFk6ZGbnIj27sEFJz9bBy93ZMEFgQtItRMScRlauDjl5ucjO1SE7TwddXi5y8nV4pWNXw13nfznzB97b+iXy9LnI0+uQq9chX+QiX+iQL3R4/cmRWPLmIADA1sPxGLp7EPSyXAgzHYRZLoRcB8h1gDwf3TALv86YDgD4+fQlvPxLmxLfS4cLE/Gi7yIAwPWUO9iW+e6/Lz5wCuzsP40NPzvb2/zbROnNIMu1g1le4ekvhd4WHg3+vcD5MUdbPCebDTu1LewtbFHfyhb1bWzRwNYWrg52aOrqaKhVK82R/3FSmf8sPnj5hTLXEhFRxZAJIYRUO/f19YW3tzdWrFhhWNayZUv069cP8+fPL1I/adIk7NmzBwkJ/359OTQ0FPHx8YiNjQUABAUFQavVYt++fYaaHj16wMHBAZs2bSrTfoUQaNiwIcLCwjBp0iQAgE6ng7OzMxYuXIhRo0aV6f1ptVrY2dlBo9HA1tb24SuYIClVg0s3UpHxn5GVrP+MrrzWpSNaujcAAOyOuYD1R3+BriAXunwddPk65BbkIregsEmZ+WIoXu5U2GB8suNXLDjyMQpELvKhg16WiwLooDcr/Hly28/xUXAfAMDEbyOw+MbLJWYc7rAaa8aOAADM3rQPMy73KrF2oOUX2D5xDABg6Q9HMOZ/JV/YGyCfh58/DAcAbPz1fxh6xKfEWv+CKTg2ey4AIPL0FQTseAoyvQqyAhVkQgkzvQpmQgkzoUKgSzB2TgoDAFz56za6LRkFc5kKFnJr2CjsYKuyhZ3aFg6WtujUvBWGde8AoHDk6/z1f9Cwni0c7SwNc/4QEVHNZMrnt2QjUrm5uTh9+jQmT55stDwgIAAxMTHFrhMbG4uAgACjZYGBgVi9ejXy8vKgUCgQGxuL8ePHF6n57LPPyrzfxMREpKSkGO1LpVKhS5cuiImJKbGR0ul00On+nWVYqy3mq90VZOy332K3bkKJr1uqfkZL98L8Eb/FYEf2uH9fNPv/x/9PfXPyandDI3UtNaVwcsAS3Eq/Z/hZqVAULdDLgXwVZHol5Gb/nlJ6rF49WNxrBzlUkAsl5FDBXKaCuUwJhUyFNk88bqj1aeoB/5NToJKroJQroTZXQWmuhIVCBQuFCoFt2xlqn3/qCSzLOgILpRJWahUsVUpYq1WwUithbaGCs/2/p6W6+zSD8Mks8b39V7NG9fHXku1lqjWXm+Gppq5lqiUiotpFskYqLS0NBQUFcHZ2Nlru7OyMlJSUYtdJSUkptj4/Px9paWlwdXUtseb+Nsuy3/v/La7mzz//LPE9zZ8/H7NmzSrx9YpkrbYCtLaFIyv6whGV+6Mrcqhgb/XvN6+e8ngcUUeCoDBTQSFTQilXQWFW+F+VXIVnWjQ31Ab5+yH3yFqozZWwUKqgVihhqVLBSqWChVKJTq3+bXgm9g/Aq7dSYGOpgo2FCtYWyhK/cv5moC/eDPxfmd6bn5e7YRTpYVzqWWP0i8+WqZaIiKiiSX6xuUxmfBpECFFk2cPqH1xelm1WVM1/hYeHY8KEf0eJtFot3NzcSqx/FBvC3sIGlO0bUhP6d8OE/t3KVNu5jSc6t/F8eCEKvynGr5oTEVFdJlkj5ejoCLlcXmT0KTU1tchI0H0uLi7F1pubm6N+/fql1tzfZln26+LiAqBwZMrV1bXYmuKoVCqoVKoSXyciIqLa5eETw1QSpVIJHx8fREZGGi2PjIyEv79/sev4+fkVqT9w4ADat28Pxf9fr1NSzf1tlmW/np6ecHFxMarJzc1FdHR0idmIiIioDhIS2rx5s1AoFGL16tXi4sWLIiwsTFhZWYnr168LIYSYPHmyCA4ONtRfu3ZNWFpaivHjx4uLFy+K1atXC4VCIbZv326oOXbsmJDL5WLBggUiISFBLFiwQJibm4vjx4+Xeb9CCLFgwQJhZ2cnduzYIc6dOydeffVV4erqKrRabZnfn0ajEQCERqN5lMNEREREVciUz29JGykhhFi2bJnw8PAQSqVSeHt7i+joaMNrw4YNE126dDGqj4qKEu3atRNKpVI0btxYrFixosg2t23bJpo3by4UCoVo0aKFiIiIMGm/Qgih1+vFjBkzhIuLi1CpVKJz587i3LlzJr03NlJEREQ1jymf35LOI1XbVeY8UkRERFQ5TPn8luwaKSIiIqKajo0UERERUTmxkSIiIiIqJzZSREREROXERoqIiIionNhIEREREZUTGykiIiKicmIjRURERFRObKSIiIiIyslc6gC12f1J47VarcRJiIiIqKzuf26X5eYvbKQqUXp6OgDAzc1N4iRERERkqvT0dNjZ2ZVaw3vtVSK9Xo+bN2/CxsYGMpmsQret1Wrh5uaGGzdu8D5+D8FjVXY8VmXHY1V2PFZlx2NVdpV5rIQQSE9PR8OGDWFmVvpVUByRqkRmZmZo1KhRpe7D1taW/7OVEY9V2fFYlR2PVdnxWJUdj1XZVdaxethI1H282JyIiIionNhIEREREZUTG6kaSqVSYcaMGVCpVFJHqfZ4rMqOx6rseKzKjseq7Hisyq66HCtebE5ERERUThyRIiIiIionNlJERERE5cRGioiIiKic2EgRERERlRMbqVpEp9PhqaeegkwmQ1xcnNRxqqWXXnoJ7u7uUKvVcHV1RXBwMG7evCl1rGrn+vXrePPNN+Hp6QkLCws0bdoUM2bMQG5urtTRqqW5c+fC398flpaWsLe3lzpOtbN8+XJ4enpCrVbDx8cHR44ckTpStXP48GH06dMHDRs2hEwmw65du6SOVG3Nnz8fHTp0gI2NDZycnNCvXz9cunRJsjxspGqRDz74AA0bNpQ6RrXWrVs3bN26FZcuXUJERASuXr2Kl19+WepY1c7vv/8OvV6Pr776ChcuXMCnn36KlStXYsqUKVJHq5Zyc3MxaNAgvP3221JHqXa2bNmCsLAwTJ06FWfOnEGnTp3Qs2dPJCUlSR2tWsnMzETbtm2xdOlSqaNUe9HR0XjnnXdw/PhxREZGIj8/HwEBAcjMzJQmkKBaYe/evaJFixbiwoULAoA4c+aM1JFqhN27dwuZTCZyc3OljlLtLVq0SHh6ekodo1pbs2aNsLOzkzpGtfL000+L0NBQo2UtWrQQkydPlihR9QdA7Ny5U+oYNUZqaqoAIKKjoyXZP0ekaoF//vkHISEh+O6772BpaSl1nBrjzp072LhxI/z9/aFQKKSOU+1pNBrUq1dP6hhUg+Tm5uL06dMICAgwWh4QEICYmBiJUlFto9FoAECyv5/YSNVwQggMHz4coaGhaN++vdRxaoRJkybBysoK9evXR1JSEnbv3i11pGrv6tWr+PLLLxEaGip1FKpB0tLSUFBQAGdnZ6Plzs7OSElJkSgV1SZCCEyYMAHPPvssWrduLUkGNlLV1MyZMyGTyUp9nDp1Cl9++SW0Wi3Cw8OljiyZsh6r+yZOnIgzZ87gwIEDkMvleP311yHqyAT/ph4rALh58yZ69OiBQYMGYeTIkRIlr3rlOVZUPJlMZvRcCFFkGVF5vPvuuzh79iw2bdokWQbeIqaaSktLQ1paWqk1jRs3xiuvvIIffvjB6C+lgoICyOVyDBkyBOvWravsqJIr67FSq9VFlv/1119wc3NDTEwM/Pz8KititWHqsbp58ya6desGX19frF27FmZmdeffXuX5vVq7di3CwsJw7969Sk5XM+Tm5sLS0hLbtm1D//79DcvHjRuHuLg4REdHS5iu+pLJZNi5cyf69esndZRqbcyYMdi1axcOHz4MT09PyXKYS7ZnKpWjoyMcHR0fWvfFF19gzpw5huc3b95EYGAgtmzZAl9f38qMWG2U9VgV5/6/I3Q6XUVGqrZMOVZ///03unXrBh8fH6xZs6ZONVHAo/1eUSGlUgkfHx9ERkYaNVKRkZHo27evhMmoJhNCYMyYMdi5cyeioqIkbaIANlI1nru7u9Fza2trAEDTpk3RqFEjKSJVW7/99ht+++03PPvss3BwcMC1a9cwffp0NG3atE6MRpni5s2b6Nq1K9zd3bF48WLcunXL8JqLi4uEyaqnpKQk3LlzB0lJSSgoKDDM4/b4448b/p+sqyZMmIDg4GC0b98efn5+WLVqFZKSkni93QMyMjLwxx9/GJ4nJiYiLi4O9erVK/L3fF33zjvv4Pvvv8fu3bthY2NjuN7Ozs4OFhYWVR9Iku8KUqVJTEzk9AclOHv2rOjWrZuoV6+eUKlUonHjxiI0NFT89ddfUkerdtasWSMAFPugooYNG1bssTp06JDU0aqFZcuWCQ8PD6FUKoW3t7dkX1Ovzg4dOlTs79CwYcOkjlbtlPR305o1ayTJw2ukiIiIiMqpbl30QERERFSB2EgRERERlRMbKSIiIqJyYiNFREREVE5spIiIiIjKiY0UERERUTmxkSIiIiIqJzZSRFTlunbtirCwMKljFOv27dtwcnLC9evXAQBRUVGQyWSVfv+88u5n7dq1sLe3N2mdDh06YMeOHSatQ0TFYyNFRDVecnIyXnvtNTRv3hxmZmYlNmkRERHw8vKCSqWCl5cXdu7cWaRm/vz56NOnDxo3bly5oSU0bdo0TJ48GXq9XuooRDUeGykiqvF0Oh0aNGiAqVOnom3btsXWxMbGIigoCMHBwYiPj0dwcDAGDx6MEydOGGqys7OxevVqjBw5sqqiS6J3797QaDT4+eefpY5CVOOxkSIiSd29exevv/46HBwcYGlpiZ49e+LKlStGNV9//TXc3NxgaWmJ/v37Y8mSJUansxo3bozPP/8cr7/+Ouzs7Irdz2effYbu3bsjPDwcLVq0QHh4OJ5//nl89tlnhpp9+/bB3Ny81JtY3759G6+++ioaNWoES0tLPPnkk9i0aZNRTdeuXTFmzBiEhYXBwcEBzs7OWLVqFTIzM/HGG2/AxsYGTZs2xb59+4ps/9ixY2jbti3UajV8fX1x7tw5o9fXrl0Ld3d3w7G4ffu20etXr15F37594ezsDGtra3To0AEHDx40qpHL5ejVq1eR3ERkOjZSRCSp4cOH49SpU9izZw9iY2MhhECvXr2Ql5cHoLCxCA0Nxbhx4xAXF4fu3btj7ty5Ju8nNjYWAQEBRssCAwMRExNjeH748GG0b9++1O3k5OTAx8cHP/74I86fP4+33noLwcHBRiNbALBu3To4Ojrit99+w5gxY/D2229j0KBB8Pf3x//+9z8EBgYiODgYWVlZRutNnDgRixcvxsmTJ+Hk5ISXXnrJcCxOnDiBESNGYPTo0YiLi0O3bt0wZ84co/UzMjLQq1cvHDx4EGfOnEFgYCD69OmDpKQko7qnn34aR44cKdvBI6KSSXKrZCKq07p06SLGjRsnLl++LACIY8eOGV5LS0sTFhYWYuvWrUIIIYKCgkTv3r2N1h8yZIiws7MrddsPUigUYuPGjUbLNm7cKJRKpeF53759xYgRI4xqDh06JACIu3fvlvh+evXqJd577z2jDM8++6zheX5+vrCyshLBwcGGZcnJyQKAiI2NNdrP5s2bDTW3b98WFhYWYsuWLUIIIV599VXRo0cPo30HBQWVeCzu8/LyEl9++aXRst27dwszMzNRUFBQ6rpEVDqOSBGRZBISEmBubg5fX1/Dsvr166N58+ZISEgAAFy6dAlPP/200XoPPi8rmUxm9FwIYbQsOzsbarW61G0UFBRg7ty5aNOmDerXrw9ra2scOHCgyIhPmzZtDD/L5XLUr18fTz75pGGZs7MzACA1NdVovf+eVqxXr57RsUhISChy2vHB55mZmfjggw/g5eUFe3t7WFtb4/fffy+Sz8LCAnq9HjqdrtT3S0SlM5c6ABHVXUKIEpffb3AebHZKW680Li4uSElJMVqWmppqaGgAwNHREXfv3i11O5988gk+/fRTfPbZZ3jyySdhZWWFsLAw5ObmGtUpFAqj5zKZzGjZ/fdUlm/O/fdYPMzEiRPx888/Y/HixXj88cdhYWGBl19+uUi+O3fuwNLSEhYWFg/dJhGVjCNSRCQZLy8v5OfnG11fdPv2bVy+fBktW7YEALRo0QK//fab0XqnTp0yeV9+fn6IjIw0WnbgwAH4+/sbnrdr1w4XL14sdTtHjhxB3759MXToULRt2xZNmjQpcnH8ozh+/Ljh57t37+Ly5cto0aIFgMLj9d/XH6y/n2/48OHo378/nnzySbi4uBjmxPqv8+fPw9vbu8JyE9VVbKSISDLNmjVD3759ERISgqNHjyI+Ph5Dhw7FY489hr59+wIAxowZg71792LJkiW4cuUKvvrqK+zbt6/IKFVcXBzi4uKQkZGBW7duIS4uzqgpGjduHA4cOICFCxfi999/x8KFC3Hw4EGjOacCAwNx4cKFUkelHn/8cURGRiImJgYJCQkYNWpUkZGuRzF79mz88ssvOH/+PIYPHw5HR0f069cPADB27Fjs378fixYtwuXLl7F06VLs37+/SL4dO3YgLi4O8fHxeO2114od9Tpy5EiRi++JyHRspIhIUmvWrIGPjw9efPFF+Pn5QQiBvXv3Gk6DPfPMM1i5ciWWLFmCtm3bYv/+/Rg/fnyRa5natWuHdu3a4fTp0/j+++/Rrl079OrVy/C6v78/Nm/ejDVr1qBNmzZYu3YttmzZYnR91pNPPon27dtj69atJeadNm0avL29ERgYiK5du8LFxcXQ6FSEBQsWYNy4cfDx8UFycjL27NkDpVIJAOjYsSO++eYbfPnll3jqqadw4MABfPjhh0brf/rpp3BwcIC/vz/69OmDwMDAIiNPf//9N2JiYvDGG29UWG6iukomynOxARGRhEJCQvD7779Xytf39+7di/fffx/nz5+HmVnt/LfmxIkTodFosGrVKqmjENV4vNiciKq9xYsXo3v37rCyssK+ffuwbt06LF++vFL21atXL1y5cgV///033NzcKmUfUnNycsL7778vdQyiWoEjUkRU7Q0ePBhRUVFIT09HkyZNMGbMGISGhkodi4iIjRQRERFRedXOCwCIiIiIqgAbKSIiIqJyYiNFREREVE5spIiIiIjKiY0UERERUTmxkSIiIiIqJzZSREREROXERoqIiIionNhIEREREZXT/wEstM74qGAwcwAAAABJRU5ErkJggg==",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "execution_count": 4,
+   "id": "bb5f871b",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
    "source": [
     "import numpy as np\n",
     "import pandas as pd\n",
@@ -1049,8 +887,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c8ba407",
-   "metadata": {},
+   "id": "4724185f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see here, when compared to the code which includes explicitely the\n",
     "intercept column, that our MSE value is actually smaller. This is\n",
@@ -1062,8 +902,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2fd1861",
-   "metadata": {},
+   "id": "63a40c4b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Test Function for what happens with OLS, Ridge and Lasso\n",
     "\n",
@@ -1085,38 +927,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
-   "id": "fc3a2cef",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[ 0.99762142  0.46713267 -2.86273371  4.1801709  -1.87068171]\n",
-      "Test MSE OLS\n",
-      "0.9990927157696581\n",
-      "0.001 [ 0.99884986  0.44395301 -2.76190591  4.02693858 -1.7953443 ]\n",
-      "0.001 [ 1.00497837 -0.11830757 -0.          0.          0.0064472 ]\n",
-      "0.021544346900318832 [ 1.01267007  0.18307821 -1.62725226  2.30283188 -0.94781064]\n",
-      "0.021544346900318832 [ 0.92756256 -0.         -0.         -0.         -0.        ]\n",
-      "0.46415888336127775 [ 1.02895918 -0.12952339 -0.27065633  0.25185716  0.05519999]\n",
-      "0.46415888336127775 [0.48494803 0.         0.         0.         0.        ]\n",
-      "10.0 [ 1.02077431 -0.14568914 -0.10041812  0.03846169  0.11382896]\n",
-      "10.0 [0. 0. 0. 0. 0.]\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
+   "execution_count": 5,
+   "id": "44e30f6c",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
    "source": [
     "import numpy as np\n",
     "import matplotlib.pyplot as plt\n",
@@ -1184,16 +1001,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab2848a3",
-   "metadata": {},
+   "id": "7614dd80",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "How can we understand this?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1a39d66b",
-   "metadata": {},
+   "id": "8d0b865b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Linking the regression analysis with a statistical interpretation\n",
     "\n",
@@ -1219,8 +1040,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c80ed3da",
-   "metadata": {},
+   "id": "7b3043fb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -1233,8 +1056,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd53a459",
-   "metadata": {},
+   "id": "cbfeab43",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The randomness of $\\varepsilon_i$ implies that\n",
     "$\\mathbf{y}_i$ is also a random variable. In particular,\n",
@@ -1250,8 +1075,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9d7404c8",
-   "metadata": {},
+   "id": "67514fcf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Assumptions made\n",
     "\n",
@@ -1262,8 +1089,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "113ca56d",
-   "metadata": {},
+   "id": "f7557d7a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
@@ -1272,8 +1101,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8899915",
-   "metadata": {},
+   "id": "60f32679",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We approximate this function with our model from the solution of the linear regression equations, that is our\n",
     "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with"
@@ -1281,8 +1112,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ec411c8d",
-   "metadata": {},
+   "id": "d5268089",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
@@ -1291,8 +1124,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8158c2fd",
-   "metadata": {},
+   "id": "1d98108e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Expectation value and variance\n",
     "\n",
@@ -1301,8 +1136,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8040f293",
-   "metadata": {},
+   "id": "6940b69c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -1315,8 +1152,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc6e1bab",
-   "metadata": {},
+   "id": "8c9b9e98",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "while\n",
     "its variance is"
@@ -1324,8 +1163,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "932e6746",
-   "metadata": {},
+   "id": "337a1fcc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
@@ -1345,8 +1186,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca2c3e6f",
-   "metadata": {},
+   "id": "b179e200",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
     "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)."
@@ -1354,8 +1197,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd423094",
-   "metadata": {},
+   "id": "189f6827",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Expectation value and variance for $\\boldsymbol{\\beta}$\n",
     "\n",
@@ -1364,8 +1209,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "78741c91",
-   "metadata": {},
+   "id": "d145d77b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
@@ -1374,8 +1221,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "72a9fdce",
-   "metadata": {},
+   "id": "d2c188e3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This means that the estimator of the regression parameters is unbiased.\n",
     "\n",
@@ -1386,8 +1235,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "02a6f171",
-   "metadata": {},
+   "id": "5e4e08b3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{eqnarray*}\n",
@@ -1415,8 +1266,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d435171e",
-   "metadata": {},
+   "id": "b5d610f9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have used  that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n",
     "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n",
@@ -1435,8 +1288,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c6b6fcf",
-   "metadata": {},
+   "id": "d6bfa2a5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
@@ -1445,8 +1300,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c3fa21c",
-   "metadata": {},
+   "id": "dff8f227",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see clearly that \n",
     "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n",
@@ -1456,8 +1313,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab5a68b3",
-   "metadata": {},
+   "id": "08d29968",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
@@ -1466,8 +1325,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc57cd3e",
-   "metadata": {},
+   "id": "20600c78",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n",
     "\n",
@@ -1476,8 +1337,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c6790592",
-   "metadata": {},
+   "id": "6ff509af",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
@@ -1486,8 +1349,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4cedacf8",
-   "metadata": {},
+   "id": "bf860be3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The difference is non-negative definite since each component of the\n",
     "matrix product is non-negative definite. \n",
@@ -1496,8 +1361,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8db6c289",
-   "metadata": {},
+   "id": "9ea27c12",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Deriving OLS from a probability distribution\n",
     "\n",
@@ -1517,8 +1384,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "84ddde11",
-   "metadata": {},
+   "id": "5548a408",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "y_i\\sim \\mathcal{N}(\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta}, \\sigma^2)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
@@ -1527,8 +1396,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d50973f1",
-   "metadata": {},
+   "id": "4bdaa109",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Independent and Identically Distributed (iid)\n",
     "\n",
@@ -1538,8 +1409,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "697b0e8d",
-   "metadata": {},
+   "id": "52d62b74",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(y_i, \\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]},\n",
@@ -1548,8 +1421,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8076ed56",
-   "metadata": {},
+   "id": "9cb00279",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which reads as finding the likelihood of an event $y_i$ with the input variables $\\boldsymbol{X}$ given the parameters (to be determined) $\\boldsymbol{\\beta}$.\n",
     "\n",
@@ -1558,8 +1433,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a74b5da5",
-   "metadata": {},
+   "id": "26afb900",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{y},\\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}=\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta}).\n",
@@ -1568,8 +1445,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b6cd0d8",
-   "metadata": {},
+   "id": "da41b50e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We will write this in a more compact form reserving $\\boldsymbol{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is\n",
     "in case we have a simple one-dimensional input and output case"
@@ -1577,8 +1456,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e757da3",
-   "metadata": {},
+   "id": "7719498b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})].\n",
@@ -1587,8 +1468,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6d256b3f",
-   "metadata": {},
+   "id": "f43bef91",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the more general case the various inputs should be replaced by the possible features represented by the input data set $\\boldsymbol{X}$. \n",
     "We can now rewrite the above probability as"
@@ -1596,8 +1479,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "417f2dfa",
-   "metadata": {},
+   "id": "d8c9aff8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
@@ -1606,16 +1491,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c071fba",
-   "metadata": {},
+   "id": "bdb01c2a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is a conditional probability (see below) and reads as the likelihood of a domain of events $\\boldsymbol{D}$ given a set of parameters $\\boldsymbol{\\beta}$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e3190436",
-   "metadata": {},
+   "id": "679bc6f9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Maximum Likelihood Estimation (MLE)\n",
     "\n",
@@ -1643,8 +1532,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90dafea3",
-   "metadata": {},
+   "id": "bde130e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## A new Cost Function\n",
     "\n",
@@ -1653,8 +1544,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77a99eca",
-   "metadata": {},
+   "id": "feac486a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}=-\\log{\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})}=-\\sum_{i=0}^{n-1}\\log{p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})},\n",
@@ -1663,16 +1556,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd43947c",
-   "metadata": {},
+   "id": "ad8049ca",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which becomes"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "94454b77",
-   "metadata": {},
+   "id": "14dd5b9e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}=\\frac{n}{2}\\log{2\\pi\\sigma^2}+\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}.\n",
@@ -1681,16 +1578,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "25f0b40e",
-   "metadata": {},
+   "id": "2f42fa20",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Taking the derivative of the *new* cost function with respect to the parameters $\\beta$ we recognize our familiar OLS equation, namely"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "31a7da5a",
-   "metadata": {},
+   "id": "0484474c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right) =0,\n",
@@ -1699,16 +1600,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "340ab07c",
-   "metadata": {},
+   "id": "6517f251",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which leads to the well-known OLS equation for the optimal paramters $\\beta$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6f9b8d49",
-   "metadata": {},
+   "id": "0238ae3e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n",
@@ -1717,16 +1622,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2ba639bc",
-   "metadata": {},
+   "id": "f829d004",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1ac764de",
-   "metadata": {},
+   "id": "6c68ecfe",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## More basic Statistics and Bayes' theorem\n",
     "\n",
@@ -1743,8 +1652,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ee7eb0ec",
-   "metadata": {},
+   "id": "0342e437",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X \\cup Y)= p(X)+p(Y)-p(X \\cap Y).\n",
@@ -1753,16 +1664,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a7c1c442",
-   "metadata": {},
+   "id": "ab4bc57a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "**The product rule (aka joint probability) is given by.**"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "902c9e42",
-   "metadata": {},
+   "id": "a67bcf0c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X \\cup Y)= p(X,Y)= p(X\\vert Y)p(Y)=p(Y\\vert X)p(X),\n",
@@ -1771,8 +1686,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "01385522",
-   "metadata": {},
+   "id": "2b549c11",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we read $p(X\\vert Y)$ as the likelihood of obtaining $X$ given $Y$.\n",
     "\n",
@@ -1781,8 +1698,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a3157bc",
-   "metadata": {},
+   "id": "b572a768",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Marginal Probability\n",
     "\n",
@@ -1791,8 +1710,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8bef4a71",
-   "metadata": {},
+   "id": "438ce1e4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X)=\\sum_{i=0}^{n-1}p(X,Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert Y=y_i)p(Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert y_i)p(y_i).\n",
@@ -1801,8 +1722,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e6fb4d8",
-   "metadata": {},
+   "id": "85b16691",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Conditional  Probability\n",
     "\n",
@@ -1811,8 +1734,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "72979501",
-   "metadata": {},
+   "id": "a01bf532",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)}=\\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}.\n",
@@ -1821,8 +1746,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1e9dfa57",
-   "metadata": {},
+   "id": "ba71fce0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Bayes' Theorem\n",
     "\n",
@@ -1831,8 +1758,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "911e3e33",
-   "metadata": {},
+   "id": "e49e52c9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)},\n",
@@ -1841,16 +1770,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "638834de",
-   "metadata": {},
+   "id": "fcae9855",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which we can rewrite as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "bb77868f",
-   "metadata": {},
+   "id": "b9152eec",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X\\vert Y)= \\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}=\\frac{p(Y\\vert X)p(X)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)},\n",
@@ -1859,16 +1792,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a7a9efd",
-   "metadata": {},
+   "id": "42302594",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which is Bayes' theorem. It allows us to evaluate the uncertainty in in $X$ after we have observed $Y$. We can easily interchange $X$ with $Y$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d7c9013f",
-   "metadata": {},
+   "id": "376c25b8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Interpretations of Bayes' Theorem\n",
     "\n",
@@ -1884,8 +1821,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58f244e9",
-   "metadata": {},
+   "id": "724ee74e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Example of Usage of Bayes' theorem\n",
     "\n",
@@ -1903,8 +1842,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "174309f3",
-   "metadata": {},
+   "id": "f655ed47",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X=1\\vert Y=1) =0.8.\n",
@@ -1913,8 +1854,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e31dfe4b",
-   "metadata": {},
+   "id": "f34f4249",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This obviously sounds  scary since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n",
     "It is however not correct, as the following Bayesian analysis shows."
@@ -1922,8 +1865,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb4db137",
-   "metadata": {},
+   "id": "0cd4faab",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Doing it correctly\n",
     "\n",
@@ -1933,8 +1878,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8b391eda",
-   "metadata": {},
+   "id": "8c12d2ed",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(Y=1) =0.004.\n",
@@ -1943,16 +1890,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "99f66158",
-   "metadata": {},
+   "id": "4db8a637",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f8ae73a6",
-   "metadata": {},
+   "id": "e29462d1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(X=1\\vert Y=0) =0.1.\n",
@@ -1961,16 +1912,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3e378e4",
-   "metadata": {},
+   "id": "b17d001a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2f684426",
-   "metadata": {},
+   "id": "e923ab3c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(Y=1\\vert X=1)=\\frac{p(X=1\\vert Y=1)p(Y=1)}{p(X=1\\vert Y=1)p(Y=1)+p(X=1\\vert Y=0)p(Y=0)}=\\frac{0.8\\times 0.004}{0.8\\times 0.004+0.1\\times 0.996}=0.031.\n",
@@ -1979,16 +1934,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "983982c0",
-   "metadata": {},
+   "id": "debb0024",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "That is, in case of a positive test, there is only a $3\\%$ chance of having breast cancer!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "88ef6911",
-   "metadata": {},
+   "id": "f62723b0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Bayes' Theorem and Ridge and Lasso Regression\n",
     "\n",
@@ -1999,8 +1958,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ba0657c3",
-   "metadata": {},
+   "id": "f7371f10",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})],\n",
@@ -2009,16 +1970,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8aa4829",
-   "metadata": {},
+   "id": "7a21e500",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "is given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "52adbfd2",
-   "metadata": {},
+   "id": "6dd4a97e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
@@ -2027,16 +1992,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2178fe65",
-   "metadata": {},
+   "id": "90d532d6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set $\\boldsymbol{\\beta}$ given a domain of events $\\boldsymbol{D}$?  That is, how can we define the posterior probability"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "97ce5044",
-   "metadata": {},
+   "id": "320c6012",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D}).\n",
@@ -2045,16 +2014,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afae14bb",
-   "metadata": {},
+   "id": "d80e04f2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Bayes' theorem comes to our rescue here since (omitting the normalization constant)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3a3fea10",
-   "metadata": {},
+   "id": "ded985c4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})\\propto p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})p(\\boldsymbol{\\beta}).\n",
@@ -2063,16 +2036,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aedd41ba",
-   "metadata": {},
+   "id": "c7b32590",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "fe2f2354",
-   "metadata": {},
+   "id": "234af442",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Ridge and Bayes\n",
     "\n",
@@ -2085,8 +2062,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ce5820c0",
-   "metadata": {},
+   "id": "2dfde9e5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
@@ -2095,16 +2074,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "186e4fbd",
-   "metadata": {},
+   "id": "86074376",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4e3b13c4",
-   "metadata": {},
+   "id": "269ca6a0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta\\vert\\boldsymbol{D})}=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
@@ -2113,8 +2096,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "501fe4bf",
-   "metadata": {},
+   "id": "c11f088c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can now optimize this quantity with respect to $\\boldsymbol{\\beta}$. As we\n",
     "did for OLS, this is most conveniently done by taking the negative\n",
@@ -2124,8 +2109,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37e09a0d",
-   "metadata": {},
+   "id": "e3dfd57a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{2\\tau^2}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
@@ -2134,16 +2121,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e2d62663",
-   "metadata": {},
+   "id": "75069581",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and replacing $1/2\\tau^2$ with $\\lambda$ we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "795eb5be",
-   "metadata": {},
+   "id": "b68431fc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
@@ -2152,16 +2143,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be730c19",
-   "metadata": {},
+   "id": "d98b69bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which is our Ridge cost function!  Nice, isn't it?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "755ba46f",
-   "metadata": {},
+   "id": "a09d696d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Lasso and Bayes\n",
     "\n",
@@ -2170,8 +2165,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "99493766",
-   "metadata": {},
+   "id": "e88b5d96",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
@@ -2180,16 +2177,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc738267",
-   "metadata": {},
+   "id": "95f30084",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "fd28fe2f",
-   "metadata": {},
+   "id": "e6b5f3a5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
@@ -2198,8 +2199,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4af71ac",
-   "metadata": {},
+   "id": "6beb7fe2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Taking the negative\n",
     "logarithm of the posterior probability and leaving out the\n",
@@ -2208,8 +2211,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ccfd13a",
-   "metadata": {},
+   "id": "ae55d2b3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{\\tau}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
@@ -2218,16 +2223,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d3cb4dac",
-   "metadata": {},
+   "id": "b64fe3bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and replacing $1/\\tau$ with $\\lambda$ we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6468d431",
-   "metadata": {},
+   "id": "127a43b6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
@@ -2236,16 +2245,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f08cda3",
-   "metadata": {},
+   "id": "8e5662a1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which is our Lasso cost function!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "ab2ab549",
-   "metadata": {},
+   "id": "32e29846",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Material for the active learning sessions Tuesday and Wednesday\n",
     "\n",
@@ -2254,8 +2267,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8b1b1af",
-   "metadata": {},
+   "id": "f76c0e33",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Linear Regression and  the SVD\n",
     "\n",
@@ -2264,8 +2279,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b21a8cd",
-   "metadata": {},
+   "id": "66b8fa8e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n",
@@ -2274,16 +2291,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bed0cd60",
-   "metadata": {},
+   "id": "433240d0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Since the matrices here have dimension $p\\times p$, with $p$ corresponding to the singular values, we defined last week the matrix"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "15e05a44",
-   "metadata": {},
+   "id": "d5600874",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\end{bmatrix},\n",
@@ -2292,16 +2313,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35b02cee",
-   "metadata": {},
+   "id": "85be35c4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the tilde-matrix $\\tilde{\\boldsymbol{\\Sigma}}$ is a matrix of dimension $p\\times p$ containing only the singular values $\\sigma_i$, that is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3b44d036",
-   "metadata": {},
+   "id": "ae1743d0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{\\boldsymbol{\\Sigma}}=\\begin{bmatrix} \\sigma_0 & 0 & 0 & \\dots & 0 & 0 \\\\\n",
@@ -2315,16 +2340,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb0a23d3",
-   "metadata": {},
+   "id": "8632fdab",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "meaning we can write"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "11b857c6",
-   "metadata": {},
+   "id": "001f688f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2\\boldsymbol{V}^T.\n",
@@ -2333,16 +2362,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32df1603",
-   "metadata": {},
+   "id": "89f09ada",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Multiplying from the right with $\\boldsymbol{V}$ (using the orthogonality of $\\boldsymbol{V}$) we get"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1589aacd",
-   "metadata": {},
+   "id": "f247b99d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{V}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2.\n",
@@ -2351,8 +2384,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9b878c6",
-   "metadata": {},
+   "id": "b9759ceb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## What does it mean?\n",
     "\n",
@@ -2363,8 +2398,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf01dcea",
-   "metadata": {},
+   "id": "9b78b20b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{v}_i=\\boldsymbol{v}_i\\sigma_i^2.\n",
@@ -2373,8 +2410,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8d12b2e4",
-   "metadata": {},
+   "id": "8aa0afb6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In other words, each non-zero singular value of $\\boldsymbol{X}$ is a positive\n",
     "square root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$.  It means also that\n",
@@ -2393,8 +2432,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a6ecf55",
-   "metadata": {},
+   "id": "91cc0736",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{C}[\\boldsymbol{X}]=\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X},\n",
@@ -2403,8 +2444,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "707eceb6",
-   "metadata": {},
+   "id": "5e183cb3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "meaning that every squared non-singular value of $\\boldsymbol{X}$ divided by $n$ (\n",
     "the number of samples) are the eigenvalues of the covariance\n",
@@ -2416,8 +2459,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "096cd515",
-   "metadata": {},
+   "id": "d773d376",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## And finally  $\\boldsymbol{X}\\boldsymbol{X}^T$\n",
     "\n",
@@ -2426,8 +2471,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f518ce45",
-   "metadata": {},
+   "id": "995b6807",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{U}^T.\n",
@@ -2436,16 +2483,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3aee5ef2",
-   "metadata": {},
+   "id": "be88c78f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Since the matrices here have dimension $n\\times n$, we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f57a6398",
-   "metadata": {},
+   "id": "a68de77c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\Sigma}\\boldsymbol{\\Sigma}^T = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}}  \\boldsymbol{0}\\\\ \\end{bmatrix}=\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix},\n",
@@ -2454,16 +2505,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47fbfc39",
-   "metadata": {},
+   "id": "0aacebb0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "leading to"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "ab044c32",
-   "metadata": {},
+   "id": "1d5d147c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\boldsymbol{U}^T.\n",
@@ -2472,16 +2527,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "887c7081",
-   "metadata": {},
+   "id": "47bad16c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Multiplying with $\\boldsymbol{U}$ from the right gives us the eigenvalue problem"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "aef54d23",
-   "metadata": {},
+   "id": "0b4c5d2d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U}=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}.\n",
@@ -2490,8 +2549,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58545dde",
-   "metadata": {},
+   "id": "238703f0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It means that the eigenvalues of $\\boldsymbol{X}\\boldsymbol{X}^T$ are again given by\n",
     "the non-zero singular values plus now a series of zeros.  The column\n",
@@ -2505,8 +2566,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a51460b7",
-   "metadata": {},
+   "id": "93b3eb1c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Code for SVD and Inversion of Matrices\n",
     "\n",
@@ -2517,8 +2580,11 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "b9e2d8bf",
-   "metadata": {},
+   "id": "d67219fd",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import numpy as np\n",
@@ -2528,8 +2594,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "491c3981",
-   "metadata": {},
+   "id": "ee1db513",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD."
    ]
@@ -2537,8 +2605,11 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "33f26b11",
-   "metadata": {},
+   "id": "b79ef2b0",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import numpy as np\n",
@@ -2574,8 +2645,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b95201dc",
-   "metadata": {},
+   "id": "4f696bf3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Inverse of Rectangular Matrix\n",
     "\n",
@@ -2592,8 +2665,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1b91d55",
-   "metadata": {},
+   "id": "8fede7c1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{A}_{\\mathrm{PI}}= \\boldsymbol{V}\\boldsymbol{D}_{\\mathrm{PI}}\\boldsymbol{U}^T,\n",
@@ -2602,8 +2677,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "783cf37c",
-   "metadata": {},
+   "id": "0929ddbe",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{D}_{\\mathrm{PI}}$ can be calculated by creating a diagonal matrix from $\\boldsymbol{\\Sigma}$ where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD."
    ]
@@ -2611,8 +2688,11 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "15f3c4fd",
-   "metadata": {},
+   "id": "7656d012",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import numpy as np\n",
@@ -2642,16 +2722,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cf791d68",
-   "metadata": {},
+   "id": "fa3121d3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "As you can see from this example, our own decomposition based on the SVD agrees with  the pseudoinverse algorithm provided by **Numpy**."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e0d6867b",
-   "metadata": {},
+   "id": "b543469e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Ridge and LASSO Regression\n",
     "\n",
@@ -2661,8 +2745,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a7560d5",
-   "metadata": {},
+   "id": "8a946491",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
@@ -2671,16 +2757,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a41ec2e",
-   "metadata": {},
+   "id": "039edb17",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "or we can state it as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "cd9c1584",
-   "metadata": {},
+   "id": "939dd966",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
@@ -2690,16 +2780,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a1809fd3",
-   "metadata": {},
+   "id": "32b81bbc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have used the definition of  a norm-2 vector, that is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "37d1dd03",
-   "metadata": {},
+   "id": "b554e41f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n",
@@ -2708,8 +2802,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e8f56dc",
-   "metadata": {},
+   "id": "e7a6391a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## From OLS to Ridge and Lasso\n",
     "\n",
@@ -2721,8 +2817,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d709e7cc",
-   "metadata": {},
+   "id": "8a9dd479",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
@@ -2732,8 +2830,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75ddcb47",
-   "metadata": {},
+   "id": "d89b0b9c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which leads to the Ridge regression minimization problem where we\n",
     "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n",
@@ -2744,8 +2844,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "15546e46",
-   "metadata": {},
+   "id": "88b4c066",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n",
@@ -2754,16 +2856,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "981f9eaf",
-   "metadata": {},
+   "id": "91069991",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have a new optimization equation"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "cd380e8c",
-   "metadata": {},
+   "id": "5abe220a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
@@ -2773,8 +2879,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3675c6af",
-   "metadata": {},
+   "id": "8446a073",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n",
     "\n",
@@ -2783,8 +2891,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e6a4788",
-   "metadata": {},
+   "id": "56c7a78e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n",
@@ -2793,8 +2903,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "25da0588",
-   "metadata": {},
+   "id": "0087d03a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Deriving the  Ridge Regression Equations\n",
     "\n",
@@ -2803,8 +2915,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7cc0e8b",
-   "metadata": {},
+   "id": "f9e2d48e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n",
@@ -2813,8 +2927,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5a9f8c59",
-   "metadata": {},
+   "id": "5fe2a8df",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and \n",
     "taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n",
@@ -2825,8 +2941,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ba67adb",
-   "metadata": {},
+   "id": "dea28fba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
@@ -2835,16 +2953,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3f707310",
-   "metadata": {},
+   "id": "bf9a8345",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "aa1900c6",
-   "metadata": {},
+   "id": "c71edcd0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n",
@@ -2853,16 +2975,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81277a42",
-   "metadata": {},
+   "id": "a5d23794",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "with $t$ a finite positive number."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f0a31465",
-   "metadata": {},
+   "id": "50df7fb3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Note on Scikit-Learn\n",
     "\n",
@@ -2871,8 +2997,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "15a2f869",
-   "metadata": {},
+   "id": "0c270857",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+n\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -2881,16 +3009,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56070d0e",
-   "metadata": {},
+   "id": "e176913b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In our codes where we compare our own codes with **Scikit-Learn**, we do thus not include the $1/n$ factor in the cost function."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1bbfe263",
-   "metadata": {},
+   "id": "95503781",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Comparison with OLS\n",
     "When we compare this with the ordinary least squares result we have"
@@ -2898,8 +3030,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2f4f4e30",
-   "metadata": {},
+   "id": "085a7886",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
@@ -2908,8 +3042,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e995e92",
-   "metadata": {},
+   "id": "c45cb7e8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
     "\n",
@@ -2922,8 +3058,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9935ba38",
-   "metadata": {},
+   "id": "98237c0b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## SVD analysis\n",
     "\n",
@@ -2933,8 +3071,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "66971309",
-   "metadata": {},
+   "id": "9adbd131",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta}  =\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n",
@@ -2943,16 +3083,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "083cb933",
-   "metadata": {},
+   "id": "e0fc7e75",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For Ridge regression this becomes"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2393f70a",
-   "metadata": {},
+   "id": "beeed609",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n",
@@ -2961,16 +3105,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2839df85",
-   "metadata": {},
+   "id": "1ec8688d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "725d6507",
-   "metadata": {},
+   "id": "e0e32455",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Interpreting the Ridge results\n",
     "\n",
@@ -2979,8 +3127,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3be3ed05",
-   "metadata": {},
+   "id": "d418a099",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n",
@@ -2989,8 +3139,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "42ba7d59",
-   "metadata": {},
+   "id": "8dd109b0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n",
     "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n",
@@ -3003,8 +3155,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a99bda20",
-   "metadata": {},
+   "id": "32acbbe9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## More interpretations\n",
     "\n",
@@ -3013,8 +3167,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "efa137df",
-   "metadata": {},
+   "id": "3bb2e7d3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n",
@@ -3023,16 +3179,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db30277c",
-   "metadata": {},
+   "id": "92b9e986",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In this case the standard OLS results in"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f0bff457",
-   "metadata": {},
+   "id": "0f66a3e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{n-1}\\boldsymbol{u}_i\\boldsymbol{u}_i^T\\boldsymbol{y},\n",
@@ -3041,16 +3201,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a85d40db",
-   "metadata": {},
+   "id": "337d6138",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "24c0cd1c",
-   "metadata": {},
+   "id": "94020497",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n",
@@ -3059,8 +3223,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1ff9f63d",
-   "metadata": {},
+   "id": "3c1f2724",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n",
     "the Ridge estimator converges to zero when the hyperparameter goes to\n",
@@ -3074,8 +3240,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "00573cf2",
-   "metadata": {},
+   "id": "f522846c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Deriving the  Lasso Regression Equations\n",
     "\n",
@@ -3084,8 +3252,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9fcb445",
-   "metadata": {},
+   "id": "f37b7397",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
@@ -3094,16 +3264,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b9ade8d2",
-   "metadata": {},
+   "id": "38a6da03",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Taking the derivative with respect to $\\boldsymbol{\\beta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "28ab82f3",
-   "metadata": {},
+   "id": "b733ca2b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{d \\vert \\beta\\vert}{d \\beta}=\\mathrm{sgn}(\\beta)=\\left\\{\\begin{array}{cc} 1 & \\beta > 0 \\\\-1 & \\beta < 0, \\end{array}\\right.\n",
@@ -3112,16 +3286,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "207f6e1a",
-   "metadata": {},
+   "id": "ca889f91",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have that the derivative of the cost function is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "02584f63",
-   "metadata": {},
+   "id": "1d0e3071",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C(\\boldsymbol{X},\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=-\\frac{2}{n}\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})+\\lambda sgn(\\boldsymbol{\\beta})=0,\n",
@@ -3130,16 +3308,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d493cd1f",
-   "metadata": {},
+   "id": "6ad964aa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and reordering we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d9e8a09b",
-   "metadata": {},
+   "id": "52eb19d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta}+\\lambda sgn(\\boldsymbol{\\beta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -3148,16 +3330,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d0a02f65",
-   "metadata": {},
+   "id": "a34076ac",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor $2/n$ in a redefinition of the parameter $\\lambda$. We will solve this type of problems using libraries like **scikit-learn**."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2045b02f",
-   "metadata": {},
+   "id": "530c29ac",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression\n",
     "\n",
@@ -3169,8 +3355,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0adb3c9d",
-   "metadata": {},
+   "id": "b8099a77",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2,\n",
@@ -3179,16 +3367,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca893163",
-   "metadata": {},
+   "id": "d4dfc8cc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and minimizing we have that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "873ca498",
-   "metadata": {},
+   "id": "bf7253d4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\beta}_i^{\\mathrm{OLS}} = y_i.\n",
@@ -3197,8 +3389,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f73c2649",
-   "metadata": {},
+   "id": "c0b7c255",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Ridge Regression\n",
     "\n",
@@ -3207,8 +3401,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08489c8d",
-   "metadata": {},
+   "id": "2546602c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\beta_i^2,\n",
@@ -3217,16 +3413,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "11e25a08",
-   "metadata": {},
+   "id": "c8b454a8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and minimizing we have that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2eac8387",
-   "metadata": {},
+   "id": "f1dffbcb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\beta}_i^{\\mathrm{Ridge}} = \\frac{y_i}{1+\\lambda}.\n",
@@ -3235,8 +3435,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b52a609",
-   "metadata": {},
+   "id": "14db4fe1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Lasso Regression\n",
     "\n",
@@ -3245,8 +3447,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68745c24",
-   "metadata": {},
+   "id": "5840fc8f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\vert\\beta_i\\vert=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\sqrt{\\beta_i^2},\n",
@@ -3255,16 +3459,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c77b8b18",
-   "metadata": {},
+   "id": "4a31c8b1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and minimizing we have that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f6454cc4",
-   "metadata": {},
+   "id": "80f5ea6d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "-2\\sum_{i=0}^{p-1}(y_i-\\beta_i)+\\lambda \\sum_{i=0}^{p-1}\\frac{(\\beta_i)}{\\vert\\beta_i\\vert}=0,\n",
@@ -3273,16 +3481,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7874a26a",
-   "metadata": {},
+   "id": "529ee90c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which leads to"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "539a5968",
-   "metadata": {},
+   "id": "d85cbf9e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}_i^{\\mathrm{Lasso}} = \\left\\{\\begin{array}{ccc}y_i-\\frac{\\lambda}{2} &\\mathrm{if} & y_i> \\frac{\\lambda}{2}\\\\\n",
@@ -3293,16 +3505,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0cd5e30",
-   "metadata": {},
+   "id": "e92311d7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\\beta_i$ for specific values of $\\lambda$. Ridge regression reduces on the other hand the values of $\\beta_i$ as function of $\\lambda$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "da1f1c65",
-   "metadata": {},
+   "id": "187085eb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Yet another Example\n",
     "\n",
@@ -3311,8 +3527,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c30c1157",
-   "metadata": {},
+   "id": "0eb5679e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y}=\\begin{bmatrix}4 \\\\ 2 \\\\3\\end{bmatrix},\n",
@@ -3321,16 +3539,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1fc75552",
-   "metadata": {},
+   "id": "3dd86b87",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and our inputs as a $3\\times 2$ design matrix"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "a955127d",
-   "metadata": {},
+   "id": "b9d05ed2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}=\\begin{bmatrix}2 & 0\\\\ 0 & 1 \\\\ 0 & 0\\end{bmatrix},\n",
@@ -3339,16 +3561,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2c25c0fd",
-   "metadata": {},
+   "id": "25c589cd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "meaning that we have two features and two unknown parameters $\\beta_0$ and $\\beta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "82979bed",
-   "metadata": {},
+   "id": "28186058",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The OLS case\n",
     "\n",
@@ -3357,8 +3583,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea2f8d9e",
-   "metadata": {},
+   "id": "8775423c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -3367,16 +3595,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "198468c6",
-   "metadata": {},
+   "id": "c9d8d70a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Inserting the above values we obtain that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "c54be5ed",
-   "metadata": {},
+   "id": "3e7e915b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\begin{bmatrix}2 \\\\ 2\\end{bmatrix},\n",
@@ -3385,16 +3617,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6413ac61",
-   "metadata": {},
+   "id": "131eedae",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The code which implements this simpler case is presented after the discussion of Ridge and Lasso."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3fc6d520",
-   "metadata": {},
+   "id": "83550688",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The Ridge case\n",
     "\n",
@@ -3403,8 +3639,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "97702dcd",
-   "metadata": {},
+   "id": "8942f0d0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -3413,16 +3651,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "44edbaef",
-   "metadata": {},
+   "id": "ac252166",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Inserting the above values we obtain that"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "22d02c34",
-   "metadata": {},
+   "id": "21c99ee8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\begin{bmatrix}\\frac{8}{4+\\lambda} \\\\ \\frac{2}{1+\\lambda}\\end{bmatrix},\n",
@@ -3431,8 +3673,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12cf8dc6",
-   "metadata": {},
+   "id": "689ab283",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "There is normally a constraint on the value of $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2$ via the parameter $\\lambda$.\n",
     "Let us for simplicity assume that $\\beta_0^2+\\beta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\\beta$ and $\\lambda$.\n",
@@ -3442,8 +3686,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "941de09b",
-   "metadata": {},
+   "id": "5a016164",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Writing the Cost Function\n",
     "\n",
@@ -3452,8 +3698,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "715d1ede",
-   "metadata": {},
+   "id": "10278ba8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}\\boldsymbol{\\beta}=\\begin{bmatrix} 2\\beta_0 \\\\ \\beta_1 \\\\0 \\end{bmatrix},\n",
@@ -3462,8 +3710,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0dd924b",
-   "metadata": {},
+   "id": "ae54d916",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\beta_0^2+\\beta_1^2),\n",
@@ -3472,16 +3722,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd40cb2f",
-   "metadata": {},
+   "id": "b70fda3d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and taking the derivative with respect to $\\beta_0$ we get"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f257ab54",
-   "metadata": {},
+   "id": "66e0f54b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0=\\frac{8}{4+\\lambda},\n",
@@ -3490,16 +3744,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ceb84ba5",
-   "metadata": {},
+   "id": "a2d10e26",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and for $\\beta_1$ we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d96cf277",
-   "metadata": {},
+   "id": "dce25955",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_1=\\frac{2}{1+\\lambda},\n",
@@ -3508,16 +3766,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "27b2d9c9",
-   "metadata": {},
+   "id": "28493542",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Using the constraint for $\\beta_0^2+\\beta_1^2=1$ we can constrain $\\lambda$ by solving"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d53c88da",
-   "metadata": {},
+   "id": "442d0da6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\left(\\frac{8}{4+\\lambda}\\right)^2+\\left(\\frac{2}{1+\\lambda}\\right)^2=1,\n",
@@ -3526,16 +3788,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70a82067",
-   "metadata": {},
+   "id": "c7d85d08",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which gives $\\lambda=4.571$ and $\\beta_0=0.933$ and $\\beta_1=0.359$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "cda36acb",
-   "metadata": {},
+   "id": "d944ea22",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Lasso case\n",
     "\n",
@@ -3545,8 +3811,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2faac1c",
-   "metadata": {},
+   "id": "b93efc0c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\vert\\beta_0\\vert+\\vert\\beta_1\\vert),\n",
@@ -3555,8 +3823,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "53192e1f",
-   "metadata": {},
+   "id": "418813a2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_0}=-4(4-2\\beta_0)+\\lambda\\mathrm{sgn}(\\beta_0)=0,\n",
@@ -3565,16 +3835,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67e7c28a",
-   "metadata": {},
+   "id": "486f3909",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "fd95e257",
-   "metadata": {},
+   "id": "cf29a145",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_1}=-2(2-\\beta_1)+\\lambda\\mathrm{sgn}(\\beta_1)=0.\n",
@@ -3583,8 +3857,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "29547ea5",
-   "metadata": {},
+   "id": "4e0a8c90",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We have now four cases to solve besides the trivial cases $\\beta_0$ and/or $\\beta_1$ are zero, namely\n",
     "1. $\\beta_0 > 0$ and $\\beta_1 > 0$,\n",
@@ -3598,8 +3874,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bad8636d",
-   "metadata": {},
+   "id": "d2cbf9b9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The first Case\n",
     "\n",
@@ -3608,8 +3886,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35d9a208",
-   "metadata": {},
+   "id": "c820e33a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "-4(4-2\\beta_0)+\\lambda=0,\n",
@@ -3618,16 +3898,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b8251035",
-   "metadata": {},
+   "id": "2962087b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af16cca9",
-   "metadata": {},
+   "id": "10230128",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "-2(2-\\beta_1)+\\lambda=0.\n",
@@ -3636,16 +3920,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "96723ed8",
-   "metadata": {},
+   "id": "a7f4bdaa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which yields"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9a3b9a51",
-   "metadata": {},
+   "id": "658eb8ac",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0=\\frac{16+\\lambda}{8},\n",
@@ -3654,16 +3942,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e21b3e4d",
-   "metadata": {},
+   "id": "57ebdef8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b646e896",
-   "metadata": {},
+   "id": "030eb562",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_1=\\frac{4+\\lambda}{2}.\n",
@@ -3672,16 +3964,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12c82826",
-   "metadata": {},
+   "id": "954ff0c5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Using the constraint on $\\beta_0$ and $\\beta_1$ we can then find the optimal value of $\\lambda$ for the different cases. We leave this as an exercise to you."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "98d8fd18",
-   "metadata": {},
+   "id": "1d489a2a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Simple code for solving the above problem\n",
     "\n",
@@ -3693,8 +3989,11 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "2d797bc4",
-   "metadata": {},
+   "id": "9e04847f",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import os\n",
@@ -3750,16 +4049,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c8dcf60",
-   "metadata": {},
+   "id": "ad6af984",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see here that we reach a plateau. What is actually happening?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9929cadc",
-   "metadata": {},
+   "id": "26f1c1eb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## With Lasso Regression"
    ]
@@ -3767,8 +4070,11 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "id": "2154c3ea",
-   "metadata": {},
+   "id": "6c6d1284",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import os\n",
@@ -3831,8 +4137,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24a50aa3",
-   "metadata": {},
+   "id": "ee356a3b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Another Example, now with a polynomial fit"
    ]
@@ -3840,8 +4148,11 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "5436f7d3",
-   "metadata": {},
+   "id": "246b382a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import os\n",
@@ -3925,25 +4236,7 @@
    ]
   }
  ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.9.18"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 5
 }
diff --git a/doc/src/week36/week36.do.txt b/doc/src/week36/week36.do.txt
index 41ad882ca..64d4401eb 100644
--- a/doc/src/week36/week36.do.txt
+++ b/doc/src/week36/week36.do.txt
@@ -11,8 +11,8 @@ DATE: September 2-6, 2024
   * Linear Regression, ordinary least squares, Ridge and Lasso  and links with Statistics
   * Recommended Reading: Goodfellow et al chapter 3  (till 3.11) on probability theory, see URL:"https://www.deeplearningbook.org/"
   * Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6 contain many useful hints which will also be relevant for the various projects as well.
-#  * "Video of lecture":"https://youtu.be/Kc20CFK0z7Y"  
-#  * "Whiteboard notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf"
+  * "Video of lecture":"https://youtu.be/oHjqjUB36KE"  
+  * "Whiteboard notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf"
 * Material for the active learning sessions on Tuesday and Wednesday
   * Summary from last week on discussion of SVD, Ridge and Lasso linear regression.
   * Raschka et al, chapter 4 pages 105-134 and chapter 6 pages 171-185. Chapter 4 and 6, as for the lecture discussion