From b5bd221899bdd8753e911e38fbf855045a8c7d53 Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Mon, 1 Sep 2025 07:35:01 +0200
Subject: [PATCH] update

---
 .../_build/.doctrees/environment.pickle       |  Bin 260100 -> 259418 bytes
 .../_build/.doctrees/week36.doctree           |  Bin 324710 -> 322886 bytes
 .../_build/html/_sources/week36.ipynb         |  768 +++---
 doc/LectureNotes/_build/html/searchindex.js   |    2 +-
 doc/LectureNotes/_build/html/week36.html      |   33 +-
 .../_build/jupyter_execute/week36.ipynb       |  768 +++---
 doc/LectureNotes/week36.ipynb                 |  768 +++---
 .../exercisesweek37-checkpoint.ipynb          |  294 ++
 doc/src/week36/exercisesweek37.do.txt         | 2357 +----------------
 doc/src/week36/exercisesweek37.ipynb          |  357 +++
 10 files changed, 1770 insertions(+), 3577 deletions(-)
 create mode 100644 doc/src/week36/.ipynb_checkpoints/exercisesweek37-checkpoint.ipynb
 create mode 100644 doc/src/week36/exercisesweek37.ipynb

diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index 833d5730f4a43bb1b0ae2e762b3d17fd6476f28a..ec8b6f80b71587608058625e55184deb04c5cb88 100644
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diff --git a/doc/LectureNotes/_build/html/_sources/week36.ipynb b/doc/LectureNotes/_build/html/_sources/week36.ipynb
index 8bda05b08..d0d3dee3f 100644
--- a/doc/LectureNotes/_build/html/_sources/week36.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b8f45e2d",
+   "id": "3586da6c",
    "metadata": {
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@@ -14,7 +14,7 @@
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   {
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-   "id": "b3f2ead9",
+   "id": "5afa3a45",
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@@ -27,7 +27,7 @@
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-   "id": "04553781",
+   "id": "739da2e4",
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@@ -57,7 +57,7 @@
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-   "id": "1016bcdb",
+   "id": "66bc8f20",
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@@ -67,7 +67,7 @@
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-   "id": "839e657a",
+   "id": "d6b147a8",
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@@ -81,7 +81,7 @@
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+   "id": "69ee41c8",
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@@ -93,7 +93,7 @@
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@@ -105,7 +105,7 @@
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@@ -117,7 +117,7 @@
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@@ -127,7 +127,7 @@
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@@ -139,7 +139,7 @@
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@@ -149,7 +149,7 @@
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@@ -161,7 +161,7 @@
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@@ -172,7 +172,7 @@
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@@ -184,7 +184,7 @@
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+   "id": "03c35639",
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@@ -196,7 +196,7 @@
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@@ -206,7 +206,7 @@
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+   "id": "563e4883",
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@@ -218,7 +218,7 @@
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@@ -230,7 +230,7 @@
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@@ -240,7 +240,7 @@
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@@ -252,7 +252,7 @@
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@@ -262,7 +262,7 @@
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@@ -274,7 +274,7 @@
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@@ -284,7 +284,7 @@
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@@ -324,7 +324,7 @@
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@@ -341,7 +341,7 @@
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@@ -364,7 +364,7 @@
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@@ -381,7 +381,7 @@
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@@ -400,7 +400,7 @@
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@@ -411,7 +411,7 @@
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@@ -423,7 +423,7 @@
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@@ -441,14 +441,14 @@
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-   "id": "64cdb37b",
+   "id": "7cf05588",
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    "source": [
     "has linearly dependent column vectors, we will not be able to compute the inverse\n",
     "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\theta_i$. \n",
-    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n",
+    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exists. \n",
     "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n",
     "the regression parameters $\\theta_i$ cannot be estimated.\n",
     "\n",
@@ -457,7 +457,7 @@
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@@ -492,7 +492,7 @@
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@@ -527,7 +527,7 @@
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@@ -562,7 +562,7 @@
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@@ -575,7 +575,7 @@
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@@ -646,7 +646,7 @@
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@@ -658,7 +658,7 @@
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@@ -670,7 +670,7 @@
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@@ -684,7 +684,7 @@
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@@ -696,7 +696,7 @@
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@@ -706,7 +706,7 @@
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@@ -718,7 +718,7 @@
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@@ -730,7 +730,7 @@
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@@ -742,7 +742,7 @@
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@@ -754,7 +754,7 @@
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@@ -766,7 +766,7 @@
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@@ -789,7 +789,7 @@
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@@ -867,7 +867,7 @@
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@@ -879,7 +879,7 @@
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@@ -894,7 +894,7 @@
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@@ -906,7 +906,7 @@
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@@ -916,7 +916,7 @@
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@@ -928,7 +928,7 @@
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@@ -938,7 +938,7 @@
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@@ -950,7 +950,7 @@
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@@ -962,7 +962,7 @@
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@@ -977,7 +977,7 @@
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@@ -988,7 +988,7 @@
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@@ -1008,7 +1008,7 @@
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@@ -1020,7 +1020,7 @@
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@@ -1030,7 +1030,7 @@
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@@ -1042,7 +1042,7 @@
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@@ -1071,7 +1071,7 @@
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@@ -1098,7 +1098,7 @@
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@@ -1109,7 +1109,7 @@
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@@ -1149,7 +1149,7 @@
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@@ -1166,7 +1166,7 @@
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@@ -1189,7 +1189,7 @@
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@@ -1203,7 +1203,7 @@
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@@ -1222,7 +1222,7 @@
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@@ -1232,7 +1232,7 @@
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@@ -1244,7 +1244,7 @@
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@@ -1258,7 +1258,7 @@
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@@ -1270,7 +1270,7 @@
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@@ -1280,7 +1280,7 @@
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@@ -1292,7 +1292,7 @@
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@@ -1309,7 +1309,7 @@
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@@ -1319,7 +1319,7 @@
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@@ -1335,7 +1335,7 @@
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@@ -1345,7 +1345,7 @@
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@@ -1361,7 +1361,7 @@
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@@ -1371,7 +1371,7 @@
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@@ -1387,7 +1387,7 @@
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@@ -1397,7 +1397,7 @@
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@@ -1414,7 +1414,7 @@
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@@ -1426,7 +1426,7 @@
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@@ -1438,7 +1438,7 @@
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@@ -1450,7 +1450,7 @@
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@@ -1460,7 +1460,7 @@
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@@ -1472,7 +1472,7 @@
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@@ -1484,7 +1484,7 @@
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@@ -1496,7 +1496,7 @@
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@@ -1506,7 +1506,7 @@
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@@ -1518,7 +1518,7 @@
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@@ -1528,7 +1528,7 @@
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@@ -1540,7 +1540,7 @@
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@@ -1557,7 +1557,7 @@
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@@ -1569,7 +1569,7 @@
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@@ -1581,7 +1581,7 @@
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@@ -1591,7 +1591,7 @@
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@@ -1603,7 +1603,7 @@
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@@ -1614,7 +1614,7 @@
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@@ -1626,7 +1626,7 @@
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@@ -1636,7 +1636,7 @@
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@@ -1648,7 +1648,7 @@
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@@ -1658,7 +1658,7 @@
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@@ -1670,7 +1670,7 @@
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@@ -1681,7 +1681,7 @@
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@@ -1693,7 +1693,7 @@
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@@ -1711,7 +1711,7 @@
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@@ -1724,7 +1724,7 @@
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@@ -1736,7 +1736,7 @@
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@@ -1746,7 +1746,7 @@
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@@ -1759,7 +1759,7 @@
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@@ -1769,7 +1769,7 @@
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@@ -1781,7 +1781,7 @@
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@@ -1794,7 +1794,7 @@
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@@ -1807,7 +1807,7 @@
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@@ -1819,7 +1819,7 @@
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@@ -1831,7 +1831,7 @@
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@@ -1841,7 +1841,7 @@
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@@ -1854,7 +1854,7 @@
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@@ -1866,7 +1866,7 @@
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@@ -1878,7 +1878,7 @@
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@@ -1895,7 +1895,7 @@
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@@ -1907,7 +1907,7 @@
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@@ -1917,7 +1917,7 @@
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@@ -1929,7 +1929,7 @@
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@@ -1939,7 +1939,7 @@
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@@ -1951,7 +1951,7 @@
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@@ -1963,7 +1963,7 @@
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@@ -1979,7 +1979,7 @@
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@@ -1991,7 +1991,7 @@
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@@ -2003,7 +2003,7 @@
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@@ -2013,7 +2013,7 @@
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@@ -2025,7 +2025,7 @@
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@@ -2035,7 +2035,7 @@
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-   "id": "a5a8ee8c",
+   "id": "3aeb28d8",
    "metadata": {
     "editable": true
    },
@@ -2047,7 +2047,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03c00df5",
+   "id": "59348b08",
    "metadata": {
     "editable": true
    },
@@ -2064,7 +2064,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "023042c7",
+   "id": "a3a38146",
    "metadata": {
     "editable": true
    },
@@ -2076,7 +2076,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0cd7107",
+   "id": "4de2d8d6",
    "metadata": {
     "editable": true
    },
@@ -2088,7 +2088,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c34ce5f",
+   "id": "5539ce38",
    "metadata": {
     "editable": true
    },
@@ -2098,7 +2098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d489ea51",
+   "id": "519481d2",
    "metadata": {
     "editable": true
    },
@@ -2110,7 +2110,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9921bc50",
+   "id": "433bee8f",
    "metadata": {
     "editable": true
    },
@@ -2120,7 +2120,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c568250e",
+   "id": "afea6f71",
    "metadata": {
     "editable": true
    },
@@ -2132,7 +2132,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21d0b485",
+   "id": "32b32b42",
    "metadata": {
     "editable": true
    },
@@ -2142,19 +2142,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6722483b",
+   "id": "54a0c1f6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9777f5cc",
+   "id": "ec7d68de",
    "metadata": {
     "editable": true
    },
@@ -2164,19 +2164,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23101fe6",
+   "id": "d26bacac",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4f604170",
+   "id": "ac45c3e5",
    "metadata": {
     "editable": true
    },
@@ -2186,7 +2186,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c7b9070",
+   "id": "451fbbf2",
    "metadata": {
     "editable": true
    },
@@ -2205,7 +2205,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0af8105c",
+   "id": "e0569839",
    "metadata": {
     "editable": true
    },
@@ -2225,7 +2225,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c93d44b0",
+   "id": "de0e44d6",
    "metadata": {
     "editable": true
    },
@@ -2241,7 +2241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b1c868cd",
+   "id": "bae50b3b",
    "metadata": {
     "editable": true
    },
@@ -2257,7 +2257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c257671",
+   "id": "2cc7033f",
    "metadata": {
     "editable": true
    },
@@ -2268,7 +2268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ddc77aef",
+   "id": "c4ca8926",
    "metadata": {
     "editable": true
    },
@@ -2280,7 +2280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "026777a7",
+   "id": "1eaf376f",
    "metadata": {
     "editable": true
    },
@@ -2290,7 +2290,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3707790b",
+   "id": "9c025697",
    "metadata": {
     "editable": true
    },
@@ -2302,7 +2302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "13ab6bec",
+   "id": "2d64abe7",
    "metadata": {
     "editable": true
    },
@@ -2312,7 +2312,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a6d69ef",
+   "id": "b83c29d5",
    "metadata": {
     "editable": true
    },
@@ -2324,7 +2324,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "336ec1b1",
+   "id": "cfcc809e",
    "metadata": {
     "editable": true
    },
@@ -2346,7 +2346,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3081ba44",
+   "id": "bdaa56d1",
    "metadata": {
     "editable": true
    },
@@ -2359,7 +2359,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c3faff2",
+   "id": "ffa4afe6",
    "metadata": {
     "editable": true
    },
@@ -2372,7 +2372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68fd52c0",
+   "id": "5c6b9239",
    "metadata": {
     "editable": true
    },
@@ -2382,7 +2382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e29a567",
+   "id": "5ddb446d",
    "metadata": {
     "editable": true
    },
@@ -2400,7 +2400,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f787c3a2",
+   "id": "9a0a5b04",
    "metadata": {
     "editable": true
    },
@@ -2410,7 +2410,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c5e1214",
+   "id": "6fe881ee",
    "metadata": {
     "editable": true
    },
@@ -2425,7 +2425,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cdd5ba1",
+   "id": "83b83194",
    "metadata": {
     "editable": true
    },
@@ -2435,7 +2435,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "276e07a3",
+   "id": "09e77119",
    "metadata": {
     "editable": true
    },
@@ -2449,7 +2449,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80fe7f13",
+   "id": "afdce492",
    "metadata": {
     "editable": true
    },
@@ -2459,7 +2459,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8838b8f",
+   "id": "b6e29c29",
    "metadata": {
     "editable": true
    },
@@ -2473,7 +2473,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2ed6fbd0",
+   "id": "ad6e09d7",
    "metadata": {
     "editable": true
    },
@@ -2488,7 +2488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0796429c",
+   "id": "a514df33",
    "metadata": {
     "editable": true
    },
@@ -2505,7 +2505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc44df4f",
+   "id": "ae758019",
    "metadata": {
     "editable": true
    },
@@ -2517,7 +2517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c88146a9",
+   "id": "1bcffe5e",
    "metadata": {
     "editable": true
    },
@@ -2531,7 +2531,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1b85445",
+   "id": "67467328",
    "metadata": {
     "editable": true
    },
@@ -2546,7 +2546,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8edde795",
+   "id": "4bac20b4",
    "metadata": {
     "editable": true
    },
@@ -2558,7 +2558,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f92e4583",
+   "id": "01e84611",
    "metadata": {
     "editable": true
    },
@@ -2569,7 +2569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0bf80264",
+   "id": "8f742c3b",
    "metadata": {
     "editable": true
    },
@@ -2597,7 +2597,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b6b24c7",
+   "id": "77acd12d",
    "metadata": {
     "editable": true
    },
@@ -2619,7 +2619,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e36297bd",
+   "id": "99ab46db",
    "metadata": {
     "editable": true
    },
@@ -2641,7 +2641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8ec331d",
+   "id": "5b4b6243",
    "metadata": {
     "editable": true
    },
@@ -2653,7 +2653,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b0d0c67",
+   "id": "2ba66a00",
    "metadata": {
     "editable": true
    },
@@ -2690,7 +2690,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bdfac28b",
+   "id": "c9a65dbe",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2718,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2032fb83",
+   "id": "7efa8d14",
    "metadata": {
     "editable": true
    },
@@ -2748,7 +2748,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5a00d18",
+   "id": "93d8901e",
    "metadata": {
     "editable": true
    },
@@ -2772,7 +2772,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "68f97538",
+   "id": "6a775d3c",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2785,7 +2785,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a7fbe32f",
+   "id": "b8360996",
    "metadata": {
     "editable": true
    },
@@ -2796,7 +2796,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b64af25",
+   "id": "0b4a5c74",
    "metadata": {
     "editable": true
    },
@@ -2808,7 +2808,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ef56e132",
+   "id": "f27ef437",
    "metadata": {
     "editable": true
    },
@@ -2818,7 +2818,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6d7f0c20",
+   "id": "6a7d8179",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ac315d2",
+   "id": "24b3bcb3",
    "metadata": {
     "editable": true
    },
@@ -2844,7 +2844,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "040d030b",
+   "id": "196152f9",
    "metadata": {
     "editable": true
    },
@@ -2860,7 +2860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cb9c213",
+   "id": "6b935471",
    "metadata": {
     "editable": true
    },
@@ -2870,7 +2870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f6ab192",
+   "id": "d5261a7f",
    "metadata": {
     "editable": true
    },
@@ -2882,7 +2882,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc154729",
+   "id": "398173a2",
    "metadata": {
     "editable": true
    },
@@ -2892,7 +2892,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cda14c18",
+   "id": "0f29a1a0",
    "metadata": {
     "editable": true
    },
@@ -2904,7 +2904,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffc6db98",
+   "id": "22f27a92",
    "metadata": {
     "editable": true
    },
@@ -2918,7 +2918,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5dd4f8d",
+   "id": "a2b8898e",
    "metadata": {
     "editable": true
    },
@@ -2928,7 +2928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "54cf8a09",
+   "id": "42b415d8",
    "metadata": {
     "editable": true
    },
@@ -2939,7 +2939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c06fcb28",
+   "id": "b522b609",
    "metadata": {
     "editable": true
    },
@@ -2954,7 +2954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16b2e122",
+   "id": "2e2e2e39",
    "metadata": {
     "editable": true
    },
@@ -2964,7 +2964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e48ac99",
+   "id": "6f14f45f",
    "metadata": {
     "editable": true
    },
@@ -2976,7 +2976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a536e9c",
+   "id": "d9210eed",
    "metadata": {
     "editable": true
    },
@@ -2988,7 +2988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9afa223",
+   "id": "5050ec60",
    "metadata": {
     "editable": true
    },
@@ -3003,7 +3003,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4f76d91",
+   "id": "04ac7d83",
    "metadata": {
     "editable": true
    },
@@ -3016,7 +3016,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "801ec4de",
+   "id": "6b2e19f6",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3073,45 +3073,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6c4900e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## And a corresponding example using **scikit-learn**"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "id": "d2f662be",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "# Importing various packages\n",
-    "from random import random, seed\n",
-    "import numpy as np\n",
-    "import matplotlib.pyplot as plt\n",
-    "from sklearn.linear_model import SGDRegressor\n",
-    "\n",
-    "n = 100\n",
-    "x = 2*np.random.rand(n,1)\n",
-    "y = 4+3*x+np.random.randn(n,1)\n",
-    "\n",
-    "X = np.c_[np.ones((n,1)), x]\n",
-    "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n",
-    "print(theta_linreg)\n",
-    "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n",
-    "sgdreg.fit(x,y.ravel())\n",
-    "print(sgdreg.intercept_, sgdreg.coef_)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "83bd5b44",
+   "id": "26e7815c",
    "metadata": {
     "editable": true
    },
@@ -3123,7 +3085,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c7ee80c",
+   "id": "0ae04360",
    "metadata": {
     "editable": true
    },
@@ -3135,7 +3097,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c70714f",
+   "id": "402501d6",
    "metadata": {
     "editable": true
    },
@@ -3145,7 +3107,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e170727b",
+   "id": "fdd060d5",
    "metadata": {
     "editable": true
    },
@@ -3159,7 +3121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b7b5d1ce",
+   "id": "8a061527",
    "metadata": {
     "editable": true
    },
@@ -3169,7 +3131,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e59753a",
+   "id": "72c8bb88",
    "metadata": {
     "editable": true
    },
@@ -3181,7 +3143,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d05de5f9",
+   "id": "a8e126ba",
    "metadata": {
     "editable": true
    },
@@ -3192,7 +3154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3a5c888",
+   "id": "cae81a93",
    "metadata": {
     "editable": true
    },
@@ -3207,7 +3169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52b009a7",
+   "id": "646feacb",
    "metadata": {
     "editable": true
    },
@@ -3221,7 +3183,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "430d2418",
+   "id": "e2b0d0a9",
    "metadata": {
     "editable": true
    },
@@ -3231,8 +3193,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
-   "id": "b889c65e",
+   "execution_count": 5,
+   "id": "b2354adf",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3293,7 +3255,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21e20510",
+   "id": "e95bd522",
    "metadata": {
     "editable": true
    },
@@ -3315,7 +3277,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59d61c18",
+   "id": "947549ab",
    "metadata": {
     "editable": true
    },
@@ -3327,7 +3289,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19a3af8f",
+   "id": "64b3eeb6",
    "metadata": {
     "editable": true
    },
@@ -3339,7 +3301,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0a91546",
+   "id": "8ddf1823",
    "metadata": {
     "editable": true
    },
@@ -3351,7 +3313,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ddc3a0d",
+   "id": "fe9d8eab",
    "metadata": {
     "editable": true
    },
@@ -3361,7 +3323,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2c51cc5b",
+   "id": "a6171f9f",
    "metadata": {
     "editable": true
    },
@@ -3373,7 +3335,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "754dd1c2",
+   "id": "7fed4d62",
    "metadata": {
     "editable": true
    },
@@ -3383,7 +3345,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37b8fcf8",
+   "id": "276ab416",
    "metadata": {
     "editable": true
    },
@@ -3400,7 +3362,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4103d318",
+   "id": "67074db1",
    "metadata": {
     "editable": true
    },
@@ -3410,7 +3372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dae6c763",
+   "id": "068b0864",
    "metadata": {
     "editable": true
    },
@@ -3422,7 +3384,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b13a4b0d",
+   "id": "20210169",
    "metadata": {
     "editable": true
    },
@@ -3432,7 +3394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0911661b",
+   "id": "4a45a715",
    "metadata": {
     "editable": true
    },
@@ -3444,7 +3406,7 @@
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@@ -3458,7 +3420,7 @@
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@@ -3470,7 +3432,7 @@
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@@ -3489,7 +3451,7 @@
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@@ -3502,7 +3464,7 @@
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@@ -3514,7 +3476,7 @@
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@@ -3524,7 +3486,7 @@
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@@ -3537,7 +3499,7 @@
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@@ -3547,7 +3509,7 @@
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@@ -3559,7 +3521,7 @@
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@@ -3574,7 +3536,7 @@
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@@ -3587,7 +3549,7 @@
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@@ -3601,7 +3563,7 @@
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@@ -3613,7 +3575,7 @@
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@@ -3623,7 +3585,7 @@
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@@ -3636,7 +3598,7 @@
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@@ -3648,7 +3610,7 @@
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@@ -3660,7 +3622,7 @@
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@@ -3672,7 +3634,7 @@
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@@ -3684,7 +3646,7 @@
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@@ -3698,7 +3660,7 @@
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@@ -3710,7 +3672,7 @@
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@@ -3720,7 +3682,7 @@
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@@ -3732,7 +3694,7 @@
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@@ -3742,7 +3704,7 @@
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@@ -3754,7 +3716,7 @@
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@@ -3766,7 +3728,7 @@
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@@ -3776,7 +3738,7 @@
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@@ -3787,7 +3749,7 @@
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@@ -3799,7 +3761,7 @@
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@@ -3815,7 +3777,7 @@
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@@ -3828,7 +3790,7 @@
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@@ -3840,7 +3802,7 @@
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@@ -3850,7 +3812,7 @@
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@@ -3862,7 +3824,7 @@
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@@ -3872,7 +3834,7 @@
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@@ -3884,7 +3846,7 @@
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@@ -3896,7 +3858,7 @@
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@@ -3912,7 +3874,7 @@
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@@ -3924,7 +3886,7 @@
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@@ -3936,7 +3898,7 @@
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@@ -3946,7 +3908,7 @@
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@@ -3958,7 +3920,7 @@
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@@ -3968,7 +3930,7 @@
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@@ -3980,7 +3942,7 @@
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@@ -3997,7 +3959,7 @@
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@@ -4009,7 +3971,7 @@
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@@ -4021,7 +3983,7 @@
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@@ -4031,7 +3993,7 @@
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@@ -4043,7 +4005,7 @@
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@@ -4053,7 +4015,7 @@
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@@ -4065,7 +4027,7 @@
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@@ -4075,7 +4037,7 @@
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@@ -4087,7 +4049,7 @@
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@@ -4097,7 +4059,7 @@
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@@ -4112,7 +4074,7 @@
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@@ -4124,7 +4086,7 @@
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@@ -4134,7 +4096,7 @@
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@@ -4146,7 +4108,7 @@
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@@ -4158,7 +4120,7 @@
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@@ -4170,7 +4132,7 @@
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@@ -4180,7 +4142,7 @@
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@@ -4192,7 +4154,7 @@
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@@ -4204,7 +4166,7 @@
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@@ -4216,7 +4178,7 @@
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@@ -4226,7 +4188,7 @@
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@@ -4238,7 +4200,7 @@
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@@ -4248,7 +4210,7 @@
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@@ -4262,7 +4224,7 @@
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@@ -4272,7 +4234,7 @@
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@@ -4284,7 +4246,7 @@
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@@ -4296,7 +4258,7 @@
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@@ -4306,7 +4268,7 @@
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@@ -4318,7 +4280,7 @@
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@@ -4328,7 +4290,7 @@
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@@ -4340,7 +4302,7 @@
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@@ -4352,7 +4314,7 @@
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@@ -4362,7 +4324,7 @@
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@@ -4374,7 +4336,7 @@
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@@ -4384,7 +4346,7 @@
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@@ -4396,7 +4358,7 @@
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@@ -4408,7 +4370,7 @@
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@@ -4418,7 +4380,7 @@
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@@ -4430,7 +4392,7 @@
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@@ -4443,7 +4405,7 @@
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@@ -4455,7 +4417,7 @@
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@@ -4467,7 +4429,7 @@
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@@ -4479,7 +4441,7 @@
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@@ -4489,7 +4451,7 @@
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@@ -4501,7 +4463,7 @@
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@@ -4511,7 +4473,7 @@
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@@ -4523,7 +4485,7 @@
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@@ -4533,7 +4495,7 @@
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@@ -4545,7 +4507,7 @@
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@@ -4555,7 +4517,7 @@
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@@ -4568,7 +4530,7 @@
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@@ -4580,7 +4542,7 @@
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@@ -4592,7 +4554,7 @@
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@@ -4602,7 +4564,7 @@
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@@ -4614,7 +4576,7 @@
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@@ -4631,7 +4593,7 @@
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@@ -4643,7 +4605,7 @@
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@@ -4655,7 +4617,7 @@
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@@ -4665,7 +4627,7 @@
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@@ -4677,7 +4639,7 @@
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@@ -4687,7 +4649,7 @@
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@@ -4699,7 +4661,7 @@
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@@ -4709,7 +4671,7 @@
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@@ -4721,7 +4683,7 @@
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@@ -4731,7 +4693,7 @@
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@@ -4745,8 +4707,8 @@
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@@ -4806,7 +4768,7 @@
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@@ -4816,7 +4778,7 @@
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@@ -4826,8 +4788,8 @@
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@@ -4894,7 +4856,7 @@
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@@ -4904,8 +4866,8 @@
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diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 7bcfa3374..a91c83fec 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"A Classification Tree": [[9, "a-classification-tree"]], "A Frequentist approach to data analysis": [[0, "a-frequentist-approach-to-data-analysis"], [24, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[24, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[13, "adam-optimizer"]], "Activation functions": [[12, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[24, "adding-error-analysis-and-training-set-up"], [25, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[24, "an-optimization-minimization-problem"]], "And a corresponding example using scikit-learn": [[26, "and-a-corresponding-example-using-scikit-learn"]], "And finally  \\boldsymbol{X}\\boldsymbol{X}^T": [[25, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[24, "and-what-about-using-neural-networks"]], "Another Example, now with a polynomial fit": [[26, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[18, null]], "Autocorrelation function": [[21, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[25, "back-to-ridge-and-lasso-regression"], [26, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[19, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [25, "basic-math-of-the-svd"], [26, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [25, "codes-for-the-svd"], [26, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[24, "communication-channels"]], "Compare  Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[26, "comparison-with-ols"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conditions on convex functions": [[26, "conditions-on-convex-functions"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex function": [[26, "convex-function"]], "Convex functions": [[13, "convex-functions"], [26, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[25, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [25, "correlation-matrix"]], "Correlation Matrix with Pandas": [[25, "correlation-matrix-with-pandas"]], "Course Format": [[24, "course-format"]], "Course setting": [[20, null]], "Covariance Matrix Examples": [[25, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[25, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[24, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep learning methods": [[24, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[25, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the  Lasso Regression Equations": [[25, "deriving-the-lasso-regression-equations"], [26, "deriving-the-lasso-regression-equations"], [26, "id6"]], "Deriving the  Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"], [26, "deriving-the-ridge-regression-equations"], [26, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[24, "discriminative-modeling"]], "Domains and probabilities": [[21, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[25, "economy-size-svd"], [26, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[21, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[24, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[25, "example-2"]], "Example 3": [[25, "example-3"]], "Example 4": [[25, "example-4"]], "Example Matrix": [[25, "example-matrix"], [26, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[25, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[25, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[24, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise:  Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function  with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function  with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Exercises week 36": [[17, null]], "Expectation values": [[21, "expectation-values"]], "Extending to more than one variable": [[26, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[24, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[25, "fixing-the-singularity"], [26, "fixing-the-singularity"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[26, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[25, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [25, "further-properties-important-for-our-analyses-later"], [26, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[19, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[24, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[24, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [24, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[24, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[26, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Gradient descent and Ridge": [[26, "gradient-descent-and-ridge"]], "Gradient descent example": [[26, "gradient-descent-example"]], "Grading": [[22, "grading"], [22, "id2"], [24, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[19, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[25, "important-technicalities-more-on-rescaling-data"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[22, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[24, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[24, "installing-r-c-cython-numba-etc"]], "Instructor information": [[22, "instructor-information"]], "Interpretations and optimizing our parameters": [[24, "interpretations-and-optimizing-our-parameters"], [24, "id2"], [24, "id3"], [25, "interpretations-and-optimizing-our-parameters"], [25, "id1"], [25, "id2"]], "Interpreting the Ridge results": [[25, "interpreting-the-ridge-results"], [26, "interpreting-the-ridge-results"], [26, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [25, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [18, "introduction"], [19, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[19, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[26, "lasso-regression"]], "Lasso case": [[26, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"]], "Learning outcomes": [[18, "learning-outcomes"], [24, "learning-outcomes"]], "Lectures and ComputerLab": [[24, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[19, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[25, "linear-regression-problems"], [26, "linear-regression-problems"]], "Linear Regression and  the SVD": [[26, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [25, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[23, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[24, "machine-learning"]], "Machine learning": [[18, "machine-learning"]], "Main textbooks": [[24, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[25, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[25, "material-for-exercises-week-35"]], "Material for lab sessions  sessions Tuesday and Wednesday": [[26, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[26, "material-for-lecture-monday-september-2"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [25, "mathematical-interpretation-of-ordinary-least-squares"], [26, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [25, "mathematics-of-the-svd-and-implications"], [26, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[24, "matrices-in-python"]], "Matrix  multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation  and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the  covariance!": [[21, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [25, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[25, "meet-the-hessian-matrix"]], "Meet the Pandas": [[24, "meet-the-pandas"]], "Min-Max Scaling": [[25, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[25, "more-interpretations"], [26, "more-interpretations"], [26, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[26, "more-on-steepest-descent"]], "More on convex functions": [[26, "more-on-convex-functions"]], "More preprocessing": [[25, "more-preprocessing"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Note about SVD Calculations": [[25, "note-about-svd-calculations"], [26, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[26, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[21, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[19, "numpy-and-arrays"], [24, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[24, 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\ No newline at end of file
+Search.setIndex({"alltitles": {"A Classification Tree": [[9, "a-classification-tree"]], "A Frequentist approach to data analysis": [[0, "a-frequentist-approach-to-data-analysis"], [24, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[24, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[13, "adam-optimizer"]], "Activation functions": [[12, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[24, "adding-error-analysis-and-training-set-up"], [25, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[24, "an-optimization-minimization-problem"]], "And finally  \\boldsymbol{X}\\boldsymbol{X}^T": [[25, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[24, "and-what-about-using-neural-networks"]], "Another Example, now with a polynomial fit": [[26, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[18, null]], "Autocorrelation function": [[21, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[25, "back-to-ridge-and-lasso-regression"], [26, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[19, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [25, "basic-math-of-the-svd"], [26, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [25, "codes-for-the-svd"], [26, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[24, "communication-channels"]], "Compare  Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[26, "comparison-with-ols"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conditions on convex functions": [[26, "conditions-on-convex-functions"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex function": [[26, "convex-function"]], "Convex functions": [[13, "convex-functions"], [26, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[25, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [25, "correlation-matrix"]], "Correlation Matrix with Pandas": [[25, "correlation-matrix-with-pandas"]], "Course Format": [[24, "course-format"]], "Course setting": [[20, null]], "Covariance Matrix Examples": [[25, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[25, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[24, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep learning methods": [[24, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[25, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the  Lasso Regression Equations": [[25, "deriving-the-lasso-regression-equations"], [26, "deriving-the-lasso-regression-equations"], [26, "id6"]], "Deriving the  Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"], [26, "deriving-the-ridge-regression-equations"], [26, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[24, "discriminative-modeling"]], "Domains and probabilities": [[21, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[25, "economy-size-svd"], [26, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[21, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[24, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[25, "example-2"]], "Example 3": [[25, "example-3"]], "Example 4": [[25, "example-4"]], "Example Matrix": [[25, "example-matrix"], [26, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[25, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[25, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[24, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise:  Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function  with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function  with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Exercises week 36": [[17, null]], "Expectation values": [[21, "expectation-values"]], "Extending to more than one variable": [[26, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[24, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[25, "fixing-the-singularity"], [26, "fixing-the-singularity"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[26, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[25, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [25, "further-properties-important-for-our-analyses-later"], [26, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[19, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[24, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[24, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [24, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[24, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[26, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Gradient descent and Ridge": [[26, "gradient-descent-and-ridge"]], "Gradient descent example": [[26, "gradient-descent-example"]], "Grading": [[22, "grading"], [22, "id2"], [24, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[19, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[25, "important-technicalities-more-on-rescaling-data"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[22, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[24, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[24, "installing-r-c-cython-numba-etc"]], "Instructor information": [[22, "instructor-information"]], "Interpretations and optimizing our parameters": [[24, "interpretations-and-optimizing-our-parameters"], [24, "id2"], [24, "id3"], [25, "interpretations-and-optimizing-our-parameters"], [25, "id1"], [25, "id2"]], "Interpreting the Ridge results": [[25, "interpreting-the-ridge-results"], [26, "interpreting-the-ridge-results"], [26, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [25, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [18, "introduction"], [19, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[19, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[26, "lasso-regression"]], "Lasso case": [[26, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"]], "Learning outcomes": [[18, "learning-outcomes"], [24, "learning-outcomes"]], "Lectures and ComputerLab": [[24, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[19, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[25, "linear-regression-problems"], [26, "linear-regression-problems"]], "Linear Regression and  the SVD": [[26, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [25, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[23, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[24, "machine-learning"]], "Machine learning": [[18, "machine-learning"]], "Main textbooks": [[24, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[25, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[25, "material-for-exercises-week-35"]], "Material for lab sessions  sessions Tuesday and Wednesday": [[26, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[26, "material-for-lecture-monday-september-2"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [25, "mathematical-interpretation-of-ordinary-least-squares"], [26, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [25, "mathematics-of-the-svd-and-implications"], [26, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[24, "matrices-in-python"]], "Matrix  multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation  and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the  covariance!": [[21, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [25, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[25, "meet-the-hessian-matrix"]], "Meet the Pandas": [[24, "meet-the-pandas"]], "Min-Max Scaling": [[25, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[25, "more-interpretations"], [26, "more-interpretations"], [26, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[26, "more-on-steepest-descent"]], "More on convex functions": [[26, "more-on-convex-functions"]], "More preprocessing": [[25, "more-preprocessing"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Note about SVD Calculations": [[25, "note-about-svd-calculations"], [26, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[26, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[21, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[19, "numpy-and-arrays"], [24, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[24, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[26, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[24, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[24, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Organizing our data": [[0, "organizing-our-data"], [24, "organizing-our-data"]], "Other Matrix and Vector Operations": [[19, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning  at UiO": [[24, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning  at UiO, contn": [[24, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[24, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[24, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[24, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[24, "own-code-for-ordinary-least-squares"], [25, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[24, "pandas-ai"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plans for week 35": [[25, "plans-for-week-35"]], "Plans for week 36": [[26, "plans-for-week-36"]], "Practical tips": [[13, "practical-tips"]], "Practicalities": [[22, "practicalities"], [22, "id1"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, 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"Reading material": [[24, "reading-material"]], "Reading recommendations:": [[25, "reading-recommendations"]], "Reading suggestions week 34": [[24, "reading-suggestions-week-34"]], "Recurrent neural networks": [[12, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[4, null]], "Reducing the number of degrees of freedom, overarching view": [[0, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [25, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[2, "reformulating-the-problem"]], "Regression Case": [[10, "regression-case"]], "Regression analysis, overarching aims": [[24, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[24, "regression-analysis-overarching-aims-ii"]], "Regularization": [[1, "regularization"]], "Reminder from last week": [[25, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[26, "reminder-on-newton-raphson-s-method"]], 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class=\"section-number\">16. </span>Convolutional Neural Networks", "<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view", "<span class=\"section-number\">4. </span>Ridge and Lasso Regression", "<span class=\"section-number\">5. </span>Resampling Methods", "<span class=\"section-number\">6. </span>Logistic Regression", "<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims", "<span class=\"section-number\">9. </span>Decision trees, overarching aims", "<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods", "<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)", "<span class=\"section-number\">13. </span>Neural networks", "<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm", "<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning", "Exercises week 34", "Exercises week 35", "Exercises week 36", "Applied Data Analysis and Machine Learning", "<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features", "Course setting", "<span class=\"section-number\">1. </span>Elements of Probability Theory and Statistical Data Analysis", "Teachers and Grading", "Textbooks", "Week 34: Introduction to the course, Logistics and Practicalities", "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression", "Week 36: Linear Regression and Gradient descent"], "titleterms": {"": [8, 10, 26], "1": [0, 15, 16, 17, 25], "2": [0, 15, 16, 17, 24, 25, 26], "2023": 22, "3": [0, 15, 16, 17, 25], "34": [15, 24], "35": [16, 25], "36": [17, 26], "4": [0, 15, 16, 17, 25], "5": [0, 16], "A": [0, 1, 4, 8, 9, 24], "And": [24, 25], "In": 22, "Ising": 6, "The": [0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 15, 18, 24, 25, 26], "To": 24, "With": [4, 26], "about": [24, 25, 26], "abov": 26, 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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html
index 7ad88c239..3126175bc 100644
--- a/doc/LectureNotes/_build/html/week36.html
+++ b/doc/LectureNotes/_build/html/week36.html
@@ -419,7 +419,6 @@ document.write(`
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hessian-matrix">The Hessian matrix</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-program">Simple program</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Gradient Descent Example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#and-a-corresponding-example-using-scikit-learn">And a corresponding example using <strong>scikit-learn</strong></a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-descent-and-ridge">Gradient descent and Ridge</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hessian-matrix-for-ridge-regression">The Hessian matrix for Ridge Regression</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#program-example-for-gradient-descent-with-ridge-regression">Program example for gradient descent with Ridge Regression</a></li>
@@ -631,7 +630,7 @@ This is equivalent to saying that the matrix <span class="math notranslate nohig
 \]</div>
 <p>has linearly dependent column vectors, we will not be able to compute the inverse
 of <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span> and we cannot find the parameters (estimators) <span class="math notranslate nohighlight">\(\theta_i\)</span>.
-The estimators are only well-defined if <span class="math notranslate nohighlight">\((\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\)</span> exits.
+The estimators are only well-defined if <span class="math notranslate nohighlight">\((\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\)</span> exists.
 This is more likely to happen when the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is high-dimensional. In this case it is likely to encounter a situation where
 the regression parameters <span class="math notranslate nohighlight">\(\theta_i\)</span> cannot be estimated.</p>
 <p>A cheap  <em>ad hoc</em> approach is  simply to add a small diagonal component to the matrix to invert, that is we change</p>
@@ -1239,12 +1238,12 @@ C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsym
 <p>and reordering we have</p>
 <div class="math notranslate nohighlight">
 \[
-\boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\theta}+\frac{n}{2}\lambda sgn(\boldsymbol{\theta})=2\boldsymbol{X}^T\boldsymbol{y}.
+\boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\theta}+\frac{n}{2}\lambda sgn(\boldsymbol{\theta})=\boldsymbol{X}^T\boldsymbol{y}.
 \]</div>
 <p>We can redefine <span class="math notranslate nohighlight">\(\lambda\)</span> to absorb the constant <span class="math notranslate nohighlight">\(n/2\)</span> and we rewrite the last equation as</p>
 <div class="math notranslate nohighlight">
 \[
-\boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\theta}+\lambda sgn(\boldsymbol{\theta})=2\boldsymbol{X}^T\boldsymbol{y}.
+\boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\theta}+\lambda sgn(\boldsymbol{\theta})=\boldsymbol{X}^T\boldsymbol{y}.
 \]</div>
 <p>This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms.We will discuss how to code the above methods using gradient descent methods.</p>
 </section>
@@ -1643,31 +1642,6 @@ plt.show()
 </div>
 </div>
 </section>
-<section id="and-a-corresponding-example-using-scikit-learn">
-<h2>And a corresponding example using <strong>scikit-learn</strong><a class="headerlink" href="#and-a-corresponding-example-using-scikit-learn" title="Link to this heading">#</a></h2>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
-
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
-print(theta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print(sgdreg.intercept_, sgdreg.coef_)
-</pre></div>
-</div>
-</div>
-</div>
-</section>
 <section id="gradient-descent-and-ridge">
 <h2>Gradient descent and Ridge<a class="headerlink" href="#gradient-descent-and-ridge" title="Link to this heading">#</a></h2>
 <p>We have also discussed Ridge regression where the loss function contains a regularized term given by the <span class="math notranslate nohighlight">\(L_2\)</span> norm of <span class="math notranslate nohighlight">\(\theta\)</span>,</p>
@@ -2488,7 +2462,6 @@ plt.show()
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hessian-matrix">The Hessian matrix</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-program">Simple program</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Gradient Descent Example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#and-a-corresponding-example-using-scikit-learn">And a corresponding example using <strong>scikit-learn</strong></a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-descent-and-ridge">Gradient descent and Ridge</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hessian-matrix-for-ridge-regression">The Hessian matrix for Ridge Regression</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#program-example-for-gradient-descent-with-ridge-regression">Program example for gradient descent with Ridge Regression</a></li>
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
index e9f8e90f7..e469616cf 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b8f45e2d",
+   "id": "3586da6c",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b3f2ead9",
+   "id": "5afa3a45",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04553781",
+   "id": "739da2e4",
    "metadata": {
     "editable": true
    },
@@ -57,7 +57,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1016bcdb",
+   "id": "66bc8f20",
    "metadata": {
     "editable": true
    },
@@ -67,7 +67,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "839e657a",
+   "id": "d6b147a8",
    "metadata": {
     "editable": true
    },
@@ -81,7 +81,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12dc9112",
+   "id": "69ee41c8",
    "metadata": {
     "editable": true
    },
@@ -93,7 +93,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df3e7f6d",
+   "id": "eec38b3f",
    "metadata": {
     "editable": true
    },
@@ -105,7 +105,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc1caa8e",
+   "id": "7ca66dca",
    "metadata": {
     "editable": true
    },
@@ -117,7 +117,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aaf95f40",
+   "id": "4a67a7b4",
    "metadata": {
     "editable": true
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e615b93",
+   "id": "4b3a2043",
    "metadata": {
     "editable": true
    },
@@ -139,7 +139,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f764f4e",
+   "id": "841bde50",
    "metadata": {
     "editable": true
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@@ -149,7 +149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e0d9985",
+   "id": "ebca953b",
    "metadata": {
     "editable": true
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@@ -161,7 +161,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a11e3b77",
+   "id": "7be2de84",
    "metadata": {
     "editable": true
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@@ -172,7 +172,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b175d8ad",
+   "id": "9916c24e",
    "metadata": {
     "editable": true
    },
@@ -184,7 +184,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0901c5d9",
+   "id": "03c35639",
    "metadata": {
     "editable": true
    },
@@ -196,7 +196,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7263285d",
+   "id": "63aaa0d6",
    "metadata": {
     "editable": true
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@@ -206,7 +206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c0f54e1a",
+   "id": "563e4883",
    "metadata": {
     "editable": true
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@@ -218,7 +218,7 @@
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   {
    "cell_type": "markdown",
-   "id": "035835e7",
+   "id": "207fb037",
    "metadata": {
     "editable": true
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@@ -230,7 +230,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a0dca7fe",
+   "id": "d0ebfe7a",
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     "editable": true
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@@ -240,7 +240,7 @@
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   {
    "cell_type": "markdown",
-   "id": "5808efd5",
+   "id": "e5eba895",
    "metadata": {
     "editable": true
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@@ -252,7 +252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cca67bb8",
+   "id": "20736fd2",
    "metadata": {
     "editable": true
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@@ -262,7 +262,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a189ce2b",
+   "id": "8ebaa24c",
    "metadata": {
     "editable": true
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@@ -274,7 +274,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7835a4fb",
+   "id": "9e0e8b1b",
    "metadata": {
     "editable": true
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@@ -284,7 +284,7 @@
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   {
    "cell_type": "markdown",
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+   "id": "a9dc6df7",
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     "editable": true
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@@ -324,7 +324,7 @@
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   {
    "cell_type": "markdown",
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+   "id": "4a1ebfbc",
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@@ -341,7 +341,7 @@
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   {
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+   "id": "632a74c9",
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@@ -364,7 +364,7 @@
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   {
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+   "id": "d8173cef",
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@@ -381,7 +381,7 @@
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   {
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-   "id": "1797b7e5",
+   "id": "8beab768",
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@@ -400,7 +400,7 @@
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   {
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+   "id": "7f8598e9",
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@@ -411,7 +411,7 @@
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   {
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+   "id": "55c77d47",
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@@ -423,7 +423,7 @@
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   {
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-   "id": "c269e5e9",
+   "id": "695e94a9",
    "metadata": {
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@@ -441,14 +441,14 @@
   },
   {
    "cell_type": "markdown",
-   "id": "64cdb37b",
+   "id": "7cf05588",
    "metadata": {
     "editable": true
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    "source": [
     "has linearly dependent column vectors, we will not be able to compute the inverse\n",
     "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\theta_i$. \n",
-    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n",
+    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exists. \n",
     "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n",
     "the regression parameters $\\theta_i$ cannot be estimated.\n",
     "\n",
@@ -457,7 +457,7 @@
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   {
    "cell_type": "markdown",
-   "id": "3e6e5a3e",
+   "id": "bed94018",
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@@ -469,7 +469,7 @@
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   {
    "cell_type": "markdown",
-   "id": "fd83bb4b",
+   "id": "aae1d2f2",
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@@ -479,7 +479,7 @@
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   {
    "cell_type": "markdown",
-   "id": "45b376e8",
+   "id": "d7b1e50f",
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@@ -492,7 +492,7 @@
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   {
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+   "id": "5aaa4dd5",
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@@ -504,7 +504,7 @@
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   {
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-   "id": "de110cca",
+   "id": "23b4b363",
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@@ -514,7 +514,7 @@
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   {
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-   "id": "a8f75c00",
+   "id": "0d156e68",
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@@ -527,7 +527,7 @@
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   {
    "cell_type": "markdown",
-   "id": "e3d1ecbd",
+   "id": "386fac67",
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@@ -537,7 +537,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a932ed24",
+   "id": "e0083fa6",
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@@ -549,7 +549,7 @@
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   {
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-   "id": "9ec04c56",
+   "id": "c3bf20b5",
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@@ -562,7 +562,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c191b236",
+   "id": "8a09eb45",
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@@ -575,7 +575,7 @@
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   {
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-   "id": "214affcd",
+   "id": "b9e924fa",
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@@ -587,7 +587,7 @@
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   {
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+   "id": "14e649b1",
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@@ -599,7 +599,7 @@
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   {
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-   "id": "a10a36e8",
+   "id": "2357300f",
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@@ -609,7 +609,7 @@
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   {
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-   "id": "51e17519",
+   "id": "d5b9128f",
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@@ -622,7 +622,7 @@
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   {
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-   "id": "eb706113",
+   "id": "9b8f2719",
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@@ -634,7 +634,7 @@
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   {
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-   "id": "f2252659",
+   "id": "ccaec8bc",
    "metadata": {
     "editable": true
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@@ -646,7 +646,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bcf67e2c",
+   "id": "b9bbc55c",
    "metadata": {
     "editable": true
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@@ -658,7 +658,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4053a663",
+   "id": "4c128865",
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     "editable": true
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@@ -670,7 +670,7 @@
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   {
    "cell_type": "markdown",
-   "id": "8c8d8ce5",
+   "id": "d9b05a5a",
    "metadata": {
     "editable": true
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@@ -684,7 +684,7 @@
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   {
    "cell_type": "markdown",
-   "id": "fd306ffb",
+   "id": "4c3de377",
    "metadata": {
     "editable": true
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@@ -696,7 +696,7 @@
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   {
    "cell_type": "markdown",
-   "id": "f819a5cd",
+   "id": "64b28254",
    "metadata": {
     "editable": true
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@@ -706,7 +706,7 @@
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   {
    "cell_type": "markdown",
-   "id": "95c2e048",
+   "id": "8123860e",
    "metadata": {
     "editable": true
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@@ -718,7 +718,7 @@
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   {
    "cell_type": "markdown",
-   "id": "071a983e",
+   "id": "0e327aac",
    "metadata": {
     "editable": true
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@@ -730,7 +730,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "36d40252",
+   "id": "134d6cb8",
    "metadata": {
     "editable": true
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@@ -742,7 +742,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e70d5af4",
+   "id": "97d5602d",
    "metadata": {
     "editable": true
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@@ -754,7 +754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d0becda4",
+   "id": "90137256",
    "metadata": {
     "editable": true
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@@ -766,7 +766,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8ee3aff",
+   "id": "cfd34e4d",
    "metadata": {
     "editable": true
    },
@@ -789,7 +789,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "19ea825b",
+   "id": "35da6db9",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -867,7 +867,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85080d04",
+   "id": "2c678f39",
    "metadata": {
     "editable": true
    },
@@ -879,7 +879,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cefa4cb7",
+   "id": "cf9d2b92",
    "metadata": {
     "editable": true
    },
@@ -894,7 +894,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10147fa0",
+   "id": "e1a5fd8a",
    "metadata": {
     "editable": true
    },
@@ -906,7 +906,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2405041d",
+   "id": "7f511458",
    "metadata": {
     "editable": true
    },
@@ -916,7 +916,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a1a6b707",
+   "id": "c40d1b84",
    "metadata": {
     "editable": true
    },
@@ -928,7 +928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae672df8",
+   "id": "045cf436",
    "metadata": {
     "editable": true
    },
@@ -938,7 +938,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "53ea5d3b",
+   "id": "2c532237",
    "metadata": {
     "editable": true
    },
@@ -950,7 +950,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19d5965a",
+   "id": "ac5cf334",
    "metadata": {
     "editable": true
    },
@@ -962,7 +962,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20608e44",
+   "id": "e2bd47a2",
    "metadata": {
     "editable": true
    },
@@ -977,7 +977,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a04cc8a",
+   "id": "ce9b602b",
    "metadata": {
     "editable": true
    },
@@ -988,7 +988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77a30599",
+   "id": "71339dde",
    "metadata": {
     "editable": true
    },
@@ -1008,7 +1008,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e97e9982",
+   "id": "93b3d2ca",
    "metadata": {
     "editable": true
    },
@@ -1020,7 +1020,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b89ea8c",
+   "id": "b13455c8",
    "metadata": {
     "editable": true
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@@ -1030,7 +1030,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "148873d3",
+   "id": "74108a43",
    "metadata": {
     "editable": true
    },
@@ -1042,7 +1042,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b008fef3",
+   "id": "ee5ebfa5",
    "metadata": {
     "editable": true
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@@ -1071,7 +1071,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82103acd",
+   "id": "6eb960f4",
    "metadata": {
     "editable": true
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@@ -1098,7 +1098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fea6570",
+   "id": "cc951a64",
    "metadata": {
     "editable": true
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@@ -1109,7 +1109,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "ac796dd0",
+   "id": "a08edf11",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1149,7 +1149,7 @@
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   {
    "cell_type": "markdown",
-   "id": "db53512b",
+   "id": "c5f84b6d",
    "metadata": {
     "editable": true
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@@ -1166,7 +1166,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "502d1308",
+   "id": "3e6537bc",
    "metadata": {
     "editable": true
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@@ -1189,7 +1189,7 @@
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   {
    "cell_type": "markdown",
-   "id": "8c2bb770",
+   "id": "c4b0d7d0",
    "metadata": {
     "editable": true
    },
@@ -1203,7 +1203,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ee7c0203",
+   "id": "2710e74d",
    "metadata": {
     "editable": true
    },
@@ -1222,7 +1222,7 @@
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   {
    "cell_type": "markdown",
-   "id": "672a491b",
+   "id": "d33e794c",
    "metadata": {
     "editable": true
    },
@@ -1232,7 +1232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8e3af41a",
+   "id": "6940c156",
    "metadata": {
     "editable": true
    },
@@ -1244,7 +1244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c0db076",
+   "id": "0a9fa0f9",
    "metadata": {
     "editable": true
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@@ -1258,7 +1258,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a6c05d18",
+   "id": "2ac8bc02",
    "metadata": {
     "editable": true
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@@ -1270,7 +1270,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ff6c524a",
+   "id": "f6197461",
    "metadata": {
     "editable": true
    },
@@ -1280,7 +1280,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ccca66ed",
+   "id": "7fa22eba",
    "metadata": {
     "editable": true
    },
@@ -1292,7 +1292,7 @@
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   {
    "cell_type": "markdown",
-   "id": "da2a1c44",
+   "id": "a70285f6",
    "metadata": {
     "editable": true
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@@ -1309,7 +1309,7 @@
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   {
    "cell_type": "markdown",
-   "id": "2f2704ab",
+   "id": "bf5774b8",
    "metadata": {
     "editable": true
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@@ -1319,7 +1319,7 @@
   },
   {
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-   "id": "e7286128",
+   "id": "8ca229ff",
    "metadata": {
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@@ -1335,7 +1335,7 @@
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   {
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-   "id": "560e14d4",
+   "id": "b6bf96ea",
    "metadata": {
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@@ -1345,7 +1345,7 @@
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   {
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-   "id": "5f35139c",
+   "id": "d49ba9c3",
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@@ -1361,7 +1361,7 @@
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   {
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-   "id": "a5732633",
+   "id": "183d0b9c",
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@@ -1371,7 +1371,7 @@
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   {
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-   "id": "cb54add9",
+   "id": "5442ca49",
    "metadata": {
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@@ -1387,7 +1387,7 @@
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   {
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-   "id": "f03fc8ea",
+   "id": "823779a1",
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@@ -1397,7 +1397,7 @@
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   {
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-   "id": "d686fb8e",
+   "id": "bc9b3b47",
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@@ -1414,7 +1414,7 @@
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-   "id": "3b44a880",
+   "id": "db89583c",
    "metadata": {
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@@ -1426,7 +1426,7 @@
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   {
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-   "id": "d2dcc6d7",
+   "id": "cbb0223f",
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@@ -1438,7 +1438,7 @@
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   {
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+   "id": "3388eb62",
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@@ -1450,7 +1450,7 @@
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-   "id": "1b1b3ce0",
+   "id": "37c056f5",
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@@ -1460,7 +1460,7 @@
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+   "id": "bbbad4ae",
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@@ -1472,7 +1472,7 @@
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-   "id": "98506ca6",
+   "id": "23ebbb1c",
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@@ -1484,7 +1484,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f7df719c",
+   "id": "bdcfe054",
    "metadata": {
     "editable": true
    },
@@ -1496,7 +1496,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1263babe",
+   "id": "3f26e5a9",
    "metadata": {
     "editable": true
    },
@@ -1506,7 +1506,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "13a792f3",
+   "id": "ab396718",
    "metadata": {
     "editable": true
    },
@@ -1518,7 +1518,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff69a1d7",
+   "id": "8ef06c70",
    "metadata": {
     "editable": true
    },
@@ -1528,7 +1528,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e4a31aa",
+   "id": "13417de5",
    "metadata": {
     "editable": true
    },
@@ -1540,7 +1540,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50768853",
+   "id": "0e6345d4",
    "metadata": {
     "editable": true
    },
@@ -1557,7 +1557,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e70b34dc",
+   "id": "83a8abf0",
    "metadata": {
     "editable": true
    },
@@ -1569,7 +1569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e505b123",
+   "id": "6f8e392e",
    "metadata": {
     "editable": true
    },
@@ -1581,7 +1581,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e143b570",
+   "id": "f8af87f5",
    "metadata": {
     "editable": true
    },
@@ -1591,7 +1591,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5135ebb4",
+   "id": "a00ee63c",
    "metadata": {
     "editable": true
    },
@@ -1603,7 +1603,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "07ce9ec4",
+   "id": "489ab15e",
    "metadata": {
     "editable": true
    },
@@ -1614,7 +1614,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80932acf",
+   "id": "6e6bc889",
    "metadata": {
     "editable": true
    },
@@ -1626,7 +1626,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "73eeda35",
+   "id": "06429f0f",
    "metadata": {
     "editable": true
    },
@@ -1636,7 +1636,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47aff789",
+   "id": "1ddbfde2",
    "metadata": {
     "editable": true
    },
@@ -1648,7 +1648,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2460c817",
+   "id": "3a910378",
    "metadata": {
     "editable": true
    },
@@ -1658,7 +1658,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e3b0d9c",
+   "id": "b03e080e",
    "metadata": {
     "editable": true
    },
@@ -1670,7 +1670,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fbf2cb86",
+   "id": "925e17ec",
    "metadata": {
     "editable": true
    },
@@ -1681,7 +1681,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ebce4ea0",
+   "id": "6825e5f1",
    "metadata": {
     "editable": true
    },
@@ -1693,7 +1693,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2795da2d",
+   "id": "fe851d64",
    "metadata": {
     "editable": true
    },
@@ -1711,7 +1711,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6a26ccff",
+   "id": "40e14e19",
    "metadata": {
     "editable": true
    },
@@ -1724,7 +1724,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "702137ae",
+   "id": "7a7da817",
    "metadata": {
     "editable": true
    },
@@ -1736,7 +1736,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6df222d6",
+   "id": "815d4798",
    "metadata": {
     "editable": true
    },
@@ -1746,7 +1746,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9cc370f4",
+   "id": "9d5e0b49",
    "metadata": {
     "editable": true
    },
@@ -1759,7 +1759,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a71ac65",
+   "id": "ef2a20e5",
    "metadata": {
     "editable": true
    },
@@ -1769,7 +1769,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4242498",
+   "id": "7ce0f92d",
    "metadata": {
     "editable": true
    },
@@ -1781,7 +1781,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6deb1e52",
+   "id": "5cca9be8",
    "metadata": {
     "editable": true
    },
@@ -1794,7 +1794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b9320dd",
+   "id": "fc5e1f2c",
    "metadata": {
     "editable": true
    },
@@ -1807,7 +1807,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43cca84c",
+   "id": "c9cb5c03",
    "metadata": {
     "editable": true
    },
@@ -1819,7 +1819,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91672a95",
+   "id": "3a4a16f9",
    "metadata": {
     "editable": true
    },
@@ -1831,7 +1831,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3e9c0e4",
+   "id": "de9d17d1",
    "metadata": {
     "editable": true
    },
@@ -1841,7 +1841,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0cc6fab9",
+   "id": "a25cf6f8",
    "metadata": {
     "editable": true
    },
@@ -1854,7 +1854,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71f360d2",
+   "id": "cca2f8c4",
    "metadata": {
     "editable": true
    },
@@ -1866,7 +1866,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a27d39c",
+   "id": "d02c6998",
    "metadata": {
     "editable": true
    },
@@ -1878,7 +1878,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d554f3b2",
+   "id": "19b2ac7d",
    "metadata": {
     "editable": true
    },
@@ -1895,7 +1895,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9597e67",
+   "id": "f2d4c46a",
    "metadata": {
     "editable": true
    },
@@ -1907,7 +1907,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d0c80c08",
+   "id": "c084dde9",
    "metadata": {
     "editable": true
    },
@@ -1917,7 +1917,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b33f0e5",
+   "id": "3bacabfd",
    "metadata": {
     "editable": true
    },
@@ -1929,7 +1929,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7db2a530",
+   "id": "6ac8a1a8",
    "metadata": {
     "editable": true
    },
@@ -1939,7 +1939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "177b3058",
+   "id": "43eb347b",
    "metadata": {
     "editable": true
    },
@@ -1951,7 +1951,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "304c0b90",
+   "id": "11e0a485",
    "metadata": {
     "editable": true
    },
@@ -1963,7 +1963,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "14936ac3",
+   "id": "012a0ea1",
    "metadata": {
     "editable": true
    },
@@ -1979,7 +1979,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "797ca0aa",
+   "id": "1f8bf06b",
    "metadata": {
     "editable": true
    },
@@ -1991,7 +1991,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a469a907",
+   "id": "afbc2061",
    "metadata": {
     "editable": true
    },
@@ -2003,7 +2003,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9452a64",
+   "id": "22d903c1",
    "metadata": {
     "editable": true
    },
@@ -2013,7 +2013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "699b712d",
+   "id": "09655feb",
    "metadata": {
     "editable": true
    },
@@ -2025,7 +2025,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e4af99e7",
+   "id": "976449ac",
    "metadata": {
     "editable": true
    },
@@ -2035,7 +2035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a5a8ee8c",
+   "id": "3aeb28d8",
    "metadata": {
     "editable": true
    },
@@ -2047,7 +2047,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03c00df5",
+   "id": "59348b08",
    "metadata": {
     "editable": true
    },
@@ -2064,7 +2064,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "023042c7",
+   "id": "a3a38146",
    "metadata": {
     "editable": true
    },
@@ -2076,7 +2076,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0cd7107",
+   "id": "4de2d8d6",
    "metadata": {
     "editable": true
    },
@@ -2088,7 +2088,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c34ce5f",
+   "id": "5539ce38",
    "metadata": {
     "editable": true
    },
@@ -2098,7 +2098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d489ea51",
+   "id": "519481d2",
    "metadata": {
     "editable": true
    },
@@ -2110,7 +2110,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9921bc50",
+   "id": "433bee8f",
    "metadata": {
     "editable": true
    },
@@ -2120,7 +2120,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c568250e",
+   "id": "afea6f71",
    "metadata": {
     "editable": true
    },
@@ -2132,7 +2132,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21d0b485",
+   "id": "32b32b42",
    "metadata": {
     "editable": true
    },
@@ -2142,19 +2142,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6722483b",
+   "id": "54a0c1f6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9777f5cc",
+   "id": "ec7d68de",
    "metadata": {
     "editable": true
    },
@@ -2164,19 +2164,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23101fe6",
+   "id": "d26bacac",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4f604170",
+   "id": "ac45c3e5",
    "metadata": {
     "editable": true
    },
@@ -2186,7 +2186,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c7b9070",
+   "id": "451fbbf2",
    "metadata": {
     "editable": true
    },
@@ -2205,7 +2205,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0af8105c",
+   "id": "e0569839",
    "metadata": {
     "editable": true
    },
@@ -2225,7 +2225,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c93d44b0",
+   "id": "de0e44d6",
    "metadata": {
     "editable": true
    },
@@ -2241,7 +2241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b1c868cd",
+   "id": "bae50b3b",
    "metadata": {
     "editable": true
    },
@@ -2257,7 +2257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c257671",
+   "id": "2cc7033f",
    "metadata": {
     "editable": true
    },
@@ -2268,7 +2268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ddc77aef",
+   "id": "c4ca8926",
    "metadata": {
     "editable": true
    },
@@ -2280,7 +2280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "026777a7",
+   "id": "1eaf376f",
    "metadata": {
     "editable": true
    },
@@ -2290,7 +2290,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3707790b",
+   "id": "9c025697",
    "metadata": {
     "editable": true
    },
@@ -2302,7 +2302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "13ab6bec",
+   "id": "2d64abe7",
    "metadata": {
     "editable": true
    },
@@ -2312,7 +2312,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a6d69ef",
+   "id": "b83c29d5",
    "metadata": {
     "editable": true
    },
@@ -2324,7 +2324,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "336ec1b1",
+   "id": "cfcc809e",
    "metadata": {
     "editable": true
    },
@@ -2346,7 +2346,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3081ba44",
+   "id": "bdaa56d1",
    "metadata": {
     "editable": true
    },
@@ -2359,7 +2359,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c3faff2",
+   "id": "ffa4afe6",
    "metadata": {
     "editable": true
    },
@@ -2372,7 +2372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68fd52c0",
+   "id": "5c6b9239",
    "metadata": {
     "editable": true
    },
@@ -2382,7 +2382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e29a567",
+   "id": "5ddb446d",
    "metadata": {
     "editable": true
    },
@@ -2400,7 +2400,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f787c3a2",
+   "id": "9a0a5b04",
    "metadata": {
     "editable": true
    },
@@ -2410,7 +2410,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c5e1214",
+   "id": "6fe881ee",
    "metadata": {
     "editable": true
    },
@@ -2425,7 +2425,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cdd5ba1",
+   "id": "83b83194",
    "metadata": {
     "editable": true
    },
@@ -2435,7 +2435,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "276e07a3",
+   "id": "09e77119",
    "metadata": {
     "editable": true
    },
@@ -2449,7 +2449,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80fe7f13",
+   "id": "afdce492",
    "metadata": {
     "editable": true
    },
@@ -2459,7 +2459,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8838b8f",
+   "id": "b6e29c29",
    "metadata": {
     "editable": true
    },
@@ -2473,7 +2473,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2ed6fbd0",
+   "id": "ad6e09d7",
    "metadata": {
     "editable": true
    },
@@ -2488,7 +2488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0796429c",
+   "id": "a514df33",
    "metadata": {
     "editable": true
    },
@@ -2505,7 +2505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc44df4f",
+   "id": "ae758019",
    "metadata": {
     "editable": true
    },
@@ -2517,7 +2517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c88146a9",
+   "id": "1bcffe5e",
    "metadata": {
     "editable": true
    },
@@ -2531,7 +2531,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1b85445",
+   "id": "67467328",
    "metadata": {
     "editable": true
    },
@@ -2546,7 +2546,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8edde795",
+   "id": "4bac20b4",
    "metadata": {
     "editable": true
    },
@@ -2558,7 +2558,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f92e4583",
+   "id": "01e84611",
    "metadata": {
     "editable": true
    },
@@ -2569,7 +2569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0bf80264",
+   "id": "8f742c3b",
    "metadata": {
     "editable": true
    },
@@ -2597,7 +2597,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b6b24c7",
+   "id": "77acd12d",
    "metadata": {
     "editable": true
    },
@@ -2619,7 +2619,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e36297bd",
+   "id": "99ab46db",
    "metadata": {
     "editable": true
    },
@@ -2641,7 +2641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8ec331d",
+   "id": "5b4b6243",
    "metadata": {
     "editable": true
    },
@@ -2653,7 +2653,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b0d0c67",
+   "id": "2ba66a00",
    "metadata": {
     "editable": true
    },
@@ -2690,7 +2690,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bdfac28b",
+   "id": "c9a65dbe",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2718,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2032fb83",
+   "id": "7efa8d14",
    "metadata": {
     "editable": true
    },
@@ -2748,7 +2748,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5a00d18",
+   "id": "93d8901e",
    "metadata": {
     "editable": true
    },
@@ -2772,7 +2772,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "68f97538",
+   "id": "6a775d3c",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2785,7 +2785,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a7fbe32f",
+   "id": "b8360996",
    "metadata": {
     "editable": true
    },
@@ -2796,7 +2796,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b64af25",
+   "id": "0b4a5c74",
    "metadata": {
     "editable": true
    },
@@ -2808,7 +2808,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ef56e132",
+   "id": "f27ef437",
    "metadata": {
     "editable": true
    },
@@ -2818,7 +2818,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6d7f0c20",
+   "id": "6a7d8179",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ac315d2",
+   "id": "24b3bcb3",
    "metadata": {
     "editable": true
    },
@@ -2844,7 +2844,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "040d030b",
+   "id": "196152f9",
    "metadata": {
     "editable": true
    },
@@ -2860,7 +2860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cb9c213",
+   "id": "6b935471",
    "metadata": {
     "editable": true
    },
@@ -2870,7 +2870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f6ab192",
+   "id": "d5261a7f",
    "metadata": {
     "editable": true
    },
@@ -2882,7 +2882,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc154729",
+   "id": "398173a2",
    "metadata": {
     "editable": true
    },
@@ -2892,7 +2892,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cda14c18",
+   "id": "0f29a1a0",
    "metadata": {
     "editable": true
    },
@@ -2904,7 +2904,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffc6db98",
+   "id": "22f27a92",
    "metadata": {
     "editable": true
    },
@@ -2918,7 +2918,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5dd4f8d",
+   "id": "a2b8898e",
    "metadata": {
     "editable": true
    },
@@ -2928,7 +2928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "54cf8a09",
+   "id": "42b415d8",
    "metadata": {
     "editable": true
    },
@@ -2939,7 +2939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c06fcb28",
+   "id": "b522b609",
    "metadata": {
     "editable": true
    },
@@ -2954,7 +2954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16b2e122",
+   "id": "2e2e2e39",
    "metadata": {
     "editable": true
    },
@@ -2964,7 +2964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e48ac99",
+   "id": "6f14f45f",
    "metadata": {
     "editable": true
    },
@@ -2976,7 +2976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a536e9c",
+   "id": "d9210eed",
    "metadata": {
     "editable": true
    },
@@ -2988,7 +2988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9afa223",
+   "id": "5050ec60",
    "metadata": {
     "editable": true
    },
@@ -3003,7 +3003,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4f76d91",
+   "id": "04ac7d83",
    "metadata": {
     "editable": true
    },
@@ -3016,7 +3016,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "801ec4de",
+   "id": "6b2e19f6",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3073,45 +3073,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6c4900e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## And a corresponding example using **scikit-learn**"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "id": "d2f662be",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "# Importing various packages\n",
-    "from random import random, seed\n",
-    "import numpy as np\n",
-    "import matplotlib.pyplot as plt\n",
-    "from sklearn.linear_model import SGDRegressor\n",
-    "\n",
-    "n = 100\n",
-    "x = 2*np.random.rand(n,1)\n",
-    "y = 4+3*x+np.random.randn(n,1)\n",
-    "\n",
-    "X = np.c_[np.ones((n,1)), x]\n",
-    "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n",
-    "print(theta_linreg)\n",
-    "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n",
-    "sgdreg.fit(x,y.ravel())\n",
-    "print(sgdreg.intercept_, sgdreg.coef_)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "83bd5b44",
+   "id": "26e7815c",
    "metadata": {
     "editable": true
    },
@@ -3123,7 +3085,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c7ee80c",
+   "id": "0ae04360",
    "metadata": {
     "editable": true
    },
@@ -3135,7 +3097,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c70714f",
+   "id": "402501d6",
    "metadata": {
     "editable": true
    },
@@ -3145,7 +3107,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e170727b",
+   "id": "fdd060d5",
    "metadata": {
     "editable": true
    },
@@ -3159,7 +3121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b7b5d1ce",
+   "id": "8a061527",
    "metadata": {
     "editable": true
    },
@@ -3169,7 +3131,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e59753a",
+   "id": "72c8bb88",
    "metadata": {
     "editable": true
    },
@@ -3181,7 +3143,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d05de5f9",
+   "id": "a8e126ba",
    "metadata": {
     "editable": true
    },
@@ -3192,7 +3154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3a5c888",
+   "id": "cae81a93",
    "metadata": {
     "editable": true
    },
@@ -3207,7 +3169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52b009a7",
+   "id": "646feacb",
    "metadata": {
     "editable": true
    },
@@ -3221,7 +3183,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "430d2418",
+   "id": "e2b0d0a9",
    "metadata": {
     "editable": true
    },
@@ -3231,8 +3193,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
-   "id": "b889c65e",
+   "execution_count": 5,
+   "id": "b2354adf",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3293,7 +3255,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21e20510",
+   "id": "e95bd522",
    "metadata": {
     "editable": true
    },
@@ -3315,7 +3277,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59d61c18",
+   "id": "947549ab",
    "metadata": {
     "editable": true
    },
@@ -3327,7 +3289,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19a3af8f",
+   "id": "64b3eeb6",
    "metadata": {
     "editable": true
    },
@@ -3339,7 +3301,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0a91546",
+   "id": "8ddf1823",
    "metadata": {
     "editable": true
    },
@@ -3351,7 +3313,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ddc3a0d",
+   "id": "fe9d8eab",
    "metadata": {
     "editable": true
    },
@@ -3361,7 +3323,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2c51cc5b",
+   "id": "a6171f9f",
    "metadata": {
     "editable": true
    },
@@ -3373,7 +3335,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "754dd1c2",
+   "id": "7fed4d62",
    "metadata": {
     "editable": true
    },
@@ -3383,7 +3345,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37b8fcf8",
+   "id": "276ab416",
    "metadata": {
     "editable": true
    },
@@ -3400,7 +3362,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4103d318",
+   "id": "67074db1",
    "metadata": {
     "editable": true
    },
@@ -3410,7 +3372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dae6c763",
+   "id": "068b0864",
    "metadata": {
     "editable": true
    },
@@ -3422,7 +3384,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b13a4b0d",
+   "id": "20210169",
    "metadata": {
     "editable": true
    },
@@ -3432,7 +3394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0911661b",
+   "id": "4a45a715",
    "metadata": {
     "editable": true
    },
@@ -3444,7 +3406,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a4144ae",
+   "id": "ac682db5",
    "metadata": {
     "editable": true
    },
@@ -3458,7 +3420,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9dfc5f86",
+   "id": "fe40344c",
    "metadata": {
     "editable": true
    },
@@ -3470,7 +3432,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "73be1c56",
+   "id": "4f23bc28",
    "metadata": {
     "editable": true
    },
@@ -3489,7 +3451,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9bd87051",
+   "id": "587b3612",
    "metadata": {
     "editable": true
    },
@@ -3502,7 +3464,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c720ad2b",
+   "id": "5b99abf5",
    "metadata": {
     "editable": true
    },
@@ -3514,7 +3476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2bb60a91",
+   "id": "5bf144d8",
    "metadata": {
     "editable": true
    },
@@ -3524,7 +3486,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3a65ac27",
+   "id": "cd14d016",
    "metadata": {
     "editable": true
    },
@@ -3537,7 +3499,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3dfeab84",
+   "id": "3d7e4869",
    "metadata": {
     "editable": true
    },
@@ -3547,7 +3509,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e8ad46c",
+   "id": "3fbcb18f",
    "metadata": {
     "editable": true
    },
@@ -3559,7 +3521,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "500a62cd",
+   "id": "ba1bd0a8",
    "metadata": {
     "editable": true
    },
@@ -3574,7 +3536,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "edf77f5d",
+   "id": "52f96491",
    "metadata": {
     "editable": true
    },
@@ -3587,7 +3549,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9d308fc9",
+   "id": "5cd4001c",
    "metadata": {
     "editable": true
    },
@@ -3601,7 +3563,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fccb6caa",
+   "id": "d1ac4860",
    "metadata": {
     "editable": true
    },
@@ -3613,7 +3575,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f1fa209f",
+   "id": "73822d93",
    "metadata": {
     "editable": true
    },
@@ -3623,7 +3585,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ae99f60",
+   "id": "678b5041",
    "metadata": {
     "editable": true
    },
@@ -3636,7 +3598,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "36cbbd77",
+   "id": "cdb9b585",
    "metadata": {
     "editable": true
    },
@@ -3648,7 +3610,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d2d67dee",
+   "id": "decbfc68",
    "metadata": {
     "editable": true
    },
@@ -3660,7 +3622,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1e84f6b",
+   "id": "27d418d8",
    "metadata": {
     "editable": true
    },
@@ -3672,7 +3634,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d0806a88",
+   "id": "c59d8340",
    "metadata": {
     "editable": true
    },
@@ -3684,7 +3646,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5217eb80",
+   "id": "1f9d8ae6",
    "metadata": {
     "editable": true
    },
@@ -3698,7 +3660,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26476c82",
+   "id": "eb92751f",
    "metadata": {
     "editable": true
    },
@@ -3710,7 +3672,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "865ed23a",
+   "id": "c8924cc6",
    "metadata": {
     "editable": true
    },
@@ -3720,7 +3682,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "267f405a",
+   "id": "b859ce0a",
    "metadata": {
     "editable": true
    },
@@ -3732,7 +3694,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "128c8fe1",
+   "id": "51db1890",
    "metadata": {
     "editable": true
    },
@@ -3742,7 +3704,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "025774a5",
+   "id": "96b9eb8a",
    "metadata": {
     "editable": true
    },
@@ -3754,7 +3716,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3d2ecae",
+   "id": "2e183c0f",
    "metadata": {
     "editable": true
    },
@@ -3766,7 +3728,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f619bb0",
+   "id": "45613b55",
    "metadata": {
     "editable": true
    },
@@ -3776,7 +3738,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa1c8647",
+   "id": "6e6ba17e",
    "metadata": {
     "editable": true
    },
@@ -3787,7 +3749,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f5a4138",
+   "id": "984b9f74",
    "metadata": {
     "editable": true
    },
@@ -3799,7 +3761,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "357000de",
+   "id": "cf9858ef",
    "metadata": {
     "editable": true
    },
@@ -3815,7 +3777,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2716f2d3",
+   "id": "7240aa01",
    "metadata": {
     "editable": true
    },
@@ -3828,7 +3790,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f1150a1",
+   "id": "08167266",
    "metadata": {
     "editable": true
    },
@@ -3840,7 +3802,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d5ee456d",
+   "id": "4a94d17e",
    "metadata": {
     "editable": true
    },
@@ -3850,7 +3812,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50b60351",
+   "id": "bf4561b9",
    "metadata": {
     "editable": true
    },
@@ -3862,7 +3824,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b349aa76",
+   "id": "bc19b151",
    "metadata": {
     "editable": true
    },
@@ -3872,7 +3834,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0cf92ff7",
+   "id": "adbd1d84",
    "metadata": {
     "editable": true
    },
@@ -3884,7 +3846,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63ab8407",
+   "id": "eb629ae2",
    "metadata": {
     "editable": true
    },
@@ -3896,7 +3858,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1c8b499",
+   "id": "e19f5e29",
    "metadata": {
     "editable": true
    },
@@ -3912,7 +3874,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38a3cc6b",
+   "id": "3bc97428",
    "metadata": {
     "editable": true
    },
@@ -3924,7 +3886,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6c9a1e21",
+   "id": "5955ee37",
    "metadata": {
     "editable": true
    },
@@ -3936,7 +3898,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77cebd55",
+   "id": "bcc364dd",
    "metadata": {
     "editable": true
    },
@@ -3946,7 +3908,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3397dc1",
+   "id": "2ec6f51e",
    "metadata": {
     "editable": true
    },
@@ -3958,7 +3920,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8b36e38d",
+   "id": "0d497cde",
    "metadata": {
     "editable": true
    },
@@ -3968,7 +3930,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c6262cc",
+   "id": "26fea14b",
    "metadata": {
     "editable": true
    },
@@ -3980,7 +3942,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "18a58f70",
+   "id": "5a26f674",
    "metadata": {
     "editable": true
    },
@@ -3997,7 +3959,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bdadc0f1",
+   "id": "0e4d2ab0",
    "metadata": {
     "editable": true
    },
@@ -4009,7 +3971,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a482e79b",
+   "id": "953d2d2d",
    "metadata": {
     "editable": true
    },
@@ -4021,7 +3983,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c1f3504",
+   "id": "b2a43ed6",
    "metadata": {
     "editable": true
    },
@@ -4031,7 +3993,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c7ff16c0",
+   "id": "5a63ea86",
    "metadata": {
     "editable": true
    },
@@ -4043,7 +4005,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df7b2ee6",
+   "id": "1f7bb354",
    "metadata": {
     "editable": true
    },
@@ -4053,7 +4015,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "768d2256",
+   "id": "70f7bdaa",
    "metadata": {
     "editable": true
    },
@@ -4065,7 +4027,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0d5cbe4",
+   "id": "789c944f",
    "metadata": {
     "editable": true
    },
@@ -4075,7 +4037,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5403622a",
+   "id": "fabf113f",
    "metadata": {
     "editable": true
    },
@@ -4087,7 +4049,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75c85a12",
+   "id": "52ecdca6",
    "metadata": {
     "editable": true
    },
@@ -4097,7 +4059,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4591459d",
+   "id": "0a7eebc3",
    "metadata": {
     "editable": true
    },
@@ -4112,7 +4074,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24cc16cd",
+   "id": "bab27602",
    "metadata": {
     "editable": true
    },
@@ -4124,7 +4086,7 @@
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   {
    "cell_type": "markdown",
-   "id": "86889b7d",
+   "id": "2a41eb77",
    "metadata": {
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@@ -4134,7 +4096,7 @@
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   {
    "cell_type": "markdown",
-   "id": "965e8e09",
+   "id": "1d649e9b",
    "metadata": {
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@@ -4146,7 +4108,7 @@
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   {
    "cell_type": "markdown",
-   "id": "5cffcb19",
+   "id": "2b6fbe08",
    "metadata": {
     "editable": true
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@@ -4158,7 +4120,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6671e517",
+   "id": "3dc6dd52",
    "metadata": {
     "editable": true
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@@ -4170,7 +4132,7 @@
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   {
    "cell_type": "markdown",
-   "id": "252e152e",
+   "id": "bca29a61",
    "metadata": {
     "editable": true
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@@ -4180,7 +4142,7 @@
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   {
    "cell_type": "markdown",
-   "id": "26d20ede",
+   "id": "58cd9edd",
    "metadata": {
     "editable": true
    },
@@ -4192,7 +4154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c25ce321",
+   "id": "6aa14ea2",
    "metadata": {
     "editable": true
    },
@@ -4204,7 +4166,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c036a923",
+   "id": "901815a2",
    "metadata": {
     "editable": true
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@@ -4216,7 +4178,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6f10942f",
+   "id": "e7101eb4",
    "metadata": {
     "editable": true
    },
@@ -4226,7 +4188,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c175b28",
+   "id": "08f2c952",
    "metadata": {
     "editable": true
    },
@@ -4238,7 +4200,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb7d2fa1",
+   "id": "2364880d",
    "metadata": {
     "editable": true
    },
@@ -4248,7 +4210,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "352e9f9a",
+   "id": "b66a92bf",
    "metadata": {
     "editable": true
    },
@@ -4262,7 +4224,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d627bc4",
+   "id": "b88f4eed",
    "metadata": {
     "editable": true
    },
@@ -4272,7 +4234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e0d9b39f",
+   "id": "eb15fecd",
    "metadata": {
     "editable": true
    },
@@ -4284,7 +4246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f84e1a77",
+   "id": "6116ecb2",
    "metadata": {
     "editable": true
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@@ -4296,7 +4258,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7845b77c",
+   "id": "ce3ab1f1",
    "metadata": {
     "editable": true
    },
@@ -4306,7 +4268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da3ca301",
+   "id": "fb60d251",
    "metadata": {
     "editable": true
    },
@@ -4318,7 +4280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ac98fa0a",
+   "id": "8206088f",
    "metadata": {
     "editable": true
    },
@@ -4328,7 +4290,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26e6c40d",
+   "id": "458a83dc",
    "metadata": {
     "editable": true
    },
@@ -4340,7 +4302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa9e70a1",
+   "id": "975eba5a",
    "metadata": {
     "editable": true
    },
@@ -4352,7 +4314,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a1a46c84",
+   "id": "5be10b6e",
    "metadata": {
     "editable": true
    },
@@ -4362,7 +4324,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f14d9318",
+   "id": "83ba42b9",
    "metadata": {
     "editable": true
    },
@@ -4374,7 +4336,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b71a9481",
+   "id": "af29af81",
    "metadata": {
     "editable": true
    },
@@ -4384,7 +4346,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "57629d3b",
+   "id": "928670e3",
    "metadata": {
     "editable": true
    },
@@ -4396,7 +4358,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3062396e",
+   "id": "52bc99df",
    "metadata": {
     "editable": true
    },
@@ -4408,7 +4370,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2879fd54",
+   "id": "c4ce97d1",
    "metadata": {
     "editable": true
    },
@@ -4418,7 +4380,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10e94837",
+   "id": "5cc4b855",
    "metadata": {
     "editable": true
    },
@@ -4430,7 +4392,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d449327b",
+   "id": "7ef6b391",
    "metadata": {
     "editable": true
    },
@@ -4443,7 +4405,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52e8baa1",
+   "id": "2b0bcf3c",
    "metadata": {
     "editable": true
    },
@@ -4455,7 +4417,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea867985",
+   "id": "12ae48d6",
    "metadata": {
     "editable": true
    },
@@ -4467,7 +4429,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "864b48cf",
+   "id": "ee553c21",
    "metadata": {
     "editable": true
    },
@@ -4479,7 +4441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f7ed5ba",
+   "id": "c9e48708",
    "metadata": {
     "editable": true
    },
@@ -4489,7 +4451,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d607e924",
+   "id": "ff8584b1",
    "metadata": {
     "editable": true
    },
@@ -4501,7 +4463,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3bd4f95f",
+   "id": "fe9c5ce2",
    "metadata": {
     "editable": true
    },
@@ -4511,7 +4473,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8940052",
+   "id": "d13b4d09",
    "metadata": {
     "editable": true
    },
@@ -4523,7 +4485,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81bbbb0e",
+   "id": "22960198",
    "metadata": {
     "editable": true
    },
@@ -4533,7 +4495,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca83ff01",
+   "id": "5d949f02",
    "metadata": {
     "editable": true
    },
@@ -4545,7 +4507,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b5ac9e9",
+   "id": "59be0f8c",
    "metadata": {
     "editable": true
    },
@@ -4555,7 +4517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19ab9b33",
+   "id": "1e85b855",
    "metadata": {
     "editable": true
    },
@@ -4568,7 +4530,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59135778",
+   "id": "117c5f1d",
    "metadata": {
     "editable": true
    },
@@ -4580,7 +4542,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8d40d8ab",
+   "id": "d3842015",
    "metadata": {
     "editable": true
    },
@@ -4592,7 +4554,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c0401f9",
+   "id": "5619e7d3",
    "metadata": {
     "editable": true
    },
@@ -4602,7 +4564,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa075ce0",
+   "id": "e091b536",
    "metadata": {
     "editable": true
    },
@@ -4614,7 +4576,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e465dee",
+   "id": "924ee170",
    "metadata": {
     "editable": true
    },
@@ -4631,7 +4593,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7af63418",
+   "id": "49434aeb",
    "metadata": {
     "editable": true
    },
@@ -4643,7 +4605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0fff0234",
+   "id": "4b7e1e31",
    "metadata": {
     "editable": true
    },
@@ -4655,7 +4617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32328a87",
+   "id": "374578e8",
    "metadata": {
     "editable": true
    },
@@ -4665,7 +4627,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b6403d7",
+   "id": "5e76c1eb",
    "metadata": {
     "editable": true
    },
@@ -4677,7 +4639,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8823b897",
+   "id": "ece41e8c",
    "metadata": {
     "editable": true
    },
@@ -4687,7 +4649,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "319984ba",
+   "id": "28ef4cad",
    "metadata": {
     "editable": true
    },
@@ -4699,7 +4661,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2382a622",
+   "id": "0c826f0e",
    "metadata": {
     "editable": true
    },
@@ -4709,7 +4671,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24b75d16",
+   "id": "9ef53895",
    "metadata": {
     "editable": true
    },
@@ -4721,7 +4683,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ba2f29c",
+   "id": "91cd7b4f",
    "metadata": {
     "editable": true
    },
@@ -4731,7 +4693,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd65fa2e",
+   "id": "fedfb550",
    "metadata": {
     "editable": true
    },
@@ -4745,8 +4707,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
-   "id": "6f044cbb",
+   "execution_count": 6,
+   "id": "1d8e47a3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -4806,7 +4768,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d69f7030",
+   "id": "b88f84eb",
    "metadata": {
     "editable": true
    },
@@ -4816,7 +4778,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a8f9315a",
+   "id": "b0c77f0b",
    "metadata": {
     "editable": true
    },
@@ -4826,8 +4788,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
-   "id": "041fe053",
+   "execution_count": 7,
+   "id": "570be9c4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -4894,7 +4856,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47db85e5",
+   "id": "3e4210bb",
    "metadata": {
     "editable": true
    },
@@ -4904,8 +4866,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
-   "id": "400b4a2a",
+   "execution_count": 8,
+   "id": "b4ad5452",
    "metadata": {
     "collapsed": false,
     "editable": true
diff --git a/doc/LectureNotes/week36.ipynb b/doc/LectureNotes/week36.ipynb
index 8bda05b08..d0d3dee3f 100644
--- a/doc/LectureNotes/week36.ipynb
+++ b/doc/LectureNotes/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b8f45e2d",
+   "id": "3586da6c",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b3f2ead9",
+   "id": "5afa3a45",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04553781",
+   "id": "739da2e4",
    "metadata": {
     "editable": true
    },
@@ -57,7 +57,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1016bcdb",
+   "id": "66bc8f20",
    "metadata": {
     "editable": true
    },
@@ -67,7 +67,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "839e657a",
+   "id": "d6b147a8",
    "metadata": {
     "editable": true
    },
@@ -81,7 +81,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12dc9112",
+   "id": "69ee41c8",
    "metadata": {
     "editable": true
    },
@@ -93,7 +93,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df3e7f6d",
+   "id": "eec38b3f",
    "metadata": {
     "editable": true
    },
@@ -105,7 +105,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc1caa8e",
+   "id": "7ca66dca",
    "metadata": {
     "editable": true
    },
@@ -117,7 +117,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aaf95f40",
+   "id": "4a67a7b4",
    "metadata": {
     "editable": true
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e615b93",
+   "id": "4b3a2043",
    "metadata": {
     "editable": true
    },
@@ -139,7 +139,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f764f4e",
+   "id": "841bde50",
    "metadata": {
     "editable": true
    },
@@ -149,7 +149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e0d9985",
+   "id": "ebca953b",
    "metadata": {
     "editable": true
    },
@@ -161,7 +161,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a11e3b77",
+   "id": "7be2de84",
    "metadata": {
     "editable": true
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@@ -172,7 +172,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b175d8ad",
+   "id": "9916c24e",
    "metadata": {
     "editable": true
    },
@@ -184,7 +184,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0901c5d9",
+   "id": "03c35639",
    "metadata": {
     "editable": true
    },
@@ -196,7 +196,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7263285d",
+   "id": "63aaa0d6",
    "metadata": {
     "editable": true
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@@ -206,7 +206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c0f54e1a",
+   "id": "563e4883",
    "metadata": {
     "editable": true
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@@ -218,7 +218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "035835e7",
+   "id": "207fb037",
    "metadata": {
     "editable": true
    },
@@ -230,7 +230,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0dca7fe",
+   "id": "d0ebfe7a",
    "metadata": {
     "editable": true
    },
@@ -240,7 +240,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5808efd5",
+   "id": "e5eba895",
    "metadata": {
     "editable": true
    },
@@ -252,7 +252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cca67bb8",
+   "id": "20736fd2",
    "metadata": {
     "editable": true
    },
@@ -262,7 +262,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a189ce2b",
+   "id": "8ebaa24c",
    "metadata": {
     "editable": true
    },
@@ -274,7 +274,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7835a4fb",
+   "id": "9e0e8b1b",
    "metadata": {
     "editable": true
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@@ -284,7 +284,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1ba547c3",
+   "id": "a9dc6df7",
    "metadata": {
     "editable": true
    },
@@ -324,7 +324,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55bc2c91",
+   "id": "4a1ebfbc",
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     "editable": true
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@@ -341,7 +341,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c702f5e0",
+   "id": "632a74c9",
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@@ -364,7 +364,7 @@
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   {
    "cell_type": "markdown",
-   "id": "86cc8cb6",
+   "id": "d8173cef",
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     "editable": true
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@@ -381,7 +381,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1797b7e5",
+   "id": "8beab768",
    "metadata": {
     "editable": true
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@@ -400,7 +400,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dc73a5a2",
+   "id": "7f8598e9",
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@@ -411,7 +411,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20b19ac5",
+   "id": "55c77d47",
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@@ -423,7 +423,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c269e5e9",
+   "id": "695e94a9",
    "metadata": {
     "editable": true
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@@ -441,14 +441,14 @@
   },
   {
    "cell_type": "markdown",
-   "id": "64cdb37b",
+   "id": "7cf05588",
    "metadata": {
     "editable": true
    },
    "source": [
     "has linearly dependent column vectors, we will not be able to compute the inverse\n",
     "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\theta_i$. \n",
-    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n",
+    "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exists. \n",
     "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n",
     "the regression parameters $\\theta_i$ cannot be estimated.\n",
     "\n",
@@ -457,7 +457,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e6e5a3e",
+   "id": "bed94018",
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@@ -469,7 +469,7 @@
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   {
    "cell_type": "markdown",
-   "id": "fd83bb4b",
+   "id": "aae1d2f2",
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@@ -479,7 +479,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "45b376e8",
+   "id": "d7b1e50f",
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@@ -492,7 +492,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90bb63a4",
+   "id": "5aaa4dd5",
    "metadata": {
     "editable": true
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@@ -504,7 +504,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "de110cca",
+   "id": "23b4b363",
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@@ -514,7 +514,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a8f75c00",
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@@ -527,7 +527,7 @@
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@@ -537,7 +537,7 @@
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@@ -549,7 +549,7 @@
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@@ -562,7 +562,7 @@
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@@ -575,7 +575,7 @@
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@@ -587,7 +587,7 @@
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@@ -599,7 +599,7 @@
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@@ -609,7 +609,7 @@
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@@ -622,7 +622,7 @@
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@@ -634,7 +634,7 @@
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@@ -646,7 +646,7 @@
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@@ -658,7 +658,7 @@
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@@ -670,7 +670,7 @@
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@@ -684,7 +684,7 @@
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@@ -696,7 +696,7 @@
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@@ -706,7 +706,7 @@
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@@ -718,7 +718,7 @@
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@@ -730,7 +730,7 @@
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@@ -742,7 +742,7 @@
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@@ -754,7 +754,7 @@
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@@ -766,7 +766,7 @@
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@@ -789,7 +789,7 @@
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@@ -867,7 +867,7 @@
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@@ -879,7 +879,7 @@
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@@ -894,7 +894,7 @@
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@@ -906,7 +906,7 @@
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@@ -916,7 +916,7 @@
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@@ -928,7 +928,7 @@
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@@ -938,7 +938,7 @@
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@@ -950,7 +950,7 @@
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@@ -962,7 +962,7 @@
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@@ -977,7 +977,7 @@
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@@ -988,7 +988,7 @@
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@@ -1008,7 +1008,7 @@
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@@ -1020,7 +1020,7 @@
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@@ -1030,7 +1030,7 @@
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@@ -1042,7 +1042,7 @@
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@@ -1071,7 +1071,7 @@
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@@ -1098,7 +1098,7 @@
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@@ -1109,7 +1109,7 @@
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@@ -1149,7 +1149,7 @@
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@@ -1166,7 +1166,7 @@
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@@ -1189,7 +1189,7 @@
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@@ -1203,7 +1203,7 @@
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@@ -1222,7 +1222,7 @@
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@@ -1232,7 +1232,7 @@
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@@ -1244,7 +1244,7 @@
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@@ -1258,7 +1258,7 @@
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@@ -1270,7 +1270,7 @@
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@@ -1280,7 +1280,7 @@
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@@ -1292,7 +1292,7 @@
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@@ -1309,7 +1309,7 @@
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@@ -1319,7 +1319,7 @@
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@@ -1335,7 +1335,7 @@
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@@ -1345,7 +1345,7 @@
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@@ -1361,7 +1361,7 @@
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@@ -1371,7 +1371,7 @@
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@@ -1387,7 +1387,7 @@
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@@ -1397,7 +1397,7 @@
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@@ -1414,7 +1414,7 @@
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@@ -1426,7 +1426,7 @@
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@@ -1438,7 +1438,7 @@
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@@ -1450,7 +1450,7 @@
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@@ -1460,7 +1460,7 @@
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@@ -1472,7 +1472,7 @@
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@@ -1484,7 +1484,7 @@
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@@ -1496,7 +1496,7 @@
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@@ -1506,7 +1506,7 @@
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@@ -1518,7 +1518,7 @@
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@@ -1528,7 +1528,7 @@
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@@ -1540,7 +1540,7 @@
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@@ -1557,7 +1557,7 @@
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@@ -1569,7 +1569,7 @@
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@@ -1581,7 +1581,7 @@
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@@ -1591,7 +1591,7 @@
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@@ -1603,7 +1603,7 @@
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@@ -1614,7 +1614,7 @@
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@@ -1626,7 +1626,7 @@
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@@ -1636,7 +1636,7 @@
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@@ -1648,7 +1648,7 @@
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@@ -1658,7 +1658,7 @@
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@@ -1670,7 +1670,7 @@
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@@ -1681,7 +1681,7 @@
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@@ -1693,7 +1693,7 @@
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@@ -1711,7 +1711,7 @@
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@@ -1724,7 +1724,7 @@
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@@ -1736,7 +1736,7 @@
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@@ -1746,7 +1746,7 @@
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@@ -1759,7 +1759,7 @@
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@@ -1769,7 +1769,7 @@
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@@ -1781,7 +1781,7 @@
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@@ -1794,7 +1794,7 @@
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@@ -1807,7 +1807,7 @@
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@@ -1819,7 +1819,7 @@
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@@ -1831,7 +1831,7 @@
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@@ -1841,7 +1841,7 @@
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@@ -1854,7 +1854,7 @@
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@@ -1866,7 +1866,7 @@
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@@ -1878,7 +1878,7 @@
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@@ -1895,7 +1895,7 @@
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@@ -1907,7 +1907,7 @@
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@@ -1917,7 +1917,7 @@
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@@ -1929,7 +1929,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7db2a530",
+   "id": "6ac8a1a8",
    "metadata": {
     "editable": true
    },
@@ -1939,7 +1939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "177b3058",
+   "id": "43eb347b",
    "metadata": {
     "editable": true
    },
@@ -1951,7 +1951,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "304c0b90",
+   "id": "11e0a485",
    "metadata": {
     "editable": true
    },
@@ -1963,7 +1963,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "14936ac3",
+   "id": "012a0ea1",
    "metadata": {
     "editable": true
    },
@@ -1979,7 +1979,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "797ca0aa",
+   "id": "1f8bf06b",
    "metadata": {
     "editable": true
    },
@@ -1991,7 +1991,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a469a907",
+   "id": "afbc2061",
    "metadata": {
     "editable": true
    },
@@ -2003,7 +2003,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9452a64",
+   "id": "22d903c1",
    "metadata": {
     "editable": true
    },
@@ -2013,7 +2013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "699b712d",
+   "id": "09655feb",
    "metadata": {
     "editable": true
    },
@@ -2025,7 +2025,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e4af99e7",
+   "id": "976449ac",
    "metadata": {
     "editable": true
    },
@@ -2035,7 +2035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a5a8ee8c",
+   "id": "3aeb28d8",
    "metadata": {
     "editable": true
    },
@@ -2047,7 +2047,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03c00df5",
+   "id": "59348b08",
    "metadata": {
     "editable": true
    },
@@ -2064,7 +2064,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "023042c7",
+   "id": "a3a38146",
    "metadata": {
     "editable": true
    },
@@ -2076,7 +2076,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0cd7107",
+   "id": "4de2d8d6",
    "metadata": {
     "editable": true
    },
@@ -2088,7 +2088,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c34ce5f",
+   "id": "5539ce38",
    "metadata": {
     "editable": true
    },
@@ -2098,7 +2098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d489ea51",
+   "id": "519481d2",
    "metadata": {
     "editable": true
    },
@@ -2110,7 +2110,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9921bc50",
+   "id": "433bee8f",
    "metadata": {
     "editable": true
    },
@@ -2120,7 +2120,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c568250e",
+   "id": "afea6f71",
    "metadata": {
     "editable": true
    },
@@ -2132,7 +2132,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21d0b485",
+   "id": "32b32b42",
    "metadata": {
     "editable": true
    },
@@ -2142,19 +2142,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6722483b",
+   "id": "54a0c1f6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\frac{n}{2}\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9777f5cc",
+   "id": "ec7d68de",
    "metadata": {
     "editable": true
    },
@@ -2164,19 +2164,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23101fe6",
+   "id": "d26bacac",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=2\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta}+\\lambda sgn(\\boldsymbol{\\theta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4f604170",
+   "id": "ac45c3e5",
    "metadata": {
     "editable": true
    },
@@ -2186,7 +2186,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c7b9070",
+   "id": "451fbbf2",
    "metadata": {
     "editable": true
    },
@@ -2205,7 +2205,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0af8105c",
+   "id": "e0569839",
    "metadata": {
     "editable": true
    },
@@ -2225,7 +2225,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c93d44b0",
+   "id": "de0e44d6",
    "metadata": {
     "editable": true
    },
@@ -2241,7 +2241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b1c868cd",
+   "id": "bae50b3b",
    "metadata": {
     "editable": true
    },
@@ -2257,7 +2257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c257671",
+   "id": "2cc7033f",
    "metadata": {
     "editable": true
    },
@@ -2268,7 +2268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ddc77aef",
+   "id": "c4ca8926",
    "metadata": {
     "editable": true
    },
@@ -2280,7 +2280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "026777a7",
+   "id": "1eaf376f",
    "metadata": {
     "editable": true
    },
@@ -2290,7 +2290,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3707790b",
+   "id": "9c025697",
    "metadata": {
     "editable": true
    },
@@ -2302,7 +2302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "13ab6bec",
+   "id": "2d64abe7",
    "metadata": {
     "editable": true
    },
@@ -2312,7 +2312,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a6d69ef",
+   "id": "b83c29d5",
    "metadata": {
     "editable": true
    },
@@ -2324,7 +2324,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "336ec1b1",
+   "id": "cfcc809e",
    "metadata": {
     "editable": true
    },
@@ -2346,7 +2346,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3081ba44",
+   "id": "bdaa56d1",
    "metadata": {
     "editable": true
    },
@@ -2359,7 +2359,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c3faff2",
+   "id": "ffa4afe6",
    "metadata": {
     "editable": true
    },
@@ -2372,7 +2372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68fd52c0",
+   "id": "5c6b9239",
    "metadata": {
     "editable": true
    },
@@ -2382,7 +2382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e29a567",
+   "id": "5ddb446d",
    "metadata": {
     "editable": true
    },
@@ -2400,7 +2400,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f787c3a2",
+   "id": "9a0a5b04",
    "metadata": {
     "editable": true
    },
@@ -2410,7 +2410,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c5e1214",
+   "id": "6fe881ee",
    "metadata": {
     "editable": true
    },
@@ -2425,7 +2425,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cdd5ba1",
+   "id": "83b83194",
    "metadata": {
     "editable": true
    },
@@ -2435,7 +2435,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "276e07a3",
+   "id": "09e77119",
    "metadata": {
     "editable": true
    },
@@ -2449,7 +2449,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80fe7f13",
+   "id": "afdce492",
    "metadata": {
     "editable": true
    },
@@ -2459,7 +2459,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8838b8f",
+   "id": "b6e29c29",
    "metadata": {
     "editable": true
    },
@@ -2473,7 +2473,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2ed6fbd0",
+   "id": "ad6e09d7",
    "metadata": {
     "editable": true
    },
@@ -2488,7 +2488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0796429c",
+   "id": "a514df33",
    "metadata": {
     "editable": true
    },
@@ -2505,7 +2505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc44df4f",
+   "id": "ae758019",
    "metadata": {
     "editable": true
    },
@@ -2517,7 +2517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c88146a9",
+   "id": "1bcffe5e",
    "metadata": {
     "editable": true
    },
@@ -2531,7 +2531,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1b85445",
+   "id": "67467328",
    "metadata": {
     "editable": true
    },
@@ -2546,7 +2546,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8edde795",
+   "id": "4bac20b4",
    "metadata": {
     "editable": true
    },
@@ -2558,7 +2558,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f92e4583",
+   "id": "01e84611",
    "metadata": {
     "editable": true
    },
@@ -2569,7 +2569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0bf80264",
+   "id": "8f742c3b",
    "metadata": {
     "editable": true
    },
@@ -2597,7 +2597,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b6b24c7",
+   "id": "77acd12d",
    "metadata": {
     "editable": true
    },
@@ -2619,7 +2619,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e36297bd",
+   "id": "99ab46db",
    "metadata": {
     "editable": true
    },
@@ -2641,7 +2641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8ec331d",
+   "id": "5b4b6243",
    "metadata": {
     "editable": true
    },
@@ -2653,7 +2653,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b0d0c67",
+   "id": "2ba66a00",
    "metadata": {
     "editable": true
    },
@@ -2690,7 +2690,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bdfac28b",
+   "id": "c9a65dbe",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2718,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2032fb83",
+   "id": "7efa8d14",
    "metadata": {
     "editable": true
    },
@@ -2748,7 +2748,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5a00d18",
+   "id": "93d8901e",
    "metadata": {
     "editable": true
    },
@@ -2772,7 +2772,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "68f97538",
+   "id": "6a775d3c",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2785,7 +2785,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a7fbe32f",
+   "id": "b8360996",
    "metadata": {
     "editable": true
    },
@@ -2796,7 +2796,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b64af25",
+   "id": "0b4a5c74",
    "metadata": {
     "editable": true
    },
@@ -2808,7 +2808,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ef56e132",
+   "id": "f27ef437",
    "metadata": {
     "editable": true
    },
@@ -2818,7 +2818,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6d7f0c20",
+   "id": "6a7d8179",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ac315d2",
+   "id": "24b3bcb3",
    "metadata": {
     "editable": true
    },
@@ -2844,7 +2844,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "040d030b",
+   "id": "196152f9",
    "metadata": {
     "editable": true
    },
@@ -2860,7 +2860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cb9c213",
+   "id": "6b935471",
    "metadata": {
     "editable": true
    },
@@ -2870,7 +2870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f6ab192",
+   "id": "d5261a7f",
    "metadata": {
     "editable": true
    },
@@ -2882,7 +2882,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc154729",
+   "id": "398173a2",
    "metadata": {
     "editable": true
    },
@@ -2892,7 +2892,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cda14c18",
+   "id": "0f29a1a0",
    "metadata": {
     "editable": true
    },
@@ -2904,7 +2904,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffc6db98",
+   "id": "22f27a92",
    "metadata": {
     "editable": true
    },
@@ -2918,7 +2918,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5dd4f8d",
+   "id": "a2b8898e",
    "metadata": {
     "editable": true
    },
@@ -2928,7 +2928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "54cf8a09",
+   "id": "42b415d8",
    "metadata": {
     "editable": true
    },
@@ -2939,7 +2939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c06fcb28",
+   "id": "b522b609",
    "metadata": {
     "editable": true
    },
@@ -2954,7 +2954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16b2e122",
+   "id": "2e2e2e39",
    "metadata": {
     "editable": true
    },
@@ -2964,7 +2964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e48ac99",
+   "id": "6f14f45f",
    "metadata": {
     "editable": true
    },
@@ -2976,7 +2976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a536e9c",
+   "id": "d9210eed",
    "metadata": {
     "editable": true
    },
@@ -2988,7 +2988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9afa223",
+   "id": "5050ec60",
    "metadata": {
     "editable": true
    },
@@ -3003,7 +3003,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4f76d91",
+   "id": "04ac7d83",
    "metadata": {
     "editable": true
    },
@@ -3016,7 +3016,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "801ec4de",
+   "id": "6b2e19f6",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3073,45 +3073,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6c4900e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## And a corresponding example using **scikit-learn**"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "id": "d2f662be",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "# Importing various packages\n",
-    "from random import random, seed\n",
-    "import numpy as np\n",
-    "import matplotlib.pyplot as plt\n",
-    "from sklearn.linear_model import SGDRegressor\n",
-    "\n",
-    "n = 100\n",
-    "x = 2*np.random.rand(n,1)\n",
-    "y = 4+3*x+np.random.randn(n,1)\n",
-    "\n",
-    "X = np.c_[np.ones((n,1)), x]\n",
-    "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n",
-    "print(theta_linreg)\n",
-    "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n",
-    "sgdreg.fit(x,y.ravel())\n",
-    "print(sgdreg.intercept_, sgdreg.coef_)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "83bd5b44",
+   "id": "26e7815c",
    "metadata": {
     "editable": true
    },
@@ -3123,7 +3085,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c7ee80c",
+   "id": "0ae04360",
    "metadata": {
     "editable": true
    },
@@ -3135,7 +3097,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c70714f",
+   "id": "402501d6",
    "metadata": {
     "editable": true
    },
@@ -3145,7 +3107,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e170727b",
+   "id": "fdd060d5",
    "metadata": {
     "editable": true
    },
@@ -3159,7 +3121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b7b5d1ce",
+   "id": "8a061527",
    "metadata": {
     "editable": true
    },
@@ -3169,7 +3131,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e59753a",
+   "id": "72c8bb88",
    "metadata": {
     "editable": true
    },
@@ -3181,7 +3143,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d05de5f9",
+   "id": "a8e126ba",
    "metadata": {
     "editable": true
    },
@@ -3192,7 +3154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3a5c888",
+   "id": "cae81a93",
    "metadata": {
     "editable": true
    },
@@ -3207,7 +3169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52b009a7",
+   "id": "646feacb",
    "metadata": {
     "editable": true
    },
@@ -3221,7 +3183,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "430d2418",
+   "id": "e2b0d0a9",
    "metadata": {
     "editable": true
    },
@@ -3231,8 +3193,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
-   "id": "b889c65e",
+   "execution_count": 5,
+   "id": "b2354adf",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3293,7 +3255,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21e20510",
+   "id": "e95bd522",
    "metadata": {
     "editable": true
    },
@@ -3315,7 +3277,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59d61c18",
+   "id": "947549ab",
    "metadata": {
     "editable": true
    },
@@ -3327,7 +3289,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19a3af8f",
+   "id": "64b3eeb6",
    "metadata": {
     "editable": true
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@@ -3339,7 +3301,7 @@
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@@ -3351,7 +3313,7 @@
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@@ -3361,7 +3323,7 @@
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@@ -3373,7 +3335,7 @@
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@@ -3383,7 +3345,7 @@
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@@ -3400,7 +3362,7 @@
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@@ -3410,7 +3372,7 @@
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@@ -3422,7 +3384,7 @@
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@@ -3432,7 +3394,7 @@
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@@ -3444,7 +3406,7 @@
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@@ -3458,7 +3420,7 @@
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@@ -3470,7 +3432,7 @@
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@@ -3489,7 +3451,7 @@
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@@ -3502,7 +3464,7 @@
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@@ -3514,7 +3476,7 @@
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@@ -3524,7 +3486,7 @@
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@@ -3537,7 +3499,7 @@
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@@ -3547,7 +3509,7 @@
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@@ -3559,7 +3521,7 @@
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@@ -3574,7 +3536,7 @@
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@@ -3587,7 +3549,7 @@
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@@ -3601,7 +3563,7 @@
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@@ -3613,7 +3575,7 @@
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@@ -3623,7 +3585,7 @@
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@@ -3636,7 +3598,7 @@
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@@ -3648,7 +3610,7 @@
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@@ -3660,7 +3622,7 @@
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@@ -3672,7 +3634,7 @@
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@@ -3684,7 +3646,7 @@
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@@ -3698,7 +3660,7 @@
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@@ -3710,7 +3672,7 @@
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@@ -3720,7 +3682,7 @@
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@@ -3732,7 +3694,7 @@
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@@ -3742,7 +3704,7 @@
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@@ -3754,7 +3716,7 @@
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@@ -3766,7 +3728,7 @@
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@@ -3776,7 +3738,7 @@
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@@ -3787,7 +3749,7 @@
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@@ -3799,7 +3761,7 @@
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@@ -3815,7 +3777,7 @@
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@@ -3828,7 +3790,7 @@
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@@ -3840,7 +3802,7 @@
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@@ -3850,7 +3812,7 @@
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@@ -3862,7 +3824,7 @@
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@@ -3872,7 +3834,7 @@
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@@ -3884,7 +3846,7 @@
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@@ -3896,7 +3858,7 @@
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@@ -3912,7 +3874,7 @@
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@@ -3924,7 +3886,7 @@
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@@ -3936,7 +3898,7 @@
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@@ -3946,7 +3908,7 @@
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@@ -3958,7 +3920,7 @@
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@@ -3968,7 +3930,7 @@
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@@ -3980,7 +3942,7 @@
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@@ -3997,7 +3959,7 @@
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@@ -4009,7 +3971,7 @@
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@@ -4021,7 +3983,7 @@
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@@ -4031,7 +3993,7 @@
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@@ -4043,7 +4005,7 @@
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@@ -4053,7 +4015,7 @@
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@@ -4065,7 +4027,7 @@
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@@ -4075,7 +4037,7 @@
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@@ -4087,7 +4049,7 @@
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@@ -4097,7 +4059,7 @@
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@@ -4112,7 +4074,7 @@
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@@ -4124,7 +4086,7 @@
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@@ -4134,7 +4096,7 @@
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@@ -4146,7 +4108,7 @@
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@@ -4158,7 +4120,7 @@
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@@ -4170,7 +4132,7 @@
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@@ -4180,7 +4142,7 @@
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@@ -4192,7 +4154,7 @@
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@@ -4204,7 +4166,7 @@
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@@ -4216,7 +4178,7 @@
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@@ -4226,7 +4188,7 @@
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@@ -4238,7 +4200,7 @@
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@@ -4248,7 +4210,7 @@
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@@ -4262,7 +4224,7 @@
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@@ -4272,7 +4234,7 @@
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@@ -4284,7 +4246,7 @@
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@@ -4296,7 +4258,7 @@
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@@ -4306,7 +4268,7 @@
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@@ -4318,7 +4280,7 @@
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@@ -4328,7 +4290,7 @@
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@@ -4340,7 +4302,7 @@
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@@ -4352,7 +4314,7 @@
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@@ -4362,7 +4324,7 @@
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@@ -4374,7 +4336,7 @@
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@@ -4384,7 +4346,7 @@
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@@ -4396,7 +4358,7 @@
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@@ -4408,7 +4370,7 @@
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@@ -4418,7 +4380,7 @@
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@@ -4430,7 +4392,7 @@
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@@ -4443,7 +4405,7 @@
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@@ -4455,7 +4417,7 @@
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@@ -4467,7 +4429,7 @@
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@@ -4479,7 +4441,7 @@
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@@ -4489,7 +4451,7 @@
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@@ -4501,7 +4463,7 @@
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@@ -4511,7 +4473,7 @@
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@@ -4523,7 +4485,7 @@
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@@ -4533,7 +4495,7 @@
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     "editable": true
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@@ -4545,7 +4507,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b5ac9e9",
+   "id": "59be0f8c",
    "metadata": {
     "editable": true
    },
@@ -4555,7 +4517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19ab9b33",
+   "id": "1e85b855",
    "metadata": {
     "editable": true
    },
@@ -4568,7 +4530,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59135778",
+   "id": "117c5f1d",
    "metadata": {
     "editable": true
    },
@@ -4580,7 +4542,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8d40d8ab",
+   "id": "d3842015",
    "metadata": {
     "editable": true
    },
@@ -4592,7 +4554,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c0401f9",
+   "id": "5619e7d3",
    "metadata": {
     "editable": true
    },
@@ -4602,7 +4564,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa075ce0",
+   "id": "e091b536",
    "metadata": {
     "editable": true
    },
@@ -4614,7 +4576,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e465dee",
+   "id": "924ee170",
    "metadata": {
     "editable": true
    },
@@ -4631,7 +4593,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7af63418",
+   "id": "49434aeb",
    "metadata": {
     "editable": true
    },
@@ -4643,7 +4605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0fff0234",
+   "id": "4b7e1e31",
    "metadata": {
     "editable": true
    },
@@ -4655,7 +4617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32328a87",
+   "id": "374578e8",
    "metadata": {
     "editable": true
    },
@@ -4665,7 +4627,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b6403d7",
+   "id": "5e76c1eb",
    "metadata": {
     "editable": true
    },
@@ -4677,7 +4639,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8823b897",
+   "id": "ece41e8c",
    "metadata": {
     "editable": true
    },
@@ -4687,7 +4649,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "319984ba",
+   "id": "28ef4cad",
    "metadata": {
     "editable": true
    },
@@ -4699,7 +4661,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2382a622",
+   "id": "0c826f0e",
    "metadata": {
     "editable": true
    },
@@ -4709,7 +4671,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24b75d16",
+   "id": "9ef53895",
    "metadata": {
     "editable": true
    },
@@ -4721,7 +4683,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ba2f29c",
+   "id": "91cd7b4f",
    "metadata": {
     "editable": true
    },
@@ -4731,7 +4693,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd65fa2e",
+   "id": "fedfb550",
    "metadata": {
     "editable": true
    },
@@ -4745,8 +4707,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
-   "id": "6f044cbb",
+   "execution_count": 6,
+   "id": "1d8e47a3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -4806,7 +4768,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d69f7030",
+   "id": "b88f84eb",
    "metadata": {
     "editable": true
    },
@@ -4816,7 +4778,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a8f9315a",
+   "id": "b0c77f0b",
    "metadata": {
     "editable": true
    },
@@ -4826,8 +4788,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
-   "id": "041fe053",
+   "execution_count": 7,
+   "id": "570be9c4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -4894,7 +4856,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47db85e5",
+   "id": "3e4210bb",
    "metadata": {
     "editable": true
    },
@@ -4904,8 +4866,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
-   "id": "400b4a2a",
+   "execution_count": 8,
+   "id": "b4ad5452",
    "metadata": {
     "collapsed": false,
     "editable": true
diff --git a/doc/src/week36/.ipynb_checkpoints/exercisesweek37-checkpoint.ipynb b/doc/src/week36/.ipynb_checkpoints/exercisesweek37-checkpoint.ipynb
new file mode 100644
index 000000000..dcef81b55
--- /dev/null
+++ b/doc/src/week36/.ipynb_checkpoints/exercisesweek37-checkpoint.ipynb
@@ -0,0 +1,294 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "2c329a04",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
+    "doconce format html exercisesweek37.do.txt  -->\n",
+    "<!-- dom:TITLE: Week 37: Linear Regression and Gradient descent -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "432a2e5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Week 37: Linear Regression and Gradient descent\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
+    "\n",
+    "Date: **September 1-5, 2025**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8352513e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge regression and a new Synthetic Dataset\n",
+    "\n",
+    "We create a synthetic linear regression dataset with a sparse\n",
+    "underlying relationship. This means we have many features but only a\n",
+    "few of them actually contribute to the target. In our example, we’ll\n",
+    "use 10 features with only 3 non-zero weights in the true model. This\n",
+    "way, the target is generated as a linear combination of a few features\n",
+    "(with known coefficients) plus some random noise. The steps we include are:\n",
+    "\n",
+    "Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
+    "Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
+    "\n",
+    "Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
+    "Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
+    "\n",
+    "Below is the code to generate the dataset:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d7eebc41",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "# Set random seed for reproducibility\n",
+    "np.random.seed(0)\n",
+    "\n",
+    "# Define dataset size\n",
+    "n_samples = 100\n",
+    "n_features = 10\n",
+    "\n",
+    "# Define true coefficients (sparse linear relationship)\n",
+    "theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
+    "\n",
+    "# Generate feature matrix X (n_samples x n_features) with random values\n",
+    "X = np.random.randn(n_samples, n_features)  # standard normal distribution\n",
+    "\n",
+    "# Generate target values y with a linear combination of X and theta_true, plus noise\n",
+    "noise = 0.5 * np.random.randn(n_samples)    # Gaussian noise\n",
+    "y = X.dot @ theta_true + noise"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1640310e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This code produces a dataset where only features 0, 1, and 6\n",
+    "significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
+    "coefficient, so they only contribute noise. For example, feature 0 has\n",
+    "a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
+    "the expected relationship is:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "60bce352",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "y \\approx 5 \\times X_0 \\;-\\; 3 \\times X_1 \\;+\\; 2 \\times X_6 \\;+\\; \\text{noise}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "75322f31",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Before fitting a regression model, it is good practice to normalize or\n",
+    "standardize the features. This ensures all features are on a\n",
+    "comparable scale, which is especially important when using\n",
+    "regularization. Here we will perform standardization, scaling each\n",
+    "feature to have mean 0 and standard deviation 1:\n",
+    "\n",
+    "Compute the mean and standard deviation of each column (feature) in $bm{X}$.\n",
+    "Subtract the mean and divide by the standard deviation for each feature.\n",
+    "\n",
+    "We also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
+    "(and each feature) means the model won’t require a separate intercept\n",
+    "term – the data is shifted such that the intercept is effectively 0\n",
+    ". (In practice, one could include an intercept in the model and not\n",
+    "penalize it, but here we simplify by centering.)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "4b3f945a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Standardize features (zero mean, unit variance for each feature)\n",
+    "X_mean = X.mean(axis=0)\n",
+    "X_std = X.std(axis=0)\n",
+    "X_std[X_std == 0] = 1  # safeguard to avoid division by zero for constant features\n",
+    "X_norm = (X - X_mean) / X_std\n",
+    "\n",
+    "# Center the target to zero mean (optional, to simplify intercept handling)\n",
+    "y_mean = y.mean()\n",
+    "y_centered = y - y_mean"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12aec8b3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "After this preprocessing, each column of $\\boldsymbol{X}_norm$ has mean zero and standard deviation $1$\n",
+    "and $\\boldsymbol{y}_centered$ has mean 0. This makes the optimization landscape\n",
+    "nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
+    "\\beta_j^2$ treats each coefficient fairly (since features are on the\n",
+    "same scale)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "ffdecd03",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Set regularization parameter\n",
+    "lam = 1.0\n",
+    "\n",
+    "# Closed-form Ridge solution: w = (X^T X + lam * I)^{-1} X^T y\n",
+    "I = np.eye(n_features)\n",
+    "w_closed_form = np.linalg.inv(X_norm.T.dot(X_norm) + lam * I).dot(X_norm.T).dot(y_centered)\n",
+    "\n",
+    "print(\"Closed-form Ridge coefficients:\", w_closed_form)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33ff39a5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This computes the ridge regression coefficients directly. The identity\n",
+    "matrix $I$ has the same size as $X^T X$ (which is n_features x\n",
+    "n_features), and lam * I adds $\\lambda$ to the diagonal of $X^T X. We\n",
+    "then invert this matrix and multiply by $X^T y. The result\n",
+    "for $\\boldsymbol{\\theta}$  is a NumPy array of shape (n_features,) containing the\n",
+    "fitted weights.\n",
+    "\n",
+    "Alternatively, we can fit the ridge regression model using gradient\n",
+    "descent. This is useful to visualize the iterative convergence and is\n",
+    "necessary if $n$ and $p$ are so large that the closed-form might be\n",
+    "too slow or memory-intensive. We derive the gradients from the cost\n",
+    "function defined above. The gradient of the ridge cost with respect to\n",
+    "the weight vector $w$ is:\n",
+    "\n",
+    "Below is the code for gradient descent implementation of ridge:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "5b72dc82",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Gradient descent parameters\n",
+    "alpha = 0.1\n",
+    "num_iters = 1000\n",
+    "\n",
+    "# Initialize weights for gradient descent\n",
+    "theta = np.zeros(n_features)\n",
+    "\n",
+    "# Arrays to store history for plotting\n",
+    "cost_history = np.zeros(num_iters)\n",
+    "\n",
+    "# Gradient descent loop\n",
+    "m = n_samples  # number of examples\n",
+    "for t in range(num_iters):\n",
+    "    # Compute prediction error\n",
+    "    error = X_norm.dot(theta) - y_centered  # shape (m,)\n",
+    "    # Compute cost (MSE + regularization) for monitoring\n",
+    "    cost = (1/(2*m)) * np.dot(error, error) + (lam/(2*m)) * np.dot(theta, theta)\n",
+    "    cost_history[t] = cost\n",
+    "    # Compute gradient\n",
+    "    grad = (1/m) * (X_norm.T.dot(error) + lam * theta)\n",
+    "    # Update weights\n",
+    "    theta = theta - alpha * grad\n",
+    "\n",
+    "# After the loop, theta contains the fitted coefficients\n",
+    "theta_gd = theta\n",
+    "print(\"Gradient Descent Ridge coefficients:\", theta_gd)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1fc5b8d0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us confirm that the two approaches (closed-form and gradient\n",
+    "descent) give similar results, and then evaluate the model. First,\n",
+    "compare the learned coefficients to the true coefficients:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "a510e4e6",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "print(\"True coefficients:\", theta_true)\n",
+    "print(\"Closed-form learned coefficients:\", theta_closed_form)\n",
+    "print(\"Gradient descent learned coefficients:\", theta_gd)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07081487",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If everything worked correctly, the learned coefficients should be\n",
+    "close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
+    "generate the data. Keep in mind that due to regularization and noise,\n",
+    "the learned values will not exactly equal the true ones, but they\n",
+    "should be in the same ballpark."
+   ]
+  }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/doc/src/week36/exercisesweek37.do.txt b/doc/src/week36/exercisesweek37.do.txt
index 8a08f81e4..114fec64d 100644
--- a/doc/src/week36/exercisesweek37.do.txt
+++ b/doc/src/week36/exercisesweek37.do.txt
@@ -1,1561 +1,9 @@
-TITLE: Week 36: Linear Regression and Gradient descent
+TITLE: Week 37: Linear Regression and Gradient descent
 AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo, Norway
 DATE: September 1-5, 2025
 
 
-!split
-===== Plans for week 36 =====
 
-_Material for the lecture on Monday September 1:_
-o Linear Regression, ordinary least squares (OLS), Ridge and Lasso and mathematical analysis
-o Derivation of Gradient descent and discussion of implementations for
-#  * "Video of lecture":"https://youtu.be/oHjqjUB36KE"  
-#  * "Whiteboard notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember2.pdf"
-_Material for the lab sessions on Tuesday and Wednesday (see at the end of these slides):_
-o Technicalities concerning Ridge and Lasso linear regression.
-o Presentation and discussion of the first project
-# * "Video of lab session":"https://youtu.be/ZrIdZdZtHe0"
-
-_Reading suggestion:_
-o Goodfellow et al, Deep Learning, introduction to gradient descent, see chapter 4.3 at https://www.deeplearningbook.org/contents/numerical.html
-o Rashcka et al, pages 37-44 and pages 278-283 with focus on linear regression.
-o Video on gradient descent at URL:"https://www.youtube.com/watch?v=sDv4f4s2SB8"
-!split
-===== Material for lecture Monday September 2 =====
-
-
-!split
-===== Mathematical Interpretation of Ordinary Least Squares =====
-
-What is presented here is a mathematical analysis of various regression algorithms (ordinary least  squares, Ridge and Lasso Regression). The analysis is based on an important algorithm in linear algebra, the so-called Singular Value Decomposition (SVD). 
-
-
-We have shown that in ordinary least squares the optimal parameters $\theta$ are given by
-
-!bt
-\[
-\hat{\bm{\theta}} = \left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-
-The _hat_ over $\bm{\theta}$ means we have the optimal parameters after minimization of the cost function.
-
-This means that our best model is defined as
-
-!bt
-\[
-\tilde{\bm{y}}=\bm{X}\hat{\bm{\theta}} = \bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-
-We now define a matrix
-!bt
-\[
-\bm{A}=\bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T.
-\]
-!et
-
-We can rewrite
-!bt
-\[
-\tilde{\bm{y}}=\bm{X}\hat{\bm{\theta}} = \bm{A}\bm{y}.
-\]
-!et
-
-The matrix $\bm{A}$ has the important property that $\bm{A}^2=\bm{A}$. This is the definition of a projection matrix.
-We can then interpret our optimal model $\tilde{\bm{y}}$ as being represented  by an orthogonal  projection of $\bm{y}$ onto a space defined by the column vectors of $\bm{X}$.  In our case here the matrix $\bm{A}$ is a square matrix. If it is a general rectangular matrix we have an oblique projection matrix.
-
-
-
-!split
-===== Residual Error =====
-
-We have defined the residual error as
-!bt
-\[
-\bm{\epsilon}=\bm{y}-\tilde{\bm{y}}=\left[\bm{I}-\bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\right]\bm{y}.
-\]
-!et
-
-The residual errors are then the projections of $\bm{y}$ onto the orthogonal component of the space defined by the column vectors of $\bm{X}$.
-
-!split
-===== Simple case =====
-
-If the matrix $\bm{X}$ is an orthogonal (or unitary in case of complex values) matrix, we have
-
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{X}\bm{X}^T = \bm{I}.
-\]
-!et
-
-In this case the matrix $\bm{A}$ becomes
-!bt
-\[
-\bm{A}=\bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T)=\bm{I},
-\]
-!et
-and we have the obvious case
-!bt
-\[
-\bm{\epsilon}=\bm{y}-\tilde{\bm{y}}=0.
-\]
-!et
-
-This serves also as a useful test of our codes. 
-
-
-
-!split
-===== The singular value decomposition  =====
-
-!bblock
-
-The examples we have looked at so far are cases where we normally can
-invert the matrix $\bm{X}^T\bm{X}$. Using a polynomial expansion where we fit of various functions leads to
-row vectors of the design matrix which are essentially orthogonal due
-to the polynomial character of our model. Obtaining the inverse of the
-design matrix is then often done via a so-called LU, QR or Cholesky
-decomposition.
-
-
-As we will also see in the first project, 
-this may
-however not the be case in general and a standard matrix inversion
-algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below and in other examples.
-
-There is however a way to circumvent this problem and also
-gain some insights about the ordinary least squares approach, and
-later shrinkage methods like Ridge and Lasso regressions.
-
-This is given by the _Singular Value Decomposition_ (SVD) algorithm,
-perhaps the most powerful linear algebra algorithm.  The SVD provides
-a numerically stable matrix decomposition that is used in a large
-swath oc applications and the decomposition is always stable
-numerically.
-
-In machine learning it plays a central role in dealing with for
-example design matrices that may be near singular or singular.
-Furthermore, as we will see here, the singular values can be related
-to the covariance matrix (and thereby the correlation matrix) and in
-turn the variance of a given quantity. It plays also an important role
-in the principal component analysis where high-dimensional data can be
-reduced to the statistically relevant features.
-
-
-!eblock
-
-
-
-!split
-===== Linear Regression Problems =====
-
-One of the typical problems we encounter with linear regression, in particular 
-when the matrix $\bm{X}$ (our so-called design matrix) is high-dimensional, 
-are problems with near singular or singular matrices. The column vectors of $\bm{X}$ 
-may be linearly dependent, normally referred to as super-collinearity.  
-This means that the matrix may be rank deficient and it is basically impossible to 
-to model the data using linear regression. As an example, consider the matrix
-!bt
-\begin{align*}
-\mathbf{X} & =  \left[
-\begin{array}{rrr}
-1 & -1 & 2
-\\
-1 & 0 & 1
-\\
-1 & 2  & -1
-\\
-1 & 1  & 0
-\end{array} \right]
-\end{align*}
-!et
-
-The columns of $\bm{X}$ are linearly dependent. We see this easily since the 
-the first column is the row-wise sum of the other two columns. The rank (more correct,
-the column rank) of a matrix is the dimension of the space spanned by the
-column vectors. Hence, the rank of $\mathbf{X}$ is equal to the number
-of linearly independent columns. In this particular case the matrix has rank 2.
-
-Super-collinearity of an $(n \times p)$-dimensional design matrix $\mathbf{X}$ implies
-that the inverse of the matrix $\bm{X}^T\bm{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
-!bt
-\begin{align*}
-\bm{X} & =  \left[
-\begin{array}{rr}
-1 & -1
-\\
-1 & -1
-\end{array} \right].
-\end{align*}
-!et
-We see easily that  $\mbox{det}(\bm{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0$. Hence, $\mathbf{X}$ is singular and its inverse is undefined.
-This is equivalent to saying that the matrix $\bm{X}$ has at least an eigenvalue which is zero.
-
-
-!split
-===== Fixing the singularity =====
-
-If our design matrix $\bm{X}$ which enters the linear regression problem
-!bt
-\begin{align}
-\bm{\theta} & =  (\bm{X}^{T} \bm{X})^{-1} \bm{X}^{T} \bm{y},
-\end{align}
-!et
-has linearly dependent column vectors, we will not be able to compute the inverse
-of $\bm{X}^T\bm{X}$ and we cannot find the parameters (estimators) $\theta_i$. 
-The estimators are only well-defined if $(\bm{X}^{T}\bm{X})^{-1}$ exists. 
-This is more likely to happen when the matrix $\bm{X}$ is high-dimensional. In this case it is likely to encounter a situation where 
-the regression parameters $\theta_i$ cannot be estimated.
-
-A cheap  *ad hoc* approach is  simply to add a small diagonal component to the matrix to invert, that is we change
-!bt
-\[
-\bm{X}^{T} \bm{X} \rightarrow \bm{X}^{T} \bm{X}+\lambda \bm{I},
-\]
-!et 
-where $\bm{I}$ is the identity matrix.  When we discuss _Ridge_ regression this is actually what we end up evaluating. The parameter $\lambda$ is called a hyperparameter. More about this later. 
-
-
-
-
-!split
-===== Ridge and LASSO Regression =====
-
-Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
-our optimization problem is
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\theta}\right)^T\left(\bm{y}-\bm{X}\bm{\theta}\right)\right\}.
-\]
-!et
-or we can state it as
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2,
-\]
-!et
-where we have used the definition of  a norm-2 vector, that is
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
-\]
-!et
-
-By minimizing the above equation with respect to the parameters
-$\bm{\theta}$ we could then obtain an analytical expression for the
-parameters $\bm{\theta}$.  We can add a regularization parameter $\lambda$ by
-defining a new cost function to be optimized, that is
-
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_2^2
-\]
-!et
-
-which leads to the Ridge regression minimization problem where we
-require that $\vert\vert \bm{\theta}\vert\vert_2^2\le t$, where $t$ is
-a finite number larger than zero. By defining
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1,
-\]
-!et
-
-we have a new optimization equation
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1
-\]
-!et
-which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. 
-
-Here we have defined the norm-1 as 
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
-\]
-!et
-
-
-!split
-===== Deriving the  Ridge Regression Equations =====
-
-Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\left\{(\bm{y}-\bm{X}\bm{\theta})^T(\bm{y}-\bm{X}\bm{\theta})\right\}+\lambda\bm{\theta}^T\bm{\theta},
-\]
-!et
-and 
-taking the derivatives with respect to $\bm{\theta}$ we obtain then
-a slightly modified matrix inversion problem which for finite values
-of $\lambda$ does not suffer from singularity problems. We obtain
-the optimal parameters
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y},
-\]
-!et
-
-with $\bm{I}$ being a $p\times p$ identity matrix with the constraint that
-
-!bt
-\[
-\sum_{i=0}^{p-1} \theta_i^2 \leq t,
-\]
-!et
-
-with $t$ a finite positive number. 
-
-If we keep the $1/n$ factor, the equation for the optimal $\theta$ changes to
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+n\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-
-In many textbooks the $1/n$ term is often omitted. Note that a library like _Scikit-Learn_ does not include the $1/n$ factor in the setup of the cost function.
-
-When we compare this with the ordinary least squares result we have
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{OLS}} = \left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y},
-\]
-!et
-which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\bm{X}^T\bm{X}$.
-
-
-We see that Ridge regression is nothing but the standard OLS with a
-modified diagonal term added to $\bm{X}^T\bm{X}$. The consequences, in
-particular for our discussion of the bias-variance tradeoff are rather
-interesting. We will see that for specific values of $\lambda$, we may
-even reduce the variance of the optimal parameters $\bm{\theta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
-
-When we have discussed the singular value decomposition of the design
-matrix $\bm{X}$, we will in turn perform a more rigorous mathematical
-discussion of Ridge regression.
-
-
-The code here is a simple demonstration of how to implement Ridge regression with our own code and compare this with scikit-learn.
-
-!bc pycod
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree))
-#We include explicitely the intercept column
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-p = Maxpolydegree
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 6
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 2, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeTheta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
-    # Note: we include the intercept column and no scaling
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    # and then make the prediction
-    ytildeOwnRidge = X_train @ OwnRidgeTheta
-    ypredictOwnRidge = X_test @ OwnRidgeTheta
-    ytildeRidge = RegRidge.predict(X_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Theta values for own Ridge implementation")
-    print(OwnRidgeTheta)
-    print("Theta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print("MSE values for own Ridge implementation")
-    print(MSEOwnRidgePredict[i])
-    print("MSE values for Scikit-Learn Ridge implementation")
-    print(MSERidgePredict[i])
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-!ec
-
-The results here agree when we force _Scikit-Learn_'s Ridge function to include the first column in our design matrix.
-We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
-What happens if we do not include the intercept in our fit? We will discuss this in more detail next week.
-
-
-
-!split
-===== Basic math of the SVD =====
-
-
-From standard linear algebra we know that a square matrix $\bm{X}$ can be diagonalized if and only if it is 
-a so-called "normal matrix":"https://en.wikipedia.org/wiki/Normal_matrix", that is if $\bm{X}\in {\mathbb{R}}^{n\times n}$
-we have $\bm{X}\bm{X}^T=\bm{X}^T\bm{X}$ or if $\bm{X}\in {\mathbb{C}}^{n\times n}$ we have $\bm{X}\bm{X}^{\dagger}=\bm{X}^{\dagger}\bm{X}$.
-The matrix has then a set of eigenpairs 
-
-!bt
-\[
-(\lambda_1,\bm{u}_1),\dots, (\lambda_n,\bm{u}_n),
-!et
-and the eigenvalues are given by the diagonal matrix
-!bt
-\[
-\bm{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n).
-\]
-!et
-The matrix $\bm{X}$ can be written in terms of an orthogonal/unitary transformation $\bm{U}$
-!bt
-\[
-\bm{X} = \bm{U}\bm{\Sigma}\bm{V}^T,
-\]
-!et
-with $\bm{U}\bm{U}^T=\bm{I}$ or $\bm{U}\bm{U}^{\dagger}=\bm{I}$.
-
-Not all square matrices are diagonalizable. A matrix like the one discussed above
-!bt
-\[
-\bm{X} = \begin{bmatrix} 
-1&  -1 \\
-1& -1\\
-\end{bmatrix} 
-\]
-!et
-is not diagonalizable, it is a so-called "defective matrix":"https://en.wikipedia.org/wiki/Defective_matrix". It is easy to see that the condition
-$\bm{X}\bm{X}^T=\bm{X}^T\bm{X}$ is not fulfilled. 
-
-
-!split
-===== The SVD, a Fantastic Algorithm =====
-
-
-However, and this is the strength of the SVD algorithm, any general
-matrix $\bm{X}$ can be decomposed in terms of a diagonal matrix and
-two orthogonal/unitary matrices.  The "Singular Value Decompostion
-(SVD) theorem":"https://en.wikipedia.org/wiki/Singular_value_decomposition"
-states that a general $m\times n$ matrix $\bm{X}$ can be written in
-terms of a diagonal matrix $\bm{\Sigma}$ of dimensionality $m\times n$
-and two orthognal matrices $\bm{U}$ and $\bm{V}$, where the first has
-dimensionality $m \times m$ and the last dimensionality $n\times n$.
-We have then
-
-!bt
-\[ 
-\bm{X} = \bm{U}\bm{\Sigma}\bm{V}^T 
-\] 
-!et 
-
-As an example, the above defective matrix can be decomposed as
-
-!bt
-\[
-\bm{X} = \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 0\\ \end{bmatrix}    \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  -1 \\ 1& 1\\ \end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T, 
-\]
-!et
-
-with eigenvalues $\sigma_1=2$ and $\sigma_2=0$. 
-The SVD exits always! 
-
-The SVD
-decomposition (singular values) gives eigenvalues 
-$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the
-eigenvalues (singular values) are zero.
-
-In the general case, where our design matrix $\bm{X}$ has dimension
-$n\times p$, the matrix is thus decomposed into an $n\times n$
-orthogonal matrix $\bm{U}$, a $p\times p$ orthogonal matrix $\bm{V}$
-and a diagonal matrix $\bm{\Sigma}$ with $r=\mathrm{min}(n,p)$
-singular values $\sigma_i\geq 0$ on the main diagonal and zeros filling
-the rest of the matrix.  There are at most $p$ singular values
-assuming that $n > p$. In our regression examples for the nuclear
-masses and the equation of state this is indeed the case, while for
-the Ising model we have $p > n$. These are often cases that lead to
-near singular or singular matrices.
-
-The columns of $\bm{U}$ are called the left singular vectors while the columns of $\bm{V}$ are the right singular vectors.
-
-!split
-===== Economy-size SVD =====
-
-If we assume that $n > p$, then our matrix $\bm{U}$ has dimension $n
-\times n$. The last $n-p$ columns of $\bm{U}$ become however
-irrelevant in our calculations since they are multiplied with the
-zeros in $\bm{\Sigma}$.
-
-The economy-size decomposition removes extra rows or columns of zeros
-from the diagonal matrix of singular values, $\bm{\Sigma}$, along with the columns
-in either $\bm{U}$ or $\bm{V}$ that multiply those zeros in the expression. 
-Removing these zeros and columns can improve execution time
-and reduce storage requirements without compromising the accuracy of
-the decomposition.
-
-If $n > p$, we keep only the first $p$ columns of $\bm{U}$ and $\bm{\Sigma}$ has dimension $p\times p$. 
-If $p > n$, then only the first $n$ columns of $\bm{V}$ are computed and $\bm{\Sigma}$ has dimension $n\times n$.
-The $n=p$ case is obvious, we retain the full SVD. 
-In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.
-
-!split
-=====  Codes for the SVD =====
-
-!bc pycod
-import numpy as np
-# SVD inversion
-def SVD(A):
-    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
-    SVD is numerically more stable than the inversion algorithms provided by
-    numpy and scipy.linalg at the cost of being slower.
-    '''
-    U, S, VT = np.linalg.svd(A,full_matrices=True)
-    print('test U')
-    print( (np.transpose(U) @ U - U @np.transpose(U)))
-    print('test VT')
-    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
-    print(U)
-    print(S)
-    print(VT)
-
-    D = np.zeros((len(U),len(VT)))
-    for i in range(0,len(VT)):
-        D[i,i]=S[i]
-    return U @ D @ VT
-
-
-X = np.array([ [1.0,-1.0], [1.0,-1.0]])
-#X = np.array([[1, 2], [3, 4], [5, 6]])
-
-print(X)
-C = SVD(X)
-# Print the difference between the original matrix and the SVD one
-print(C-X)
-!ec
-
-The matrix $\bm{X}$ has columns that are linearly dependent. The first
-column is the row-wise sum of the other two columns. The rank of a
-matrix (the column rank) is the dimension of space spanned by the
-column vectors. The rank of the matrix is the number of linearly
-independent columns, in this case just $2$. We see this from the
-singular values when running the above code. Running the standard
-inversion algorithm for matrix inversion with $\bm{X}^T\bm{X}$ results
-in the program terminating due to a singular matrix.
-
-
-!split
-=====  Note about SVD Calculations =====
-
-The $U$, $S$, and $V$ matrices returned from the _svd()_ function
-cannot be multiplied directly.
-
-As you can see from the code, the $S$ vector must be converted into a
-diagonal matrix. This may cause a problem as the size of the matrices
-do not fit the rules of matrix multiplication, where the number of
-columns in a matrix must match the number of rows in the subsequent
-matrix.
-
-If you wish to include the zero singular values, you will need to
-resize the matrices and set up a diagonal matrix as done in the above
-example
-
-
-
-
-
-!split
-===== Mathematics of the SVD and implications =====
-
-Let us take a closer look at the mathematics of the SVD and the various implications for machine learning studies.
-
-Our starting point is our design matrix $\bm{X}$ of dimension $n\times p$
-!bt
-\[
-\bm{X}=\begin{bmatrix}
-x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
-x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
-x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
-\dots & \dots & \dots & \dots \dots & \dots \\
-x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
-x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
-\end{bmatrix}.
-\]
-!et
-
-We can SVD decompose our matrix as
-!bt
-\[
-\bm{X}=\bm{U}\bm{\Sigma}\bm{V}^T,
-\]
-!et
-where $\bm{U}$ is an orthogonal matrix of dimension $n\times n$, meaning that $\bm{U}\bm{U}^T=\bm{U}^T\bm{U}=\bm{I}_n$. Here $\bm{I}_n$ is the unit matrix of dimension $n \times n$.
-
-Similarly, $\bm{V}$ is an orthogonal matrix of dimension $p\times p$, meaning that $\bm{V}\bm{V}^T=\bm{V}^T\bm{V}=\bm{I}_p$. Here $\bm{I}_n$ is the unit matrix of dimension $p \times p$.
-
-Finally $\bm{\Sigma}$ contains the singular values $\sigma_i$. This matrix has dimension $n\times p$ and the singular values $\sigma_i$ are all positive. The non-zero values are ordered in descending order, that is
-
-!bt
-\[
-\sigma_0 > \sigma_1 > \sigma_2 > \dots > \sigma_{p-1} > 0. 
-\]
-!et
-
-All values beyond $p-1$ are all zero.
-
-!split
-===== Example Matrix =====
-
-As an example, consider the following $3\times 2$ example for the matrix $\bm{\Sigma}$
-
-!bt
-\[
-\bm{\Sigma}=
-\begin{bmatrix}
-2& 0 \\
-0 & 1 \\
-0 & 0 \\
-\end{bmatrix}
-\]
-!et
-
-The singular values are $\sigma_0=2$ and $\sigma_1=1$. It is common to rewrite the matrix $\bm{\Sigma}$ as
-
-!bt
-\[
-\bm{\Sigma}=
-\begin{bmatrix}
-\bm{\tilde{\Sigma}}\\
-\bm{0}\\
-\end{bmatrix},
-\]
-!et
-
-where
-!bt
-\[
-\bm{\tilde{\Sigma}}=
-\begin{bmatrix}
-2& 0 \\
-0 & 1 \\
-\end{bmatrix},
-\]
-!et
-contains only the singular values.   Note also (and we will use this below) that
-
-!bt
-\[
-\bm{\Sigma}^T\bm{\Sigma}=
-\begin{bmatrix}
-4& 0 \\
-0 & 1 \\
-\end{bmatrix},
-\]
-!et
-which is a $2\times 2 $ matrix while
-!bt
-\[
-\bm{\Sigma}\bm{\Sigma}^T=
-\begin{bmatrix}
-4& 0 & 0\\
-0 & 1 & 0\\
-0 & 0 & 0\\
-\end{bmatrix},
-\]
-!et
-
-is a $3\times 3 $ matrix. The last row and column of this last matrix
-contain only zeros. This will have important consequences for our SVD
-decomposition of the design matrix.
-
-
-!split
-=====  Setting up the Matrix to be inverted =====
-
-The matrix that may cause problems for us is $\bm{X}^T\bm{X}$. Using the SVD we can rewrite this matrix as
-
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{U}\bm{\Sigma}\bm{V}^T,
-\]
-!et
-and using the orthogonality of the matrix $\bm{U}$ we have
-
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T.
-\]
-!et
-We define $\bm{\Sigma}^T\bm{\Sigma}=\tilde{\bm{\Sigma}}^2$ which is  a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
-
-
-We can now insert the result for the matrix $\bm{X}^T\bm{X}$ into our equation for ordinary least squares where
-
-!bt
-\[
-\tilde{y}_{\mathrm{OLS}}=\bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y},
-\]
-!et
-and using our SVD decomposition of $\bm{X}$ we have
-
-!bt
-\[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\left(\bm{V}\tilde{\bm{\Sigma}}^{2}(\bm{V}^T\right)^{-1}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
-\]
-!et
-which gives us, using the orthogonality of the matrix $\bm{V}$,
-
-!bt
-\[
-\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_i\bm{y},
-\]
-!et
-which is not the same as $\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}$, which due to the orthogonality of $\bm{U}$ would have given us that the model equals the output.
-
-It means that the ordinary least square model (with the optimal
-parameters) $\bm{\tilde{y}}$, corresponds to an orthogonal
-transformation of the output (or target) vector $\bm{y}$ by the
-vectors of the matrix $\bm{U}$. _Note that the summation ends at_
-$p-1$, that is $\bm{\tilde{y}}\ne \bm{y}$. We can thus not use the
-orthogonality relation for the matrix $\bm{U}$. 
-
-!split
-===== Further properties (important for our analyses later) =====
-
-Let us study again $\bm{X}^T\bm{X}$ in terms of our SVD,
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{U}\bm{\Sigma}\bm{V}^T=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T. 
-\]
-!et
-
-If we now multiply from the right with $\bm{V}$ (using the orthogonality of $\bm{V}$) we get
-!bt
-\[
-\left(\bm{X}^T\bm{X}\right)\bm{V}=\bm{V}\bm{\Sigma}^T\bm{\Sigma}. 
-\]
-!et
-This means the vectors $\bm{v}_i$ of the orthogonal matrix $\bm{V}$ are the eigenvectors of the matrix $\bm{X}^T\bm{X}$
-with eigenvalues given by the singular values squared, that is
-!bt
-\[
-\left(\bm{X}^T\bm{X}\right)\bm{v}_i=\bm{v}_i\sigma_i^2. 
-\]
-!et
-
-Similarly, if we use the SVD decomposition for the matrix $\bm{X}\bm{X}^T$, we have
-!bt
-\[
-\bm{X}\bm{X}^T=\bm{U}\bm{\Sigma}\bm{V}^T\bm{V}\bm{\Sigma}^T\bm{U}^T=\bm{U}\bm{\Sigma}\bm{\Sigma}^T\bm{U}^T. 
-\]
-!et
-
-If we now multiply from the right with $\bm{U}$ (using the orthogonality of $\bm{U}$) we get
-!bt
-\[
-\left(\bm{X}\bm{X}^T\right)\bm{U}=\bm{U}\bm{\Sigma}\bm{\Sigma}^T. 
-\]
-!et
-This means the vectors $\bm{u}_i$ of the orthogonal matrix $\bm{U}$ are the eigenvectors of the matrix $\bm{X}\bm{X}^T$
-with eigenvalues given by the singular values squared, that is
-!bt
-\[
-\left(\bm{X}\bm{X}^T\right)\bm{u}_i=\bm{u}_i\sigma_i^2. 
-\]
-!et
-
-_Important note_: we have defined our design matrix $\bm{X}$ to be an
-$n\times p$ matrix. In most supervised learning cases we have that $n
-\ge p$, and quite often we have $n >> p$. For linear algebra based methods like ordinary least squares or Ridge regression, this leads to a matrix $\bm{X}^T\bm{X}$ which is small and thereby easier to handle from a computational point of view (in terms of number of floating point operations).
-
-In our lectures, the number of columns will
-always refer to the number of features in our data set, while the
-number of rows represents the number of data inputs. Note that in
-other texts you may find the opposite notation. This has consequences
-for the definition of for example the covariance matrix and its relation to the SVD.
-
-
-
-!split
-===== Back to Ridge and LASSO Regression =====
-
-Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
-our optimization problem is
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\theta}\right)^T\left(\bm{y}-\bm{X}\bm{\theta}\right)\right\}.
-\]
-!et
-or we can state it as
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2,
-\]
-!et
-where we have used the definition of  a norm-2 vector, that is
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
-\]
-!et
-
-By minimizing the above equation with respect to the parameters
-$\bm{\theta}$ we could then obtain an analytical expression for the
-parameters $\bm{\theta}$.  We can add a regularization parameter $\lambda$ by
-defining a new cost function to be optimized, that is
-
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_2^2
-\]
-!et
-
-which leads to the Ridge regression minimization problem where we
-require that $\vert\vert \bm{\theta}\vert\vert_2^2\le t$, where $t$ is
-a finite number larger than zero. By defining
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1,
-\]
-!et
-
-we have a new optimization equation
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1
-\]
-!et
-which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. 
-
-Here we have defined the norm-1 as 
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
-\]
-!et
-
-
-Ridge regression, as discussed above,  is nothing but the standard OLS with a
-modified diagonal term added to $\bm{X}^T\bm{X}$. The consequences, in
-particular for our discussion of the bias-variance tradeoff are rather
-interesting. We will see that for specific values of $\lambda$, we may
-even reduce the variance of the optimal parameters $\bm{\theta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
-
-Using our insights about the SVD of the design matrix $\bm{X}$ 
-We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\bm{U}$ as
-!bt
-\[
-\tilde{\bm{y}}_{\mathrm{OLS}}=\bm{X}\bm{\theta}  =\bm{U}\bm{U}^T\bm{y}.
-\]
-!et
-
-
-For Ridge regression this becomes
-
-!bt
-\[
-\tilde{\bm{y}}_{\mathrm{Ridge}}=\bm{X}\bm{\theta}_{\mathrm{Ridge}} = \bm{U\Sigma V^T}\left(\bm{V}\bm{\Sigma}^2\bm{V}^T+\lambda\bm{I} \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\sum_{j=0}^{p-1}\bm{u}_j\bm{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\bm{y},
-\]
-!et
-
-with the vectors $\bm{u}_j$ being the columns of $\bm{U}$ from the SVD of the matrix $\bm{X}$. 
-
-!split
-===== Interpreting the Ridge results =====
-
-Since $\lambda \geq 0$, it means that compared to OLS, we have 
-
-!bt
-\[
-\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. 
-\]
-!et
-
-Ridge regression finds the coordinates of $\bm{y}$ with respect to the
-orthonormal basis $\bm{U}$, it then shrinks the coordinates by
-$\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has
-eigenvalues ordered in a descending way, that is $\sigma_i \geq
-\sigma_{i+1}$.
-
-For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
-
-
-!split
-===== More interpretations =====
-
-For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
-
-!bt
-\[
-\bm{X}^T\bm{X}=(\bm{X}^T\bm{X})^{-1} =\bm{I}. 
-\]
-!et
-
-In this case the standard OLS results in 
-!bt
-\[
-\bm{\theta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{n-1}\bm{u}_i\bm{u}_i^T\bm{y},
-\]
-!et
-
-and
-
-!bt
-\[
-\bm{\theta}^{\mathrm{Ridge}} = \left(\bm{I}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}=\left(1+\lambda\right)^{-1}\bm{\theta}^{\mathrm{OLS}},
-\]
-!et
-
-that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\lambda$, and
-the Ridge estimator converges to zero when the hyperparameter goes to
-infinity.
-
-We will come back to more interpreations after we have gone through some of the statistical analysis part. 
-
-For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
-Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
-
-!split
-===== Deriving the  Lasso Regression Equations =====
-
-Using the matrix-vector expression for Lasso regression, we have the following _cost_ function
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\frac{1}{n}\left\{(\bm{y}-\bm{X}\bm{\theta})^T(\bm{y}-\bm{X}\bm{\theta})\right\}+\lambda\vert\vert\bm{\theta}\vert\vert_1,
-\]
-!et
-
-Taking the derivative with respect to $\bm{\theta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicty)
-!bt
-\[
-\frac{d \vert \theta\vert}{d \theta}=\mathrm{sgn}(\theta)=\left\{\begin{array}{cc} 1 & \theta > 0 \\-1 & \theta < 0, \end{array}\right.
-\]
-!et
-we have that the derivative of the cost function is
-
-!bt
-\[
-\frac{\partial C(\bm{X},\bm{\theta})}{\partial \bm{\theta}}=-\frac{2}{n}\bm{X}^T(\bm{y}-\bm{X}\bm{\theta})+\lambda sgn(\bm{\theta})=0,
-\]
-!et
-and reordering we have
-!bt
-\[
-\bm{X}^T\bm{X}\bm{\theta}+\frac{n}{2}\lambda sgn(\bm{\theta})=\bm{X}^T\bm{y}.
-\]
-!et
-We can redefine $\lambda$ to absorb the constant $n/2$ and we rewrite the last equation as
-!bt
-\[
-\bm{X}^T\bm{X}\bm{\theta}+\lambda sgn(\bm{\theta})=\bm{X}^T\bm{y}.
-\]
-!et
-
-
-This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms.We will discuss how to code the above methods using gradient descent methods.
-
-!split
-===== Optimization and gradient descent, the central part of any Machine Learning algortithm =====
-
-
-Almost every problem in machine learning and data science starts with
-a dataset $X$, a model $g(\theta)$, which is a function of the
-parameters $\theta$ and a cost function $C(X, g(\theta))$ that allows
-us to judge how well the model $g(\theta)$ explains the observations
-$X$. The model is fit by finding the values of $\theta$ that minimize
-the cost function. Ideally we would be able to solve for $\theta$
-analytically, however this is not possible in general and we must use
-some approximative/numerical method to compute the minimum.
-
-
-
-!split
-===== Reminder on Newton-Raphson's method =====
-
-Let us quickly remind ourselves how we derive the above method.
-
-Perhaps the most celebrated of all one-dimensional root-finding
-routines is Newton's method, also called the Newton-Raphson
-method. This method  requires the evaluation of both the
-function $f$ and its derivative $f'$ at arbitrary points. 
-If you can only calculate the derivative
-numerically and/or your function is not of the smooth type, we
-normally discourage the use of this method.
-
-!split
-===== The equations =====
-
-The Newton-Raphson formula consists geometrically of extending the
-tangent line at a current point until it crosses zero, then setting
-the next guess to the abscissa of that zero-crossing.  The mathematics
-behind this method is rather simple. Employing a Taylor expansion for
-$x$ sufficiently close to the solution $s$, we have
-
-
-!bt
-\[
-    f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
-    \label{eq:taylornr}
-\]
-!et
-
-For small enough values of the function and for well-behaved
-functions, the terms beyond linear are unimportant, hence we obtain
-
-
-!bt
-\[
-   f(x)+(s-x)f'(x)\approx 0,
-\]
-!et
-yielding
-!bt
-\[
-   s\approx x-\frac{f(x)}{f'(x)}.
-\]
-!et
-
-Having in mind an iterative procedure, it is natural to start iterating with
-!bt
-\[
-   x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
-\]
-!et
-
-!split
-===== Simple geometric interpretation =====
-
-The above is Newton-Raphson's method. It has a simple geometric
-interpretation, namely $x_{n+1}$ is the point where the tangent from
-$(x_n,f(x_n))$ crosses the $x$-axis.  Close to the solution,
-Newton-Raphson converges fast to the desired result. However, if we
-are far from a root, where the higher-order terms in the series are
-important, the Newton-Raphson formula can give grossly inaccurate
-results. For instance, the initial guess for the root might be so far
-from the true root as to let the search interval include a local
-maximum or minimum of the function.  If an iteration places a trial
-guess near such a local extremum, so that the first derivative nearly
-vanishes, then Newton-Raphson may fail totally
-
-
-!split
-===== Extending to more than one variable =====
-
-Newton's method can be generalized to systems of several non-linear equations
-and variables. Consider the case with two equations
-!bt
-\[
-   \begin{array}{cc} f_1(x_1,x_2) &=0\\
-                     f_2(x_1,x_2) &=0,\end{array}
-\]
-!et
-which we Taylor expand to obtain
-
-!bt
-\[
-   \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
-                     \partial f_1/\partial x_1+h_2
-                     \partial f_1/\partial x_2+\dots\\
-                     0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
-                     \partial f_2/\partial x_1+h_2
-                     \partial f_2/\partial x_2+\dots
-                       \end{array}.
-\]
-!et
-Defining the Jacobian matrix ${\bf \bm{J}}$ we have
-!bt
-\[
- {\bf \bm{J}}=\left( \begin{array}{cc}
-                         \partial f_1/\partial x_1  & \partial f_1/\partial x_2 \\
-                          \partial f_2/\partial x_1     &\partial f_2/\partial x_2
-             \end{array} \right),
-\]
-!et
-we can rephrase Newton's method as
-!bt
-\[
-\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
-\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
-\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
-\]
-!et
-where we have defined
-!bt
-\[
-   \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
-   -{\bf \bm{J}}^{-1}
-   \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
-\]
-!et
-We need thus to compute the inverse of the Jacobian matrix and it
-is to understand that difficulties  may
-arise in case ${\bf \bm{J}}$ is nearly singular.
-
-It is rather straightforward to extend the above scheme to systems of
-more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. 
-
-
-
-!split
-===== Steepest descent =====
-
-The basic idea of gradient descent is
-that a function $F(\mathbf{x})$, 
-$\mathbf{x} \equiv (x_1,\cdots,x_n)$, decreases fastest if one goes from $\bf {x}$ in the
-direction of the negative gradient $-\nabla F(\mathbf{x})$.
-
-It can be shown that if 
-!bt
-\[
-\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
-\]
-!et
-with $\gamma_k > 0$.
-
-For $\gamma_k$ small enough, then $F(\mathbf{x}_{k+1}) \leq
-F(\mathbf{x}_k)$. This means that for a sufficiently small $\gamma_k$
-we are always moving towards smaller function values, i.e a minimum.
-
-!split 
-===== More on Steepest descent =====
-
-The previous observation is the basis of the method of steepest
-descent, which is also referred to as just gradient descent (GD). One
-starts with an initial guess $\mathbf{x}_0$ for a minimum of $F$ and
-computes new approximations according to
-
-!bt
-\[
-\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
-\]
-!et
-
-The parameter $\gamma_k$ is often referred to as the step length or
-the learning rate within the context of Machine Learning.
-
-!split 
-===== The ideal =====
-
-Ideally the sequence $\{\mathbf{x}_k \}_{k=0}$ converges to a global
-minimum of the function $F$. In general we do not know if we are in a
-global or local minimum. In the special case when $F$ is a convex
-function, all local minima are also global minima, so in this case
-gradient descent can converge to the global solution. The advantage of
-this scheme is that it is conceptually simple and straightforward to
-implement. However the method in this form has some severe
-limitations:
-
-In machine learing we are often faced with non-convex high dimensional
-cost functions with many local minima. Since GD is deterministic we
-will get stuck in a local minimum, if the method converges, unless we
-have a very good intial guess. This also implies that the scheme is
-sensitive to the chosen initial condition.
-
-Note that the gradient is a function of $\mathbf{x} =
-(x_1,\cdots,x_n)$ which makes it expensive to compute numerically.
-
-
-!split 
-===== The sensitiveness of the gradient descent ===== 
-
-The gradient descent method 
-is sensitive to the choice of learning rate $\gamma_k$. This is due
-to the fact that we are only guaranteed that $F(\mathbf{x}_{k+1}) \leq
-F(\mathbf{x}_k)$ for sufficiently small $\gamma_k$. The problem is to
-determine an optimal learning rate. If the learning rate is chosen too
-small the method will take a long time to converge and if it is too
-large we can experience erratic behavior.
-
-Many of these shortcomings can be alleviated by introducing
-randomness. One such method is that of Stochastic Gradient Descent
-(SGD), to be discussed next week.
-
-
-!split 
-===== Convex functions ===== 
-
-Ideally we want our cost/loss function to be convex(concave).
-
-First we give the definition of a convex set: A set $C$ in
-$\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and
-all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to
-C. Geometrically this means that every point on the line segment
-connecting $x$ and $y$ is in $C$ as discussed below.
-
-The convex subsets of $\mathbb{R}$ are the intervals of
-$\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the
-regular polygons (triangles, rectangles, pentagons, etc...).
-
-!split
-===== Convex function =====
-
-_Convex function_: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.
-
-!split
-===== Conditions on convex functions =====
-
-In the following we state first and second-order conditions which
-ensures convexity of a function $f$. We write $D_f$ to denote the
-domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more
-details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/, 2004".
-
-!bblock First order condition
-Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for
-all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$
-is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
-for all $x,y \in D_f$. This condition means that for a convex function
-the first order Taylor expansion (right hand side above) at any point
-a global under estimator of the function. To convince yourself you can
-make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and
-note that it is always below the graph.  
-!eblock
-
-!bblock Second order condition 
-Assume that $f$ is twice
-differentiable, i.e the Hessian matrix exists at each point in
-$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its
-Hessian is positive semi-definite for all $x\in D_f$. For a
-single-variable function this reduces to $f''(x) \geq 0$. Geometrically this means that $f$ has nonnegative curvature
-everywhere.
-!eblock
-
-This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
-
-!split
-===== More on convex functions =====
-
-The next result is of great importance to us and the reason why we are
-going on about convex functions. In machine learning we frequently
-have to minimize a loss/cost function in order to find the best
-parameters for the model we are considering. 
-
-Ideally we want the
-global minimum (for high-dimensional models it is hard to know
-if we have local or global minimum). However, if the cost/loss function
-is convex the following result provides invaluable information:
-
-!bblock Any minimum is global for convex functions
-Consider the problem of finding $x \in \mathbb{R}^n$ such that $f(x)$
-is minimal, where $f$ is convex and differentiable. Then, any point
-$x^*$ that satisfies $\nabla f(x^*) = 0$ is a global minimum.
-!eblock
-
-This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
-
-!split
-===== Some simple problems =====
-
-o Show that $f(x)=x^2$ is convex for $x \in \mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \in D_f$ and any $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
-
-o Using the second order condition show that the following functions are convex on the specified domain.
- * $f(x) = e^x$ is convex for $x \in \mathbb{R}$.
- * $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$.
-o Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \in \mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.
-
-o A norm is any function that satisfy the following properties
- * $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$.
- * $f(x+y) \leq f(x) + f(y)$
- * $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$
-
-Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
-
-
-
-
-
-
-!split 
-===== Revisiting Ordinary Least Squares =====
-
-We will use linear regression as a case study for the gradient descent
-methods. Linear regression is a great test case for the gradient
-descent methods discussed in the lectures since it has several
-desirable properties such as:
-
-o An analytical solution (recall homework sets for week 35).
-o The gradient can be computed analytically.
-o The cost function is convex which guarantees that gradient descent converges for small enough learning rates
-
-We revisit an example similar to what we had in the first homework set. We had a function  of the type
-
-!bc pycod
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
-!ec
-with $x_i \in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\cal {N}(0,1)$. 
-The linear regression model is given by 
-!bt
-\[
-h_\theta(x) = \bm{y} = \theta_0 + \theta_1 x,
-\] 
-!et
-such that 
-!bt
-\[
-\bm{y}_i = \theta_0 + \theta_1 x_i.
-\]
-!et
-
-!split 
-===== Gradient descent example =====
-
-Let $\mathbf{y} = (y_1,\cdots,y_n)^T$, $\mathbf{\bm{y}} = (\bm{y}_1,\cdots,\bm{y}_n)^T$ and $\theta = (\theta_0, \theta_1)^T$
-
-It is convenient to write $\mathbf{\bm{y}} = X\theta$ where $X \in \mathbb{R}^{100 \times 2} $ is the design matrix given by (we keep the intercept here)
-!bt
-\[
-X \equiv \begin{bmatrix}
-1 & x_1  \\
-\vdots & \vdots  \\
-1 & x_{100} &  \\
-\end{bmatrix}.
-\]
-!et
-The cost/loss/risk function is given by (
-!bt
-\[
-C(\theta) = \frac{1}{n}||X\theta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\theta_0 + \theta_1 x_i)^2 - 2 y_i (\theta_0 + \theta_1 x_i) + y_i^2\right] 
-\]
-!et
-and we want to find $\theta$ such that $C(\theta)$ is minimized.
-
-!split
-===== The derivative of the cost/loss function =====
-
-Computing $\partial C(\theta) / \partial \theta_0$ and $\partial C(\theta) / \partial \theta_1$ we can show  that the gradient can be written as
-!bt
-\[
-\nabla_{\theta} C(\theta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\theta_0+\theta_1x_i-y_i\right) \\
-\sum_{i=1}^{100}\left( x_i (\theta_0+\theta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} = \frac{2}{n}X^T(X\theta - \mathbf{y}), 
-\]
-!et
-where $X$ is the design matrix defined above.
-
-!split
-===== The Hessian matrix =====
-The Hessian matrix of $C(\theta)$ is given by 
-!bt
-\[
-\bm{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\theta)}{\partial \theta_0^2} & \frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1}  \\
-\frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1} & \frac{\partial^2 C(\theta)}{\partial \theta_1^2} &  \\
-\end{bmatrix} = \frac{2}{n}X^T X.
-\]
-!et
-This result implies that $C(\theta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.
-
-
-
-
-!split
-===== Simple program =====
-
-We can now write a program that minimizes $C(\theta)$ using the gradient descent method with a constant learning rate $\gamma$ according to 
-!bt
-\[
-\theta_{k+1} = \theta_k - \gamma \nabla_\theta C(\theta_k), \ k=0,1,\cdots 
-\]
-!et
-
-We can use the expression we computed for the gradient and let use a
-$\theta_0$ be chosen randomly and let $\gamma = 0.001$. Stop iterating
-when $||\nabla_\theta C(\theta_k) || \leq \epsilon = 10^{-8}$. _Note that the code below does not include the latter stop criterion_.
-
-And finally we can compare our solution for $\theta$ with the analytic result given by 
-$\theta= (X^TX)^{-1} X^T \mathbf{y}$.
-
-!split
-===== Gradient Descent Example =====
-
-Here our simple example
-!bc pycod
-
-# Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
-
-# the number of datapoints
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-# Hessian matrix
-H = (2.0/n)* X.T @ X
-# Get the eigenvalues
-EigValues, EigVectors = np.linalg.eig(H)
-print(f"Eigenvalues of Hessian Matrix:{EigValues}")
-
-theta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y
-print(theta_linreg)
-theta = np.random.randn(2,1)
-
-eta = 1.0/np.max(EigValues)
-Niterations = 1000
-
-for iter in range(Niterations):
-    gradient = (2.0/n)*X.T @ (X @ theta-y)
-    theta -= eta*gradient
-
-print(theta)
-xnew = np.array([[0],[2]])
-xbnew = np.c_[np.ones((2,1)), xnew]
-ypredict = xbnew.dot(theta)
-ypredict2 = xbnew.dot(theta_linreg)
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Gradient descent example')
-plt.show()
-
-!ec
-
-
-!split 
-===== Gradient descent and Ridge =====
-
-We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\theta$, 
-!bt
-\[
-C_{\text{ridge}}(\theta) = \frac{1}{n}||X\theta -\mathbf{y}||^2 + \lambda ||\theta||^2, \ \lambda \geq 0.
-\]
-!et
-
-In order to minimize $C_{\text{ridge}}(\theta)$ using GD we adjust the gradient as follows 
-!bt
-\[
-\nabla_\theta C_{\text{ridge}}(\theta)  = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\theta_0+\theta_1x_i-y_i\right) \\
-\sum_{i=1}^{100}\left( x_i (\theta_0+\theta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} + 2\lambda\begin{bmatrix} \theta_0 \\ \theta_1\end{bmatrix} = 2 (\frac{1}{n}X^T(X\theta - \mathbf{y})+\lambda \theta).
-\]
-!et
-
-We can easily extend our program to minimize $C_{\text{ridge}}(\theta)$ using gradient descent and compare with the analytical solution given by 
-!bt
-\[
-\theta_{\text{ridge}} = \left(X^T X + n\lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
-\]
-!et
-
-!split
-===== The Hessian matrix for Ridge Regression =====
-The Hessian matrix of Ridge Regression for our simple example  is given by 
-!bt
-\[
-\bm{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\theta)}{\partial \theta_0^2} & \frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1}  \\
-\frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1} & \frac{\partial^2 C(\theta)}{\partial \theta_1^2} &  \\
-\end{bmatrix} = \frac{2}{n}X^T X+2\lambda\bm{I}.
-\]
-!et
-This implies that the Hessian matrix  is positive definite, hence the stationary point is a
-minimum.
-Note that the Ridge cost function is convex being  a sum of two convex
-functions. Therefore, the stationary point is a global
-minimum of this function.
-
-
-!split
-===== Program example for gradient descent with Ridge Regression =====
-!bc pycod
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
-
-# the number of datapoints
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-XT_X = X.T @ X
-
-#Ridge parameter lambda
-lmbda  = 0.001
-Id = n*lmbda* np.eye(XT_X.shape[0])
-
-# Hessian matrix
-H = (2.0/n)* XT_X+2*lmbda* np.eye(XT_X.shape[0])
-# Get the eigenvalues
-EigValues, EigVectors = np.linalg.eig(H)
-print(f"Eigenvalues of Hessian Matrix:{EigValues}")
-
-
-theta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y
-print(theta_linreg)
-# Start plain gradient descent
-theta = np.random.randn(2,1)
-
-eta = 1.0/np.max(EigValues)
-Niterations = 100
-
-for iter in range(Niterations):
-    gradients = 2.0/n*X.T @ (X @ (theta)-y)+2*lmbda*theta
-    theta -= eta*gradients
-
-print(theta)
-ypredict = X @ theta
-ypredict2 = X @ theta_linreg
-plt.plot(x, ypredict, "r-")
-plt.plot(x, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Gradient descent example for Ridge')
-plt.show()
-!ec
-
-
-!split
 ===== Ridge regression and a new Synthetic Dataset =====
 
 
@@ -1596,7 +44,7 @@ y = X.dot @ theta_true + noise
 !ec
 
 This code produces a dataset where only features 0, 1, and 6
-significantly influence y. The rest of the features have zero true
+significantly influence $\bm{y}$. The rest of the features have zero true
 coefficient, so they only contribute noise. For example, feature 0 has
 a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so
 the expected relationship is:
@@ -1608,19 +56,23 @@ y \approx 5 \times X_0 \;-\; 3 \times X_1 \;+\; 2 \times X_6 \;+\; \text{noise}.
 
 
 
-Before fitting a regression model, it’s good practice to normalize or
+Before fitting a regression model, it is good practice to normalize or
 standardize the features. This ensures all features are on a
 comparable scale, which is especially important when using
 regularization. Here we will perform standardization, scaling each
 feature to have mean 0 and standard deviation 1:
 
-Compute the mean and standard deviation of each column (feature) in $bm{X}X.
+Compute the mean and standard deviation of each column (feature) in $bm{X}$.
 Subtract the mean and divide by the standard deviation for each feature.
 
 
-We also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$ (and each feature) means the model won’t require a separate intercept term – the data is shifted such that the intercept is effectively 0 . (In practice, one could include an intercept in the model and not penalize it, but here we simplify by centering.)
+We also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$
+(and each feature) means the model won’t require a separate intercept
+term – the data is shifted such that the intercept is effectively 0
+. (In practice, one could include an intercept in the model and not
+penalize it, but here we simplify by centering.)
 
-!bc pyco
+!bc pycod
 # Standardize features (zero mean, unit variance for each feature)
 X_mean = X.mean(axis=0)
 X_std = X.std(axis=0)
@@ -1658,7 +110,12 @@ fitted weights.
 
 
 
-Alternatively, we can fit the ridge regression model using gradient descent. This is useful to visualize the iterative convergence and is necessary if $n$ and $p$ are so large that the closed-form might be too slow or memory-intensive. We derive the gradients from the cost function defined above. The gradient of the ridge cost with respect to the weight vector $w$ is:
+Alternatively, we can fit the ridge regression model using gradient
+descent. This is useful to visualize the iterative convergence and is
+necessary if $n$ and $p$ are so large that the closed-form might be
+too slow or memory-intensive. We derive the gradients from the cost
+function defined above. The gradient of the ridge cost with respect to
+the weight vector $w$ is:
 
 
 
@@ -1693,792 +150,18 @@ print("Gradient Descent Ridge coefficients:", theta_gd)
 !ec
 
 
-Let uss confirm that the two approaches (closed-form and gradient descent) give similar results, and then evaluate the model. First, compare the learned coefficients to the true coefficients:
+Let us confirm that the two approaches (closed-form and gradient
+descent) give similar results, and then evaluate the model. First,
+compare the learned coefficients to the true coefficients:
 
 !bc pycod
 print("True coefficients:", theta_true)
 print("Closed-form learned coefficients:", theta_closed_form)
 print("Gradient descent learned coefficients:", theta_gd)
-
+!ec
 If everything worked correctly, the learned coefficients should be
 close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to
 generate the data. Keep in mind that due to regularization and noise,
 the learned values will not exactly equal the true ones, but they
 should be in the same ballpark.
 
-
-
-
-!split
-===== Using gradient descent methods, limitations =====
-
-* _Gradient descent (GD) finds local minima of our function_. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
-
-* _GD is sensitive to initial conditions_. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
-
-* _Gradients are computationally expensive to calculate for large datasets_. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called ``mini batches''. This has the added benefit of introducing stochasticity into our algorithm.
-
-* _GD is very sensitive to choices of learning rates_. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.
-
-* _GD treats all directions in parameter space uniformly._ Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. 
-	
-* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.
-	
-
-
-
-
-
-!split
-===== Material for lab sessions  sessions Tuesday and Wednesday =====
-
-The material here contains a summary of the lecture on Monday and discussion of SVD, Ridge and Lasso regression with examples 
-
-!split
-===== Linear Regression and  the SVD =====
-
-We used the SVD to analyse the matrix to invert in ordinary lineat regression
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{U}\bm{\Sigma}\bm{V}^T=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T. 
-\]
-!et
-Since the matrices here have dimension $p\times p$, with $p$ corresponding to the singular values, we defined last week the matrix
-!bt
-\[
-\bm{\Sigma}^T\bm{\Sigma} = \begin{bmatrix} \tilde{\bm{\Sigma}} & \bm{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\bm{\Sigma}} \\ \bm{0}\end{bmatrix},
-\]
-!et
-where the tilde-matrix $\tilde{\bm{\Sigma}}$ is a matrix of dimension $p\times p$ containing only the singular values $\sigma_i$, that is
-
-!bt
-\[
-\tilde{\bm{\Sigma}}=\begin{bmatrix} \sigma_0 & 0 & 0 & \dots & 0 & 0 \\
-                                    0 & \sigma_1 & 0 & \dots & 0 & 0 \\
-				    0 & 0 & \sigma_2 & \dots & 0 & 0 \\
-				    0 & 0 & 0 & \dots & \sigma_{p-2} & 0 \\
-				    0 & 0 & 0 & \dots & 0 & \sigma_{p-1} \\
-\end{bmatrix},
-\]
-!et
-meaning we can write
-!bt
-\[
-\bm{X}^T\bm{X}=\bm{V}\tilde{\bm{\Sigma}}^2\bm{V}^T. 
-\]
-!et
-Multiplying from the right with $\bm{V}$ (using the orthogonality of $\bm{V}$) we get
-!bt
-\[
-\left(\bm{X}^T\bm{X}\right)\bm{V}=\bm{V}\tilde{\bm{\Sigma}}^2. 
-\]
-!et
-
-!split
-===== What does it mean? =====
-
-This means the vectors $\bm{v}_i$ of the orthogonal matrix $\bm{V}$
-are the eigenvectors of the matrix $\bm{X}^T\bm{X}$ with eigenvalues
-given by the singular values squared, that is
-
-!bt
-\[
-\left(\bm{X}^T\bm{X}\right)\bm{v}_i=\bm{v}_i\sigma_i^2. 
-\]
-!et
-
-In other words, each non-zero singular value of $\bm{X}$ is a positive
-square root of an eigenvalue of $\bm{X}^T\bm{X}$.  It means also that
-the columns of $\bm{V}$ are the eigenvectors of
-$\bm{X}^T\bm{X}$. Since we have ordered the singular values of
-$\bm{X}$ in a descending order, it means that the column vectors
-$\bm{v}_i$ are hierarchically ordered by how much correlation they
-encode from the columns of $\bm{X}$. 
-
-
-Note that these are also the eigenvectors and eigenvalues of the
-Hessian matrix.
-
-!split
-===== Ridge and LASSO Regression =====
-
-Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
-our optimization problem is
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\theta}\right)^T\left(\bm{y}-\bm{X}\bm{\theta}\right)\right\}.
-\]
-!et
-or we can state it as
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2,
-\]
-!et
-where we have used the definition of  a norm-2 vector, that is
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
-\]
-!et
-
-!split
-===== From OLS to Ridge and Lasso =====
-
-By minimizing the above equation with respect to the parameters
-$\bm{\theta}$ we could then obtain an analytical expression for the
-parameters $\bm{\theta}$.  We can add a regularization parameter $\lambda$ by
-defining a new cost function to be optimized, that is
-
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_2^2
-\]
-!et
-
-which leads to the Ridge regression minimization problem where we
-require that $\vert\vert \bm{\theta}\vert\vert_2^2\le t$, where $t$ is
-a finite number larger than zero. We do not include such a constraints in the discussions here.
-
-By defining
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1,
-\]
-!et
-
-we have a new optimization equation
-!bt
-\[
-{\displaystyle \min_{\bm{\theta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\theta}\vert\vert_2^2+\lambda\vert\vert \bm{\theta}\vert\vert_1
-\]
-!et
-which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. 
-
-Here we have defined the norm-1 as 
-!bt
-\[
-\vert\vert \bm{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
-\]
-!et
-
-
-!split
-===== Deriving the  Ridge Regression Equations =====
-
-Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\left\{(\bm{y}-\bm{X}\bm{\theta})^T(\bm{y}-\bm{X}\bm{\theta})\right\}+\lambda\bm{\theta}^T\bm{\theta},
-\]
-!et
-and 
-taking the derivatives with respect to $\bm{\theta}$ we obtain then
-a slightly modified matrix inversion problem which for finite values
-of $\lambda$ does not suffer from singularity problems. We obtain
-the optimal parameters
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y},
-\]
-!et
-
-with $\bm{I}$ being a $p\times p$ identity matrix with the constraint that
-
-
-
-!bt
-\[
-\sum_{i=0}^{p-1} \theta_i^2 \leq t,
-\]
-!et
-
-with $t$ a finite positive number. 
-
-
-!split
-===== Note on Scikit-Learn =====
-
-Note well that a library like _Scikit-Learn_ does not include the $1/n$ factor in the expression for the mean-squared error. If you include it, the optimal parameter $\theta$ becomes
-
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+n\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-
-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.
-
-
-!split
-===== Comparison with OLS =====
-When we compare this with the ordinary least squares result we have
-!bt
-\[
-\hat{\bm{\theta}}_{\mathrm{OLS}} = \left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y},
-\]
-!et
-which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\bm{X}^T\bm{X}$.
-
-
-We see that Ridge regression is nothing but the standard OLS with a
-modified diagonal term added to $\bm{X}^T\bm{X}$. The consequences, in
-particular for our discussion of the bias-variance tradeoff are rather
-interesting. We will see that for specific values of $\lambda$, we may
-even reduce the variance of the optimal parameters $\bm{\theta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
-
-!split
-===== SVD analysis =====
-
-Using our insights about the SVD of the design matrix $\bm{X}$ 
-We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\bm{U}$ as
-!bt
-\[
-\tilde{\bm{y}}_{\mathrm{OLS}}=\bm{X}\bm{\theta}  =\bm{U}\bm{U}^T\bm{y}.
-\]
-!et
-
-
-For Ridge regression this becomes
-
-!bt
-\[
-\tilde{\bm{y}}_{\mathrm{Ridge}}=\bm{X}\bm{\theta}_{\mathrm{Ridge}} = \bm{U\Sigma V^T}\left(\bm{V}\bm{\Sigma}^2\bm{V}^T+\lambda\bm{I} \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\sum_{j=0}^{p-1}\bm{u}_j\bm{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\bm{y},
-\]
-!et
-
-with the vectors $\bm{u}_j$ being the columns of $\bm{U}$ from the SVD of the matrix $\bm{X}$. 
-
-!split
-===== Interpreting the Ridge results =====
-
-Since $\lambda \geq 0$, it means that compared to OLS, we have 
-
-!bt
-\[
-\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. 
-\]
-!et
-
-Ridge regression finds the coordinates of $\bm{y}$ with respect to the
-orthonormal basis $\bm{U}$, it then shrinks the coordinates by
-$\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has
-eigenvalues ordered in a descending way, that is $\sigma_i \geq
-\sigma_{i+1}$.
-
-For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. 
-
-
-!split
-===== More interpretations =====
-
-For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
-
-!bt
-\[
-\bm{X}^T\bm{X}=(\bm{X}^T\bm{X})^{-1} =\bm{I}. 
-\]
-!et
-
-In this case the standard OLS results in 
-!bt
-\[
-\bm{\theta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{n-1}\bm{u}_i\bm{u}_i^T\bm{y},
-\]
-!et
-
-and
-
-!bt
-\[
-\bm{\theta}^{\mathrm{Ridge}} = \left(\bm{I}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}=\left(1+\lambda\right)^{-1}\bm{\theta}^{\mathrm{OLS}},
-\]
-!et
-
-that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\lambda$, and
-the Ridge estimator converges to zero when the hyperparameter goes to
-infinity.
-
-We will come back to more interpreations after we have gone through some of the statistical analysis part. 
-
-For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
-Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
-
-!split
-===== Deriving the  Lasso Regression Equations =====
-
-Using the matrix-vector expression for Lasso regression, we have the following _cost_ function
-
-!bt
-\[
-C(\bm{X},\bm{\theta})=\frac{1}{n}\left\{(\bm{y}-\bm{X}\bm{\theta})^T(\bm{y}-\bm{X}\bm{\theta})\right\}+\lambda\vert\vert\bm{\theta}\vert\vert_1,
-\]
-!et
-
-Taking the derivative with respect to $\bm{\theta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)
-!bt
-\[
-\frac{d \vert \theta\vert}{d \theta}=\mathrm{sgn}(\theta)=\left\{\begin{array}{cc} 1 & \theta > 0 \\-1 & \theta < 0, \end{array}\right.
-\]
-!et
-we have that the derivative of the cost function is
-
-!bt
-\[
-\frac{\partial C(\bm{X},\bm{\theta})}{\partial \bm{\theta}}=-\frac{2}{n}\bm{X}^T(\bm{y}-\bm{X}\bm{\theta})+\lambda sgn(\bm{\theta})=0,
-\]
-!et
-and reordering we have
-!bt
-\[
-\bm{X}^T\bm{X}\bm{\theta}+\lambda sgn(\bm{\theta})=\bm{X}^T\bm{y}.
-\]
-!et
-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_ and using our own gradient descent code in project 1.
-
-
-
-
-!split
-===== Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression =====
-
-Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the
-diagonal. In this case we have an equal number of rows and columns $n=p$.
-
-Our model approximation is just $\tilde{\bm{y}}=\bm{\theta}$ and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term $1/n$) 
-!bt
-\[
-C(\bm{\theta})=\sum_{i=0}^{p-1}(y_i-\theta_i)^2,
-\]
-!et
-and minimizing we have that
-!bt
-\[
-\hat{\theta}_i^{\mathrm{OLS}} = y_i.
-\]
-!et
-
-!split
-===== Ridge Regression =====
-
-For Ridge regression our cost function is
-!bt
-\[
-C(\bm{\theta})=\sum_{i=0}^{p-1}(y_i-\theta_i)^2+\lambda\sum_{i=0}^{p-1}\theta_i^2,
-\]
-!et
-and minimizing we have that
-!bt
-\[
-\hat{\theta}_i^{\mathrm{Ridge}} = \frac{y_i}{1+\lambda}.
-\]
-!et
-
-
-!split
-===== Lasso Regression =====
-
-For Lasso regression our cost function is
-!bt
-\[
-C(\bm{\theta})=\sum_{i=0}^{p-1}(y_i-\theta_i)^2+\lambda\sum_{i=0}^{p-1}\vert\theta_i\vert=\sum_{i=0}^{p-1}(y_i-\theta_i)^2+\lambda\sum_{i=0}^{p-1}\sqrt{\theta_i^2},
-\]
-!et
-and minimizing we have that
-!bt
-\[
--2\sum_{i=0}^{p-1}(y_i-\theta_i)+\lambda \sum_{i=0}^{p-1}\frac{(\theta_i)}{\vert\theta_i\vert}=0,
-\]
-!et
-which leads to 
-!bt
-\[
-\hat{\bm{\theta}}_i^{\mathrm{Lasso}} = \left\{\begin{array}{ccc}y_i-\frac{\lambda}{2} &\mathrm{if} & y_i> \frac{\lambda}{2}\\
-                                                          y_i+\frac{\lambda}{2} &\mathrm{if} & y_i< -\frac{\lambda}{2}\\
-							  0 &\mathrm{if} & \vert y_i\vert\le  \frac{\lambda}{2}\end{array}\right.\\.
-\]
-!et
-
-Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\theta_i$ for specific values of $\lambda$. Ridge regression reduces on the other hand the values of $\theta_i$ as function of $\lambda$.
-
-
-
-!split
-===== Yet another Example =====
-
-Let us assume we have a data set with outputs/targets given by the vector
-
-!bt
-\[
-\bm{y}=\begin{bmatrix}4 \\ 2 \\3\end{bmatrix},
-\]
-!et
-and our inputs as a $3\times 2$ design matrix
-!bt
-\[
-\bm{X}=\begin{bmatrix}2 & 0\\ 0 & 1 \\ 0 & 0\end{bmatrix},
-\]
-!et
-meaning that we have two features and two unknown parameters $\theta_0$ and $\theta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression.
-
-!split
-===== The OLS case =====
-
-For ordinary least squares (OLS) we know that the optimal solution is
-
-!bt
-\[
-\hat{\bm{\theta}}^{\mathrm{OLS}}=\left( \bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-Inserting the above values we obtain that 
-
-!bt
-\[
-\hat{\bm{\theta}}^{\mathrm{OLS}}=\begin{bmatrix}2 \\ 2\end{bmatrix},
-\]
-!et
-
-The code which implements this simpler case is presented after the discussion of Ridge and Lasso.
-
-!split
-===== The Ridge case =====
-
-For Ridge regression we have
-
-!bt
-\[
-\hat{\bm{\theta}}^{\mathrm{Ridge}}=\left( \bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}.
-\]
-!et
-Inserting the above values we obtain that 
-
-!bt
-\[
-\hat{\bm{\theta}}^{\mathrm{Ridge}}=\begin{bmatrix}\frac{8}{4+\lambda} \\ \frac{2}{1+\lambda}\end{bmatrix},
-\]
-!et
-
-There is normally a constraint on the value of $\vert\vert \bm{\theta}\vert\vert_2$ via the parameter $\lambda$.
-Let us for simplicity assume that $\theta_0^2+\theta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\theta$ and $\lambda$.
-
-To see this, let us write the cost function for Ridge regression.  
-
-
-!split
-===== Writing the Cost Function =====
-
-We define the MSE without the $1/n$ factor and have then, using that
-!bt
-\[
-\bm{X}\bm{\theta}=\begin{bmatrix} 2\theta_0 \\ \theta_1 \\0 \end{bmatrix},
-\]
-!et
-
-!bt
-\[
-C(\bm{\theta})=(4-2\theta_0)^2+(2-\theta_1)^2+\lambda(\theta_0^2+\theta_1^2),
-\]
-!et
-and taking the derivative with respect to $\theta_0$ we get
-!bt
-\[
-\theta_0=\frac{8}{4+\lambda},
-\]
-!et
-and for $\theta_1$ we obtain
-!bt
-\[
-\theta_1=\frac{2}{1+\lambda},
-\]
-!et
-
-Using the constraint for $\theta_0^2+\theta_1^2=1$ we can constrain $\lambda$ by solving
-!bt
-\[
-\left(\frac{8}{4+\lambda}\right)^2+\left(\frac{2}{1+\lambda}\right)^2=1,
-\]
-!et
-which gives $\lambda=4.571$ and $\theta_0=0.933$ and $\theta_1=0.359$.
-
-!split
-===== Lasso case =====
-
-For Lasso we need now, keeping a  constraint on $\vert\theta_0\vert+\vert\theta_1\vert=1$,  to take the derivative of the absolute values of $\theta_0$
-and $\theta_1$. This gives us the following derivatives of the cost function
-!bt
-\[
-C(\bm{\theta})=(4-2\theta_0)^2+(2-\theta_1)^2+\lambda(\vert\theta_0\vert+\vert\theta_1\vert),
-\]
-!et
-
-!bt
-\[
-\frac{\partial C(\bm{\theta})}{\partial \theta_0}=-4(4-2\theta_0)+\lambda\mathrm{sgn}(\theta_0)=0,
-\]
-!et
-and
-!bt
-\[
-\frac{\partial C(\bm{\theta})}{\partial \theta_1}=-2(2-\theta_1)+\lambda\mathrm{sgn}(\theta_1)=0.
-\]
-!et
-We have now four cases to solve besides the trivial cases $\theta_0$ and/or $\theta_1$ are zero, namely
-o $\theta_0 > 0$ and $\theta_1 > 0$,
-o $\theta_0 > 0$ and $\theta_1 < 0$,
-o $\theta_0 < 0$ and $\theta_1 > 0$,
-o $\theta_0 < 0$ and $\theta_1 < 0$.
-
-!split
-===== The first Case =====
-
-If we consider the first case, we have then
-!bt
-\[
--4(4-2\theta_0)+\lambda=0,
-\]
-!et
-and
-!bt
-\[
--2(2-\theta_1)+\lambda=0.
-\]
-!et
-which yields
-
-!bt
-\[
-\theta_0=\frac{16+\lambda}{8},
-\]
-!et
-and
-!bt
-\[
-\theta_1=\frac{4+\lambda}{2}.
-\]
-!et
-
-Using the constraint on $\theta_0$ and $\theta_1$ we can then find the optimal value of $\lambda$ for the different cases. We leave this as an exercise to you.
-
-!split
-===== Simple code for solving the above problem =====
-
-Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of $\lambda$, meaning that we need to perform a search in order to find the optimal values.
-
-First we study and compare the OLS and Ridge results.  The next code compares all three methods.
-
-
-!bc pycod
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-
-X = np.array( [ [ 2, 0], [0, 1], [0,0]])
-y = np.array( [4, 2, 3])
-
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y,ytildeOLS))
-ypredictOLS = X @ OLSbeta
-
-# Repeat now for Ridge regression and various values of the regularization parameter
-I = np.eye(2,2)
-# Decide which values of lambda to use
-nlambdas = 100
-MSEPredict = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 4, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
-#    print(Ridgebeta)
-    # and then make the prediction
-    ypredictRidge = X @ Ridgebeta
-    MSEPredict[i] = MSE(y,ypredictRidge)
-#    print(MSEPredict[i])
-    # Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-!ec
-
-We see here that we reach a plateau. What is actually happening?
-
-
-!split
-===== With Lasso Regression =====
-
-!bc pycod
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn import linear_model
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-
-X = np.array( [ [ 2, 0], [0, 1], [0,0]])
-y = np.array( [4, 2, 3])
-
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y,ytildeOLS))
-ypredictOLS = X @ OLSbeta
-
-# Repeat now for Ridge regression and various values of the regularization parameter
-I = np.eye(2,2)
-# Decide which values of lambda to use
-nlambdas = 100
-MSERidgePredict = np.zeros(nlambdas)
-MSELassoPredict = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 4, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
-    print(Ridgebeta)
-    # and then make the prediction
-    ypredictRidge = X @ Ridgebeta
-    MSERidgePredict[i] = MSE(y,ypredictRidge)
-    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
-    RegLasso.fit(X,y)
-    ypredictLasso = RegLasso.predict(X)
-    print(RegLasso.coef_)
-    MSELassoPredict[i] = MSE(y,ypredictLasso)
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')
-plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-!ec
-
-!split
-=====  Another Example, now with a polynomial fit =====
-
-!bc pycod
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-# number of features p (here degree of polynomial
-p = 3
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),p))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x*x
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-# Repeat now for Lasso and Ridge regression and various values of the regularization parameter
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 100
-MSEPredict = np.zeros(nlambdas)
-MSETrain = np.zeros(nlambdas)
-MSELassoPredict = np.zeros(nlambdas)
-MSELassoTrain = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 4, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
-    # include lasso using Scikit-Learn
-    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
-    RegLasso.fit(X_train,y_train)
-    # and then make the prediction
-    ytildeRidge = X_train @ Ridgebeta
-    ypredictRidge = X_test @ Ridgebeta
-    ytildeLasso = RegLasso.predict(X_train)
-    ypredictLasso = RegLasso.predict(X_test)
-    MSEPredict[i] = MSE(y_test,ypredictRidge)
-    MSETrain[i] = MSE(y_train,ytildeRidge)
-    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
-    MSELassoTrain[i] = MSE(y_train,ytildeLasso)
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')
-plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')
-plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-!ec
-
-
-
-
-
-
diff --git a/doc/src/week36/exercisesweek37.ipynb b/doc/src/week36/exercisesweek37.ipynb
new file mode 100644
index 000000000..75e3df763
--- /dev/null
+++ b/doc/src/week36/exercisesweek37.ipynb
@@ -0,0 +1,357 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "2c329a04",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
+    "doconce format html exercisesweek37.do.txt  -->\n",
+    "<!-- dom:TITLE: Week 37: Linear Regression and Gradient descent -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "432a2e5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Week 37: Linear Regression and Gradient descent\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
+    "\n",
+    "Date: **September 1-5, 2025**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8352513e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge regression and a new Synthetic Dataset\n",
+    "\n",
+    "We create a synthetic linear regression dataset with a sparse\n",
+    "underlying relationship. This means we have many features but only a\n",
+    "few of them actually contribute to the target. In our example, we’ll\n",
+    "use 10 features with only 3 non-zero weights in the true model. This\n",
+    "way, the target is generated as a linear combination of a few features\n",
+    "(with known coefficients) plus some random noise. The steps we include are:\n",
+    "\n",
+    "Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
+    "Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
+    "\n",
+    "Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
+    "Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
+    "\n",
+    "Below is the code to generate the dataset:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "d7eebc41",
+   "metadata": {
+    "collapsed": false,
+    "editable": true,
+    "jupyter": {
+     "outputs_hidden": false
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "# Set random seed for reproducibility\n",
+    "np.random.seed(0)\n",
+    "\n",
+    "# Define dataset size\n",
+    "n_samples = 100\n",
+    "n_features = 10\n",
+    "\n",
+    "# Define true coefficients (sparse linear relationship)\n",
+    "theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
+    "\n",
+    "# Generate feature matrix X (n_samples x n_features) with random values\n",
+    "X = np.random.randn(n_samples, n_features)  # standard normal distribution\n",
+    "\n",
+    "# Generate target values y with a linear combination of X and theta_true, plus noise\n",
+    "noise = 0.5 * np.random.randn(n_samples)    # Gaussian noise\n",
+    "y = X @ theta_true + noise"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1640310e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This code produces a dataset where only features 0, 1, and 6\n",
+    "significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
+    "coefficient, so they only contribute noise. For example, feature 0 has\n",
+    "a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
+    "the expected relationship is:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "60bce352",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "y \\approx 5 \\times X_0 \\;-\\; 3 \\times X_1 \\;+\\; 2 \\times X_6 \\;+\\; \\text{noise}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "75322f31",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Before fitting a regression model, it is good practice to normalize or\n",
+    "standardize the features. This ensures all features are on a\n",
+    "comparable scale, which is especially important when using\n",
+    "regularization. Here we will perform standardization, scaling each\n",
+    "feature to have mean 0 and standard deviation 1:\n",
+    "\n",
+    "Compute the mean and standard deviation of each column (feature) in $bm{X}$.\n",
+    "Subtract the mean and divide by the standard deviation for each feature.\n",
+    "\n",
+    "We also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
+    "(and each feature) means the model won’t require a separate intercept\n",
+    "term – the data is shifted such that the intercept is effectively 0\n",
+    ". (In practice, one could include an intercept in the model and not\n",
+    "penalize it, but here we simplify by centering.)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "4b3f945a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true,
+    "jupyter": {
+     "outputs_hidden": false
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Standardize features (zero mean, unit variance for each feature)\n",
+    "X_mean = X.mean(axis=0)\n",
+    "X_std = X.std(axis=0)\n",
+    "X_std[X_std == 0] = 1  # safeguard to avoid division by zero for constant features\n",
+    "X_norm = (X - X_mean) / X_std\n",
+    "\n",
+    "# Center the target to zero mean (optional, to simplify intercept handling)\n",
+    "y_mean = y.mean()\n",
+    "y_centered = y - y_mean"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12aec8b3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "After this preprocessing, each column of $\\boldsymbol{X}_norm$ has mean zero and standard deviation $1$\n",
+    "and $\\boldsymbol{y}_centered$ has mean 0. This makes the optimization landscape\n",
+    "nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
+    "\\beta_j^2$ treats each coefficient fairly (since features are on the\n",
+    "same scale)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "ffdecd03",
+   "metadata": {
+    "collapsed": false,
+    "editable": true,
+    "jupyter": {
+     "outputs_hidden": false
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Closed-form Ridge coefficients: [ 4.98041921 -2.85683951 -0.01549982  0.15658961 -0.07810233 -0.05094777\n",
+      "  1.75377987  0.00679113  0.0343071  -0.04883458]\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Set regularization parameter\n",
+    "lam = 1.0\n",
+    "\n",
+    "# Closed-form Ridge solution: w = (X^T X + lam * I)^{-1} X^T y\n",
+    "I = np.eye(n_features)\n",
+    "theta_closed_form = np.linalg.inv(X_norm.T.dot(X_norm) + lam * I).dot(X_norm.T).dot(y_centered)\n",
+    "\n",
+    "print(\"Closed-form Ridge coefficients:\", theta_closed_form)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33ff39a5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This computes the ridge regression coefficients directly. The identity\n",
+    "matrix $I$ has the same size as $X^T X$ (which is n_features x\n",
+    "n_features), and lam * I adds $\\lambda$ to the diagonal of $X^T X. We\n",
+    "then invert this matrix and multiply by $X^T y. The result\n",
+    "for $\\boldsymbol{\\theta}$  is a NumPy array of shape (n_features,) containing the\n",
+    "fitted weights.\n",
+    "\n",
+    "Alternatively, we can fit the ridge regression model using gradient\n",
+    "descent. This is useful to visualize the iterative convergence and is\n",
+    "necessary if $n$ and $p$ are so large that the closed-form might be\n",
+    "too slow or memory-intensive. We derive the gradients from the cost\n",
+    "function defined above. The gradient of the ridge cost with respect to\n",
+    "the weight vector $w$ is:\n",
+    "\n",
+    "Below is the code for gradient descent implementation of ridge:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "5b72dc82",
+   "metadata": {
+    "collapsed": false,
+    "editable": true,
+    "jupyter": {
+     "outputs_hidden": false
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Gradient Descent Ridge coefficients: [ 4.98041921 -2.85683951 -0.01549982  0.15658961 -0.07810233 -0.05094777\n",
+      "  1.75377987  0.00679113  0.0343071  -0.04883458]\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Gradient descent parameters\n",
+    "alpha = 0.1\n",
+    "num_iters = 1000\n",
+    "\n",
+    "# Initialize weights for gradient descent\n",
+    "theta = np.zeros(n_features)\n",
+    "\n",
+    "# Arrays to store history for plotting\n",
+    "cost_history = np.zeros(num_iters)\n",
+    "\n",
+    "# Gradient descent loop\n",
+    "m = n_samples  # number of examples\n",
+    "for t in range(num_iters):\n",
+    "    # Compute prediction error\n",
+    "    error = X_norm.dot(theta) - y_centered  # shape (m,)\n",
+    "    # Compute cost (MSE + regularization) for monitoring\n",
+    "    cost = (1/(2*m)) * np.dot(error, error) + (lam/(2*m)) * np.dot(theta, theta)\n",
+    "    cost_history[t] = cost\n",
+    "    # Compute gradient\n",
+    "    grad = (1/m) * (X_norm.T.dot(error) + lam * theta)\n",
+    "    # Update weights\n",
+    "    theta = theta - alpha * grad\n",
+    "\n",
+    "# After the loop, theta contains the fitted coefficients\n",
+    "theta_gd = theta\n",
+    "print(\"Gradient Descent Ridge coefficients:\", theta_gd)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1fc5b8d0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us confirm that the two approaches (closed-form and gradient\n",
+    "descent) give similar results, and then evaluate the model. First,\n",
+    "compare the learned coefficients to the true coefficients:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "a510e4e6",
+   "metadata": {
+    "collapsed": false,
+    "editable": true,
+    "jupyter": {
+     "outputs_hidden": false
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "True coefficients: [ 5. -3.  0.  0.  0.  0.  2.  0.  0.  0.]\n",
+      "Closed-form learned coefficients: [ 4.98041921 -2.85683951 -0.01549982  0.15658961 -0.07810233 -0.05094777\n",
+      "  1.75377987  0.00679113  0.0343071  -0.04883458]\n",
+      "Gradient descent learned coefficients: [ 4.98041921 -2.85683951 -0.01549982  0.15658961 -0.07810233 -0.05094777\n",
+      "  1.75377987  0.00679113  0.0343071  -0.04883458]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(\"True coefficients:\", theta_true)\n",
+    "print(\"Closed-form learned coefficients:\", theta_closed_form)\n",
+    "print(\"Gradient descent learned coefficients:\", theta_gd)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07081487",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If everything worked correctly, the learned coefficients should be\n",
+    "close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
+    "generate the data. Keep in mind that due to regularization and noise,\n",
+    "the learned values will not exactly equal the true ones, but they\n",
+    "should be in the same ballpark."
+   ]
+  }
+ ],
+ "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": "3.9.15"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}