diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index 7b060ebdd..01be86968 100644
Binary files a/doc/LectureNotes/_build/.doctrees/environment.pickle and b/doc/LectureNotes/_build/.doctrees/environment.pickle differ
diff --git a/doc/LectureNotes/_build/.doctrees/week35.doctree b/doc/LectureNotes/_build/.doctrees/week35.doctree
index 72605c4d9..466c7947a 100644
Binary files a/doc/LectureNotes/_build/.doctrees/week35.doctree and b/doc/LectureNotes/_build/.doctrees/week35.doctree differ
diff --git a/doc/LectureNotes/_build/html/_sources/week35.ipynb b/doc/LectureNotes/_build/html/_sources/week35.ipynb
index 01ae64c94..b5da95d0a 100644
--- a/doc/LectureNotes/_build/html/_sources/week35.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/week35.ipynb
@@ -2,7 +2,7 @@
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     "### Reading recommendations:\n",
     "\n",
     "1. These lecture notes\n",
-    "<!-- o [Video of lecture](https://youtu.be/VKakN-e4aUA) -->\n",
-    "<!-- o [Video for exercises week 35](https://youtu.be/yiY0OltU1s8) -->\n",
     "\n",
-    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n",
+    "2. Video of lecture at <https://youtu.be/2mvizAQFST8>\n",
     "\n",
-    "3. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n",
+    "3. Whiteboard notes  at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2025/FYSSTKweek35.pdf>\n",
     "\n",
-    "4. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week."
+    "4. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n",
+    "\n",
+    "5. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n",
+    "\n",
+    "6. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week."
    ]
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@@ -1191,7 +1193,7 @@
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@@ -1203,7 +1205,7 @@
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@@ -1213,7 +1215,7 @@
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@@ -1225,7 +1227,7 @@
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@@ -1243,7 +1245,7 @@
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@@ -1255,7 +1257,7 @@
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@@ -1267,7 +1269,7 @@
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@@ -1277,7 +1279,7 @@
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@@ -1289,7 +1291,7 @@
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@@ -1299,7 +1301,7 @@
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@@ -1311,7 +1313,7 @@
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@@ -1321,7 +1323,7 @@
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@@ -1335,7 +1337,7 @@
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@@ -1347,7 +1349,7 @@
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@@ -1358,7 +1360,7 @@
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@@ -1371,7 +1373,7 @@
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@@ -1398,7 +1400,7 @@
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@@ -1409,7 +1411,7 @@
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@@ -1422,7 +1424,7 @@
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@@ -1436,7 +1438,7 @@
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@@ -1449,7 +1451,7 @@
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@@ -1460,7 +1462,7 @@
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@@ -1472,7 +1474,7 @@
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@@ -1483,7 +1485,7 @@
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@@ -1499,7 +1501,7 @@
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@@ -1510,7 +1512,7 @@
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@@ -1524,7 +1526,7 @@
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@@ -1545,7 +1547,7 @@
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@@ -1556,7 +1558,7 @@
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@@ -1609,7 +1611,7 @@
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@@ -1620,7 +1622,7 @@
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@@ -1645,7 +1647,7 @@
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@@ -1657,7 +1659,7 @@
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@@ -1686,7 +1688,7 @@
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@@ -1711,7 +1713,7 @@
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@@ -1731,7 +1733,7 @@
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@@ -1758,7 +1760,7 @@
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@@ -1771,7 +1773,7 @@
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@@ -1783,7 +1785,7 @@
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@@ -1794,7 +1796,7 @@
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@@ -1810,7 +1812,7 @@
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@@ -1844,7 +1846,7 @@
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@@ -1854,7 +1856,7 @@
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@@ -1869,7 +1871,7 @@
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@@ -1881,7 +1883,7 @@
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@@ -1891,7 +1893,7 @@
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@@ -1907,7 +1909,7 @@
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@@ -1924,7 +1926,7 @@
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@@ -1937,7 +1939,7 @@
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@@ -1984,7 +1986,7 @@
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@@ -1998,7 +2000,7 @@
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@@ -2010,7 +2012,7 @@
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@@ -2022,7 +2024,7 @@
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@@ -2034,7 +2036,7 @@
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@@ -2044,7 +2046,7 @@
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@@ -2056,7 +2058,7 @@
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@@ -2066,7 +2068,7 @@
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@@ -2078,7 +2080,7 @@
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@@ -2089,7 +2091,7 @@
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@@ -2101,7 +2103,7 @@
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@@ -2113,7 +2115,7 @@
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@@ -2123,7 +2125,7 @@
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@@ -2135,7 +2137,7 @@
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@@ -2147,7 +2149,7 @@
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@@ -2157,7 +2159,7 @@
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@@ -2169,7 +2171,7 @@
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@@ -2179,7 +2181,7 @@
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@@ -2191,7 +2193,7 @@
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@@ -2201,7 +2203,7 @@
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@@ -2241,7 +2243,7 @@
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@@ -2258,7 +2260,7 @@
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@@ -2281,7 +2283,7 @@
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@@ -2298,7 +2300,7 @@
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@@ -2317,7 +2319,7 @@
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@@ -2328,7 +2330,7 @@
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@@ -2340,7 +2342,7 @@
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@@ -2358,7 +2360,7 @@
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@@ -2374,7 +2376,7 @@
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@@ -2386,7 +2388,7 @@
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@@ -2396,7 +2398,7 @@
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@@ -2409,7 +2411,7 @@
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@@ -2421,7 +2423,7 @@
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@@ -2431,7 +2433,7 @@
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@@ -2444,7 +2446,7 @@
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@@ -2454,7 +2456,7 @@
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@@ -2466,7 +2468,7 @@
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@@ -2479,7 +2481,7 @@
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@@ -2492,7 +2494,7 @@
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@@ -2504,7 +2506,7 @@
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@@ -2516,7 +2518,7 @@
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@@ -2526,7 +2528,7 @@
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@@ -2539,7 +2541,7 @@
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@@ -2551,7 +2553,7 @@
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@@ -2563,7 +2565,7 @@
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@@ -2575,7 +2577,7 @@
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@@ -2587,7 +2589,7 @@
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@@ -2601,7 +2603,7 @@
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@@ -2613,7 +2615,7 @@
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@@ -2623,7 +2625,7 @@
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@@ -2635,7 +2637,7 @@
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@@ -2647,7 +2649,7 @@
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@@ -2659,7 +2661,7 @@
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@@ -2671,7 +2673,7 @@
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@@ -2683,7 +2685,7 @@
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@@ -2706,7 +2708,7 @@
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@@ -2782,7 +2784,7 @@
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@@ -2794,7 +2796,7 @@
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@@ -2809,7 +2811,7 @@
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@@ -2821,7 +2823,7 @@
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@@ -2831,7 +2833,7 @@
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@@ -2843,7 +2845,7 @@
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@@ -2853,7 +2855,7 @@
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@@ -2865,7 +2867,7 @@
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@@ -2877,7 +2879,7 @@
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@@ -2892,7 +2894,7 @@
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@@ -2903,7 +2905,7 @@
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@@ -2923,7 +2925,7 @@
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@@ -2935,7 +2937,7 @@
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@@ -2945,7 +2947,7 @@
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@@ -2957,7 +2959,7 @@
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@@ -2986,7 +2988,7 @@
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@@ -3013,7 +3015,7 @@
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@@ -3024,7 +3026,7 @@
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@@ -3064,7 +3066,7 @@
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@@ -3081,7 +3083,7 @@
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@@ -3104,7 +3106,7 @@
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@@ -3118,7 +3120,7 @@
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@@ -3137,7 +3139,7 @@
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@@ -3147,7 +3149,7 @@
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@@ -3159,7 +3161,7 @@
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@@ -3173,7 +3175,7 @@
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@@ -3185,7 +3187,7 @@
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@@ -3195,7 +3197,7 @@
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@@ -3207,7 +3209,7 @@
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@@ -3224,7 +3226,7 @@
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@@ -3234,7 +3236,7 @@
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@@ -3250,7 +3252,7 @@
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@@ -3260,7 +3262,7 @@
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@@ -3276,7 +3278,7 @@
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@@ -3286,7 +3288,7 @@
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@@ -3302,7 +3304,7 @@
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@@ -3312,7 +3314,7 @@
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@@ -3329,7 +3331,7 @@
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@@ -3341,7 +3343,7 @@
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@@ -3353,7 +3355,7 @@
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@@ -3365,7 +3367,7 @@
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@@ -3375,7 +3377,7 @@
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@@ -3387,7 +3389,7 @@
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@@ -3399,7 +3401,7 @@
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@@ -3411,7 +3413,7 @@
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@@ -3421,7 +3423,7 @@
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@@ -3433,7 +3435,7 @@
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@@ -3443,7 +3445,7 @@
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@@ -3455,7 +3457,7 @@
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@@ -3472,7 +3474,7 @@
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@@ -3484,7 +3486,7 @@
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@@ -3496,7 +3498,7 @@
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@@ -3506,7 +3508,7 @@
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@@ -3518,7 +3520,7 @@
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@@ -3529,7 +3531,7 @@
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@@ -3541,7 +3543,7 @@
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@@ -3551,7 +3553,7 @@
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@@ -3563,7 +3565,7 @@
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@@ -3573,7 +3575,7 @@
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@@ -3585,7 +3587,7 @@
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@@ -3596,7 +3598,7 @@
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@@ -3608,7 +3610,7 @@
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@@ -3626,7 +3628,7 @@
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@@ -3642,7 +3644,7 @@
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@@ -3654,7 +3656,7 @@
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@@ -3666,7 +3668,7 @@
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@@ -3678,7 +3680,7 @@
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@@ -3691,7 +3693,7 @@
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@@ -3707,7 +3709,7 @@
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@@ -3721,7 +3723,7 @@
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@@ -3731,7 +3733,7 @@
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@@ -3743,7 +3745,7 @@
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@@ -3753,7 +3755,7 @@
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@@ -3765,7 +3767,7 @@
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@@ -3775,7 +3777,7 @@
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@@ -3789,7 +3791,7 @@
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@@ -3806,7 +3808,7 @@
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@@ -3822,7 +3824,7 @@
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@@ -3834,7 +3836,7 @@
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@@ -3847,7 +3849,7 @@
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@@ -3861,7 +3863,7 @@
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@@ -3871,7 +3873,7 @@
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@@ -3884,7 +3886,7 @@
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@@ -3903,7 +3905,7 @@
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@@ -3915,7 +3917,7 @@
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@@ -3927,7 +3929,7 @@
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@@ -3937,7 +3939,7 @@
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@@ -3949,7 +3951,7 @@
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@@ -3962,7 +3964,7 @@
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@@ -3981,7 +3983,7 @@
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@@ -3991,7 +3993,7 @@
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@@ -4010,7 +4012,7 @@
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@@ -4028,7 +4030,7 @@
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@@ -4042,7 +4044,7 @@
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@@ -4057,7 +4059,7 @@
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@@ -4078,7 +4080,7 @@
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@@ -4095,7 +4097,7 @@
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@@ -4127,7 +4129,7 @@
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@@ -4141,7 +4143,7 @@
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@@ -4154,7 +4156,7 @@
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-   "id": "81d407da",
+   "id": "488f60ab",
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@@ -4179,7 +4181,7 @@
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@@ -4191,7 +4193,7 @@
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@@ -4203,7 +4205,7 @@
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@@ -4213,7 +4215,7 @@
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@@ -4230,7 +4232,7 @@
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@@ -4926,7 +4928,7 @@
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@@ -5043,7 +5045,7 @@
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@@ -5055,13 +5057,856 @@
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    "source": [
     "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 using for example the Python package [CVXOPT](https://cvxopt.org/). We will discuss how to code LASSO regression next week, when we have introduced gradient methods."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c47881af",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for exercises week 35"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de625607",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Important technicalities: More on Rescaling data\n",
+    "\n",
+    "When you are comparing your own code with for example **Scikit-Learn**'s\n",
+    "library, there are some technicalities to keep in mind.  The examples\n",
+    "here demonstrate some of these aspects with potential pitfalls.\n",
+    "\n",
+    "The discussion here focuses on the role of the intercept, how we can\n",
+    "set up the design matrix, what scaling we should use and other topics\n",
+    "which tend  confuse us.\n",
+    "\n",
+    "The intercept can be interpreted as the expected value of our\n",
+    "target/output variables when all other predictors are set to zero.\n",
+    "Thus, if we cannot assume that the expected outputs/targets are zero\n",
+    "when all predictors are zero (the columns in the design matrix), it\n",
+    "may be a bad idea to implement a model which penalizes the intercept.\n",
+    "Furthermore, in for example Ridge and Lasso regression, the default solutions\n",
+    "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n",
+    "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
+    "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
+    "\n",
+    "If our predictors represent different scales, then it is important to\n",
+    "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
+    "column from the corresponding column and dividing the column with its\n",
+    "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n",
+    "the results may differ. \n",
+    "\n",
+    "The\n",
+    "[Standardscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n",
+    "function in **Scikit-Learn** does this for us.  For the data sets we\n",
+    "have been studying in our various examples, the data are in many cases\n",
+    "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform  a\n",
+    "survey of your data, with a critical assessment of them in case you need to scale the data.\n",
+    "\n",
+    "If you need to scale the data, not doing so will give an *unfair*\n",
+    "penalization of the parameters since their magnitude depends on the\n",
+    "scale of their corresponding predictor.\n",
+    "\n",
+    "The **Scikit-Learn** site <https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section> has a good discussion of different ways of preprocessing data.\n",
+    "\n",
+    "Suppose as an example that you \n",
+    "you have an input variable given by the heights of different persons.\n",
+    "Human height might be measured in inches or meters or\n",
+    "kilometers. If measured in kilometers, a standard linear regression\n",
+    "model with this predictor would probably give a much bigger\n",
+    "coefficient term, than if measured in millimeters.\n",
+    "This can clearly lead to problems in evaluating the cost/loss functions.\n",
+    "\n",
+    "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
+    "on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "c00a9805",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "\"\"\"\n",
+    "#Model training, we compute the mean value of y and X\n",
+    "y_train_mean = np.mean(y_train)\n",
+    "X_train_mean = np.mean(X_train,axis=0)\n",
+    "X_train = X_train - X_train_mean\n",
+    "y_train = y_train - y_train_mean\n",
+    "\n",
+    "# The we fit our model with the training data\n",
+    "trained_model = some_model.fit(X_train,y_train)\n",
+    "\n",
+    "\n",
+    "#Model prediction, we need also to transform our data set used for the prediction.\n",
+    "X_test = X_test - X_train_mean #Use mean from training data\n",
+    "y_pred = trained_model(X_test)\n",
+    "y_pred = y_pred + y_train_mean\n",
+    "\"\"\""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b3fb9183",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us try to understand what this may imply mathematically when we\n",
+    "subtract the mean values, also known as *zero centering*. For\n",
+    "simplicity, we will focus on  ordinary regression, as done in the above example.\n",
+    "\n",
+    "The cost/loss function  for regression is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07a54955",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1ac9c31d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Recall also that we use the squared value. This expression can lead to an\n",
+    "increased penalty for higher differences between predicted and\n",
+    "output/target values.\n",
+    "\n",
+    "What we have done is to single out the $\\beta_0$ term in the\n",
+    "definition of the mean squared error (MSE).  The design matrix $X$\n",
+    "does in this case not contain any intercept column.  When we take the\n",
+    "derivative with respect to $\\beta_0$, we want the derivative to obey"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ee6c6a4c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "beb40c81",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "for all $j$. For $\\beta_0$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b051946e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "885a6b08",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying away the constant $2/n$, we obtain"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5774d6be",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "73ff6979",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
+    "Our result for $\\beta_0$ simplifies then to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "792128bf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2be70ed5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We obtain then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d0c4ac1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2d4a5936",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If we define"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d30abc39",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7f945f8b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and the mean value of the outputs as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dff5bf57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f222bcca",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f1658ccd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "394ef3c4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f1dd1af",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0cafc2d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We can rewrite the latter equation as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2c878f98",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1dab41db",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have defined"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cf2f2908",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "20bac2a4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
+    "\n",
+    "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0285afd3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ffff8823",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eea815a7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0d935c60",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+    "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
+    "\n",
+    "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb4eabb8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1da18d81",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
+    "\n",
+    "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to  Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
+    "Note also that we do not split the data into training and test."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "a11f0699",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "from sklearn.linear_model import LinearRegression\n",
+    "\n",
+    "\n",
+    "np.random.seed(2021)\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "def fit_beta(X, y):\n",
+    "    return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
+    "\n",
+    "\n",
+    "true_beta = [2, 0.5, 3.7]\n",
+    "\n",
+    "x = np.linspace(0, 1, 11)\n",
+    "y = np.sum(\n",
+    "    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n",
+    ") + 0.1 * np.random.normal(size=len(x))\n",
+    "\n",
+    "degree = 3\n",
+    "X = np.zeros((len(x), degree))\n",
+    "\n",
+    "# Include the intercept in the design matrix\n",
+    "for p in range(degree):\n",
+    "    X[:, p] = x ** p\n",
+    "\n",
+    "beta = fit_beta(X, y)\n",
+    "\n",
+    "# Intercept is included in the design matrix\n",
+    "skl = LinearRegression(fit_intercept=False).fit(X, y)\n",
+    "\n",
+    "print(f\"True beta: {true_beta}\")\n",
+    "print(f\"Fitted beta: {beta}\")\n",
+    "print(f\"Sklearn fitted beta: {skl.coef_}\")\n",
+    "ypredictOwn = X @ beta\n",
+    "ypredictSKL = skl.predict(X)\n",
+    "print(f\"MSE with intercept column\")\n",
+    "print(MSE(y,ypredictOwn))\n",
+    "print(f\"MSE with intercept column from SKL\")\n",
+    "print(MSE(y,ypredictSKL))\n",
+    "\n",
+    "\n",
+    "plt.figure()\n",
+    "plt.scatter(x, y, label=\"Data\")\n",
+    "plt.plot(x, X @ beta, label=\"Fit\")\n",
+    "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n",
+    "\n",
+    "\n",
+    "# Do not include the intercept in the design matrix\n",
+    "X = np.zeros((len(x), degree - 1))\n",
+    "\n",
+    "for p in range(degree - 1):\n",
+    "    X[:, p] = x ** (p + 1)\n",
+    "\n",
+    "# Intercept is not included in the design matrix\n",
+    "skl = LinearRegression(fit_intercept=True).fit(X, y)\n",
+    "\n",
+    "# Use centered values for X and y when computing coefficients\n",
+    "y_offset = np.average(y, axis=0)\n",
+    "X_offset = np.average(X, axis=0)\n",
+    "\n",
+    "beta = fit_beta(X - X_offset, y - y_offset)\n",
+    "intercept = np.mean(y_offset - X_offset @ beta)\n",
+    "\n",
+    "print(f\"Manual intercept: {intercept}\")\n",
+    "print(f\"Fitted beta (without intercept): {beta}\")\n",
+    "print(f\"Sklearn intercept: {skl.intercept_}\")\n",
+    "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n",
+    "ypredictOwn = X @ beta\n",
+    "ypredictSKL = skl.predict(X)\n",
+    "print(f\"MSE with Manual intercept\")\n",
+    "print(MSE(y,ypredictOwn+intercept))\n",
+    "print(f\"MSE with Sklearn intercept\")\n",
+    "print(MSE(y,ypredictSKL))\n",
+    "\n",
+    "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n",
+    "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n",
+    "plt.grid()\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed61dd49",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The intercept is the value of our output/target variable\n",
+    "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
+    "\n",
+    "Printing the MSE, we see first that both methods give the same MSE, as\n",
+    "they should.  However, when we move to for example Ridge regression,\n",
+    "the way we treat the intercept may give a larger or smaller MSE,\n",
+    "meaning that the MSE can be penalized by the value of the\n",
+    "intercept. Not including the intercept in the fit, means that the\n",
+    "regularization term does not include $\\beta_0$. For different values\n",
+    "of $\\lambda$, this may lead to different MSE values. \n",
+    "\n",
+    "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5de45190",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f53e8c1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "but when we take out the intercept, this equation becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b0ffc74",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10305919",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "For Lasso regression we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b2ce8814",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e08c9fd7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It means that, when scaling the design matrix and the outputs/targets,\n",
+    "by subtracting the mean values, we have an optimization problem which\n",
+    "is not penalized by the intercept. The MSE value can then be smaller\n",
+    "since it focuses only on the remaining quantities. If we however bring\n",
+    "back the intercept, we will get a MSE which then contains the\n",
+    "intercept.\n",
+    "\n",
+    "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known  vanilla data set."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "5b7e1c63",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(3155)\n",
+    "\n",
+    "n = 100\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+    "\n",
+    "Maxpolydegree = 20\n",
+    "X = np.zeros((n,Maxpolydegree))\n",
+    "#We include explicitely the intercept column\n",
+    "for degree in range(Maxpolydegree):\n",
+    "    X[:,degree] = x**degree\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "p = Maxpolydegree\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 6\n",
+    "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 2, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+    "    # Note: we include the intercept column and no scaling\n",
+    "    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ytildeOwnRidge = X_train @ OwnRidgeBeta\n",
+    "    ypredictOwnRidge = X_test @ OwnRidgeBeta\n",
+    "    ytildeRidge = RegRidge.predict(X_train)\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    print(\"Beta values for own Ridge implementation\")\n",
+    "    print(OwnRidgeBeta)\n",
+    "    print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+    "    print(RegRidge.coef_)\n",
+    "    print(\"MSE values for own Ridge implementation\")\n",
+    "    print(MSEOwnRidgePredict[i])\n",
+    "    print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+    "    print(MSERidgePredict[i])\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
+    "\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bbf9639e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
+    "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
+    "What happens if we do not include the intercept in our fit?\n",
+    "Let us see how we can change this code by zero centering."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "3a82ee91",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "from sklearn.preprocessing import StandardScaler\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(315)\n",
+    "\n",
+    "n = 100\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+    "\n",
+    "Maxpolydegree = 20\n",
+    "X = np.zeros((n,Maxpolydegree-1))\n",
+    "\n",
+    "for degree in range(1,Maxpolydegree): #No intercept column\n",
+    "    X[:,degree-1] = x**(degree)\n",
+    "\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
+    "X_train_mean = np.mean(X_train,axis=0)\n",
+    "#Center by removing mean from each feature\n",
+    "X_train_scaled = X_train - X_train_mean \n",
+    "X_test_scaled = X_test - X_train_mean\n",
+    "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n",
+    "#Remove the intercept from the training data.\n",
+    "y_scaler = np.mean(y_train)           \n",
+    "y_train_scaled = y_train - y_scaler   \n",
+    "\n",
+    "p = Maxpolydegree-1\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 6\n",
+    "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "\n",
+    "lambdas = np.logspace(-4, 2, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n",
+    "    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n",
+    "    #Add intercept to prediction\n",
+    "    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
+    "    RegRidge = linear_model.Ridge(lmb)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    print(\"Beta values for own Ridge implementation\")\n",
+    "    print(OwnRidgeBeta) #Intercept is given by mean of target variable\n",
+    "    print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+    "    print(RegRidge.coef_)\n",
+    "    print('Intercept from own implementation:')\n",
+    "    print(intercept_)\n",
+    "    print('Intercept from Scikit-Learn Ridge implementation')\n",
+    "    print(RegRidge.intercept_)\n",
+    "    print(\"MSE values for own Ridge implementation\")\n",
+    "    print(MSEOwnRidgePredict[i])\n",
+    "    print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+    "    print(MSERidgePredict[i])\n",
+    "\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c061da66",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see here, when compared to the code which includes explicitely the\n",
+    "intercept column, that our MSE value is actually smaller. This is\n",
+    "because the regularization term does not include the intercept value\n",
+    "$\\beta_0$ in the fitting.  This applies to Lasso regularization as\n",
+    "well.  It means that our optimization is now done only with the\n",
+    "centered matrix and/or vector that enter the fitting procedure."
+   ]
   }
  ],
  "metadata": {},
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 8a9f7b94f..005f6fa10 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 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, 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"]], "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"]], "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"]], "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"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex functions": [[13, "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"]], "Deriving the  Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"]], "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"]], "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"]], "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]], "Expectation values": [[21, "expectation-values"]], "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"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "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"]], "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 boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "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"]], "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"]], "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"]], "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"]], "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"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [25, "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"]], "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"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "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"]], "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, 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 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</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 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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, 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"]], "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"]], "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"]], "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"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex functions": [[13, "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"]], "Deriving the  Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"]], "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"]], "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"]], "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]], "Expectation values": [[21, "expectation-values"]], "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"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "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"]], "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 boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "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"]], "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"]], "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"]], "Linear Regression, basic elements": [[0, 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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", "Weekly Exercises 3", "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, 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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index 353391878..3f4cb1e47 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -440,6 +440,8 @@ document.write(`
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#interpreting-the-ridge-results">Interpreting the Ridge results</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-interpretations">More interpretations</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-the-lasso-regression-equations">Deriving the  Lasso Regression Equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-exercises-week-35">Material for exercises week 35</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#important-technicalities-more-on-rescaling-data">Important technicalities: More on Rescaling data</a></li>
 </ul>
             </nav>
         </div>
@@ -471,10 +473,8 @@ doconce format html week35.do.txt --no_mako -->
 <h3>Reading recommendations:<a class="headerlink" href="#reading-recommendations" title="Link to this heading">#</a></h3>
 <ol class="arabic simple">
 <li><p>These lecture notes</p></li>
-</ol>
-<!-- o [Video of lecture](https://youtu.be/VKakN-e4aUA) -->
-<!-- o [Video for exercises week 35](https://youtu.be/yiY0OltU1s8) -->
-<ol class="arabic simple" start="2">
+<li><p>Video of lecture at <a class="reference external" href="https://youtu.be/2mvizAQFST8">https://youtu.be/2mvizAQFST8</a></p></li>
+<li><p>Whiteboard notes  at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2025/FYSSTKweek35.pdf">CompPhysics/MachineLearning</a></p></li>
 <li><p>Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra</p></li>
 <li><p>Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.</p></li>
 <li><p>For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL”<a class="reference external" href="https://mml-book.github.io/">https://mml-book.github.io/</a>” (section 5.5 on derivatives) is very useful for exercise 1 this coming week.</p></li>
@@ -2354,6 +2354,452 @@ C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsym
 \]</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 using for example the Python package <a class="reference external" href="https://cvxopt.org/">CVXOPT</a>. We will discuss how to code LASSO regression next week, when we have introduced gradient methods.</p>
 </section>
+<section id="material-for-exercises-week-35">
+<h2>Material for exercises week 35<a class="headerlink" href="#material-for-exercises-week-35" title="Link to this heading">#</a></h2>
+</section>
+<section id="important-technicalities-more-on-rescaling-data">
+<h2>Important technicalities: More on Rescaling data<a class="headerlink" href="#important-technicalities-more-on-rescaling-data" title="Link to this heading">#</a></h2>
+<p>When you are comparing your own code with for example <strong>Scikit-Learn</strong>’s
+library, there are some technicalities to keep in mind.  The examples
+here demonstrate some of these aspects with potential pitfalls.</p>
+<p>The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend  confuse us.</p>
+<p>The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library <strong>Scikit-Learn</strong> (when not shrinking <span class="math notranslate nohighlight">\(\beta_0\)</span>) for the unknown parameters
+<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>, are derived under the assumption that both <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> and
+<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> are zero centered, that is we subtract the mean values.</p>
+<p>If our predictors represent different scales, then it is important to
+standardize the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.</p>
+<p>The
+<a class="reference external" href="https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html">Standardscaler</a>
+function in <strong>Scikit-Learn</strong> does this for us.  For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform  a
+survey of your data, with a critical assessment of them in case you need to scale the data.</p>
+<p>If you need to scale the data, not doing so will give an <em>unfair</em>
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.</p>
+<p>The <strong>Scikit-Learn</strong> site <a class="reference external" href="https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section">https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section</a> has a good discussion of different ways of preprocessing data.</p>
+<p>Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.</p>
+<p>Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>&quot;&quot;&quot;
+#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+&quot;&quot;&quot;
+</pre></div>
+</div>
+</div>
+</div>
+<p>Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as <em>zero centering</em>. For
+simplicity, we will focus on  ordinary regression, as done in the above example.</p>
+<p>The cost/loss function  for regression is</p>
+<div class="math notranslate nohighlight">
+\[
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+\]</div>
+<p>Recall also that we use the squared value. This expression can lead to an
+increased penalty for higher differences between predicted and
+output/target values.</p>
+<p>What we have done is to single out the <span class="math notranslate nohighlight">\(\beta_0\)</span> term in the
+definition of the mean squared error (MSE).  The design matrix <span class="math notranslate nohighlight">\(X\)</span>
+does in this case not contain any intercept column.  When we take the
+derivative with respect to <span class="math notranslate nohighlight">\(\beta_0\)</span>, we want the derivative to obey</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{\partial C}{\partial \beta_j} = 0,
+\]</div>
+<p>for all <span class="math notranslate nohighlight">\(j\)</span>. For <span class="math notranslate nohighlight">\(\beta_0\)</span> we have</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+\]</div>
+<p>Multiplying away the constant <span class="math notranslate nohighlight">\(2/n\)</span>, we obtain</p>
+<div class="math notranslate nohighlight">
+\[
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
+\]</div>
+<p>Let us specialize first to the case where we have only two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.
+Our result for <span class="math notranslate nohighlight">\(\beta_0\)</span> simplifies then to</p>
+<div class="math notranslate nohighlight">
+\[
+n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
+\]</div>
+<p>We obtain then</p>
+<div class="math notranslate nohighlight">
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
+\]</div>
+<p>If we define</p>
+<div class="math notranslate nohighlight">
+\[
+\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
+\]</div>
+<p>and the mean value of the outputs as</p>
+<div class="math notranslate nohighlight">
+\[
+\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
+\]</div>
+<p>we have</p>
+<div class="math notranslate nohighlight">
+\[
+\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
+\]</div>
+<p>In the general case with more parameters than <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>, we have</p>
+<div class="math notranslate nohighlight">
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
+\]</div>
+<p>We can rewrite the latter equation as</p>
+<div class="math notranslate nohighlight">
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
+\]</div>
+<p>where we have defined</p>
+<div class="math notranslate nohighlight">
+\[
+\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
+\]</div>
+<p>the mean value for all elements of the column vector <span class="math notranslate nohighlight">\(\boldsymbol{x}_j\)</span>.</p>
+<p>Replacing <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(y_i - y_i - \overline{\boldsymbol{y}}\)</span> and centering also our design matrix results in a cost function (in vector-matrix disguise)</p>
+<div class="math notranslate nohighlight">
+\[
+C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+\]</div>
+<p>If we minimize with respect to <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we have then</p>
+<div class="math notranslate nohighlight">
+\[
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}\)</span>
+and <span class="math notranslate nohighlight">\(\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}\)</span>.</p>
+<p>For Ridge regression we need to add <span class="math notranslate nohighlight">\(\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}\)</span> to the cost function and get then</p>
+<div class="math notranslate nohighlight">
+\[
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+\]</div>
+<p>What does this mean? And why do we insist on all this? Let us look at some examples.</p>
+<p>This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (<em>code example thanks to  Øyvind Sigmundson Schøyen</em>). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
+import matplotlib.pyplot as plt
+
+from sklearn.linear_model import LinearRegression
+
+
+np.random.seed(2021)
+
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+def fit_beta(X, y):
+    return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
+) + 0.1 * np.random.normal(size=len(x))
+
+degree = 3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p in range(degree):
+    X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f&quot;True beta: {true_beta}&quot;)
+print(f&quot;Fitted beta: {beta}&quot;)
+print(f&quot;Sklearn fitted beta: {skl.coef_}&quot;)
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f&quot;MSE with intercept column&quot;)
+print(MSE(y,ypredictOwn))
+print(f&quot;MSE with intercept column from SKL&quot;)
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label=&quot;Data&quot;)
+plt.plot(x, X @ beta, label=&quot;Fit&quot;)
+plt.plot(x, skl.predict(X), label=&quot;Sklearn (fit_intercept=False)&quot;)
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree - 1))
+
+for p in range(degree - 1):
+    X[:, p] = x ** (p + 1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f&quot;Manual intercept: {intercept}&quot;)
+print(f&quot;Fitted beta (without intercept): {beta}&quot;)
+print(f&quot;Sklearn intercept: {skl.intercept_}&quot;)
+print(f&quot;Sklearn fitted beta (without intercept): {skl.coef_}&quot;)
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f&quot;MSE with Manual intercept&quot;)
+print(MSE(y,ypredictOwn+intercept))
+print(f&quot;MSE with Sklearn intercept&quot;)
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, &quot;--&quot;, label=&quot;Fit (manual intercept)&quot;)
+plt.plot(x, skl.predict(X), &quot;--&quot;, label=&quot;Sklearn (fit_intercept=True)&quot;)
+plt.grid()
+plt.legend()
+
+plt.show()
+</pre></div>
+</div>
+</div>
+</div>
+<p>The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the <span class="math notranslate nohighlight">\(y\)</span>-axis (for a one-dimensional case).</p>
+<p>Printing the MSE, we see first that both methods give the same MSE, as
+they should.  However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include <span class="math notranslate nohighlight">\(\beta_0\)</span>. For different values
+of <span class="math notranslate nohighlight">\(\lambda\)</span>, this may lead to different MSE values.</p>
+<p>To remind the reader, the regularization term, with the intercept in Ridge regression, is given by</p>
+<div class="math notranslate nohighlight">
+\[
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
+\]</div>
+<p>but when we take out the intercept, this equation becomes</p>
+<div class="math notranslate nohighlight">
+\[
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
+\]</div>
+<p>For Lasso regression we have</p>
+<div class="math notranslate nohighlight">
+\[
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
+\]</div>
+<p>It means that, when scaling the design matrix and the outputs/targets,
+by subtracting the mean values, we have an optimization problem which
+is not penalized by the intercept. The MSE value can then be smaller
+since it focuses only on the remaining quantities. If we however bring
+back the intercept, we will get a MSE which then contains the
+intercept.</p>
+<p>Armed with this wisdom, we attempt first to simply set the intercept equal to <strong>False</strong> in our implementation of Ridge regression for our well-known  vanilla data set.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>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]
+    OwnRidgeBeta = 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 @ OwnRidgeBeta
+    ypredictOwnRidge = X_test @ OwnRidgeBeta
+    ytildeRidge = RegRidge.predict(X_train)
+    ypredictRidge = RegRidge.predict(X_test)
+    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    print(&quot;Beta values for own Ridge implementation&quot;)
+    print(OwnRidgeBeta)
+    print(&quot;Beta values for Scikit-Learn Ridge implementation&quot;)
+    print(RegRidge.coef_)
+    print(&quot;MSE values for own Ridge implementation&quot;)
+    print(MSEOwnRidgePredict[i])
+    print(&quot;MSE values for Scikit-Learn Ridge implementation&quot;)
+    print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, &#39;r&#39;, label = &#39;MSE own Ridge Test&#39;)
+plt.plot(np.log10(lambdas), MSERidgePredict, &#39;g&#39;, label = &#39;MSE Ridge Test&#39;)
+
+plt.xlabel(&#39;log10(lambda)&#39;)
+plt.ylabel(&#39;MSE&#39;)
+plt.legend()
+plt.show()
+</pre></div>
+</div>
+</div>
+</div>
+<p>The results here agree when we force <strong>Scikit-Learn</strong>’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?
+Let us see how we can change this code by zero centering.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>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
+from sklearn.preprocessing import StandardScaler
+
+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(315)
+
+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-1))
+
+for degree in range(1,Maxpolydegree): #No intercept column
+    X[:,degree-1] = 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)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean 
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)           
+y_train_scaled = y_train - y_scaler   
+
+p = Maxpolydegree-1
+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]
+    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
+    #Add intercept to prediction
+    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler 
+    RegRidge = linear_model.Ridge(lmb)
+    RegRidge.fit(X_train,y_train)
+    ypredictRidge = RegRidge.predict(X_test)
+    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    print(&quot;Beta values for own Ridge implementation&quot;)
+    print(OwnRidgeBeta) #Intercept is given by mean of target variable
+    print(&quot;Beta values for Scikit-Learn Ridge implementation&quot;)
+    print(RegRidge.coef_)
+    print(&#39;Intercept from own implementation:&#39;)
+    print(intercept_)
+    print(&#39;Intercept from Scikit-Learn Ridge implementation&#39;)
+    print(RegRidge.intercept_)
+    print(&quot;MSE values for own Ridge implementation&quot;)
+    print(MSEOwnRidgePredict[i])
+    print(&quot;MSE values for Scikit-Learn Ridge implementation&quot;)
+    print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, &#39;b--&#39;, label = &#39;MSE own Ridge Test&#39;)
+plt.plot(np.log10(lambdas), MSERidgePredict, &#39;g--&#39;, label = &#39;MSE SL Ridge Test&#39;)
+plt.xlabel(&#39;log10(lambda)&#39;)
+plt.ylabel(&#39;MSE&#39;)
+plt.legend()
+plt.show()
+</pre></div>
+</div>
+</div>
+</div>
+<p>We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+<span class="math notranslate nohighlight">\(\beta_0\)</span> in the fitting.  This applies to Lasso regularization as
+well.  It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure.</p>
+</section>
 </section>
 
     <script type="text/x-thebe-config">
@@ -2484,6 +2930,8 @@ C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsym
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#interpreting-the-ridge-results">Interpreting the Ridge results</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-interpretations">More interpretations</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-the-lasso-regression-equations">Deriving the  Lasso Regression Equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-exercises-week-35">Material for exercises week 35</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#important-technicalities-more-on-rescaling-data">Important technicalities: More on Rescaling data</a></li>
 </ul>
   </nav></div>
 
diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
index d1052d6cf..cf1481a67 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "f0622ebb",
+   "id": "d3666266",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3fe98f20",
+   "id": "f9417425",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8e604e3c",
+   "id": "aaba55da",
    "metadata": {
     "editable": true
    },
@@ -49,7 +49,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c7f0b40a",
+   "id": "c4e2f992",
    "metadata": {
     "editable": true
    },
@@ -57,19 +57,21 @@
     "### Reading recommendations:\n",
     "\n",
     "1. These lecture notes\n",
-    "<!-- o [Video of lecture](https://youtu.be/VKakN-e4aUA) -->\n",
-    "<!-- o [Video for exercises week 35](https://youtu.be/yiY0OltU1s8) -->\n",
     "\n",
-    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n",
+    "2. Video of lecture at <https://youtu.be/2mvizAQFST8>\n",
     "\n",
-    "3. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n",
+    "3. Whiteboard notes  at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2025/FYSSTKweek35.pdf>\n",
     "\n",
-    "4. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week."
+    "4. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n",
+    "\n",
+    "5. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n",
+    "\n",
+    "6. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "507709a4",
+   "id": "f0040522",
    "metadata": {
     "editable": true
    },
@@ -102,7 +104,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bb3b5f67",
+   "id": "d64fcc78",
    "metadata": {
     "editable": true
    },
@@ -118,7 +120,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "14444be5",
+   "id": "a1ada6ce",
    "metadata": {
     "editable": true
    },
@@ -130,7 +132,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4edc5258",
+   "id": "5645817b",
    "metadata": {
     "editable": true
    },
@@ -140,7 +142,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d6f9c43",
+   "id": "30bee0bb",
    "metadata": {
     "editable": true
    },
@@ -152,7 +154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb52f7f5",
+   "id": "e6e20e11",
    "metadata": {
     "editable": true
    },
@@ -173,7 +175,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fff6b39",
+   "id": "7bfd72c1",
    "metadata": {
     "editable": true
    },
@@ -185,7 +187,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "54a4cabb",
+   "id": "69275ea0",
    "metadata": {
     "editable": true
    },
@@ -198,7 +200,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d383a507",
+   "id": "976bb1cb",
    "metadata": {
     "editable": true
    },
@@ -210,7 +212,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "498fe31e",
+   "id": "0d85580a",
    "metadata": {
     "editable": true
    },
@@ -222,7 +224,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7bbbdcd2",
+   "id": "5a8ac559",
    "metadata": {
     "editable": true
    },
@@ -232,7 +234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "405a61ac",
+   "id": "64f74e34",
    "metadata": {
     "editable": true
    },
@@ -244,7 +246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7c07f57",
+   "id": "8d503698",
    "metadata": {
     "editable": true
    },
@@ -257,7 +259,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6054782f",
+   "id": "787c6923",
    "metadata": {
     "editable": true
    },
@@ -269,7 +271,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b55104f",
+   "id": "54c2b6dc",
    "metadata": {
     "editable": true
    },
@@ -279,7 +281,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4635525e",
+   "id": "a8114ec8",
    "metadata": {
     "editable": true
    },
@@ -291,7 +293,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "11f6ffc7",
+   "id": "650e695e",
    "metadata": {
     "editable": true
    },
@@ -303,7 +305,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "820ddc3f",
+   "id": "885a3110",
    "metadata": {
     "editable": true
    },
@@ -314,7 +316,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa4b864",
+   "id": "9fbaf50f",
    "metadata": {
     "editable": true
    },
@@ -326,7 +328,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c542e149",
+   "id": "2376fa59",
    "metadata": {
     "editable": true
    },
@@ -345,7 +347,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7be13194",
+   "id": "937df3c9",
    "metadata": {
     "editable": true
    },
@@ -358,7 +360,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aad2641f",
+   "id": "e066a9bc",
    "metadata": {
     "editable": true
    },
@@ -368,7 +370,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9dfe7e22",
+   "id": "af821847",
    "metadata": {
     "editable": true
    },
@@ -380,7 +382,7 @@
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@@ -390,7 +392,7 @@
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@@ -402,7 +404,7 @@
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@@ -412,7 +414,7 @@
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@@ -424,7 +426,7 @@
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@@ -435,7 +437,7 @@
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@@ -447,7 +449,7 @@
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@@ -457,7 +459,7 @@
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@@ -469,7 +471,7 @@
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@@ -479,7 +481,7 @@
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@@ -491,7 +493,7 @@
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@@ -511,7 +513,7 @@
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@@ -538,7 +540,7 @@
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@@ -550,7 +552,7 @@
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@@ -562,7 +564,7 @@
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@@ -578,7 +580,7 @@
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@@ -596,7 +598,7 @@
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@@ -608,7 +610,7 @@
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@@ -620,7 +622,7 @@
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@@ -643,7 +645,7 @@
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@@ -653,7 +655,7 @@
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@@ -665,7 +667,7 @@
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@@ -679,7 +681,7 @@
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@@ -691,7 +693,7 @@
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@@ -703,7 +705,7 @@
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@@ -715,7 +717,7 @@
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@@ -725,7 +727,7 @@
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@@ -737,7 +739,7 @@
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@@ -749,7 +751,7 @@
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@@ -761,7 +763,7 @@
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@@ -775,7 +777,7 @@
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@@ -787,7 +789,7 @@
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@@ -799,7 +801,7 @@
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@@ -811,7 +813,7 @@
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@@ -821,7 +823,7 @@
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@@ -833,7 +835,7 @@
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@@ -843,7 +845,7 @@
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@@ -855,7 +857,7 @@
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@@ -865,7 +867,7 @@
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@@ -877,7 +879,7 @@
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@@ -889,7 +891,7 @@
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@@ -901,7 +903,7 @@
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@@ -916,7 +918,7 @@
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@@ -928,7 +930,7 @@
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@@ -938,7 +940,7 @@
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@@ -950,7 +952,7 @@
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@@ -960,7 +962,7 @@
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@@ -972,7 +974,7 @@
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@@ -982,7 +984,7 @@
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@@ -994,7 +996,7 @@
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@@ -1006,7 +1008,7 @@
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@@ -1018,7 +1020,7 @@
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@@ -1028,7 +1030,7 @@
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@@ -1040,7 +1042,7 @@
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@@ -1053,7 +1055,7 @@
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@@ -1065,7 +1067,7 @@
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@@ -1075,7 +1077,7 @@
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@@ -1087,7 +1089,7 @@
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@@ -1097,7 +1099,7 @@
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@@ -1109,7 +1111,7 @@
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@@ -1119,7 +1121,7 @@
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@@ -1131,7 +1133,7 @@
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@@ -1141,7 +1143,7 @@
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@@ -1153,7 +1155,7 @@
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@@ -1163,7 +1165,7 @@
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@@ -1175,7 +1177,7 @@
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@@ -1191,7 +1193,7 @@
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@@ -1203,7 +1205,7 @@
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@@ -1213,7 +1215,7 @@
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@@ -1225,7 +1227,7 @@
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@@ -1243,7 +1245,7 @@
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@@ -1255,7 +1257,7 @@
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@@ -1267,7 +1269,7 @@
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@@ -1277,7 +1279,7 @@
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@@ -1289,7 +1291,7 @@
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@@ -1299,7 +1301,7 @@
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@@ -1311,7 +1313,7 @@
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@@ -1321,7 +1323,7 @@
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@@ -1335,7 +1337,7 @@
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@@ -1347,7 +1349,7 @@
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@@ -1358,7 +1360,7 @@
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@@ -1371,7 +1373,7 @@
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@@ -1398,7 +1400,7 @@
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@@ -1409,7 +1411,7 @@
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@@ -1422,7 +1424,7 @@
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@@ -1436,7 +1438,7 @@
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@@ -1449,7 +1451,7 @@
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@@ -1460,7 +1462,7 @@
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@@ -1472,7 +1474,7 @@
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@@ -1483,7 +1485,7 @@
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@@ -1499,7 +1501,7 @@
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@@ -1510,7 +1512,7 @@
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@@ -1524,7 +1526,7 @@
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@@ -1545,7 +1547,7 @@
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@@ -1556,7 +1558,7 @@
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@@ -1609,7 +1611,7 @@
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@@ -1620,7 +1622,7 @@
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@@ -1645,7 +1647,7 @@
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@@ -1657,7 +1659,7 @@
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@@ -1686,7 +1688,7 @@
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@@ -1711,7 +1713,7 @@
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@@ -1731,7 +1733,7 @@
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@@ -1758,7 +1760,7 @@
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@@ -1771,7 +1773,7 @@
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@@ -1783,7 +1785,7 @@
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@@ -1794,7 +1796,7 @@
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@@ -1810,7 +1812,7 @@
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@@ -1844,7 +1846,7 @@
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@@ -1854,7 +1856,7 @@
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@@ -1869,7 +1871,7 @@
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@@ -1881,7 +1883,7 @@
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@@ -1891,7 +1893,7 @@
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@@ -1907,7 +1909,7 @@
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@@ -1924,7 +1926,7 @@
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@@ -1937,7 +1939,7 @@
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@@ -1984,7 +1986,7 @@
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@@ -1998,7 +2000,7 @@
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@@ -2010,7 +2012,7 @@
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@@ -2022,7 +2024,7 @@
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@@ -2034,7 +2036,7 @@
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@@ -2044,7 +2046,7 @@
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@@ -2056,7 +2058,7 @@
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@@ -2066,7 +2068,7 @@
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@@ -2078,7 +2080,7 @@
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@@ -2089,7 +2091,7 @@
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@@ -2101,7 +2103,7 @@
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@@ -2113,7 +2115,7 @@
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@@ -2123,7 +2125,7 @@
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@@ -2135,7 +2137,7 @@
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@@ -2147,7 +2149,7 @@
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@@ -2157,7 +2159,7 @@
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@@ -2169,7 +2171,7 @@
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@@ -2179,7 +2181,7 @@
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@@ -2191,7 +2193,7 @@
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@@ -2201,7 +2203,7 @@
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@@ -2241,7 +2243,7 @@
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@@ -2258,7 +2260,7 @@
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@@ -2281,7 +2283,7 @@
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@@ -2298,7 +2300,7 @@
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@@ -2317,7 +2319,7 @@
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@@ -2328,7 +2330,7 @@
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@@ -2340,7 +2342,7 @@
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@@ -2358,7 +2360,7 @@
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@@ -2374,7 +2376,7 @@
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@@ -2386,7 +2388,7 @@
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@@ -2396,7 +2398,7 @@
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@@ -2409,7 +2411,7 @@
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@@ -2421,7 +2423,7 @@
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@@ -2431,7 +2433,7 @@
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@@ -2444,7 +2446,7 @@
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@@ -2454,7 +2456,7 @@
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@@ -2466,7 +2468,7 @@
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@@ -2479,7 +2481,7 @@
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@@ -2492,7 +2494,7 @@
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@@ -2504,7 +2506,7 @@
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@@ -2516,7 +2518,7 @@
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@@ -2526,7 +2528,7 @@
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@@ -2539,7 +2541,7 @@
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@@ -2551,7 +2553,7 @@
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@@ -2563,7 +2565,7 @@
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@@ -2575,7 +2577,7 @@
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@@ -2587,7 +2589,7 @@
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@@ -2601,7 +2603,7 @@
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@@ -2613,7 +2615,7 @@
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@@ -2623,7 +2625,7 @@
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@@ -2635,7 +2637,7 @@
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@@ -2647,7 +2649,7 @@
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@@ -2659,7 +2661,7 @@
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@@ -2671,7 +2673,7 @@
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@@ -2683,7 +2685,7 @@
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@@ -2706,7 +2708,7 @@
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@@ -2782,7 +2784,7 @@
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@@ -2794,7 +2796,7 @@
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@@ -2809,7 +2811,7 @@
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@@ -2821,7 +2823,7 @@
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@@ -2831,7 +2833,7 @@
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@@ -2843,7 +2845,7 @@
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@@ -2853,7 +2855,7 @@
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@@ -2865,7 +2867,7 @@
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@@ -2877,7 +2879,7 @@
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@@ -2892,7 +2894,7 @@
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@@ -2903,7 +2905,7 @@
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@@ -2923,7 +2925,7 @@
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@@ -2935,7 +2937,7 @@
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@@ -2945,7 +2947,7 @@
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@@ -2957,7 +2959,7 @@
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@@ -2986,7 +2988,7 @@
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@@ -3013,7 +3015,7 @@
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@@ -3024,7 +3026,7 @@
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@@ -3064,7 +3066,7 @@
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@@ -3081,7 +3083,7 @@
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@@ -3104,7 +3106,7 @@
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@@ -3118,7 +3120,7 @@
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@@ -3137,7 +3139,7 @@
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@@ -3147,7 +3149,7 @@
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@@ -3159,7 +3161,7 @@
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@@ -3173,7 +3175,7 @@
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@@ -3185,7 +3187,7 @@
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@@ -3195,7 +3197,7 @@
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@@ -3207,7 +3209,7 @@
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@@ -3224,7 +3226,7 @@
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@@ -3234,7 +3236,7 @@
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@@ -3250,7 +3252,7 @@
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@@ -3260,7 +3262,7 @@
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@@ -3276,7 +3278,7 @@
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@@ -3286,7 +3288,7 @@
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@@ -3302,7 +3304,7 @@
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@@ -3312,7 +3314,7 @@
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@@ -3329,7 +3331,7 @@
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@@ -3341,7 +3343,7 @@
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@@ -3353,7 +3355,7 @@
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@@ -3365,7 +3367,7 @@
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@@ -3375,7 +3377,7 @@
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@@ -3387,7 +3389,7 @@
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@@ -3399,7 +3401,7 @@
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@@ -3411,7 +3413,7 @@
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@@ -3421,7 +3423,7 @@
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@@ -3433,7 +3435,7 @@
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@@ -3443,7 +3445,7 @@
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@@ -3455,7 +3457,7 @@
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@@ -3472,7 +3474,7 @@
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@@ -3484,7 +3486,7 @@
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@@ -3496,7 +3498,7 @@
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@@ -3506,7 +3508,7 @@
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@@ -3518,7 +3520,7 @@
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@@ -3529,7 +3531,7 @@
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@@ -3541,7 +3543,7 @@
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@@ -3551,7 +3553,7 @@
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@@ -3563,7 +3565,7 @@
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@@ -3573,7 +3575,7 @@
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@@ -3585,7 +3587,7 @@
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@@ -3596,7 +3598,7 @@
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@@ -3608,7 +3610,7 @@
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@@ -3626,7 +3628,7 @@
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@@ -3642,7 +3644,7 @@
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@@ -3654,7 +3656,7 @@
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@@ -3666,7 +3668,7 @@
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@@ -3678,7 +3680,7 @@
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@@ -3691,7 +3693,7 @@
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@@ -3707,7 +3709,7 @@
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@@ -3721,7 +3723,7 @@
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@@ -3731,7 +3733,7 @@
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@@ -3743,7 +3745,7 @@
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@@ -3753,7 +3755,7 @@
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@@ -3765,7 +3767,7 @@
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@@ -3775,7 +3777,7 @@
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@@ -3789,7 +3791,7 @@
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@@ -3806,7 +3808,7 @@
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@@ -3822,7 +3824,7 @@
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@@ -3834,7 +3836,7 @@
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@@ -3847,7 +3849,7 @@
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@@ -3861,7 +3863,7 @@
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@@ -3871,7 +3873,7 @@
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@@ -3884,7 +3886,7 @@
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@@ -3903,7 +3905,7 @@
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@@ -3915,7 +3917,7 @@
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@@ -3927,7 +3929,7 @@
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@@ -3937,7 +3939,7 @@
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@@ -3949,7 +3951,7 @@
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@@ -3962,7 +3964,7 @@
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@@ -3981,7 +3983,7 @@
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@@ -3991,7 +3993,7 @@
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@@ -4010,7 +4012,7 @@
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@@ -4028,7 +4030,7 @@
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@@ -4042,7 +4044,7 @@
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@@ -4057,7 +4059,7 @@
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@@ -4078,7 +4080,7 @@
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@@ -4095,7 +4097,7 @@
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@@ -4127,7 +4129,7 @@
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@@ -4141,7 +4143,7 @@
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@@ -4154,7 +4156,7 @@
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@@ -4179,7 +4181,7 @@
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@@ -4191,7 +4193,7 @@
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@@ -4203,7 +4205,7 @@
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@@ -4213,7 +4215,7 @@
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@@ -4230,7 +4232,7 @@
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@@ -4240,7 +4242,7 @@
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@@ -4255,7 +4257,7 @@
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@@ -4265,7 +4267,7 @@
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@@ -4279,7 +4281,7 @@
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@@ -4291,7 +4293,7 @@
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@@ -4303,7 +4305,7 @@
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@@ -4315,7 +4317,7 @@
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@@ -4325,7 +4327,7 @@
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@@ -4337,7 +4339,7 @@
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@@ -4347,7 +4349,7 @@
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@@ -4364,7 +4366,7 @@
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@@ -4374,7 +4376,7 @@
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@@ -4386,7 +4388,7 @@
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@@ -4396,7 +4398,7 @@
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@@ -4408,7 +4410,7 @@
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@@ -4422,7 +4424,7 @@
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@@ -4434,7 +4436,7 @@
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@@ -4456,7 +4458,7 @@
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@@ -4468,7 +4470,7 @@
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@@ -4483,7 +4485,7 @@
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@@ -4495,7 +4497,7 @@
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@@ -4507,7 +4509,7 @@
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@@ -4517,7 +4519,7 @@
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@@ -4529,7 +4531,7 @@
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@@ -4539,7 +4541,7 @@
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@@ -4551,7 +4553,7 @@
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@@ -4561,7 +4563,7 @@
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@@ -4573,7 +4575,7 @@
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@@ -4590,7 +4592,7 @@
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-   "id": "a2af3e04",
+   "id": "43987fb6",
    "metadata": {
     "editable": true
    },
@@ -4603,7 +4605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0342c08c",
+   "id": "78741f5d",
    "metadata": {
     "editable": true
    },
@@ -4615,7 +4617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bdd5faeb",
+   "id": "ecf46f39",
    "metadata": {
     "editable": true
    },
@@ -4625,7 +4627,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35873dc6",
+   "id": "f98d1ea2",
    "metadata": {
     "editable": true
    },
@@ -4638,7 +4640,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ff6ab78",
+   "id": "56a4e22a",
    "metadata": {
     "editable": true
    },
@@ -4648,7 +4650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77286b5f",
+   "id": "3087fc2c",
    "metadata": {
     "editable": true
    },
@@ -4660,7 +4662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0ade2616",
+   "id": "8d386a35",
    "metadata": {
     "editable": true
    },
@@ -4673,7 +4675,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da22a168",
+   "id": "c8d58182",
    "metadata": {
     "editable": true
    },
@@ -4686,7 +4688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "caa18da2",
+   "id": "15fe7dd6",
    "metadata": {
     "editable": true
    },
@@ -4698,7 +4700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f31c8084",
+   "id": "042bbee6",
    "metadata": {
     "editable": true
    },
@@ -4710,7 +4712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "53dc71a2",
+   "id": "911f8e5c",
    "metadata": {
     "editable": true
    },
@@ -4720,7 +4722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "753442d5",
+   "id": "40d4bbda",
    "metadata": {
     "editable": true
    },
@@ -4733,7 +4735,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e0b2f58a",
+   "id": "8a00d027",
    "metadata": {
     "editable": true
    },
@@ -4745,7 +4747,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e4af8e7e",
+   "id": "ff109317",
    "metadata": {
     "editable": true
    },
@@ -4757,7 +4759,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e353a3a",
+   "id": "4e5859eb",
    "metadata": {
     "editable": true
    },
@@ -4774,7 +4776,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "66a4056b",
+   "id": "75a8694b",
    "metadata": {
     "editable": true
    },
@@ -4786,7 +4788,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea5b3d9f",
+   "id": "80c47724",
    "metadata": {
     "editable": true
    },
@@ -4796,7 +4798,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db50585a",
+   "id": "6b385376",
    "metadata": {
     "editable": true
    },
@@ -4808,7 +4810,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afd3cd57",
+   "id": "32963935",
    "metadata": {
     "editable": true
    },
@@ -4818,7 +4820,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6606774b",
+   "id": "de8c6756",
    "metadata": {
     "editable": true
    },
@@ -4830,7 +4832,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bcfd3adb",
+   "id": "76b587d1",
    "metadata": {
     "editable": true
    },
@@ -4842,7 +4844,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "46d7227b",
+   "id": "8a1fc10c",
    "metadata": {
     "editable": true
    },
@@ -4858,7 +4860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a214b881",
+   "id": "a6377518",
    "metadata": {
     "editable": true
    },
@@ -4870,7 +4872,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "727c87de",
+   "id": "367d1837",
    "metadata": {
     "editable": true
    },
@@ -4882,7 +4884,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5083858e",
+   "id": "2fa779fe",
    "metadata": {
     "editable": true
    },
@@ -4892,7 +4894,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "973ad2fa",
+   "id": "e0f99ac6",
    "metadata": {
     "editable": true
    },
@@ -4904,7 +4906,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fbd0028",
+   "id": "d2134909",
    "metadata": {
     "editable": true
    },
@@ -4914,7 +4916,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bee26901",
+   "id": "09b0f22e",
    "metadata": {
     "editable": true
    },
@@ -4926,7 +4928,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da771d96",
+   "id": "1d1987fc",
    "metadata": {
     "editable": true
    },
@@ -4943,7 +4945,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ecb8a62",
+   "id": "16f94ea4",
    "metadata": {
     "editable": true
    },
@@ -4955,7 +4957,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5cf08e3d",
+   "id": "21c224fd",
    "metadata": {
     "editable": true
    },
@@ -4967,7 +4969,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d984ce12",
+   "id": "80c9611e",
    "metadata": {
     "editable": true
    },
@@ -4977,7 +4979,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "837a69b7",
+   "id": "ca8706ca",
    "metadata": {
     "editable": true
    },
@@ -4989,7 +4991,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21dde5ad",
+   "id": "6bcce614",
    "metadata": {
     "editable": true
    },
@@ -4999,7 +5001,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d49c112",
+   "id": "857b9ab1",
    "metadata": {
     "editable": true
    },
@@ -5011,7 +5013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b4eabcf",
+   "id": "bb8c1546",
    "metadata": {
     "editable": true
    },
@@ -5021,7 +5023,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "28dcb3c8",
+   "id": "2f7636ba",
    "metadata": {
     "editable": true
    },
@@ -5033,7 +5035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f315a306",
+   "id": "07e308e9",
    "metadata": {
     "editable": true
    },
@@ -5043,7 +5045,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b4f990c",
+   "id": "7a20da6b",
    "metadata": {
     "editable": true
    },
@@ -5055,13 +5057,856 @@
   },
   {
    "cell_type": "markdown",
-   "id": "197daa81",
+   "id": "404a4814",
    "metadata": {
     "editable": true
    },
    "source": [
     "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 using for example the Python package [CVXOPT](https://cvxopt.org/). We will discuss how to code LASSO regression next week, when we have introduced gradient methods."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c47881af",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for exercises week 35"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de625607",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Important technicalities: More on Rescaling data\n",
+    "\n",
+    "When you are comparing your own code with for example **Scikit-Learn**'s\n",
+    "library, there are some technicalities to keep in mind.  The examples\n",
+    "here demonstrate some of these aspects with potential pitfalls.\n",
+    "\n",
+    "The discussion here focuses on the role of the intercept, how we can\n",
+    "set up the design matrix, what scaling we should use and other topics\n",
+    "which tend  confuse us.\n",
+    "\n",
+    "The intercept can be interpreted as the expected value of our\n",
+    "target/output variables when all other predictors are set to zero.\n",
+    "Thus, if we cannot assume that the expected outputs/targets are zero\n",
+    "when all predictors are zero (the columns in the design matrix), it\n",
+    "may be a bad idea to implement a model which penalizes the intercept.\n",
+    "Furthermore, in for example Ridge and Lasso regression, the default solutions\n",
+    "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n",
+    "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
+    "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
+    "\n",
+    "If our predictors represent different scales, then it is important to\n",
+    "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
+    "column from the corresponding column and dividing the column with its\n",
+    "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n",
+    "the results may differ. \n",
+    "\n",
+    "The\n",
+    "[Standardscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n",
+    "function in **Scikit-Learn** does this for us.  For the data sets we\n",
+    "have been studying in our various examples, the data are in many cases\n",
+    "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform  a\n",
+    "survey of your data, with a critical assessment of them in case you need to scale the data.\n",
+    "\n",
+    "If you need to scale the data, not doing so will give an *unfair*\n",
+    "penalization of the parameters since their magnitude depends on the\n",
+    "scale of their corresponding predictor.\n",
+    "\n",
+    "The **Scikit-Learn** site <https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section> has a good discussion of different ways of preprocessing data.\n",
+    "\n",
+    "Suppose as an example that you \n",
+    "you have an input variable given by the heights of different persons.\n",
+    "Human height might be measured in inches or meters or\n",
+    "kilometers. If measured in kilometers, a standard linear regression\n",
+    "model with this predictor would probably give a much bigger\n",
+    "coefficient term, than if measured in millimeters.\n",
+    "This can clearly lead to problems in evaluating the cost/loss functions.\n",
+    "\n",
+    "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
+    "on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "c00a9805",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "\"\"\"\n",
+    "#Model training, we compute the mean value of y and X\n",
+    "y_train_mean = np.mean(y_train)\n",
+    "X_train_mean = np.mean(X_train,axis=0)\n",
+    "X_train = X_train - X_train_mean\n",
+    "y_train = y_train - y_train_mean\n",
+    "\n",
+    "# The we fit our model with the training data\n",
+    "trained_model = some_model.fit(X_train,y_train)\n",
+    "\n",
+    "\n",
+    "#Model prediction, we need also to transform our data set used for the prediction.\n",
+    "X_test = X_test - X_train_mean #Use mean from training data\n",
+    "y_pred = trained_model(X_test)\n",
+    "y_pred = y_pred + y_train_mean\n",
+    "\"\"\""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b3fb9183",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us try to understand what this may imply mathematically when we\n",
+    "subtract the mean values, also known as *zero centering*. For\n",
+    "simplicity, we will focus on  ordinary regression, as done in the above example.\n",
+    "\n",
+    "The cost/loss function  for regression is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07a54955",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1ac9c31d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Recall also that we use the squared value. This expression can lead to an\n",
+    "increased penalty for higher differences between predicted and\n",
+    "output/target values.\n",
+    "\n",
+    "What we have done is to single out the $\\beta_0$ term in the\n",
+    "definition of the mean squared error (MSE).  The design matrix $X$\n",
+    "does in this case not contain any intercept column.  When we take the\n",
+    "derivative with respect to $\\beta_0$, we want the derivative to obey"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ee6c6a4c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "beb40c81",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "for all $j$. For $\\beta_0$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b051946e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "885a6b08",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying away the constant $2/n$, we obtain"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5774d6be",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "73ff6979",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
+    "Our result for $\\beta_0$ simplifies then to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "792128bf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2be70ed5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We obtain then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d0c4ac1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2d4a5936",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If we define"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d30abc39",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7f945f8b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and the mean value of the outputs as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dff5bf57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f222bcca",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f1658ccd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "394ef3c4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f1dd1af",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0cafc2d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We can rewrite the latter equation as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2c878f98",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1dab41db",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have defined"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cf2f2908",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "20bac2a4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
+    "\n",
+    "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0285afd3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ffff8823",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eea815a7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0d935c60",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+    "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
+    "\n",
+    "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb4eabb8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1da18d81",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
+    "\n",
+    "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to  Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
+    "Note also that we do not split the data into training and test."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "a11f0699",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "from sklearn.linear_model import LinearRegression\n",
+    "\n",
+    "\n",
+    "np.random.seed(2021)\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "def fit_beta(X, y):\n",
+    "    return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
+    "\n",
+    "\n",
+    "true_beta = [2, 0.5, 3.7]\n",
+    "\n",
+    "x = np.linspace(0, 1, 11)\n",
+    "y = np.sum(\n",
+    "    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n",
+    ") + 0.1 * np.random.normal(size=len(x))\n",
+    "\n",
+    "degree = 3\n",
+    "X = np.zeros((len(x), degree))\n",
+    "\n",
+    "# Include the intercept in the design matrix\n",
+    "for p in range(degree):\n",
+    "    X[:, p] = x ** p\n",
+    "\n",
+    "beta = fit_beta(X, y)\n",
+    "\n",
+    "# Intercept is included in the design matrix\n",
+    "skl = LinearRegression(fit_intercept=False).fit(X, y)\n",
+    "\n",
+    "print(f\"True beta: {true_beta}\")\n",
+    "print(f\"Fitted beta: {beta}\")\n",
+    "print(f\"Sklearn fitted beta: {skl.coef_}\")\n",
+    "ypredictOwn = X @ beta\n",
+    "ypredictSKL = skl.predict(X)\n",
+    "print(f\"MSE with intercept column\")\n",
+    "print(MSE(y,ypredictOwn))\n",
+    "print(f\"MSE with intercept column from SKL\")\n",
+    "print(MSE(y,ypredictSKL))\n",
+    "\n",
+    "\n",
+    "plt.figure()\n",
+    "plt.scatter(x, y, label=\"Data\")\n",
+    "plt.plot(x, X @ beta, label=\"Fit\")\n",
+    "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n",
+    "\n",
+    "\n",
+    "# Do not include the intercept in the design matrix\n",
+    "X = np.zeros((len(x), degree - 1))\n",
+    "\n",
+    "for p in range(degree - 1):\n",
+    "    X[:, p] = x ** (p + 1)\n",
+    "\n",
+    "# Intercept is not included in the design matrix\n",
+    "skl = LinearRegression(fit_intercept=True).fit(X, y)\n",
+    "\n",
+    "# Use centered values for X and y when computing coefficients\n",
+    "y_offset = np.average(y, axis=0)\n",
+    "X_offset = np.average(X, axis=0)\n",
+    "\n",
+    "beta = fit_beta(X - X_offset, y - y_offset)\n",
+    "intercept = np.mean(y_offset - X_offset @ beta)\n",
+    "\n",
+    "print(f\"Manual intercept: {intercept}\")\n",
+    "print(f\"Fitted beta (without intercept): {beta}\")\n",
+    "print(f\"Sklearn intercept: {skl.intercept_}\")\n",
+    "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n",
+    "ypredictOwn = X @ beta\n",
+    "ypredictSKL = skl.predict(X)\n",
+    "print(f\"MSE with Manual intercept\")\n",
+    "print(MSE(y,ypredictOwn+intercept))\n",
+    "print(f\"MSE with Sklearn intercept\")\n",
+    "print(MSE(y,ypredictSKL))\n",
+    "\n",
+    "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n",
+    "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n",
+    "plt.grid()\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed61dd49",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The intercept is the value of our output/target variable\n",
+    "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
+    "\n",
+    "Printing the MSE, we see first that both methods give the same MSE, as\n",
+    "they should.  However, when we move to for example Ridge regression,\n",
+    "the way we treat the intercept may give a larger or smaller MSE,\n",
+    "meaning that the MSE can be penalized by the value of the\n",
+    "intercept. Not including the intercept in the fit, means that the\n",
+    "regularization term does not include $\\beta_0$. For different values\n",
+    "of $\\lambda$, this may lead to different MSE values. \n",
+    "\n",
+    "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5de45190",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f53e8c1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "but when we take out the intercept, this equation becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b0ffc74",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10305919",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "For Lasso regression we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b2ce8814",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e08c9fd7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It means that, when scaling the design matrix and the outputs/targets,\n",
+    "by subtracting the mean values, we have an optimization problem which\n",
+    "is not penalized by the intercept. The MSE value can then be smaller\n",
+    "since it focuses only on the remaining quantities. If we however bring\n",
+    "back the intercept, we will get a MSE which then contains the\n",
+    "intercept.\n",
+    "\n",
+    "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known  vanilla data set."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "5b7e1c63",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(3155)\n",
+    "\n",
+    "n = 100\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+    "\n",
+    "Maxpolydegree = 20\n",
+    "X = np.zeros((n,Maxpolydegree))\n",
+    "#We include explicitely the intercept column\n",
+    "for degree in range(Maxpolydegree):\n",
+    "    X[:,degree] = x**degree\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "p = Maxpolydegree\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 6\n",
+    "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 2, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+    "    # Note: we include the intercept column and no scaling\n",
+    "    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ytildeOwnRidge = X_train @ OwnRidgeBeta\n",
+    "    ypredictOwnRidge = X_test @ OwnRidgeBeta\n",
+    "    ytildeRidge = RegRidge.predict(X_train)\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    print(\"Beta values for own Ridge implementation\")\n",
+    "    print(OwnRidgeBeta)\n",
+    "    print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+    "    print(RegRidge.coef_)\n",
+    "    print(\"MSE values for own Ridge implementation\")\n",
+    "    print(MSEOwnRidgePredict[i])\n",
+    "    print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+    "    print(MSERidgePredict[i])\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
+    "\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bbf9639e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
+    "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
+    "What happens if we do not include the intercept in our fit?\n",
+    "Let us see how we can change this code by zero centering."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "3a82ee91",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "from sklearn.preprocessing import StandardScaler\n",
+    "\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(315)\n",
+    "\n",
+    "n = 100\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+    "\n",
+    "Maxpolydegree = 20\n",
+    "X = np.zeros((n,Maxpolydegree-1))\n",
+    "\n",
+    "for degree in range(1,Maxpolydegree): #No intercept column\n",
+    "    X[:,degree-1] = x**(degree)\n",
+    "\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
+    "X_train_mean = np.mean(X_train,axis=0)\n",
+    "#Center by removing mean from each feature\n",
+    "X_train_scaled = X_train - X_train_mean \n",
+    "X_test_scaled = X_test - X_train_mean\n",
+    "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n",
+    "#Remove the intercept from the training data.\n",
+    "y_scaler = np.mean(y_train)           \n",
+    "y_train_scaled = y_train - y_scaler   \n",
+    "\n",
+    "p = Maxpolydegree-1\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 6\n",
+    "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "\n",
+    "lambdas = np.logspace(-4, 2, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n",
+    "    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n",
+    "    #Add intercept to prediction\n",
+    "    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
+    "    RegRidge = linear_model.Ridge(lmb)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    print(\"Beta values for own Ridge implementation\")\n",
+    "    print(OwnRidgeBeta) #Intercept is given by mean of target variable\n",
+    "    print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+    "    print(RegRidge.coef_)\n",
+    "    print('Intercept from own implementation:')\n",
+    "    print(intercept_)\n",
+    "    print('Intercept from Scikit-Learn Ridge implementation')\n",
+    "    print(RegRidge.intercept_)\n",
+    "    print(\"MSE values for own Ridge implementation\")\n",
+    "    print(MSEOwnRidgePredict[i])\n",
+    "    print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+    "    print(MSERidgePredict[i])\n",
+    "\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c061da66",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see here, when compared to the code which includes explicitely the\n",
+    "intercept column, that our MSE value is actually smaller. This is\n",
+    "because the regularization term does not include the intercept value\n",
+    "$\\beta_0$ in the fitting.  This applies to Lasso regularization as\n",
+    "well.  It means that our optimization is now done only with the\n",
+    "centered matrix and/or vector that enter the fitting procedure."
+   ]
   }
  ],
  "metadata": {},