diff --git a/doc/pub/week34/html/._week34-bs004.html b/doc/pub/week34/html/._week34-bs004.html
index 0c5e407f3..c00fd6297 100644
--- a/doc/pub/week34/html/._week34-bs004.html
+++ b/doc/pub/week34/html/._week34-bs004.html
@@ -363,7 +363,7 @@ MathJax.Hub.Config({
diff --git a/doc/pub/week34/html/week34-reveal.html b/doc/pub/week34/html/week34-reveal.html
index cdedc490b..cf5266a0d 100644
--- a/doc/pub/week34/html/week34-reveal.html
+++ b/doc/pub/week34/html/week34-reveal.html
@@ -266,7 +266,7 @@ MathJax.Hub.Config({
diff --git a/doc/pub/week34/html/week34-solarized.html b/doc/pub/week34/html/week34-solarized.html
index fe1baaee7..ae165af13 100644
--- a/doc/pub/week34/html/week34-solarized.html
+++ b/doc/pub/week34/html/week34-solarized.html
@@ -364,7 +364,7 @@ MathJax.Hub.Config({
diff --git a/doc/pub/week34/html/week34.html b/doc/pub/week34/html/week34.html
index 032efe5e9..7464e7368 100644
--- a/doc/pub/week34/html/week34.html
+++ b/doc/pub/week34/html/week34.html
@@ -441,7 +441,7 @@ MathJax.Hub.Config({
diff --git a/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz b/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz
index 15fe95bbb..1e9c95a60 100644
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diff --git a/doc/pub/week34/ipynb/week34.ipynb b/doc/pub/week34/ipynb/week34.ipynb
index 4707ced88..d83dee88c 100644
--- a/doc/pub/week34/ipynb/week34.ipynb
+++ b/doc/pub/week34/ipynb/week34.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
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+ "id": "48c00eab",
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@@ -14,7 +14,7 @@
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@@ -102,12 +102,12 @@
"\n",
"* Chat and communications via \n",
"\n",
- "* **Discord** channel at "
+ "* **Discord** channel at "
]
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@@ -513,7 +513,7 @@
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@@ -741,7 +741,7 @@
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@@ -872,7 +872,7 @@
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@@ -966,7 +966,7 @@
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@@ -977,7 +977,7 @@
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@@ -1060,7 +1060,7 @@
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@@ -1077,7 +1077,7 @@
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@@ -1103,7 +1103,7 @@
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@@ -1153,7 +1153,7 @@
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@@ -1168,7 +1168,7 @@
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@@ -1182,7 +1182,7 @@
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@@ -1193,7 +1193,7 @@
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@@ -1234,7 +1234,7 @@
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@@ -1245,7 +1245,7 @@
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@@ -1456,7 +1456,7 @@
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@@ -1482,7 +1482,7 @@
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@@ -1952,7 +1952,7 @@
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@@ -2012,7 +2012,7 @@
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{
"cell_type": "markdown",
- "id": "761b015a",
+ "id": "cff1c1f8",
"metadata": {
"editable": true
},
@@ -3639,7 +3639,7 @@
},
{
"cell_type": "markdown",
- "id": "33405808",
+ "id": "f320c0b3",
"metadata": {
"editable": true
},
@@ -3651,7 +3651,7 @@
},
{
"cell_type": "markdown",
- "id": "0ec2af93",
+ "id": "a8ba839e",
"metadata": {
"editable": true
},
@@ -3661,7 +3661,7 @@
},
{
"cell_type": "markdown",
- "id": "0fa818cb",
+ "id": "90be5e7e",
"metadata": {
"editable": true
},
@@ -3673,7 +3673,7 @@
},
{
"cell_type": "markdown",
- "id": "7776a06f",
+ "id": "a1afcaf0",
"metadata": {
"editable": true
},
@@ -3694,7 +3694,7 @@
},
{
"cell_type": "markdown",
- "id": "c8486fa2",
+ "id": "ccc68760",
"metadata": {
"editable": true
},
@@ -3708,7 +3708,7 @@
},
{
"cell_type": "markdown",
- "id": "88560171",
+ "id": "b607fcf4",
"metadata": {
"editable": true
},
@@ -3719,7 +3719,7 @@
},
{
"cell_type": "markdown",
- "id": "4c2381b1",
+ "id": "09de2082",
"metadata": {
"editable": true
},
@@ -3731,7 +3731,7 @@
},
{
"cell_type": "markdown",
- "id": "f8c0511a",
+ "id": "ba195c19",
"metadata": {
"editable": true
},
@@ -3741,7 +3741,7 @@
},
{
"cell_type": "markdown",
- "id": "f68b7193",
+ "id": "3aeb0e05",
"metadata": {
"editable": true
},
@@ -3753,7 +3753,7 @@
},
{
"cell_type": "markdown",
- "id": "dd1de710",
+ "id": "e76e08d7",
"metadata": {
"editable": true
},
@@ -3763,7 +3763,7 @@
},
{
"cell_type": "markdown",
- "id": "0c7d7741",
+ "id": "ccd8129b",
"metadata": {
"editable": true
},
@@ -3775,7 +3775,7 @@
},
{
"cell_type": "markdown",
- "id": "cccaf815",
+ "id": "143d2185",
"metadata": {
"editable": true
},
@@ -3787,7 +3787,7 @@
},
{
"cell_type": "markdown",
- "id": "b490b68c",
+ "id": "310f3d39",
"metadata": {
"editable": true
},
@@ -3801,7 +3801,7 @@
{
"cell_type": "code",
"execution_count": 37,
- "id": "7f74487d",
+ "id": "e0541d11",
"metadata": {
"collapsed": false,
"editable": true
@@ -3816,7 +3816,7 @@
},
{
"cell_type": "markdown",
- "id": "523bd8d0",
+ "id": "7148e6e9",
"metadata": {
"editable": true
},
@@ -3827,7 +3827,7 @@
{
"cell_type": "code",
"execution_count": 38,
- "id": "a4d7eedd",
+ "id": "d289b356",
"metadata": {
"collapsed": false,
"editable": true
@@ -3840,7 +3840,7 @@
},
{
"cell_type": "markdown",
- "id": "1655538a",
+ "id": "6ef4cefc",
"metadata": {
"editable": true
},
@@ -3851,7 +3851,7 @@
{
"cell_type": "code",
"execution_count": 39,
- "id": "85a500e6",
+ "id": "41ca4b7b",
"metadata": {
"collapsed": false,
"editable": true
@@ -3874,7 +3874,7 @@
},
{
"cell_type": "markdown",
- "id": "f6ad59d6",
+ "id": "3fb76ce8",
"metadata": {
"editable": true
},
@@ -3888,7 +3888,7 @@
{
"cell_type": "code",
"execution_count": 40,
- "id": "4ee168d2",
+ "id": "851247d0",
"metadata": {
"collapsed": false,
"editable": true
@@ -3901,7 +3901,7 @@
},
{
"cell_type": "markdown",
- "id": "ed6ec813",
+ "id": "58a747ce",
"metadata": {
"editable": true
},
@@ -3912,7 +3912,7 @@
{
"cell_type": "code",
"execution_count": 41,
- "id": "ff7101e3",
+ "id": "c503eb37",
"metadata": {
"collapsed": false,
"editable": true
@@ -3924,7 +3924,7 @@
},
{
"cell_type": "markdown",
- "id": "1654e703",
+ "id": "4e36645f",
"metadata": {
"editable": true
},
@@ -3935,7 +3935,7 @@
{
"cell_type": "code",
"execution_count": 42,
- "id": "7a71b2d7",
+ "id": "9e0d51bd",
"metadata": {
"collapsed": false,
"editable": true
@@ -3951,7 +3951,7 @@
},
{
"cell_type": "markdown",
- "id": "963be797",
+ "id": "ef1e78b7",
"metadata": {
"editable": true
},
@@ -3962,7 +3962,7 @@
{
"cell_type": "code",
"execution_count": 43,
- "id": "f6e90610",
+ "id": "6979c3f8",
"metadata": {
"collapsed": false,
"editable": true
@@ -3976,7 +3976,7 @@
},
{
"cell_type": "markdown",
- "id": "897bf983",
+ "id": "5a8caf43",
"metadata": {
"editable": true
},
@@ -3998,7 +3998,7 @@
},
{
"cell_type": "markdown",
- "id": "3972780e",
+ "id": "cf2b742e",
"metadata": {
"editable": true
},
@@ -4010,7 +4010,7 @@
},
{
"cell_type": "markdown",
- "id": "ddaf293e",
+ "id": "7b652644",
"metadata": {
"editable": true
},
@@ -4020,7 +4020,7 @@
},
{
"cell_type": "markdown",
- "id": "2ba1a0ac",
+ "id": "68eef44c",
"metadata": {
"editable": true
},
@@ -4032,7 +4032,7 @@
},
{
"cell_type": "markdown",
- "id": "52b3a243",
+ "id": "d4354161",
"metadata": {
"editable": true
},
@@ -4044,7 +4044,7 @@
},
{
"cell_type": "markdown",
- "id": "3084e52e",
+ "id": "4a9278ab",
"metadata": {
"editable": true
},
@@ -4054,7 +4054,7 @@
},
{
"cell_type": "markdown",
- "id": "e8b81bc2",
+ "id": "85dcc719",
"metadata": {
"editable": true
},
@@ -4066,7 +4066,7 @@
},
{
"cell_type": "markdown",
- "id": "606abbaf",
+ "id": "af633e0f",
"metadata": {
"editable": true
},
@@ -4076,7 +4076,7 @@
},
{
"cell_type": "markdown",
- "id": "0fc6a6fc",
+ "id": "abc9731a",
"metadata": {
"editable": true
},
@@ -4088,7 +4088,7 @@
},
{
"cell_type": "markdown",
- "id": "257ab024",
+ "id": "b98c9215",
"metadata": {
"editable": true
},
@@ -4098,7 +4098,7 @@
},
{
"cell_type": "markdown",
- "id": "ef10e546",
+ "id": "ef75f143",
"metadata": {
"editable": true
},
@@ -4110,7 +4110,7 @@
},
{
"cell_type": "markdown",
- "id": "711417a9",
+ "id": "018ae997",
"metadata": {
"editable": true
},
@@ -4122,7 +4122,7 @@
},
{
"cell_type": "markdown",
- "id": "bf7ef733",
+ "id": "2b484604",
"metadata": {
"editable": true
},
@@ -4132,7 +4132,7 @@
},
{
"cell_type": "markdown",
- "id": "480bcfb2",
+ "id": "b2d59b73",
"metadata": {
"editable": true
},
@@ -4144,7 +4144,7 @@
},
{
"cell_type": "markdown",
- "id": "e30125a8",
+ "id": "43af0c2c",
"metadata": {
"editable": true
},
@@ -4154,7 +4154,7 @@
},
{
"cell_type": "markdown",
- "id": "c8b55590",
+ "id": "9786e84c",
"metadata": {
"editable": true
},
@@ -4166,7 +4166,7 @@
},
{
"cell_type": "markdown",
- "id": "be1b8071",
+ "id": "699e2de9",
"metadata": {
"editable": true
},
@@ -4178,7 +4178,7 @@
},
{
"cell_type": "markdown",
- "id": "c5d84196",
+ "id": "95f061ba",
"metadata": {
"editable": true
},
@@ -4190,7 +4190,7 @@
},
{
"cell_type": "markdown",
- "id": "904ae1cf",
+ "id": "b47033f4",
"metadata": {
"editable": true
},
@@ -4200,7 +4200,7 @@
},
{
"cell_type": "markdown",
- "id": "43af5112",
+ "id": "cd936233",
"metadata": {
"editable": true
},
@@ -4212,7 +4212,7 @@
},
{
"cell_type": "markdown",
- "id": "6ece0163",
+ "id": "024ec512",
"metadata": {
"editable": true
},
@@ -4222,7 +4222,7 @@
},
{
"cell_type": "markdown",
- "id": "75240b86",
+ "id": "5f0fbdc7",
"metadata": {
"editable": true
},
@@ -4234,7 +4234,7 @@
},
{
"cell_type": "markdown",
- "id": "75ab537c",
+ "id": "c48c6461",
"metadata": {
"editable": true
},
@@ -4244,7 +4244,7 @@
},
{
"cell_type": "markdown",
- "id": "cf3be8df",
+ "id": "65334c5f",
"metadata": {
"editable": true
},
@@ -4256,7 +4256,7 @@
},
{
"cell_type": "markdown",
- "id": "1e323640",
+ "id": "98641b90",
"metadata": {
"editable": true
},
@@ -4267,7 +4267,7 @@
},
{
"cell_type": "markdown",
- "id": "b98ec1cd",
+ "id": "fd6c1d45",
"metadata": {
"editable": true
},
@@ -4279,7 +4279,7 @@
},
{
"cell_type": "markdown",
- "id": "5960f894",
+ "id": "4817bd05",
"metadata": {
"editable": true
},
@@ -4289,7 +4289,7 @@
},
{
"cell_type": "markdown",
- "id": "e12c2cd4",
+ "id": "312805de",
"metadata": {
"editable": true
},
@@ -4301,7 +4301,7 @@
},
{
"cell_type": "markdown",
- "id": "0669ce10",
+ "id": "c3c18f3d",
"metadata": {
"editable": true
},
@@ -4311,7 +4311,7 @@
},
{
"cell_type": "markdown",
- "id": "32b20573",
+ "id": "7d4ff47d",
"metadata": {
"editable": true
},
@@ -4323,7 +4323,7 @@
},
{
"cell_type": "markdown",
- "id": "ca4d3e37",
+ "id": "389b398a",
"metadata": {
"editable": true
},
@@ -4336,7 +4336,7 @@
},
{
"cell_type": "markdown",
- "id": "6b26c4e9",
+ "id": "6cafaa0a",
"metadata": {
"editable": true
},
@@ -4348,7 +4348,7 @@
},
{
"cell_type": "markdown",
- "id": "785c3035",
+ "id": "c2472e8d",
"metadata": {
"editable": true
},
@@ -4360,7 +4360,7 @@
},
{
"cell_type": "markdown",
- "id": "8557f71e",
+ "id": "9fca2f2a",
"metadata": {
"editable": true
},
@@ -4372,7 +4372,7 @@
},
{
"cell_type": "markdown",
- "id": "25a50c1c",
+ "id": "0b4b7322",
"metadata": {
"editable": true
},
@@ -4384,7 +4384,7 @@
},
{
"cell_type": "markdown",
- "id": "b76b6837",
+ "id": "81aa7109",
"metadata": {
"editable": true
},
@@ -4396,7 +4396,7 @@
},
{
"cell_type": "markdown",
- "id": "ec2bc31e",
+ "id": "d55ffc3b",
"metadata": {
"editable": true
},
@@ -4406,7 +4406,7 @@
},
{
"cell_type": "markdown",
- "id": "3bc1c2e3",
+ "id": "950b1dd1",
"metadata": {
"editable": true
},
@@ -4418,7 +4418,7 @@
},
{
"cell_type": "markdown",
- "id": "0354e692",
+ "id": "b9c8909c",
"metadata": {
"editable": true
},
@@ -4430,7 +4430,7 @@
},
{
"cell_type": "markdown",
- "id": "d65dd254",
+ "id": "e89e9876",
"metadata": {
"editable": true
},
@@ -4443,7 +4443,7 @@
},
{
"cell_type": "markdown",
- "id": "6574f03f",
+ "id": "2f677804",
"metadata": {
"editable": true
},
@@ -4470,7 +4470,7 @@
},
{
"cell_type": "markdown",
- "id": "19e3b7a0",
+ "id": "42d4d665",
"metadata": {
"editable": true
},
@@ -4481,7 +4481,7 @@
{
"cell_type": "code",
"execution_count": 44,
- "id": "9d354722",
+ "id": "e6057957",
"metadata": {
"collapsed": false,
"editable": true
@@ -4577,7 +4577,7 @@
},
{
"cell_type": "markdown",
- "id": "68a978aa",
+ "id": "03aaa217",
"metadata": {
"editable": true
},
@@ -4592,7 +4592,7 @@
},
{
"cell_type": "markdown",
- "id": "69bfb6ca",
+ "id": "fa5b22cd",
"metadata": {
"editable": true
},
@@ -4614,7 +4614,7 @@
{
"cell_type": "code",
"execution_count": 45,
- "id": "a8cf549e",
+ "id": "80b4e2fd",
"metadata": {
"collapsed": false,
"editable": true
@@ -4689,7 +4689,7 @@
},
{
"cell_type": "markdown",
- "id": "edd97676",
+ "id": "ef276791",
"metadata": {
"editable": true
},
@@ -4701,7 +4701,7 @@
},
{
"cell_type": "markdown",
- "id": "8b1a182a",
+ "id": "ecac9f49",
"metadata": {
"editable": true
},
@@ -4770,7 +4770,7 @@
},
{
"cell_type": "markdown",
- "id": "3a973417",
+ "id": "7394d9c5",
"metadata": {
"editable": true
},
@@ -4784,7 +4784,7 @@
{
"cell_type": "code",
"execution_count": 46,
- "id": "c3295eaa",
+ "id": "a65b2934",
"metadata": {
"collapsed": false,
"editable": true
@@ -4797,7 +4797,7 @@
},
{
"cell_type": "markdown",
- "id": "7a4fca8c",
+ "id": "82b37eaf",
"metadata": {
"editable": true
},
@@ -4811,7 +4811,7 @@
},
{
"cell_type": "markdown",
- "id": "13316f63",
+ "id": "85929fd6",
"metadata": {
"editable": true
},
@@ -4824,7 +4824,7 @@
},
{
"cell_type": "markdown",
- "id": "820a0264",
+ "id": "3c0ecece",
"metadata": {
"editable": true
},
@@ -4835,7 +4835,7 @@
},
{
"cell_type": "markdown",
- "id": "abfb8260",
+ "id": "8e30f349",
"metadata": {
"editable": true
},
@@ -4847,7 +4847,7 @@
},
{
"cell_type": "markdown",
- "id": "98edfb0f",
+ "id": "d61feb53",
"metadata": {
"editable": true
},
@@ -4857,7 +4857,7 @@
},
{
"cell_type": "markdown",
- "id": "40e5092f",
+ "id": "958c619a",
"metadata": {
"editable": true
},
@@ -4869,7 +4869,7 @@
},
{
"cell_type": "markdown",
- "id": "29f0cb53",
+ "id": "f1e51afb",
"metadata": {
"editable": true
},
@@ -4880,7 +4880,7 @@
},
{
"cell_type": "markdown",
- "id": "db5058c8",
+ "id": "5b7a1dd2",
"metadata": {
"editable": true
},
@@ -4899,7 +4899,7 @@
{
"cell_type": "code",
"execution_count": 47,
- "id": "38f7c5e6",
+ "id": "5a5612a3",
"metadata": {
"collapsed": false,
"editable": true
@@ -4915,7 +4915,7 @@
},
{
"cell_type": "markdown",
- "id": "31fd7bd4",
+ "id": "894f4f46",
"metadata": {
"editable": true
},
@@ -4925,7 +4925,7 @@
},
{
"cell_type": "markdown",
- "id": "c99adcd3",
+ "id": "fc7ca802",
"metadata": {
"editable": true
},
@@ -4936,7 +4936,7 @@
},
{
"cell_type": "markdown",
- "id": "d56769ba",
+ "id": "d1212a10",
"metadata": {
"editable": true
},
@@ -4948,7 +4948,7 @@
},
{
"cell_type": "markdown",
- "id": "9a4867aa",
+ "id": "c051f42f",
"metadata": {
"editable": true
},
diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb
index c2d22256b..c981143f7 100644
--- a/doc/pub/week35/ipynb/week35.ipynb
+++ b/doc/pub/week35/ipynb/week35.ipynb
@@ -3,9 +3,7 @@
{
"cell_type": "markdown",
"id": "2070762b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"\n",
@@ -15,9 +13,7 @@
{
"cell_type": "markdown",
"id": "3a35ce03",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"# Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression\n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
@@ -28,9 +24,7 @@
{
"cell_type": "markdown",
"id": "f250b29c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Plans for week 35\n",
"\n",
@@ -50,9 +44,7 @@
{
"cell_type": "markdown",
"id": "157d1840",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"### Reading recommendations:\n",
"\n",
@@ -66,9 +58,7 @@
{
"cell_type": "markdown",
"id": "1f0e5f64",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week\n",
"\n",
@@ -100,9 +90,7 @@
{
"cell_type": "markdown",
"id": "e15c5dd7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## The equations for ordinary least squares\n",
"\n",
@@ -116,9 +104,7 @@
{
"cell_type": "markdown",
"id": "7205f9ed",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"y_i=f(x_i)+\\epsilon_i,\n",
@@ -128,9 +114,7 @@
{
"cell_type": "markdown",
"id": "58035123",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"or in general"
]
@@ -138,9 +122,7 @@
{
"cell_type": "markdown",
"id": "5c054dc2",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{y}=f(\\boldsymbol{x})+\\boldsymbol{\\epsilon},\n",
@@ -150,9 +132,7 @@
{
"cell_type": "markdown",
"id": "1ccbf405",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $\\boldsymbol{\\epsilon}$ represents some noise which is normally assumed to\n",
"be distributed via a normal probability distribution with zero mean\n",
@@ -171,9 +151,7 @@
{
"cell_type": "markdown",
"id": "4fc747be",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n",
@@ -183,9 +161,7 @@
{
"cell_type": "markdown",
"id": "324815b4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and in order to find the optimal parameters $\\beta_i$ we defined a function which\n",
"gives a measure of the spread between the values $y_i$ (which\n",
@@ -196,9 +172,7 @@
{
"cell_type": "markdown",
"id": "c31fc801",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## The cost/loss function\n",
"\n",
@@ -208,9 +182,7 @@
{
"cell_type": "markdown",
"id": "5f390ec4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n",
@@ -220,9 +192,7 @@
{
"cell_type": "markdown",
"id": "a7b23937",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"or using the matrix $\\boldsymbol{X}$ and in a more compact matrix-vector notation as"
]
@@ -230,9 +200,7 @@
{
"cell_type": "markdown",
"id": "db0cea3d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
@@ -242,9 +210,7 @@
{
"cell_type": "markdown",
"id": "936418d7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"This function represents one of many possible ways to define the so-called cost function.\n",
"\n",
@@ -255,9 +221,7 @@
{
"cell_type": "markdown",
"id": "3b2a6714",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{2n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2,\n",
@@ -267,9 +231,7 @@
{
"cell_type": "markdown",
"id": "54120755",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"since when taking the first derivative with respect to the unknown parameters $\\beta$, the factor of $2$ cancels out."
]
@@ -277,9 +239,7 @@
{
"cell_type": "markdown",
"id": "f8854f6e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Interpretations and optimizing our parameters\n",
"\n",
@@ -289,9 +249,7 @@
{
"cell_type": "markdown",
"id": "468fd216",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\},\n",
@@ -301,9 +259,7 @@
{
"cell_type": "markdown",
"id": "cd5e2caa",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. \n",
"When linking (see the discussions next week) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value"
@@ -312,9 +268,7 @@
{
"cell_type": "markdown",
"id": "32e08e08",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"y_{i}=\\langle y_i \\rangle = \\beta_0x_{i,0}+\\beta_1x_{i,1}+\\beta_2x_{i,2}+\\dots+\\beta_{n-1}x_{i,n-1}+\\epsilon_i,\n",
@@ -324,9 +278,7 @@
{
"cell_type": "markdown",
"id": "af8df8e6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $\\langle y_i \\rangle$ is the mean value. Keep in mind also that\n",
"till now we have treated $y_i$ as the exact value. Normally, the\n",
@@ -343,9 +295,7 @@
{
"cell_type": "markdown",
"id": "89f55eba",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
@@ -356,9 +306,7 @@
{
"cell_type": "markdown",
"id": "3594bcb9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"In practical terms it means we will require"
]
@@ -366,9 +314,7 @@
{
"cell_type": "markdown",
"id": "1e45992c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)^2\\right]=0,\n",
@@ -378,9 +324,7 @@
{
"cell_type": "markdown",
"id": "90198ba6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which results in"
]
@@ -388,9 +332,7 @@
{
"cell_type": "markdown",
"id": "78c4cc43",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_{ij}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)\\right]=0,\n",
@@ -400,9 +342,7 @@
{
"cell_type": "markdown",
"id": "8d138d4c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"or in a matrix-vector form as (multiplying away the factor $-2/n$, see derivation below)"
]
@@ -410,9 +350,7 @@
{
"cell_type": "markdown",
"id": "ba40b8d7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}^T} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right).\n",
@@ -422,9 +360,7 @@
{
"cell_type": "markdown",
"id": "e8f749fe",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Interpretations and optimizing our parameters\n",
"We can rewrite, see the derivations below,"
@@ -433,9 +369,7 @@
{
"cell_type": "markdown",
"id": "c679efad",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}^T} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right),\n",
@@ -445,9 +379,7 @@
{
"cell_type": "markdown",
"id": "f1aaeaca",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"as"
]
@@ -455,9 +387,7 @@
{
"cell_type": "markdown",
"id": "fd4ae817",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{X}^T\\boldsymbol{y} = \\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta},\n",
@@ -467,9 +397,7 @@
{
"cell_type": "markdown",
"id": "d09f9a77",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and if the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ is invertible we have the solution"
]
@@ -477,9 +405,7 @@
{
"cell_type": "markdown",
"id": "09ec2766",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{\\beta} =\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
@@ -489,9 +415,7 @@
{
"cell_type": "markdown",
"id": "ad758d65",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n",
"{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n",
@@ -509,9 +433,7 @@
{
"cell_type": "markdown",
"id": "9d72460a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Some useful matrix and vector expressions\n",
"\n",
@@ -536,9 +458,7 @@
{
"cell_type": "markdown",
"id": "318afac7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{y}=f(\\boldsymbol{x}).\n",
@@ -548,9 +468,7 @@
{
"cell_type": "markdown",
"id": "e885156f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## The Jacobian\n",
"\n",
@@ -560,9 +478,7 @@
{
"cell_type": "markdown",
"id": "cd7e8abb",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{J}=\\frac{\\partial \\boldsymbol{y}}{\\partial \\boldsymbol{x}}=\\begin{bmatrix} \\frac{\\partial y_0}{\\partial x_0} & \\frac{\\partial y_0}{\\partial x_1} & \\frac{\\partial y_0}{\\partial x_2} & \\dots & \\dots & \\frac{\\partial y_0}{\\partial x_{n-1}} \\\\ \\frac{\\partial y_0}{\\partial x_0} & \\frac{\\partial y_1}{\\partial x_1} & \\frac{\\partial y_1}{\\partial x_2} & \\dots & \\dots & \\frac{\\partial y_1}{\\partial x_{n-1}} \\\\\n",
@@ -576,9 +492,7 @@
{
"cell_type": "markdown",
"id": "6a7946a6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which is an $m\\times n$ matrix. If $\\boldsymbol{x}$ is a scalar, then the\n",
"Jacobian is only a single-column vector, or an $m\\times 1$ matrix. If\n",
@@ -594,9 +508,7 @@
{
"cell_type": "markdown",
"id": "bc7dac5b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Derivatives, example 1\n",
"\n",
@@ -606,9 +518,7 @@
{
"cell_type": "markdown",
"id": "fb7b7c7a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"y_i = \\sum_{j=0}^{n-1}a_{ij}x_j,\n",
@@ -618,9 +528,7 @@
{
"cell_type": "markdown",
"id": "fdab3f90",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"with $\\forall i=0,1,2,\\dots,m-1$. The individual matrix elements of $\\boldsymbol{A}$ are given by the symbol $a_{ij}$.\n",
"It follows that the partial derivatives of $y_i$ with respect to $x_k$"
@@ -629,9 +537,7 @@
{
"cell_type": "markdown",
"id": "d398d440",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial y_i }{\\partial x_k}= a_{ik} \\forall i=0,1,2,\\dots,m-1.\n",
@@ -641,9 +547,7 @@
{
"cell_type": "markdown",
"id": "f90d2059",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"From this we have, using the definition of the Jacobian"
]
@@ -651,9 +555,7 @@
{
"cell_type": "markdown",
"id": "3872052b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\boldsymbol{y} }{\\partial \\boldsymbol{x}}= \\boldsymbol{A}.\n",
@@ -663,9 +565,7 @@
{
"cell_type": "markdown",
"id": "535cf990",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Example 2\n",
"\n",
@@ -677,9 +577,7 @@
{
"cell_type": "markdown",
"id": "197d3e9f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\boldsymbol{y}^T\\boldsymbol{A}\\boldsymbol{x},\n",
@@ -689,9 +587,7 @@
{
"cell_type": "markdown",
"id": "e2c7a4f9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"with $\\boldsymbol{y}$ a vector of length $m$, $\\boldsymbol{A}$ an $m\\times n$ matrix and $\\boldsymbol{x}$ a vector of length $n$. We assume also that $\\boldsymbol{A}$ does not depend on any of the two vectors.\n",
"In order to find the derivative of $\\alpha$ with respect to the two vectors, we define an intermediate vector $\\boldsymbol{z}$. We define first\n",
@@ -701,9 +597,7 @@
{
"cell_type": "markdown",
"id": "834ecb73",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\boldsymbol{z}^T\\boldsymbol{x},\n",
@@ -713,9 +607,7 @@
{
"cell_type": "markdown",
"id": "89e5b0d3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which means that (using our previous example) we have"
]
@@ -723,9 +615,7 @@
{
"cell_type": "markdown",
"id": "f7e022ca",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{x}} = \\boldsymbol{z}=bm{A}^T\\boldsymbol{y}.\n",
@@ -735,9 +625,7 @@
{
"cell_type": "markdown",
"id": "0f9fa0cd",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"Note that the resulting vector elements are the same for $\\boldsymbol{z}^T$ and $\\boldsymbol{z}$, the only difference is that one if just the transpose of the other.\n",
"\n",
@@ -747,9 +635,7 @@
{
"cell_type": "markdown",
"id": "7207e78d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{y}} = \\boldsymbol{z}^T=\\boldsymbol{x}^T\\boldsymbol{A}^T.\n",
@@ -759,9 +645,7 @@
{
"cell_type": "markdown",
"id": "a58d233a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Example 3\n",
"\n",
@@ -773,9 +657,7 @@
{
"cell_type": "markdown",
"id": "08676c15",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x},\n",
@@ -785,9 +667,7 @@
{
"cell_type": "markdown",
"id": "0c4417ed",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"with $\\boldsymbol{x}$ a vector of length $n$.\n",
"\n",
@@ -797,9 +677,7 @@
{
"cell_type": "markdown",
"id": "9f43d971",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\sum_{i=0}^{n-1}\\sum_{j=0}^{n-1}x_i a_{ij}x_j,\n",
@@ -809,9 +687,7 @@
{
"cell_type": "markdown",
"id": "b0fc3f13",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"taking the derivative of $\\alpha$ with respect to a given component $x_k$ we get the two sums"
]
@@ -819,9 +695,7 @@
{
"cell_type": "markdown",
"id": "105376e3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial x_k} = \\sum_{i=0}^{n-1}a_{ik}x_i+\\sum_{j=0}^{n-1}a_{kj}x_j,\n",
@@ -831,9 +705,7 @@
{
"cell_type": "markdown",
"id": "6aa6fe7f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"for $\\forall k =0,1,2,\\dots,n-1$. We identify these sums as"
]
@@ -841,9 +713,7 @@
{
"cell_type": "markdown",
"id": "7fecb158",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{x}} = \\boldsymbol{x}^T\\left(\\boldsymbol{A}^T+\\boldsymbol{A}\\right).\n",
@@ -853,9 +723,7 @@
{
"cell_type": "markdown",
"id": "d0e52ab7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"If the matrix $\\boldsymbol{A}$ is symmetric, that is $\\boldsymbol{A}=\\boldsymbol{A}^T$, we have"
]
@@ -863,9 +731,7 @@
{
"cell_type": "markdown",
"id": "12dd776d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{x}} = 2\\boldsymbol{x}^T\\boldsymbol{A}.\n",
@@ -875,9 +741,7 @@
{
"cell_type": "markdown",
"id": "78b3f176",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Example 4\n",
"\n",
@@ -887,9 +751,7 @@
{
"cell_type": "markdown",
"id": "6b03b52f",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\boldsymbol{y}^T\\boldsymbol{x},\n",
@@ -899,9 +761,7 @@
{
"cell_type": "markdown",
"id": "7e9addc3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where both $\\boldsymbol{y}$ and $\\boldsymbol{x}$ have the same length $n$, or if we\n",
"wish to think of them as column vectors, they have dimensions $n\\times\n",
@@ -914,9 +774,7 @@
{
"cell_type": "markdown",
"id": "6338444a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\alpha = \\sum_{i=0}^{n-1}y_ix_i,\n",
@@ -926,9 +784,7 @@
{
"cell_type": "markdown",
"id": "415c5e3d",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and the partial derivative"
]
@@ -936,9 +792,7 @@
{
"cell_type": "markdown",
"id": "a1cbb5b8",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial z_k} = \\sum_{i=0}^{n-1}\\left(x_i\\frac{\\partial y_i}{\\partial z_k}+y_i\\frac{\\partial x_i}{\\partial z_k}\\right),\n",
@@ -948,9 +802,7 @@
{
"cell_type": "markdown",
"id": "43d23259",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"for $\\forall k =0,1,2,\\dots,n-1$. We can rewrite the partial derivative in a more compact form as"
]
@@ -958,9 +810,7 @@
{
"cell_type": "markdown",
"id": "2171f18c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{z}} = \\boldsymbol{x}^T\\frac{\\partial \\boldsymbol{y}}{\\partial \\boldsymbol{z}}+\\boldsymbol{y}^T\\frac{\\partial \\boldsymbol{x}}{\\partial \\boldsymbol{z}},\n",
@@ -970,9 +820,7 @@
{
"cell_type": "markdown",
"id": "292802ce",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and if $\\boldsymbol{y}=\\boldsymbol{x}$ we have"
]
@@ -980,9 +828,7 @@
{
"cell_type": "markdown",
"id": "b1fa893a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\alpha}{\\partial \\boldsymbol{z}} = 2\\boldsymbol{x}^T\\frac{\\partial \\boldsymbol{x}}{\\partial \\boldsymbol{z}}.\n",
@@ -992,9 +838,7 @@
{
"cell_type": "markdown",
"id": "d5a8e55b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## The mean squared error and its derivative\n",
"\n",
@@ -1004,9 +848,7 @@
{
"cell_type": "markdown",
"id": "475b3f93",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n",
@@ -1016,9 +858,7 @@
{
"cell_type": "markdown",
"id": "aff37293",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"or using the design/feature matrix $\\boldsymbol{X}$ we have the more compact matrix-vector"
]
@@ -1026,9 +866,7 @@
{
"cell_type": "markdown",
"id": "4fbe21f5",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
@@ -1038,9 +876,7 @@
{
"cell_type": "markdown",
"id": "3febdaf6",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n",
"We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n",
@@ -1051,9 +887,7 @@
{
"cell_type": "markdown",
"id": "81b55f55",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{w}=\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n",
@@ -1063,9 +897,7 @@
{
"cell_type": "markdown",
"id": "19b56c51",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"which depends on $\\boldsymbol{\\beta}$. We rewrite the cost function as"
]
@@ -1073,9 +905,7 @@
{
"cell_type": "markdown",
"id": "97f789ce",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta})=\\frac{1}{n}\\boldsymbol{w}^T\\boldsymbol{w},\n",
@@ -1085,9 +915,7 @@
{
"cell_type": "markdown",
"id": "5a3a13b2",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"with partial derivative"
]
@@ -1095,9 +923,7 @@
{
"cell_type": "markdown",
"id": "94a125c0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=\\frac{2}{n}\\boldsymbol{w}^T\\frac{\\partial \\boldsymbol{w}}{\\partial \\boldsymbol{\\beta}},\n",
@@ -1107,9 +933,7 @@
{
"cell_type": "markdown",
"id": "91987be9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and using that"
]
@@ -1117,9 +941,7 @@
{
"cell_type": "markdown",
"id": "bf3d4757",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\boldsymbol{w}}{\\partial \\boldsymbol{\\beta}}=-\\boldsymbol{X},\n",
@@ -1129,9 +951,7 @@
{
"cell_type": "markdown",
"id": "73a98618",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where we used the result from example two above. Inserting the last expression we obtain"
]
@@ -1139,9 +959,7 @@
{
"cell_type": "markdown",
"id": "d45f357b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=-\\frac{2}{n}\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\boldsymbol{X},\n",
@@ -1151,9 +969,7 @@
{
"cell_type": "markdown",
"id": "4f286236",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"or as"
]
@@ -1161,9 +977,7 @@
{
"cell_type": "markdown",
"id": "8e44e034",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}^T}=-\\frac{2}{n}\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right).\n",
@@ -1173,9 +987,7 @@
{
"cell_type": "markdown",
"id": "2b173bf4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Other useful relations\n",
"\n",
@@ -1185,9 +997,7 @@
{
"cell_type": "markdown",
"id": "44c07349",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial (\\boldsymbol{b}^T\\boldsymbol{a})}{\\partial \\boldsymbol{a}} = \\boldsymbol{b},\n",
@@ -1197,9 +1007,7 @@
{
"cell_type": "markdown",
"id": "9655d305",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial tr(\\boldsymbol{B}\\boldsymbol{A})}{\\partial \\boldsymbol{A}} = \\boldsymbol{B}^T,\n",
@@ -1209,9 +1017,7 @@
{
"cell_type": "markdown",
"id": "c6f9029b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n",
@@ -1221,9 +1027,7 @@
{
"cell_type": "markdown",
"id": "c9bdbee5",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Meet the Hessian Matrix\n",
"\n",
@@ -1237,9 +1041,7 @@
{
"cell_type": "markdown",
"id": "37e20b51",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\frac{\\partial}{\\partial \\boldsymbol{\\beta}}\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}^T} =\\frac{\\partial}{\\partial \\boldsymbol{\\beta}}\\left[-\\frac{2}{n}\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right]=\\frac{2}{n}\\boldsymbol{X}^T\\boldsymbol{X}.\n",
@@ -1249,9 +1051,7 @@
{
"cell_type": "markdown",
"id": "44d0f0c2",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"The Hessian matrix plays an important role and is defined here as"
]
@@ -1259,9 +1059,7 @@
{
"cell_type": "markdown",
"id": "f041c827",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{H}=\\boldsymbol{X}^T\\boldsymbol{X}.\n",
@@ -1271,9 +1069,7 @@
{
"cell_type": "markdown",
"id": "b954f008",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"For ordinary least squares, it is inversely proportional (derivation\n",
"next week) with the variance of the optimal parameters\n",
@@ -1289,9 +1085,7 @@
{
"cell_type": "markdown",
"id": "123e7a0b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Interpretations and optimizing our parameters\n",
"\n",
@@ -1301,9 +1095,7 @@
{
"cell_type": "markdown",
"id": "ff1d71b0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n",
@@ -1313,9 +1105,7 @@
{
"cell_type": "markdown",
"id": "d50b1eff",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and with"
]
@@ -1323,9 +1113,7 @@
{
"cell_type": "markdown",
"id": "b653da9a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n",
@@ -1335,9 +1123,7 @@
{
"cell_type": "markdown",
"id": "6763c705",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"we have"
]
@@ -1345,9 +1131,7 @@
{
"cell_type": "markdown",
"id": "4814c60a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n",
@@ -1357,9 +1141,7 @@
{
"cell_type": "markdown",
"id": "b311b8f0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"meaning that the solution for $\\boldsymbol{\\beta}$ is the one which minimizes the residuals."
]
@@ -1367,9 +1149,7 @@
{
"cell_type": "markdown",
"id": "024997c9",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Example relevant for the exercises\n",
"\n",
@@ -1381,9 +1161,7 @@
{
"cell_type": "markdown",
"id": "fd049343",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"\\tilde{y}_i = \\beta_0+\\beta_1x_i+\\beta_2x_i^2+\\beta_3x_i^3+\\beta_4x_i^4.\n",
@@ -1393,9 +1171,7 @@
{
"cell_type": "markdown",
"id": "2b2c756c",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n",
"$n\\times p$ matrix $\\boldsymbol{X}$."
@@ -1404,9 +1180,7 @@
{
"cell_type": "markdown",
"id": "71e03c6e",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Own code for Ordinary Least Squares\n",
"\n",
@@ -1415,38 +1189,42 @@
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 21,
"id": "49ebb09f",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[2.00000000e+00 5.08482145e-14 5.00000000e+00]\n"
+ ]
+ }
+ ],
"source": [
"# matrix inversion to find beta\n",
"# First we set up the data\n",
"import numpy as np\n",
"x = np.random.rand(100)\n",
- "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+ "y = 2.0+5*x*x#+0.1*np.random.randn(100)\n",
"# and then the design matrix X including the intercept\n",
"# The design matrix now as function of a fourth-order polynomial\n",
- "X = np.zeros((len(x),5))\n",
+ "X = np.zeros((len(x),3))\n",
"X[:,0] = 1.0\n",
"X[:,1] = x\n",
"X[:,2] = x**2\n",
- "X[:,3] = x**3\n",
- "X[:,4] = x**4\n",
+ "#X[:,3] = x**3\n",
+ "#X[:,4] = x**4\n",
"beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
"# and then make the prediction\n",
- "ytilde = X @ beta"
+ "ytilde = X @ beta\n",
+ "print(beta)"
]
},
{
"cell_type": "markdown",
"id": "015ed20b",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"Alternatively, you can use the least squares functionality in **Numpy** as"
]
@@ -1455,10 +1233,7 @@
"cell_type": "code",
"execution_count": 2,
"id": "7fcdbb07",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"fit = np.linalg.lstsq(X, y, rcond =None)[0]\n",
@@ -1468,9 +1243,7 @@
{
"cell_type": "markdown",
"id": "e4dc0c88",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Adding error analysis and training set up\n",
"\n",
@@ -1482,10 +1255,7 @@
"cell_type": "code",
"execution_count": 3,
"id": "4e8dd283",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"def R2(y_data, y_model):\n",
@@ -1495,9 +1265,7 @@
{
"cell_type": "markdown",
"id": "8e7038f4",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and we would be using it as"
]
@@ -1506,10 +1274,7 @@
"cell_type": "code",
"execution_count": 4,
"id": "3c9ff2b4",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"print(R2(y,ytilde))"
@@ -1518,22 +1283,25 @@
{
"cell_type": "markdown",
"id": "b7d31cfc",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"We can easily add our **MSE** score as"
]
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 22,
"id": "56b5aa44",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "3.7981483218902363e-28\n"
+ ]
+ }
+ ],
"source": [
"def MSE(y_data,y_model):\n",
" n = np.size(y_model)\n",
@@ -1545,9 +1313,7 @@
{
"cell_type": "markdown",
"id": "418766ca",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"and finally the relative error as"
]
@@ -1556,10 +1322,7 @@
"cell_type": "code",
"execution_count": 6,
"id": "9ce8738f",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"def RelativeError(y_data,y_model):\n",
@@ -1570,9 +1333,7 @@
{
"cell_type": "markdown",
"id": "f70a5810",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Splitting our Data in Training and Test data\n",
"\n",
@@ -1591,22 +1352,33 @@
{
"cell_type": "markdown",
"id": "02999e0a",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## The complete code with a simple data set"
]
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 23,
"id": "44b7ab0c",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[ 2.04266637 -0.12695475 4.66800499 1.30798223 -0.96738204]\n",
+ "Training R2\n",
+ "0.9956006550968114\n",
+ "Training MSE\n",
+ "0.009753726082528709\n",
+ "Test R2\n",
+ "0.9985674179471976\n",
+ "Test MSE\n",
+ "0.003282890787616731\n"
+ ]
+ }
+ ],
"source": [
"%matplotlib inline\n",
"\n",
@@ -1655,9 +1427,7 @@
{
"cell_type": "markdown",
"id": "8bf25c52",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Making your own test-train splitting"
]
@@ -1666,10 +1436,7 @@
"cell_type": "code",
"execution_count": 8,
"id": "f1617c0f",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"# equivalently in numpy\n",
@@ -1691,9 +1458,7 @@
{
"cell_type": "markdown",
"id": "b6cf3134",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"But since **scikit-learn** has its own function for doing this and since\n",
"it interfaces easily with **tensorflow** and other libraries, we\n",
@@ -1703,9 +1468,7 @@
{
"cell_type": "markdown",
"id": "62a7efa0",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Reducing the number of degrees of freedom, overarching view\n",
"\n",
@@ -1732,9 +1495,7 @@
{
"cell_type": "markdown",
"id": "705a9e78",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Preprocessing our data\n",
"\n",
@@ -1757,9 +1518,7 @@
{
"cell_type": "markdown",
"id": "a28e9f54",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Functionality in Scikit-Learn\n",
"\n",
@@ -1777,9 +1536,7 @@
{
"cell_type": "markdown",
"id": "4ef6e8e7",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## More preprocessing\n",
"\n",
@@ -1804,9 +1561,7 @@
{
"cell_type": "markdown",
"id": "792a7f87",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Frequently used scaling functions\n",
"\n",
@@ -1817,9 +1572,7 @@
{
"cell_type": "markdown",
"id": "be85a0d3",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"$$\n",
"x_j^{(i)} \\rightarrow \\frac{x_j^{(i)} - \\overline{x}_j}{\\sigma(x_j)},\n",
@@ -1829,9 +1582,7 @@
{
"cell_type": "markdown",
"id": "dc73c283",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"where $\\overline{x}_j$ and $\\sigma(x_j)$ are the mean and standard deviation, respectively, of the feature $x_j$.\n",
"This ensures that each feature has zero mean and unit standard deviation. For data sets where we do not have the standard deviation or don't wish to calculate it, it is then common to simply set it to one."
@@ -1840,9 +1591,7 @@
{
"cell_type": "markdown",
"id": "0fc3e620",
- "metadata": {
- "editable": true
- },
+ "metadata": {},
"source": [
"## Example of own Standard scaling\n",
"\n",
@@ -1854,13 +1603,419 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 11,
"id": "8781e6d8",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
+ "metadata": {},
+ "outputs": [
+ {
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+ "
\n",
+ " \n",
+ " | 9 | \n",
+ " -1.557795 | \n",
+ " -1.066837 | \n",
+ " 0.401842 | \n",
+ " -1.213743 | \n",
+ " 1.138775 | \n",
+ "
\n",
+ " \n",
+ "
\n",
+ "
\n",
+ "\n",
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\n",
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+ " 3 | \n",
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\n",
+ " \n",
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\n",
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\n",
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\n",
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\n",
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\n",
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+ " 0.0 | \n",
+ " 0.0 | \n",
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