From 59449d4e7610ccd8ab9cc180990a8d2be4b216a0 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 14 Sep 2023 15:27:57 +0200 Subject: [PATCH] update book --- .../_build/.doctrees/environment.pickle | Bin 307525 -> 308178 bytes .../_build/.doctrees/week37.doctree | Bin 586981 -> 715254 bytes .../_build/html/_images/week37_144_0.png | Bin 22076 -> 22031 bytes .../_build/html/_images/week37_162_3.png | Bin 0 -> 29329 bytes .../_build/html/_images/week37_176_1.png | Bin 0 -> 33782 bytes .../_build/html/_images/week37_208_1.png | Bin 0 -> 33784 bytes .../_build/html/_sources/week37.ipynb | 1002 ++++++++++++--- doc/LectureNotes/_build/html/searchindex.js | 2 +- doc/LectureNotes/_build/html/week37.html | 424 ++++++- .../_build/jupyter_execute/week37.ipynb | 1110 ++++++++++++++--- .../_build/jupyter_execute/week37.py | 367 +++++- .../_build/jupyter_execute/week37_144_0.png | Bin 22076 -> 22031 bytes .../_build/jupyter_execute/week37_162_3.png | Bin 0 -> 29329 bytes 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3d6a1c1b9..96c6b8d8c 100644 --- a/doc/LectureNotes/_build/html/_sources/week37.ipynb +++ b/doc/LectureNotes/_build/html/_sources/week37.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "5d85a8ac", + "id": "9d1d9c58", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "82315184", + "id": "c1bdb9de", "metadata": { "editable": true }, @@ -22,7 +22,7 @@ "# Week 37: Statitsitcal interpretations and Resampling Methods\n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University\n", "\n", - "Date: **Sep 11, 2023**\n", + "Date: **Sep 14, 2023**\n", "\n", "Copyright 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -31,7 +31,7 @@ }, { "cell_type": "markdown", - "id": "7132df07", + "id": "ccaa7ab9", "metadata": { "editable": true }, @@ -46,29 +46,35 @@ "\n", " * Work on project 1\n", "\n", - " * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session.\n", + " * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.\n", "\n", " * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", "\n", " \n", "**Material for the lecture on Thursday September 7.**\n", "\n", - " * Statistical interpretation of Ridge and Lasso regression\n", + " * [Video of Lecture](https://youtu.be/YOBBr_toYxc)\n", + "\n", + " * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf)\n", "\n", " * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff\n", "\n", - " * Reads and Videos:\n", + " * Statistical interpretation of Ridge and Lasso regression\n", + "\n", + " * Readings and Videos:\n", "\n", " * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). \n", "\n", " * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)\n", "\n", + " * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)\n", + "\n", " * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)" ] }, { "cell_type": "markdown", - "id": "0928fd34", + "id": "1c5c27bf", "metadata": { "editable": true }, @@ -78,7 +84,7 @@ }, { "cell_type": "markdown", - "id": "9ee88de5", + "id": "656c8db6", "metadata": { "editable": true }, @@ -107,7 +113,7 @@ }, { "cell_type": "markdown", - "id": "fd2470fe", + "id": "4c35f924", "metadata": { "editable": true }, @@ -123,7 +129,7 @@ }, { "cell_type": "markdown", - "id": "8be57b04", + "id": "9087bc37", "metadata": { "editable": true }, @@ -142,7 +148,7 @@ }, { "cell_type": "markdown", - "id": "ee33fde6", + "id": "8a7bb057", "metadata": { "editable": true }, @@ -156,7 +162,7 @@ }, { "cell_type": "markdown", - "id": "14124ae5", + "id": "68392e55", "metadata": { "editable": true }, @@ -168,7 +174,7 @@ }, { "cell_type": "markdown", - "id": "02600bff", + "id": "27cd8fb1", "metadata": { "editable": true }, @@ -179,7 +185,7 @@ }, { "cell_type": "markdown", - "id": "7df4b213", + "id": "f0344126", "metadata": { "editable": true }, @@ -191,7 +197,7 @@ }, { "cell_type": "markdown", - "id": "996d8e2d", + "id": "d23667d2", "metadata": { "editable": true }, @@ -203,7 +209,7 @@ }, { "cell_type": "markdown", - "id": "815b60ed", + "id": "63687c52", "metadata": { "editable": true }, @@ -219,7 +225,7 @@ }, { "cell_type": "markdown", - "id": "a93b64c7", + "id": "54e07f66", "metadata": { "editable": true }, @@ -230,7 +236,7 @@ }, { "cell_type": "markdown", - "id": "61c06c9b", + "id": "de6492ed", "metadata": { "editable": true }, @@ -253,7 +259,7 @@ }, { "cell_type": "markdown", - "id": "bb1eb3a5", + "id": "a487e7d3", "metadata": { "editable": true }, @@ -264,7 +270,7 @@ }, { "cell_type": "markdown", - "id": "ea804d82", + "id": "dbac7b3f", "metadata": { "editable": true }, @@ -276,7 +282,7 @@ }, { "cell_type": "markdown", - "id": "896c9968", + "id": "edc35885", "metadata": { "editable": true }, @@ -288,7 +294,7 @@ }, { "cell_type": "markdown", - "id": "27adffa2", + "id": "5b79b613", "metadata": { "editable": true }, @@ -302,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "b3cb4e94", + "id": "b435eaee", "metadata": { "editable": true }, @@ -311,13 +317,13 @@ "\\begin{eqnarray*}\n", "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", "\\\\\n", - "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n", + "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} - \\boldsymbol{\\beta}]^{T} \\}\n", "\\\\\n", - "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", "% \\\\\n", - "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{y} \\, \\mathbf{y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", "% \\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{y} \\, \\mathbf{y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", "\\\\\n", "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", "% \\\\\n", @@ -333,12 +339,12 @@ }, { "cell_type": "markdown", - "id": "98964900", + "id": "5aaad09e", "metadata": { "editable": true }, "source": [ - "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", + "where we have used that $\\mathbb{E} (\\mathbf{y} \\mathbf{y}^{T}) =\n", "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", @@ -355,44 +361,44 @@ }, { "cell_type": "markdown", - "id": "ae46385d", + "id": "276770f2", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", + "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "862ec8d1", + "id": "ac99320c", "metadata": { "editable": true }, "source": [ "We see clearly that \n", - "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n", + "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n", "\n", "We can also compute the variance as" ] }, { "cell_type": "markdown", - "id": "1c05ca70", + "id": "9bb35322", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", + "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", "$$" ] }, { "cell_type": "markdown", - "id": "a714192d", + "id": "83113c02", "metadata": { "editable": true }, @@ -404,19 +410,19 @@ }, { "cell_type": "markdown", - "id": "bd80e6ae", + "id": "2dea05a4", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", + "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "272d7c5c", + "id": "f8a821f1", "metadata": { "editable": true }, @@ -430,7 +436,7 @@ }, { "cell_type": "markdown", - "id": "c87b5aa5", + "id": "ff9cbfd6", "metadata": { "editable": true }, @@ -440,7 +446,7 @@ }, { "cell_type": "markdown", - "id": "c946f771", + "id": "2592817d", "metadata": { "editable": true }, @@ -463,7 +469,7 @@ }, { "cell_type": "markdown", - "id": "e6b4e7dd", + "id": "b134aee7", "metadata": { "editable": true }, @@ -475,7 +481,7 @@ }, { "cell_type": "markdown", - "id": "8169d91a", + "id": "2040959c", "metadata": { "editable": true }, @@ -488,7 +494,7 @@ }, { "cell_type": "markdown", - "id": "99e90d11", + "id": "130fdf44", "metadata": { "editable": true }, @@ -500,7 +506,7 @@ }, { "cell_type": "markdown", - "id": "0b4adc26", + "id": "2e162d22", "metadata": { "editable": true }, @@ -512,7 +518,7 @@ }, { "cell_type": "markdown", - "id": "9ca9e238", + "id": "a2e8bbf0", "metadata": { "editable": true }, @@ -524,7 +530,7 @@ }, { "cell_type": "markdown", - "id": "23dcb3a5", + "id": "92c5755a", "metadata": { "editable": true }, @@ -535,7 +541,7 @@ }, { "cell_type": "markdown", - "id": "67cb2d01", + "id": "f3b2d261", "metadata": { "editable": true }, @@ -547,7 +553,7 @@ }, { "cell_type": "markdown", - "id": "80d807da", + "id": "9d4d6500", "metadata": { "editable": true }, @@ -558,7 +564,7 @@ }, { "cell_type": "markdown", - "id": "14b1cbfd", + "id": "9a18f7aa", "metadata": { "editable": true }, @@ -570,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "cf86395e", + "id": "ead724f7", "metadata": { "editable": true }, @@ -580,7 +586,7 @@ }, { "cell_type": "markdown", - "id": "4eee417a", + "id": "7aeab260", "metadata": { "editable": true }, @@ -611,7 +617,7 @@ }, { "cell_type": "markdown", - "id": "02893846", + "id": "43afa9cd", "metadata": { "editable": true }, @@ -623,7 +629,7 @@ }, { "cell_type": "markdown", - "id": "098674aa", + "id": "2bbe6c3a", "metadata": { "editable": true }, @@ -635,7 +641,7 @@ }, { "cell_type": "markdown", - "id": "76b18e95", + "id": "1106ec56", "metadata": { "editable": true }, @@ -645,7 +651,7 @@ }, { "cell_type": "markdown", - "id": "52c82f5e", + "id": "cc4efbb2", "metadata": { "editable": true }, @@ -657,7 +663,7 @@ }, { "cell_type": "markdown", - "id": "46e101f5", + "id": "031ab659", "metadata": { "editable": true }, @@ -667,7 +673,7 @@ }, { "cell_type": "markdown", - "id": "e8527450", + "id": "09f14189", "metadata": { "editable": true }, @@ -679,7 +685,7 @@ }, { "cell_type": "markdown", - "id": "a421d3d5", + "id": "775ed960", "metadata": { "editable": true }, @@ -689,7 +695,7 @@ }, { "cell_type": "markdown", - "id": "22137f8a", + "id": "f0e242cf", "metadata": { "editable": true }, @@ -701,7 +707,7 @@ }, { "cell_type": "markdown", - "id": "4eb8b2fa", + "id": "aa1d0f4e", "metadata": { "editable": true }, @@ -711,7 +717,7 @@ }, { "cell_type": "markdown", - "id": "8752defc", + "id": "04b1875f", "metadata": { "editable": true }, @@ -731,7 +737,7 @@ }, { "cell_type": "markdown", - "id": "4befff39", + "id": "fd5fd7d3", "metadata": { "editable": true }, @@ -743,7 +749,7 @@ }, { "cell_type": "markdown", - "id": "d18ce941", + "id": "dbb5f8dd", "metadata": { "editable": true }, @@ -753,7 +759,7 @@ }, { "cell_type": "markdown", - "id": "0a4fd449", + "id": "7cda70f2", "metadata": { "editable": true }, @@ -765,7 +771,7 @@ }, { "cell_type": "markdown", - "id": "ec27160e", + "id": "ec080569", "metadata": { "editable": true }, @@ -777,7 +783,7 @@ }, { "cell_type": "markdown", - "id": "cd11746f", + "id": "3b12ffc4", "metadata": { "editable": true }, @@ -789,7 +795,7 @@ }, { "cell_type": "markdown", - "id": "b966cbbf", + "id": "b347871f", "metadata": { "editable": true }, @@ -801,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "bccd1443", + "id": "7af990c8", "metadata": { "editable": true }, @@ -813,7 +819,7 @@ }, { "cell_type": "markdown", - "id": "7b6361b1", + "id": "ff6fb728", "metadata": { "editable": true }, @@ -825,7 +831,7 @@ }, { "cell_type": "markdown", - "id": "f1ff2fe1", + "id": "9b9c9cc4", "metadata": { "editable": true }, @@ -837,7 +843,7 @@ }, { "cell_type": "markdown", - "id": "ccc6b096", + "id": "294e9ad9", "metadata": { "editable": true }, @@ -849,7 +855,7 @@ }, { "cell_type": "markdown", - "id": "86507503", + "id": "8021f77b", "metadata": { "editable": true }, @@ -859,7 +865,7 @@ }, { "cell_type": "markdown", - "id": "bee1fbb6", + "id": "78fbc86a", "metadata": { "editable": true }, @@ -871,7 +877,7 @@ }, { "cell_type": "markdown", - "id": "ae2e7deb", + "id": "ab52abe5", "metadata": { "editable": true }, @@ -881,7 +887,7 @@ }, { "cell_type": "markdown", - "id": "8e262661", + "id": "1f809753", "metadata": { "editable": true }, @@ -900,7 +906,7 @@ }, { "cell_type": "markdown", - "id": "5de65e5b", + "id": "908364e7", "metadata": { "editable": true }, @@ -921,7 +927,7 @@ }, { "cell_type": "markdown", - "id": "6f1ba40c", + "id": "6e5c83b1", "metadata": { "editable": true }, @@ -933,7 +939,7 @@ }, { "cell_type": "markdown", - "id": "0e4cdf38", + "id": "5653360e", "metadata": { "editable": true }, @@ -944,7 +950,7 @@ }, { "cell_type": "markdown", - "id": "76584b5c", + "id": "e617edc1", "metadata": { "editable": true }, @@ -957,7 +963,7 @@ }, { "cell_type": "markdown", - "id": "a879a6bd", + "id": "ecb9f426", "metadata": { "editable": true }, @@ -969,7 +975,7 @@ }, { "cell_type": "markdown", - "id": "dd0274e0", + "id": "eb2347e9", "metadata": { "editable": true }, @@ -979,7 +985,7 @@ }, { "cell_type": "markdown", - "id": "684fbc25", + "id": "ce975ccd", "metadata": { "editable": true }, @@ -991,7 +997,7 @@ }, { "cell_type": "markdown", - "id": "54cd9a76", + "id": "4b0ad706", "metadata": { "editable": true }, @@ -1001,7 +1007,7 @@ }, { "cell_type": "markdown", - "id": "051338d6", + "id": "249c79a0", "metadata": { "editable": true }, @@ -1013,7 +1019,7 @@ }, { "cell_type": "markdown", - "id": "e5f3d505", + "id": "6497aa62", "metadata": { "editable": true }, @@ -1023,7 +1029,7 @@ }, { "cell_type": "markdown", - "id": "ef60f6e2", + "id": "36ac6a6c", "metadata": { "editable": true }, @@ -1037,7 +1043,7 @@ }, { "cell_type": "markdown", - "id": "dde5ce68", + "id": "0667ce18", "metadata": { "editable": true }, @@ -1049,7 +1055,7 @@ }, { "cell_type": "markdown", - "id": "428b7cfc", + "id": "23e6bbe1", "metadata": { "editable": true }, @@ -1059,7 +1065,7 @@ }, { "cell_type": "markdown", - "id": "95ad5eba", + "id": "1303e4fa", "metadata": { "editable": true }, @@ -1071,7 +1077,7 @@ }, { "cell_type": "markdown", - "id": "6e6b706c", + "id": "07a72af1", "metadata": { "editable": true }, @@ -1081,7 +1087,7 @@ }, { "cell_type": "markdown", - "id": "6e6a60cb", + "id": "a0fdb3bd", "metadata": { "editable": true }, @@ -1093,7 +1099,7 @@ }, { "cell_type": "markdown", - "id": "c7127db8", + "id": "0c191fc5", "metadata": { "editable": true }, @@ -1103,7 +1109,7 @@ }, { "cell_type": "markdown", - "id": "60288e6d", + "id": "05a216b4", "metadata": { "editable": true }, @@ -1115,7 +1121,7 @@ }, { "cell_type": "markdown", - "id": "0e8d9dfd", + "id": "21d80a1c", "metadata": { "editable": true }, @@ -1125,7 +1131,7 @@ }, { "cell_type": "markdown", - "id": "c0450362", + "id": "06073a94", "metadata": { "editable": true }, @@ -1141,7 +1147,7 @@ }, { "cell_type": "markdown", - "id": "5911b76d", + "id": "aaf3ac26", "metadata": { "editable": true }, @@ -1153,7 +1159,7 @@ }, { "cell_type": "markdown", - "id": "6029916b", + "id": "5ff662ce", "metadata": { "editable": true }, @@ -1163,7 +1169,7 @@ }, { "cell_type": "markdown", - "id": "b704a6fd", + "id": "2c210fdb", "metadata": { "editable": true }, @@ -1175,7 +1181,7 @@ }, { "cell_type": "markdown", - "id": "60c7948c", + "id": "ebbc15d5", "metadata": { "editable": true }, @@ -1188,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "51dcd9c3", + "id": "b9123830", "metadata": { "editable": true }, @@ -1200,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "c6512c09", + "id": "9fb1e84e", "metadata": { "editable": true }, @@ -1210,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "2c5ca7b5", + "id": "c48c43d0", "metadata": { "editable": true }, @@ -1222,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "1c530682", + "id": "ed19a572", "metadata": { "editable": true }, @@ -1232,7 +1238,7 @@ }, { "cell_type": "markdown", - "id": "81771a49", + "id": "65ea42da", "metadata": { "editable": true }, @@ -1244,7 +1250,7 @@ }, { "cell_type": "markdown", - "id": "c2aac5ba", + "id": "587ea365", "metadata": { "editable": true }, @@ -1256,7 +1262,7 @@ }, { "cell_type": "markdown", - "id": "d2e548d7", + "id": "55422904", "metadata": { "editable": true }, @@ -1266,7 +1272,7 @@ }, { "cell_type": "markdown", - "id": "2901f9c7", + "id": "55e99fd7", "metadata": { "editable": true }, @@ -1278,7 +1284,7 @@ }, { "cell_type": "markdown", - "id": "8b0dc761", + "id": "fb8e8c1e", "metadata": { "editable": true }, @@ -1290,7 +1296,7 @@ }, { "cell_type": "markdown", - "id": "81709208", + "id": "72c3b3dc", "metadata": { "editable": true }, @@ -1302,7 +1308,7 @@ }, { "cell_type": "markdown", - "id": "757cbc8c", + "id": "348550b2", "metadata": { "editable": true }, @@ -1312,7 +1318,7 @@ }, { "cell_type": "markdown", - "id": "337a92ec", + "id": "6c3f6965", "metadata": { "editable": true }, @@ -1324,7 +1330,7 @@ }, { "cell_type": "markdown", - "id": "170ec18c", + "id": "24532f00", "metadata": { "editable": true }, @@ -1334,7 +1340,7 @@ }, { "cell_type": "markdown", - "id": "1919001a", + "id": "7634be1d", "metadata": { "editable": true }, @@ -1353,7 +1359,7 @@ }, { "cell_type": "markdown", - "id": "861c96a1", + "id": "5dcc9cb2", "metadata": { "editable": true }, @@ -1381,7 +1387,7 @@ }, { "cell_type": "markdown", - "id": "9e8d9abb", + "id": "0971b192", "metadata": { "editable": true }, @@ -1407,7 +1413,7 @@ }, { "cell_type": "markdown", - "id": "cca36e61", + "id": "00c350e4", "metadata": { "editable": true }, @@ -1424,7 +1430,7 @@ }, { "cell_type": "markdown", - "id": "060a4427", + "id": "aa3988c6", "metadata": { "editable": true }, @@ -1444,7 +1450,7 @@ }, { "cell_type": "markdown", - "id": "bc631501", + "id": "9868a98b", "metadata": { "editable": true }, @@ -1473,7 +1479,7 @@ }, { "cell_type": "markdown", - "id": "65212453", + "id": "5015585e", "metadata": { "editable": true }, @@ -1498,7 +1504,7 @@ }, { "cell_type": "markdown", - "id": "ce801e18", + "id": "92d9273d", "metadata": { "editable": true }, @@ -1518,7 +1524,7 @@ }, { "cell_type": "markdown", - "id": "927e2be5", + "id": "cd12bb37", "metadata": { "editable": true }, @@ -1530,7 +1536,7 @@ }, { "cell_type": "markdown", - "id": "fe6c7817", + "id": "cd8fc890", "metadata": { "editable": true }, @@ -1540,7 +1546,7 @@ }, { "cell_type": "markdown", - "id": "b0ae80c3", + "id": "265ab8d1", "metadata": { "editable": true }, @@ -1555,7 +1561,7 @@ }, { "cell_type": "markdown", - "id": "b8bf320d", + "id": "cdcb3e67", "metadata": { "editable": true }, @@ -1568,7 +1574,7 @@ }, { "cell_type": "markdown", - "id": "62def127", + "id": "89a036fe", "metadata": { "editable": true }, @@ -1581,7 +1587,7 @@ }, { "cell_type": "markdown", - "id": "71d882fb", + "id": "db8fc317", "metadata": { "editable": true }, @@ -1593,7 +1599,7 @@ }, { "cell_type": "markdown", - "id": "37f348c8", + "id": "8f878995", "metadata": { "editable": true }, @@ -1606,7 +1612,7 @@ }, { "cell_type": "markdown", - "id": "0b949710", + "id": "705c200b", "metadata": { "editable": true }, @@ -1617,7 +1623,7 @@ }, { "cell_type": "markdown", - "id": "673d5c06", + "id": "594730bc", "metadata": { "editable": true }, @@ -1631,7 +1637,7 @@ }, { "cell_type": "markdown", - "id": "81c0beaf", + "id": "50cedd29", "metadata": { "editable": true }, @@ -1641,7 +1647,7 @@ }, { "cell_type": "markdown", - "id": "e83202fb", + "id": "6d7b9a46", "metadata": { "editable": true }, @@ -1655,7 +1661,7 @@ }, { "cell_type": "markdown", - "id": "27744da9", + "id": "ffeff7ca", "metadata": { "editable": true }, @@ -1668,7 +1674,7 @@ }, { "cell_type": "markdown", - "id": "420c8688", + "id": "caf853d3", "metadata": { "editable": true }, @@ -1681,7 +1687,7 @@ }, { "cell_type": "markdown", - "id": "591e95ca", + "id": "b0237a6a", "metadata": { "editable": true }, @@ -1691,7 +1697,7 @@ }, { "cell_type": "markdown", - "id": "a044c8a4", + "id": "4475d0be", "metadata": { "editable": true }, @@ -1704,7 +1710,7 @@ }, { "cell_type": "markdown", - "id": "22b21609", + "id": "1ba48e74", "metadata": { "editable": true }, @@ -1714,7 +1720,7 @@ }, { "cell_type": "markdown", - "id": "461ba192", + "id": "6561a4db", "metadata": { "editable": true }, @@ -1727,7 +1733,7 @@ }, { "cell_type": "markdown", - "id": "97d21082", + "id": "a0416060", "metadata": { "editable": true }, @@ -1739,7 +1745,7 @@ }, { "cell_type": "markdown", - "id": "1231f2f0", + "id": "5b86aaee", "metadata": { "editable": true }, @@ -1758,7 +1764,7 @@ }, { "cell_type": "markdown", - "id": "def2d3ca", + "id": "4620502c", "metadata": { "editable": true }, @@ -1771,7 +1777,7 @@ }, { "cell_type": "markdown", - "id": "acdeb582", + "id": "03867622", "metadata": { "editable": true }, @@ -1783,7 +1789,7 @@ }, { "cell_type": "markdown", - "id": "ad0ce315", + "id": "c4908940", "metadata": { "editable": true }, @@ -1796,7 +1802,7 @@ }, { "cell_type": "markdown", - "id": "6be519fc", + "id": "c74f36e7", "metadata": { "editable": true }, @@ -1816,7 +1822,7 @@ }, { "cell_type": "markdown", - "id": "72497b42", + "id": "e497bcf9", "metadata": { "editable": true }, @@ -1839,7 +1845,7 @@ }, { "cell_type": "markdown", - "id": "7d663e8f", + "id": "60541de0", "metadata": { "editable": true }, @@ -1854,7 +1860,7 @@ }, { "cell_type": "markdown", - "id": "4925ba6e", + "id": "62737ab8", "metadata": { "editable": true }, @@ -1866,7 +1872,7 @@ }, { "cell_type": "markdown", - "id": "9fbe8108", + "id": "96722498", "metadata": { "editable": true }, @@ -1886,7 +1892,7 @@ }, { "cell_type": "markdown", - "id": "b6b2b234", + "id": "c1242f98", "metadata": { "editable": true }, @@ -1906,7 +1912,7 @@ }, { "cell_type": "markdown", - "id": "82bd2034", + "id": "9d6890f4", "metadata": { "editable": true }, @@ -1930,7 +1936,7 @@ }, { "cell_type": "markdown", - "id": "61a94d24", + "id": "eaebe344", "metadata": { "editable": true }, @@ -1951,7 +1957,7 @@ }, { "cell_type": "markdown", - "id": "fd22fd14", + "id": "17ee9799", "metadata": { "editable": true }, @@ -1981,7 +1987,7 @@ }, { "cell_type": "markdown", - "id": "3497401f", + "id": "ea87812a", "metadata": { "editable": true }, @@ -2005,7 +2011,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "3ef8772f", + "id": "a21ecc6d", "metadata": { "collapsed": false, "editable": true @@ -2044,7 +2050,7 @@ }, { "cell_type": "markdown", - "id": "ffe9e08b", + "id": "79da869e", "metadata": { "editable": true }, @@ -2054,7 +2060,7 @@ }, { "cell_type": "markdown", - "id": "c8770096", + "id": "ce9b96bc", "metadata": { "editable": true }, @@ -2065,7 +2071,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "ac35a65b", + "id": "1e35dddb", "metadata": { "collapsed": false, "editable": true @@ -2085,7 +2091,7 @@ }, { "cell_type": "markdown", - "id": "e4ad71a5", + "id": "3bdc7fb8", "metadata": { "editable": true }, @@ -2103,7 +2109,7 @@ }, { "cell_type": "markdown", - "id": "edcf8b5c", + "id": "8854ebad", "metadata": { "editable": true }, @@ -2115,7 +2121,7 @@ }, { "cell_type": "markdown", - "id": "77b70dca", + "id": "677d1f32", "metadata": { "editable": true }, @@ -2132,7 +2138,7 @@ }, { "cell_type": "markdown", - "id": "07cccff6", + "id": "4ffc6245", "metadata": { "editable": true }, @@ -2144,7 +2150,7 @@ }, { "cell_type": "markdown", - "id": "b04fa9bd", + "id": "6059ec46", "metadata": { "editable": true }, @@ -2154,7 +2160,7 @@ }, { "cell_type": "markdown", - "id": "c79570c2", + "id": "1d29cec4", "metadata": { "editable": true }, @@ -2166,7 +2172,7 @@ }, { "cell_type": "markdown", - "id": "9a8a6a3b", + "id": "7ef17636", "metadata": { "editable": true }, @@ -2183,7 +2189,7 @@ }, { "cell_type": "markdown", - "id": "a9a892c2", + "id": "17963e0a", "metadata": { "editable": true }, @@ -2195,7 +2201,7 @@ }, { "cell_type": "markdown", - "id": "0057120a", + "id": "58693061", "metadata": { "editable": true }, @@ -2205,7 +2211,7 @@ }, { "cell_type": "markdown", - "id": "9e9b0c0f", + "id": "ed10ae30", "metadata": { "editable": true }, @@ -2217,7 +2223,7 @@ }, { "cell_type": "markdown", - "id": "4b28d5f5", + "id": "2ca576c8", "metadata": { "editable": true }, @@ -2227,7 +2233,7 @@ }, { "cell_type": "markdown", - "id": "3fbb0a03", + "id": "de708fa8", "metadata": { "editable": true }, @@ -2239,7 +2245,7 @@ }, { "cell_type": "markdown", - "id": "c367d950", + "id": "45ce65a1", "metadata": { "editable": true }, @@ -2249,7 +2255,7 @@ }, { "cell_type": "markdown", - "id": "e71acf60", + "id": "ca73b735", "metadata": { "editable": true }, @@ -2265,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "93070f01", + "id": "a460cbc4", "metadata": { "editable": true }, @@ -2276,7 +2282,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "a46b37f7", + "id": "520c4af0", "metadata": { "collapsed": false, "editable": true @@ -2341,7 +2347,7 @@ }, { "cell_type": "markdown", - "id": "152a6857", + "id": "e3e8854d", "metadata": { "editable": true }, @@ -2352,7 +2358,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "c385948e", + "id": "642a16bf", "metadata": { "collapsed": false, "editable": true @@ -2409,7 +2415,7 @@ }, { "cell_type": "markdown", - "id": "a6ddc62c", + "id": "143e1777", "metadata": { "editable": true }, @@ -2447,7 +2453,7 @@ }, { "cell_type": "markdown", - "id": "3ff69bcd", + "id": "0ddb1890", "metadata": { "editable": true }, @@ -2458,7 +2464,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "16bd46aa", + "id": "74c7f5dc", "metadata": { "collapsed": false, "editable": true @@ -2540,7 +2546,7 @@ }, { "cell_type": "markdown", - "id": "d898e22e", + "id": "8f89de90", "metadata": { "editable": true }, @@ -2565,7 +2571,7 @@ }, { "cell_type": "markdown", - "id": "b3c4bc74", + "id": "9465a17d", "metadata": { "editable": true }, @@ -2593,7 +2599,7 @@ }, { "cell_type": "markdown", - "id": "35277af5", + "id": "d0e4b197", "metadata": { "editable": true }, @@ -2606,7 +2612,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "6e8a01e6", + "id": "8a0b7b89", "metadata": { "collapsed": false, "editable": true @@ -2706,7 +2712,7 @@ }, { "cell_type": "markdown", - "id": "addf22bf", + "id": "a41beafb", "metadata": { "editable": true }, @@ -2717,7 +2723,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "f1af3e2c", + "id": "e3038a58", "metadata": { "collapsed": false, "editable": true @@ -2806,7 +2812,7 @@ }, { "cell_type": "markdown", - "id": "c05b2a3a", + "id": "19f4f20f", "metadata": { "editable": true }, @@ -2816,7 +2822,7 @@ }, { "cell_type": "markdown", - "id": "be50bfa2", + "id": "98b035c3", "metadata": { "editable": true }, @@ -2829,7 +2835,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "33fb2f51", + "id": "6ce294be", "metadata": { "collapsed": false, "editable": true @@ -2904,6 +2910,626 @@ "plt.legend()\n", "plt.show()" ] + }, + { + "cell_type": "markdown", + "id": "63a7497b", + "metadata": { + "editable": true + }, + "source": [ + "## Notes on scaling with examples\n", + "\n", + "The programs here use both ordinrary least squares (OLS) and Ridge\n", + "regression with one value only for the hyperparameter $\\lambda$. The\n", + "first example has no scaling and includes the intercept as well and we\n", + "are trying to fit a second-order polynomial. The second code takes out\n", + "the intercept and subtracts the mean values of each column of the\n", + "design matrix and the mean value of the outputs.\n", + "\n", + "The third and final code uses **Scikit-Learn** as library in order to\n", + "calculate the optimal parameters for OLS and Ridge regression. Note\n", + "that it is highly recommended to not include the intercept in Ridge\n", + "and Lasso regression, in order to avoid penalizing the optimization by\n", + "the intercept. The second and third codes do thus not include the\n", + "intercept. In the second code we do the scaling ourselves while the\n", + "last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where\n", + "we subtract the mean values)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "938f039a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.linear_model import LinearRegression\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.model_selection import train_test_split\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", + "\n", + "def OLS_fit_beta(X, y):\n", + " return np.linalg.pinv(X.T @ X) @ X.T @ y\n", + "\n", + "def Ridge_fit_beta(X, y,L,d):\n", + " I = np.eye(d,d)\n", + " return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y\n", + "\n", + "# Same random numbers for each test.\n", + "np.random.seed(2018)\n", + "n = 100\n", + "d = 3\n", + "# hyperparameter lambda\n", + "Lambda = 0.01\n", + "\n", + "# Make data set, simple second-order polynomial\n", + "x = np.linspace(-3, 3, n)\n", + "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", + "\n", + "# The design matrix X includes the intercept and no scaling is made\n", + "X = np.zeros((len(x), d))\n", + "for p in range(d): \n", + " X[:, p] = x ** (p) \n", + "\n", + "\n", + "#Split data, no scaling is used and we include the intercept\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "\n", + "#Calculate beta, own code\n", + "beta_OLS = OLS_fit_beta(X_train, y_train)\n", + "beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)\n", + "print(beta_OLS)\n", + "print(beta_Ridge)\n", + "#predict value\n", + "ytilde_test_OLS = X_test @ beta_OLS\n", + "ytilde_test_Ridge = X_test @ beta_Ridge\n", + "\n", + "#Calculate MSE\n", + "print(\" \")\n", + "print(\"test MSE of OLS:\")\n", + "print(MSE(y_test,ytilde_test_OLS))\n", + "print(\" \")\n", + "print(\"test MSE of Ridge\")\n", + "print(MSE(y_test,ytilde_test_Ridge))\n", + "\n", + "plt.scatter(x,y,label='Data')\n", + "plt.plot(x, X @ beta_OLS,'*', label=\"OLS_Fit\")\n", + "plt.plot(x, X @ beta_Ridge, label=\"Ridge_Fit\")\n", + "plt.grid()\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e732961c", + "metadata": { + "editable": true + }, + "source": [ + "In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).\n", + "see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).\n", + "It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.\n", + "\n", + "Before we discuss the code, we repeat some of the basic math from the slides of week 36.\n", + "\n", + "Let us try to understand what this may imply mathematically when we\n", + "subtract the mean values, also known as *zero centering* or simply *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": "4c89151d", + "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": "52318e4d", + "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": "b8dc7f6d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0ea24a0d", + "metadata": { + "editable": true + }, + "source": [ + "for all $j$. For $\\beta_0$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "47719be0", + "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": "4be82ca9", + "metadata": { + "editable": true + }, + "source": [ + "Multiplying away the constant $2/n$, we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "778ad469", + "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": "c8690ebb", + "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": "19422394", + "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": "bc1ec19a", + "metadata": { + "editable": true + }, + "source": [ + "We obtain then" + ] + }, + { + "cell_type": "markdown", + "id": "61370bdd", + "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": "9ff08be6", + "metadata": { + "editable": true + }, + "source": [ + "If we define" + ] + }, + { + "cell_type": "markdown", + "id": "9cd5def5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b080c860", + "metadata": { + "editable": true + }, + "source": [ + "and the mean value of the outputs as" + ] + }, + { + "cell_type": "markdown", + "id": "10fc1a84", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "df6a1ef4", + "metadata": { + "editable": true + }, + "source": [ + "we have" + ] + }, + { + "cell_type": "markdown", + "id": "c62e999c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e16831f8", + "metadata": { + "editable": true + }, + "source": [ + "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have" + ] + }, + { + "cell_type": "markdown", + "id": "863e58fa", + "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": "b146fe2b", + "metadata": { + "editable": true + }, + "source": [ + "We can rewrite the latter equation as" + ] + }, + { + "cell_type": "markdown", + "id": "89ca9b85", + "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": "323f217d", + "metadata": { + "editable": true + }, + "source": [ + "where we have defined" + ] + }, + { + "cell_type": "markdown", + "id": "ce63593a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "07c90e67", + "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": "467273e3", + "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": "eaebf350", + "metadata": { + "editable": true + }, + "source": [ + "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then" + ] + }, + { + "cell_type": "markdown", + "id": "bdb72a09", + "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": "68e0b318", + "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": "9c298056", + "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": "fa2668d8", + "metadata": { + "editable": true + }, + "source": [ + "Now we try to implement this." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "010ff40f", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "np.random.seed(2018)\n", + "n = 100\n", + "# we do not include the intercept\n", + "d = 2\n", + "Lambda = 0.01\n", + "\n", + "# Make data set.\n", + "x = np.linspace(-3, 3, n)\n", + "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", + "\n", + "#Design matrix X does not include the intercept. \n", + "X = np.zeros((len(x), d))\n", + "for p in range(d): \n", + " X[:, p] = x ** (p+1)\n", + "\n", + "\n", + "#Split data in train and test\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "# Scale data by subtracting mean value,own implementation\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, note)\n", + "y_scaler = np.mean(y_train)\n", + "y_train_scaled = y_train - y_scaler\n", + "\n", + "\n", + "#Calculate beta\n", + "beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)\n", + "beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)\n", + "print(beta_OLS)\n", + "print(beta_Ridge)\n", + "# calculate intercepts and print them\n", + "interceptOLS = y_scaler - X_train_mean @ beta_OLS\n", + "interceptRidge = y_scaler - X_train_mean @ beta_Ridge\n", + "print(interceptOLS)\n", + "print(interceptRidge)\n", + "\n", + "#predict value with intercept\n", + "ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler\n", + "ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler\n", + "\n", + "\n", + "#Calculate MSE\n", + "\n", + "print(\" \")\n", + "print(\"test MSE of OLS:\")\n", + "print(MSE(y_test,ytilde_test_OLS))\n", + "print(\" \")\n", + "print(\"test MSE of Ridge\")\n", + "print(MSE(y_test,ytilde_test_Ridge))\n", + "\n", + "plt.scatter(x,y,label='Data')\n", + "plt.plot(x, X @ beta_OLS+interceptOLS,'*', label=\"OLS_Fit\")\n", + "plt.plot(x, X @ beta_Ridge+interceptRidge, label=\"Ridge_Fit\")\n", + "plt.grid()\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e857615a", + "metadata": { + "editable": true + }, + "source": [ + "Finally, instead of using our own function we repeat the same example\n", + "using the **standardscaler** functionality of the library\n", + "**Scikit-Learn**. Here we limit ourselves to Ridge regression only." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "a3ad5ac3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "from sklearn import linear_model\n", + "np.random.seed(2018)\n", + "n = 10\n", + "d = 2\n", + "Lambda = 0.01\n", + "\n", + "# Make data set.\n", + "x = np.linspace(-3, 3, n)\n", + "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n", + "\n", + "# Design matrix X does not include the intercept. \n", + "X = np.zeros((n, d))\n", + "for p in range(d): \n", + " X[:, p] = x ** (p+1)\n", + "\n", + "#Split data in train and test\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "# Scale data by subtracting mean value of the input using scikit-learn\n", + "scaler = StandardScaler(with_std=False)\n", + "scaler.fit(X_train)\n", + "X_train_mean = np.mean(X_train,axis=0)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "# We scale also the output, here by our own code\n", + "y_scaler = np.mean(y_train)\n", + "y_train_scaled = y_train - y_scaler\n", + "y_test_scaled = y_test- y_scaler\n", + "\n", + "#Calculate beta\n", + "OLS = LinearRegression()\n", + "betaOLS=OLS.fit(X_train_scaled,y_train_scaled)\n", + "ypredictOLS = OLS.predict(X_test_scaled)\n", + "linear_model.Ridge(Lambda)\n", + "RegRidge.fit(X_train_scaled,y_train_scaled)\n", + "ypredictRidge = RegRidge.predict(X_test_scaled)\n", + "betaOLS = OLS.coef_\n", + "betaRidge = RegRidge.coef_\n", + "print(betaOLS)\n", + "print(betaRidge)\n", + "interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS\n", + "interceptRidge = y_scaler - X_train_mean @ betaRidge\n", + "print(interceptOLS)\n", + "print(interceptRidge)\n", + "#predict value \n", + "ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler\n", + "ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler\n", + "\n", + "#Calculate MSE\n", + "print(\" \")\n", + "print(\"test MSE of OLS\")\n", + "print(MSE(y_test,ytilde_test_OLS))\n", + "print(\" \")\n", + "print(\"test MSE of Ridge\")\n", + "print(MSE(y_test,ytilde_test_Ridge))\n", + "plt.scatter(x,y,label='Data')\n", + "plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label=\"Ridge_Fit\")\n", + "plt.grid()\n", + "plt.legend()\n", + "plt.show()" + ] } ], "metadata": {}, diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js index 4db0feb6b..b9b7db8bf 100644 --- a/doc/LectureNotes/_build/html/searchindex.js +++ b/doc/LectureNotes/_build/html/searchindex.js @@ -1 +1 @@ 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Teaching schedule with links to material","1. 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques","Week 37: Statitsitcal interpretations and Resampling 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/html/week37.html b/doc/LectureNotes/_build/html/week37.html index ba5623255..4851c040d 100644 --- a/doc/LectureNotes/_build/html/week37.html +++ b/doc/LectureNotes/_build/html/week37.html @@ -642,6 +642,11 @@ const thebe_selector_output = ".output, .cell_output" The same example but now with cross-validation +

  • + + Notes on scaling with examples + +
    • @@ -933,6 +938,11 @@ const thebe_selector_output = ".output, .cell_output" The same example but now with cross-validation +
  • + + Notes on scaling with examples + +
    • @@ -947,7 +957,7 @@ doconce format html week37.do.txt --no_mako -->

    Week 37: Statitsitcal interpretations and Resampling Methods¶

    Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University

    -

    Date: Sep 11, 2023

    +

    Date: Sep 14, 2023

    Copyright 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license

    Plans for week 37¶

    @@ -956,17 +966,20 @@ doconce format html week37.do.txt --no_mako -->

  • Lecture from last week on calculations of expectation values

  • Exercise for week 37

  • Work on project 1

    • -

  • See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session.

    • +

  • See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.

  • For more discussions of Ridge regression and calculation of averages, Wessel van Wieringen’s article is highly recommended.

    • Material for the lecture on Thursday September 7.

      -

    • Statistical interpretation of Ridge and Lasso regression

      • +

    • Video of Lecture

      • +

    • Whiteboard notes

    • Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff

      • -

    • Reads and Videos:

      +

    • Statistical interpretation of Ridge and Lasso regression

      • +

    • Readings and Videos:

      • @@ -1073,13 +1086,13 @@ mean value \(\boldsymbol{X}\boldsymbo \begin{eqnarray*} \mbox{Var}(\boldsymbol{\hat{\beta}}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} \\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} +& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}]^{T} \} \\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} % \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} \, \mathbf{y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} % \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{y} \, \mathbf{y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \\ & = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} % \\ @@ -1091,7 +1104,7 @@ mean value \(\boldsymbol{X}\boldsymbo \, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, \end{eqnarray*} \end{split}\]
    -

    where we have used that \(\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = +

    where we have used that \(\mathbb{E} (\mathbf{y} \mathbf{y}^{T}) = \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \, \mathbf{I}_{nn}\). From \(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}\), one obtains an estimate of the @@ -1104,20 +1117,20 @@ when we employ Ridge regression, allowing us again to define a confidence interv

    It is rather straightforward to show that

    \[ -\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. +\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}. \]

    We see clearly that -\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\) for any \(\lambda > 0\).

    +\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\) for any \(\lambda > 0\).

    We can also compute the variance as

    \[ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, +\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, \]

    and it is easy to see that if the parameter \(\lambda\) goes to infinity then the variance of Ridge parameters \(\boldsymbol{\beta}\) goes to zero.

    With this, we can compute the difference

    \[ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. +\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. \]

    The difference is non-negative definite since each component of the matrix product is non-negative definite. @@ -1754,7 +1767,7 @@ theorem.

    Bootstrap Statistics :
 original           bias      std. error
- 100.232  14.9426        100.231        0.148895
+ 99.9713   14.943         99.972        0.149583
    @@ -2021,7 +2034,9 @@ Error: 0.021592704588021178 Bias^2: 0.010516485576646504 Var: 0.01107621901137467 0.021592704588021178 >= 0.010516485576646504 + 0.01107621901137467 = 0.021592704588021174 -Polynomial degree: 11 +
    + +
    Polynomial degree: 11
 Error: 0.07160048164232538
 Bias^2: 0.014436800088896381
 Var: 0.05716368155342902
@@ -2038,7 +2053,7 @@ Var: 0.20867052175003364
 0.2284246870217162 >= 0.01975416527168255 + 0.20867052175003364 = 0.2284246870217162
    -_images/week37_162_2.png +_images/week37_162_3.png @@ -2458,12 +2473,12 @@ Mean squared error on test data: 238.16356503 Degree of polynomial: 20 Mean squared error on training data: 0.00140849 Mean squared error on test data: 1345.68592431 -Degree of polynomial: 21 -Mean squared error on training data: 0.00119699 -Mean squared error on test data: 1836.21110005 -
    Degree of polynomial:  22
+
    Degree of polynomial:  21
+Mean squared error on training data: 0.00119699
+Mean squared error on test data: 1836.21110005
+Degree of polynomial:  22
 Mean squared error on training data: 0.00092904
 Mean squared error on test data: 1182.64316482
 Degree of polynomial:  23
@@ -2478,12 +2493,12 @@ Mean squared error on test data: 7697.35412147
 Degree of polynomial:  26
 Mean squared error on training data: 0.00075597
 Mean squared error on test data: 1078.81597834
-Degree of polynomial:  27
-Mean squared error on training data: 0.00068088
-Mean squared error on test data: 3189.20355156
 
    
 
    
-
    Degree of polynomial:  28
+
    Degree of polynomial:  27
+Mean squared error on training data: 0.00068088
+Mean squared error on test data: 3189.20355156
+Degree of polynomial:  28
 Mean squared error on training data: 0.00063364
 Mean squared error on test data: 692.24085321
 Degree of polynomial:  29
@@ -2491,9 +2506,9 @@ Mean squared error on training data: 0.00063862
 Mean squared error on test data: 3073.63180447
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_7768/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_31560/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_7768/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_31560/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 
    
 
    
@@ -2578,7 +2593,7 @@ Mean squared error on test data: 3073.63180447
 
    
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_7768/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_31560/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 
    
 
    
@@ -2586,6 +2601,361 @@ Mean squared error on test data: 3073.63180447
 
    
 
    
 
    
+
    
+
    Notes on scaling with examples¶
    
+
    The programs here use both ordinrary least squares (OLS) and Ridge
+regression with one value only for the hyperparameter \(\lambda\). The
+first example has no scaling and includes the intercept as well and we
+are trying to fit a second-order polynomial. The second code takes out
+the intercept and subtracts the mean values of each column of the
+design matrix and the mean value of the outputs.
    
+
    The third and final code uses Scikit-Learn as library in order to
+calculate the optimal parameters for OLS and Ridge regression. Note
+that it is highly recommended to not include the intercept in Ridge
+and Lasso regression, in order to avoid penalizing the optimization by
+the intercept. The second and third codes do thus not include the
+intercept. In the second code we do the scaling ourselves while the
+last code uses the standard scaler option included in Scikit-Learn, known as centering (where
+we subtract the mean values).
    
+
    
+
    
+
    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.linear_model import LinearRegression
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.model_selection import train_test_split
+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
+
+def OLS_fit_beta(X, y):
+    return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+def Ridge_fit_beta(X, y,L,d):
+    I = np.eye(d,d)
+    return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y
+
+# Same random numbers for each test.
+np.random.seed(2018)
+n = 100
+d = 3
+# hyperparameter lambda
+Lambda = 0.01
+
+# Make data set, simple second-order polynomial
+x = np.linspace(-3, 3, n)
+y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
+
+# The design matrix X includes the intercept and no scaling is made
+X = np.zeros((len(x), d))
+for p in range(d):     
+    X[:, p] = x ** (p) 
+
+
+#Split data, no scaling is used and we include the intercept
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+
+#Calculate beta, own code
+beta_OLS = OLS_fit_beta(X_train, y_train)
+beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)
+print(beta_OLS)
+print(beta_Ridge)
+#predict value
+ytilde_test_OLS = X_test @ beta_OLS
+ytilde_test_Ridge = X_test @ beta_Ridge
+
+#Calculate MSE
+print("  ")
+print("test MSE of OLS:")
+print(MSE(y_test,ytilde_test_OLS))
+print("  ")
+print("test MSE of Ridge")
+print(MSE(y_test,ytilde_test_Ridge))
+
+plt.scatter(x,y,label='Data')
+plt.plot(x, X @ beta_OLS,'*', label="OLS_Fit")
+plt.plot(x, X @ beta_Ridge, label="Ridge_Fit")
+plt.grid()
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    [1.79934087 0.47179152 5.01549939]
+[1.79909592 0.47176716 5.01550546]
+  
+test MSE of OLS:
+1.13943111290393
+  
+test MSE of Ridge
+1.1395235273363686
+
    
+
    
+_images/week37_176_1.png
+
    
+
    
+
    In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the lecture material.
+see also the weekly slides for week 36.
+It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.
    
+
    Before we discuss the code, we repeat some of the basic math from the slides of week 36.
    
+
    Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as zero centering or simply centering. For
+simplicity, we will focus on  ordinary regression, as done in the above example.
    
+
    The cost/loss function  for regression is
    
+
    
+\[
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+\]
    
+
    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.
    
+
    What we have done is to single out the \(\beta_0\) term in the
+definition of the mean squared error (MSE).  The design matrix \(X\)
+does in this case not contain any intercept column.  When we take the
+derivative with respect to \(\beta_0\), we want the derivative to obey
    
+
    
+\[
+\frac{\partial C}{\partial \beta_j} = 0,
+\]
    
+
    for all \(j\). For \(\beta_0\) we have
    
+
    
+\[
+\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+\]
    
+
    Multiplying away the constant \(2/n\), we obtain
    
+
    
+\[
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
+\]
    
+
    Let us specialize first to the case where we have only two parameters \(\beta_0\) and \(\beta_1\).
+Our result for \(\beta_0\) simplifies then to
    
+
    
+\[
+n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
+\]
    
+
    We obtain then
    
+
    
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
+\]
    
+
    If we define
    
+
    
+\[
+\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
+\]
    
+
    and the mean value of the outputs as
    
+
    
+\[
+\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
+\]
    
+
    we have
    
+
    
+\[
+\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
+\]
    
+
    In the general case with more parameters than \(\beta_0\) and \(\beta_1\), we have
    
+
    
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
+\]
    
+
    We can rewrite the latter equation as
    
+
    
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
+\]
    
+
    where we have defined
    
+
    
+\[
+\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
+\]
    
+
    the mean value for all elements of the column vector \(\boldsymbol{x}_j\).
    
+
    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)
    
+
    
+\[
+C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+\]
    
+
    If we minimize with respect to \(\boldsymbol{\beta}\) we have then
    
+
    
+\[
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
+\]
    
+
    where \(\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}\)
+and \(\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}\).
    
+
    For Ridge regression we need to add \(\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}\) to the cost function and get then
    
+
    
+\[
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+\]
    
+
    Now we try to implement this.
    
+
    
+
    
+
    np.random.seed(2018)
+n = 100
+# we do not include the intercept
+d = 2
+Lambda = 0.01
+
+# Make data set.
+x = np.linspace(-3, 3, n)
+y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
+
+#Design matrix X does not include the intercept. 
+X = np.zeros((len(x), d))
+for p in range(d):     
+    X[:, p] = x ** (p+1)
+
+
+#Split data in train and test
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+# Scale data by subtracting mean value,own implementation
+#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, note)
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+
+
+#Calculate beta
+beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)
+beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)
+print(beta_OLS)
+print(beta_Ridge)
+# calculate intercepts and print them
+interceptOLS = y_scaler - X_train_mean @ beta_OLS
+interceptRidge = y_scaler - X_train_mean @ beta_Ridge
+print(interceptOLS)
+print(interceptRidge)
+
+#predict value with intercept
+ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler
+ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler
+
+
+#Calculate MSE
+
+print("  ")
+print("test MSE of OLS:")
+print(MSE(y_test,ytilde_test_OLS))
+print("  ")
+print("test MSE of Ridge")
+print(MSE(y_test,ytilde_test_Ridge))
+
+plt.scatter(x,y,label='Data')
+plt.plot(x, X @ beta_OLS+interceptOLS,'*', label="OLS_Fit")
+plt.plot(x, X @ beta_Ridge+interceptRidge, label="Ridge_Fit")
+plt.grid()
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    [0.47179152 5.01549939]
+[0.47176783 5.01542292]
+1.7993408651198877
+1.7995707762668065
+  
+test MSE of OLS:
+1.1394311129039245
+  
+test MSE of Ridge
+1.1395084586525954
+
    
+
    
+_images/week37_208_1.png
+
    
+
    
+
    Finally, instead of using our own function we repeat the same example
+using the standardscaler functionality of the library
+Scikit-Learn.  Here we limit ourselves to Ridge regression only.
    
+
    
+
    
+
    from sklearn import linear_model
+np.random.seed(2018)
+n = 10
+d = 2
+Lambda = 0.01
+
+# Make data set.
+x = np.linspace(-3, 3, n)
+y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
+
+# Design matrix X does not include the intercept. 
+X = np.zeros((n, d))
+for p in range(d):     
+    X[:, p] = x ** (p+1)
+
+#Split data in train and test
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+# Scale data by subtracting mean value of the input using scikit-learn
+scaler = StandardScaler(with_std=False)
+scaler.fit(X_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# We scale also the output, here by our own code
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+y_test_scaled = y_test- y_scaler
+
+#Calculate beta
+OLS = LinearRegression()
+betaOLS=OLS.fit(X_train_scaled,y_train_scaled)
+ypredictOLS = OLS.predict(X_test_scaled)
+linear_model.Ridge(Lambda)
+RegRidge.fit(X_train_scaled,y_train_scaled)
+ypredictRidge = RegRidge.predict(X_test_scaled)
+betaOLS = OLS.coef_
+betaRidge = RegRidge.coef_
+print(betaOLS)
+print(betaRidge)
+interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS
+interceptRidge = y_scaler - X_train_mean @ betaRidge
+print(interceptOLS)
+print(interceptRidge)
+#predict value 
+ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler
+ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler
+
+#Calculate MSE
+print("  ")
+print("test MSE of OLS")
+print(MSE(y_test,ytilde_test_OLS))
+print("  ")
+print("test MSE of Ridge")
+print(MSE(y_test,ytilde_test_Ridge))
+plt.scatter(x,y,label='Data')
+plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label="Ridge_Fit")
+plt.grid()
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    ---------------------------------------------------------------------------
+NameError                                 Traceback (most recent call last)
+Input In [11], in <cell line: 34>()
+     32 ypredictOLS = OLS.predict(X_test_scaled)
+     33 linear_model.Ridge(Lambda)
+---> 34 RegRidge.fit(X_train_scaled,y_train_scaled)
+     35 ypredictRidge = RegRidge.predict(X_test_scaled)
+     36 betaOLS = OLS.coef_
+
+NameError: name 'RegRidge' is not defined
+
    
+
    
+
    
+
    
+