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-44,7 +44,7 @@ }, { "cell_type": "markdown", - "id": "3c35d277", + "id": "53c4e245", "metadata": { "editable": true }, @@ -62,7 +62,7 @@ }, { "cell_type": "markdown", - "id": "3fea75a2", + "id": "e5813e09", "metadata": { "editable": true }, @@ -74,7 +74,7 @@ }, { "cell_type": "markdown", - "id": "b4a2ad5f", + "id": "f8af4862", "metadata": { "editable": true }, @@ -84,7 +84,7 @@ }, { "cell_type": "markdown", - "id": "52645879", + "id": "8a114760", "metadata": { "editable": true }, @@ -97,7 +97,7 @@ }, { "cell_type": "markdown", - "id": "254fb658", + "id": "c4610a33", "metadata": { "editable": true }, @@ -107,7 +107,7 @@ }, { "cell_type": "markdown", - "id": "f5a7c9bb", + "id": "a3608d5c", "metadata": { "editable": true }, @@ -119,7 +119,7 @@ }, { "cell_type": "markdown", - "id": "5dfb00b0", + "id": "5dcf5daa", "metadata": { "editable": true }, @@ -134,7 +134,7 @@ }, { "cell_type": "markdown", - "id": "f3a06e4a", + "id": "0cbd93fd", "metadata": { "editable": true }, @@ -147,7 +147,7 @@ }, { "cell_type": "markdown", - "id": "5ebd015a", + "id": "6fba5d18", "metadata": { "editable": true }, @@ -159,7 +159,7 @@ }, { "cell_type": "markdown", - "id": "1b1a8ec0", + "id": "bec3e21d", "metadata": { "editable": true }, @@ -171,7 +171,7 @@ }, { "cell_type": "markdown", - "id": "03028b12", + "id": "380e526b", "metadata": { "editable": true }, @@ -183,7 +183,7 @@ }, { "cell_type": "markdown", - "id": "f69937ca", + "id": "97a04d2d", "metadata": { "editable": true }, @@ -193,7 +193,7 @@ }, { "cell_type": "markdown", - "id": "5f6c92bf", + "id": "a378590f", "metadata": { "editable": true }, @@ -205,7 +205,7 @@ }, { "cell_type": "markdown", - "id": "49ff5bbe", + "id": "f2d2d7ae", "metadata": { "editable": true }, @@ -217,7 +217,7 @@ }, { "cell_type": "markdown", - "id": "af2180fd", + "id": "b2ffd80f", "metadata": { "editable": true }, @@ -229,19 +229,19 @@ }, { "cell_type": "markdown", - "id": "444ff356", + "id": "3c3a9191", "metadata": { "editable": true }, "source": [ "### b) The singular value decomposition\n", "\n", - "Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" + "Here we will use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" ] }, { "cell_type": "markdown", - "id": "9763b84e", + "id": "4e3f0bb8", "metadata": { "editable": true }, @@ -253,12 +253,13 @@ }, { "cell_type": "markdown", - "id": "3b151e0d", + "id": "040a6f88", "metadata": { "editable": true }, "source": [ - "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", + "to study properties of Ridge regression and ordinary least squares regression.\n", + "Here $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", "$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n", "$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n", "\n", @@ -268,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "22e9035e", + "id": "0107597b", "metadata": { "editable": true }, @@ -280,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "15d6527a", + "id": "55fd1a9a", "metadata": { "editable": true }, @@ -290,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9a0a76fa", + "id": "8a05634c", "metadata": { "editable": true }, @@ -302,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "9407f726", + "id": "5d46ab65", "metadata": { "editable": true }, @@ -314,7 +315,7 @@ }, { "cell_type": "markdown", - "id": "c36ff1cb", + "id": "478a39d8", "metadata": { "editable": true }, @@ -338,7 +339,7 @@ }, { "cell_type": "markdown", - "id": "409205f5", + "id": "ed28dbd2", "metadata": { "editable": true }, @@ -350,7 +351,7 @@ }, { "cell_type": "markdown", - "id": "fa81481d", + "id": "1939c401", "metadata": { "editable": true }, @@ -360,7 +361,7 @@ }, { "cell_type": "markdown", - "id": "c10bc842", + "id": "10877c2d", "metadata": { "editable": true }, @@ -372,7 +373,7 @@ }, { "cell_type": "markdown", - "id": "d42231f2", + "id": "a7adfe54", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/_build/html/exercisesweek36.html b/doc/LectureNotes/_build/html/exercisesweek36.html index 8bb8345c5..a534ed99f 100644 --- a/doc/LectureNotes/_build/html/exercisesweek36.html +++ b/doc/LectureNotes/_build/html/exercisesweek36.html @@ -445,7 +445,7 @@ const thebe_selector_output = ".output, .cell_output"
+doconce format html exercisesweek36.do.txt --no_mako -->

Exercises week 36

September 4-8, 2023

@@ -516,12 +516,13 @@ a finite number larger than zero. We will not implement that here.

b) The singular value decomposition

-

Use the singular value decomposition of an \(n\times p\) matrix \(\boldsymbol{X}\) (our design matrix)

+

Here we will use the singular value decomposition of an \(n\times p\) matrix \(\boldsymbol{X}\) (our design matrix)

\[ \boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, \]
-

where \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are orthogonal matrices of dimensions +

to study properties of Ridge regression and ordinary least squares regression. +Here \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are orthogonal matrices of dimensions \(n\times n\) and \(p\times p\), respectively, and \(\boldsymbol{\Sigma}\) is an \(n\times p\) matrix which contains the singular values only. This material was discussed during the lectures of week 35.

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\ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb index 271509869..80f26812f 100644 --- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb @@ -2,19 +2,19 @@ "cells": [ { "cell_type": "markdown", - "id": "c6c4a1a9", + "id": "fb4d27e1", "metadata": { "editable": true }, "source": [ "\n", + "doconce format html exercisesweek36.do.txt --no_mako -->\n", "" ] }, { "cell_type": "markdown", - "id": "b9e17125", + "id": "c3ef468d", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "1d1f3e67", + "id": "b81273ff", "metadata": { "editable": true }, @@ -44,7 +44,7 @@ }, { "cell_type": "markdown", - "id": "3c35d277", + "id": "53c4e245", "metadata": { "editable": true }, @@ -62,7 +62,7 @@ }, { "cell_type": "markdown", - "id": "3fea75a2", + "id": "e5813e09", "metadata": { "editable": true }, @@ -74,7 +74,7 @@ }, { "cell_type": "markdown", - "id": "b4a2ad5f", + "id": "f8af4862", "metadata": { "editable": true }, @@ -84,7 +84,7 @@ }, { "cell_type": "markdown", - "id": "52645879", + "id": "8a114760", "metadata": { "editable": true }, @@ -97,7 +97,7 @@ }, { "cell_type": "markdown", - "id": "254fb658", + "id": "c4610a33", "metadata": { "editable": true }, @@ -107,7 +107,7 @@ }, { "cell_type": "markdown", - "id": "f5a7c9bb", + "id": "a3608d5c", "metadata": { "editable": true }, @@ -119,7 +119,7 @@ }, { "cell_type": "markdown", - "id": "5dfb00b0", + "id": "5dcf5daa", "metadata": { "editable": true }, @@ -134,7 +134,7 @@ }, { "cell_type": "markdown", - "id": "f3a06e4a", + "id": "0cbd93fd", "metadata": { "editable": true }, @@ -147,7 +147,7 @@ }, { "cell_type": "markdown", - "id": "5ebd015a", + "id": "6fba5d18", "metadata": { "editable": true }, @@ -159,7 +159,7 @@ }, { "cell_type": "markdown", - "id": "1b1a8ec0", + "id": "bec3e21d", "metadata": { "editable": true }, @@ -171,7 +171,7 @@ }, { "cell_type": "markdown", - "id": "03028b12", + "id": "380e526b", "metadata": { "editable": true }, @@ -183,7 +183,7 @@ }, { "cell_type": "markdown", - "id": "f69937ca", + "id": "97a04d2d", "metadata": { "editable": true }, @@ -193,7 +193,7 @@ }, { "cell_type": "markdown", - "id": "5f6c92bf", + "id": "a378590f", "metadata": { "editable": true }, @@ -205,7 +205,7 @@ }, { "cell_type": "markdown", - "id": "49ff5bbe", + "id": "f2d2d7ae", "metadata": { "editable": true }, @@ -217,7 +217,7 @@ }, { "cell_type": "markdown", - "id": "af2180fd", + "id": "b2ffd80f", "metadata": { "editable": true }, @@ -229,19 +229,19 @@ }, { "cell_type": "markdown", - "id": "444ff356", + "id": "3c3a9191", "metadata": { "editable": true }, "source": [ "### b) The singular value decomposition\n", "\n", - "Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" + "Here we will use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" ] }, { "cell_type": "markdown", - "id": "9763b84e", + "id": "4e3f0bb8", "metadata": { "editable": true }, @@ -253,12 +253,13 @@ }, { "cell_type": "markdown", - "id": "3b151e0d", + "id": "040a6f88", "metadata": { "editable": true }, "source": [ - "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", + "to study properties of Ridge regression and ordinary least squares regression.\n", + "Here $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", "$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n", "$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n", "\n", @@ -268,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "22e9035e", + "id": "0107597b", "metadata": { "editable": true }, @@ -280,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "15d6527a", + "id": "55fd1a9a", "metadata": { "editable": true }, @@ -290,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9a0a76fa", + "id": "8a05634c", "metadata": { "editable": true }, @@ -302,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "9407f726", + "id": "5d46ab65", "metadata": { "editable": true }, @@ -314,7 +315,7 @@ }, { "cell_type": "markdown", - "id": "c36ff1cb", + "id": "478a39d8", "metadata": { "editable": true }, @@ -338,7 +339,7 @@ }, { "cell_type": "markdown", - "id": "409205f5", + "id": "ed28dbd2", "metadata": { "editable": true }, @@ -350,7 +351,7 @@ }, { "cell_type": "markdown", - "id": "fa81481d", + "id": "1939c401", "metadata": { "editable": true }, @@ -360,7 +361,7 @@ }, { "cell_type": "markdown", - "id": "c10bc842", + "id": "10877c2d", "metadata": { "editable": true }, @@ -372,7 +373,7 @@ }, { "cell_type": "markdown", - "id": "d42231f2", + "id": "a7adfe54", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/exercisesweek36.ipynb b/doc/LectureNotes/exercisesweek36.ipynb index c4ffc89b3..405343ad2 100644 --- a/doc/LectureNotes/exercisesweek36.ipynb +++ b/doc/LectureNotes/exercisesweek36.ipynb @@ -2,19 +2,19 @@ "cells": [ { "cell_type": "markdown", - "id": "c6c4a1a9", + "id": "fb4d27e1", "metadata": { "editable": true }, "source": [ "\n", + "doconce format html exercisesweek36.do.txt --no_mako -->\n", "" ] }, { "cell_type": "markdown", - "id": "b9e17125", + "id": "c3ef468d", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "1d1f3e67", + "id": "b81273ff", "metadata": { "editable": true }, @@ -44,7 +44,7 @@ }, { "cell_type": "markdown", - "id": "3c35d277", + "id": "53c4e245", "metadata": { "editable": true }, @@ -62,7 +62,7 @@ }, { "cell_type": "markdown", - "id": "3fea75a2", + "id": "e5813e09", "metadata": { "editable": true }, @@ -74,7 +74,7 @@ }, { "cell_type": "markdown", - "id": "b4a2ad5f", + "id": "f8af4862", "metadata": { "editable": true }, @@ -84,7 +84,7 @@ }, { "cell_type": "markdown", - "id": "52645879", + "id": "8a114760", "metadata": { "editable": true }, @@ -97,7 +97,7 @@ }, { "cell_type": "markdown", - "id": "254fb658", + "id": "c4610a33", "metadata": { "editable": true }, @@ -107,7 +107,7 @@ }, { "cell_type": "markdown", - "id": "f5a7c9bb", + "id": "a3608d5c", "metadata": { "editable": true }, @@ -119,7 +119,7 @@ }, { "cell_type": "markdown", - "id": "5dfb00b0", + "id": "5dcf5daa", "metadata": { "editable": true }, @@ -134,7 +134,7 @@ }, { "cell_type": "markdown", - "id": "f3a06e4a", + "id": "0cbd93fd", "metadata": { "editable": true }, @@ -147,7 +147,7 @@ }, { "cell_type": "markdown", - "id": "5ebd015a", + "id": "6fba5d18", "metadata": { "editable": true }, @@ -159,7 +159,7 @@ }, { "cell_type": "markdown", - "id": "1b1a8ec0", + "id": "bec3e21d", "metadata": { "editable": true }, @@ -171,7 +171,7 @@ }, { "cell_type": "markdown", - "id": "03028b12", + "id": "380e526b", "metadata": { "editable": true }, @@ -183,7 +183,7 @@ }, { "cell_type": "markdown", - "id": "f69937ca", + "id": "97a04d2d", "metadata": { "editable": true }, @@ -193,7 +193,7 @@ }, { "cell_type": "markdown", - "id": "5f6c92bf", + "id": "a378590f", "metadata": { "editable": true }, @@ -205,7 +205,7 @@ }, { "cell_type": "markdown", - "id": "49ff5bbe", + "id": "f2d2d7ae", "metadata": { "editable": true }, @@ -217,7 +217,7 @@ }, { "cell_type": "markdown", - "id": "af2180fd", + "id": "b2ffd80f", "metadata": { "editable": true }, @@ -229,19 +229,19 @@ }, { "cell_type": "markdown", - "id": "444ff356", + "id": "3c3a9191", "metadata": { "editable": true }, "source": [ "### b) The singular value decomposition\n", "\n", - "Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" + "Here we will use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" ] }, { "cell_type": "markdown", - "id": "9763b84e", + "id": "4e3f0bb8", "metadata": { "editable": true }, @@ -253,12 +253,13 @@ }, { "cell_type": "markdown", - "id": "3b151e0d", + "id": "040a6f88", "metadata": { "editable": true }, "source": [ - "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", + "to study properties of Ridge regression and ordinary least squares regression.\n", + "Here $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", "$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n", "$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n", "\n", @@ -268,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "22e9035e", + "id": "0107597b", "metadata": { "editable": true }, @@ -280,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "15d6527a", + "id": "55fd1a9a", "metadata": { "editable": true }, @@ -290,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9a0a76fa", + "id": "8a05634c", "metadata": { "editable": true }, @@ -302,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "9407f726", + "id": "5d46ab65", "metadata": { "editable": true }, @@ -314,7 +315,7 @@ }, { "cell_type": "markdown", - "id": "c36ff1cb", + "id": "478a39d8", "metadata": { "editable": true }, @@ -338,7 +339,7 @@ }, { "cell_type": "markdown", - "id": "409205f5", + "id": "ed28dbd2", "metadata": { "editable": true }, @@ -350,7 +351,7 @@ }, { "cell_type": "markdown", - "id": "fa81481d", + "id": "1939c401", "metadata": { "editable": true }, @@ -360,7 +361,7 @@ }, { "cell_type": "markdown", - "id": "c10bc842", + "id": "10877c2d", "metadata": { "editable": true }, @@ -372,7 +373,7 @@ }, { "cell_type": "markdown", - "id": "d42231f2", + "id": "a7adfe54", "metadata": { "editable": true }, diff --git a/doc/src/week36/._exercisesweek36-bs000.html b/doc/src/week36/._exercisesweek36-bs000.html new file mode 100644 index 000000000..b03841a64 --- /dev/null +++ b/doc/src/week36/._exercisesweek36-bs000.html @@ -0,0 +1,285 @@ + + + + + + + +Exercises week 36 + + + + + + + + + + + + + + + + + + +

+
+ + Exercises week 36 +
+
+ +
+
+
+
+

 

 

 

+ + +
+
+

Exercises week 36

+
+ + +
+September 4-8, 2023 +
+ +
+
+

Deadline is Sunday September 10 at midnight

+
+
+ + +
+

Overarching aims of the exercises this week

+ +

This set of exercises form an important part of the first project. The +analytical exercises deal with the material covered last week on the +mathematical interpretations of ordinary least squares and of Ridge +regression. The numerical exercises can be seen as a continuation of +exercise 3 from week 35, with the inclusion of Ridge regression. This +material enters also the discussions of the first project. +

+ + +

Exercise 1: Analytical exercises

+ +

The aim here is to derive the expression for the optimal parameters +using Ridge regression. Furthermore, using the singular value +decomposition, we will analyze the difference between the ordinary +least squares approach and Ridge regression. +

+ +

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the +optimization problem +

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

which we can also write as

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +$$ + +

where we have used the definition of a norm-2 vector, that is

+$$ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +$$ + +

By minimizing the above equation with respect to the parameters +\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the +parameters \( \boldsymbol{\beta} \). +

+ +

We can add a regularization parameter \( \lambda \) by +defining a new cost function to be optimized, that is +

+ +$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +$$ + +

which leads to the Ridge regression minimization problem. One can require as part of the optimization problem +that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is +a finite number larger than zero. We will not implement that here. +

+

a) Expression for Ridge regression

+

Show that the optimal parameters

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that

+ +$$ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +$$ + +

with \( t \) a finite positive number. In the optimization, we will not require that the latter is satisfied.

+ +

The ordinary least squares result is

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

b) The singular value decomposition

+ +

Here we will use the singular value decomposition of an \( n\times p \) matrix \( \boldsymbol{X} \) (our design matrix)

+$$ +\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +$$ + +

to study properties of Ridge regression and ordinary least squares regression. +Here \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal matrices of dimensions +\( n\times n \) and \( p\times p \), respectively, and \( \boldsymbol{\Sigma} \) is an +\( n\times p \) matrix which contains the singular values only. This material was discussed during the lectures of week 35. +

+ +

Show that you can write the +OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix \( \boldsymbol{U} \) as +

+$$ +\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}. +$$ + +

For Ridge regression, show that the corresponding equation is

+ +$$ +\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +$$ + +

with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \).

+ +

Give an interpretation of the results. Section 3.4 of Hastie et al's textbook gives a good discussion of the above results.

+ + + + +

Exercise 2: Adding Ridge Regression

+ +

This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to +generate our data set, still staying with a simple function \( y(x) \) +which we want to fit using linear regression, but now extending the +analysis to include the Ridge regression method. +

+ +

In this exercise you need to include the same elements from last week, that is

+
    +
  1. scale your data by subtracting the mean value from each column in the design matrix.
    2. +
  2. perform a split of the data in a training set and a test set.
    3. +
+

The addition to the analysis this time is the introduction of the hyperparameter \( \lambda \) when introducing Ridge regression.

+ +

Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \( \lambda \). The optimal parameters \( \hat{\beta} \) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +

The ordinary least squares result you encoded last week is given by

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

Use these results to compute the mean squared error for ordinary least +squares and Ridge regression first for a polynomial of degree five +with \( n=100 \) data points and five selected values of +\( \lambda=[0.0001,0.001, 0.01,0.1,1.0] \). Compute thereafter the mean +squared error for the same values of \( \lambda \) for polynomials of degree ten +and \( 15 \). Discuss your results for the training MSE and test MSE with +Ridge regression and ordinary least squares. +

+ + +

+ +

    +
  • 1
    • +
+ +
+ + + + + + + diff --git a/doc/src/week36/README.txt b/doc/src/week36/README.txt new file mode 100644 index 000000000..251fd7184 --- /dev/null +++ b/doc/src/week36/README.txt @@ -0,0 +1,2 @@ +This IPython notebook exercisesweek36.ipynb does not require any additional +programs. diff --git a/doc/src/week36/exercisesweek36-bs.html b/doc/src/week36/exercisesweek36-bs.html new file mode 100644 index 000000000..b03841a64 --- /dev/null +++ b/doc/src/week36/exercisesweek36-bs.html @@ -0,0 +1,285 @@ + + + + + + + +Exercises week 36 + + + + + + + + + + + + + + + + + + +
+
+ + Exercises week 36 +
+
+ +
+
+
+
+

 

 

 

+ + +
+
+

Exercises week 36

+
+ + +
+September 4-8, 2023 +
+ +
+
+

Deadline is Sunday September 10 at midnight

+
+
+ + +
+

Overarching aims of the exercises this week

+ +

This set of exercises form an important part of the first project. The +analytical exercises deal with the material covered last week on the +mathematical interpretations of ordinary least squares and of Ridge +regression. The numerical exercises can be seen as a continuation of +exercise 3 from week 35, with the inclusion of Ridge regression. This +material enters also the discussions of the first project. +

+ + +

Exercise 1: Analytical exercises

+ +

The aim here is to derive the expression for the optimal parameters +using Ridge regression. Furthermore, using the singular value +decomposition, we will analyze the difference between the ordinary +least squares approach and Ridge regression. +

+ +

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the +optimization problem +

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

which we can also write as

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +$$ + +

where we have used the definition of a norm-2 vector, that is

+$$ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +$$ + +

By minimizing the above equation with respect to the parameters +\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the +parameters \( \boldsymbol{\beta} \). +

+ +

We can add a regularization parameter \( \lambda \) by +defining a new cost function to be optimized, that is +

+ +$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +$$ + +

which leads to the Ridge regression minimization problem. One can require as part of the optimization problem +that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is +a finite number larger than zero. We will not implement that here. +

+

a) Expression for Ridge regression

+

Show that the optimal parameters

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that

+ +$$ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +$$ + +

with \( t \) a finite positive number. In the optimization, we will not require that the latter is satisfied.

+ +

The ordinary least squares result is

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

b) The singular value decomposition

+ +

Here we will use the singular value decomposition of an \( n\times p \) matrix \( \boldsymbol{X} \) (our design matrix)

+$$ +\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +$$ + +

to study properties of Ridge regression and ordinary least squares regression. +Here \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal matrices of dimensions +\( n\times n \) and \( p\times p \), respectively, and \( \boldsymbol{\Sigma} \) is an +\( n\times p \) matrix which contains the singular values only. This material was discussed during the lectures of week 35. +

+ +

Show that you can write the +OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix \( \boldsymbol{U} \) as +

+$$ +\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}. +$$ + +

For Ridge regression, show that the corresponding equation is

+ +$$ +\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +$$ + +

with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \).

+ +

Give an interpretation of the results. Section 3.4 of Hastie et al's textbook gives a good discussion of the above results.

+ + + + +

Exercise 2: Adding Ridge Regression

+ +

This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to +generate our data set, still staying with a simple function \( y(x) \) +which we want to fit using linear regression, but now extending the +analysis to include the Ridge regression method. +

+ +

In this exercise you need to include the same elements from last week, that is

+
    +
  1. scale your data by subtracting the mean value from each column in the design matrix.
    2. +
  2. perform a split of the data in a training set and a test set.
    3. +
+

The addition to the analysis this time is the introduction of the hyperparameter \( \lambda \) when introducing Ridge regression.

+ +

Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \( \lambda \). The optimal parameters \( \hat{\beta} \) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +

The ordinary least squares result you encoded last week is given by

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

Use these results to compute the mean squared error for ordinary least +squares and Ridge regression first for a polynomial of degree five +with \( n=100 \) data points and five selected values of +\( \lambda=[0.0001,0.001, 0.01,0.1,1.0] \). Compute thereafter the mean +squared error for the same values of \( \lambda \) for polynomials of degree ten +and \( 15 \). Discuss your results for the training MSE and test MSE with +Ridge regression and ordinary least squares. +

+ + +

+ +

    +
  • 1
    • +
+ +
+ + + + + + + diff --git a/doc/src/week36/exercisesweek36-reveal.html b/doc/src/week36/exercisesweek36-reveal.html new file mode 100644 index 000000000..210d5df59 --- /dev/null +++ b/doc/src/week36/exercisesweek36-reveal.html @@ -0,0 +1,497 @@ + + + + + + + + +Exercises week 36 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + + + + +
+ +
+

Exercises week 36

+
+ + +
+September 4-8, 2023 +
+ +
+
+

Deadline is Sunday September 10 at midnight

+
+
+

Overarching aims of the exercises this week

+ +

This set of exercises form an important part of the first project. The +analytical exercises deal with the material covered last week on the +mathematical interpretations of ordinary least squares and of Ridge +regression. The numerical exercises can be seen as a continuation of +exercise 3 from week 35, with the inclusion of Ridge regression. This +material enters also the discussions of the first project. +

+ + +

Exercise 1: Analytical exercises

+ +

The aim here is to derive the expression for the optimal parameters +using Ridge regression. Furthermore, using the singular value +decomposition, we will analyze the difference between the ordinary +least squares approach and Ridge regression. +

+ +

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the +optimization problem +

+

 
+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ +

 
+ +

which we can also write as

+

 
+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +$$ +

 
+ +

where we have used the definition of a norm-2 vector, that is

+

 
+$$ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +$$ +

 
+ +

By minimizing the above equation with respect to the parameters +\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the +parameters \( \boldsymbol{\beta} \). +

+ +

We can add a regularization parameter \( \lambda \) by +defining a new cost function to be optimized, that is +

+ +

 
+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +$$ +

 
+ +

which leads to the Ridge regression minimization problem. One can require as part of the optimization problem +that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is +a finite number larger than zero. We will not implement that here. +

+

a) Expression for Ridge regression

+

Show that the optimal parameters

+

 
+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

 
+ +

with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that

+ +

 
+$$ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +$$ +

 
+ +

with \( t \) a finite positive number. In the optimization, we will not require that the latter is satisfied.

+ +

The ordinary least squares result is

+

 
+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

 
+

b) The singular value decomposition

+ +

Here we will use the singular value decomposition of an \( n\times p \) matrix \( \boldsymbol{X} \) (our design matrix)

+

 
+$$ +\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +$$ +

 
+ +

to study properties of Ridge regression and ordinary least squares regression. +Here \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal matrices of dimensions +\( n\times n \) and \( p\times p \), respectively, and \( \boldsymbol{\Sigma} \) is an +\( n\times p \) matrix which contains the singular values only. This material was discussed during the lectures of week 35. +

+ +

Show that you can write the +OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix \( \boldsymbol{U} \) as +

+

 
+$$ +\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}. +$$ +

 
+ +

For Ridge regression, show that the corresponding equation is

+ +

 
+$$ +\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +$$ +

 
+ +

with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \).

+ +

Give an interpretation of the results. Section 3.4 of Hastie et al's textbook gives a good discussion of the above results.

+ + + + +

Exercise 2: Adding Ridge Regression

+ +

This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to +generate our data set, still staying with a simple function \( y(x) \) +which we want to fit using linear regression, but now extending the +analysis to include the Ridge regression method. +

+ +

In this exercise you need to include the same elements from last week, that is

+
    +

  1. scale your data by subtracting the mean value from each column in the design matrix.
    2. +

  2. perform a split of the data in a training set and a test set.
    3. +
+

+

The addition to the analysis this time is the introduction of the hyperparameter \( \lambda \) when introducing Ridge regression.

+ +

Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \( \lambda \). The optimal parameters \( \hat{\beta} \) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations

+

 
+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ +

 
+ +

The ordinary least squares result you encoded last week is given by

+

 
+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

 
+ +

Use these results to compute the mean squared error for ordinary least +squares and Ridge regression first for a polynomial of degree five +with \( n=100 \) data points and five selected values of +\( \lambda=[0.0001,0.001, 0.01,0.1,1.0] \). Compute thereafter the mean +squared error for the same values of \( \lambda \) for polynomials of degree ten +and \( 15 \). Discuss your results for the training MSE and test MSE with +Ridge regression and ordinary least squares. +

+ + +
+ + + +
+
+ + + + + + + + + + + + + diff --git a/doc/src/week36/exercisesweek36-solarized.html b/doc/src/week36/exercisesweek36-solarized.html new file mode 100644 index 000000000..7af9f0f51 --- /dev/null +++ b/doc/src/week36/exercisesweek36-solarized.html @@ -0,0 +1,238 @@ + + + + + + + +Exercises week 36 + + + + + + + + + + + + + + + + + + +
+

Exercises week 36

+
+ + +
+September 4-8, 2023 +
+ +
+
+

Deadline is Sunday September 10 at midnight

+
+
+

Overarching aims of the exercises this week

+ +

This set of exercises form an important part of the first project. The +analytical exercises deal with the material covered last week on the +mathematical interpretations of ordinary least squares and of Ridge +regression. The numerical exercises can be seen as a continuation of +exercise 3 from week 35, with the inclusion of Ridge regression. This +material enters also the discussions of the first project. +

+ + +

Exercise 1: Analytical exercises

+ +

The aim here is to derive the expression for the optimal parameters +using Ridge regression. Furthermore, using the singular value +decomposition, we will analyze the difference between the ordinary +least squares approach and Ridge regression. +

+ +

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the +optimization problem +

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

which we can also write as

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +$$ + +

where we have used the definition of a norm-2 vector, that is

+$$ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +$$ + +

By minimizing the above equation with respect to the parameters +\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the +parameters \( \boldsymbol{\beta} \). +

+ +

We can add a regularization parameter \( \lambda \) by +defining a new cost function to be optimized, that is +

+ +$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +$$ + +

which leads to the Ridge regression minimization problem. One can require as part of the optimization problem +that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is +a finite number larger than zero. We will not implement that here. +

+

a) Expression for Ridge regression

+

Show that the optimal parameters

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that

+ +$$ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +$$ + +

with \( t \) a finite positive number. In the optimization, we will not require that the latter is satisfied.

+ +

The ordinary least squares result is

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

b) The singular value decomposition

+ +

Here we will use the singular value decomposition of an \( n\times p \) matrix \( \boldsymbol{X} \) (our design matrix)

+$$ +\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +$$ + +

to study properties of Ridge regression and ordinary least squares regression. +Here \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal matrices of dimensions +\( n\times n \) and \( p\times p \), respectively, and \( \boldsymbol{\Sigma} \) is an +\( n\times p \) matrix which contains the singular values only. This material was discussed during the lectures of week 35. +

+ +

Show that you can write the +OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix \( \boldsymbol{U} \) as +

+$$ +\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}. +$$ + +

For Ridge regression, show that the corresponding equation is

+ +$$ +\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +$$ + +

with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \).

+ +

Give an interpretation of the results. Section 3.4 of Hastie et al's textbook gives a good discussion of the above results.

+ + + + +

Exercise 2: Adding Ridge Regression

+ +

This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to +generate our data set, still staying with a simple function \( y(x) \) +which we want to fit using linear regression, but now extending the +analysis to include the Ridge regression method. +

+ +

In this exercise you need to include the same elements from last week, that is

+
    +
  1. scale your data by subtracting the mean value from each column in the design matrix.
    2. +
  2. perform a split of the data in a training set and a test set.
    3. +
+

The addition to the analysis this time is the introduction of the hyperparameter \( \lambda \) when introducing Ridge regression.

+ +

Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \( \lambda \). The optimal parameters \( \hat{\beta} \) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +

The ordinary least squares result you encoded last week is given by

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

Use these results to compute the mean squared error for ordinary least +squares and Ridge regression first for a polynomial of degree five +with \( n=100 \) data points and five selected values of +\( \lambda=[0.0001,0.001, 0.01,0.1,1.0] \). Compute thereafter the mean +squared error for the same values of \( \lambda \) for polynomials of degree ten +and \( 15 \). Discuss your results for the training MSE and test MSE with +Ridge regression and ordinary least squares. +

+ + + + + + diff --git a/doc/src/week36/exercisesweek36.do.txt b/doc/src/week36/exercisesweek36.do.txt index 5a60bde40..8c3e79626 100644 --- a/doc/src/week36/exercisesweek36.do.txt +++ b/doc/src/week36/exercisesweek36.do.txt @@ -85,14 +85,14 @@ The ordinary least squares result is === b) The singular value decomposition === -Use the singular value decomposition of an $n\times p$ matrix $\bm{X}$ (our design matrix) +Here we will use the singular value decomposition of an $n\times p$ matrix $\bm{X}$ (our design matrix) !bt \[ \bm{X}=\bm{U}\bm{\Sigma}\bm{V}^T, \] !et - -where $\bm{U}$ and $\bm{V}$ are orthogonal matrices of dimensions +to study properties of Ridge regression and ordinary least squares regression. +Here $\bm{U}$ and $\bm{V}$ are orthogonal matrices of dimensions $n\times n$ and $p\times p$, respectively, and $\bm{\Sigma}$ is an $n\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35. diff --git a/doc/src/week36/exercisesweek36.html b/doc/src/week36/exercisesweek36.html new file mode 100644 index 000000000..927ce428f --- /dev/null +++ b/doc/src/week36/exercisesweek36.html @@ -0,0 +1,315 @@ + + + + + + + +Exercises week 36 + + + + + + + + + + + + + + +
+

Exercises week 36

+
+ + +
+September 4-8, 2023 +
+ +
+
+

Deadline is Sunday September 10 at midnight

+
+
+

Overarching aims of the exercises this week

+ +

This set of exercises form an important part of the first project. The +analytical exercises deal with the material covered last week on the +mathematical interpretations of ordinary least squares and of Ridge +regression. The numerical exercises can be seen as a continuation of +exercise 3 from week 35, with the inclusion of Ridge regression. This +material enters also the discussions of the first project. +

+ + +

Exercise 1: Analytical exercises

+ +

The aim here is to derive the expression for the optimal parameters +using Ridge regression. Furthermore, using the singular value +decomposition, we will analyze the difference between the ordinary +least squares approach and Ridge regression. +

+ +

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the +optimization problem +

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

which we can also write as

+$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +$$ + +

where we have used the definition of a norm-2 vector, that is

+$$ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +$$ + +

By minimizing the above equation with respect to the parameters +\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the +parameters \( \boldsymbol{\beta} \). +

+ +

We can add a regularization parameter \( \lambda \) by +defining a new cost function to be optimized, that is +

+ +$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +$$ + +

which leads to the Ridge regression minimization problem. One can require as part of the optimization problem +that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is +a finite number larger than zero. We will not implement that here. +

+

a) Expression for Ridge regression

+

Show that the optimal parameters

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that

+ +$$ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +$$ + +

with \( t \) a finite positive number. In the optimization, we will not require that the latter is satisfied.

+ +

The ordinary least squares result is

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ +

b) The singular value decomposition

+ +

Here we will use the singular value decomposition of an \( n\times p \) matrix \( \boldsymbol{X} \) (our design matrix)

+$$ +\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +$$ + +

to study properties of Ridge regression and ordinary least squares regression. +Here \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal matrices of dimensions +\( n\times n \) and \( p\times p \), respectively, and \( \boldsymbol{\Sigma} \) is an +\( n\times p \) matrix which contains the singular values only. This material was discussed during the lectures of week 35. +

+ +

Show that you can write the +OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix \( \boldsymbol{U} \) as +

+$$ +\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta} = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}. +$$ + +

For Ridge regression, show that the corresponding equation is

+ +$$ +\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +$$ + +

with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \) from the SVD of the matrix \( \boldsymbol{X} \).

+ +

Give an interpretation of the results. Section 3.4 of Hastie et al's textbook gives a good discussion of the above results.

+ + + + +

Exercise 2: Adding Ridge Regression

+ +

This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to +generate our data set, still staying with a simple function \( y(x) \) +which we want to fit using linear regression, but now extending the +analysis to include the Ridge regression method. +

+ +

In this exercise you need to include the same elements from last week, that is

+
    +
  1. scale your data by subtracting the mean value from each column in the design matrix.
    2. +
  2. perform a split of the data in a training set and a test set.
    3. +
+

The addition to the analysis this time is the introduction of the hyperparameter \( \lambda \) when introducing Ridge regression.

+ +

Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \( \lambda \). The optimal parameters \( \hat{\beta} \) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +

The ordinary least squares result you encoded last week is given by

+$$ +\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +$$ + +

Use these results to compute the mean squared error for ordinary least +squares and Ridge regression first for a polynomial of degree five +with \( n=100 \) data points and five selected values of +\( \lambda=[0.0001,0.001, 0.01,0.1,1.0] \). Compute thereafter the mean +squared error for the same values of \( \lambda \) for polynomials of degree ten +and \( 15 \). Discuss your results for the training MSE and test MSE with +Ridge regression and ordinary least squares. +

+ + + + + + diff --git a/doc/src/week36/ipynb-exercisesweek36-src.tar.gz b/doc/src/week36/ipynb-exercisesweek36-src.tar.gz new file mode 100644 index 0000000000000000000000000000000000000000..2688a9e97e7e74fca0ca8dc5dd9c66fdbf440dc8 GIT binary patch literal 197 zcmV;$06PC4iwFP~r}kt31MSbj3W7io24K%V#XNzmx~u0P4;_MtUSQ2JH*?o^SJJmn z6Y5l*B!c}eKLbC*%syMKw^?9&>n(;5l28g$rU`o_iRSqsj1mYbO&N<~8ZjVY%qa1I zEO*jNXDrv_DNS`oX;8lFjkU7;uxEY+p7|#Zl`yc`4YpDeq_UVRRRi{@kHVn2A=Ysc zh<1=U6bjAM0gcyAXap_?;6)* (<%= grunt.template.today("yyyy-mm-dd, HH:MM") %>)\n' + + ' * http://lab.hakim.se/reveal-js\n' + + ' * MIT licensed\n' + + ' *\n' + + ' * Copyright (C) 2014 Hakim El Hattab, http://hakim.se\n' + + ' */' + }, + + qunit: { + files: [ 'test/*.html' ] + }, + + uglify: { + options: { + banner: '<%= meta.banner %>\n' + }, + build: { + src: 'js/reveal.js', + dest: 'js/reveal.min.js' + } + }, + + cssmin: { + compress: { + files: { + 'css/reveal.min.css': [ 'css/reveal.css' ] + } + } + }, + + sass: { + main: { + files: { + 'css/theme/darkgray.css': 'css/theme/source/darkgray.scss', + 'css/theme/beigesmall.css': 'css/theme/source/beigesmall.scss', + 'css/theme/cbc.css': 'css/theme/source/cbc.scss', + 'css/theme/default.css': 'css/theme/source/default.scss', + 'css/theme/beige.css': 'css/theme/source/beige.scss', + 'css/theme/night.css': 'css/theme/source/night.scss', + 'css/theme/serif.css': 'css/theme/source/serif.scss', + 'css/theme/simple.css': 'css/theme/source/simple.scss', + 'css/theme/sky.css': 'css/theme/source/sky.scss', + 'css/theme/moon.css': 'css/theme/source/moon.scss', + 'css/theme/solarized.css': 'css/theme/source/solarized.scss', + 'css/theme/blood.css': 'css/theme/source/blood.scss' + } + } + }, + + jshint: { + options: { + curly: false, + eqeqeq: true, + immed: true, + latedef: true, + newcap: true, + noarg: true, + sub: true, + undef: true, + eqnull: true, + browser: true, + expr: true, + globals: { + head: false, + module: false, + console: false, + unescape: false + } + }, + files: [ 'Gruntfile.js', 'js/reveal.js' ] + }, + + connect: { + server: { + options: { + port: port, + base: '.' + } + } + }, + + zip: { + 'reveal-js-presentation.zip': [ + 'index.html', + 'css/**', + 'js/**', + 'lib/**', + 'images/**', + 'plugin/**' + ] + }, + + watch: { + main: { + files: [ 'Gruntfile.js', 'js/reveal.js', 'css/reveal.css' ], + tasks: 'default' + }, + theme: { + files: [ 'css/theme/source/*.scss', 'css/theme/template/*.scss' ], + tasks: 'themes' + } + } + + }); + + // Dependencies + grunt.loadNpmTasks( 'grunt-contrib-qunit' ); + grunt.loadNpmTasks( 'grunt-contrib-jshint' ); + grunt.loadNpmTasks( 'grunt-contrib-cssmin' ); + grunt.loadNpmTasks( 'grunt-contrib-uglify' ); + grunt.loadNpmTasks( 'grunt-contrib-watch' ); + grunt.loadNpmTasks( 'grunt-contrib-sass' ); + grunt.loadNpmTasks( 'grunt-contrib-connect' ); + grunt.loadNpmTasks( 'grunt-zip' ); + + // Default task + grunt.registerTask( 'default', [ 'jshint', 'cssmin', 'uglify', 'qunit' ] ); + + // Theme task + grunt.registerTask( 'themes', [ 'sass' ] ); + + // Package presentation to archive + grunt.registerTask( 'package', [ 'default', 'zip' ] ); + + // Serve presentation locally + grunt.registerTask( 'serve', [ 'connect', 'watch' ] ); + + // Run tests + grunt.registerTask( 'test', [ 'jshint', 'qunit' ] ); + +}; diff --git a/doc/src/week36/reveal.js/LICENSE b/doc/src/week36/reveal.js/LICENSE new file mode 100644 index 000000000..09623076f --- /dev/null +++ b/doc/src/week36/reveal.js/LICENSE @@ -0,0 +1,19 @@ +Copyright (C) 2015 Hakim El Hattab, http://hakim.se + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in +all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN +THE SOFTWARE. \ No newline at end of file diff --git a/doc/src/week36/reveal.js/README.md b/doc/src/week36/reveal.js/README.md new file mode 100644 index 000000000..573b19597 --- /dev/null +++ b/doc/src/week36/reveal.js/README.md @@ -0,0 +1,1052 @@ +# reveal.js [![Build Status](https://travis-ci.org/hakimel/reveal.js.svg?branch=master)](https://travis-ci.org/hakimel/reveal.js) + +A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/). + +reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere. + + +#### More reading: +- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer. +- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history. +- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own! +- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks. +- [Plugins](https://github.com/hakimel/reveal.js/wiki/Plugins,-Tools-and-Hardware): A list of plugins that can be used to extend reveal.js. + +## Online Editor + +Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com). + + +## Instructions + +### Markup + +Markup hierarchy needs to be ``
`` where the ``
`` represents one slide and can be repeated indefinitely. If you place multiple ``
``'s inside of another ``
`` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example: + +```html +
+
+
Single Horizontal Slide
+
+
Vertical Slide 1
+
Vertical Slide 2
+
+
+
+``` + +### Markdown + +It's possible to write your slides using Markdown. To enable Markdown, add the ```data-markdown``` attribute to your ```
``` elements and wrap the contents in a ``` +
+``` + +#### External Markdown + +You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file. + +When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup). + +```html +
+
+``` + +#### Element Attributes + +Special syntax (in html comment) is available for adding attributes to Markdown elements. This is useful for fragments, amongst other things. + +```html +
+ +
+``` + +#### Slide Attributes + +Special syntax (in html comment) is available for adding attributes to the slide `
` elements generated by your Markdown. + +```html +
+ +
+``` + + +### Configuration + +At the end of your page you need to initialize reveal by running the following code. Note that all config values are optional and will default as specified below. + +```javascript +Reveal.initialize({ + + // Display controls in the bottom right corner + controls: true, + + // Display a presentation progress bar + progress: true, + + // Display the page number of the current slide + slideNumber: false, + + // Push each slide change to the browser history + history: false, + + // Enable keyboard shortcuts for navigation + keyboard: true, + + // Enable the slide overview mode + overview: true, + + // Vertical centering of slides + center: true, + + // Enables touch navigation on devices with touch input + touch: true, + + // Loop the presentation + loop: false, + + // Change the presentation direction to be RTL + rtl: false, + + // Turns fragments on and off globally + fragments: true, + + // Flags if the presentation is running in an embedded mode, + // i.e. contained within a limited portion of the screen + embedded: false, + + // Flags if we should show a help overlay when the questionmark + // key is pressed + help: true, + + // Number of milliseconds between automatically proceeding to the + // next slide, disabled when set to 0, this value can be overwritten + // by using a data-autoslide attribute on your slides + autoSlide: 0, + + // Stop auto-sliding after user input + autoSlideStoppable: true, + + // Enable slide navigation via mouse wheel + mouseWheel: false, + + // Hides the address bar on mobile devices + hideAddressBar: true, + + // Opens links in an iframe preview overlay + previewLinks: false, + + // Transition style + transition: 'default', // none/fade/slide/convex/concave/zoom + + // Transition speed + transitionSpeed: 'default', // default/fast/slow + + // Transition style for full page slide backgrounds + backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom + + // Number of slides away from the current that are visible + viewDistance: 3, + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" + + // Amount to move parallax background (horizontal and vertical) on slide change + // Number, e.g. 100 + parallaxBackgroundHorizontal: '', + parallaxBackgroundVertical: '' + +}); +``` + + +The configuration can be updated after initialization using the ```configure``` method: + +```javascript +// Turn autoSlide off +Reveal.configure({ autoSlide: 0 }); + +// Start auto-sliding every 5s +Reveal.configure({ autoSlide: 5000 }); +``` + + +### Dependencies + +Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example: + +```javascript +Reveal.initialize({ + dependencies: [ + // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/ + { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } }, + + // Interpret Markdown in
elements + { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + + // Syntax highlight for elements + { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } }, + + // Zoom in and out with Alt+click + { src: 'plugin/zoom-js/zoom.js', async: true }, + + // Speaker notes + { src: 'plugin/notes/notes.js', async: true }, + + // Remote control your reveal.js presentation using a touch device + { src: 'plugin/remotes/remotes.js', async: true }, + + // MathJax + { src: 'plugin/math/math.js', async: true } + ] +}); +``` + +You can add your own extensions using the same syntax. The following properties are available for each dependency object: +- **src**: Path to the script to load +- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false +- **callback**: [optional] Function to execute when the script has loaded +- **condition**: [optional] Function which must return true for the script to be loaded + + +### Ready Event + +A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`. + +```javascript +Reveal.addEventListener( 'ready', function( event ) { + // event.currentSlide, event.indexh, event.indexv +} ); +``` + + +### Presentation Size + +All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport. + +See below for a list of configuration options related to sizing, including default values: + +```javascript +Reveal.initialize({ + + ... + + // The "normal" size of the presentation, aspect ratio will be preserved + // when the presentation is scaled to fit different resolutions. Can be + // specified using percentage units. + width: 960, + height: 700, + + // Factor of the display size that should remain empty around the content + margin: 0.1, + + // Bounds for smallest/largest possible scale to apply to content + minScale: 0.2, + maxScale: 1.5 + +}); +``` + + +### Auto-sliding + +Presentations can be configured to progress through slides automatically, without any user input. To enable this you will need to tell the framework how many milliseconds it should wait between slides: + +```javascript +// Slide every five seconds +Reveal.configure({ + autoSlide: 5000 +}); +``` +When this is turned on a control element will appear that enables users to pause and resume auto-sliding. Alternatively, sliding can be paused or resumed by pressing »a« on the keyboard. Sliding is paused automatically as soon as the user starts navigating. You can disable these controls by specifying ```autoSlideStoppable: false``` in your reveal.js config. + +You can also override the slide duration for individual slides and fragments by using the ```data-autoslide``` attribute: + +```html +
+

After 2 seconds the first fragment will be shown.

+

After 10 seconds the next fragment will be shown.

+

Now, the fragment is displayed for 2 seconds before the next slide is shown.

+
+``` + +Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired. + + +### Keyboard Bindings + +If you're unhappy with any of the default keyboard bindings you can override them using the ```keyboard``` config option: + +```javascript +Reveal.configure({ + keyboard: { + 13: 'next', // go to the next slide when the ENTER key is pressed + 27: function() {}, // do something custom when ESC is pressed + 32: null // don't do anything when SPACE is pressed (i.e. disable a reveal.js default binding) + } +}); +``` + +### Lazy Loading + +When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option. + +To enable lazy loading all you need to do is change your "src" attributes to "data-src" as shown below. This is supported for image, video, audio and iframe elements. Lazy loaded iframes will also unload when the containing slide is no longer visible. + +```html +
+ + + +
+``` + + +### API + +The ``Reveal`` object exposes a JavaScript API for controlling navigation and reading state: + +```javascript +// Navigation +Reveal.slide( indexh, indexv, indexf ); +Reveal.left(); +Reveal.right(); +Reveal.up(); +Reveal.down(); +Reveal.prev(); +Reveal.next(); +Reveal.prevFragment(); +Reveal.nextFragment(); + +// Toggle presentation states, optionally pass true/false to force on/off +Reveal.toggleOverview(); +Reveal.togglePause(); +Reveal.toggleAutoSlide(); + +// Change a config value at runtime +Reveal.configure({ controls: true }); + +// Returns the present configuration options +Reveal.getConfig(); + +// Fetch the current scale of the presentation +Reveal.getScale(); + +// Retrieves the previous and current slide elements +Reveal.getPreviousSlide(); +Reveal.getCurrentSlide(); + +Reveal.getIndices(); // { h: 0, v: 0 } } +Reveal.getProgress(); // 0-1 +Reveal.getTotalSlides(); + +// State checks +Reveal.isFirstSlide(); +Reveal.isLastSlide(); +Reveal.isOverview(); +Reveal.isPaused(); +Reveal.isAutoSliding(); +``` + +### Slide Changed Event + +A 'slidechanged' event is fired each time the slide is changed (regardless of state). The event object holds the index values of the current slide as well as a reference to the previous and current slide HTML nodes. + +Some libraries, like MathJax (see [#226](https://github.com/hakimel/reveal.js/issues/226#issuecomment-10261609)), get confused by the transforms and display states of slides. Often times, this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'slidechanged', function( event ) { + // event.previousSlide, event.currentSlide, event.indexh, event.indexv +} ); +``` + +### Presentation State + +The presentation's current state can be fetched by using the `getState` method. A state object contains all of the information required to put the presentation back as it was when `getState` was first called. Sort of like a snapshot. It's a simple object that can easily be stringified and persisted or sent over the wire. + +```javascript +Reveal.slide( 1 ); +// we're on slide 1 + +var state = Reveal.getState(); + +Reveal.slide( 3 ); +// we're on slide 3 + +Reveal.setState( state ); +// we're back on slide 1 +``` + +### Slide States + +If you set ``data-state="somestate"`` on a slide ``
``, "somestate" will be applied as a class on the document element when that slide is opened. This allows you to apply broad style changes to the page based on the active slide. + +Furthermore you can also listen to these changes in state via JavaScript: + +```javascript +Reveal.addEventListener( 'somestate', function() { + // TODO: Sprinkle magic +}, false ); +``` + +### Slide Backgrounds + +Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your ```
``` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples. + +```html +
+

All CSS color formats are supported, like rgba() or hsl().

+
+
+

This slide will have a full-size background image.

+
+
+

This background image will be sized to 100px and repeated.

+
+
+

Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.

+
+
+

Embeds a web page as a background. Note that the page won't be interactive.

+
+``` + +Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition. + + +### Parallax Background + +If you want to use a parallax scrolling background, set the first two config properties below when initializing reveal.js (the other two are optional). + +```javascript +Reveal.initialize({ + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto) + + // Amount of pixels to move the parallax background per slide step, + // a value of 0 disables movement along the given axis + // These are optional, if they aren't specified they'll be calculated automatically + parallaxBackgroundHorizontal: 200, + parallaxBackgroundVertical: 50 + +}); +``` + +Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px). + + + +### Slide Transitions +The global presentation transition is set using the ```transition``` config value. You can override the global transition for a specific slide by using the ```data-transition``` attribute: + +```html +
+

This slide will override the presentation transition and zoom!

+
+ +
+

Choose from three transition speeds: default, fast or slow!

+
+``` + +You can also use different in and out transitions for the same slide: + +```html +
+ The train goes on … +
+
+ and on … +
+
+ and stops. +
+
+ (Passengers entering and leaving) +
+
+ And it starts again. +
+``` + + +Note that this does not work with the page and cube transitions. + + +### Internal links + +It's easy to link between slides. The first example below targets the index of another slide whereas the second targets a slide with an ID attribute (```
```): + +```html +Link +Link +``` + +You can also add relative navigation links, similar to the built in reveal.js controls, by appending one of the following classes on any element. Note that each element is automatically given an ```enabled``` class when it's a valid navigation route based on the current slide. + +```html + + + + + + +``` + + +### Fragments +Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://lab.hakim.se/reveal-js/#/fragments + +The default fragment style is to start out invisible and fade in. This style can be changed by appending a different class to the fragment: + +```html +
+

grow

+

shrink

+

fade-out

+

visible only once

+

blue only once

+

highlight-red

+

highlight-green

+

highlight-blue

+
+``` + +Multiple fragments can be applied to the same element sequentially by wrapping it, this will fade in the text on the first step and fade it back out on the second. + +```html +
+ + I'll fade in, then out + +
+``` + +The display order of fragments can be controlled using the ```data-fragment-index``` attribute. + +```html +
+

Appears last

+

Appears first

+

Appears second

+
+``` + +### Fragment events + +When a slide fragment is either shown or hidden reveal.js will dispatch an event. + +Some libraries, like MathJax (see #505), get confused by the initially hidden fragment elements. Often times this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'fragmentshown', function( event ) { + // event.fragment = the fragment DOM element +} ); +Reveal.addEventListener( 'fragmenthidden', function( event ) { + // event.fragment = the fragment DOM element +} ); +``` + +### Code syntax highlighting + +By default, Reveal is configured with [highlight.js](http://softwaremaniacs.org/soft/highlight/en/) for code syntax highlighting. Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present surrounding whitespace is automatically removed. + +```html +
+ 

+(def lazy-fib
+  (concat
+   [0 1]
+   ((fn rfib [a b]
+        (lazy-cons (+ a b) (rfib b (+ a b)))) 0 1)))
+
+
+``` + +### Slide number +If you would like to display the page number of the current slide you can do so using the ```slideNumber``` configuration value. + +```javascript +// Shows the slide number using default formatting +Reveal.configure({ slideNumber: true }); + +// Slide number formatting can be configured using these variables: +// h: current slide's horizontal index +// v: current slide's vertical index +// c: current slide index (flattened) +// t: total number of slides (flattened) +Reveal.configure({ slideNumber: 'c / t' }); + +``` + + +### Overview mode + +Press "Esc" or "o" keys to toggle the overview mode on and off. While you're in this mode, you can still navigate between slides, +as if you were at 1,000 feet above your presentation. The overview mode comes with a few API hooks: + +```javascript +Reveal.addEventListener( 'overviewshown', function( event ) { /* ... */ } ); +Reveal.addEventListener( 'overviewhidden', function( event ) { /* ... */ } ); + +// Toggle the overview mode programmatically +Reveal.toggleOverview(); +``` + +### Fullscreen mode +Just press »F« on your keyboard to show your presentation in fullscreen mode. Press the »ESC« key to exit fullscreen mode. + + +### Embedded media +Embedded HTML5 `