Merge branch 'master' of https://github.com/CompPhysics/MachineLearning

2020-09-16 13:32:21 +02:00
parent 9e27a3567c 9754fc6e6d
commit db013fc185
69 changed files with 6930 additions and 0 deletions
@@ -0,0 +1 @@
+{'year': '1999-2020', 'holder': ['Morten Hjorth-Jensen'], 'cite doconce': False, 'license': 'Released under CC Attribution-NonCommercial 4.0 license'}
@@ -0,0 +1,243 @@
+translating doconce text in Regression.do.txt to ipynb
+*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+Failed to remove ans_at_end environment
+Failed to remove sol_at_end environment
+output in Regression.ipynb
+translating doconce text in Regression.do.txt to ipynb
+*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+Failed to remove ans_at_end environment
+Failed to remove sol_at_end environment
+output in Regression.ipynb
+translating doconce text in Regression.do.txt to ipynb
+*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
+    Stick to \[ ... \], equation, equation*, align, or align*
+    environments in math environments.
+
+Failed to remove ans_at_end environment
+Failed to remove sol_at_end environment
+output in Regression.ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,888 @@
+TITLE: Week 37: Ridge and Lasso Regression
+AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+DATE: today
+
+!split
+=====  Plans for week 37 =====
+
+* Thursday September 10: Motivation for shrinkage methods and the Singular Value Decompostion theorem
+* Friday September 11: Ridge and Lasso regression
+
+!split
+=====  A Bayesian approach to develop intuition about skrinkage methods =====
+
+
+!split
+===== The singular value decomposition  =====
+
+!bblock
+
+The examples we have looked at so far are cases where we normally can
+invert the matrix $\bm{X}^T\bm{X}$. Using a polynomial expansion as we
+did both for the masses and the fitting of the equation of state,
+leads to row vectors of the design matrix which are essentially
+orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
+
+
+
+This may
+however not the be case in general and a standard matrix inversion
+algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.
+
+There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions. 
+
+This is given by the _Singular Value Decomposition_ algorithm, perhaps
+the most powerful linear algebra algorithm.  Let us look at a
+different example where we may have problems with the standard matrix
+inversion algorithm. Thereafter we dive into the math of the SVD.
+
+!eblock
+
+
+
+!split
+===== Linear Regression Problems =====
+
+One of the typical problems we encounter with linear regression, in particular 
+when the matrix $\bm{X}$ (our so-called design matrix) is high-dimensional, 
+are problems with near singular or singular matrices. The column vectors of $\bm{X}$ 
+may be linearly dependent, normally referred to as super-collinearity.  
+This means that the matrix may be rank deficient and it is basically impossible to 
+to model the data using linear regression. As an example, consider the matrix
+!bt
+\begin{align*}
+\mathbf{X} & =  \left[
+\begin{array}{rrr}
+1 & -1 & 2
+\\
+1 & 0 & 1
+\\
+1 & 2  & -1
+\\
+1 & 1  & 0
+\end{array} \right]
+\end{align*}
+!et
+
+The columns of $\bm{X}$ are linearly dependent. We see this easily since the 
+the first column is the row-wise sum of the other two columns. The rank (more correct,
+the column rank) of a matrix is the dimension of the space spanned by the
+column vectors. Hence, the rank of $\mathbf{X}$ is equal to the number
+of linearly independent columns. In this particular case the matrix has rank 2.
+
+Super-collinearity of an $(n \times p)$-dimensional design matrix $\mathbf{X}$ implies
+that the inverse of the matrix $\bm{X}^T\bm{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+!bt
+\begin{align*}
+\bm{X} & =  \left[
+\begin{array}{rr}
+1 & -1
+\\
+1 & -1
+\end{array} \right].
+\end{align*}
+!et
+We see easily that  $\mbox{det}(\bm{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0$. Hence, $\mathbf{X}$ is singular and its inverse is undefined.
+This is equivalent to saying that the matrix $\bm{X}$ has at least an eigenvalue which is zero.
+
+
+!split
+===== Fixing the singularity =====
+
+If our design matrix $\bm{X}$ which enters the linear regression problem
+!bt
+\begin{align}
+\bm{\beta} & =  (\bm{X}^{T} \bm{X})^{-1} \bm{X}^{T} \bm{y},
+\end{align}
+!et
+has linearly dependent column vectors, we will not be able to compute the inverse
+of $\bm{X}^T\bm{X}$ and we cannot find the parameters (estimators) $\beta_i$. 
+The estimators are only well-defined if $(\bm{X}^{T}\bm{X})^{-1}$ exits. 
+This is more likely to happen when the matrix $\bm{X}$ is high-dimensional. In this case it is likely to encounter a situation where 
+the regression parameters $\beta_i$ cannot be estimated.
+
+A cheap  *ad hoc* approach is  simply to add a small diagonal component to the matrix to invert, that is we change
+!bt
+\[
+\bm{X}^{T} \bm{X} \rightarrow \bm{X}^{T} \bm{X}+\lambda \bm{I},
+\]
+!et 
+where $\bm{I}$ is the identity matrix.  When we discuss _Ridge_ regression this is actually what we end up evaluating. The parameter $\lambda$ is called a hyperparameter. More about this later. 
+
+
+
+!split
+===== Basic math of the SVD =====
+
+
+From standard linear algebra we know that a square matrix $\bm{X}$ can be diagonalized if and only it is 
+a so-called "normal matrix":"https://en.wikipedia.org/wiki/Normal_matrix", that is if $\bm{X}\in {\mathbb{R}}^{n\times n}$
+we have $\bm{X}\bm{X}^T=\bm{X}^T\bm{X}$ or if $\bm{X}\in {\mathbb{C}}^{n\times n}$ we have $\bm{X}\bm{X}^{\dagger}=\bm{X}^{\dagger}\bm{X}$.
+The matrix has then a set of eigenpairs 
+
+!bt
+\[
+(\lambda_1,\bm{u}_1),\dots, (\lambda_n,\bm{u}_n),
+!et
+and the eigenvalues are given by the diagonal matrix
+!bt
+\[
+\bm{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n).
+\]
+!et
+The matrix $\bm{X}$ can be written in terms of an orthogonal/unitary transformation $\bm{U}$
+!bt
+\[
+\bm{X} = \bm{U}\bm{\Sigma}\bm{V}^T,
+\]
+!et
+with $\bm{U}\bm{U}^T=\bm{I}$ or $\bm{U}\bm{U}^{\dagger}=\bm{I}$.
+
+Not all square matrices are diagonalizable. A matrix like the one discussed above
+!bt
+\[
+\bm{X} = \begin{bmatrix} 
+1&  -1 \\
+1& -1\\
+\end{bmatrix} 
+\]
+!et
+is not diagonalizable, it is a so-called "defective matrix":"https://en.wikipedia.org/wiki/Defective_matrix". It is easy to see that the condition
+$\bm{X}\bm{X}^T=\bm{X}^T\bm{X}$ is not fulfilled. 
+
+
+!split
+===== The SVD, a Fantastic Algorithm =====
+
+
+However, and this is the strength of the SVD algorithm, any general
+matrix $\bm{X}$ can be decomposed in terms of a diagonal matrix and
+two orthogonal/unitary matrices.  The "Singular Value Decompostion
+(SVD) theorem":"https://en.wikipedia.org/wiki/Singular_value_decomposition"
+states that a general $m\times n$ matrix $\bm{X}$ can be written in
+terms of a diagonal matrix $\bm{\Sigma}$ of dimensionality $m\times n$
+and two orthognal matrices $\bm{U}$ and $\bm{V}$, where the first has
+dimensionality $m \times m$ and the last dimensionality $n\times n$.
+We have then
+
+!bt
+\[ 
+\bm{X} = \bm{U}\bm{\Sigma}\bm{V}^T 
+\] 
+!et 
+
+As an example, the above defective matrix can be decomposed as
+
+!bt
+\[
+\bm{X} = \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 0\\ \end{bmatrix}    \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  -1 \\ 1& 1\\ \end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T, 
+\]
+!et
+
+with eigenvalues $\sigma_1=2$ and $\sigma_2=0$. 
+The SVD exits always! 
+
+The SVD
+decomposition (singular values) gives eigenvalues 
+$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the
+eigenvalues (singular values) are zero.
+
+In the general case, where our design matrix $\bm{X}$ has dimension
+$n\times p$, the matrix is thus decomposed into an $n\times n$
+orthogonal matrix $\bm{U}$, a $p\times p$ orthogonal matrix $\bm{V}$
+and a diagonal matrix $\bm{\Sigma}$ with $r=\mathrm{min}(n,p)$
+singular values $\sigma_i\geq 0$ on the main diagonal and zeros filling
+the rest of the matrix.  There are at most $p$ singular values
+assuming that $n > p$. In our regression examples for the nuclear
+masses and the equation of state this is indeed the case, while for
+the Ising model we have $p > n$. These are often cases that lead to
+near singular or singular matrices.
+
+The columns of $\bm{U}$ are called the left singular vectors while the columns of $\bm{V}$ are the right singular vectors.
+
+!split
+===== Economy-size SVD =====
+
+If we assume that $n > p$, then our matrix $\bm{U}$ has dimension $n
+\times n$. The last $n-p$ columns of $\bm{U}$ become however
+irrelevant in our calculations since they are multiplied with the
+zeros in $\bm{\Sigma}$.
+
+The economy-size decomposition removes extra rows or columns of zeros
+from the diagonal matrix of singular values, $\bm{\Sigma}$, along with the columns
+in either $\bm{U}$ or $\bm{V}$ that multiply those zeros in the expression. 
+Removing these zeros and columns can improve execution time
+and reduce storage requirements without compromising the accuracy of
+the decomposition.
+
+If $n > p$, we keep only the first $p$ columns of $\bm{U}$ and $\bm{\Sigma}$ has dimension $p\times p$. 
+If $p > n$, then only the first $n$ columns of $\bm{V}$ are computed and $\bm{\Sigma}$ has dimension $n\times n$.
+The $n=p$ case is obvious, we retain the full SVD. 
+In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.
+
+!split
+=====  Codes for the SVD =====
+
+!bc pycod
+import numpy as np
+# SVD inversion
+def SVDinv(A):
+    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
+    SVD is numerically more stable than the inversion algorithms provided by
+    numpy and scipy.linalg at the cost of being slower.
+    '''
+    U, s, VT = np.linalg.svd(A)
+#    print('test U')
+#    print( (np.transpose(U) @ U - U @np.transpose(U)))
+#    print('test VT')
+#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
+    print(U)
+    print(s)
+    print(VT)
+
+    D = np.zeros((len(U),len(VT)))
+    for i in range(0,len(VT)):
+        D[i,i]=s[i]
+    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
+    return np.matmul(V,np.matmul(invD,UT))
+
+
+X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+print(X)
+A = np.transpose(X) @ X
+print(A)
+# Brute force inversion of super-collinear matrix
+#B = np.linalg.inv(A)
+#print(B)
+C = SVDinv(A)
+print(C)
+
+!ec
+
+The matrix $\bm{X}$ has columns that are linearly dependent. The first
+column is the row-wise sum of the other two columns. The rank of a
+matrix (the column rank) is the dimension of space spanned by the
+column vectors. The rank of the matrix is the number of linearly
+independent columns, in this case just $2$. We see this from the
+singular values when running the above code. Running the standard
+inversion algorithm for matrix inversion with $\bm{X}^T\bm{X}$ results
+in the program terminating due to a singular matrix.
+
+
+
+!split
+===== Mathematical Properties =====
+
+There are several interesting mathematical properties which will be
+relevant when we are going to discuss the differences between say
+ordinary least squares (OLS) and _Ridge_ regression.
+
+We have from OLS that the parameters of the linear approximation are given by
+!bt
+\[
+\bm{\tilde{y}} = \bm{X}\bm{\beta} = \bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}. 
+\]
+!et
+
+The matrix to invert can be rewritten in terms of our SVD decomposition as
+
+!bt
+\[
+\bm{X}^T\bm{X} = \bm{V}\bm{\Sigma}^T\bm{U}^T\bm{U}\bm{\Sigma}\bm{V}^T.
+\]
+!et
+Using the orthogonality properties of $\bm{U}$ we have
+
+!bt
+\[
+\bm{X}^T\bm{X} = \bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T =  \bm{V}\bm{D}\bm{V}^T,
+\]
+!et
+with $\bm{D}$ being a diagonal matrix with values along the diagonal given by the singular values squared. 
+
+This means that
+!bt
+\[
+(\bm{X}^T\bm{X})\bm{V} = \bm{V}\bm{D},
+\]
+!et
+that is the eigenvectors of $(\bm{X}^T\bm{X})$ are given by the columns of the right singular matrix of $\bm{X}$ and the eigenvalues are the squared singular values.  It is easy to show (show this) that
+!bt
+\[
+(\bm{X}\bm{X}^T)\bm{U} = \bm{U}\bm{D},
+\]
+!et
+that is, the eigenvectors of $(\bm{X}\bm{X})^T$ are the columns of the left singular matrix and the eigenvalues are the same. 
+
+Going back to our OLS equation we have
+!bt
+\[
+\bm{X}\bm{\beta} = \bm{X}\left(\bm{V}\bm{D}\bm{V}^T \right)^{-1}\bm{X}^T\bm{y}=\bm{U\Sigma V^T}\left(\bm{V}\bm{D}\bm{V}^T \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\bm{U}\bm{U}^T\bm{y}.
+\]
+!et
+We will come back to this expression when we discuss Ridge regression. 
+
+
+$$ \tilde{y}^{OLS}={\bf X}\hat{\beta}^{OLS}=\sum_{j=1}^p {\bf u}_j{\bf u}_j^T{\bf y}$$ and for Ridge we have 
+
+$$ \tilde{y}^{Ridge}={\bf X}\hat{\beta}^{Ridge}=\sum_{j=1}^p {\bf u}_j\frac{\sigma_j^2}{\sigma_j^2+\lambda}{\bf u}_j^T{\bf y}$$ . 
+
+It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$. 
+
+Here we have that $${\bf X} = {\bf U}{\bf \Sigma}{\bf V}^T$$, with $$\Sigma$$ being an $$ n\times p$$ matrix and $${\bf V}$$ being a $$ p\times p$$ matrix. We also have assumed here that $$ n > p$$. 
+
+
+!split
+===== Ridge and LASSO Regression =====
+
+Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
+our optimization problem is
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\beta}\right)^T\left(\bm{y}-\bm{X}\bm{\beta}\right)\right\}.
+\]
+!et
+or we can state it as
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2,
+\]
+!et
+where we have used the definition of  a norm-2 vector, that is
+!bt
+\[
+\vert\vert \bm{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
+\]
+!et
+
+By minimizing the above equation with respect to the parameters
+$\bm{\beta}$ we could then obtain an analytical expression for the
+parameters $\bm{\beta}$.  We can add a regularization parameter $\lambda$ by
+defining a new cost function to be optimized, that is
+
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2+\lambda\vert\vert \bm{\beta}\vert\vert_2^2
+\]
+!et
+
+which leads to the Ridge regression minimization problem where we
+require that $\vert\vert \bm{\beta}\vert\vert_2^2\le t$, where $t$ is
+a finite number larger than zero. By defining
+
+!bt
+\[
+C(\bm{X},\bm{\beta})=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2+\lambda\vert\vert \bm{\beta}\vert\vert_1,
+\]
+!et
+
+we have a new optimization equation
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2+\lambda\vert\vert \bm{\beta}\vert\vert_1
+\]
+!et
+which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. 
+
+Here we have defined the norm-1 as 
+!bt
+\[
+\vert\vert \bm{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
+\]
+!et
+
+
+!split
+===== More on Ridge Regression =====
+
+Using the matrix-vector expression for Ridge regression,
+
+!bt
+\[
+C(\bm{X},\bm{\beta})=\frac{1}{n}\left\{(\bm{y}-\bm{X}\bm{\beta})^T(\bm{y}-\bm{X}\bm{\beta})\right\}+\lambda\bm{\beta}^T\bm{\beta},
+\]
+!et
+
+by taking the derivatives with respect to $\bm{\beta}$ we obtain then
+a slightly modified matrix inversion problem which for finite values
+of $\lambda$ does not suffer from singularity problems. We obtain
+
+!bt
+\[
+\bm{\beta}^{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y},
+\]
+!et
+
+with $\bm{I}$ being a $p\times p$ identity matrix with the constraint that
+
+!bt
+\[
+\sum_{i=0}^{p-1} \beta_i^2 \leq t,
+\]
+!et
+
+with $t$ a finite positive number. 
+
+We see that Ridge regression is nothing but the standard
+OLS with a modified diagonal term added to $\bm{X}^T\bm{X}$. The
+consequences, in particular for our discussion of the bias-variance tradeoff 
+are rather interesting.
+
+Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had
+!bt
+\[
+(\bm{X}\bm{X}^T)\bm{U} = \bm{U}\bm{D}.
+\]
+!et
+
+We can  analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\bm{U}$ as
+!bt
+\[
+\bm{X}\bm{\beta} = \bm{X}\left(\bm{V}\bm{D}\bm{V}^T \right)^{-1}\bm{X}^T\bm{y}=\bm{U\Sigma V^T}\left(\bm{V}\bm{D}\bm{V}^T \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\bm{U}\bm{U}^T\bm{y}
+\]
+!et
+
+
+For Ridge regression this becomes
+
+!bt
+\[
+\bm{X}\bm{\beta}^{\mathrm{Ridge}} = \bm{U\Sigma V^T}\left(\bm{V}\bm{D}\bm{V}^T+\lambda\bm{I} \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\sum_{j=0}^{p-1}\bm{u}_j\bm{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\bm{y},
+\]
+!et
+
+with the vectors $\bm{u}_j$ being the columns of $\bm{U}$. 
+
+!split
+===== Interpreting the Ridge results =====
+
+Since $\lambda \geq 0$, it means that compared to OLS, we have 
+
+!bt
+\[
+\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. 
+\]
+!et
+
+Ridge regression finds the coordinates of $\bm{y}$ with respect to the
+orthonormal basis $\bm{U}$, it then shrinks the coordinates by
+$\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has
+eigenvalues ordered in a descending way, that is $\sigma_i \geq
+\sigma_{i+1}$.
+
+For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
+Actually, calculating the variance of $\bm{X}\bm{v}_j$ shows that this quantity is equal to $\sigma_j^2/n$.
+With a parameter $\lambda$ we can thus shrink the role of specific parameters.  
+
+
+!split
+===== More interpretations =====
+
+For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
+
+!bt
+\[
+\bm{X}^T\bm{X}=(\bm{X}^T\bm{X})^{-1} =\bm{I}. 
+\]
+!et
+
+In this case the standard OLS results in 
+!bt
+\[
+\bm{\beta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_j\bm{u}_j^T\bm{y},
+\]
+!et
+
+and
+
+!bt
+\[
+\bm{\beta}^{\mathrm{Ridge}} = \left(\bm{I}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y}=\left(1+\lambda\right)^{-1}\bm{\beta}^{\mathrm{OLS}},
+\]
+!et
+
+that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\lambda$, and
+the Ridge estimator converges to zero when the hyperparameter goes to
+infinity.
+
+We will come back to more interpreations after we have gone through some of the statistical analysis part. 
+
+For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
+Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
+
+
+!split 
+===== A better understanding of regularization =====
+
+The parameter $\lambda$ that we have introduced in the Ridge (and
+Lasso as well) regression is often called a regularization parameter
+or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?
+
+Here we will first look at how to analyze the difference between the
+standard OLS equations and the Ridge expressions in terms of a linear
+algebra analysis using the SVD algorithm. Thereafter, we will link
+(see the material on the bias-variance tradeoff below) these
+observation to the statisical analysis of the results. In particular
+we consider how the variance of the parameters $\bm{\beta}$ is
+affected by changing the parameter $\lambda$.
+
+!split
+===== Decomposing the OLS and Ridge expressions =====
+
+We have our design matrix
+ $\bm{X}\in {\mathbb{R}}^{n\times p}$. With the SVD we decompose it as 
+
+!bt 
+\[
+\bm{X} = \bm{U\Sigma V^T},
+\]
+!et
+
+with $\bm{U}\in {\mathbb{R}}^{n\times n}$, $\bm{\Sigma}\in {\mathbb{R}}^{n\times p}$
+and $\bm{V}\in {\mathbb{R}}^{p\times p}$.
+
+The matrices $\bm{U}$ and $\bm{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\bm{U}^T\bm{U}=\bm{U}\bm{U}^T=\bm{I}$ and $\bm{V}^T\bm{V}=\bm{V}\bm{V}^T=\bm{I}$.
+
+
+
+!split
+===== Introducing the Covariance and Correlation functions  =====
+
+Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about
+the definition of the covariance and the correlation function. These are quantities 
+
+Suppose we have defined two vectors
+$\hat{x}$ and $\hat{y}$ with $n$ elements each. The covariance matrix $\bm{C}$ is defined as 
+!bt
+\[
+\bm{C}[\bm{x},\bm{y}] = \begin{bmatrix} \mathrm{cov}[\bm{x},\bm{x}] & \mathrm{cov}[\bm{x},\bm{y}] \\
+                              \mathrm{cov}[\bm{y},\bm{x}] & \mathrm{cov}[\bm{y},\bm{y}] \\
+             \end{bmatrix},
+\]
+!et
+where for example
+!bt
+\[
+\mathrm{cov}[\bm{x},\bm{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
+\]
+!et
+With this definition and recalling that the variance is defined as
+!bt
+\[
+\mathrm{var}[\bm{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
+\]
+!et
+we can rewrite the covariance matrix as 
+!bt
+\[
+\bm{C}[\bm{x},\bm{y}] = \begin{bmatrix} \mathrm{var}[\bm{x}] & \mathrm{cov}[\bm{x},\bm{y}] \\
+                              \mathrm{cov}[\bm{x},\bm{y}] & \mathrm{var}[\bm{y}] \\
+             \end{bmatrix}.
+\]
+!et
+
+The covariance takes values between zero and infinity and may thus
+lead to problems with loss of numerical precision for particularly
+large values. It is common to scale the covariance matrix by
+introducing instead the correlation matrix defined via the so-called
+correlation function
+
+!bt
+\[
+\mathrm{corr}[\bm{x},\bm{y}]=\frac{\mathrm{cov}[\bm{x},\bm{y}]}{\sqrt{\mathrm{var}[\bm{x}] \mathrm{var}[\bm{y}]}}.
+\]
+!et
+
+The correlation function is then given by values $\mathrm{corr}[\bm{x},\bm{y}]
+\in [-1,1]$. This avoids eventual problems with too large values. We
+can then define the correlation matrix for the two vectors $\bm{x}$
+and $\bm{y}$ as
+
+!bt
+\[
+\bm{K}[\bm{x},\bm{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\bm{x},\bm{y}] \\
+                              \mathrm{corr}[\bm{y},\bm{x}] & 1 \\
+             \end{bmatrix},
+\]
+!et
+
+In the above example this is the function we constructed using _pandas_.
+
+!split
+===== Correlation Function and Design/Feature Matrix =====
+
+In our derivation of the various regression algorithms like _Ordinary Least Squares_ or _Ridge regression_
+we defined the design/feature matrix $\bm{X}$ as
+
+!bt
+\[
+\bm{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+\]
+!et
+with $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$  refering to the column numbers and the
+entries $n$ being the row elements.
+We can rewrite the design/feature matrix in terms of its column vectors as
+!bt
+\[
+\bm{X}=\begin{bmatrix} \bm{x}_0 & \bm{x}_1 & \bm{x}_2 & \dots & \dots & \bm{x}_{p-1}\end{bmatrix},
+\]
+!et
+with a given vector
+!bt
+\[
+\bm{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
+\]
+!et
+
+With these definitions, we can now rewrite our $2\times 2$
+correaltion/covariance matrix in terms of a moe general design/feature
+matrix $\bm{X}\in {\mathbb{R}}^{n\times p}$. This leads to a $p\times p$
+covariance matrix for the vectors $\bm{x}_i$ with $i=0,1,\dots,p-1$
+
+!bt
+\[
+\bm{C}[\bm{x}] = \begin{bmatrix}
+\mathrm{var}[\bm{x}_0] & \mathrm{cov}[\bm{x}_0,\bm{x}_1]  & \mathrm{cov}[\bm{x}_0,\bm{x}_2] & \dots & \dots & \mathrm{cov}[\bm{x}_0,\bm{x}_{p-1}]\\
+\mathrm{cov}[\bm{x}_1,\bm{x}_0] & \mathrm{var}[\bm{x}_1]  & \mathrm{cov}[\bm{x}_1,\bm{x}_2] & \dots & \dots & \mathrm{cov}[\bm{x}_1,\bm{x}_{p-1}]\\
+\mathrm{cov}[\bm{x}_2,\bm{x}_0]   & \mathrm{cov}[\bm{x}_2,\bm{x}_1] & \mathrm{var}[\bm{x}_2] & \dots & \dots & \mathrm{cov}[\bm{x}_2,\bm{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{cov}[\bm{x}_{p-1},\bm{x}_0]   & \mathrm{cov}[\bm{x}_{p-1},\bm{x}_1] & \mathrm{cov}[\bm{x}_{p-1},\bm{x}_{2}]  & \dots & \dots  & \mathrm{var}[\bm{x}_{p-1}]\\
+\end{bmatrix},
+\]
+!et
+and the correlation matrix
+!bt
+\[
+\bm{K}[\bm{x}] = \begin{bmatrix}
+1 & \mathrm{corr}[\bm{x}_0,\bm{x}_1]  & \mathrm{corr}[\bm{x}_0,\bm{x}_2] & \dots & \dots & \mathrm{corr}[\bm{x}_0,\bm{x}_{p-1}]\\
+\mathrm{corr}[\bm{x}_1,\bm{x}_0] & 1  & \mathrm{corr}[\bm{x}_1,\bm{x}_2] & \dots & \dots & \mathrm{corr}[\bm{x}_1,\bm{x}_{p-1}]\\
+\mathrm{corr}[\bm{x}_2,\bm{x}_0]   & \mathrm{corr}[\bm{x}_2,\bm{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\bm{x}_2,\bm{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{corr}[\bm{x}_{p-1},\bm{x}_0]   & \mathrm{corr}[\bm{x}_{p-1},\bm{x}_1] & \mathrm{corr}[\bm{x}_{p-1},\bm{x}_{2}]  & \dots & \dots  & 1\\
+\end{bmatrix},
+\]
+!et
+
+
+!split
+===== Covariance Matrix Examples =====
+
+
+The Numpy function _np.cov_ calculates the covariance elements using
+the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have
+the exact mean values.  The following simple function uses the
+_np.vstack_ function which takes each vector of dimension $1\times n$
+and produces a $2\times n$ matrix $\bm{W}$
+
+
+!bt
+\[
+\bm{W} = \begin{bmatrix} x_0 & y_0 \\
+                          x_1 & y_1 \\
+                          x_2 & y_2\\
+                          \dots & \dots \\
+                          x_{n-2} & y_{n-2}\\
+                          x_{n-1} & y_{n-1} & 
+             \end{bmatrix},
+\]
+!et
+
+which in turn is converted into into the $2\times 2$ covariance matrix
+$\bm{C}$ via the Numpy function _np.cov()_. We note that we can also calculate
+the mean value of each set of samples $\bm{x}$ etc using the Numpy
+function _np.mean(x)_. We can also extract the eigenvalues of the
+covariance matrix through the _np.linalg.eig()_ function.
+
+!bc pycod
+# Importing various packages
+import numpy as np
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
+!ec
+
+!split
+===== Correlation Matrix  =====
+
+The previous example can be converted into the correlation matrix by
+simply scaling the matrix elements with the variances.  We should also
+subtract the mean values for each column. This leads to the following
+code which sets up the correlations matrix for the previous example in
+a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\times 2$ correlation matrix (since we have only two vectors). 
+
+!bc pycod
+import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+!ec
+
+We see that the matrix elements along the diagonal are one as they
+should be and that the matrix is symmetric. Furthermore, diagonalizing
+this matrix we easily see that it is a positive definite matrix.
+
+The above procedure with _numpy_ can be made more compact if we use _pandas_.
+
+!split
+===== Correlation Matrix with Pandas =====
+
+We whow here how we can set up the correlation matrix using _pandas_, as done in this simple code
+!bc pycod
+import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+!ec
+
+
+We expand this model to the Franke function discussed above.
+
+!split
+===== Correlation Matrix with Pandas and the Franke function =====
+
+!bc pycod
+# Common imports
+import numpy as np
+import pandas as pd
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+!ec
+
+We note here that the covariance is zero for the first rows and
+columns since all matrix elements in the design matrix were set to one
+(we are fitting the function in terms of a polynomial of degree $n$).
+
+This means that the variance for these elements will be zero and will
+cause problems when we set up the correlation matrix.  We can simply
+drop these elements and construct a correlation
+matrix without these elements. 
+
+
+!split
+===== Rewriting the Covariance and/or Correlation Matrix =====
+
+We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\bm{X}$ as 
+!bt
+\[
+\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}^T\bm{X}= \mathbb{E}[\bm{X}^T\bm{X}].
+\]
+!et
+
+To see this let us simply look at a design matrix $\bm{X}\in {\mathbb{R}}^{2\times 2}$
+!bt
+\[
+\bm{X}=\begin{bmatrix}
+x_{00} & x_{01}\\
+x_{10} & x_{11}\\
+\end{bmatrix}=\begin{bmatrix}
+\bm{x}_{0} & \bm{x}_{1}\\
+\end{bmatrix}.
+\]
+!et
+
+If we then compute the expectation value
+!bt
+\[
+\mathbb{E}[\bm{X}^T\bm{X}] = \frac{1}{n}\bm{X}^T\bm{X}=\begin{bmatrix}
+x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
+x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
+\end{bmatrix},
+\]
+!et
+which is just 
+!bt
+\[
+\bm{C}[\bm{x}_0,\bm{x}_1] = \bm{C}[\bm{x}]=\begin{bmatrix} \mathrm{var}[\bm{x}_0] & \mathrm{cov}[\bm{x}_0,\bm{x}_1] \\
+                              \mathrm{cov}[\bm{x}_1,\bm{x}_0] & \mathrm{var}[\bm{x}_1] \\
+             \end{bmatrix},
+\]
+!et
+where we wrote $$\bm{C}[\bm{x}_0,\bm{x}_1] = \bm{C}[\bm{x}]$$ to indicate that this the covariance of the vectors $\bm{x}$ of the design/feature matrix $\bm{X}$.
+
+It is easy to generalize this to a matrix $\bm{X}\in {\mathbb{R}}^{n\times p}$.
+
+
+!split
+===== Linking with SVD  =====
+
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,552 @@
+TITLE: Data Analysis and Machine Learning: Logistic Regression
+AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+DATE: today
+
+
+!split
+===== Plans for week 38 =====
+
+* Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples
+* Friday: Logistic Regression
+
+
+!split
+=====  Thursday: =====
+
+Material will added during Wednesday 16
+
+!split
+=====  Friday: Intro to Logistic Regression =====
+
+
+!split 
+===== Logistic Regression =====
+
+In linear regression our main interest was centered on learning the
+coefficients of a functional fit (say a polynomial) in order to be
+able to predict the response of a continuous variable on some unseen
+data. The fit to the continuous variable $y_i$ is based on some
+independent variables $\hat{x}_i$. Linear regression resulted in
+analytical expressions for standard ordinary Least Squares or Ridge
+regression (in terms of matrices to invert) for several quantities,
+ranging from the variance and thereby the confidence intervals of the
+parameters $\hat{\beta}$ to the mean squared error. If we can invert
+the product of the design matrices, linear regression gives then a
+simple recipe for fitting our data.
+
+!split 
+===== Classification problems =====
+
+
+Classification problems, however, are concerned with outcomes taking
+the form of discrete variables (i.e. categories). We may for example,
+on the basis of DNA sequencing for a number of patients, like to find
+out which mutations are important for a certain disease; or based on
+scans of various patients' brains, figure out if there is a tumor or
+not; or given a specific physical system, we'd like to identify its
+state, say whether it is an ordered or disordered system (typical
+situation in solid state physics); or classify the status of a
+patient, whether she/he has a stroke or not and many other similar
+situations.
+
+The most common situation we encounter when we apply logistic
+regression is that of two possible outcomes, normally denoted as a
+binary outcome, true or false, positive or negative, success or
+failure etc.
+
+!split
+===== Optimization and Deep learning =====
+
+Logistic regression will also serve as our stepping stone towards
+neural network algorithms and supervised deep learning. For logistic
+learning, the minimization of the cost function leads to a non-linear
+equation in the parameters $\hat{\beta}$. The optimization of the
+problem calls therefore for minimization algorithms. This forms the
+bottle neck of all machine learning algorithms, namely how to find
+reliable minima of a multi-variable function. This leads us to the
+family of gradient descent methods. The latter are the working horses
+of basically all modern machine learning algorithms.
+
+We note also that many of the topics discussed here on logistic 
+regression are also commonly used in modern supervised Deep Learning
+models, as we will see later.
+
+
+!split 
+===== Basics =====
+
+We consider the case where the dependent variables, also called the
+responses or the outcomes, $y_i$ are discrete and only take values
+from $k=0,\dots,K-1$ (i.e. $K$ classes).
+
+The goal is to predict the
+output classes from the design matrix $\hat{X}\in\mathbb{R}^{n\times p}$
+made of $n$ samples, each of which carries $p$ features or predictors. The
+primary goal is to identify the classes to which new unseen samples
+belong.
+ 
+Let us specialize to the case of two classes only, with outputs
+$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a
+credit card user that could default or not on her/his credit card
+debt. That is
+
+
+!bt
+\[
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\  1 & \mathrm{yes} \end{bmatrix}.
+\]
+!et
+
+
+
+!split
+===== Linear classifier =====
+
+Before moving to the logistic model, let us try to use our linear
+regression model to classify these two outcomes. We could for example
+fit a linear model to the default case if $y_i > 0.5$ and the no
+default case $y_i \leq 0.5$.
+
+We would then have our 
+weighted linear combination, namely 
+!bt
+\begin{equation}
+\hat{y} = \hat{X}^T\hat{\beta} +  \hat{\epsilon},
+\end{equation}
+!et
+where $\hat{y}$ is a vector representing the possible outcomes, $\hat{X}$ is our
+$n\times p$ design matrix and $\hat{\beta}$ represents our estimators/predictors.
+
+!split
+===== Some selected properties =====
+
+The main problem with our function is that it takes values on the
+entire real axis. In the case of logistic regression, however, the
+labels $y_i$ are discrete variables. A typical example is the credit
+card data discussed below here, where we can set the state of
+defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons
+in the data set (see the full example below).
+
+One simple way to get a discrete output is to have sign
+functions that map the output of a linear regressor to values $\{0,1\}$,
+$f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise. 
+We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
+literature. This model is extremely simple. However, in many cases it is more
+favorable to use a ``soft" classifier that outputs
+the probability of a given category. This leads us to the logistic function.
+
+
+!split
+===== The logistic function =====
+
+The perceptron is an example of a ``hard classification'' model. We
+will encounter this model when we discuss neural networks as
+well. Each datapoint is deterministically assigned to a category (i.e
+$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a ``soft''
+classifier that outputs the probability of a given category rather
+than a single value. For example, given $x_i$, the classifier
+outputs the probability of being in a category $k$.  Logistic regression
+is the most common example of a so-called soft classifier. In logistic
+regression, the probability that a data point $x_i$
+belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, 
+!bt
+\[
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+\]
+!et
+Note that $1-p(t)= p(-t)$.
+
+!split
+===== Examples of likelihood functions used in logistic regression and nueral networks =====
+
+
+The following code plots the logistic function, the step function and other functions we will encounter from here and on.
+
+
+!bc pycod
+"""The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""tanh Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()
+!ec
+
+
+
+
+
+
+
+!split
+=====  Two parameters =====
+
+We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\beta$ in our fitting of the Sigmoid function, that is we define probabilities 
+!bt
+\begin{align*}
+p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
+\end{align*}
+!et
+where $\hat{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. 
+
+Note that we used
+!bt
+\[
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
+\]
+!et
+
+!split 
+===== Maximum likelihood =====
+
+In order to define the total likelihood for all possible outcomes from a  
+dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels
+$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called "Maximum Likelihood Estimation":"https://en.wikipedia.org/wiki/Maximum_likelihood_estimation" (MLE) principle. 
+We aim thus at maximizing 
+the probability of seeing the observed data. We can then approximate the 
+likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is 
+!bt
+\begin{align*}
+P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\
+\end{align*}
+!et
+from which we obtain the log-likelihood and our _cost/loss_ function
+!bt
+\[
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
+\]
+!et
+
+!split
+===== The cost function rewritten =====
+
+Reordering the logarithms, we can rewrite the _cost/loss_ function as
+!bt
+\[
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+\]
+!et
+
+The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$.
+Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
+!bt
+\[
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+\]
+!et
+This equation is known in statistics as the _cross entropy_. Finally, we note that just as in linear regression, 
+in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.
+
+!split
+=====  Minimizing the cross entropy =====
+
+The cross entropy is a convex function of the weights $\hat{\beta}$ and,
+therefore, any local minimizer is a global minimizer. 
+
+
+Minimizing this
+cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain
+
+!bt
+\[
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n  \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
+\]
+!et
+and 
+!bt
+\[
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n  \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\]
+!et
+
+!split
+=====  A more compact expression =====
+
+Let us now define a vector $\hat{y}$ with $n$ elements $y_i$, an
+$n\times p$ matrix $\hat{X}$ which contains the $x_i$ values and a
+vector $\hat{p}$ of fitted probabilities $p(y_i\vert x_i,\hat{\beta})$. We can rewrite in a more compact form the first
+derivative of cost function as
+
+!bt
+\[
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). 
+\]
+!et
+
+If we in addition define a diagonal matrix $\hat{W}$ with elements 
+$p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})$, we can obtain a compact expression of the second derivative as 
+
+!bt
+\[
+\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. 
+\]
+!et
+
+!split
+===== Extending to more predictors =====
+
+Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors
+!bt
+\[
+\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
+\]
+!et
+Here we defined $\hat{x}=[1,x_1,x_2,\dots,x_p]$ and $\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to
+!bt
+\[
+p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
+\]
+!et
+
+!split
+===== Including more classes =====
+
+Till now we have mainly focused on two classes, the so-called binary
+system. Suppose we wish to extend to $K$ classes.  Let us for the sake
+of simplicity assume we have only two predictors. We have then
+following model
+
+!bt
+\[
+\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
+\]
+!et
+!bt
+\[
+\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
+\]
+!et
+and so on till the class $C=K-1$ class
+!bt
+\[
+\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
+\]
+!et
+
+and the model is specified in term of $K-1$ so-called log-odds or
+_logit_ transformations.
+
+
+!split
+===== More classes =====
+
+In our discussion of neural networks we will encounter the above again
+in terms of a slightly modified function, the so-called _Softmax_ function.
+
+The softmax function is used in various multiclass classification
+methods, such as multinomial logistic regression (also known as
+softmax regression), multiclass linear discriminant analysis, naive
+Bayes classifiers, and artificial neural networks.  Specifically, in
+multinomial logistic regression and linear discriminant analysis, the
+input to the function is the result of $K$ distinct linear functions,
+and the predicted probability for the $k$-th class given a sample
+vector $\hat{x}$ and a weighting vector $\hat{\beta}$ is (with two
+predictors):
+
+!bt
+\[
+p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
+\]
+!et
+It is easy to extend to more predictors. The final class is 
+!bt
+\[
+p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
+\]
+!et
+
+and they sum to one. Our earlier discussions were all specialized to
+the case with two classes only. It is easy to see from the above that
+what we derived earlier is compatible with these equations.
+
+To find the optimal parameters we would typically use a gradient
+descent method.  Newton's method and gradient descent methods are
+discussed in the material on "optimization
+methods":"https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html".
+
+
+
+
+!split
+===== A simple classification problem =====
+!bc pycod
+import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
+
+
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
+
+
+def visualize(X, y, clf):
+    plot_decision_boundary(lambda x: clf.predict(x), X, y)
+
+def plot_decision_boundary(pred_func, X, y):
+    # Set min and max values and give it some padding
+    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
+    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
+    h = 0.01
+    # Generate a grid of points with distance h between them
+    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
+    # Predict the function value for the whole gid
+    Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
+    Z = Z.reshape(xx.shape)
+    # Plot the contour and training examples
+    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
+    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
+    plt.show()
+
+
+def classify(X, y):
+    clf = linear_model.LogisticRegressionCV()
+    clf.fit(X, y)
+    return clf
+
+
+def main():
+    X, y = generate_data()
+    # visualize(X, y)
+    clf = classify(X, y)
+    visualize(X, y, clf)
+
+if __name__ == "__main__":
+    main()
+!ec
+
+
+
+!split
+=====  Cancer Data again now with Decision Trees and other Methods =====
+!bc pycod
+import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.linear_model import LogisticRegression
+
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+logreg.fit(X_train, y_train)
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+!ec
+
+
+!split
+===== Other measures in classification studies: Cancer Data  again =====
+!bc pycod 
+import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.linear_model import LogisticRegression
+
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+logreg.fit(X_train, y_train)
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+
+
+from sklearn.preprocessing import LabelEncoder
+from sklearn.model_selection import cross_validate
+#Cross validation
+accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Logistic Regression  and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+
+
+import scikitplot as skplt
+y_pred = logreg.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+plt.show()
+y_probas = logreg.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+plt.show()
+
+!ec
+
+
+
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
@@ -0,0 +1,3 @@
+#!/bin/sh
+doconce clean
+rm -rf *.pdf *.tex ipynb*.tar.gz *.html ._*.html *~ reveal.js Trash README.txt
@@ -0,0 +1,79 @@
+#!/bin/sh
+set -x
+
+function system {
+  "$@"
+  if [ $? -ne 0 ]; then
+    echo "make.sh: unsuccessful command $@"
+    echo "abort!"
+    exit 1
+  fi
+}
+
+if [ $# -eq 0 ]; then
+echo 'bash make.sh slides1|slides2'
+exit 1
+fi
+
+name=$1
+rm -f *.tar.gz
+
+opt="--encoding=utf-8"
+# Note: Makefile examples contain constructions like ${PROG} which
+# looks like Mako constructions, but they are not. Use --no_mako
+# to turn off Mako processing.
+opt="--no_mako"
+
+rm -f *.aux
+
+
+html=${name}-reveal
+system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt
+system doconce slides_html $html reveal --html_slide_theme=beige
+
+# Plain HTML documents
+
+html=${name}-solarized
+system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+html=${name}
+system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt
+system doconce split_html $html.html --method=space10
+
+# Bootstrap style
+html=${name}-bs
+system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
+system doconce split_html $html.html --method=split --pagination --nav_button=bottom
+
+# IPython notebook
+system doconce format ipynb $name $opt
+
+
+# Publish
+dest=../../pub
+if [ ! -d $dest/$name ]; then
+mkdir $dest/$name
+mkdir $dest/$name/html
+mkdir $dest/$name/ipynb
+fi
+cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html
+
+# Figures: cannot just copy link, need to physically copy the files
+if [ -d fig-${name} ]; then
+if [ ! -d $dest/$name/html/fig-$name ]; then
+mkdir $dest/$name/html/fig-$name
+fi
+cp -r fig-${name}/* $dest/$name/html/fig-$name
+fi
+
+cp ${name}.ipynb $dest/$name/ipynb
+ipynb_tarfile=ipynb-${name}-src.tar.gz
+if [ ! -f ${ipynb_tarfile} ]; then
+cat > README.txt <<EOF
+This IPython notebook ${name}.ipynb does not require any additional
+programs.
+EOF
+tar czf ${ipynb_tarfile} README.txt
+fi
+cp ${ipynb_tarfile} $dest/$name/ipynb
				`@@ -0,0 +1 @@`
				`{'year': '1999-2020', 'holder': ['Morten Hjorth-Jensen'], 'cite doconce': False, 'license': 'Released under CC Attribution-NonCommercial 4.0 license'}`