feat: Add lua script for matrix product representation

.gitattributes

@@ -0,0 +1 @@
+main.pdf filter=lfs diff=lfs merge=lfs -text

content/chapters/part1/1.tex

@@ -1,34 +1,22 @@
+\chapter{Linear Model}
 
-
+\section{Simple Linear Regression}
-\section{Generalized Linear Model}
 
 \[
-    g(\EE(Y)) = X \beta
+    Y_i = \beta_0 + \beta_1 X_i  + \varepsilon_i
 \]
-with $g$ being
+\[
-\begin{itemize}
+    \Y = \X \beta + \varepsilon.
-    \item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
+\]
-    \item Poission regression: $g(v) = \log(v)$, for instance for discrete variables. 
+\[
-\end{itemize}
-
-\subsection{Penalized Regression}
-
-When the number of variables is large, e.g, when the number of explanatory variable is above the number of observations, if $p >> n$ ($p$: the number of explanatory variable, $n$ is the number of observations), we cannot estimate the parameters.
-In order to estimate the parameters, we can use penalties (additional terms).
-
-Lasso regression, Elastic Net, etc.
-
-\subsection{Simple Linear Model}
-
-\begin{align*}
-    \Y &= \X \beta + \varepsilon \\
     \begin{pmatrix}
         Y_1 \\
         Y_2 \\
         \vdots \\
         Y_n
     \end{pmatrix}
-    &= \begin{pmatrix}
+    = 
+    \begin{pmatrix}
         1 & X_1 \\
         1 & X_2 \\
         \vdots & \vdots \\
@@ -42,17 +30,35 @@ Lasso regression, Elastic Net, etc.
     \begin{pmatrix}
         \varepsilon_1 \\
         \varepsilon_2 \\
-        \vdots \\
+        \vdots
         \varepsilon_n
     \end{pmatrix}
-\end{align*}
+\]
 
-\subsection{Assumptions}
+\paragraph*{Assumptions}
+\begin{enumerate}[label={\color{primary}{($A_\arabic*$)}}]
+    \item $\varepsilon_i$ are independent;
+    \item $\varepsilon_i$ are identically distributed;
+    \item $\varepsilon_i$ are i.i.d $\sim \Norm(0, \sigma^2)$ (homoscedasticity).
+\end{enumerate}
 
+\section{Generalized Linear Model}
+
+\[
+    g(\EE(Y)) = X \beta
+\]
+with $g$ being
 \begin{itemize}
-    \item 
+    \item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
+    \item Poisson regression: $g(v) = \log(v)$, for instance for discrete variables. 
 \end{itemize}
 
+\subsection{Penalized Regression}
+
+When the number of variables is large, e.g, when the number of explanatory variable is above the number of observations, if $p >> n$ ($p$: the number of explanatory variable, $n$ is the number of observations), we cannot estimate the parameters.
+In order to estimate the parameters, we can use penalties (additional terms).
+
+Lasso regression, Elastic Net, etc.
 
 \subsection{Statistical Analysis Workflow}
 
@@ -95,9 +101,9 @@ is noted equivalently as
 
 \subsection{General Case}
 
-If $\X^\T\X$ is invertible, the OLS estimator is:
+If $\X^T\X$ is invertible, the OLS estimator is:
 \begin{equation}
-\hat{\beta} = (\X^\T\X)^{-1} \X^\T \Y
+\hat{\beta} = (\X^T\X)^{-1} \X^T \Y
 \end{equation}
 
 \subsection{Ordinary Least Square Algorithm}
@@ -112,12 +118,12 @@ We want to minimize the distance between $\X\beta$ and $\Y$:
     \Rightarrow& \forall v \in w,\, vy = v proj^w(y)\\
     \Rightarrow& \forall i: \\
     & \X_i \Y = \X_i X\hat{\beta} \qquad \text{where $\hat{\beta}$ is the estimator of $\beta$} \\
-    \Rightarrow& \X^\T \Y = \X^\T \X \hat{\beta} \\
+    \Rightarrow& \X^T \Y = \X^T \X \hat{\beta} \\
-    \Rightarrow& {\color{gray}(\X^\T \X)^{-1}} \X^\T \Y = {\color{gray}(\X^\T \X)^{-1}} (\X^\T\X) \hat{\beta} \\
+    \Rightarrow& {\color{gray}(\X^T \X)^{-1}} \X^T \Y = {\color{gray}(\X^T \X)^{-1}} (\X^T\X) \hat{\beta} \\
-    \Rightarrow& \hat{\beta} = (X^\T\X)^{-1} \X^\T \Y
+    \Rightarrow& \hat{\beta} = (\X^T\X)^{-1} \X^T \Y
 \end{align*}
 
-This formula comes from the orthogonal projection of $\Y$ on the subspace define by the explanatory variables $\X$
+This formula comes from the orthogonal projection of $\Y$ on the vector subspace defined by the explanatory variables $\X$
 
 $\X \hat{\beta}$ is the closest point to $\Y$ in the subspace generated by $\X$.
 

content/chapters/part1/2.tex

@@ -31,12 +31,12 @@ Let $u = \begin{pmatrix}
     We may use $\scalar{u, v}$ or $u \cdot v$ notations.
 \end{definition}
 \paragraph{Dot product properties}
-\begin{itemize}
+\begin{description}
-    \item $\scalar{u, v} = \scalar{v, u}$
+    \item[Commutative] $\scalar{u, v} = \scalar{v, u}$
-    \item $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
+    \item[Distributive] $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
-    \item $\scalar{u, v}$
+    \item $\scalar{u, v} = \norm{u} \times \norm{v} \times \cos(\widehat{u, v})$
-    \item $\scalar{\vec{u}, \vec{v}} = \norm{\vec{u}} \times \norm{\vec{v}} \times \cos(\widehat{\vec{u}, \vec{v}})$
+    \item $\scalar{a, a} = \norm{a}^2$
-\end{itemize}
+\end{description}
 
 \begin{definition}[Norm]
     Length of the vector.
@@ -99,7 +99,7 @@ Let $u = \begin{pmatrix}
     \end{align*}
 \end{proof}
 
-\begin{theorem}{Pythagorean theorem}
+\begin{theorem}[Pythagorean theorem]
     If $u \perp v$, then $\norm{u+v}^2 = \norm{u}^2 + \norm{v}^2$ .
 \end{theorem}
 
@@ -110,7 +110,7 @@ Let $y = \begin{pmatrix}
         y_1 \\
         .   \\
         y_n
-    \end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$
+    \end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$.
 $\mathcal{Y}$ can be written as the orthogonal projection of $y$ on $w$:
 \[
     \mathcal{Y} = proj^w(y) + z,
@@ -178,8 +178,25 @@ The scalar product between $z$ and (?) is zero.
             x_3 \\
             x_4
         \end{pmatrix}
-        & = \begin{pmatrix}
+        =
-                a x_1 + b x_2 + c x_3 \ldots
+        \begin{pmatrix} 
+            \luadirect{
+                local matrix_product = require("scripts.matrix_product")
+                local m1 = {
+                    {"a", "b", "c", "d"}, 
+                    {"e", "f", "g", "h"},
+                    {"i", "j", "k", "l"}
+                }
+                local m2 = {
+                    {"x_1"},
+                    {"x_2"},
+                    {"x_3"},
+                    {"x_4"}
+                }
+                local product_matrix = matrix_product.matrix_product_repr(m1,m2)
+                local matrix_dump = matrix_product.dump_matrix(product_matrix)
+                tex.print(matrix_dump)
+            }
         \end{pmatrix}
     \end{align*}
 \end{example}
@@ -190,7 +207,7 @@ The number of columns has to be the same as the dimension of the vector to which
     Let $A = \begin{pmatrix}
         a & b \\
         c & d
-    \end{pmatrix}$, then $A^\T =  \begin{pmatrix}
+    \end{pmatrix}$, then $A^T =  \begin{pmatrix}
         a & c \\
         b & d
     \end{pmatrix}$

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figures/schemes/regression_plan_3D.tex

@@ -0,0 +1,26 @@
+\documentclass[tikz,border=3.14mm]{standalone}
+\usepackage{tikz-3dplot}
+\begin{document}
+\tdplotsetmaincoords{105}{-30}
+\usetikzlibrary{patterns}
+\begin{tikzpicture}[tdplot_main_coords,font=\sffamily]
+ \tdplotsetrotatedcoords{00}{30}{0}
+ \begin{scope}[tdplot_rotated_coords]
+  \begin{scope}[canvas is xy plane at z=0]
+    \draw[fill opacity=0,pattern=north west lines,pattern color=gray] (-2,-3) rectangle (2,3); 
+    \draw[gray,fill=lightgray,fill opacity=0.75] (-2,-3) rectangle (2,3); 
+   \draw[very thick] (-2,0) -- (2,0);
+   \path (-150:2) coordinate (H) (-1.5,0) coordinate(X);
+   \pgflowlevelsynccm
+   \draw[very thick,-stealth,gray] (0,0) -- (-30:1.5);
+  \end{scope} 
+  \draw[stealth-] (H) -- ++ (-1,0,0.2) node[pos=1.3]{$H$};
+  \draw[stealth-] (X) -- ++ (0,1,0.2) node[pos=1.3]{$X$};
+  \draw[very thick,-stealth] (0,0,0) coordinate (O) -- (0,0,3) node[right]{$p$};
+ \end{scope}
+ \pgfmathsetmacro{\Radius}{1.5}
+ \draw[-stealth]  (O)-- (2.5*\Radius,0,0) node[pos=1.15] {$x$};
+ \draw[-stealth] (O) -- (0,3.5*\Radius,0) node[pos=1.15] {$z$};
+ \draw[-stealth] (O) -- (0,0,2.5*\Radius) node[pos=1.05] {$y$};
+\end{tikzpicture} 
+\end{document} 

preamble.tex

@@ -3,4 +3,4 @@
 \usepackage{standalone}
 \usepackage{tikz-3dplot}
 \usepackage{tkz-euclide}
-\usepackage{mathtools}
+\usepackage{nicematrix}

scripts/matrix_product.lua

@@ -0,0 +1,57 @@
+local function matrix_product_repr(m1, m2)
+    if #m1[1] ~= #m2 then -- inner matrix-dimensions must agree
+        return nil      
+    end 
+
+    local res = {}
+    
+    for i = 1, #m1 do
+        res[i] = {}
+        for j = 1, #m2[1] do
+            res[i][j] = " "
+            for k = 1, #m2 do
+                if k ~= 1 then
+                    res[i][j] = res[i][j] .. " + "
+                end
+                res[i][j] = res[i][j] ..  m1[i][k] .. " " .. m2[k][j]
+            end
+        end
+    end
+    return res
+end
+
+local function dump_matrix(matrix)
+    local repr = ""
+    for i, row in ipairs(matrix) do
+        for j, cell in ipairs(row) do
+            repr = repr .. " " .. cell
+            if j ~= #row then
+                repr = repr .. " & "
+            end
+        end
+        if i ~= #matrix then
+            repr = repr .. [[ \\ ]]
+        end
+        repr = repr .. "\n"
+    end
+    return repr
+end
+
+local m1 = {
+    {"a", "b", "c", "d"}, 
+    {"e", "f", "g", "h"},
+    {"i", "j", "k", "l"}
+}
+local m2 = {
+    {"x_1"},
+    {"x_2"},
+    {"x_3"},
+    {"x_4"}
+}
+
+print(dump_matrix(matrix_product_repr(m1, m2)))
+
+return {
+    matrix_product_repr = matrix_product_repr,
+    dump_matrix = dump_matrix
+}