feat: Add ordinary least square figure

content/chapters/part1/1.tex

@@ -135,4 +135,11 @@ If $H$ is the projection matrix of the subspace generated by $\X$, $X\Y$ is the
     \includestandalone{figures/schemes/orthogonal_projection}
     \caption{Orthogonal projection of $\Y$ on plan generated by the base described by $\X$. $\color{blue}a$ corresponds to $\norm{\X\hat{\beta} - \bar{\Y}}^2$ and $\color{blue}b$ corresponds to $\norm{\Y - \hat{\beta}\X}^2$}
     \label{fig:scheme-orthogonal-projection}
-\end{figure}
+\end{figure}
+
+\begin{figure}
+    \centering
+    \includestandalone{figures/schemes/ordinary_least_squares}
+    \caption{Ordinary least squares and regression line with simulated data.}
+    \label{fig:ordinary-least-squares}
+\end{figure}

content/chapters/part1/2.tex

@@ -69,7 +69,7 @@ Let $u = \begin{pmatrix}
 \begin{figure}
     \centering
     \includestandalone{figures/schemes/vector_orthogonality}
-    \caption{Illustration for the scalar product of two orthogonal vectors.}
+    \caption{Scalar product of two orthogonal vectors.}
     \label{fig:scheme-orthogonal-scalar-product}
 \end{figure}
 

figures/schemes/ordinary_least_squares.tex

@@ -0,0 +1,45 @@
+\documentclass[margin=0.5cm]{standalone}
+\usepackage{tikz}
+\usepackage{luacode}
+
+
+\begin{document}
+
+\begin{tikzpicture}
+    % Draw axes
+    \draw[->] (0,0) -- (5,0);
+    \draw[->] (0,0) -- (0,5);
+
+    \directlua{
+        function runif(min, max)
+            return min + (max - min) * math.random()
+        end
+        math.randomseed(42)
+        x_min = 0
+        x_max = 5
+        error_min = -1
+        error_max = 1
+        beta0 = 2
+        beta1 = 1/5
+        x_values = {}
+        y_values = {}
+        for i=1,42 do
+            x = runif(x_min, x_max)
+            epsilon = runif(error_min, error_max)
+            y_hat = beta0 + beta1 * x
+            y = y_hat + epsilon
+            tex.print("\\draw[-,very thin, lightgray] ("..x..","..y_hat..") -- ("..x..","..y..") ;")
+            x_values[i] = x
+            y_values[i] = y
+        end
+        for i=1,42 do
+            x = x_values[i]
+            y = y_values[i]
+            tex.print("\\node[black] at ("..x..","..y..") {.};")
+        end
+    }
+    % Draw least square line
+    \draw[-,blue,thick] (0,2) -- (5,\directlua{tex.print(5*beta1+beta0)});
+    % Draw square norm
+\end{tikzpicture}
+\end{document}