feat(typst): backup_slides

slides/backup_slides.typ

@@ -0,0 +1,64 @@
+#import "@preview/polylux:0.3.1": *
+#import themes.clean: *
+#import "utils.typ": *
+
+#new-section-slide("Backup Slides")
+
+#slide(
+  title: [How the Normal distribution arose
+    #footnote[Origins can be traced back to Abraham de Moivre in 1738. A better explanation
+      can be found by
+      #link(
+        "http://www.stat.yale.edu/~pollard/Courses/241.fall2014/notes2014/Bin.Normal.pdf",
+      )[clicking here].]],
+)[
+  #text(
+    size: 16pt,
+  )[
+
+    $
+      "Binomial"(n, k)                           &= binom(n, k) p^k (1-p)^(n-k) \
+      n!                                         &≈ sqrt(2 π n) (n / e)^n \
+      lim_(n → oo) binom(n, k) p^k (1-p)^(n-k) &=
+      1 / (sqrt(2 π n p q)) e^(-((k - n p)^2) / (2 n p q))
+    $
+
+    We know that in the binomial: $op("E") = n p$ and $op("Var") = n p q$; hence
+    replacing $op("E")$ by $μ$ and $op("Var")$ by $σ^2$:
+
+    $
+      lim_(n → oo) binom(n, k) p^k (1-p)^(n-k) = 1 / (σ sqrt(2 π)) e^(-((k - μ)^2) / (σ^2))
+    $
+  ]
+]
+
+#slide(
+  title: [QR Decomposition],
+)[
+  #text(
+    size: 13pt,
+  )[
+    In Linear Algebra 101, we learn that any matrix (even non-square ones) can be
+    decomposed into a product of two matrices:
+
+    - $bold(Q)$: an orthogonal matrix (its columns are orthogonal unit vectors, i.e. $bold(Q)^T = bold(Q)^(-1)$)
+    - $bold(R)$: an upper-triangular matrix
+
+    Now, we incorporate the QR decomposition into the linear regression model. Here,
+    I am going to use the "thin" QR instead of the "fat", which scales $bold(Q)$ and $bold(R)$ matrices
+    by a factor of
+    $sqrt(n - 1)$ where $n$ is the number of rows in $bold(X)$. In practice, it is
+    better to implement the thin QR, than the fat QR decomposition. It is more
+    numerical stable. Mathematically speaking, the thing QR decomposition is:
+
+    $
+      bold(X)   &= bold(Q)^* bold(R)^* \
+      bold(Q)^* &= bold(Q) dot sqrt(n - 1) \
+      bold(R)^* &= 1 / (sqrt(n - 1)) dot bold(R) \
+      bold(μ)  &= α + bold(X) dot bold(β) + σ \
+                &= α + bold(Q)^* dot bold(R)^* dot bold(β) + σ \
+                &= α + bold(Q)^* dot (bold(R)^* dot bold(β)) + σ \
+                &= α + bold(Q)^* dot tilde(bold(β)) + σ
+    $
+  ]
+]

slides/slides.typ

@@ -72,6 +72,7 @@
 // #include "11-hierarchical_models.typ"
 // #include "12-mcmc.typ"
 // #include "13-model_comparison.typ"
+#include "backup_slides.typ"
 
 #slide(title: "Bibliography")[
   #bibliography(title: none, "references.yml")