diff --git a/docs/pca_1/pca_1.html b/docs/pca_1/pca_1.html
index 9bad878b..696bee60 100644
--- a/docs/pca_1/pca_1.html
+++ b/docs/pca_1/pca_1.html
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@@ -812,13 +812,11 @@ <h3 data-number="24.4.2" class="anchored" data-anchor-id="deriving-pca-as-error-
 <p><span class="math display">\[\begin{aligned}
 L(z,w) &amp;= \frac{1}{n}\sum_{i=1}^{n}(X_i - z_{i}w)(X_i - z_{i}w)^T \\
 &amp;= \frac{1}{n}\sum_{i=1}^{n}(X_{i}X_{i}^T - 2z_{i}X_{i}w^T + z_{i}^{2}ww^T) &amp; \text{(expand the loss)} \\
-= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) &amp; \text{(First term is constant and }ww^T=1\text{ by orthonormality)}
+= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) &amp; \text{(First term is constant and }ww^T=1\text{ by orthonormality)} \\
 \end{aligned}\]</span></p>
-<p>\end{aligned}$$</p>
 <p>Now, we can take the derivative with respect to <span class="math inline">\(Z_i\)</span>. <span class="math display">\[\begin{aligned}
 \frac{\partial{L(Z,W)}}{\partial{z_i}} &amp;= \frac{1}{n}(-2X_{i}w^T + 2z_{i}) \\
-z_i &amp;= X_iw^T &amp; \text{(Setting derivative equal to 0 and solving for }z_i\text{)}
-\end{aligned}\]</span></p>
+z_i &amp;= X_iw^T &amp; \text{(Setting derivative equal to 0 and solving for }z_i\text{)}\end{aligned}\]</span></p>
 <p>We can now substitute our solution for <span class="math inline">\(z_i\)</span> in our loss function:</p>
 <p><span class="math display">\[\begin{aligned}
 L(z,w) &amp;= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) \\
diff --git a/index.log b/index.log
index d407c463..9bdd2d4b 100644
--- a/index.log
+++ b/index.log
@@ -1,4 +1,4 @@
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+This is XeTeX, Version 3.141592653-2.6-0.999995 (TeX Live 2023) (preloaded format=xelatex 2024.3.3)  17 APR 2024 16:26
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
diff --git a/index.pdf b/index.pdf
index 7f6fca76..b17b7582 100644
Binary files a/index.pdf and b/index.pdf differ
diff --git a/index.tex b/index.tex
index b0ca02f9..9e07b520 100644
--- a/index.tex
+++ b/index.tex
@@ -220,7 +220,7 @@
 
 \begin{document}
 \maketitle
-\ifdefined\Shaded\renewenvironment{Shaded}{\begin{tcolorbox}[enhanced, borderline west={3pt}{0pt}{shadecolor}, interior hidden, boxrule=0pt, frame hidden, breakable, sharp corners]}{\end{tcolorbox}}\fi
+\ifdefined\Shaded\renewenvironment{Shaded}{\begin{tcolorbox}[breakable, boxrule=0pt, borderline west={3pt}{0pt}{shadecolor}, sharp corners, interior hidden, frame hidden, enhanced]}{\end{tcolorbox}}\fi
 
 \renewcommand*\contentsname{Table of contents}
 {
@@ -262,7 +262,7 @@ \section*{About the Course Notes}\label{about-the-course-notes}}
 \hypertarget{introduction}{%
 \chapter{Introduction}\label{introduction}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -316,7 +316,7 @@ \chapter{Introduction}\label{introduction}}
 allowing you to take data and produce useful insights on the world's
 most challenging and ambiguous problems.
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Course Goals}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Course Goals}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -334,7 +334,7 @@ \chapter{Introduction}\label{introduction}}
 
 \end{tcolorbox}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Some Topics We'll Cover}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Some Topics We'll Cover}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -364,7 +364,7 @@ \chapter{Introduction}\label{introduction}}
 
 \end{tcolorbox}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Prerequisites}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Prerequisites}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 To ensure that you can get the most out of the course content, please
 make sure that you are familiar with:
@@ -602,7 +602,7 @@ \section{Conclusion}\label{conclusion}}
 \hypertarget{pandas-i}{%
 \chapter{Pandas I}\label{pandas-i}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -3276,7 +3276,7 @@ \section{Parting Note}\label{parting-note}}
 \hypertarget{pandas-ii}{%
 \chapter{Pandas II}\label{pandas-ii}}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -4334,7 +4334,7 @@ \section{Parting Note}\label{parting-note-1}}
 \hypertarget{pandas-iii}{%
 \chapter{Pandas III}\label{pandas-iii}}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -20061,7 +20061,7 @@ \chapter{Data Cleaning and EDA}\label{data-cleaning-and-eda}}
 \end{Highlighting}
 \end{Shaded}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -22623,7 +22623,7 @@ \subsection{EDA and Data Wrangling}\label{eda-and-data-wrangling}}
 \hypertarget{regular-expressions}{%
 \chapter{Regular Expressions}\label{regular-expressions}}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -23911,7 +23911,7 @@ \section{Limitations of Regular
 \hypertarget{visualization-i}{%
 \chapter{Visualization I}\label{visualization-i}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -24749,7 +24749,7 @@ \subsubsection{Evaluating Histograms}\label{evaluating-histograms}}
 \hypertarget{visualization-ii}{%
 \chapter{Visualization II}\label{visualization-ii}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -26295,7 +26295,7 @@ \subsection{Harnessing Context}\label{harnessing-context}}
 \hypertarget{sampling}{%
 \chapter{Sampling}\label{sampling}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -27012,7 +27012,7 @@ \section{Summary}\label{summary-1}}
 \hypertarget{introduction-to-modeling}{%
 \chapter{Introduction to Modeling}\label{introduction-to-modeling}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -27375,7 +27375,7 @@ \subsection{Derivation}\label{derivation}}
   \(\hat{a} = \text{average of }y - \text{slope}\cdot\text{average of }x\)
 \end{itemize}
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 Proof:
 
@@ -27727,7 +27727,7 @@ \section{Fitting the Model}\label{fitting-the-model}}
 \chapter{Constant Model, Loss, and
 Transformations}\label{constant-model-loss-and-transformations}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -29065,7 +29065,7 @@ \section{Bonus: Calculating Constant Model MSE Using an Algebraic
 \hypertarget{ordinary-least-squares}{%
 \chapter{Ordinary Least Squares}\label{ordinary-least-squares}}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -29239,7 +29239,7 @@ \subsection{Multiple Linear
 \hypertarget{linear-algebra-approach}{%
 \subsection{Linear Algebra Approach}\label{linear-algebra-approach}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Vector Dot Product}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Vector Dot Product}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 The \textbf{dot product (or inner product)} is a vector operation that:
 
@@ -29337,7 +29337,7 @@ \subsection{Linear Algebra Approach}\label{linear-algebra-approach}}
 \(\mathbb{Y}\) is also a vector with \(n\) elements
 (\(\mathbb{Y} \in \mathbb{R}^{n}\)).
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Linearity}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Linearity}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 An expression is \textbf{linear in \(\theta\)} (a set of parameters) if
 it is a linear combination of the elements of the set. Checking if an
@@ -29382,7 +29382,7 @@ \subsection{Mean Squared Error}\label{mean-squared-error}}
 indication of how ``far away'' the predictions are from the true values,
 on average.
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: L2 Norm}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: L2 Norm}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 When working with vectors, this idea of ``distance'' or the vector's
 \textbf{size/length} is represented by the \textbf{norm}. More
@@ -29530,7 +29530,7 @@ \subsection{A Note on Terminology for Multiple Linear
 \hypertarget{geometric-derivation}{%
 \section{Geometric Derivation}\label{geometric-derivation}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: Span}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: Span}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 Recall that the \textbf{span} or \textbf{column space} of a matrix
 \(\mathbb{X}\) (denoted \(span(\mathbb{X})\)) is the set of all possible
@@ -29542,7 +29542,7 @@ \section{Geometric Derivation}\label{geometric-derivation}}
 
 \end{tcolorbox}
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: Matrix-Vector Multiplication}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 There are 2 ways we can think about matrix-vector multiplication
 
@@ -29631,7 +29631,7 @@ \section{Geometric Derivation}\label{geometric-derivation}}
 visualize this as the vector created by dropping a perpendicular line
 from \(\mathbb{Y}\) onto the span of \(\mathbb{X}\).
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: Orthogonality}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra: Orthogonality}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 Recall that two vectors \(\vec{a}\) and \(\vec{b}\) are orthogonal if
 their dot product is zero: \(\vec{a}^{T}\vec{b} = 0\).
@@ -29754,7 +29754,7 @@ \section{OLS Properties}\label{ols-properties}}
 
 \[\mathbb{X}^Te = 0 \]
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 Proof:
 
@@ -29792,7 +29792,7 @@ \section{OLS Properties}\label{ols-properties}}
 
 \[\sum_i^n e_i = 0\]
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 Proof:
 
@@ -29858,7 +29858,7 @@ \section{Bonus: Uniqueness of the
 The Least Squares estimate \(\hat{\theta}\) is \textbf{unique} if and
 only if \(\mathbb{X}\) is \textbf{full column rank}.
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 Proof:
 
@@ -29948,7 +29948,7 @@ \section{Bonus: Uniqueness of the
 \chapter{sklearn and Gradient
 Descent}\label{sklearn-and-gradient-descent}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -31134,7 +31134,7 @@ \subsection{Batch Gradient Descent and Stochastic Gradient
 \hypertarget{feature-engineering}{%
 \chapter{Feature Engineering}\label{feature-engineering}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -31731,7 +31731,7 @@ \section{Complexity and Overfitting}\label{complexity-and-overfitting}}
 we can improve model performance by designing increasingly complex
 models.
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Math Fact: Polynomial Degrees}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Math Fact: Polynomial Degrees}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 Given \(N\) overlapping data points, we can always find a polynomial of
 degree \(N-1\) that goes through all those points.
@@ -31827,7 +31827,7 @@ \chapter{Case Study in Human Contexts and
 understand the material. The course notes will have the same broader
 structure but are by no means comprehensive.
 
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+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -32052,7 +32052,7 @@ \section{The Response: Cook County Open Data
 \subsection{1. Question/Problem
 Formulation}\label{questionproblem-formulation}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -32092,7 +32092,7 @@ \subsection{1. Question/Problem
 comfortable. We can determine some metrics of success and frame a social
 problem as a data science problem.
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Definitions: Fairness and Transparency}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Definitions: Fairness and Transparency}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 The definitions, as given by the Cook County Assessor's Office, are
 given below:
@@ -32185,7 +32185,7 @@ \subsection{1. Question/Problem
 \subsection{2. Data Acquisition and
 Cleaning}\label{data-acquisition-and-cleaning}}
 
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 \begin{itemize}
 \tightlist
@@ -32262,7 +32262,7 @@ \subsection{2. Data Acquisition and
 \subsection{3. Exploratory Data
 Analysis}\label{exploratory-data-analysis}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
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 \begin{itemize}
 \tightlist
@@ -32308,7 +32308,7 @@ \subsection{3. Exploratory Data
 \subsection{4. Prediction and
 Inference}\label{prediction-and-inference}}
 
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 \begin{itemize}
 \tightlist
@@ -32370,7 +32370,7 @@ \subsection{4. Prediction and
 \hypertarget{results-and-conclusions}{%
 \subsection{5. Results and Conclusions}\label{results-and-conclusions}}
 
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 \begin{itemize}
 \tightlist
@@ -32516,7 +32516,7 @@ \section{Key Takeaways}\label{key-takeaways}}
 \chapter{Cross Validation and
 Regularization}\label{cross-validation-and-regularization}}
 
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 \begin{itemize}
 \tightlist
@@ -33314,7 +33314,7 @@ \section{Regression Summary}\label{regression-summary}}
 \hypertarget{random-variables}{%
 \chapter{Random Variables}\label{random-variables}}
 
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 \begin{itemize}
 \tightlist
@@ -33360,7 +33360,7 @@ \chapter{Random Variables}\label{random-variables}}
 implications for modeling, but if you want to go a step further, take
 Data 140, CS 70, and/or EECS 126.
 
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 Recall the following concepts from Data 8:
 
@@ -33719,7 +33719,7 @@ \subsection{Variance}\label{variance}}
 calculations, as \(\mathbb{E}[X^2] = \text{Var}(X)\) if \(X\) is
 centered and \(E(X)=0\).
 
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 \[\begin{align}
    \text{Var}(X) &= \mathbb{E}[(X-\mathbb{E}[X])^2] \\
@@ -33747,7 +33747,7 @@ \subsection{Example: Die}\label{example-die}}
       0, \text{otherwise} 
    \end{cases}\]
 
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 \[ \begin{align} 
          \mathbb{E}[X] &= 1\big(\frac{1}{6}\big) + 2\big(\frac{1}{6}\big) + 3\big(\frac{1}{6}\big) + 4\big(\frac{1}{6}\big) + 5\big(\frac{1}{6}\big) + 6\big(\frac{1}{6}\big) \\
@@ -33757,7 +33757,7 @@ \subsection{Example: Die}\label{example-die}}
 
 \end{tcolorbox}
 
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 Using Approach 1 (definition): \[\begin{align} 
       \text{Var}(X) &= \big(\frac{1}{6}\big)((1 - \frac{7}{2})^2 + (2 - \frac{7}{2})^2 + (3 - \frac{7}{2})^2 + (4 - \frac{7}{2})^2 + (5 - \frac{7}{2})^2 + (6 - \frac{7}{2})^2) \\
@@ -33820,7 +33820,7 @@ \subsection{Properties of Expectation}\label{properties-of-expectation}}
 
 \[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\]
 
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 \[\begin{align}
         \mathbb{E}[aX+b] &= \sum_{x} (ax + b) P(X=x) \\
@@ -33841,7 +33841,7 @@ \subsection{Properties of Expectation}\label{properties-of-expectation}}
 
 \[\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]\]
 
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 \[\begin{align}
     \mathbb{E}[X+Y] &= \sum_{s} (X+Y)(s) P(s) \\
@@ -33892,7 +33892,7 @@ \subsection{Properties of Variance}\label{properties-of-variance}}
   \(X\) by \(b\) units.
 \end{itemize}
 
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 We know that \[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\]
 
@@ -33931,7 +33931,7 @@ \subsection{Properties of Variance}\label{properties-of-variance}}
   \[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text{ independent}\]
 \end{enumerate}
 
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 The variance of a sum is affected by the dependence between the two
 random variables that are being added. Let's expand the definition of
@@ -34103,7 +34103,7 @@ \subsection{Summary}\label{summary-2}}
 \chapter{Estimators, Bias, and
 Variance}\label{estimators-bias-and-variance}}
 
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 \begin{itemize}
 \tightlist
@@ -34255,7 +34255,7 @@ \subsection{Example}\label{example}}
 
 C. \(Y_C = 20 * X_1\)
 
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 Let \(X_1, X_2, ... X_{20}\) be 20 i.i.d Bernoulli(0.5) random
 variables. Since the \(X_i\)'s are independent,
@@ -34416,7 +34416,7 @@ \subsection{Using the Sample Mean to Estimate the Population
 \textbf{unbiased estimator} of the population mean and will explore this
 idea more in the next lecture.
 
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 The square root law
 (\href{https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html\#the-square-root-law}{Data
@@ -34539,7 +34539,7 @@ \subsubsection{Estimating a Linear
 modeled by \[Y = g(x) + \epsilon\]
 \[ f_{\theta}(x) = Y = \theta_0 + \sum_{j=1}^p \theta_j x_j + \epsilon\]
 
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 In our two equations above, the true relationship
 \(g(x) = \theta_0 + \sum_{j=1}^p \theta_j x_j\) is not random, but
@@ -34554,7 +34554,7 @@ \subsubsection{Estimating a Linear
 use it to train a model and obtain an estimate of \(\hat{\theta}\)
 \[\hat{Y}(x) = f_{\hat{\theta}}(x) = \hat{\theta_0} + \sum_{j=1}^p \hat{\theta_j} x_j\]
 
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 In our estimating equation above, our sample \(\Bbb{X}\), \(\Bbb{Y}\)
 are random (often due to human error). Hence, the estimates we calculate
@@ -34745,7 +34745,7 @@ \subsubsection{Model Bias}\label{model-bias}}
 \(g(x)\); if it's negative, our model tends to underestimate \(g(x)\).
 And if it's 0, we can say that our model is \textbf{unbiased}.
 
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 An \textbf{unbiased model} has a \(\text{model bias } = 0\). In other
 words, our model predicts \(g(x)\) on average.
@@ -34842,7 +34842,7 @@ \section{{[}Bonus{]} Proof of Bias-Variance
 Decomposition in the Bias-Variance Tradeoff section above, and this
 content is out of scope.
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 \textbf{Click to show}\vspace{2mm}
 
@@ -34991,7 +34991,7 @@ \subsection{Step 4: Bias-Variance
 \chapter{Causal Inference and
 Confounding}\label{causal-inference-and-confounding}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -35120,7 +35120,7 @@ \section{Review: Bootstrap
   accurately model the variability of our estimates.
 \end{itemize}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-warning-color}{\faExclamationTriangle}\hspace{0.5em}{Why must we resample \emph{with replacement}?}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-warning-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-warning-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-warning-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-warning-color}{\faExclamationTriangle}\hspace{0.5em}{Why must we resample \emph{with replacement}?}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-warning-color-frame]
 
 Given an original sample of size \(n\), we want a resample that has the
 same size \(n\) as the original. Sampling \emph{without} replacement
@@ -35784,7 +35784,7 @@ \subsection{How to perform causal
 \hypertarget{sql-i}{%
 \chapter{SQL I}\label{sql-i}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -36665,7 +36665,7 @@ \section{Summary}\label{summary-3}}
 \hypertarget{sql-ii}{%
 \chapter{SQL II}\label{sql-ii}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -37637,7 +37637,7 @@ \subsection{Common Table Expressions}\label{common-table-expressions}}
 \hypertarget{logistic-regression-i}{%
 \chapter{Logistic Regression I}\label{logistic-regression-i}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -38096,7 +38096,7 @@ \subsubsection{3. Putting it Together}\label{putting-it-together}}
 
 \[\sigma(t) = \frac{1}{1+e^{-t}}\]
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Properties of the Sigmoid}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Properties of the Sigmoid}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -38170,7 +38170,7 @@ \section{The Logistic Regression
 \hat{P}_{\theta}(Y = 1 | x) = \sigma(x^{\top}\theta)
 \end{align}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Properties of the Logistic Model}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Properties of the Logistic Model}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 Consider a logistic regression model with one feature and an intercept
 term:
@@ -38195,7 +38195,7 @@ \section{The Logistic Regression
 
 \end{tcolorbox}
 
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, toprule=.15mm, breakable, arc=.35mm, rightrule=.15mm, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, left=2mm, opacityback=0, leftrule=.75mm, arc=.35mm, colback=white, rightrule=.15mm, bottomrule=.15mm, toprule=.15mm]
 
 \textbf{Example Calculation}\vspace{2mm}
 
@@ -38646,7 +38646,7 @@ \subsection{Likelihood of Data}\label{likelihood-of-data}}
 \hypertarget{logistic-regression-ii}{%
 \chapter{Logistic Regression II}\label{logistic-regression-ii}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -39275,7 +39275,7 @@ \subsection{Stochastic Gradient Descent Update
 \hypertarget{pca-i}{%
 \chapter{PCA I}\label{pca-i}}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-note-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-note-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame]
 
 \begin{itemize}
 \tightlist
@@ -39528,7 +39528,7 @@ \section{Matrix Decomposition
   The output is the cost of each bowl at each store.
 \end{itemize}
 
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Matrix Multiplication}, breakable, coltitle=black, opacitybacktitle=0.6, colframe=quarto-callout-tip-color-frame, leftrule=.75mm, bottomtitle=1mm, arc=.35mm, titlerule=0mm, toptitle=1mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, colback=white, left=2mm, opacityback=0, bottomrule=.15mm]
+\begin{tcolorbox}[enhanced jigsaw, breakable, titlerule=0mm, colbacktitle=quarto-callout-tip-color!10!white, toprule=.15mm, bottomtitle=1mm, opacitybacktitle=0.6, arc=.35mm, left=2mm, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Linear Algebra Review: Matrix Multiplication}, opacityback=0, leftrule=.75mm, toptitle=1mm, colback=white, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-tip-color-frame]
 
 In general, there are two ways to interpret matrix multiplication:
 
@@ -39702,16 +39702,13 @@ \subsection{Deriving PCA as Error
 \[\begin{aligned}
 L(z,w) &= \frac{1}{n}\sum_{i=1}^{n}(X_i - z_{i}w)(X_i - z_{i}w)^T \\
 &= \frac{1}{n}\sum_{i=1}^{n}(X_{i}X_{i}^T - 2z_{i}X_{i}w^T + z_{i}^{2}ww^T) & \text{(expand the loss)} \\
-= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \text{(First term is constant and }ww^T=1\text{ by orthonormality)}
+= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \text{(First term is constant and }ww^T=1\text{ by orthonormality)} \\
 \end{aligned}\]
 
-\textbackslash end\{aligned\}\$\$
-
 Now, we can take the derivative with respect to \(Z_i\).
 \[\begin{aligned}
 \frac{\partial{L(Z,W)}}{\partial{z_i}} &= \frac{1}{n}(-2X_{i}w^T + 2z_{i}) \\
-z_i &= X_iw^T & \text{(Setting derivative equal to 0 and solving for }z_i\text{)}
-\end{aligned}\]
+z_i &= X_iw^T & \text{(Setting derivative equal to 0 and solving for }z_i\text{)}\end{aligned}\]
 
 We can now substitute our solution for \(z_i\) in our loss function:
 
diff --git a/pca_1/pca_1.qmd b/pca_1/pca_1.qmd
index ee269923..18ffcbcc 100644
--- a/pca_1/pca_1.qmd
+++ b/pca_1/pca_1.qmd
@@ -250,16 +250,13 @@ In our derivation, let's first work with the case where $k=1$. Here Z will be an
 $$\begin{aligned}
 L(z,w) &= \frac{1}{n}\sum_{i=1}^{n}(X_i - z_{i}w)(X_i - z_{i}w)^T \\
 &= \frac{1}{n}\sum_{i=1}^{n}(X_{i}X_{i}^T - 2z_{i}X_{i}w^T + z_{i}^{2}ww^T) & \text{(expand the loss)} \\
-= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \text{(First term is constant and }ww^T=1\text{ by orthonormality)}
-\end{aligned}$$
-
+= \frac{1}{n}\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \text{(First term is constant and }ww^T=1\text{ by orthonormality)} \\
 \end{aligned}$$
 
 Now, we can take the derivative with respect to $Z_i$. 
 $$\begin{aligned}
 \frac{\partial{L(Z,W)}}{\partial{z_i}} &= \frac{1}{n}(-2X_{i}w^T + 2z_{i}) \\
-z_i &= X_iw^T & \text{(Setting derivative equal to 0 and solving for }z_i\text{)}
-\end{aligned}$$
+z_i &= X_iw^T & \text{(Setting derivative equal to 0 and solving for }z_i\text{)}\end{aligned}$$
 
 We can now substitute our solution for $z_i$ in our loss function:
 
@@ -293,3 +290,4 @@ This result implies that:
 This derivation can inductively be used for the next (second) principal component (not shown).
 
 The final takeaway from this derivation is that the **principal components** are the **eigenvectors** with the **largest eigenvalues** of the **covariance matrix**. These are the **directions** of the **maximum variance** of the data.
+