From 4f427f345fb703c5db7ac01eb440a69cce09872b Mon Sep 17 00:00:00 2001
From: Justin Bedo <cu@cua0.org>
Date: Mon, 21 Nov 2022 09:39:44 +1100
Subject: init

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+\documentclass[aspectratio=169,UKenglish]{beamer}
+
+\usetheme{metropolis}
+\usepackage[sfdefault]{FiraSans}
+\usefonttheme{professionalfonts}
+\setbeamerfont{footnote}{size=
+  \tiny}
+
+\usepackage{microtype}
+
+\usepackage{tikz}
+\usetikzlibrary{shapes}
+\usetikzlibrary{bayesnet}
+\usepackage{stmaryrd}
+
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\bx}{\mathbf{x}}
+\DeclareMathOperator{\alr}{alr}
+\DeclareMathOperator{\clr}{clr}
+
+\usepackage[natbib=true,url=false,style=verbose-ibid]{biblatex}
+\addbibresource{slides.bib}
+\AtBeginBibliography{\small}
+
+\author{Justin Bed\H{o}}
+\title{Exploration of deep mutational scanning data with unsupervised methods}
+\date{December 13, 2022}
+
+\begin{document}
+
+  \maketitle
+
+  \section{Deep Mutational Scanning (DMS) data}
+
+  \begin{frame}{Deep Mutational Scanning (DMS) data} Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
+    Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.
+    Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.
+    Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
+  \end{frame}
+
+  \section{Compositional data}
+
+  \begin{frame}{Basics}
+    \begin{definition}[Compositional data] Data \(X \in \R^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex
+      \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\}    \]
+      for constant \(\kappa > 0\).
+    \end{definition} Information is therefore given only by the ratios of components and any composition can be normalised to the standard simplex where \(\kappa = 1\) (c.f., dividing by library size).
+  \end{frame}
+
+  \begin{frame}{Isomorphisms to Euclidean vector spaces} The simplex forms a \(d-1\) dimensional Euclidean vector space
+    \footfullcite{Aitchison1982}:
+    \begin{definition}[Additive logratio transform]
+      \[\alr(\bx)_i = \log \frac{x_i}{x_0}    \]
+    \end{definition}
+    \begin{definition}[Center logratio transform]
+      \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}}    \]
+    \end{definition}
+  \end{frame}
+
+  \begin{frame}{PCA on DMS data}
+    \begin{block}{Transformation approach}
+      \begin{enumerate}
+        \item Map DMS data to Euclidean space via ALR/CLR
+        \item Apply standard PCA
+      \end{enumerate}
+    \end{block}
+    \begin{block}{Problems}
+      \begin{itemize}
+        \item Zeros:
+        \begin{enumerate}
+          \item geometric mean is \(0\) \(\Rightarrow\) CLR is undefined
+          \item ALR is undefined for unobserved components
+        \end{enumerate}
+        \item Interpretation:
+        \begin{enumerate}
+          \item ALR is not isometry
+          \item CLR is degenerate
+        \end{enumerate}
+      \end{itemize}
+    \end{block}
+  \end{frame}
+
+  \section{Bregman divergences}
+
+\end{document}
-- 
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