\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}
