fat_equations.tex

% !TEX TS-program = xelatex
%
% Created by Ryan Batt on 2016-05-23.
% Copyright (c) 2016 .
\documentclass{article}

\usepackage{polyglossia}
\usepackage{hyperref}

\hypersetup
{
  pdftitle   = {FatEquations},
  pdfsubject = {Extremes},
  pdfauthor  = {Ryan Batt}
}

\title{Equations for Extreme Events in Ecosystem Time Series}
\author{Ryan Batt}

\begin{document}

\maketitle

\begin{abstract}
    Abstract
\end{abstract}

\section{Equations}

% GEV CDF
GEV cummulative distribution function:  
\[F\left( x;\mu ,\sigma ,\xi  \right)=\left\{ \begin{matrix}
   \exp \left\{ -{{[1+\xi (x-\mu )/\sigma ]}^{-1/\xi }} \right\} & \xi \ne 0  \\
   \exp \left\{ -\exp \left[ -(x-\mu )/\sigma  \right] \right\} & \xi =0  \\
\end{matrix} \right.\]

% GEV constraint in-text
$1+\xi (x-\mu )/\sigma >0$

% GEV Support
\[\begin{array}{*{35}{l}}
   x\in [\mu -\sigma /\xi ,+\infty ) & when\text{ }\xi >0,  \\
   x\in (-\infty ,+\infty ) & when\text{ }\xi =0,  \\
   x\in (-\infty ,\mu -\sigma /\xi ] & when\text{ }\xi <0  \\
\end{array}\]

% Heuristic Population Eq1 
\[{{X}_{T}}\left( t+\Delta t \right)=B\left( \varphi \left( t \right),{{X}_{T}} \right)S\left( \gamma \left( t \right),{{X}_{T}} \right){{X}_{T}}\left( t \right)\]

% Heuristic Population, in-text term 1
\[B\left( \varphi \left( t \right),{{X}_{T}} \right)\]

%  Heuristic Population, in-text term 2
\[S\left( \gamma \left( t \right),{{X}_{T}} \right)\]

% Heuristic Population Eq2
\[{{X}_{T+1}}={{X}_{T}}\prod\limits_{i=1}^{1/\Delta t}{B\left( \varphi \left( i\Delta t \right),{{X}_{T}} \right)S\left( \gamma \left( i\Delta t \right),{{X}_{T}} \right)}\]

% Heuristic Population Eq3
\[\begin{array}{*{35}{l}}
   {{x}_{T+1}} & ={{x}_{T}}+\sum\limits_{i=1}^{1/\Delta t}{\left\{ b\left( \varphi \left( i\Delta t \right),{{x}_{T}} \right)+s\left( \gamma \left( i\Delta t \right),{{x}_{T}} \right) \right\}}  \\
   {} & ={{x}_{T}}+r\left( T,{{x}_{T}} \right)+\varepsilon \left( T,{{x}_{T}} \right)  \\
\end{array}\]

% Heuristic Population in-text term 3 (Fig params part1)
\[B\left( \varphi  \right)=\max \left[ 1,{{e}^{-b\left( 1+\varphi  \right)}} \right]\]

% Heuristic Population in-text term 4 (Fig params part2)
$S\left( \gamma ,{{X}_{T}} \right)={{\left[ \left( 1+{{e}^{-s\left( 1+\gamma  \right)}} \right){{\left( 1+a{{X}_{T}} \right)}^{1/\Delta t}} \right]}^{-1}}$

% ARMA Equation
(x_t - \mu) = \sum_{i=1}^p \varphi_i (x_{t-i} - \mu) +  \sum_{j=0}^q \theta_j \epsilon_{t-j}


\end{document}