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+\documentclass[14pt,aspectratio=169]{beamer}
+\usepackage{polyglossia}
+\setdefaultlanguage{english}
+\usetheme{SaintPetersburg}
+
+\usepackage{amsthm}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{mathtools}
+\usepackage{listings}
+\usepackage{booktabs}
+\usepackage{graphicx}
+\usepackage{tikz}
+
+\graphicspath{{figures/}}
+
+\newcommand{\Fourier}[1]{\mathcal{F}\left\{#1\right\}}
+\newcommand{\InverseFourier}[1]{\mathcal{F}^{-1}\left\{#1\right\}}
+\newcommand{\Var}[1]{\sigma_{#1}^2}
+
+\AtBeginSection[]{
+% \iffirstsection
+% \begin{frame}{Plan}
+% \tableofcontents
+% \end{frame}
+% \firstsectionfalse
+% \fi
+% \begin{frame}{Plan}
+% \tableofcontents[currentsection]
+% \end{frame}
+ \frame{\sectionpage}
+}
+
+\title{Generating standing and propagating ocean waves with three-dimensional ARMA model}
+\subtitle{Technical report}
+\author{Ivan Gankevich}
+\date{Aug 26, 2016}
+
+\begin{document}
+
+ \frame{\maketitle}
+
+ \section{Two methods of finding wave's ACF}
+
+ \begin{frame}
+ \frametitle{Analytic method}
+ \only<1>{%
+ Apply Wiener---Khinchin theorem to a wave profile $\zeta$ to get ACF $K$:
+ \begin{equation*}
+ K(t) = \Fourier{\left| \zeta(t) \right|^2}.
+ \label{eq:wiener-khinchin}
+ \end{equation*}%
+ }
+ \only<2>{%
+ \begin{example}
+ Standing wave profile:
+ \begin{equation*}
+ \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq:standing-wave}
+ \end{equation*}
+ Standing wave ACF:
+ \begin{equation*}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos \beta t
+ \cos \left[ \beta x + \beta y \right].
+ \label{eq:standing-wave-acf}
+ \end{equation*}
+ \end{example}%
+ }
+ \only<3>{%
+ \begin{example}
+ Propagating wave profile:
+ \begin{equation*}
+ \zeta(t, x, y) = A \cos (\sigma t + k_x x + k_y y).
+ \label{eq:propagating-wave}
+ \end{equation*}
+ Propagating wave ACF:
+ \begin{equation*}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos\left[\beta (t+x+y) \right].
+ \label{eq:propagating-wave-acf}
+ \end{equation*}
+ \end{example}%
+ }
+ \only<4>{%
+ Some observations:
+ \begin{itemize}
+ \item Taking Fourier transform of sine/cosine wave profile requires
+ multiplying it by an decaying exponent to produce useful ACF.
+ \item Fourier Transform of squared exponent (Gaussian) is another Gaussian.
+ \end{itemize}
+ \vfill\centering%
+ \alert{Why use Fourier transform at all?}%
+ }
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Empirical method}
+ The algorithm:
+ \begin{enumerate}
+ \item Multiply wave profile by an decaying exponent.
+ \item Adjust sine/cosine phase to move maximum value to the origin
+ (or substitute sine with cosine to get the same effect).
+ \end{enumerate}
+ \vfill%
+ In case of plain waves result is the same as for analitic method.
+ \end{frame}
+
+ \section{Governing equations for 3-dimensional ARMA process}
+
+ \begin{frame}
+ \frametitle{3-D ARMA process}
+ Three-dimensional autoregressive moving average process is defined by
+ \begin{equation*}
+ \zeta_{i,j,k} =
+ \sum\limits_{l=0}^{p_1}
+ \sum\limits_{m=0}^{p_2}
+ \sum\limits_{n=0}^{p_3}
+ \Phi_{l,m,n} \zeta_{i-l,j-m,k-n}
+ +
+ \sum\limits_{l=0}^{q_1}
+ \sum\limits_{m=0}^{q_2}
+ \sum\limits_{n=0}^{q_3}
+ \Theta_{l,m,n} \epsilon_{i-l,j-m,k-n}
+ ,
+ \label{eq:arma-process}
+ \end{equation*}
+ \small{%
+ where $\zeta$ --- wave elevation, $\Phi$ --- AR coefficients, $\Theta$ --- MA
+ coefficients, \newline$\epsilon$ --- white noise with Gaussian distribution,
+ $(p_1,p_2,p_3)$ --- AR process order, $(q_1,q_2,q_3)$ --- MA process order, and
+ $\Phi_{0,0,0} \equiv 0$, $\Theta_{0,0,0} \equiv 0$.%
+ }
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Determining coefficients}
+ \framesubtitle{AR process}
+ \small%
+ Solve linear system of equations (3-D Yule---Walker equations) for $\Phi$:
+ \begin{equation*}
+ \Gamma
+ \left[
+ \begin{array}{l}
+ \Phi_{0,0,0}\\
+ \Phi_{0,0,1}\\
+ \vdotswithin{\Phi_{0,0,0}}\\
+ \Phi_{p_1,p_2,p_3}
+ \end{array}
+ \right]
+ =
+ \left[
+ \begin{array}{l}
+ K_{0,0,0}-\Var{\epsilon}\\
+ K_{0,0,1}\\
+ \vdotswithin{K_{0,0,0}}\\
+ K_{p_1,p_2,p_3}
+ \end{array}
+ \right],
+ \qquad
+ \Gamma=
+ \left[
+ \begin{array}{llll}
+ \Gamma_0 & \Gamma_1 & \cdots & \Gamma_{p_1} \\
+ \Gamma_1 & \Gamma_0 & \ddots & \vdotswithin{\Gamma_0} \\
+ \vdotswithin{\Gamma_0} & \ddots & \ddots & \Gamma_1 \\
+ \Gamma_{p_1} & \cdots & \Gamma_1 & \Gamma_0
+ \end{array}
+ \right],
+ \end{equation*}
+ \begin{equation*}
+ \Gamma_i =
+ \left[
+ \begin{array}{llll}
+ \Gamma^0_i & \Gamma^1_i & \cdots & \Gamma^{p_2}_i \\
+ \Gamma^1_i & \Gamma^0_i & \ddots & \vdotswithin{\Gamma^0_i} \\
+ \vdotswithin{\Gamma^0_i} & \ddots & \ddots & \Gamma^1_i \\
+ \Gamma^{p_2}_i & \cdots & \Gamma^1_i & \Gamma^0_i
+ \end{array}
+ \right]
+ \qquad
+ \Gamma_i^j=
+ \left[
+ \begin{array}{llll}
+ K_{i,j,0} & K_{i,j,1} & \cdots & K_{i,j,p_3} \\
+ K_{i,j,1} & K_{i,j,0} & \ddots &x \vdotswithin{K_{i,j,0}} \\
+ \vdotswithin{K_{i,j,0}} & \ddots & \ddots & K_{i,j,1} \\
+ K_{i,j,p_3} & \cdots & K_{i,j,1} & K_{i,j,0}
+ \end{array}
+ \right].
+ \end{equation*}
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Determining coefficients}
+ \framesubtitle{MA process}
+ \small%
+ Solve non-linear system of equations for $\Theta$:
+ \begin{equation*}
+ K_{i,j,k} =
+ \left[
+ \displaystyle
+ \sum\limits_{l=i}^{q_1}
+ \sum\limits_{m=j}^{q_2}
+ \sum\limits_{n=k}^{q_3}
+ \Theta_{l,m,n}\Theta_{l-i,m-j,n-k}
+ \right]
+ \Var{\epsilon}
+ \end{equation*}
+ via fixed-point iteration method:
+ \begin{equation*}
+ \theta_{i,j,k} =
+ -\frac{K_{0,0,0}}{\Var{\epsilon}}
+ +
+ \sum\limits_{l=i}^{q_1}
+ \sum\limits_{m=j}^{q_2}
+ \sum\limits_{n=k}^{q_3}
+ \Theta_{l,m,n} \Theta_{l-i,m-j,n-k}.
+ \end{equation*}
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Determining coefficients}
+ \framesubtitle{ARMA process}
+ To mix processes one needs to divide ACF between processes, and
+ recompute one of the parts to match process properties (mean,
+ variance etc.).
+ \vfill%
+ \begin{center}
+ \alert{There is no recomputation formula for 3-D proccess.}
+ \end{center}
+ \end{frame}
+
+ \begin{frame}
+ \frametitle{Our approach}
+ Use AR process for standing waves and MA process for
+ propagating waves.
+ \vfill%
+ Supporting experimental results:
+ \begin{itemize}
+ \item It works that way in practice.
+ \item It does not work the other way round
+ (processes diverge).
+ \item Wavy surface integral characteristics
+ match the ones of real ocean waves.
+ \end{itemize}
+ \end{frame}
+
+ \section{Evaluation and verification}
+
+ \begin{frame}
+ \frametitle{Experiment setup}
+ \begin{itemize}
+ \item Generate standing/propagating waves with
+ AR/MA processes respectively.
+ \item Estimate distributions of integral
+ characteristics.
+ \item Compare estimated distributions to the
+ known ones via QQ plots.
+ \end{itemize}
+ \vfill%
+ \begin{center}
+ \small
+ \begin{tabular}{ll}
+ \toprule
+ Characteristic & Weibull shape ($k$) \\
+ \midrule
+ Wave height & 2 \\
+ Wave length & 2.3 \\
+ Crest length & 2.3 \\
+ Wave period & 3 \\
+ Wave slope & 2.5 \\
+ Three-dimensionality & 2.5 \\
+ \bottomrule
+ \end{tabular}%
+ \end{center}
+ \end{frame}
+
+% \begin{frame}
+% \frametitle{Input ACFs (time slices)}
+% Standing wave ACF:
+% \vfill%
+% \begin{tabular}{llll}%
+% \includegraphics[scale=0.45]{standing-acf-0} &
+% \includegraphics[scale=0.45]{standing-acf-1} &
+% \includegraphics[scale=0.45]{standing-acf-3} &
+% \includegraphics[scale=0.45]{standing-acf-4} \\
+% \end{tabular}
+% \vfill%
+% Propagating wave ACF:
+% \vfill%
+% \begin{tabular}{llll}%
+% \includegraphics[scale=0.45]{propagating-acf-00} &
+% \includegraphics[scale=0.45]{propagating-acf-01} &
+% \includegraphics[scale=0.45]{propagating-acf-03} &
+% \includegraphics[scale=0.45]{propagating-acf-04} \\
+% \end{tabular}
+% \end{frame}
+
+ \begin{frame}
+ \frametitle{Verification results (QQ plots)}
+ \small%
+ \centering
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \centering%
+ Standing waves
+ \begin{tabular}{ll}
+ \includegraphics[scale=0.5]{standing-elevation} &
+ \includegraphics[scale=0.5]{standing-wave-height-x} \\
+ \addlinespace
+ \includegraphics[scale=0.5]{standing-wave-length-x} &
+ \includegraphics[scale=0.5]{standing-wave-period} \\
+ \end{tabular}
+ \end{column}
+ \begin{column}{0.5\textwidth}
+ \centering%
+ Propagating waves
+ \begin{tabular}{ll}
+ \includegraphics[scale=0.5]{propagating-elevation} &
+ \includegraphics[scale=0.5]{propagating-wave-height-x} \\
+ \addlinespace
+ \includegraphics[scale=0.5]{propagating-wave-length-x} &
+ \includegraphics[scale=0.5]{propagating-wave-period} \\
+ \end{tabular}
+ \end{column}
+ \end{columns}
+ \end{frame}
+
+\end{document}