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diff --git a/Master/texmf-dist/doc/latex/beamertheme-saintpetersburg/example.tex b/Master/texmf-dist/doc/latex/beamertheme-saintpetersburg/example.tex new file mode 100644 index 00000000000..bf1a1846d94 --- /dev/null +++ b/Master/texmf-dist/doc/latex/beamertheme-saintpetersburg/example.tex @@ -0,0 +1,334 @@ +\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} |