258 lines
8.5 KiB
TeX
258 lines
8.5 KiB
TeX
\documentclass[8pt,landscape]{article}
|
||
\usepackage{multicol}
|
||
\usepackage{calc}
|
||
\usepackage{bookmark}
|
||
\usepackage{ifthen}
|
||
\usepackage[a4paper, landscape]{geometry}
|
||
\usepackage{hyperref}
|
||
\usepackage{ccicons}
|
||
\usepackage{amsmath, amsfonts, amssymb, amsthm}
|
||
\usepackage{listings}
|
||
\usepackage{graphicx}
|
||
\usepackage{fontawesome5}
|
||
\usepackage{xcolor}
|
||
\usepackage{float}
|
||
\usepackage[
|
||
type={CC},
|
||
modifier={by-sa},
|
||
version={3.0}
|
||
]{doclicense}
|
||
|
||
\graphicspath{{./img/}}
|
||
|
||
\definecolor{codegreen}{rgb}{0,0.6,0}
|
||
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
|
||
\definecolor{codepurple}{rgb}{0.58,0,0.82}
|
||
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
|
||
|
||
\lstdefinestyle{mystyle}{
|
||
backgroundcolor=\color{backcolour},
|
||
commentstyle=\color{codegreen},
|
||
keywordstyle=\color{magenta},
|
||
numberstyle=\tiny\color{codegray},
|
||
stringstyle=\color{codepurple},
|
||
basicstyle=\ttfamily\footnotesize,
|
||
breakatwhitespace=false,
|
||
breaklines=true,
|
||
captionpos=b,
|
||
keepspaces=true,
|
||
numbers=left,
|
||
numbersep=5pt,
|
||
showspaces=false,
|
||
showstringspaces=false,
|
||
showtabs=false,
|
||
tabsize=2
|
||
}
|
||
|
||
\lstset{style=mystyle}
|
||
|
||
% To make this come out properly in landscape mode, do one of the following
|
||
% 1.
|
||
% pdflatex latexsheet.tex
|
||
%
|
||
% 2.
|
||
% latex latexsheet.tex
|
||
% dvips -P pdf -t landscape latexsheet.dvi
|
||
% ps2pdf latexsheet.ps
|
||
|
||
|
||
% If you're reading this, be prepared for confusion. Making this was
|
||
% a learning experience for me, and it shows. Much of the placement
|
||
% was hacked in; if you make it better, let me know...
|
||
|
||
|
||
% 2008-04
|
||
% Changed page margin code to use the geometry package. Also added code for
|
||
% conditional page margins, depending on paper size. Thanks to Uwe Ziegenhagen
|
||
% for the suggestions.
|
||
|
||
% 2006-08
|
||
% Made changes based on suggestions from Gene Cooperman. <gene at ccs.neu.edu>
|
||
|
||
|
||
% To Do:
|
||
% \listoffigures \listoftables
|
||
% \setcounter{secnumdepth}{0}
|
||
|
||
|
||
% This sets page margins to .5 inch if using letter paper, and to 1cm
|
||
% if using A4 paper. (This probably isn't strictly necessary.)
|
||
% If using another size paper, use default 1cm margins.
|
||
\ifthenelse{\lengthtest { \paperwidth = 11in}}
|
||
{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
|
||
{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
|
||
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
|
||
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
|
||
}
|
||
|
||
% Turn off header and footer
|
||
\pagestyle{empty}
|
||
|
||
|
||
% Redefine section commands to use less space
|
||
\makeatletter
|
||
\renewcommand{\section}{\@startsection{section}{1}{0mm}%
|
||
{-1ex plus -.5ex minus -.2ex}%
|
||
{0.5ex plus .2ex}%x
|
||
{\normalfont\large\bfseries}}
|
||
\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
|
||
{-1explus -.5ex minus -.2ex}%
|
||
{0.5ex plus .2ex}%
|
||
{\normalfont\normalsize\bfseries}}
|
||
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
|
||
{-1ex plus -.5ex minus -.2ex}%
|
||
{1ex plus .2ex}%
|
||
{\normalfont\small\bfseries}}
|
||
|
||
|
||
\makeatother
|
||
|
||
% Define BibTeX command
|
||
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
|
||
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
|
||
|
||
% Don't print section numbers
|
||
% \setcounter{secnumdepth}{0}
|
||
|
||
|
||
\setlength{\parindent}{0pt}
|
||
\setlength{\parskip}{0pt plus 0.5ex}
|
||
|
||
% -----------------------------------------------------------------------
|
||
|
||
\begin{document}
|
||
|
||
\raggedright
|
||
\footnotesize
|
||
\begin{multicols*}{3}
|
||
|
||
|
||
% multicol parameters
|
||
% These lengths are set only within the two main columns
|
||
%\setlength{\columnseprule}{0.25pt}
|
||
\setlength{\premulticols}{1pt}
|
||
\setlength{\postmulticols}{1pt}
|
||
\setlength{\multicolsep}{1pt}
|
||
\setlength{\columnsep}{2pt}
|
||
|
||
\begin{center}
|
||
\Large{Applied Time Series } \\
|
||
\small{\href{http://vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?semkez=2021S&ansicht=LEHRVERANSTALTUNGEN&lerneinheitId=149645&lang=de}{401-6624-11L}} \\
|
||
\small{Jannis Portmann \the\year} \\
|
||
{\ccbysa}
|
||
\rule{\linewidth}{0.25pt}
|
||
\end{center}
|
||
|
||
\section{Mathematical Concepts}
|
||
|
||
For the \textbf{time series process}, we have to assume the following
|
||
|
||
\subsection{Stochastic Model}
|
||
From the lecture
|
||
\begin{quote}
|
||
A time series process is a set $\{X_t, t \in T\}$ of random variables, where $T$ is the set of times. Each of the random variables $X_t,t \in t$ has a univariate probability distribution $F_t$.
|
||
\end{quote}
|
||
\begin{itemize}
|
||
\item If we exclusively consider time series processes with
|
||
equidistant time intervals, we can enumerate $\{T = 1,2,3,...\}$
|
||
\item An observed time series is a realization of $X = \{X_1 ,..., X_n\}$,
|
||
and is denoted with small letters as $x = (x_1 ,... , x_n)$.
|
||
\item We have a multivariate distribution, but only 1 observation
|
||
(i.e. 1 realization from this distribution) is available. In order
|
||
to perform “statistics”, we require some additional structure.
|
||
\end{itemize}
|
||
|
||
\subsection{Stationarity}
|
||
\subsubsection{Strict}
|
||
For being able to do statistics with time series, we require that the
|
||
series “doesn’t change its probabilistic character” over time. This is
|
||
mathematically formulated by strict stationarity.
|
||
|
||
\begin{quote}
|
||
A time series $\{X_t, t \in T\}$ is strictly stationary, if the joint distribution of the random vector $(X_t ,... , X_{t+k})$ is equal to the one of $(X_s ,... , X_{s+k})$ for all combinations of $t,s$ and $k$
|
||
\end{quote}
|
||
|
||
\begin{tabular}{ll}
|
||
$X_t \sim F$ & all $X_t$ are identically distributed \\
|
||
$E[X_t] = \mu$ & all $X_t$ have identical expected value \\
|
||
$Var(X_t) = \sigma^2$ & all $X_t$ have identical variance \\
|
||
$Cov[X_t,X_{t+h}] = \gamma_h$ & autocovariance depends only on lag $h$ \\
|
||
\end{tabular}
|
||
|
||
\subsubsection{Weak}
|
||
It is impossible to „prove“ the theoretical concept of stationarity from data. We can only search for evidence in favor or against it. \\
|
||
\vspace{0.1cm}
|
||
However, with strict stationarity, even finding evidence only is too difficult. We thus resort to the concept of weak stationarity.
|
||
|
||
\begin{quote}
|
||
A time series $\{X_t , t \in T\}$ is said to be weakly stationary, if \\
|
||
$E[X_t] = \mu$ \\
|
||
$Cov(X_t,X_{t+h} = \gamma_h)$, for all lags $h$ \\
|
||
and thus $Var(X_t) = \sigma^2$
|
||
\end{quote}
|
||
|
||
\subsubsection{Testing stationarity}
|
||
\begin{itemize}
|
||
\item In time series analysis, we need to verify whether the series has arisen from a stationary process or not. Be careful: stationarity is a property of the process, and not of the data.
|
||
\item Treat stationarity as a hypothesis! We may be able to reject it when the data strongly speak against it. However, we can never prove stationarity with data. At best, it is plausible.
|
||
\item Formal tests for stationarity do exist. We discourage their use due to their low power for detecting general non-stationarity, as well as their complexity.
|
||
\end{itemize}
|
||
|
||
\textbf{Evidence for non-stationarity}
|
||
\begin{itemize}
|
||
\item Trend, i.e. non-constant expected value
|
||
\item Seasonality, i.e. deterministic, periodical oscillations
|
||
\item Non-constant variance, i.e. multiplicative error
|
||
\item Non-constant dependency structure
|
||
\end{itemize}
|
||
|
||
\textbf{Strategies for Detecting Non-Stationarity}
|
||
\begin{itemize}
|
||
\item Time series plot
|
||
\subitem - non-constant expected value (trend/seasonal effect)
|
||
\subitem - changes in the dependency structure
|
||
\subitem - non-constant variance
|
||
\item Correlogram (presented later...)
|
||
\subitem - non-constant expected value (trend/seasonal effect)
|
||
\subitem - changes in the dependency structure
|
||
\end{itemize}
|
||
A (sometimes) useful trick, especially when working with the correlogram, is to split up the series in two or more parts, and producing plots for each of the pieces separately.
|
||
|
||
\subsection{Examples}
|
||
\begin{figure}[H]
|
||
\centering
|
||
\includegraphics[width=.25\textwidth]{stationary.png}
|
||
\caption{Stationary Series}
|
||
\label{fig:stationary}
|
||
\end{figure}
|
||
|
||
\begin{figure}[H]
|
||
\centering
|
||
\includegraphics[width=.25\textwidth]{non-stationary.png}
|
||
\caption{Non-stationary Series}
|
||
\label{fig:non-stationary}
|
||
\end{figure}
|
||
|
||
\scriptsize
|
||
|
||
\section*{Copyleft}
|
||
|
||
\doclicenseImage \\
|
||
Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\
|
||
\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp/ats-zf} \\
|
||
\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/ats-zf} \\
|
||
Jannis Portmann, FS21
|
||
|
||
\section*{Referenzen}
|
||
\begin{enumerate}
|
||
\item Skript
|
||
\end{enumerate}
|
||
|
||
\section*{Bildquellen}
|
||
\begin{itemize}
|
||
\item Bild
|
||
\end{itemize}
|
||
|
||
\end{multicols*}
|
||
|
||
\end{document}
|