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\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{First} +\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 @@ -153,15 +240,17 @@ \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}} \\ +\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*}