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\begin{document}

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\footnotesize
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\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}
