Create mathematical concepts chapter

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 \begin{center}
-     \Large{ats-zf} \\
+     \Large{Applied Time Series } \\
-    \small{\href{http://www.vvz.ethz.ch/}{VL-NUMMER}} \\
+    \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} \\
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-\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
 
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 \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*}