Finish ARMA

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jannisp 2021-08-23 16:22:03 +02:00
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\usepackage{ifthen} \usepackage{ifthen}
\usepackage[a4paper, landscape]{geometry} \usepackage[a4paper, landscape]{geometry}
\usepackage{hyperref} \usepackage{hyperref}
\usepackage{ccicons}
\usepackage{amsmath, amsfonts, amssymb, amsthm} \usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{listings} \usepackage{listings}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{fontawesome5} \usepackage{fontawesome5}
\usepackage{xcolor} \usepackage{xcolor}
\usepackage{float} \usepackage{float}
\usepackage[
type={CC},
modifier={by-sa},
version={3.0}
]{doclicense}
\graphicspath{{./img/}} \graphicspath{{./img/}}
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\Large{Applied Time Series } \\ \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{\href{http://vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?semkez=2021S&ansicht=LEHRVERANSTALTUNGEN&lerneinheitId=149645&lang=de}{401-6624-11L}} \\
\small{Jannis Portmann \the\year} \\ \small{Jannis Portmann \the\year} \\
{\ccbysa}
\rule{\linewidth}{0.25pt} \rule{\linewidth}{0.25pt}
\end{center} \end{center}
@ -784,7 +777,7 @@ As with AR(p) models, there are three main steps:
\end{itemize} \end{itemize}
\end{enumerate} \end{enumerate}
\subsubsection{Parameter estimation} \subsubsection{Parameter estimation}\label{ma-parameter-estimation}
The simplest idea is to exploit the relation between model parameters and autocorrelation coefficients («Yule-Walker») after the global mean $m$ has been estimated and subtracted. \\ The simplest idea is to exploit the relation between model parameters and autocorrelation coefficients («Yule-Walker») after the global mean $m$ has been estimated and subtracted. \\
In contrast to the Yule-Walker method for AR(p) models, this yields an inefficient estimator that generally generates poor results and hence should not be used in practice. In contrast to the Yule-Walker method for AR(p) models, this yields an inefficient estimator that generally generates poor results and hence should not be used in practice.
@ -862,8 +855,48 @@ May be more difficult in reality than in theory:
\item In many cases, an AIC grid search over all $ARMA(p,q)$ with $p+q < 5$ may help to identify promising models. \item In many cases, an AIC grid search over all $ARMA(p,q)$ with $p+q < 5$ may help to identify promising models.
\end{itemize} \end{itemize}
\subsubsection{Parameter estimation}
See \ref{ma-parameter-estimation}, with
$$E_0 = E_{-1} = E_{-2} = \dots = 0$$
and
$$X_t = \alpha_1 X_{t-1} + \dots + \alpha_p X_{t-p} + E_t + \beta_1 E_{t-1} + \dots + \beta_q X_{t-q}$$
respectively.
\subsubsection{R example}
\begin{lstlisting}[language=R]
> fit0 <- arima(nao, order=c(1,0,1));
Coefficients:
ar1 ma1 intercept
0.3273 -0.1285 -0.0012
s.e. 0.1495 0.1565 0.0446
sigma^2=0.9974; log-likelihood=-1192.28, aic=2392.55
\end{lstlisting}
\subsubsection{Residual analysis}
See \ref{residual-analysis} again
\subsubsection{AIC-based model choice}
In R, finding the AIC-minimizing $ARMA(p,q)$-model is convenient with the use of \verb|auto.arima()| from \verb|library(forecast)|. \\
\vspace{.2cm}
\textbf{Beware}: Handle this function with care! It will always identify a «best fitting» $ARMA(p,q)$, but there is no guarantee that this model provides an adequate fit! \\
\vspace{.2cm}
Using \verb|auto.arima()| should always be complemented by visual inspection of the time series for assessing stationarity, verifying the ACF/PACF plots for a second thought on suitable models. Finally, model diagnostics with the usual residual plots will decide whether the model is useful in practice.
\section{General concepts}
\subsection{AIC}
The \textit{Akaike-information-criterion} is useful for determining the order of an $ARMA(p,q)$ model. The formula is as follows:
$$AIC = -2 \log (L) + 2(p+q+k+1)$$
where
\begin{itemize}
\item $\log(L)$: Goodness-of-fit criterion: Log-likelihood function
\item $p+q+k+1$: Penalty for model complexity: $p, q$ are the $AR$- resp. $MA$-orders; $k = 1$ if a global mean is in use, else $0$ . The final $+1$ is for the innovation variance
\end{itemize}
For small samples $n$, often a corrected version is used:
$$AICc = AIC + \frac{2(p + q + k + 1)(p + q + k + 2)}{n - p - q - k - 2}$$
\scriptsize \scriptsize
\newpage
\section*{Copyright} \section*{Copyright}
Nearly everything is copy paste from the slides or the script. Copyright belongs to M. Dettling \\ Nearly everything is copy paste from the slides or the script. Copyright belongs to M. Dettling \\
\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp/ats-zf} \\ \faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp/ats-zf} \\