Finish ARMA

main.tex

@@ -5,18 +5,12 @@
 \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/}} 
 
@@ -139,7 +133,6 @@
      \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}
 
@@ -784,7 +777,7 @@ As with AR(p) models, there are three main steps:
     \end{itemize}
 \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. \\
 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.
 \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
 
+\newpage
+
 \section*{Copyright}
 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} \\