diff --git a/main.tex b/main.tex index a4d8594..d80367a 100644 --- a/main.tex +++ b/main.tex @@ -179,8 +179,8 @@ mathematically formulated by strict stationarity. $Cov[X_t,X_{t+h}] = \gamma_h$ & 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. \\ +\subsubsection{Weak} \label{weak-stationarity} + 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. @@ -393,7 +393,7 @@ fit <- gam(log(maine) ~ s(tnum) + mm) \end{lstlisting} \section{Autocorrelation} -For most of the rest of this course, we will deal with (weakly) stationary time series. See \ref{} +For most of the rest of this course, we will deal with (weakly) stationary time series. See \ref{weak-stationarity} \\ \vspace{.2cm} Definition of autocorrelation at lag $k$ $$Cor(X_{t+k},X_t) = \frac{Cov(X_{k+t},X_t)}{\sqrt{Var(X_{k+t})\cdot Var(X_t)}} = \rho(k)$$ @@ -410,25 +410,134 @@ We assume $\rho(k) = 0.7$ \item From this we can also conclude that any $\rho(k) < 0.4$ is not a strong association, i.e. has a small effect on the next observation only. \end{itemize} +\subsection{Lagged scatterplot approach} +Create a plot of $(x_t, x_{t+k}) \, \forall \, t = 1,...,n-k$ and compute the canonical Pearson correlation coefficient of these pairs and use it as an estimation for the autocorrelation $\tilde{\rho}(k)$ + +\begin{lstlisting}[language=R] +lag.plot(wave, do.lines=FALSE, pch=20) +\end{lstlisting} + +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{lagged-scatterplot.png} + \caption{Lagged scatterplot example for $k=1$} + \label{fig:lagged-scatterplot} +\end{figure} + +\subsection{Plug-in estimation} +Plug-in estimation relies on the canonical covariance estimator: +$$\hat{\rho}(k) = \frac{Cov(X_t,X_{t+k})}{Var(X_t)}$$ +Plug-in estimates are biased, i.e. shrunken towards zero for large lags $k$. Nevertheless, they are generally more reliable and precise. + +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{lagged-scatterplot-vs-plug-in.png} + \caption{Lagged scatterplot estimation vs. plug-in estimation} + \label{fig:lagged-scatterplot-vs-plug-in} +\end{figure} + +\subsection{Important points on ACF estimation} +\begin{itemize} + \item Correlations measure linear association and usually fail if there are non-linear associations between the variables. + \item The bigger the lag $k$ for which $\rho(k)$ is estimated, the fewer data pairs remain. Hence the higher the lag, the bigger the variability in $\hat{\rho}(k)$ . + \item To avoid spurious autocorrelation, the plug-in approach shrinks $\hat{\rho}(k)$ for large $k$ towards zero. This creates a bias, but pays off in terms of mean squared error. + \item Autocorrelations are only computed and inspected for lags up to $10 \log_{10}(n)$, where they have less bias/variance +\end{itemize} + +\subsection{Correlogram} +\begin{lstlisting}[language=R] +acf(wave, ylim=c(-1,1)) +\end{lstlisting} + +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{correlogram.png} + \caption{Example correlogram} + \label{fig:correlogram} +\end{figure} + + +\subsubsection{Confidence Bands} +Even for an i.i.d. series $X_t$ without autocorrelation, i.e. $\rho(k) = 0 \, \forall \, k$, the estimates will be different from zero: $\hat{\rho}(k) \neq 0$ \\ +\textbf{Question}: Which $\hat{\rho}(k)$ are significantly different from zero? + +$$\hat{\rho}(k) \sim N(0,1/n), \; \mathrm{for \, large} \, n$$ +\begin{itemize} + \item Under the null hypothesis of an i.i.d. series, a 95\% acceptance region for the null is given by the interval $\pm 1.96 / \sqrt{n}$ + \item For any stationary series, $\hat{\rho}(k)$ within the confidence bands are considered to be different from 0 only by chance, while those outside are considered to be truly different from zero. +\end{itemize} +\textbf{Type I Errors} \\ +For iid series, we need to expect 5\% of type I errors, i.e. $\hat{\rho}(k)$ that go beyond the confidence bands by chance. \\ +\textbf{Non i.i.d. series} \\ +The confidence bands are asymptotic for i.i.d. series. Real finite length non-i.i.d. series have different (unknown) properties. + +\subsection{Ljung-box test} +The Ljung-Box approach tests the null hypothesis that a number of autocorrelation coefficients are simultaneously equal to zero. \\ +Thus, it tests for significant autocorrelation in a series. The test statistic is: + +$$Q(h) = n(n+2)\sum_{k=1}^h \frac{\hat{\rho}^2}{n-k} \sim \chi_h^2$$ + +\begin{lstlisting}[language=R] +Box.test(wave, lag=10, type="Ljung-Box") +\end{lstlisting} + +\subsection{ACF and outliers} +The estimates $\hat{\rho}(k)$ are sensitive to outliers. They can be diagnosed using the lagged scatterplot, where every single outlier appears twice. \\ +\vspace{.2cm} +\textbf{Some basic strategies for dealing with outliers} +\begin{itemize} + \item if it is bad data point: delete the observation + \item most (if not all) R functions can deal with missing data + \item if complete data are required, replace missing values with + \begin{itemize} + \item global mean of the series + \item local mean of the series, e.g. $\pm 3$ observations + \item fit a time series model and predict the missing value + \end{itemize} +\end{itemize} + +\subsection{Properties of estimated ACF} +\begin{itemize} + \item Appearance of the series $\Rightarrow$ Appearance of the ACF \\ Appearance of the series $\nLeftarrow$ Appearance of the ACF + \item The compensation issue: \\ $\sum_{k=1}^{n-1}\hat{\rho}(k) = -1/2$ \\ All estimable autocorrelation coefficients sum up to -1/2 + \item For large lags $k$ , there are only few data pairs for estimating $\rho(k)$. This leads to higher variability and hence the plug-in estimates are shrunken towards zero. +\end{itemize} + +\subsection{Application: Variance of the arithmetic mean} +We need to estimate the mean of a realized/observed time series. We would like to attach a standard error +\begin{itemize} + \item If we estimate the mean of a time series without taking into account the dependency, the standard error will be flawed. + \item This leads to misinterpretation of tests and confidence intervals and therefore needs to be corrected. + \item The standard error of the mean can both be over-, but also underestimated. This depends on the ACF of the series. +\end{itemize} + +\subsubsection{Confidence interval} +For a 95\% CI: +$$\hat{\mu} \pm 1.96 \sqrt{\frac{\gamma(0)}{n^2} \bigg(n + 2 \cdot \sum_{k=1}^{10log_{10}(n)}(n-k)\rho(k) \bigg)}$$ + +In R we can use +\begin{lstlisting}[language=R] +n <- length(b) +var.ts <- 1/n^2*acf(b,lag=0,type="cov")$acf[1]*(n+2*sum(((n-1):(n-10))*acf(b,10)$acf[-1])) +mean(b) + c(-1.96,1.96)*sqrt(var.ts) +\end{lstlisting} + \scriptsize -\section*{Copyleft} - -\doclicenseImage \\ -Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\ +\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} \\ \faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/ats-zf} \\ Jannis Portmann, FS21 -\section*{Referenzen} +\section*{References} \begin{enumerate} - \item Skript + \item ATSA\_Script\_v219219.docx, M. Dettling + \item ATSA\_Slides\_v219219.pptx, M. Dettling \end{enumerate} -\section*{Bildquellen} -\begin{itemize} - \item Bild -\end{itemize} +\section*{Image sources} +All pictures are taken from the slides or the script mentioned above. \end{multicols*}