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zBW8H>B|O^p8Vb_(8YXZg5^~%GMjPFFc?^lJofH&uRBdd;Si}tdK`hs(EzGz>x%93eO%7g0l_S10NXlvm~Bci@hp*VH;qiS-@SmheVG!nty{g z4^U~JAUDdYtE=nN)TC`XSn5UWX=+kAM&gC%3>e`7gF(}>0i4a(*l!|o`~+>ZXY`DW zpO?4Zn+Osyy}tIT%TSyDl(Ba;AOF(D2>?>8D1$Vy-*DSZ!#Z4dfq4p5a}nz zx^q%zK!`@`&9ec?a40tJsQwQmfW9D-t8ssLG+_pY;?4LE0a^iG@4?Y6jAUj#DGoaD zzJ)%0dKa-xk{wR literal 0 HcmV?d00001 diff --git a/main.tex b/main.tex index ab5b13a..32f3e9c 100644 --- a/main.tex +++ b/main.tex @@ -544,6 +544,189 @@ pacf(wave, ylim=c(1,1)) \label{fig:pacf} \end{figure} +\section{Basics of modelling} +\subsection{White noise} +\begin{quote} + A time series $(W_1, W_2,..., W_n)$ is a \textbf{White Noise} series if the random variables $W_1 , W_2,...$ are i.i.d with mean zero. +\end{quote} +This implies that all $W_t$ have the same variance $\sigma_W^2$ and +$$Cov(W_i,W_j) = 0 \, \forall \, i \neq j$$ +Thus, there is no autocorrelation either: $\rho_k = 0 \, \forall \, k \neq 0$. \\ +\vspace{.2cm} +If in addition, the variables also follow a Gaussian distribution, i.e. $W_t \sim N(0, \sigma_W^2)$, the series is called \textbf{Gaussian White Noise}. The term White Noise is due to the analogy to white light (all wavelengths are equally distributed). + +\subsection{Autoregressive models (AR)} +In an $AR(p)$ process, the random variable $X_t$ depends on an autoregressive linear combination of the preceding $X_{t-1},..., X_{t-p}$, plus a „completely independent“ term called innovation $E_t$. +$$X_t = \alpha_1 X_{t-1} + ... + \alpha_p X_{t-p} + E_t$$ +Here, $p$ is called the order of the AR model. Hence, we abbreviate by $AR(p)$. An alternative notation is with the backshift operator $B$: +$$(1-\alpha_1 B - \alpha_2 B^2 - ... \alpha_p B^p) X_t = E_t \Leftrightarrow \Phi(B)X_t = E_t$$ +Here, $\Phi(B)$ is called the characteristic polynomial of the $AR(p)$. It determines most of the relevant properties of the process. + +\subsubsection{AR(1)-Model}\label{ar-1} +$$X_t = \alpha_1 X_{t-1} + E_t$$ +where $E_t$ is i.i.d. with $E[E_t] = 0$ and $Var(E_t) = \sigma_E^2$. We also require that $E_t$ is independent of $X_s, s p$. The behavior before lag $p$ can be arbitrary. +\end{itemize} +If what we observe is fundamentally different from the above, it is unlikely that the series was generated from an $AR(p)$-process. We thus need other models, maybe more sophisticated ones. + +\subsubsection{Parameter estimation} +Observed time series are rarely centered. Then, it is inappropriate to fit a pure $AR(p)$ process. All R routines by default assume the shifted process $Y_t = m + X_t$. Thus, we face the problem: +$$(Y_t - m) = \alpha_1(Y_{t-1} - m) + ... + \alpha_p(Y_{t-p} - m) + E_t$$ +The goal is to estimate the global mean m , the AR-coefficients $\alpha_1 ,..., \alpha_p$, and some parameters defining the distribution of the innovation $E_t$. We usually assume a Gaussian, hence this is $\sigma_E^2$.\\ +\vspace{.2cm} +We will discuss 4 methods for estimating the parameters:\\ +\vspace{.2cm} + +\textbf{OLS Estimation} \\ +If we rethink the previously stated problem, we recognize a multiple linear regression problem without +intercept on the centered observations. What we do is: +\begin{enumerate} + \item Estimate $\hat{m} = \bar{y}$ and $x_t = y_t - m$ + \item Run a regression without intercept on $x_t$ to obtain $\hat{\alpha_1},\dots,\hat{\alpha_p}$ + \item For $\hat{\sigma_E^2}$, take the residual standard error from the output +\end{enumerate} + +\vspace{.2cm} + +\textbf{Burg's algorithm} \\ +While OLS works, the first $p$ instances are never evaluated as responses. This is cured by Burg’s algorithm, which uses the property of time-reversal in stochastic processes. We thus evaluate the RSS of forward and backward prediction errors: +$$\sum_{t=p+1}^n \bigg[\bigg(X_t - \sum_{k=1}^p \alpha_k X_{t-k}\bigg)^2 + \bigg(X_{t-p} - \sum_{k=1}^p \alpha_k X_{t-p+k}\bigg)^2 \bigg]$$ +In contrast to OLS, there is no explicit solution and numerical optimization is required. This is done with a recursive method called the Durbin-Levison algorithm (implemented in R). + +\begin{lstlisting}[language=R] +f.burg <- ar.burg(llynx, aic=F, order.max=2) +\end{lstlisting} + +\vspace{.2cm} + +\textbf{Yule-Walker Equations} \\ +The Yule-Walker-Equations yield a LES that connects the true ACF with the true AR-model parameters. We plug-in the estimated ACF coefficients: +$$\hat{\rho}(k) = \hat{\alpha_k}\hat{\rho}(k-1) + \dots + \hat{\alpha_p}\hat{\rho}(k-p), \, \mathrm{for} \, k=1,\dots,p$$ +and solve the LES to obtain the AR-parameter estimates.\\ +\vspace{.2cm} +In R we can use \verb|ar.yw()| \\ + +\vspace{.2cm} + +\textbf{Maximum-likelihood-estimation} \\ +Idea: Determine the parameters such that, given the observed time series $(y_1 ,\dots, y_n)$, the resulting model is the most plausible (i.e. the most likely) one. \\ +This requires the choice of a probability model for the time series. By assuming Gaussian innovations, $E_t \sim N (0,\sigma_E^2)$ , any $AR(p)$ process has a multivariate normal distribution: +$$Y = (Y_1,\dots,Y_n) \sim N(m \cdot \vec{1},V)$$ +with $V$ depending on $\vec{\alpha},\sigma_E^2$ \\ +MLE then provides simultaneous estimates by optimizing: +$$L(\alpha,m,\sigma_E^2) \propto \exp \bigg( \sum_{t=1}^n(x_t - \hat{x_t}) \bigg)$$ + +\begin{lstlisting}[language=R] +> f.ar.mle +Call: arima(x = log(lynx), order = c(2, 0, 0)) +\end{lstlisting} + +\vspace{.2cm} + +\textbf{Some remarks} \\ +\begin{itemize} + \item All 4 estimation methods are asymptotically equivalent and even on finite samples, the differences are usually small. + \item All 4 estimation methods are non-robust against outliers and perform best on data that are approximately Gaussian. + \item Function \verb|arima()| provides standard errors for $\hat{m}; \hat{\alpha}_1 ,\dots, \hat{\alpha}_p$ so that statements about significance become feasible and confidence intervals for the parameters can be built. + \item \verb|ar.ols()|, \verb|ar.yw()| and \verb|ar.burg()| allow for convenient choice of the optimal model order $p$ using the AIC criterion. Among these methods, \verb|ar.burg()| is usually preferred. + +\end{itemize} + +\subsection{Model diagnostics} +\subsubsection{Residual analysis} +"residuals" = "estimated innovations" +$$\hat{E_t} = (y_t - \hat{m}) - (\hat{\alpha_1}(y_{t-1} - \hat{m}) - \dots - \hat{\alpha}_p(y_{t-1} - \hat{m}))$$ +With assumptions as in Chapter \ref{ar-1} \\ + +\vspace{.2cm} +We can check these, using (in R: \verb|tsdisplay(resid(fit))|) +\begin{itemize} + \item Time-series plot of $\hat{E}_t$ + \item ACF/PACF-plot of $\hat{E}_t$ + \item QQ-plot of $\hat{E}_t$ +\end{itemize} + +The time-series should look like white-noise \\ +\vspace{.2cm} +\textbf{Alternative} \\ +Using \verb|checkresiduals()|: \\ +A convenient alternative for residual analysis is this function from \verb|library(forecast)|. It only works correctly when fitting with \verb|arima()|, though. + +\begin{lstlisting}[language=R] +> f.arima <- arima(log(lynx), c(11,0,0)) +> checkresiduals(f.arima) +Ljung-Box test +data: Residuals from ARIMA(11,0,0) with non-zero mean +Q* = 4.7344, df = 3, p-value = 0.1923 +Model df: 12. Total lags used: 15 +\end{lstlisting} + +The function carries out a Ljung-Box test to check whether residuals are still correlated. It also provides a graphical output: +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{checkresiduals.png} + \caption{Example output from above code} + \label{fig:checkresiduals} +\end{figure} + +\subsubsection{Diagsnostic by simulation} +As a last check before a model is called appropriate, simulating from the estimated coefficients and visually inspecting the resulting series (without any prejudices) to the original one can be beneficial. +\begin{itemize} + \item The simulated series should "look like" the original. If this is not the case, the model failed to capture (some of) the properties in the original data. + \item A larger or more sophisticated model may be necessary in cases where simulation does not recapture the features in the original data. +\end{itemize} + +\subsection{Moving average models (MA)} +Whereas for $AR(p)$-models, the current observation of a series is written as a linear combination of its own past, $MA(q)$-models can be seen as an extension of the "pure" process +$$X_t = E_t$$ +in the sense that the last q innovation terms $E_{t-1} , E_{t-2} ,...$ are included, too. We call this a moving average model: +$$X_t = E_t + \beta_1 E_{t-1} + \beta_2 E_{t-2} + \dots + \beta_q E_{t-q}$$ +This is a time series process that is stationary, but not i.i.d. In many aspects, $MA(q)$ models are complementary to $AR(p)$. + \scriptsize