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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 From e0c5fad908579158a9204db8e76f5e9bc1c772ef Mon Sep 17 00:00:00 2001 From: jannisp Date: Sun, 22 Aug 2021 16:16:28 +0200 Subject: [PATCH 3/5] MA-models --- main.tex | 89 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 87 insertions(+), 2 deletions(-) diff --git a/main.tex b/main.tex index 32f3e9c..a268864 100644 --- a/main.tex +++ b/main.tex @@ -180,7 +180,7 @@ mathematically formulated by strict stationarity. \end{tabular} \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. \\ + 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. @@ -677,7 +677,7 @@ Call: arima(x = log(lynx), order = c(2, 0, 0)) \end{itemize} \subsection{Model diagnostics} -\subsubsection{Residual analysis} +\subsubsection{Residual analysis}\label{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} \\ @@ -727,6 +727,91 @@ in the sense that the last q innovation terms $E_{t-1} , E_{t-2} ,...$ are inclu $$X_t = E_t + \beta_1 E_{t-1} + \beta_2 E_{t-2} + \dots + \beta_q E_{t-q}$$ This is a time series process that is stationary, but not i.i.d. In many aspects, $MA(q)$ models are complementary to $AR(p)$. +\subsubsection{Stationarity of MA models} +We first restrict ourselves to the simple $MA(1)$-model: +$$X_t = E_t + \beta_1 E_{t-1}$$ +The series $X_t$ is always weakly stationary, no matter what the choice of the parameter $\beta_1$ is. + +\subsubsection{ACF/PACF of MA processes} +For the ACF +$$\rho(1) = \frac{\gamma(1)}{\gamma(0)} = \frac{\beta_1}{1+\beta_1^2} < 0.5$$ +and +$$\rho(k) = 0 \, \forall \, k > 1$$ + +Thus, we have a «cut-off» situation, i.e. a similar behavior to the one of the PACF in an $AR(1)$ process. This is why and how $AR(1)$ and $MA(1)$ are complementary. + +\subsubsection{Invertibility} +Without additional assumptions, the ACF of an $MA(1)$ does not allow identification of the generating model. +$$X_t = E_t + 0.5 E_{t-1}$$ +$$U_t = E_t + 2 E_{t-1}$$ +have identical ACF! +$$\rho(1) = \frac{\beta_{1}}{1+\beta_1^2} = \frac{1/\beta_1}{1+(1/\beta_1^2)}$$ + +\begin{itemize} + \item An $MA(1)$-, or in general an $MA(q)$-process is said to be invertible if the roots of the characteristic polynomial $\Theta(B)$ exceed one in absolute value. + \item Under this condition, there exists only one $MA(q)$-process for any given ACF. But please note that any $MA(q)$ is stationary, no matter if it is invertible or not. + \item The condition on the characteristic polynomial translates to restrictions on the coefficients. For any MA(1)-model, $|\beta_1| < 1$ is required. + \item R function \verb|polyroot()| can be used for finding the roots. +\end{itemize} + +\textbf{Practical importance:} \\ +The condition of invertibility is not only a technical issue, but has important practical meaning. All invertible $MA(q)$ processes can be expressed in terms of an $AR(\infty)$, e.g. for an $MA(1)$: +\begin{align*} +X_t &= E_t + \beta_1 E_{t-1} \\ + &= E_t + \beta_1(X_{t-1} - \beta_1 E_{t-2}) \\ + &= \dots \\ + &= E_t + \beta_1 X_{t-1} - \beta_1^2 X_{t-2} + \beta_1^3X_{t-3} + \dots \\ + &= E_t + \sum_{i=1}^\infty \psi_i X_{t-i} +\end{align*} + +\subsection{Fitting MA(q)-models to data} +As with AR(p) models, there are three main steps: +\begin{enumerate} + \item Model identification + \begin{itemize} + \item Is the series stationary? + \item Do the properties of ACF/PACF match? + \item Derive order $q$ from the cut-off in the ACF + \end{itemize} + \item Parameter estimation + \begin{itemize} + \item How to determine estimates for $m, \beta_1 ,\dots, \beta_q, \sigma_E^2$? + \item Conditional Sum of Squares or MLE + \end{itemize} + \item Model diagnostics + \begin{itemize} + \item With the same tools/techniques as for AR(p) models + \end{itemize} +\end{enumerate} + +\subsubsection{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. + +\vspace{.2cm} +It is better to use \textbf{Conditional sum of squares}:\\ +This is based on the fundamental idea of expressing $\sum E_t^2$ in terms of $X_1 ,..., X_n$ and $\beta_1 ,\dots, \beta_q$, as the innovations themselves are unobservable. This is possible for any invertible $MA(q)$, e.g. the $MA(1)$: +$$E_t = X_t = \beta_1 X_{t-1} + \beta_1^2 X_{t-2} + \dots + (-\beta)^{t-1} X_1 + \beta_1^t E_0$$ +Conditional on the assumption of $E_0 = 0$ , it is possible to rewrite $\sum E_t^2$ for any $MA(1)$ using $X_1 ,\dots, X_n $ and $\beta_1$. \\ +\vspace{.2cm} +Numerical optimization is required for finding the optimal parameter $\beta_1$, but is available in R function \verb|arima()| with: +\begin{lstlisting}[language=R] +> arima(..., order=c(...), method="CSS") +\end{lstlisting} + +\textbf{Maximium-likelihood estimation} +\begin{lstlisting}[language=R] +> arima(..., order=c(...), method="CSS-ML") +\end{lstlisting} +This is the default methods in R, which is based on finding starting values for MLE using the CSS approach. If assuming Gaussian innovations, then: +$$X_t = E_t + \beta_1 E_{t-1} + \beta_q E_{t-q}$$ +will follow a Gaussian distribution as well, and we have: +$$X = (X_1, \dots, X_n) \sim N(0,V)$$ +Hence it is possible to derive the likelihood function and simultaneously estimate the parameters $m;\beta_1,\dots,\beta_q;\sigma_E^2$. + +\subsubsection{Residual analysis} +See \ref{residual-analysis} + \scriptsize From f2900a1fb739b95642da7530bf2fda172e3e75b6 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 23 Aug 2021 09:34:46 +0200 Subject: [PATCH 4/5] Changes for new deployment --- Jenkinsfile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Jenkinsfile b/Jenkinsfile index ac8f2ec..43acc44 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -12,6 +12,6 @@ node { } stage('Publish PDF') { - sh 'scp -i /root/.ssh/id_rsa ats-zf.pdf jannis@192.168.178.45:/var/www/html/download/latex-previews/ats-zf.pdf' + sh 'scp -i /root/.ssh/id_rsa ats-zf.pdf thisfro@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/ats-zf.pdf' } } \ No newline at end of file From 1a87ddc913dc84fa6a2529d94aacf54f0167631d Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 23 Aug 2021 09:35:12 +0200 Subject: [PATCH 5/5] Start ARMA --- main.tex | 49 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 49 insertions(+) diff --git a/main.tex b/main.tex index a268864..e1b3ba4 100644 --- a/main.tex +++ b/main.tex @@ -812,6 +812,55 @@ Hence it is possible to derive the likelihood function and simultaneously estima \subsubsection{Residual analysis} See \ref{residual-analysis} +\subsection{ARMA(p,q)-models} +An $ARMA(p,q)$ model combines $AR(p)$ and $MA(q)$: +$$X_t = \alpha_1 X_{t-1} + \dots + \alpha_p X_{t-p} + E_t + \beta_1 E_{t-1} + \dots + \beta_q E{t-q}$$ +where $E_t$ are i.i.d. innovations (=a white noise process).\\ +\vspace{.2cm} +It‘s easier to write $ARMA(p,q)$’s with the characteristic polynomials: \\ +\vspace{.2cm} +$\Phi(B)X_t = \Theta(B)E_t$, where \\ +$\Phi(z) = 1 - \alpha_1 z - \dots - \alpha_p z^p$, is the cP of the $AR$-part, and \\ +$\Theta(z) = 1 + \beta_1 z + \dots + \beta_1 z^q$ is the cP of the $MA$-part + +\subsubsection{Properties of ARMA(p,q)-Models} +The stationarity is determined by the $AR(p)$-part of the model:\\ +If the roots of the characteristic polynomial $\Phi(B)$ exceed one in absolute value, the process is stationary.\\ +\vspace{.2cm} +The invertibility is determined by the $MA(q)$-part of the model:\\ +If the roots of the characteristic polynomial $\Theta(B)$ exceed one in absolute value, the process is invertible.\\ +\vspace{.2cm} +Any stationary and invertible $ARMA(p,q)$ can either be rewritten in the form of a non-parsimonious $AR(\infty)$ or an $MA(\infty)$.\\ +In practice, we mostly consider shifted $ARMA(p,q)$: $Y_t = m + X_t$ + +\begin{table}[H] + \centering + \begin{tabular}{l|l|l} + & ACF & PACF \\ + \hline + $AR(p)$ & exponential decay & cut-off at lag $p$ \\ + $MA(q)$ & cut-off at lag $q$ & exponential decay \\ + $ARMA(p,q)$ & mix decay/cut-off & mix decay/cut-off \\ + \end{tabular} + \caption{Comparison of $AR$-,$MA$-, $ARMA$-models} +\end{table} + +\begin{itemize} + \item In an $ARMA(p,q)$, depending on the coefficients of the model, either the $AR(p)$ or the $MA(q)$ part can dominate the ACF/PACF characteristics. + \item In an $ARMA(p,q)$, depending on the coefficients of the model, either the $AR(p)$ or the $MA(q)$ part can dominate the ACF/PACF characteristics. + +\end{itemize} + +\subsubsection{Fitting ARMA-models to data} +See $AR$- and $MA$-modelling + +\subsubsection{Identification of order (p,q)} +May be more difficult in reality than in theory: +\begin{itemize} + \item We only have one single realization of the time series with finite length. The ACF/PACF plots are not «facts», but are estimates with uncertainty. The superimposed cut-offs may be difficult to identify from the ACF/PACF plots. + \item $ARMA(p,q)$ models are parsimonius, but can usually be replaced by high-order pure $AR(p)$ or $MA(q)$ models. This is not a good idea in practice, however! + \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} \scriptsize