--- title: "Lecture 8" subtitle: "ARMA Models" date: "2/13/2018" fontsize: 11pt output: beamer_presentation: theme: metropolis highlight: pygments fig_caption: false latex_engine: xelatex includes: in_header: ../settings.tex --- {r setup, include=FALSE} library(dplyr) library(ggplot2) library(forecast)  {r config, include=FALSE} knitr::opts_chunkset( collapse = TRUE, fig.width=7, fig.height=4.5, out.width="\\textwidth", fig.align="center", echo=TRUE, warning=FALSE ) ggplot2::theme_set(ggplot2::theme_bw()) source("../util.R")  # AR models ## AR(p) models {.t} We can generalize from an AR(1) to an AR(p) model by simply adding additional autoregressive terms to the model. \begin{aligned} AR(p): \quad y_t &= \delta + \phi_1 \, y_{t-1} + \phi_2 \, y_{t-2} + \cdots + \phi_p \, y_{t-p} + w_t \\ &= \delta + w_t + \sum_{i=1}^p \phi_i \, y_{t-i} \end{aligned} . . . What are the properities ofAR(p)$, 1. Expected value? 2. Autocovariance / autocorrelation? 3. Stationarity conditions? ## Lag operator The lag operator is convenience notation for writing out AR (and other) time series models. We define the lag operator$Las follows, $$L \, y_t = y_{t-1}$$ . . . this can be generalized where, \begin{aligned} L^2 y_t &= L\,(L\, y_{t})\\ &= L \, y_{t-1} \\ &= y_{t-2} \\ \end{aligned} therefore, $$L^k \, y_t = y_{t-k}$$ ## Lag polynomial {.t} Lets rewrite theAR(p)model using the lag operator, \begin{aligned} y_t &= \delta + \phi_1 \, y_{t-1} + \phi_2 \, y_{t-2} + \cdots + \phi_p \, y_{t-p} + w_t \\ y_t &= \delta + \phi_1 \, L \, y_t + \phi_2 \, L^2 \, y_t + \cdots + \phi_p \, L^p \, y_t + w_t \end{aligned} . . . \begin{aligned} y_t - \phi_1 \, L \, y_t - \phi_2 \, L^2 \, y_t - \cdots - \phi_p \, L^p \, y_t &= \delta + w_t \\ (1 - \phi_1 \, L - \phi_2 \, L^2 - \cdots - \phi_p \, L^p) \, y_t &= \delta + w_t \end{aligned} . . . $$~$$ This polynomial of the lags $$\phi_p(L) = (1 - \phi_1 \, L - \phi_2 \, L^2 - \cdots - \phi_p \, L^p)$$ is called the characteristic polynomial of the AR process. ## Stationarity ofAR(p)$processes {.t} **Claim**: An$AR(p)$process is stationary if the roots of the characteristic polynomial lay *outside* the complex unit circle . . . If we define$\lambda = 1/L$then we can rewrite the characteristic polynomial as $$(\lambda^p - \phi_1 \lambda^{p-1} - \phi_2 \lambda^{p-2} - \cdots - \phi_{p-1} \lambda - \phi_p)$$ then as a corollary of our claim the$AR(p)$process is stationary if the roots of this new polynomial are *inside* the complex unit circle (i.e.$|\lambda| < 1$). ## Example AR(1) ## Example AR(2) ## AR(2) Stationarity Conditions \begin{center} \includegraphics[width=0.85\textwidth]{figs/ar2_conditions.png} \end{center} \tiny{From \url{Shumway & Stofer 4th ed.}} ## Proof Sketch We can rewrite the$AR(p)$model into an$AR(1)form using matrix notation \begin{aligned} y_t &= \delta + \phi_1 \, y_{t-1} + \phi_2 \, y_{t-2} + \cdots + \phi_p \, y_{t-p} + w_t \\ \symbf\xi_t &= \symbf{\delta} + \symbf{F} \, \symbf\xi_{t-1} + \symbf w_t \end{aligned} where \begin{aligned} \begin{bmatrix} y_t \\ y_{t-1} \\ y_{t-2} \\ \vdots \\ y_{t-p+1} \end{bmatrix} &= \begin{bmatrix} \delta \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} + \begin{bmatrix} \phi_1 & \phi_2 & \phi_3 & \cdots & \phi_{p-1} & \phi_p \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ \end{bmatrix} \begin{bmatrix} y_{t-1} \\ y_{t-2} \\ y_{t-3} \\ \vdots \\ y_{t-p} \end{bmatrix} + \begin{bmatrix} w_t \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \\ &= \begin{bmatrix} \delta + w_t + \sum_{i=1}^p \phi_i \, y_{t-i} \\ y_{t-1} \\ y_{t-2} \\ \vdots \\ y_{t-p+1} \end{bmatrix} \end{aligned} ## Proof sketch (cont.) {.t} So just like the originalAR(1)we can expand out the autoregressive equation \begin{aligned} \symbf\xi_t &= \symbf{\delta} + \symbf w_t + \symbf{F} \, \symbf\xi_{t-1} \\ &= \symbf{\delta} + \symbf w_t + \symbf{F} \, (\symbf\delta+\symbf w_{t-1}) + \symbf{F}^2 \, (\symbf \delta+\symbf w_{t-2}) + \cdots \\ &\qquad\qquad~\,\,\, + \symbf{F}^{t-1} \, (\symbf \delta+\symbf w_{1}) + \symbf{F}^t \, (\symbf \delta+\symbf w_0) \\ &= (\sum_{i=0}^t F^i)\symbf{\delta} + \sum_{i=0}^t F^i \, w_{t-i} \end{aligned} and therefore we need\underset{t\to\infty}{\lim} F^t \to 0$. ## Proof sketch (cont.) {.t} We can find the eigen decomposition such that$\symbf F = \symbf Q \symbf \Lambda \symbf Q^{-1}$where the columns of$\symbf Q$are the eigenvectors of$\symbf F$and$\symbf \Lambdais a diagonal matrix of the corresponding eigenvalues. A useful property of the eigen decomposition is that $$\symbf{F}^i = \symbf Q \symbf \Lambda^i \symbf Q^{-1}$$ . . . Using this property we can rewrite our equation from the previous slide as \begin{aligned} \symbf\xi_t &= (\sum_{i=0}^t F^i)\symbf{\delta} + \sum_{i=0}^t F^i \, w_{t-i} \\ &= (\sum_{i=0}^t \symbf Q \symbf \Lambda^i \symbf Q^{-1})\symbf{\delta} + \sum_{i=0}^t \symbf Q \symbf \Lambda^i \symbf Q^{-1} \, w_{t-i} \end{aligned} ## Proof sketch (cont.) {.t} $$\symbf \Lambda^i = \begin{bmatrix} \lambda_1^i & 0 & \cdots & 0 \\ 0 & \lambda_2^i & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_p^i \\ \end{bmatrix}$$ Therefore, $$\underset{t\to\infty}{\lim} F^t \to 0$$ when $$\underset{t\to\infty}{\lim} \Lambda^t \to 0$$ which requires that $$|\lambda_i| < 1 \;\; \text{ for all} \; i$$ ## Proof sketch (cont.) Eigenvalues are defined such that for\symbf \lambda$, $$\det (\symbf{F}-\symbf\lambda\,\symbf{I}) = 0$$ based on our definition of$\symbf F$our eigenvalues will therefore be the roots of $$\lambda^p -\phi_1\,\lambda^{p-1}-\phi_2\,\lambda^{p-2} - \cdots - \phi_{p_1} \, \lambda^1 - \phi_p = 0$$ . . . which if we multiply by$1/\lambda^p$where$L = 1/\lambda$gives $$1 -\phi_1\,L-\phi_2\,L^2 - \cdots - \phi_{p_1} \, L^{p-1} - \phi_p \, L^p = 0$$ ## Properties of$AR(2)${.t} For a *stationary*$AR(2)$process where$w_t$has$E(w_t) = 0$and$Var(w_t) = \sigma_w^2$## Properties of$AR(p)${.t} For a *stationary*$AR(p)$process where$w_t$has$E(w_t) = 0$and$Var(w_t) = \sigma_w^2$$E(Y_t) = \frac{\delta}{1-\phi_1 -\phi_2-\cdots-\phi_p}$$ \begin{aligned} \gamma(0) = \phi_1\gamma_1 + \phi_2\gamma_2 + \cdots + \phi_p\gamma_p + \sigma_w^2 \\ \gamma(h) = \phi_1\gamma_{j-1} + \phi_2\gamma_{j-2} + \cdots + \phi_p\gamma_{j-p} \end{aligned} $$\rho(h) = \phi_1\rho_{j-1} + \phi_2\rho_{j-2} + \cdots + \phi_p\rho_{j-p}$$ # Moving Average (MA) Processes ## MA(1) {.t} A moving average process is similar to an AR process, except that the autoregression is on the error term. $$MA(1): \qquad y_t = \delta + w_t + \theta \, w_{t-1}$$ Properties: ## Time series {r echo=FALSE, dev="png", dpi=150} ma = data_frame( t = 1:200, "θ=0.1" = arima.sim(n=200, list(ma=c(0.1))) %>% strip_attrs(), "θ=0.8" = arima.sim(n=200, list(ma=c(0.8))) %>% strip_attrs(), "θ=2.0" = arima.sim(n=200, list(ma=c(2.0))) %>% strip_attrs(), "θ=-0.1" = arima.sim(n=200, list(ma=c(-0.1))) %>% strip_attrs(), "θ=-0.8" = arima.sim(n=200, list(ma=c(-0.8))) %>% strip_attrs(), "θ=-2.0" = arima.sim(n=200, list(ma=c(-2.0))) %>% strip_attrs() ) ma %>% tidyr::gather(model, y, -t) %>% mutate(model = forcats::as_factor(model)) %>% ggplot(aes(x=t,y=y)) + geom_line() + facet_wrap(~model)  ## ACF {r echo=FALSE} par(mfrow=c(2,3)) acf(maθ=0.1, lag.max = 10, main=expression(paste(theta,"=0.1"))) acf(ma$θ=0.8, lag.max = 10, main=expression(paste(theta,"=0.8"))) acf(ma$θ=2.0, lag.max = 10, main=expression(paste(theta,"=2.0"))) pacf(ma$θ=0.1, lag.max = 10, main=expression(paste(theta,"=0.1"))) pacf(ma$θ=0.8, lag.max = 10, main=expression(paste(theta,"=0.8"))) pacf(maθ=2.0, lag.max = 10, main=expression(paste(theta,"=2.0")))  ## MA(q) {.t} $$MA(q): \qquad y_t = \delta + w_t + \theta_1 \, w_{t-1} + \theta_2 \, w_{t-2} + \cdots + \theta_q \, w_{t-q}$$ Properties: $$E(y_t) = \delta$$ \begin{aligned} \gamma(0) &= (1 + \theta_1^2+\theta_2^2 + \cdots + \theta_q^2) \, \sigma_w^2 \\ \gamma(h) &= \begin{cases} -\theta_k + \theta_1\theta_{k+1} + \theta_2\theta_{k+2} + \cdots + \theta_{q+k}\theta_{q} & \text{if } |k| \in \{1,\ldots,q\} \\ 0 & \text{otherwise} \end{cases} \end{aligned} ## Example series {r echo=FALSE} ma_q = data_frame( t = 1:100, "θ={-1.5}" = arima.sim(n=100, list(ma=c(-1.5))) %>% strip_attrs(), "θ={-1.5, -1}" = arima.sim(n=100, list(ma=c(-1.5, -1))) %>% strip_attrs(), "θ={-1.5, -1, 2}" = arima.sim(n=100, list(ma=c(-1.5, -1, 2))) %>% strip_attrs(), "θ={-1.5, -1, 2, 3}" = arima.sim(n=100, list(ma=c(-1.5, -1, 2, 3))) %>% strip_attrs() ) ma_q %>% tidyr::gather(model, y, -t) %>% mutate(model = forcats::as_factor(model)) %>% ggplot(aes(x=t,y=y)) + geom_line() + facet_wrap(~model)  ## ACF {r echo=FALSE} par(mfrow=c(2,2)) acf(ma_q[[2]], lag.max = 10, main=expression(paste(theta,"={-1.5}" ))) acf(ma_q[[3]], lag.max = 10, main=expression(paste(theta,"={-1.5, -1}" ))) acf(ma_q[[4]], lag.max = 10, main=expression(paste(theta,"={-1.5, -1, 2}" ))) acf(ma_q[[5]], lag.max = 10, main=expression(paste(theta,"={-1.5, -1, 2, 3}")))  # ARMA Model ## ARMA Model {.t} An ARMA model is a composite of AR and MA processes,ARMA(p,q)$: \footnoteoutput $$y_t = \delta + \phi_1 \, y_{t-1} + \cdots \phi_p \, y_{t-p} + w_{t} + \theta_1 w_{t-1} + \cdots + \theta_q w_{t_q}$$ $$\phi_p(L) y_t = \delta + \theta_q(L)w_t$$ Since all$MA$processes are stationary, we only need to examine the$AR$aspect to determine stationarity (roots of$\phi_p(L)$lie outside the complex unit circle). ## Time series {r echo=FALSE, fig.height=5} arma = data_frame( t = 1:100, "φ={0.9}, θ={-}" = arima.sim(n=100, list(ar=c( 0.9), ma=c() )) %>% strip_attrs(), "φ={-0.9}, θ={-}" = arima.sim(n=100, list(ar=c(-0.9), ma=c() )) %>% strip_attrs(), "φ={-}, θ={0.9}" = arima.sim(n=100, list(ar=c(), ma=c(0.9))) %>% strip_attrs(), "φ={-}, θ={-0.9}" = arima.sim(n=100, list(ar=c(), ma=c(-0.9))) %>% strip_attrs(), "φ={0.9}, θ={0.9}" = arima.sim(n=100, list(ar=c( 0.9), ma=c(0.9))) %>% strip_attrs(), "φ={-0.9}, θ={0.9}" = arima.sim(n=100, list(ar=c(-0.9), ma=c(0.9))) %>% strip_attrs(), "φ={0.9}, θ={-0.9}" = arima.sim(n=100, list(ar=c( 0.9), ma=c(-0.9))) %>% strip_attrs(), "φ={-0.9}, θ={-0.9}" = arima.sim(n=100, list(ar=c(-0.9), ma=c(0.9))) %>% strip_attrs() ) arma %>% tidyr::gather(model, y, -t) %>% mutate(model = forcats::as_factor(model)) %>% ggplot(aes(x=t,y=y)) + geom_line() + facet_wrap(~model, ncol=4)  ##$\phi=0.9$,$\theta=0${r echo=FALSE} forecast::tsdisplay(arma[[2]], main=expression(paste(phi,"={0.9}, ",theta,"={0}")))  ##$\phi=0$,$\theta=0.9${r echo=FALSE} forecast::tsdisplay(arma[[4]], main=expression(paste(phi,"={0}, ",theta,"={0.9}")))  ##$\phi=0.9$,$\theta=0.9\$ {r echo=FALSE} forecast::tsdisplay(arma[[6]], main=expression(paste(phi,"={0.9}, ",theta,"={0.9}")))