Lec12 – Time Series Analysis - ARIMA

Acf armaacfarc15 75 ma0 24 ar2pacf armaacfarc15 75

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Unformatted text preview: ion of xh on {xh-1, xh-2,…,x1} i.e. and xh-10 denotes the regression of x0 on {x1,…, xh-2, xh-1} PACF is the autocorrelation between xt and xt+h with the linear dependence of xt+1 through to xt+h-1 removed; The PACF of an AR(p) process is zero at lag p + 1 and greater. CIE/IE 500 Transportation Analytics - Fall 2012 33 Examples of PACF- AR(2) > ar2 = arima.sim(list(order=c(2,0,0), ar=c(1.5,-.75)), n = 100) > ts.plot(ar2) xt = 1.5 xt −1 − 0.75 xt − 2 + wt > ar2.acf = ARMAacf(ar=c(1.5,-.75), ma=0, 24) > ar2.pacf = ARMAacf(ar=c(1.5,-.75), ma=0, 24, pacf=T) > par(mfrow=c(2,1)) > acf(ar2, lwd=2) > lines(0:24,ar2.acf, type="p",lwd=3, col="red") > pacf(ar2,lwd=2) > lines(ar2.pacf, type="p",lwd=3, col="red") ARMAacf(…) Compute the theoretical ACF or PACF CIE/IE 500 Transportation Analytics - Fall 2012 34 Examples of PACF- AR(2) 95% significance bounds Bars: Sample ACF and PACF Red dots: Theoretic ACF and PACF lag=1 Question: What happen to PACF when lag >2? CIE/IE 500 Transportation Analytics - Fall 2012 35 Examples of PACF-MA(2) > ma2 = arima.sim(list(order=c(0,0,2), ma=c(1.5,-.75)), n = 100) > ts.plot(ma2) xt = wt + 1.5wt −1 − 0.75wt − 2 > ma2.acf = ARMAacf(ma=c(1,1), ar=0, 24) > ma2.pacf = ARMAacf(ma=c(1,1), ar=0, 24, pacf=T) > par(mfrow=c(2,1)) > acf(ma2, lwd=2) > lines(0:24,ma2.acf, type="p",lwd=3, col="red") > pacf(ma2,lwd=2) > lines(ma2.pacf, type="p",lwd=3, col="red") CIE/IE 500 Transportation Analytics - Fall 2012 36 Examples of PACF-MA(2) Bars: Sample ACF and PACF Red dots: Theoretic ACF and PACF lag=1 Question: What happen to ACF when lag >2? CIE/IE 500 Transportation Analytics - Fall 2012 37 Summary Table: ACF and PACF for Causal and Invertible ARMA Models ACF PACF AR(p) Tails off Cuts off after lag p MA(q) Cuts off after lag q Tails off CIE/IE 500 Transportation Analytics - Fall 2012 ARMA(p,q) Tails off Tails off 38 Example: Model Selection, AR(p), MA(q) or ARMA(p,q) AR(p) MA(q) ARMA(p,q) ACF Tails off Cuts off after lag q Tails off PACF Cuts off after lag p Tails off Tails off 2-min Quiz: Propose and rank 3 models (best first) according to the following ACF and PACF correlogram. starting with lag =1 starting with lag =0 1.AR(2) 2.MA(3) 2.ARMA(1,2) how to generate ARMA(1,2) ??? Ans: see Rule 3 in the next slide 95% significance bounds CIE/IE 500 Transportation Analytics - Fall 2012 39 this means we have might mildly overdifferencing the data Some Rules for ARMA Model Selection Rule 1: If the PACF of the differenced series displays a sharp cutoff and/or the lag-1 autocorrelation is positive, then consider adding an AR term to the model. The lag at which the PACF cuts off is the indicated number of AR terms. Rule 2: If the ACF of the differenced series displays a sharp cutoff and/or the lag-1 autocorrelation is negative, then consider adding an MA term to the model. The lag at which the ACF cuts off is the indicated number of MA terms. In most cases, the be...
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This document was uploaded on 02/12/2014.

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