Lec12 – Time Series Analysis - ARIMA

Rule 2 if the autocorrelation at the seasonal period

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Unformatted text preview: B 2 s + ... + ΘQ B Qs CIE/IE 500 Transportation Analytics - Fall 2012 54 Example: A SARIMA Model What is the equation for model ARIMA(0,1,1) ×(0,1,1)12? Let’s write the equation for model ARIMA(0,1,1) (1 − B) xt = xt − xt −1 = (1 + θB) wt = wt + θwt −1 How about ARIMA(0,1,1) ×(0,1,1)12? xt = xt −1 + wt + θwt −1 2-min Quiz! (1 − B12 )(1 − B) xt = (1 + ΘB12 )(1 + θB) wt (1 − B − B12 + B13 ) xt = (1 + θB + ΘB12 + ΘθB13 ) wt xt = xt −1 + xt −12 − xt −13 + wt + θwt −1 + Θwt −12 + Θθwt −13 CIE/IE 500 Transportation Analytics - Fall 2012 55 Rules to Select Seasonal Terms Rule 1: If the series has a strong and consistent seasonal pattern, then you should use an order of seasonal differencing--but never use more than one order of seasonal differencing or more than 2 orders of total differencing (seasonal+nonseasonal). Rule 2: If the autocorrelation at the seasonal period is positive, consider adding an SAR term to the model. If the autocorrelation at the seasonal period is negative, consider adding an SMA term to the model. Do not mix SAR and SMA terms in the same model, and avoid using more than one of either kind. Source http://www.duke.edu/~rnau/arimrule.htm CIE/IE 500 Transportation Analytics - Fall 2012 56 Example: 30-min Volume Data m_data <- read.table("C:/Docs_Qing/Courses/Transportation Analytics/data/lec12_ARIMA/time_series_volspd.csv",sep=',',header = T) m_data$idx <- floor(m_data$Count/10) ts_vol_all <-aggregate(Volume_3min~idx, data= m_data, FUN=sum, na.rm=TRUE)[,2] plot.ts(ts_vol_all) 30-min data, 48 per day, so select s=48, Strong seasonal pattern, set D = 1 acf(ts_vol_all,5) ACF>0 when s=48, so add a SAR term, P=1 CIE/IE 500 Transportation Analytics - Fall 2012 57 Build Model ARIMA(0,1,1)X(1,1,0)48 > vol_011110<-arima(ts_vol_all, order=c(0,1,1),seasonal=list(order=c(1,1,0), period=48)) > vol_011110 Call: arima(x = ts_vol_all, order = c(0, 1, 1), seasonal = list(order = c(1, 1, 0), period = 48)) Coefficients: Compared to ARIMA, ma1 sar1 SARIMA performs 0.0236 -0.5506 much better s.e. 0.0439 0.0434 sigma^2 estimated as 25198: log likelihood = -3744.72, aic = 7495.44 > vol_011<-arima(ts_vol_all, order=c(0,1,1)) > vol_011 Call: arima(x = ts_vol_all, order = c(0, 1, 1)) Coefficients: ma1 0.1251 s.e. 0.0409 sigma^2 estimated as 22410: log likelihood = -4010.81, aic = 8025.62 CIE/IE 500 Transportation Analytics - Fall 2012 58 Our Textbook for Time Series Analysis R.H. Shumway & D.S. Stoffer, Time Series Analysis and Its Applications: With R Examples (Third Edition), Springer 2010 Chapter 1-3 are corresponding to our lecture 10-12. CIE/IE 500 Transportation Analytics - Fall 2012 59...
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This document was uploaded on 02/12/2014.

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