Lec12 – Time Series Analysis - ARIMA

Dukeedurnau411arim3htm cieie 500 transportation

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Unformatted text preview: st model turns out a model that uses either only AR terms or only MA terms, although in some cases a "mixed" model with both AR and MA terms may provide the best fit to the data. However, care must be exercised when fitting mixed models. It is possible for an AR term and an MA term to cancel each other's effects, even though both may appear significant in the model Rule 3: It is possible for an AR term and an MA term to cancel each other's effects, so if a mixed AR-MA model seems to fit the data, also try a model with one fewer AR term and one fewer MA term Source: http://www.duke.edu/~rnau/411arim3.htm CIE/IE 500 Transportation Analytics - Fall 2012 40 ARIMA Definition Definition (ARIMA) A process is said to be ARIMA(p,d,q) if ∇ d xt = (1 − B) d xt is ARMA(p; q). Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ ( B )(1 − B ) d xt = θ ( B ) wt Claim that most time series can be parsimoniously represented by the ARIMA class of models ARIMA(p,d,q)- Models attempt to describe the systematic pattern of a time series by 3 parameters * p: Number of autoregressive terms (AR-terms) in a time series * d: Number of differences to achieve stationarity of a time series * q: Number of moving average terms (MA-terms) in a time series CIE/IE 500 Transportation Analytics - Fall 2012 41 AIC: smaller, better. Can be negative The Box-Jenkins Methodology Model Identification Data Preparation • Transform time series for stationarity (i.e. ln()) • Difference time series for stationarity Model selection • Examine ACF & PACF • Identify potential Models (p,q)(sq,sp) Model Estimation and Testing Model Estimation • Estimate parameters in potential models • Select best model using suitable criterion (AIC) Model Diagnostics / Testing • Check ACF & PACF of residuals -> if white noise? • Run portmanteau test (Ljung-Box test) of residuals Model Application Mode Application • Use selected model to forecast CIE/IE 500 Transportation Analytics - Fall 2012 42 Model Diagnostics / Testing ˆ xt − xtt −1 or standardized residuals Investigate the residuals ˆ ˆ et = xt − xtt −1 / Pt t −1 ˆ where xtt −1 is one-step-ahead prediction at time step t-1 and ˆ Pt t −1 is the estimated one-step-ahead error variance. If the model fits well, the standardized residuals should behave like an iid sequence with mean zero and variance one. Diagnostic Checks 1. Check departures from normality : plot of Standardized residuals, or Q-Q plot 2. Check randomness (if the residuals are uncorrelated): sample ACF of the residuals 3. Check collective autocorrelations: Use the Ljung-Box-Pierce Q-statistic to measure collective autocorrelative (not just significance at a single lag). The null hypothesis says residuals from the ARIMA model have no autocorrelation. p greater, better CIE/IE 500 Transportation Analytics - Fall 2012 43 Example – Time Series of 3-min Traffic Volume Data m_data <- read.table("C:/Docs_Qing/Courses/Transportation Analytics/data/lec12_ARIMA/time_s...
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This document was uploaded on 02/12/2014.

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