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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 ARMA 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 (ARterms) in a time series
* d: Number of differences to achieve stationarity of a time series
* q: Number of moving average terms (MAterms) in a time series
CIE/IE 500 Transportation Analytics  Fall 2012 41 AIC: smaller, better. Can be
negative The BoxJenkins 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 (LjungBox 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 onestepahead prediction at time step t1 and
ˆ
Pt t −1 is the estimated onestepahead 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 QQ plot
2. Check randomness (if the residuals are uncorrelated): sample ACF of the residuals
3. Check collective autocorrelations: Use the LjungBoxPierce Qstatistic 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 3min Traffic Volume Data
m_data < read.table("C:/Docs_Qing/Courses/Transportation
Analytics/data/lec12_ARIMA/time_s...
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 Fall '09

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