Lec12 – Time Series Analysis - ARIMA

And initial condition cieie 500 transportation

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Unformatted text preview: , for j = 1, . . . , q, and ψj = 0, For the general case of ARMA(p, q) models, the task of solving for the ψ-weights is much more complicated, To solve for the ψweights in general, we must match the coefficients in ψ(z)φ(z) = θ(z): CIE/IE 500 Transportation Analytics - Fall 2012 24 ψ-weights for a Causal ARMA(p,q) cont’d The first few values are where we would take φj = 0 for j > p, and θj = 0 for j > q. and initial condition CIE/IE 500 Transportation Analytics - Fall 2012 25 ACF of MA(q) Recall ACF of MA(1) Brushing over the computations, the ACF of MA(q) with representation xt = µ + wt + θ1wt −1 + θ 2 wt − 2 + ... + θ q wt − q is given as No statistical significance in the correlogram(ACF plot) after lag q indicates the process may be an MA(q) process CIE/IE 500 Transportation Analytics - Fall 2012 26 Autocovariance Function of an ARMA(p,q) The computation of the autocovariance function of the ARMA model φ(B)xt = θ(B)wt begins as CIE/IE 500 Transportation Analytics - Fall 2012 27 Autocovariance Function of a Causal ARMA(p,q) For a causal ARMA(p,q) model, we have So that Therefore, CIE/IE 500 Transportation Analytics - Fall 2012 28 Autocovariance Recurrence Note that θj = 0 for j > q, so we have the following recurrence relationship With initial condition Dividing through by γ(0) gives a similar recursion on ρ(h)= γ(h)/ γ(0) CIE/IE 500 Transportation Analytics - Fall 2012 29 Example: ACF of Causal ARMA(1,1) Start with the causal ARMA(1,1) given by xt = φxt-1 + θwt-1 + wt. From the previous slide, the following recursion holds So the general solution is We use the initial condition from the previous slide to determine the constant c. To continue, we need to know the values of ψ0 and ψ1. Recall ψ0 =1, ψ1= φ1+θ1 = φ+θ, This gives two equations and two unknowns (γ(0) and γ(1) ) CIE/IE 500 Transportation Analytics - Fall 2012 30 Example: ACF of Causal ARMA(1,1) cont’d We can continue to solve for these unknowns to produce To solve for c, note that γ(1) = cφ, in which case c = γ(1)/φ. Hence, the specific solution is Finally, dividing through by γ(0) yields the ACF CIE/IE 500 Transportation Analytics - Fall 2012 31 Motivation of Partial Autocorrelation Function (PACF) As we learned, ACF can detect a MA(q) process since NO statistical significance in the correlogram(ACF plot) after lag q indicates the process may be an MA(q) process Thus, the ACF provides a considerable amount of information about the order of the dependence when the process is a moving average (MA) process. Question: How to detect an AR(p) process? Recall the ACF of an AR(1) process: so a geometric decay of the ACF would indicate AR(1). But this is hard to observe, Other method to clearly indicate AR(1) or AR(p)? The PACF! CIE/IE 500 Transportation Analytics - Fall 2012 32 PACF Definition The partial autocorrelation function (PACF) of a stationary process, xt, denoted φhh, for h = 1, 2, . . . , is and where xh-1h denotes the regress...
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This document was uploaded on 02/12/2014.

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