STA 6857 Forecasting ( 3.5 cont.)

Transcription

1 STA 6857 Forecasting ( 3.5 cont.)

2 Outline 1 Forecasting 2 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 2/ 20

3 Outline 1 Forecasting 2 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 3/ 20

4 Best Linear Predictor We will focus on linear predictors of the form n xn+m n = α 0 + α k x k k=1 ( ) Theorem (Best Linear Prediction for Stationary Processes) Given data x 1, x 2,..., x n, the best linear predictor of x n+m for m 1 is found by solving E [ (x n+m xn+m)x n ] k = 0, k = 0, 1,..., n where x 0 = 1. Equations ( ) are called the prediction equations, and they provide a useful tool in computing the best linear predictor x n n+m. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 4/ 20

5 Prediction Equation Picture E [ (x n+m x n n+m)x k ] = 0, k = 0, 1,..., n Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 5/ 20

6 Using the Prediction Equations Assume the time series is mean zero and write the best linear predictor as n xn+m n = φ (m) n1 x n + φ (m) n2 x n φ (m) nn x 1 = φ (m) nj x n+1 j From the prediction equations, we have for k = 1,..., n, n E x n+m x n+1 k j=1 φ (m) nj x n+1 j n = E (x n+m x n+1 k ) E = γ(k + m 1) = 0 n j=1 j=1 φ (m) nj φ (m) nj γ(k j) j=1 x n+1 j x n+1 k Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 6/ 20

7 Matrix Formulation of the Prediction Equations From the equations n φ (m) nj γ(k j) = γ(k + m 1), k = 1,..., n j=1 we have the following compact representation Γ n φ (m) n = γ (m) n where γ(0) γ(1) γ(n 1) Γ n = {γ(k j)} n j,k=1 = γ(1) γ(0) γ(n 2) , γ(n 1) γ(n 2) γ(0) ( ) φ (m) n = φ (m) n1, φ(m) n2,..., φ(m) nn and γ (m) n = (γ(m), γ(m + 1),..., γ(m + n 1)). Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 7/ 20

8 A Solution by Matrix Inversion If Γ n is invertible, then from we can solve φ (m) n to be Γ n φ (m) n φ (m) n = γ (m) n = Γ 1 n γ n (m) When n is large and Γ n is difficult to invert, the Durbin-Levinson recursive algorithm can be used. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 8/ 20

9 Durbin-Levinson Algorithm Algorithm (Durbin-Levinson Algorithm) The algorithm is initiated with φ 00 = 0 P 0 1 = γ(0). For n 1, φ nn = ρ(n) n 1 k=1 φ n 1,kρ(n k) 1 n 1 k=1 φ n 1,kρ(k) P n n+1 = Pn 1 n (1 φ 2 nn) ( ) where, for n 2, φ nk = φ n 1,k φ nn φ n 1,n k, k = 1, 2,..., n 1. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 9/ 20

10 Durbin-Levinson Algorithm Algorithm (Durbin-Levinson Algorithm) The algorithm is initiated with φ 00 = 0 P 0 1 = γ(0). For n 1, φ nn = ρ(n) n 1 k=1 φ n 1,kρ(n k) 1 n 1 k=1 φ n 1,kρ(k) P n n+1 = Pn 1 n (1 φ 2 nn) ( ) where, for n 2, φ nk = φ n 1,k φ nn φ n 1,n k, k = 1, 2,..., n 1. Bonus: The PACF of a stationary process x t is given by φ nn in ( ). Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 9/ 20

11 Using the Complete Past The prediction x n+m = E(x n+m x n, x n 1,..., x 1 ) can be approximated by the prediction derived from conditioning on the infinite past given as x n+m = E(x n+m x n, x n 1,...) Using this approximation and replacing the unknown values of x 0, x 1,... with 0, we construct the forecast for a general ARMA process as where x n+m = φ 1 x n+m 1 + φ p x n+m p + θ 1 w n+m θ q w n+m q { 0, t 0 x t = x, 1 t n { 0, t 0 or t > n w t = φ(b) x t θ 1 w t 1 θ q w t q, 1 t n. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 10/ 20

12 Prediction Error The mean square m-step-ahead prediction error is P n n+m = E [ (x n+m x n n+m) 2] = γ(0) γ (m) n The prediction error using the complete past is given as P n n+m = E [ (x n+m x n+m ) 2] m 1 = σ 2 j=0 Γ 1 n γ n (m) ψ 2 j Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 11/ 20

13 Prediction Intervals (1 α) prediction intervals are of the form x n n+m ± z α/2 P n n+m So for an approximate 95% prediction interval, one can use z Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 12/ 20

14 Back to the Recruitment Series > setwd("c:/users/berg.ufad/documents/sta 6857/R") > rec = ts(scan("mydata/recruit.dat"), start=1950, frequency=12) Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 13/ 20

15 Recruitment Series (cont.) > par(mfrow=c(2,1)) > acf(rec, 48) > pacf(rec, 48) Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 14/ 20

16 Recruitment Series (cont.) > (fit = ar.ols(rec,order=2,demean=f,intercept=t)) Call: ar.ols(x = rec, order.max = 2, demean = F, intercept = T) Coefficients: Intercept: (1.111) Order selected 2 sigma^2 estimated as > fit$asy.se $x.mean [1] $ar [1] Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 15/ 20

17 Recruitment Series (cont.) > rec.pr = predict(rec.yw, n.ahead=24) > U = rec.pr$pred + rec.pr$se > L = rec.pr$pred - rec.pr$se > minx = min(rec,l) > maxx = max(rec,u) > ts.plot(rec, rec.pr$pred, xlim=c(1980,1990), ylim=c(minx,maxx)) > lines(rec.pr$pred, col="red", type="o") > lines(u, col="blue", lty="dashed") > lines(l, col="blue", lty="dashed") Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 16/ 20

18 A Look Ahead Class discussion on setting the time of the Midterm Exam. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 17/ 20

19 Outline 1 Forecasting 2 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 18/ 20

20 Do Your Homework! You should be up to 3.6 on the reading. Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 19/ 20

21 Textbook Problems Do the following exercise from the textbook 3.15 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 20/ 20