Lecture 6: Forecasting of time series
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1 Lecture 6, page 1 Lecture 6: Forecasting of time series Outline of lesson 6 (chapter 5) Forecasting Very complicated and untidy in the book A lot of theory developed (in which we will not dwell). Only chapter Sketch of the theory Practical example C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 1 of 1
2 Lecture 6, page 2 Forecasting Given a time series {X 1, X 2,, X n } Want to forecast X n+k How do we do that? Many methods; Let s start with X n+1 We want to find X n+1 such that: (X n+1 a 0 a 1 X n a 2 X n-1 a n X 1 ) 2 is minimised. As {X 1, X 2,, X n } is known, the a i s can be found by derivation and setting the expression equal to zero => Least squares procedure. However, when in the Box-Jenkins context, the a i s already have a known parametric form (estimated coefficients). The procedure thus reduces to Estimate the time series Use the coefficients to estimate X n+1 For k>1, the new, estimated values will be part of the conditional expectation Generalised for X n+k, we have: n p q n+ k = ϕ i Pn X n+ k 1 + θ n+ k 1, j ( X n+ k j i= 1 j= k P X Xˆ ) on the n first observartions n+ k j, where P n is the estimate based - We use the sum of the AR-coefficients multiplied by the past observations p lags back in time. - We add the noise ( X Xˆ ) multiplied by MA-coefficients if the k is less than q. C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 2 of 2
3 Lecture 6, page 3 In Splus this is implemented in the arima.forecast routine. (The Kalman-filter is the practical iteration procedure) Venables and Ripley (1994) suggest that the first (long) part of the time series can be used to forecast the latter (short) part of the time series. -> By this approach you will quickly get an idea about how well your model performs, and not the least for how long you can trust your forecast. C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 3 of 3
4 Lecture 6, page 4 # Australian red wine tmp <- read.table("e:\\programfiler\\itsm96\\wine_b.dat") # tmp <- read.table("c:\\kyrre\\studier\\drgrad\\kurs\\series\\tsdata\\wine_b.dat") vin <- ts(tmp[,1], frequency=12, start=c(1980,1)) vin89 <- window(vin, end=c(1989,12)) # Using first part of series to predict latter part par(mfrow=c(2,1)) ts.plot(vin, main="australian red wine consume", xlab="year", ylab="litres") ts.points(vin89, pch=28, col=8) legend(locator(1), legend=c("wine "), marks=28, col=8) Australian red wine consume litres Wine Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 Jan 92 Year Special features? Season Trend Increasing variance C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 4 of 4
5 # Taking log to stabilise variance vin89 <- log(vin89) ts.plot(vin89, main="australian red wine consum", xlab="year", ylab="ln(litres)") ts.points(vin89, pch=28, col=8) Lecture 6, page 5 Australian red wine consum ln(litres) Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 Year C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 5 of 5
6 Lecture 6, page 6 # Deseasonlising the data tsp(vin89) # [1] vin.ln.stl <- stl(vin89, "periodic") par(mfrow=c(2,1)) ts.plot(vin.ln.stl$seas, main="seasonal components", ylab="", xlab="") ts.plot(vin.ln.stl$rem, main="remainder") Seasonal components Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 Remainder Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 6 of 6
7 Lecture 6, page 7 # Removing the trend y <- 1:length(vin.ln.stl$rem) vin.trend.lin <- ts(lm(vin.ln.stl$rem ~ y)$fitted.values, frequency=12, start=c(1980,1)) par(mfrow=c(2,2)) ts.plot(vin.ln.stl$rem, vin.trend.lin, lty=c(1,1), main="residual time series w/linear trend ( )", xlab="year", ylab="ln(litres)") vin.resid <- vin.ln.stl$rem - vin.trend.lin ts.plot(vin.resid, main="detrended Red wine time series", xlab="year", ylab="ln(litres)") acf(vin.resid) acf(vin.resid, type="p") Residual time series w/linear trend ( ) Detrended Red wine time series ln(litres) ln(litres) Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 Year Series : vin.resid Jan 80 Jan 82 Jan 84 Jan 86 Jan 88 Jan 90 Year Series : vin.resid ACF Partial ACF Lag Lag Have arrived at a time series where the autoregression damps down reasonably fast. C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 7 of 7
8 Lecture 6, page 8 # AR-method library(mass) par(mfrow=c(2,2)) cpgram(vin.resid) ar(vin.resid)$aic [1] [5] [9] [13] [17] [21] length(ar(vin.resid)$aic) [1] 21 plot(0:20, ar(vin.resid)$aic, xlab="order", ylab="aic", main="aic for AR(p)") Series: vin.resid frequency AIC AIC for AR(p) order C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 8 of 8
9 Lecture 6, page 9 vin89.ar2 <- arima.mle(vin.resid, model=list(order=c(2,0,0))) # vin89.ar2$model$ar # [1] vin89.fore <- arima.forecast(vin.resid, n=24, model=vin89.ar2$model) par(mfrow=c(2,1)) ts.plot(vin89.fore$mean, main="forecast ") Forecast # Adding seasonal component vin89.fore$mean <- vin89.fore$mean + vin.ln.stl$sea[1:24] par(mfrow=c(2,1)) ts.plot(vin89.fore$mean, main="forecast w/seasonal pattern ") Forecast w/seasonal pattern C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 9 of 9
10 Lecture 6, page 10 # Adding linear trend lm(vin.ln.stl$rem ~ y) Call: lm(formula = vin.ln.stl$rem ~ y) Coefficients: (Intercept) y Degrees of freedom: 120 total; 118 residual Residual standard error: y2 <- seq( * , * , length=24) vin89.fore$mean <- vin89.fore$mean + y2 par(mfrow=c(2,1)) ts.plot(window(log(vin), start=c(1990,1)), vin89.fore$mean, vin89.fore$mean *vin89.fore$std.err, vin89.fore$mean *vin89.fore$std.err, col=c(1,3,1,1), lty=c(1,1,6,6)) ts.points(window(log(vin), start=c(1990,1)), pch=4) title("ar-method") ts.plot(exp(window(log(vin), start=c(1990,1))), exp(vin89.fore$mean), exp(vin89.fore$mean *vin89.fore$std.err), exp(vin89.fore$mean *vin89.fore$std.err), col=c(1,3,1,1), lty=c(1,1,6,6)) ts.points(exp(window(log(vin), start=c(1990,1))), pch=4) title("on a normal scale") AR-method On a normal scale C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 10 of 10
11 Lecture 6, page 11 # ARMA modelling vin89.arma <- arima.mle(vin.resid, model=list(order=c(2,0,1)), n.cond=6) vin89.arma$model $order: [1] $ar: [1] $ndiff: [1] 0 $ma: [1] > length(vin.resid) [1] 120 > vin89.arma <- arima.mle(vin.resid, model=list(order=c(2,0,0)), n.cond=6) > aicc(vin89.arma$loglik, 2, 0, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(2,0,1)), n.cond=6) > aicc(vin89.arma$loglik, 2, 1, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(2,0,2)), n.cond=6) > aicc(vin89.arma$loglik, 2, 2, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(1,0,1)), n.cond=6) > aicc(vin89.arma$loglik, 1, 1, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(1,0,2)), n.cond=6) > aicc(vin89.arma$loglik, 1, 2, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(0,0,1)), n.cond=6) > aicc(vin89.arma$loglik, 0, 1, 120) [1] > vin89.arma <- arima.mle(vin.resid, model=list(order=c(0,0,2)), n.cond=6) > aicc(vin89.arma$loglik, 0, 2, 120) [1] vin89.arma <- arima.mle(vin.resid, model=list(order=c(1,0,1)), n.cond=6) vin89.fore <- arima.forecast(vin.resid, n=24, model=vin89.arma$model) # Adding seasonal component vin89.fore$mean <- vin89.fore$mean + vin.ln.stl$sea[1:24] # Adding linear trend vin89.fore$mean <- vin89.fore$mean + y2 ts.plot(window(log(vin), start=c(1990,1)), vin89.fore$mean, vin89.fore$mean *vin89.fore$std.err, vin89.fore$mean *vin89.fore$std.err, col=c(1,3,1,1), lty=c(1,1,6,6)) ts.points(window(log(vin), start=c(1990,1)), pch=4) title("arma-method") ts.plot(exp(window(log(vin), start=c(1990,1))), exp(vin89.fore$mean), exp(vin89.fore$mean *vin89.fore$std.err), exp(vin89.fore$mean *vin89.fore$std.err), col=c(1,3,1,1), lty=c(1,1,6,6)) ts.points(exp(window(log(vin), start=c(1990,1))), pch=4) title("on a normal scale") C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 11 of 11
12 Lecture 6, page 12 ARMA-method On a normal scale C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 12 of 12
13 Lecture 6, page 13 Forecasting: Future values are found based on previous observations (by way of the estimated parameters) (In practice, a least squares approach by way of the Kalman Filter is used in the estimation.) Use the first part of the time seires to estimate the parameters of the model. Use the obtained coefficients to estimate X n+1 For k>1 a recursive procedure is used. (Remember to preprosess the data) The performance of the actual forecast (model) decays quickly to the mean. "Back-tranformation" must be done to display the forecast. C:\Kyrre\studier\drgrad\Kurs\series\lecture doc, KL, , page 13 of 13
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