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  Cobb, G. W. (1978) The problem of the Nile: conditional solution to a
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  Cobb, G. W. (1978) The problem of the Nile: conditional solution to a
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  change-point problem.  \emph{Biometrika} \bold{65}, 243--51.
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  change-point problem.  \emph{Biometrika} \bold{65}, 243--51.
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}
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}
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\examples{
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\examples{
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require(stats); require(graphics)
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require(stats); require(graphics)
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par(mfrow = c(2,2))
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par(mfrow = c(2, 2))
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plot(Nile)
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plot(Nile)
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acf(Nile)
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acf(Nile)
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pacf(Nile)
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pacf(Nile)
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ar(Nile) # selects order 2
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ar(Nile) # selects order 2
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cpgram(ar(Nile)$resid)
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cpgram(ar(Nile)$resid)
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par(mfrow = c(1,1))
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par(mfrow = c(1, 1))
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arima(Nile, c(2, 0, 0))
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arima(Nile, c(2, 0, 0))
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## Now consider missing values, following Durbin & Koopman
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## Now consider missing values, following Durbin & Koopman
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NileNA <- Nile
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NileNA <- Nile
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NileNA[c(21:40, 61:80)] <- NA
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NileNA[c(21:40, 61:80)] <- NA
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arima(NileNA, c(2, 0, 0))
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arima(NileNA, c(2, 0, 0))
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plot(NileNA)
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plot(NileNA)
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pred <-
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pred <-
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   predict(arima(window(NileNA, 1871, 1890), c(2,0,0)), n.ahead = 20)
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   predict(arima(window(NileNA, 1871, 1890), c(2, 0, 0)), n.ahead = 20)
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lines(pred$pred, lty = 3, col = "red")
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lines(pred$pred, lty = 3, col = "red")
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lines(pred$pred + 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred + 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred - 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred - 2*pred$se, lty = 2, col = "blue")
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pred <-
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pred <-
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   predict(arima(window(NileNA, 1871, 1930), c(2,0,0)), n.ahead = 20)
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   predict(arima(window(NileNA, 1871, 1930), c(2, 0, 0)), n.ahead = 20)
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lines(pred$pred, lty = 3, col = "red")
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lines(pred$pred, lty = 3, col = "red")
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lines(pred$pred + 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred + 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred - 2*pred$se, lty = 2, col = "blue")
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lines(pred$pred - 2*pred$se, lty = 2, col = "blue")
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## Structural time series models
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## Structural time series models