\HeaderA{Nile}{Flow of the River Nile}{Nile}
\keyword{datasets}{Nile}
\begin{Description}\relax
Measurements of the annual flow of the river Nile at Ashwan 1871--1970.
\end{Description}
\begin{Usage}
\begin{verbatim}Nile\end{verbatim}
\end{Usage}
\begin{Format}\relax
A time series of length 100.
\end{Format}
\begin{Source}\relax
Durbin, J. and Koopman, S. J. (2001) \emph{Time Series Analysis by
State Space Methods.}  Oxford University Press.
\url{http://www.ssfpack.com/dkbook/}
\end{Source}
\begin{References}\relax
Balke, N. S. (1993) Detecting level shifts in time series.
\emph{Journal of Business and Economic Statistics} \bold{11}, 81--92.

Cobb, G. W. (1978) The problem of the Nile: conditional solution to a
change-point problem.  \emph{Biometrika} \bold{65}, 243--51.
\end{References}
\begin{Examples}
\begin{ExampleCode}
require(stats)
par(mfrow = c(2,2))
plot(Nile)
acf(Nile)
pacf(Nile)
ar(Nile) # selects order 2
cpgram(ar(Nile)$resid)
par(mfrow = c(1,1))
arima(Nile, c(2, 0, 0))

## Now consider missing values, following Durbin & Koopman
NileNA <- Nile
NileNA[c(21:40, 61:80)] <- NA
arima(NileNA, c(2, 0, 0))
plot(NileNA)
pred <- predict(arima(window(NileNA, 1871, 1890), c(2,0,0)), n.ahead = 20)
lines(pred$pred, lty = 3, col = "red")
lines(pred$pred + 2*pred$se, lty=2, col="blue")
lines(pred$pred - 2*pred$se, lty=2, col="blue")
pred <- predict(arima(window(NileNA, 1871, 1930), c(2,0,0)), n.ahead = 20)
lines(pred$pred, lty = 3, col = "red")
lines(pred$pred + 2*pred$se, lty=2, col="blue")
lines(pred$pred - 2*pred$se, lty=2, col="blue")

## Structural time series models
par(mfrow = c(3, 1))
plot(Nile)
## local level model
(fit <- StructTS(Nile, type = "level"))
lines(fitted(fit), lty = 2)              # contempareneous smoothing
lines(tsSmooth(fit), lty = 2, col = 4)   # fixed-interval smoothing
plot(residuals(fit)); abline(h = 0, lty = 3)
## local trend model
(fit2 <- StructTS(Nile, type = "trend")) ## constant trend fitted
pred <- predict(fit, n.ahead = 30)
## with 50% confidence interval
ts.plot(Nile, pred$pred, pred$pred + 0.67*pred$se, pred$pred -0.67*pred$se)

## Now consider missing values
plot(NileNA)
(fit3 <- StructTS(NileNA, type = "level"))
lines(fitted(fit3), lty = 2)
lines(tsSmooth(fit3), lty = 3)
plot(residuals(fit3)); abline(h = 0, lty = 3)
\end{ExampleCode}
\end{Examples}