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% File src/library/datasets/man/Nile.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2018 R Core Team
% Distributed under GPL 2 or later

\name{Nile}
\docType{data}
\alias{Nile}
\title{Flow of the River Nile}
\usage{Nile}
\description{
  Measurements of the annual flow of the river Nile at Aswan (formerly
  \code{Assuan}), 1871--1970, in \eqn{10^8 m^3},
  \dQuote{with apparent changepoint near 1898} (Cobb(1978), Table 1, p.249).
}
\format{
  A time series of length 100.
}
\source{
  Durbin, J. and Koopman, S. J. (2001).
  \emph{Time Series Analysis by State Space Methods}.
  Oxford University Press.
  \url{http://www.ssfpack.com/DKbook.html}
}
\references{
  Balke, N. S. (1993).
  Detecting level shifts in time series.
  \emph{Journal of Business and Economic Statistics}, \bold{11}, 81--92.
  \doi{10.2307/1391308}.

  Cobb, G. W. (1978).
  The problem of the Nile: conditional solution to a change-point
  problem.
  \emph{Biometrika} \bold{65}, 243--51.
  \doi{10.2307/2335202}.
}
\examples{
require(stats); require(graphics)
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)              # contemporaneous 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)
}
\keyword{datasets}