Flow of the River Nile
Measurements of the annual flow of the river Nile at Aswan (formerly
Assuan), 1871–1970, in 10^8 m^3,
“with apparent changepoint near 1898” (Cobb(1978), Table 1, p.249).
Usage
Nile
Format
A time series of length 100.
Source
Durbin, J. and Koopman, S. J. (2001) Time Series Analysis by
State Space Methods. Oxford University Press.
http://www.ssfpack.com/DKbook.html
References
Balke, N. S. (1993) Detecting level shifts in time series.
Journal of Business and Economic Statistics 11, 81–92.
Cobb, G. W. (1978) The problem of the Nile: conditional solution to a
change-point problem. Biometrika 65, 243–51.
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)