# Nile

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Documentation

## 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)

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