\HeaderA{AirPassengers}{Monthly Airline Passenger Numbers 1949-1960}{AirPassengers}
\keyword{datasets}{AirPassengers}
\begin{Description}\relax
The classic Box \& Jenkins airline data.  Monthly totals of
international airline passengers, 1949 to 1960.
\end{Description}
\begin{Usage}
\begin{verbatim}AirPassengers\end{verbatim}
\end{Usage}
\begin{Format}\relax
A monthly time series, in thousands.
\end{Format}
\begin{Source}\relax
Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1976)
\emph{Time Series Analysis, Forecasting and Control.}
Third Edition. Holden-Day. Series G.
\end{Source}
\begin{Examples}
\begin{ExampleCode}
## Not run: 
## These are quite slow and so not run by example(AirPassengers)

## The classic 'airline model', by full ML
(fit <- arima(log10(AirPassengers), c(0, 1, 1),
              seasonal = list(order=c(0, 1 ,1), period=12)))
update(fit, method = "CSS")
update(fit, x=window(log10(AirPassengers), start = 1954))
pred <- predict(fit, n.ahead = 24)
tl <- pred$pred - 1.96 * pred$se
tu <- pred$pred + 1.96 * pred$se
ts.plot(AirPassengers, 10^tl, 10^tu, log = "y", lty = c(1,2,2))

## full ML fit is the same if the series is reversed, CSS fit is not
ap0 <- rev(log10(AirPassengers))
attributes(ap0) <- attributes(AirPassengers)
arima(ap0, c(0, 1, 1), seasonal = list(order=c(0, 1 ,1), period=12))
arima(ap0, c(0, 1, 1), seasonal = list(order=c(0, 1 ,1), period=12),
      method = "CSS")

## Structural Time Series
ap <- log10(AirPassengers) - 2
(fit <- StructTS(ap, type= "BSM"))
par(mfrow=c(1,2))
plot(cbind(ap, fitted(fit)), plot.type = "single")
plot(cbind(ap, tsSmooth(fit)), plot.type = "single")
## End(Not run)\end{ExampleCode}
\end{Examples}