Airline Passenger Analysis
A smooth rising trend multiplied by an annual cycle whose amplitude grows with the trend.
Key takeaways
- – Passenger numbers grow steadily across the whole period, at roughly 2.7% per year.
- – A twelve-month cycle is present throughout and is the dominant short-term structure.
- – The amplitude of that cycle grows with the level of the series, which rules out a purely additive decomposition.
- – Forecast uncertainty widens with the horizon; twelve months ahead the interval is already wide enough to matter.
Executive summary
The series is well described by three components: a smooth long-term trend, an annual periodic component, and an interaction between the two that makes the seasonal swing larger as the trend rises. Residual variation is small and shows no obvious remaining structure.
Dataset
Monthly totals of international airline passengers between January 1949 and December 1960, 144 observations, no missing values. This is the Box–Jenkins airline series, used here because it is public, well understood, and the example the original Automatic Statistician research reported on.
Data quality
Complete series, regular monthly spacing, no duplicated periods, no zero or negative values. No cleaning was required beyond checking the date index.
Observed trend
The level roughly triples across twelve years. Fitting a smooth trend component alone leaves a strongly periodic residual, which is the first indication that the seasonal structure must be modelled explicitly rather than differenced away.
Periodicity
A twelve-month cycle dominates, with a secondary summer peak. The cycle is stable in phase — peaks fall in the same months every year — but not in amplitude.
Structural components
| Component | Structure | Interpretation |
|---|---|---|
| 1 | SE | Smooth long-term trend |
| 2 | PER | Annual periodic structure |
| 3 | SE × PER | Seasonal amplitude increasing with the trend |
| 4 | WN | Residual variation |
TABLE 1 — Discovered components and their interpretation
Model
k(x,x’) = k_{SE} + k_{SE}\\,k_{PER} + k_{WN}
The composed covariance function
The multiplicative term is what produces seasonality whose amplitude grows with the level. An additive-only model fits the early years well and underestimates the later swings.
Predictions
Extrapolating twelve months beyond the end of the series reproduces the growing seasonal pattern and continues the trend. The point forecast should be read together with its interval rather than on its own.
Uncertainty
The 95% interval is narrow where observations are dense and widens steadily beyond the last observation. At a twelve-month horizon the interval spans a range large enough that decisions sensitive to the exact value should not be made from the point forecast.
Model criticism and limitations
The model assumes the growth pattern continues unchanged. It has no mechanism for the structural breaks that a real airline market produces — fuel shocks, deregulation, capacity changes. On a series that included such an event the changepoint component seen in the unemployment analysis would be required.
Methodology
Model structures were built from a small kernel grammar and compared by marginal likelihood, following the approach described on the research page. Full criteria are in the methodology.
References
- Box, G. E. P. & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control. Holden-Day.
- Lloyd, J. R., Duvenaud, D., Grosse, R., Tenenbaum, J. B. & Ghahramani, Z. (2014). Automatic Construction and Natural-Language Description of Nonparametric Regression Models. AAAI 2014.