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A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 453

When we view the univariate phase map movie for this data set, we immediately recognize that this time senes is a great deal morę complex than the simpler synthetic series we generated earlier. In particular we notę that there is a very strong pattem in the movie that moves in location (in one direction only) and "grows" over time, see Figurę 13. An extract of two distinct cycles from the univariate phase map movie is presented in Figurę 12 below, for one of the early years and one of the later years.

If we look at observations 13 to 24 and 85 to 96 of the typical presentation of the time series data in Figurę 11, we see that there is a similarity in the number of peaks and the overall progression of the peaks during these two time periods. However it is less readily apparent that the change in size and relationship of the intermediate peaks from observations 13 through 24 and 85 through 96 is not due to some shift in the passenger travel pattem.

The univariate phase map movie extracts in Figurę 12 clearly indicate that despite the change in location on the plot (growth), that there is considerable underlying similarity in the underlying pattem produced by the univariate phase map movies. There have been only minor, probably random, changes in the pattem in the phase map over time and so the change in the relative sizes and placements of the peaks do not appear to be a sign of a changed underlying model.

Of interest to the analyst is that, sińce the pattem in the extract is so strong, any single point significantly breaking the established pattem may be a

Figurę 12. The airline passenger data. On the left is an extract from the univariate phase map movie for observations 13 to 24 and on the right an extract of observations 85 to 96. The movie extract is restricted to 12 lags which represents the expected seasonality. Notę that the vertical axis is inverted.