00444 )a588943785e7aaa29215489e95475a

A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 449 retain on the screen at the same time. A good univariate phase map movie program will provide the user with control over the amount of history to be displayed on the screen at one time. The amount of history for the movie extracts in this paper are controlled morę by the amount of apparent rotation and clutter than anything else. In a few cases to be discussed later, the amount of history is controlled to elucidate expected seasonality.

If too much univariate phase map movie history is presented at one time, the dominant pattems ąuickly clutter the screen, obscuring the pattems. The extract pattems are also observable when points are plotted as scatterplots but they are harder to see as there is a strong tendency to present much morę data (history) at one time. In either case, if all the data is presented as was done to form the typical scatterplot in Figurę 1, the pattems are completely lost.

Synthetic Autoregressive Time Senes with Positive Coefficients

The next set of series to be examined are synthetic time series of the same orders as before but with positive coefficients. As before the series were generated using the same seed and they were allowed time to settle before observations were recorded. The series examined are as follows:

X, = 0.95X,__l +e,

X, = 0.05X,_, + 0.95X,_₂+e_l X, =0.0LV^r,_₁ + 0.0LY,_₂ + 0.95X_t_₃ +e,

The fourth series that we examine is the synthetic AR(1) series with positive coefficients. The traditional time order plot of the series is displayed on the left side of Figurę 8. A typical extract from the univariate phase map movie is presented on the right side of Figurę 8. In the extract from the movie, the plotted linę segments are observed to move back and forth along a linę extending out from the origin. In this particular case, the iterates might be said to "coil" around the linę.

The fifth series that is examined is the synthetic AR(2) time series with positive coefficients. A time order plot of the series is presented on the left side of Figurę 9 below. A typical extract from the movie is presented on the right side of Figurę 9 below. At first glance this extract appears to be similar to that of the AR(1) synthetic time series with negative coefficients. However on closer examination, the extract is seen in the movie to be moving back and forth along a linę extending from the origin as was the case for the