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A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 443 would fail to recognize the high quality of the data — and the data would be treated as if it were noisy. The analyst may be able to introduce corrections in the form of transfer functions, if any are known to exist, to assist with fitting the data in such cases. As this demonstrates, it is preferable to work with the underlying true physical models themselves, provided they are known and simple enough.

In many actual time series modeling situations of dynamical processes it is reasonable to assume that there may be underlying changes in the process itself over time, reÄ…uiring modifications both in the parameters being fit and the number of lags of autocorrelations exhibited by the iterates. The analyst in such cases needs to be able to determine if the approximating traditional model that would be fitted to the data needs to change over time and how.

The Uniyariate Phase Map

Drawing upon the concept of strange attractors from chaotic systcms, it might be inferred that there is a strong possibility that the univariate phase map can show evidence of such an attractor. An attractor would become evident siÅ„ce orbits of the iterates that are near an attractor tend to remain near the attractor. In fact, phase maps are commonly used to assist in finding attractors of dynamical systems.

One enhancement to the traditional scatter plot form of a univariate phase map is to connect successive points with lines (Russell, 1991). This is particularly helpful if the system that is being investigated is not "stable." However, if too many successive points are plotted and connected, then the resulting plot may become very messy. To reduce the clutter, we can present the connected points in smaller groups. Since we don't know what the optimal number of iterates is for this type of presentation for any given data set, we can present the uniyariate phase map as a movie with a fixed (but changeable) number of successive points in the uniyariate State space being connected at any one time.

Based on the prior discussion about dynamical systems, and realizing that traditional time series models will function primarily as approximations to the real underlying processes, there are a number of Ä…uestions we need to consider. We would like to know, for any given autoregressive time series, if the uniyariate phase maps are stable and in what sense. We would also like to know if the uniyariate phase maps are distinctive enough to allow the analyst to determine which model underlies the time series data.

If there is some stability in the phase maps and the uniyariate phase map movies of different models appear distinctiye, then the uniyariate phase