00437 ”69e9ec3c68434b76555b02df43db31

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Russell

at all suggests that dynami cal systems can sometimes be decomposed to repeated trials of simpler systems that may be understood both statistically and physically. In fact many mathematical models that produce deterministic chaos are actually formed through the recursion of simple equations.

Implications for Times Senes Analysis

The existence of dynamical systems that have some oibits of iterates which either are or appear to be chaotic has implications for the statistician trying to cope with data from those systems. Some questions posed to the analyst may be easily answered by providing common statistics of bulk system properties such as means and standard deviations. These properties are sufFicient for instance in Ä…uantifying our long term expectations for the Quincunx and random number generators.

However in many cases, such overall bulk distribution properties are not sufficient. Consider for instance a process control setting in which one wants to determine in advance which bin a bali is going to drop in for a Quincunx. For a smali number of rods, it might be quite reasonable to build either a statistical or physical model, or perhaps a combination of both if the number of rods is a little bit too large. As another case, consider a pseudo-random number generator — the data contains no noise at all and yet if the seed is not known (or worse, the algorithm), determining the next number in the sequence is not normally a tractable problem.

Notice also, if we were trying to fit the population growth parameter, r, for the Verhulst model, that if all that was known of the true underlying generating model was the output values (iterates) of a periodic orbit for some unknown but fixed growth ratÄ™ r₀, then any traditional time senes model developed for this data would be in considerable difficulty if the value of r₀ shifts by even a tiny amount in some cases. This is particularly true if the value or r₀ is near 3.57 or higher. A slight shift in r₀ could appear as a change in the underlying order or the seasonality of the approximating time series model. In other cases, it may appear as a simple shift in the fitted value of r. Such a shift might be caused, in a population growth model, by the temporaiy development of either a morÄ™ favorable or less favorable climate leading to a higher or lower growth ratÄ™ than previously existed.

So we see that even if a traditional time series model produced a perfect fit to the above data, the model would still be unable to cope well with smali changes in the apparent underlying system. On the other hand, suppose that the data is actually perfect data, with no noise at all, but the growth parameter is shifling due to some cause, then a traditional modeling approach