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Russell

then retums to the "bow-tie" pattem as in the plot on the right side of Figurę 7 below. This phenomena needs morę investigation to be fully understood.

Even with rather casual examination of the univariate phase map movie extracts for the synthetic time senes with negative coefficients examined in this paper - it is relatively easy to see the differences between series of different orders. One of the morę surprising results is that the observed pattems apparent in the univariate phase map movies are not unconditionally stable but appear to remain distinct. This is most apparent in the higher order models examined in this paper.

There appear to be occasions, perhaps corresponding to data points at which the noise term dominated, upon which the pattem of the phase map tends to change to a similar pattem. Compare the extracts in right side of figurę 6 and the left side of Figurę 7. In addition there are cases in which the dominant pattem is temporarily abandoned, see the right side of Figurę 7. The impact of this lack of stability on the usefulness of the univariate phase map to identify autoregressive time series in noisy data remains to be evaluated.

It is also rapidly discovered in practice that, for each time series viewed, there is an optimal amount of univariate phase map movie history to

Autoregressive Time Series: AR(3), Coefficients Negative

Figurę 7. Left - An extract, from the univariate phase map movie for the AR(3) series with negative coefficients, of observations 275 to 300. Right - An extract, from the univariate phase map movie for the same series, of observations 235 to 250. Notice that the pattem on the right is not characteristic for the synthetic data in this data set. Notę that the vertical axes for the plots are inverted.