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A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 451

The sixth senes that we examine is a synthetic AR(3) time senes with positive coefficients. As before, the traditional time order plot of the series is presented on the left side of FigurÄ™ 10. In addition, a typical extract from the univariate phase map movie is presented on the right side of FigurÄ™ 10.

In a relationship that appears to be similar to that of the AR(2) series with positive coefficients appearing like the AR(1) series with negative coefficients, this series appears to be similar to the AR(2) series with negative coefficients. However, again similar to the former case, the "triangles" formed by the AR(3) series with positive coefficients move back and forth along a linÄ™ extending from the origin.

Even with simple casual examination of the univariate phase map movies for the synthetic time series with positive coefficients, it is again easy to see that there is a distinct difference between the autoregressive models of differcnt orders. It is also apparent that there is a possible relationship between the extract character of synthetic series with positive coefficients and those with negative coefficients of one order lower. In addition we should notÄ™ that the models with positive coefficients appear to have phase map movies that move back and forth along a linÄ™ extending from the origin. In contrast, the models with the negative coefficients appear to remain in place and may possibly rotate about some center.

Autoregressive Time Series AR(3), Coefficients Positive

AR03nSS ^\m^ Plot

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FigurÄ™ 10. Left - the time order plot of the AR(3) synthetic time series with positive coefficients. Right - An extract, from the univariate phase map movie for the same series, of observations 280 to 300. NotÄ™ that the vertical axis for the plot on the right is inverted.