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Russell

Autoregressive Time Senes AR(1), Coeflicients Negative

ARO Ip95 Umm Plot

FigurÄ™ 4. Left - the time order plot of the AR(1) synthetic time series with a negative coefficient. Right - An extract, from the univariate phase map movie for the same series, of observations 150 to 170. NotÄ™ that the vertical axis for the plot on the right is inverted.

itself these lines essentially tracÄ™ over themselves. The length of the lines in the extract are directly related to the excursions between the points in the time series. So in regions of the time series with shorter excursions between successive points in the time series, we see shorter lines connecting points in the movie extract.

The second series that we examine is the AR(2) series with negative coefficients. A time order plot of this series is provided in FigurÄ™ 5 as is the univariate phase map movie extract. The movie extract differs from the AR(1) with negative coefficients in that the successive lines connecting plotted points roughly form "triangles." Over time, in the movie, the "triangles" rotate around a rough center.

The third times series that we examine is the AR(3) time series with negative coefficients. As before a time order plot of the series and a phase map movie extract are presented in FigurÄ™ 6. This movie extract differs from the previous movie extracts in that a typical extract shows a "bow-tie" type of pattem. Hence the AR(1), AR(2), and AR(3) time series with negative coefficients have movie extracts that are distinctly different.

One other notÄ™ that is readily apparent in the movie extract for the AR(3) with negative coefficients is that not only can the "bow-tie" shape appear to rotate, it may have sides that have different lengths but that still roughly form the same "bow-tie" shape as in the plot on the left side of FigurÄ™ 7 below.