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A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 44S

²    (^-i)jr_Å‚+1+#_ajr,

and

(#,-l)jr_i+I+#₂Jr,

^m\ =--——-

It becomes obvious that tÅ‚iis becomes worse with even higher order autoregressive processes. And worse yet if the noise terms are included in the eÄ…uations.

Synthetic Autoregressive Time Series with Negative CoefTicients

To overcome our lack of intuition about the properties of the univariate phase map movies based on recursive eÄ…uations in the presence of noise, we turn to simulated processes. In this paper six simulated autoregressive time series will be examined, and portions of the univariate phase map movies are presented. Only a smali percentage of noise was included in the synthetic data series to enhance the understanding of the underlying properties. It is fair to point out that in the author's experience that even when substantially morÄ™ noise was introduced through the noise term, the major features of these series were still identifiable. However morÄ™ data has to be viewed to get a graphical sense of the data trends.

In the set of synthetic time series with negative coefficients, three highly autocorrelated time series are examined. In each case, the seed was the same. Also over 100 iterations of the simulated process producing the series were discarded to allow the series time to stabilize. The series are

X,=-0.95X,__x+e,

X, = -0.95X,_,-0.95X,_₂+*,

X, = -0.95A^r,_₁ -0.95X,_₂ - 0.95X,_3 + e,

The first synthetic series that we examine is the AR(1) series with negative coefficients. A time order plot of the series and a phase map movie extract are presented in FigurÄ™ 4. Notice that in the movie extract, the lines connecting the points lie essentially in a smali region of the plot. In the movie