Spurious Dimension Estimates From Time Series Of Geomagnetic Indice

time series. Notice the non–stationary mean and variance, especially the elevated mean during the fourth day, and the relatively quiet conditions during the later part of the fifth day. These short periods of high correlation may be leading to a spurious dimension estimate. Next we tried estimating the dimension of the attractor using the method of Termonia and Alexandrowicz (1983), but no scaling region was found. Negative results were also obtained by trying to estimate the information dimension using the slopes of log C 1. Because of these inconsistent results we decided to break to the data set in half and analyze each half separately (3600 points each.) For the first half of the data set we see no convergence of slopes, but for the second half (Figure 5) we see a convergence of slopes to a value of roughly 2.8. Lyapunov exponents measure the rate at which nearby phase space trajectories diverge; a chaotic system must have at least one positive Lyapunov exponent. Both the Wolf et al. (1985) and Eckmann et al. (1986) methods were used to estimate the largest Lyapunov exponent of the various time series. The Eckmann method gave inconclusive results in all cases. The Wolf method gave estimates in the range 0.005 to 0.04 bits/min, but the results are not stable over changes in the analysis parameters (see Wolf et al.) and so should not be trusted. Further, analysis of the surrogate data show very similar results suggesting that these estimates are spurious. Conclusion For the data sets examined we see no evidence for the presence of low dimensional chaos in either AE or AL time series, except in the April 1983 data. For this set we do find a finite correlation dimension using the modified Grassberger and Procaccia method, but other methods did not lead to finite dimensions. Because of the lack of agreement we distrust the results of the Grassberger and Procaccia method. We also were unable to determine a reliable estimate for the largest Lyapunov exponent for this data set, suggesting that the dimension estimator was 'fooled' due to the non–stationarity of the data set. This sort of behavior leads us to believe that AE and AL time series are essentially random with some nonlinear structure. Casdagli et al. (1992a) have shown this is the sort of behavior one would expect for a driven system with a random forcing function (we find …