Astron. Astrophys. 357, 337-350 (2000) 4. Results4.1. Correlation dimension analysisThe application of the stationarity test led to the result, that only two of the preselected events did not reveal stationary subsections fulfilling Eq. 13. These events were excluded from further analysis. For each of the other events, the longest stationary subsections analyzed are listed in Table 1 and Table 2. If more stationary subsections were found, for the analysis the longest three were selected, and non overlapping or only partially overlapping subsections were preferred. The correlation dimension analysis of the stationary subsections did not reveal a finite dimension for any of the events. We identify mainly three cases, in which negative results occurred: no convergence , no scaling region , and deformed scaling region . These three cases are of course ideal ones, and the practice quite often revealed a mixture of it. Therefore, for the single event sections analyzed, we do not specify the different reasons causing the negative result, but describe and discuss the principal cases in the following subsections in a general frame. 4.1.1. No convergenceIf the converge with increasing embedding dimension m, this gives indications for the existence of a low-dimensional attractor underlying a time series. In the analyzed data sets, for which the curves revealed a clear scaling region, no convergence with increasing m occurred. As an example, Fig. 5 shows the divergent behavior for a type IV event. Even for rather high embedding dimensions a distinct scaling region exists. However, as the curves of the local slopes clearly reveal, the plateau of the scaling region moves to higher values for increasing m, indicating that the are divergent. The meaning of such a divergent behavior can be manifold, being related to the physical state of the system as well as to restrictions regarding the analysis methodology:
4.1.2. No scaling regionThere exist cases, in which for increasing embedding dimension m the scaling region disappears. In the curves of the correlation integral no linear range can be detected and, correspondingly, in the graphs of the local slopes no plateau region exists. In such cases the quantity is not defined. An example is shown in Fig. 6. The main reason for such a behavior might be that the analyzed data set is too short and/or too strongly contaminated by noise. In such a case the deviations from ideal scaling at small and large length scales r merge at intermediate r, and cause the scaling region to disappear. This effect is stronger for higher embedding dimensions, since for increasing m the attractor is sparsely covered with points, which can be verified in Fig. 6. As shown by Schreiber & Kantz (1995), even small amounts of measurement noise can conceal possible scaling behavior.
4.1.3. Deformed scaling regionThe third case, in which the correlation dimension analysis led to negative results, is, that the curves of the correlation integral are highly deformed. Such deviations from ideal scaling are mainly caused by the presence of intermittent sections, or generally, by the presence of very different amplitude scales in the data. Fig. 3 shows such deformed scaling regions for a stationary subsection of a sample type I storm, caused by an intermittent section. This can be clarified as the elimination of the intermittent section causes the deformation to disappear. Explained by the typical presence of intermittent phases in type I storms, the correlation dimension analysis rather often results in deformed plateau regions for type I burst series. 4.2. Local pointwise dimension analysisThe third set of columns in Table 1 and Table 2 contains the results of the pointwise dimension analysis for the type I and the type IV events, respectively. gives the averaged pointwise dimension with standard deviation , calculated over an embedding range , and "ok" denotes the percentage of points which passed the scaling and convergence test. gives the increase of the averaged pointwise dimension, when the embedding range is increased by two dimensions, from to . "surr." denotes the outcome of the surrogate data test, and positive means that the null hypothesis, which states that the results are caused by linearly correlated noise, can be rejected, and that significant evidence for nonlinearity in the data is given. As an example, Fig. 7 illustrates the results of the local pointwise dimension analysis obtained for a type IV event with quasi-periodic pulsations. The top panel shows the time series, the middle panel the time evolution of the local pointwise dimensions, . In the bottom panels the histograms of the local dimensions are plotted, calculated over different embedding ranges. About half of the points of the time series passed the scaling and convergence test and were used to compute the averaged pointwise dimension. However, as the histograms of the local dimensions calculated over increasing embedding dimensions (, , ) reveal, no absolute convergence exists, but a slight increase with increasing m-ranges occurs. This can be clearly seen in the right bottom panel of Fig. 7, as for higher embedding ranges the center of the histogram moves to higher dimension values. The same phenomenon occurs for all analyzed samples, as the positive values indicate, on the average .
This phenomenon, on the one hand, could result from the fact that the do not really converge with increasing m and no convergence to a low-dimensional attractor exists. On the other hand, a comparison with simulated time series from well known chaotic attractors contaminated with Gaussian noise reveals a similar behavior. For selected samples we repeated the analysis after application of a simple nonlinear noise reduction to the data (Schreiber 1993), to find out if the increase is caused by measurement noise. As it is common in nonlinear time series analysis, the used noise reduction method does not rely on frequency information but makes use of the structure in the reconstructed phase space. Fig. 8 shows an example of the application of the noise reduction method. Repeating the analysis with the times series after the application of the noise reduction led to the effect, that the dimension increase is softened but not fully eliminated though. Such result suggests that the dimension increase is not caused by measurement noise contaminating the signal of a deterministic system.
The surrogate data analysis yields a positive outcome for most of the analyzed samples, indicating that the obtained results are not an artifact, which could arise when the analysis is applied to non-stationary stochastic data. For 75% of the type I and 85% of the type IV burst time series with a significance of , we can reject the null hypothesis that the results are caused by a linear stochastic process, and have evidence for nonlinearity in the data. Fig. 9 illustrates the outcome of the surrogate data analysis for a sample type I storm. The top panel shows the averaged pointwise dimension as a function of the embedding dimension m, calculated from all points passing the scaling test. Diamonds represent the original data, crosses the surrogates. The bottom panel shows the related significance S, which reaches at the maximum a level of , giving strong evidence for nonlinearity in the data. However, corresponding to the positive values, the of the original data slightly increase with increasing m and do not show a definitive convergence to a finite dimension value.
© European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |