Astron. Astrophys. 357, 337-350 (2000)

5. Discussion

The outcome of the dimension analysis does not allow to claim low-dimensional determinism for the analyzed data sets. First, the correlation dimension analysis failed in all cases. Second, the obtained averaged pointwise dimension values  are too high to characterize a low-dimensional physical system. Third, the do not reveal an absolute convergence with increasing embedding dimension m. On the other hand though, the local pointwise dimensions obtained over a specific embedding range yield a quite distinct behavior, and the surrogate data analysis evidences nonlinearity in the data.

In Fig. 10 we show a comparison of the correlation dimension and the local pointwise dimension analysis of a sample type IV event. The correlation integral and the related local slopes do not give any evidence for low-dimensional determinism, since no significant scaling region and no convergent behavior occurs. However, the corresponding curves of the local pointwise dimension analysis reveal a distinct scaling region and a clear convergence to a finite and low dimension value for certain points . The figure clearly illustrates that the calculated local dimensions do not represent an artifact, which might arise, for instance, if the automatic scaling and convergence procedure is not well adapted to the analysis. Moreover, such comparison suggests that the local dimension analysis of a times series is more robust than the classical correlation dimension method. The main reason might be that in the correlation dimension analysis the scaling behavior itself is a global property, since all pairs of points contribute to the correlation integral, and the scaling region can possibly be smeared out by such an averaging process. Contrary to that, the local pointwise dimensions are based on the scaling behavior at each single point, and can reveal a well defined scaling at certain points.

Fig. 10. Comparison of the correlation dimension and the local pointwise dimension analysis, calculated for a type IV burst series with pulsations (March 7, 1991). The top and middle panels illustrate the correlation dimension analysis. The left top panel shows the curves of the correlation integral  for embedding dimensions to . The top panel at the right hand side and the middle panels depict the curves of the local slopes  for three different embedding dimensions. The bottom panels illustrate the outcome of the local pointwise dimension analysis for point . The left bottom panel shows the curves of the probability . In the right bottom panel we show the curves of the related local slopes .

In Table 3 we give a summary of the averaged pointwise dimensions for the different event types. For the type IV events we additionally calculated the quantities separately for the different subtypes. We list the event type, the number of events belonging to each type (or subtype), the number of events passing the surrogate test, and the average of the pointwise dimensions over the respective event types, calculated only from the samples for which the surrogate data test gave a positive result. On the average, the type I storms reveal lower  values than the type IV events and a significant smaller standard deviation, indicating that type I storms represent a comparatively homogeneous class. The large standard deviation for the type IV burst series basically reflects the different subtypes. The average of the particular type IV subtypes yields values which are significantly different: pulsations and sudden reductions reveal the lowest values, followed by the type IVs with fine structures of no particular kind and spikes, whereas the fast pulsations are characterized by the highest values.

Table 3. Statistics of the pointwise dimensions. We list the number of events belonging to a particular type or subtype (including type IV events with fine structures of no particular kind, pulsations, fast pulsations, sudden reductions, and spikes), the number of events giving a positive outcome of the surrogate data test, and the pointwise dimension averaged over the respective (sub)type.

Such statistics suggests that the  are quite well representing the different types of radio bursts under investigation. This means that even if low-dimensional determinism cannot be proved, the local dimension analysis can provide a quantitative description of the events, which is not possible with the commonly used correlation dimension. As argued by Schreiber (1999), such description has the drawback that it does not provide an invariant characterization of a system. On the other hand, it offers an alternative statistical approach for systems, for which pure determinism cannot be established. This is of particular interest in astrophysics, since astrophysical systems represent real-world systems, which cannot be influenced by the observer and are highly interconnected with their surroundings, making pure determinism rather improbable.

Moreover, we claim that on a comparative basis the retrieved dimension values are related to the degree of freedom of the system. From numerical experiments with known chaotic attractors contaminated with Gaussian noise we retrieved pointwise dimensions, which are slowly increasing with increasing embedding dimension m, i.e. the absolute convergence to the definite attractor dimension disappeared due to the contamination of the deterministic signal with a stochastic component. This behavior is similar to the one of the analyzed radio events. However, from this similarity we cannot conclude that the pointwise dimension analysis is indicative for hidden deterministic chaos in the radio burst time series, since the retrieved dimension values are too high to characterize low-dimensional determinism, and for a reliable dimension analysis of high-dimensional systems much longer time series are needed than we have at disposal (- a limitation which is intrinsic to time series representing real-world systems). Nevertheless, what we can infer from this similarity is that the retrieved dimension values, even if not representing attractor dimensions, are still indicative for the degree of freedom of the physical system underlying the time series, characterizing its complexity. Based on this fact, we make use of the retrieved dimension values to describe the complexity of the related systems in a comparative way, without claiming or supposing the presence of low-dimensional determinism.

© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
helpdesk.link@springer.de