## 5. DiscussionThe 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 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
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.
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 © European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |