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

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4. Results

4.1. Correlation dimension analysis

The 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 convergence

If the [FORMULA] 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 [FORMULA] 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 [FORMULA] clearly reveal, the plateau of the scaling region moves to higher values for increasing m, indicating that the [FORMULA] 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:

[FIGURE] Fig. 5. Top panel: Stationary subsection of a type IV event with sudden reductions (January 13, 1989). The left panel of the middle column shows the correlation integral [FORMULA] for [FORMULA] to [FORMULA]. The graphs of the local slopes [FORMULA], plotted for [FORMULA], 6, 9, 12 and 15, reveal distinct plateau regions (hatched), which for increasing m are moving to higher values, indicating that the [FORMULA] are divergent.

  1. The underlying physical system is stochastic.

  2. The signal is the output of a deterministic but high-dimensional system, with a dimension too high to be extracted from the given time series of finite length.

  3. The signal represents a system to which noise is coupled intrinsically to the dynamics.

  4. The analyzed time series is the result of different physical systems which are independently operating at the same time, e.g., different uncoupled radio burst sources simultaneously present on the sun, whose emissions sum up to the measured signal.

  5. Despite the careful data selection, the measurement noise is still too high for the kind of analysis carried out and dominates the results.

  6. The choice of the time delay [FORMULA], which is a quite critical and sensitive parameter in the correlation dimension analysis, is not optimal.

4.1.2. No scaling region

There exist cases, in which for increasing embedding dimension m the scaling region disappears. In the curves of the correlation integral [FORMULA] no linear range can be detected and, correspondingly, in the graphs of the local slopes [FORMULA] no plateau region exists. In such cases the quantity [FORMULA] 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.

[FIGURE] Fig. 6. Top panel: Stationary subsection of a type IV event (July 14, 1984). The curves of the correlation integral [FORMULA], plotted for [FORMULA] to [FORMULA], reveal only for small m a linear range. In the graphs of the local slopes [FORMULA], the plateau region (hatched) shrinks with increasing m, and for [FORMULA] no distinct scaling region exists.

4.1.3. Deformed scaling region

The 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 analysis

The 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. [FORMULA] gives the averaged pointwise dimension with standard deviation [FORMULA], calculated over an embedding range [FORMULA], and "ok" denotes the percentage of points which passed the scaling and convergence test. [FORMULA] gives the increase of the averaged pointwise dimension, when the embedding range is increased by two dimensions, from [FORMULA] to [FORMULA]. "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, [FORMULA]. 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 ([FORMULA], [FORMULA], [FORMULA]) 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 [FORMULA] values indicate, on the average [FORMULA].

[FIGURE] Fig. 7. The top panel depicts the time series of a type IV event with pulsations (March 7, 1991). The middle panel shows the evolution of the local pointwise dimensions. The left bottom panel shows the histogram of the dimension values. In the right botton panel we show the same histogram, overplotted by the histograms of dimensions calculated at higher embedding ranges.

This phenomenon, on the one hand, could result from the fact that the [FORMULA] 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.

[FIGURE] Fig. 8. Illustration of the application of a nonlinear noise reduction. The top panel shows the original time series (subsection of a type IV with pulsations, March 7, 1991). The bottom panel shows the same series after the application of the noise reduction.

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 [FORMULA] 75% of the type I and [FORMULA] 85% of the type IV burst time series with a significance of [FORMULA], 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 [FORMULA], giving strong evidence for nonlinearity in the data. However, corresponding to the positive [FORMULA] values, the [FORMULA] of the original data slightly increase with increasing m and do not show a definitive convergence to a finite dimension value.

[FIGURE] Fig. 9. Illustration of the outcome of the surrogate data test for a type I storm (January 30, 1991). The top panel shows the averaged pointwise dimensions, calculated for the original data (diamonds) and for an ensemble of 10 surrogates (crosses), for [FORMULA] to [FORMULA]. In the bottom panel the significance S is plotted, which reaches values significantly larger than the [FORMULA] level, giving evidence on nonlinearity in the data.

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Online publication: May 3, 2000