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

2. Methods

2.1. Phase space reconstruction

Generally, not all relevant parameters of the dynamics of a system are measured during an observation but only a one-dimensional time series is given. To reconstruct the phase space of the related dynamical system, techniques have to be applied to unfold the multi-dimensional attractor from a scalar time series. By the technique of time delayed coordinates (Takens 1981), from a given one-dimensional time series  an m-dimensional phase space  is built up by the prescription

where is the time delay.

According to the embedding theorem of Takens (1981), the embedding of the attractor in the m-dimensional reconstructed phase space can be ensured if , with D the dimension of the original phase space. For time series of infinite length and accuracy, can be chosen in a more or less arbitrary way without affecting the results. However, in practice not every value for the time delay  will be suitable. Too small  will build up coordinates which are too strongly correlated, while for large  the vector components show no causal connection. The choice of the time delay  in the reconstruction of the phase space usually strongly affects the quality of the analysis, and different procedures have been worked out to ensure a proper choice of . The most prominent methods use the auto-correlation time ( decay time, first zero crossing, first minimum) or the first minimum of the mutual information (Fraser & Swinney 1986).

One advantage of the mutual information over the auto-correlation function is that it takes into account nonlinear properties of the data. The mutual information is based on the Shannon entropy (Shannon & Weaver 1962), and gives the information about the state of a system at time that we already possess if we know the state at time t. The choice of the first minimum of the mutual information for the time delay  is motivated by the fact that two successive delay coordinates should be as independent as possible without making  too large.

2.2. Correlation dimension

The correlation dimension is one out of many definitions of fractal dimensions, and was introduced by Grassberger & Procaccia (1983a, 1983b) to determine fractal dimensions from time series. The correlation dimension is based on distance measurements of points in phase space. Therefore, as first step, from the time series  the phase space vectors  have to be constructed. With the reconstructed vectors, the correlation integral  can be calculated, which is given by the normalized number of pairs of points within a distance r. As the correlation dimension is based on spatial correlations in phase space, it is an important precaution to exclude serially correlated points in counting the pairs (for details see Sect. 2.4.2), and the length of the window, W, should at least cover all points within the auto-correlation time (Theiler 1986). With this correction, the correlation integral is given by

where N denotes the overall number of data points, and is the Heaviside step function. For small distances r, the correlation integral  is expected to scale with a power of r, and the scaling exponent defines the correlation dimension :

Practically one computes the correlation integral for increasing embedding dimension m and calculates the related in the scaling region. If the  reach a saturation value  for relatively small m, this gives an indication that an attractor with dimension  exists underlying the analyzed time series.

2.3. Local pointwise dimensions

The local pointwise dimension is a locally defined variant of the correlation dimension. Its definition is based on the probability  to find points in a neighborhood of a point  with size :

where gives the actual number of pairs of points in the sum. For small distances r, is expected to scale with a power of r, and the scaling exponent  gives the local pointwise dimension at point :

Averaging  over all points of the time series or a number of reference points yields the averaged pointwise dimension, , which is equivalent to the correlation dimension and gives a global description of the geometry of an attractor:

with the number of accepted reference points.

Since the are local functions and defined for each point, they are, on the one hand, a function of the position  on the attractor, characterizing its local geometry. On the other hand, the local dimensions can be interpreted as a function of time, , since they reflect the temporal evolution, in which the points  on the attractor are covered by the dynamics (Mayer-Kress 1994). Based on this fact, local dimension estimations have the interesting property that they enable to cope with non-stationary data.

An example is shown in Fig. 1, which illustrates the local pointwise dimensions calculated for a simulated time series, made up of three sections related to different attractors, the Lorenz, a limit cycle and the Rössler attractor. The evolution of the local pointwise dimensions  detects the different attractors successively operating in time. The averaged pointwise dimensions  in the respective sections deviate less than 10% from the true values. As for the calculation of the local dimension at point  the distances to all points of the time series, even those related to a different attractor, are taken into account, it is a quite striking feature that the different attractors can be disentangled by the method. Moreover, for this exemplary analysis the length of the time series was taken rather short in order to mimic the conditions of observed time series. Such a behavior reveals that the reference point , which is equivalent to a time  of the dynamical evolution of the system, dominates the local dimension calculation. However, we want to stress that small changes of the attractor dimension, as, e.g., in the case of the Lorenz and the Rössler attractor (see Fig. 1), cannot be detected.

Fig. 1. Top panel: Simulated time series, made up of three sections related to different attractors, the Lorenz, a limit cycle (Sine), and the Rössler attractor. The middle panels depict an enlargement of the respective sections. The bottom panel shows the related local pointwise dimensions . Its evolution reflects the different attractors successively operating in time.

2.4. Pitfalls in dimension estimations

The determination of fractal dimensions from time series, characterized by finite length and accuracy, includes many pitfalls, which can lead to quite spurious results. In this chapter the most prominent problems will be discussed and the strategies used by the authors to avoid such pitfalls. A brief but quite dense review with respect to critical points in the determination of the correlation dimension can be found in Grassberger et al. (1991). As the local pointwise dimensions are a variant of the correlation dimension, most of the problems occur in a similar way.

2.4.1. Noise and quality of the scaling region

As expressed in Eq. 4 for the correlation integral  and Eq. 7 for the probability , respectively, a scaling behavior is expected for . However, for real time series, which are contaminated by noise and which are of finite length, the scaling behavior is expected to occur at intermediate length scales. Noise is acting at small scales and therefore dominating the scaling behavior for small r. Deviations from ideal scaling at large length scales are due to edge effects caused by the finite length of the time series.

Fig. 2 shows such typical scaling behavior for a time series of finite length and accuracy. For better illustration we have plotted the local slopes of the correlation integral, given by the expression

Fig. 2. Scaling behavior for a time series of finite length and accuracy, calculated from a stationary subsection of a type IV event with sudden reductions (January 13, 1989). On small scales r [I], the scaling is dominated by noise. Since noise tends to fill the whole phase space: . At intermediate length scales [II], the physically relevant scaling is located: . For large r [III], deviations from the ideal scaling occur due to the finite number of data points.

In the case of ideal scaling, the , calculated for different embedding dimensions m, form straight lines parallel to the x-axes, the so-called plateau region, with the constant y-value corresponding to . However, it is typical for observational time series that at least three different parts in the curves are distinguishable, described in the caption of Fig. 2. To avoid spurious finite dimensions which might arise from a misinterpretation of the existence or non-existence of a physically relevant scaling region, we implemented an algorithm to automatically check for scaling behavior (see Sect. 2.5).

2.4.2. Temporal correlations

The correlation integral  and the probability  are a measure of spatial correlations on the attractor. They are basically calculated by counting pairs of points which are closer to each other than a given distance r. However, at small scales successive points of the time series give an additional contribution, since they are close in time. Nevertheless, such points do not reflect spatial correlations, i.e., a clustering of points in phase space, but are serially correlated by the temporal order of the time series. To avoid the spurious contribution of serially correlated points, which can cause strong artificial effects, especially for data recorded with a high sampling rate, at least all pairs of points closer than the auto-correlation time have to be excluded (Theiler 1986):

A similar phenomenon can occur when the analyzed time series is rather short. If the time series is too short to ensure that the attractor is well covered with points, then most of the points of the series are serially correlated, which again might result in spurious dimensions. Different relations have been derived for the minimum length of a time series needed for dimension estimations, as the one by Eckmann & Ruelle (1992):

with N the length of the time series. However, not only the length of the data series is of relevance but also the length with regard to the sampling rate. Applications to different kind of time series, from known chaotic attractors as well as measured time series (Brandstater & Swinney 1987; Kurths et al. 1991; Isliker 1992a; Isliker & Benz 1994a), have revealed that the analyzed time series should at least cover 50 structures - "structure" meaning a full orbit in phase space or generally a typical time scale of the analyzed time series. According to Isliker (1992a), we define the structures by the first minimum of the auto-correlation function,

where gives the number of structures, N the length of the time series, the temporal resolution, and the auto-correlation time.

2.4.3. Stationarity

As shown by Osborne et al. (1986) and Osborne & Provenzale (1989), the determination of the correlation dimension from non-stationary stochastic processes, can erroneously lead to finite dimension values. To take into account that problem, we applied a stationarity test proposed by Isliker & Kurths (1993), which is based on the binned probability distribution of the data. To check for stationarity one divides the time series into subsections, and compares the probability distribution of the section under investigation with the probability distribution of the first half of it by a -test. By the use of this test we searched for stationary subsections in the radio burst time series, and only to such stationary subsections the correlation dimension analysis was applied.

However, with the concept of local dimensions it is possible to cope with non-stationary data. One important advantage of the local dimension method is that it can enable to detect dynamical changes in a time series. To make use of this potentiality, we calculated the local pointwise dimensions from the whole time series instead of using stationary subsections. Moreover, since the statistics in the calculation of the local pointwise dimensions grows linearly with the length of the time series, whereas the correlation dimension grows with the square of the number of points, for the local pointwise dimension analysis the time series should be kept as long as possible. However, to avoid spurious results due to non-stationarities, we applied a surrogate data test (Sect. 2.6).

2.4.4. Intermittency

Intermittency describes the phenomenon that a time series is interrupted by quiet phases or phases of very low amplitudes. Such a phenomenon is quite typical for chaotic systems, but can cause problematic situations when calculating fractal dimensions from limited time series. In phase space the intermittent sections represent regions, which, in the context of the global attractor scale, degenerate to a point. This would trivially result in an erroneously low dimension value. One way to cope with this problem is just to discard intermittent phases from the analyzed time series. Since the dimension of an attractor is a geometric descriptor, which means a static quantity, it is not influenced by the serial order of points, and therefore discarding subsections is a valid strategy.

As an example, the top panel of Fig. 3 shows the subsection of a type I burst series. The middle and bottom panels show the results of the correlation dimension analysis, calculated from the whole time series (middle panels) and the time series after the intermittent section was eliminated (bottom panels). A comparison of the middle and bottom panels reveals that the intermittent phase causes a deformation in the curves of the correlation integral and the related local slopes. Eliminating the intermittent section, these deformations disappear. However, we want to stress that the used algorithm for the automatic detection of the scaling region does not identify the sink-region as a scaling region, which could result in spurious finite dimensions. Therefore, by the set up of the scaling algorithm we avoid erroneous low dimensions caused by intermittency effects.

Fig. 3. Top panel: Stationary subsection of a type I storm (January 30, 1991) with an intermittent phase present (marked by a line). The abscissa values are given in points (the same applies to following figures). Middle panel: Correlation integral  for to , and local slopes for and , calculated from the whole time series depicted. Bottom panels: Same curves, but calculated from the time series after the putative intermittent section was eliminated. The hatched ranges in the graphs of the local slopes mark the automatically determined scaling regions.

2.5. Scaling and convergence test

The practical computation of the local pointwise dimensions turns out to be more difficult than the correlation dimension, particularly as for each reference point the scaling region has to be determined and the related , which are calculated in the scaling region, have to be checked for convergence with increasing embedding dimension m. For this purpose, an automatic and fast procedure is needed. Moreover, such an automatic procedure can also be used in the correlation dimension analysis to avoid subjective influences on the scaling and convergence judgement. We implemented such an algorithm, based on the one used by Skinner et al. (1991), but modified in order to reach higher stability and significance. This automatic procedure searches for the scaling region, defined as the longest linear range in the curves, tests if the scaling region is of significant length, and finally checks if the  are converging with increasing m. Only for points , which pass the scaling and the convergence test, a local pointwise dimension  is accepted. In the following we give a description of the algorithm.

According to Eq. 6, for each reference point  and each considered embedding dimension m, we calculate the cumulative histogram of the , dividing the r-range into 100 equidistant points in the logarithmic representation, which cover the whole range from the smallest to the largest actual distance. The local slopes of the versus curves, given by

calculated in the scaling region, represent the . For the determination of the location of the scaling range, we shift a window with length 10 points through the overall r-range, and for each of these windows a least squares linear fit is applied to the versus curves. The slope of the linear fit corresponds to the average value of the local slopes in the corresponding r-range, denoted as . Starting with the first window, successive  are compared with the corresponding quantity in the first window, until the difference is larger than a certain threshold value, chosen as 20% of the initial . In this case, the position of the first point of the start window and the last point of the actual window are stored, and the procedure continues with the second window as start window. If the new positions determined cover a larger range than the old ones, the old values are overwritten by the new ones, and so on. The two stored values remaining at the end give the location of the scaling region, in which we calculate . If the determined scaling region is smaller than 20% of the overall length, it is interpreted as not significant and rejected. To avoid spurious results at large r, which usually correspond to small  values, in the whole procedure we suppress values . Fig. 4 shows a sample application of the algorithm detecting the scaling range.

Fig. 4. The curves of local slopes , with , are shown for different embedding dimensions m, calculated from a sample type I storm (August 21, 1991). For better illustration we plot versus instead of versus . The marked regions indicate the location of the scaling range, as detected by the automatic procedure. For the determined scaling region is too short (less than 20% of the overall r-range), and therefore rejected.

After the scaling region is determined for each point  and each embedding dimension m, for those points which reveal a scaling region of significant length for all considered m-values, we test the convergence of the  with increasing m. This is simply done by averaging the  over four successive m-values. If the standard deviation turns out to be less than 15% of the average value, the convergence is accepted, and the average of the  over the considered m-range is taken as local pointwise dimension . The different kind of threshold values used have been adjusted by application of the algorithm to different time series, and slight changes do not qualitatively change the outcome.

2.6. Surrogate data test

As already mentioned, local dimensions enable to deal with non-stationary data. However, to avoid spurious dimensions which can result from non-stationary stochastic processes, we applied a surrogate data test (Osborne et al. 1986; Theiler et al. 1992). For this purpose, a Fourier transform of the time series is performed, the Fourier phases are randomized, and finally to this phase-altered series the inverse Fourier tranform is applied. The phase randomization keeps the power spectrum unchanged while linear correlations in the time series are eliminated. If the results of the dimension analysis of such an ensemble of surrogate data are significantly different from those computed from the original data, the hypothesis can be rejected that the obtained results are caused by a linear stochastic process, and nonlinearity in the data is detected.

To quantify the statistical significance we make use of a hypothesis testing given in Theiler et al. (1992). The null hypothesis assumes that the original data represent linearly correlated noise. To compute a discrimination statistics we need a set of surrogate data, from which the same statistical quantities are derived as from the original time series. In particular, we use the averaged pointwise dimensions as function of the embedding dimension, . Let  denote the statistical quantity computed for the original time series, and  for the ith surrogate generated under the null hypothesis. Let  and  represent the (sample) mean and standard deviation of the distribution of . With this notation the measure of "significance", S, is given by the expression

© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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