## 2. Methods## 2.1. Phase space reconstructionGenerally, 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
where is the time delay. According to the embedding theorem of Takens (1981), the embedding
of the attractor in the One advantage of the ## 2.2. Correlation dimensionThe 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 where N denotes the overall number of data points, and
is the Heaviside step function. For
small distances Practically one computes the correlation integral for increasing
embedding dimension ## 2.3. Local pointwise dimensionsThe 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 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
## 2.4. Pitfalls in dimension estimationsThe 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 regionAs 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 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
In the case of ideal scaling, the
, calculated for different embedding
dimensions ## 2.4.2. Temporal correlationsThe 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 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. StationarityAs 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. IntermittencyIntermittency 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.
## 2.5. Scaling and convergence testThe practical computation of the local pointwise dimensions turns
out to be more difficult than the correlation dimension, particularly
as for According to Eq. 6, for each reference
point and each considered
embedding dimension 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
After the scaling region is determined for each
point and each embedding
dimension ## 2.6. Surrogate data testAs 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
© European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |