This first part of the thesis dealt with the introduction and development of an analysis framework based on windowed recurrence network analysis (wRNA) and windowed scale-specific recurrence network analysis (wssRNA) that can be used to reliably detect and characterise dynamical anomalies in time series. In particular, we focused on experimentally measured time series representing a system whose dynamics are not accessible analytically.
After a general introduction to non-linear time series analysis, we established the theory on which this thesis is based in chapter 2. That is, we (i) studied the problem of phase space reconstruction to recover the dynamics of a higher-dimensional system from a univariate measured time series, (ii) explored the concepts of recurrence network analysis and scale-specific recurrence network analysis with particular focus on sliding window approaches, and (iii) entered the field of significance testing using surrogate data.
To further study and develop the existing framework of wRNA, we compared the methods of uniform time delay embedding and derivative embedding for different ways to estimate the derivatives for the case of non-uniformly sampled and noisy data in chapter 3. Such data pose a particular challenge to the reconstruction of the system’s phase space as time delay embedding is not defined for non-uniformly sampled data and derivative embedding suffers from noise amplification for higher-dimensional embeddings. We found that using linear interpolation in combination with uniform time delay embedding performs reasonably well in many cases and its performance can sometimes be improved by using cubic spline interpolation. For derivative embedding, we found that estimating the derivatives by using the relation between the derivatives of a time series and the discrete Legendre polynomials performs best compared to the other ways of derivative estimation. In some cases, this approach can be a useful alternative to time delay embedding.
We then considered the problem of defining anomalies and their significance in wRNA and wssRNA in chapter 4. When applying sliding window approaches or when repeating an analysis for a set of analysis parameters, intrinsic correlations within the analysis results complicate the detection of anomalies as in such cases, patches of significant points instead of isolated significant points appear irrespective of whether they are true or false positives. Based on a numerical estimation of the decorrelation
length of the intrinsic correlations using surrogate data with respect to a specified null model, we introduced an areawise significance test that can identify patches of false positive significant analysis results. When applying the test in combination with wRNA and wssRNA to study a non-stationary Rössler system, we found that the areawise significance test is indeed able to considerably reduce the number of significant points. Also, using the network transitivity to infer information on the dimensionality of the system’s dynamics, we observed good agreement with the expected behaviour of the corresponding stationary system. Deviations may result from the transient nature of the studied time series. Thus, we are confident that the detected areawise significant anomalies quantify the system’s dynamics and that the areawise test improves the reliability of the framework of wRNA.
For completeness, we here summarise the analysis framework as we think it is most suitable for the detection of dynamical anomalies in real-world time series. For this, we assume that the available time series x(t) is univariate and represents the dynamics of some higher-dimensional dynamical system.
• The first step in the analysis of the time series x(t) is the reconstruction of the higher-dimensional phase space of the underlying dynamical system. For this, uniform time delay embedding or derivative embedding when estimating the derivatives with discrete Legendre polynomials is used. For time delay embedding, the delay time is estimated by the first root of the autocorrelation function or the first minimum of the mutual information. For the discrete Legendre polynomials, the number of neighbours to each side taken into account for estimating the derivatives is chosen of the order of the embedding dimension.
The embedding dimension is determined with the help of the false nearest neighbour method corrected for autocorrelation effects.
• In the second step, the embedded time series is analysed with wRNA and wssRNA for varying window widths and scales, respectively. The threshold for the recurrences is chosen adaptively by fixing a recurrence rate and distances in phase space are calculated using the maximum norm. The resulting network transitivity is used to characterise the dimensionality of the system’s dynamics.
• Then, to test whether the values of the network transitivity actually show dynamical anomalies, a pointwise significance test is applied. The surrogates can, for example, be random shuffling surrogates, but any other appropriate null model may be chosen.
• Finally, to correct for intrinsic correlations of the windowed analysis for varying window widths and scales, the areawise significance test is applied with respect to a chosen null model and the confidence level of the areawise test are determined.
We recommend to use a data-adaptive null model such as iAAFT surrogates or to use a hierarchy of null models.
• Areawise significant values of the network transitivity then characterise dynam-ical anomalies in terms of significantly higher- or lower-dimensional dynamics compared to the dynamics of the null models.
In the second part of the thesis, we apply this analysis framework to palaeoclimate proxy time series. With this, we want to assess how well wRNA is suited to detect dynamical anomalies in past climate variability from data of different palaeoclimate archives. However, the framework can in principle be applied to any type of time series data.