Stephan Zschiegner

Bio: Stephan Zschiegner is an academic researcher from University of Giessen. The author has contributed to research in topics: Multifractal system & Knudsen number. The author has an hindex of 5, co-authored 10 publications receiving 4829 citations. Previous affiliations of Stephan Zschiegner include Leipzig University & University of Marburg.

Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series

Abstract: We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

Multifractal detrended fluctuation analysis of nonstationary time series

Abstract: We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series to those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima (WTMM) method, and show that the results are equivalent.

Multifractality of river runoff and precipitation: Comparison of fluctuation analysis and wavelet methods

Abstract: We study the multifractal temporal scaling properties of river discharge and precipitation records. We compare the results for the multifractal detrended fluctuation analysis method with the results for the wavelet-transform modulus maxima technique and obtain agreement within the error margins. In contrast to previous studies, we find non-universal behaviour: on long time scales, above a crossover time scale of several weeks, the runoff records are described by fluctuation exponents varying from river to river in a wide range. Similar variations are observed for the precipitation records which exhibit weaker, but still significant multifractality. For all runoff records the type of multifractality is consistent with a modified version of the binomial multifractal model, while several precipitation records seem to require different models.

Multifractal detrended $uctuation analysis of nonstationary time series

Abstract: We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended $uctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shu6ed series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent. c

Lambert diffusion in porous media in the Knudsen regime: equivalence of self-diffusion and transport diffusion.

Abstract: We study molecular diffusion in nanopores with different types of roughness under the exclusion of mutual molecular collisions, i.e., in the so-called Knudsen regime. We show that the diffusion problem can be mapped onto Levy walks and discuss the roughness dependence of the diffusion coefficients ${D}_{s}$ and ${D}_{t}$ of self- and transport diffusion, respectively. While diffusion is normal in $d=3$, diffusion is anomalous in $d=2$ with ${D}_{s}\ensuremath{\sim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}t$ and ${D}_{t}\ensuremath{\sim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L$, where $t$ and $L$ are time and system size, respectively. Both diffusion coefficients decrease significantly when the roughness is enhanced, in remarkable disagreement with earlier findings.

Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

Abstract: Physiological and behavioural phenomena are often complex, characterized by variations in time series. Variations in time series reflect how these phenomena organize into coherent structures by interactions that span multiple scales in both time and space. The present tutorial is an introduction to multifractal analyses that can identify these scale invariant interactions within time series by its multifractal spectrum. The multifractal spectrum can be estimated directly from scale-dependent measurements or from its q-order statistics. The tutorial emphasizes the most common scale-dependent measurements defined by the wavelet transforms and the detrended fluctuation analyses. The tutorial also emphasizes common features of all multifractal analyses, like the choice of linear regression method, scaling range and elimination of spurious singularities, which are important for a robust estimation of the multifractal spectrum. The tutorial ends with two brief examples where multifractal analyses are employed to time series from multifractal models and the complex phenomena of cognitive performance. References to available software for multifractal analyses are included at the end of the tutorial. The main aim of the tutorial is to give the reader an introduction to multifractal analyses without the extensive technicalities typically provided in mathematical journals.

Multifractal detrended cross-correlation analysis for two nonstationary signals.

Abstract: We propose a method called multifractal detrended cross-correlation analysis to investigate the multifractal behaviors in the power-law cross-correlations between two time series or higher-dimensional quantities recorded simultaneously, which can be applied to diverse complex systems such as turbulence, finance, ecology, physiology, geophysics, and so on. The method is validated with cross-correlated one- and two-dimensional binomial measures and multifractal random walks. As an example, we illustrate the method by analyzing two financial time series.

Spike avalanches in vivo suggest a driven, slightly subcritical brain state

Abstract: In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore we analyzed highly parallel spike recordings from awake rats and monkeys, anaesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a melange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.

Modelling Financial Time Series

Abstract: Financial time series, in general, exhibit average behaviour at “long” time scales and stochastic behaviour at ‘short” time scales As in statistical physics, the two have to be modelled using different approaches — deterministic for trends and probabilistic for fluctuations about the trend In this talk, we will describe a new wavelet based approach to separate the trend from the fluctuations in a time series A deterministic (non-linear regression) model is then constructed for the trend using genetic algorithm We thereby obtain an explicit analytic model to describe dynamics of the trend Further the model is used to make predictions of the trend We also study statistical and scaling properties of the fluctuations The fluctuations have non-Gaussian probability distribution function and show multi-scaling behaviour Thus, our work results in a comprehensive model of trends and fluctuations of a financial time series