TY  - GEN
A1  - Ślęzak, Jakub
A1  - Metzler, Ralf
A1  - Magdziarz, Marcin
T1  - Codifference can detect ergodicity breaking and non-Gaussianity
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2  - We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 748 
KW  - diffusion
KW  - anomalous diffusion
KW  - stochastic time series
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436178
IS  - 748
ER  - 
TY  - GEN
A1  - Ślęzak, Jakub
A1  - Burnecki, Krzysztof
A1  - Metzler, Ralf
T1  - Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2  - Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as ‘superstatistics’ or ‘diffusing diffusivity’. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 765 
KW  - diffusion
KW  - Langevin equation
KW  - Brownian yet non-Gaussian diffusion
KW  - diffusing diffusivity
KW  - superstatistics
KW  - autoregressive models
KW  - time series analysis
KW  - codifference
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-437923
SN  - 1866-8372
IS  - 765
ER  - 
TY  - GEN
A1  - Sposini, Vittoria
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
T1  - Single-trajectory spectral analysis of scaled Brownian motion
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2  - Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 753 
KW  - diffusion
KW  - anomalous diffusion
KW  - power spectral analysis
KW  - single trajectory analysis
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436522
SN  - 1866-8372
IS  - 753
ER  -