@article{ThapaWyłomańskaSikoraetal.2021, author = {Thapa, Samudrajit and Wyłomańska, Agnieszka and Sikora, Grzegorz and Wagner, Caroline E. and Krapf, Diego and Kantz, Holger and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories}, series = {New Journal of Physics}, volume = {23}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges. ; IOP}, address = {Bad Honnef ; London}, issn = {1367-2630}, doi = {10.1088/1367-2630/abd50e}, pages = {22}, year = {2021}, abstract = {Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.}, language = {en} } @article{GajdaWylomanskaKantzetal.2018, author = {Gajda, J. and Wylomanska, Agnieszka and Kantz, Holger and Chechkin, Aleksei V. and Sikora, Grzegorz}, title = {Large deviations of time-averaged statistics for Gaussian processes}, series = {Statistics \& Probability Letters}, volume = {143}, journal = {Statistics \& Probability Letters}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-7152}, doi = {10.1016/j.spl.2018.07.013}, pages = {47 -- 55}, year = {2018}, abstract = {In this paper we study the large deviations of time averaged mean square displacement (TAMSD) for Gaussian processes. The theory of large deviations is related to the exponential decay of probabilities of large fluctuations in random systems. From the mathematical point of view a given statistics satisfies the large deviation principle, if the probability that it belongs to a certain range decreases exponentially. The TAMSD is one of the main statistics used in the problem of anomalous diffusion detection. Applying the theory of generalized chi-squared distribution and sub-gamma random variables we prove the upper bound for large deviations of TAMSD for Gaussian processes. As a special case we consider fractional Brownian motion, one of the most popular models of anomalous diffusion. Moreover, we derive the upper bound for large deviations of the estimator for the anomalous diffusion exponent. (C) 2018 Elsevier B.V. All rights reserved.}, language = {en} } @misc{ThapaWyłomańskaSikoraetal.2021, author = {Thapa, Samudrajit and Wyłomańska, Agnieszka and Sikora, Grzegorz and Wagner, Caroline E. and Krapf, Diego and Kantz, Holger and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1118}, issn = {1866-8372}, doi = {10.25932/publishup-49349}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-493494}, pages = {24}, year = {2021}, abstract = {Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.}, language = {en} } @article{BurneckiWylomanskaBeletskiietal.2012, author = {Burnecki, Krzysztof and Wylomanska, Agnieszka and Beletskii, Aleksei and Gonchar, Vsevolod and Chechkin, Aleksei V.}, title = {Recognition of stable distribution with levy index alpha close to 2}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {85}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.85.056711}, pages = {8}, year = {2012}, abstract = {We address the problem of recognizing alpha-stable Levy distribution with Levy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Levy to Gaussian fluctuation statistics.}, language = {en} } @misc{BurneckiWylomanskaChechkin2015, author = {Burnecki, Krzysztof and Wylomanska, Agnieszka and Chechkin, Aleksei V.}, title = {Discriminating between light- and heavy-tailed distributions with limit theorem}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {495}, issn = {1866-8372}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-408172}, pages = {23}, year = {2015}, abstract = {In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual and easy to implement, is to check whether the underlying law belongs to the domain of attraction of the Gaussian or non-Gaussian stable distribution by examining its rate of convergence. The method allows to discriminate between stable and various non-stable distributions. The test allows to differentiate between distributions, which appear the same according to standard Kolmogorov-Smirnov test. In particular, it helps to distinguish between stable and Student's t probability laws as well as between the stable and tempered stable, the cases which are considered in the literature as very cumbersome. Finally, we illustrate the procedure on plasma data to identify cases with so-called L-H transition.}, language = {en} }