Anomalous dynamics of complex physical and biological systems - stochastic modeling and statistical identification

Grant no.: NCN 2012/06/A/ST1/00258
Funding agency:
National Science Centre (NCN), Poland
Funding scheme: Maestro
Funding period: 18.04.2013-17.04.2018 (5 years)
1 759 000 PLN
Title in Polish: Anomalna dynamika złożonych systemów fizycznych i biologicznych - modelowanie stochastyczne i statystyczna identyfikacja

Research team
Principal Investigator (Kierownik)
  • Aleksander Weron - leader and coordinator of the project
Senior Investigators (Główni wykonawcy)
  • Marcin Magdziarz
  • Andrzej Fuliński
  • Krzysztof Burnecki
Investigators (Wykonawcy)
  • Janusz Gajda
  • Tomasz Żórawik
  • Grzegorz Sikora
  • Marek Teuerle
  • Jakub Ślęzak
  • Agnieszka Wyłomańska
  • Michał Balcerek
  • Hanna Loch-Olszewska
  • Karol Niczyj
  • Janusz Szwabiński
Collaborators (Współpracownicy)
  • Davide Calebiro
  • Yuval Garini
  • Denis Grebenkov
  • Eldad Kepten
  • Diego Krapf
  • Carlo Manzo
  • Aleksander A. Stanislavski
  • Titivat Sungkaworn
There is no applied mathematics in form of a ready doctrine.
It originates in the contact of mathematical thought with
the surrounding world, but only when mathematical
spirit and the matter are in a flexible state.

Hugo Steinhaus (1987-1972)

Aims and scope

The main goal of this research project is the analysis of anomalous diffusion processes. We plan to build a rigorous mathematical theory of fractional dynamics observed in molecular biology of complex systems and to propose effective statistical algorithms for the analysis of experimental data in nanoscale. The project consists of three parts. The first two parts have theoretical character, the third one is devoted to the statistical analysis of experimental data. The obtained results shall be published in respected scientific journals from ISI Master Journal List and will be presented on the international scientific conferences.

  • Analysis of Levy walks and correlated continuous-time random walks
  • FARIMA (ARFIMA) time series in the modeling of anomalous diffusion processes
  • Single particle tracking experiments - statistical identification of anomalous diffusion complex systems and their properties.

Peer-reviewed articles in JCR-listed journals

  • A.A.Stanislavsky, K.Burnecki, J.Janczura, K.Niczyj, A.Weron, "Solar X-ray variability in terms of a fractional heteroskedastic time series model", Monthly Notices of the Royal Astronomical Society, Volume 485, Issue 3, May 2019, Pages 3970–3980, (doi:10.1093/mnras/stz656).
  • A.Kumar, J.Gajda, A.Wylomanska (2019), "Stable Levy Process Delayed by Tempered Stable Subordinator", Statistics and Probability Letters 145, 284-292 (doi: 10.1016/j.spl.2018.09.008).
  • M.Balcerek, H.Loch-Olszewska, J.A.Torreno-Pina, M.F.Garcia-Parajo, A.Weron, C.Manzo, and K.Burnecki (2019) "Inhomogeneous membrane receptor diffusion explained by a fractional heteroscedastic time series model", Physical Chemistry Chemical Physics 21, 3114-3121 (doi: 10.1039/c8cp06781c).
  • A. Maćkała, M.Magdziarz (2019), "Statistical analysis of superstatistical fractional Brownian motion and applications", Physical Review E 99, 012143.
  • K.Burnecki, G.Sikora, A.Weron, M.M.Tamkun, and D.Krapf (2019), "Identifying diffusive motions in single-particle trajectorieson the plasma membrane via fractional time series models", Physical Review E (doi: 10.1103/PhysRevE.99.012101).

  • 2018
  • G. Sikora (2018), "Statistical test for fractional Brownian motion based on detrending moving average algorithm", Chaos, Solitons & Fractals 116, 54-62.
  • J. Gajda, A. Wylomanska, H. Kantz, A. Chechkin, G.Sikora (2018), "Large deviations of time-averaged statistics for Gaussian processes", Statistics & Probability Letters 143, 47-55 (doi: 10.1016/j.spl.2018.07.013).
  • J. Gajda, G. Bartnicki, K. Burnecki (2018) "Modeling of water usage by means of ARFIMA-GARCH processes", Physica A 512, 644-657.
  • H.Loch-Olszewska, J.Szwabiński (2018) "Detection of epsilon-ergodicity breaking in experimental data - a study of the dynamical functional sensibility", Journal of Chemical Physics , 148, 204105.

  • 2017
  • K.Burnecki, G.Sikora (2017) "Identification and validation of stable ARFIMA processes with application to UMTS data", Chaos, Solitons & Fractals, 102, 456-466.
  • A.Fuliński (2017) "Fractional Brownian motions: memory, diffusion velocity, and correlation functions", J. Phys. A: Math. Theor., 50, 054002.
  • M.Magdziarz, M.Teuerle (2017) "Fractional diffusion equation with distributed-order material derivative. Stochastic foundations", J. Phys. A: Math. Theor., 50, 184005.
  • M.Magdziarz, T.Żórawik, (2017) "Method of calculating densities for isotropic ballistic Lévy walks", Communications in Nonlinear Science and Numerical Simulation, 48, 462-473.
  • M.Magdziarz, T.Żórawik, (2017) "Aging ballistic Levy walks", Physical Review E, 95, 022126.
  • G.Sikora, K.Burnecki, A.Wyłomańska (2017) "Mean-squared-displacement statistical test for fractional Brownian motion", Physical Review E, 95, 032110.
  • G.Sikora, E.Kepten, A.Weron, M.Balcerek, K.Burnecki (2017) "An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics", Physical Chemistry Chemical Physics, 2017, 19, 26566-26581.
  • G.Sikora, M.Teuerle, A.Wyłomańska, D.S.Grebenkov (2017) "Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement", Physical Review E, 96, 022132.
  • G.Sikora, A.Wyłomańska, J.Gajda, L.Sole, E.J.Akin, M.M.Tamkun, D.Krapf (2017) "Elucidating distinct ion channel populations on the surface of hippocampal neurons via single-particle tracking recurrence analysis", Physical Review E, 96, 062404.
  • T.Sungkaworn, ML.Jobin, K.Burnecki, A.Weron, M.J.Lohse, D.Calebiro (2017) "Single-molecule imaging reveals receptor-G protein interactions at cell surface hot spots", Nature 550, 543-547.
  • J.Ślęzak (2017) "Asymptotic behaviour of time averages for non-ergodic Gaussian processes", Annals of Physics, 383, 285-311.
  • A.Weron, K.Burnecki, E.J.Akin, L.Sole, M.Balcerek, M.M.Tamkun, D.Krapf (2017) "Ergodicity breaking on the neuronal surface emerges from random switching between diffusive states", Scientific Reports 7, 5404.
  • A.Wyłomańska, A. Kumar, R.Połoczański, S.Sundar (2017), "Fractional Brownian motion time-changed by gamma and inverse gamma process", Physica A, 468, 648-667.

  • 2016
  • H.Loch-Olszewska, J.Janczura, A.Weron (2016) "Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes", Physical Review E, 93, 043317.
  • H.Loch-Olszewska, G.Sikora, J.Janczura, A.Weron (2016) "Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length", Physical Review E, 94, 052136.
  • M.Magdziarz, T.Żórawik, (2016) "Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times and space-time-dependent coefficients", Proc. Amer.Math.Soc., 144, 1767-1778.
  • M.Magdziarz, T.Żórawik, (2016) "Densities of scaling limits of coupled continuous time random walks", Fractional Calculus and Applied Analysis, 19, 1488-1506.
  • M.Magdziarz, T.Żórawik, (2016) "Explicit densities of multidimensional ballistic Levy walks", Physical Review E, 94, 022130.
  • A.Wyłomańska, A.Kumar, R.Połoczański, P.Vellaisamy (2016) "Inverse Gaussian and its inverse process as the subordinators of fractional Brownian motion", Phys. Rev. E, 94, 042128.

  • 2015
  • M.Annunziato, A.Borzi, M.Magdziarz, A.Weron (2015) "A fractional Fokker-Planck control framework for subdiffusion processes", Optimal Control Applications and Methods, DOI: 10.1002/oca.2168.
  • K.Burnecki, E.Kepten, Y.Garini, G.Sikora, A.Weron (2015) "Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach", Scientific Reports, 5:11306, DOI: 10.1038/srep11306.
  • K.Burnecki, A.Wyłomańska, A.Chechkin (2015) "Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem", PLoS ONE, 10(12): e0145604. doi:10.1371/journal.pone.0145604.
  • J.Gajda, A.Wyłomańska (2015) "Time-changed Ornstein-Uhlenbeck process", J. Phys. A: Math. Theor., 48 135004.
  • J.Janczura, A.Weron (2015) "Ergodicity testing for anomalous diffusion: Small sample statistics", The Journal of Chemical Physics 142, 144103.
  • E.Kepten, A.Weron, I.Bronstein, K.Burnecki, Y.Garini (2015) "Uniform Contraction-Expansion Description of Relative Centromere and Telomere Motion", Biophysical Journal 109, 1454-1462.
  • E.Kepten, A.Weron, K.Burnecki, G.Sikora, Y.Garini (2015) "Guidelines for the Fitting of Anomalous Diffusion Mean Square Displacement Graphs from Single Particle Tracking Experiments", PLOS ONE, DOI:10.1371/ journal.pone. 0117722.
  • M.Magdziarz, M.Teuerle (2015) "Asymptotic properties and numerical simulation of multidimensional Lévy walks", Communications in Nonlinear Science and Numerical Simulations, 20(2), 489-505.
  • M.Magdziarz, H.P. Scheffler, P. Straka, P. Zebrowski (2015) "Limit theorems and governing equations for Levy walks", Stoch. Proc. Appl., 125, 4021-4038.
  • A.Stanislavsky, K.Weron, A.Weron (2015) "Anomalous diffusion approach to non-exponential relaxation in complex physical systems", Communications in Nonlinear Science and Numerical Simulation.
  • J.Slezak, S.Drobczyński (2015), "Time-series methods in analysis of the optical tweezers recordings", Applied Optics, 54, 7106-7114.
  • J.Ślęzak, A.Weron (2015) "From physical linear systems to discrete-time series. A guide for analysis of the sampled experimental data", Phys Rev E, 91, 053302.

  • 2014
  • K.Burnecki, A.Weron (2014) "Algorithms for testing of fractional dynamics: a practical guide to ARFIMA modelling", J. Stat. Mech., P10036.
  • J.Gajda, M.Magdziarz (2014) "Large deviations for subordinated Brownian motion and applications", Statistics & Probability Letters, 88, 149-156.
  • M.Magdziarz, J.Gajda, T.Żórawik (2014) "Comment on Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion", Journal of Statistical Physics, 154, 1241-1250.
  • A.Stanislavsky, K.Weron, A.Weron (2014) "Anomalous diffusion with transient subordinators: A link to compound relaxation laws", Journal of Chemical Physics, 140, 054113.
  • J.Ślęzak, S.Drobczyński, K.Weron, J.Masajda (2014) "Moving average process underlying the holographic-optical-tweezers experiments", Applied Optics, Vol. 53, Issue 10, pp. B254-B258, doi: 10.1364/AO.53.00B254.
  • J.Ślęzak, M.Magdziarz (2014) "Rotational invariance of stochastic processes with application to fractional dynamics", J. Stat. Mech., P10028.
Developed software components
Calibration surfaces
P-variation testing