Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Cover of the book

2022 | OriginalPaper | Chapter

Time-Averaged Statistics-Based Methods for Anomalous Diffusive Exponent Estimation of Fractional Brownian Motion

Authors : Katarzyna Maraj, Agnieszka Wyłomańska

Published in: Nonstationary Systems: Theory and Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The anomalous diffusive processes are widely discussed in the research papers. In contrast to the diffusive processes with the linear second moment, they are characterized by the nonlinear variance. The anomalous diffusive processes exhibit many interesting properties which are not adequate to the diffusive systems and thus they have found various applications including, among others, biology, physics and environmental engineering. Very useful in the testing of the anomalous diffusive behavior are the time-averaged statistics which are based on the sample trajectory of the given process. Similar as the empirical second moment, they exhibit different behavior for anomalous diffusive and diffusive processes. Thus, they can be very effective tools for the estimation and statistical testing of the anomalous diffusive behavior. One can find different theoretical anomalous diffusive processes. One of the classical examples is the fractional Brownian motion. In this chapter, we demonstrate how the selected time-averaged statistics behave for the fractional Brownian motion and show how they can be applied in order to estimate the Hurst exponent (responsible for the anomalous diffusive behavior). By using Monte Carlo simulations, we compare the effectiveness of the presented estimation methods for the considered stochastic process. The described methodology can be applied to any other anomalous diffusive processes.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
next chapter First-Order Integer Valued AR Processes with Zero-Inflated Innovations
Literature
1.
Alessio, E., Carbone, A., Castelli, G., Frappietro, V.: Second-order moving average and scaling of stochastic time series. Phys. Condens. Matter 27, 197–200 (2002)
2.
Andreanov, A., Grebenkov, D.S.: Time-averaged MSD of Brownian motion. J. Stat. Mech. Theor. Exp. 2012(07), P07001 (2012)CrossRef
3.
Arcizet, D., Meier, B., Sackmann, E., Rädler, J.O., Heinrich, D.: Temporal analysis of active and passive transport in living cells. Phys. Rev. Lett. 101, 248103 (2008)CrossRef
4.
Bashan, A., Bartsch, R., Kantelhardt, J.W., Havlin, S.: Comparison of detrending methods for fluctuation analysis. Physica A 387, 5080–5090 (2008)CrossRef
5.
Bertseva, E., Grebenkov, D., Schmidhauser, P., Gribkova, S., Jeney, S., Forró, L.: Optical trapping microrheology in cultured human cells. Eur. Phys. J. E Soft Matter 35, 63 (2012)CrossRef
6.
7.
Bressloff, P.C., Newby, J.M.: Stochastic models of intracellular transport. Rev. Mod. Phys. 85, 135–196 (2013)CrossRef
8.
Bronstein, I., et al.: Transient anomalous diffusion of telomeres in the nucleus of mammalian cells. Phys. Rev. Lett. 103, 018102 (2009)CrossRef
9.
10.
Burnecki, K., Kepten, E., Garini, Y., Sikora, G.: Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - an alternative approach. Sci. Rep. 5, 11306 (2015)CrossRef
11.
Carbone, A., Kiyono, K.: Detrending moving average algorithm: Frequency response and scaling performances. Phys. Rev. E 93, 063309 (2016)CrossRef
12.
Chen, Z., Hu, K., Carpena, P., Bernaola-Galvan, P., Stanley, H.E., Ivanov, P.C.: Effect of nonlinear filters on detrended fluctuation analysis. Phys. Rev. E 71, 011104 (2005)CrossRef
13.
Chen, Z., Ivanov, P.C., Hu, K., Stanley, H.E.: Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E 65, 041107 (2002)CrossRef
14.
Coeurjolly, J.: Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. J. Stat. Softw. 05, 1–53 (2000)CrossRef
15.
Cohen, L.: Generalization of the Wiener-Khinchin theorem. IEEE Sig. Process. Lett. 5, 292–294 (1998)CrossRef
16.
Dieker, T.: Simulation of fractional Brownian motion. M.Sc. thesis, Vrije Universiteit Amsterdam (2013)
17.
Fanelli, D., McKane, A.J.: Diffusion in a crowded environment. Phys. Rev. E 82, 021113 (2010)CrossRef
18.
Gajda, J., Wyłomańska, A., Kantz, H., Chechkin, A., Sikora, G.: Large deviations of time-averaged statistics for gaussian processes. Statist. Probab. Lett. 143, 47–55 (2018)MathSciNetMATHCrossRef
19.
Gal, N., Lechtman-Goldstein, D., Weihs, D.: Particle tracking in living cells: a review of the mean square displacement method and beyond. Rheol. Acta 5, 425–443 (2013)CrossRef
20.
Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006)CrossRef
21.
Grebenkov, D.S.: Probability distribution of the time-averaged mean-square displacement of a gaussian process. Phys. Rev. E 84, 031124 (2011)CrossRef
22.
Grebenkov, D.S.: Optimal and suboptimal quadratic forms for noncentered gaussian processes. Phys. Rev. E 88, 032140 (2013)CrossRef
23.
Guigas, G., Kalla, C., Weiss, M.: Probing the nanoscale viscoelasticity of intracellular fluids in living cells. Biophys. J . 93, 316 (2007)CrossRef
24.
Hellmann, M., Klafter, J., Heermann, D.W., Weiss, M.: Challenges in determining anomalous diffusion in crowded fluids. J. Phys. Condens. Matter 23(23), 234113 (2011)CrossRef
25.
Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Stanley, H.E.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64, 011114 (2001)CrossRef
26.
Jeon, J.-H., Metzler, R.: Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E 81, 021103 (2010)MathSciNetCrossRef
27.
Jeon, J.H., et al.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)CrossRef
28.
Kepten, E., Bronshtein, I., Garini, Y.: Ergodicity convergence test suggests telomere motion obeys fractional dynamics. Phys. Rev. E 83, 041919 (2011)CrossRef
29.
Kepten, E., Weron, A., Sikora, G., Burnecki, K., Garini, Y.: Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments. PLOS ONE 10(2), 1–10 (2015)CrossRef
30.
Kumar, A., Gajda, J., Wyłomańska, A.: Fractional Brownian motion delayed by tempered and inverse tempered stable subordinators. Methodol. Comput. Appl. Probab. 21, 185–202 (2019)MathSciNetMATHCrossRef
31.
Kumar, A., Wyłomańska, A., Połoczański, R., Sundar, S.: Fractional Brownian motion time-changed by gamma and inverse gamma process. Phys. A 468, 648–667 (2017)MathSciNetMATHCrossRef
32.
Lanoiselée, Y., Sikora, G., Grzesiek, A., Grebenkov, D.S., Wyłomańska, A.: Optimal parameters for anomalous-diffusion-exponent estimation from noisy data. Phys. Rev. E 98, 062139 (2018)CrossRef
33.
Ma, Q.D.Y., Bartsch, R.P., Bernaola-Galván, P., Yoneyama, M., Ivanov, P.C.: Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis. Phys. Rev. E 81, 031101 (2010)CrossRef
34.
Magdziarz, M., Weron, A.: Anomalous diffusion: testing ergodicity breaking in experimental data. Phys. Rev. E: Stat., Nonlin, Soft Matter Phys. 84, 051138 (2011)CrossRef
35.
36.
Masaaki, K., Chun, M.T.: Fractional Brownian motions in financial models and their Monte Carlo simulation. In: IntechOpen (2014)
37.
Meroz, Y., Sokolov, I.M., Klafter, J.: Test for determining a subdiffusive model in ergodic systems from single trajectories. Phys. Rev. Lett. 110, 090601 (2013)CrossRef
38.
Metzler, R., et al.: Analysis of single particle trajectories: from normal to anomalous diffusion. Acta Physica Polonica Ser. B B40(5), 1315–1330 (2009)
39.
Movahed, M.S., Jafari, G.R., Ghasemi, F., Rahvar, S., Tabar, M.R.R.: Multifractal detrended fluctuation analysis of sunspot time series. J. Stat. Mech. Theor. Exp. 2006(02), P02003–P02003 (2006)MATHCrossRef
40.
Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689 (1994)CrossRef
41.
Sackmann, E., Keber, F., Heinrich, D.: Physics of cellular movements. Ann. Rev. Condens. Matter Phys. 1(1), 257–276 (2010)CrossRef
42.
Sikora, G.: Statistical test for fractional Brownian motion based on detrending moving average algorithm. Chaos, Solitons Fractals 116, 54–62 (2018)MathSciNetMATHCrossRef
43.
Sikora, G., Burnecki, K., Wyłomańska, A.: Mean-squared-displacement statistical test for fractional Brownian motion. Phys. Rev. E 95, 032110 (2017)CrossRef
44.
Sikora, G., Teuerle, M., Wyłomańska, A., Grebenkov, D.: Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement. Phys. Rev. E 96, 022132 (2017)CrossRef
45.
Subba Rao, T.: Time Series Analysis: Methods and Applications. Elsevier (2012)
46.
Szymanski, J., Weiss, M.: Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103, 038102 (2009)CrossRef
47.
Tejedor, V., et al.: Quantitative analysis of single particle trajectories: mean maximal excursion method. Biophys. J . 98, 1364–1372 (2010)CrossRef
48.
Tolić-Nørrelykke, I.M., Munteanu, E.-L., Thon, G., Oddershede, L., Berg-Sørensen, K.: Anomalous diffusion in living yeast cells. Phys. Rev. Lett. 93, 078102 (2004)
49.
Türkcan, S., Masson, J.-B.: Bayesian decision tree for the classification of the mode of motion in single-molecule trajectories. PLoS ONE 8(12), 1–14 (2013)CrossRef
50.
Wilhelm, C.: Out-of-equilibrium microrheology inside living cells. Phys. Rev. Lett. 101, 028101 (2008)CrossRef
51.
Xi, C., Zhang, S., Xiong, G., Zhao, H.: A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum. Phys. A 454, 34–50 (2016)CrossRef
52.
Xu, L., Ivanov, P.C., Hu, K., Chen, Z., Carbone, A., Stanley, H.E.: Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques. Phys. Rev. E 71, 051101 (2005)CrossRef
Metadata
Title
Time-Averaged Statistics-Based Methods for Anomalous Diffusive Exponent Estimation of Fractional Brownian Motion
Authors
Katarzyna Maraj
Agnieszka Wyłomańska
Copyright Year
2022
Book
Nonstationary Systems: Theory and Applications

Print ISBN: 978-3-030-82191-3

Electronic ISBN: 978-3-030-82110-4

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-030-82110-4_1