@article{MakaravaBenmehdiHolschneider2011,
  author    = {Makarava, Natallia and Benmehdi, Sabah and Holschneider, Matthias},
  title     = {Bayesian estimation of self-similarity exponent},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {84},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {2},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.84.021109},
  pages     = {9},
  year      = {2011},
  abstract  = {In this study we propose a Bayesian approach to the estimation of the Hurst exponent in terms of linear mixed models. Even for unevenly sampled signals and signals with gaps, our method is applicable. We test our method by using artificial fractional Brownian motion of different length and compare it with the detrended fluctuation analysis technique. The estimation of the Hurst exponent of a Rosenblatt process is shown as an example of an H-self-similar process with non-Gaussian dimensional distribution. Additionally, we perform an analysis with real data, the Dow-Jones Industrial Average closing values, and analyze its temporal variation of the Hurst exponent.},
  language  = {en}
}
@phdthesis{Makarava2012,
  author    = {Makarava, Natallia},
  title     = {Bayesian estimation of self-similarity exponent},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64099},
  school      = {Universit{\"a}t Potsdam},
  year      = {2012},
  abstract  = {Estimation of the self-similarity exponent has attracted growing interest in recent decades and became a research subject in various fields and disciplines. Real-world data exhibiting self-similar behavior and/or parametrized by self-similarity exponent (in particular Hurst exponent) have been collected in different fields ranging from finance and human sciencies to hydrologic and traffic networks. Such rich classes of possible applications obligates researchers to investigate qualitatively new methods for estimation of the self-similarity exponent as well as identification of long-range dependencies (or long memory). In this thesis I present the Bayesian estimation of the Hurst exponent. In contrast to previous methods, the Bayesian approach allows the possibility to calculate the point estimator and confidence intervals at the same time, bringing significant advantages in data-analysis as discussed in this thesis. Moreover, it is also applicable to short data and unevenly sampled data, thus broadening the range of systems where the estimation of the Hurst exponent is possible. Taking into account that one of the substantial classes of great interest in modeling is the class of Gaussian self-similar processes, this thesis considers the realizations of the processes of fractional Brownian motion and fractional Gaussian noise. Additionally, applications to real-world data, such as the data of water level of the Nile River and fixational eye movements are also discussed.},
  language  = {en}
}
@article{MakaravaBettenbuehlEngbertetal.2012,
  author    = {Makarava, Natallia and Bettenb{\"u}hl, Mario and Engbert, Ralf and Holschneider, Matthias},
  title     = {Bayesian estimation of the scaling parameter of fixational eye movements},
  series = {epl : a letters journal exploring the frontiers of physics},
  volume    = {100},
  journal   = {epl : a letters journal exploring the frontiers of physics},
  number    = {4},
  publisher = {EDP Sciences},
  address   = {Mulhouse},
  issn      = {0295-5075},
  doi       = {10.1209/0295-5075/100/40003},
  pages     = {6},
  year      = {2012},
  abstract  = {In this study we re-evaluate the estimation of the self-similarity exponent of fixational eye movements using Bayesian theory. Our analysis is based on a subsampling decomposition, which permits an analysis of the signal up to some scale factor. We demonstrate that our approach can be applied to simulated data from mathematical models of fixational eye movements to distinguish the models' properties reliably.},
  language  = {en}
}
@article{BenmehdiMakaravaBenhamidoucheetal.2011,
  author    = {Benmehdi, Sabah and Makarava, Natallia and Benhamidouche, N. and Holschneider, Matthias},
  title     = {Bayesian estimation of the self-similarity exponent of the Nile River fluctuation},
  series = {Nonlinear processes in geophysics},
  volume    = {18},
  journal   = {Nonlinear processes in geophysics},
  number    = {3},
  publisher = {Copernicus},
  address   = {G{\"o}ttingen},
  issn      = {1023-5809},
  doi       = {10.5194/npg-18-441-2011},
  pages     = {441 -- 446},
  year      = {2011},
  abstract  = {The aim of this paper is to estimate the Hurst parameter of Fractional Gaussian Noise (FGN) using Bayesian inference. We propose an estimation technique that takes into account the full correlation structure of this process. Instead of using the integrated time series and then applying an estimator for its Hurst exponent, we propose to use the noise signal directly. As an application we analyze the time series of the Nile River, where we find a posterior distribution which is compatible with previous findings. In addition, our technique provides natural error bars for the Hurst exponent.},
  language  = {en}
}
@article{MakaravaMenzThevesetal.2014,
  author    = {Makarava, Natallia and Menz, Stephan and Theves, Matthias and Huisinga, Wilhelm and Beta, Carsten and Holschneider, Matthias},
  title     = {Quantifying the degree of persistence in random amoeboid motion based on the Hurst exponent of fractional Brownian motion},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {90},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.90.042703},
  pages     = {6},
  year      = {2014},
  abstract  = {Amoebae explore their environment in a random way, unless external cues like, e. g., nutrients, bias their motion. Even in the absence of cues, however, experimental cell tracks show some degree of persistence. In this paper, we analyzed individual cell tracks in the framework of a linear mixed effects model, where each track is modeled by a fractional Brownian motion, i.e., a Gaussian process exhibiting a long-term correlation structure superposed on a linear trend. The degree of persistence was quantified by the Hurst exponent of fractional Brownian motion. Our analysis of experimental cell tracks of the amoeba Dictyostelium discoideum showed a persistent movement for the majority of tracks. Employing a sliding window approach, we estimated the variations of the Hurst exponent over time, which allowed us to identify points in time, where the correlation structure was distorted ("outliers"). Coarse graining of track data via down-sampling allowed us to identify the dependence of persistence on the spatial scale. While one would expect the (mode of the) Hurst exponent to be constant on different temporal scales due to the self-similarity property of fractional Brownian motion, we observed a trend towards stronger persistence for the down-sampled cell tracks indicating stronger persistence on larger time scales.},
  language  = {en}
}