<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Lévy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss three experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises. Even if the finer details of the estimation procedures are case specific, we hope that the suggested checklist will still have been of great use as a practical guide. In Appendices A-F we describe useful fractional dynamics identification and validation methods.</p></abstract>

In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Lévy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss three experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises. Even if the finer details of the estimation procedures are case specific, we hope that the suggested checklist will still have been of great use as a practical guide. In Appendices A-F we describe useful fractional dynamics identification and validation methods.

Mathematical Models for Dynamics of Molecular Processes in Living Biological Cells. A Single Particle Tracking Approach

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely...

Mathematical Models for Dynamics of Molecular Processes in Living Biological Cells. A Single Particle Tracking Approach

Published Online: Aug 24, 2018

Page range: 5 - 41

Received: Mar 02, 2017

Accepted: Oct 26, 2017

DOI: https://doi.org/10.1515/amsil-2017-0019

Keywords
stochastic processes, fractional particle dynamics, ARFIMA time series, identification and validation algorithms

© 2018 Aleksander Weron, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mathematical Models for Dynamics of Molecular Processes in Living Biological Cells. A Single Particle Tracking Approach

Published Online: Aug 24, 2018

Page range: 5 - 41

Received: Mar 02, 2017

Accepted: Oct 26, 2017

DOI: https://doi.org/10.1515/amsil-2017-0019

Keywordsstochastic processes, fractional particle dynamics, ARFIMA time series, identification and validation algorithms

© 2018 Aleksander Weron, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
stochastic processes, fractional particle dynamics, ARFIMA time series, identification and validation algorithms