Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks.

Dawid Szarek,Ireneusz Jabłoński,Michał Balcerek,Agnieszka Wyłomańska,Grzegorz Sikora

doi:10.3390/e22111322

Dawid Szarek, Ireneusz Jabłoński + Show 3 more

Open Access

https://doi.org/10.3390/e22111322

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Abstract

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.

Highlights

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior [1,2]
We considered the fractional Brownian motion (FBM) as an exemplary model with fractional dynamics; that is, the Gaussian process with stationary increments and the so-called self-similar property parametrized by the
To see the differences between the behavior of the trajectories corresponding to different anomalous types, in Figure 1, we demonstrate the exemplary trajectories of FBM for H = 0.3, H = 0.5, and H = 0.7

Summary

Introduction

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior [1,2]. Information retrieval from the sample ACVF vector is performed here with a learning-based scheme operating on the most informative lags, i.e., a feedforward neural network (FNN) [44] is designed and applied for solving the regression task This approach has been proven to be an effective encoder of the properties inherited in complex data [45,46,47]. The goal is to preliminarily assess (using computer simulations) the predictive properties of an FNN for the estimation of the anomalous diffusion exponent based on a short data set This exercise provides evidence of the performance of the simple version of the neural network (NN). The last section concludes the paper and presents a future study

Fractional Brownian Motion

ACVF-Based Methods for the Estimation of the Hurst Exponent

ACVF and NN-Based Methods for the Estimation of the Hurst Exponent

Simulation Study

H Intervals

Summary and Conclusions