Abstract
Anomalous diffusion behavior can be observed in many single-particle (contained in crowded environments) tracking experimental data. Numerous models can be used to describe such data. In this paper, we focus on two common processes: fractional Brownian motion (fBm) and scaled Brownian motion (sBm). We proposed novel methods for sBm anomalous diffusion parameter estimation based on the autocovariance function (ACVF). Such a function, for centered Gaussian processes, allows its unique identification. The first estimation method is based solely on theoretical calculations, and the other one additionally utilizes neural networks (NN) to achieve a more robust and well-performing estimator. Both fBm and sBm methods were compared between the theoretical estimators and the ones utilizing artificial NN. For the NN-based approaches, we used such architectures as multilayer perceptron (MLP) and long short-term memory (LSTM). Furthermore, the analysis of the additive noise influence on the estimators’ quality was conducted for NN models with and without the regularization method.
Highlights
The anomalous diffusion behavior can be observed in various environments starting from single-particle motion in crowded environments [1,2,3,4,5,6], to finance [7, 8], ecology [9], hydrology [10], biology [11, 12] as well as meteorology and geophysics [13, 14]
Fractional Brownian noise is defined as the increment process of fractional Brownian motion (fBm), namely: DBaðtÞ 1⁄4 BaðtÞ À Baðt À 1Þ; t 2 Nþ: As it was established in Sect. 2.1, fBm has stationary increments and fBn is stationary, which implies that the distribution of the process does not change in time DBaðtÞ $ Bað1Þ $ N ð0; CÞ
We explored two popular anomalous diffusion Gaussian processes used to model such behavior
Summary
The anomalous diffusion behavior can be observed in various environments starting from single-particle motion in crowded environments [1,2,3,4,5,6], to finance [7, 8], ecology [9], hydrology [10], biology [11, 12] as well as meteorology and geophysics [13, 14]. The main difficulty is fitting the model and estimating its parameters While both fBm and sBm are simple models, there are problems with reliable estimation using classic methods, especially when the measurement methods of the trajectories are inaccurate [31]. We define normal and anomalous diffusion using partial differential equations and the stochastic processes emerging from their solutions with the focus on fBm and sBm. In the third section, the estimation methods are presented, starting from the summary of the fBm identification approach (described in detail in [34]) ending with the sBm identification methodology and the description of NN architecture. The last section is a summary of the paper with propositions for further improvement of the presented methodology
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