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

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.

Detection of Anomalous Diffusion with Deep Residual Networks

“…Following Ref. [ 77 ], we decided to check if the autocorrelation function taken as additional input improves the accuracy of the model. We combined the raw trajectories with their autocorrelations calculated at lags 8, 16, and 24 into a single tensor structure and used it as input to the model.…”

Neural network-based anomalous diffusion parameter estimation approaches for Gaussian processes

“…The architecture of the proposed neural network will be simple and similar to one in [34]. However, one significant change will be done that allows for variable input vector size.…”

“…The RNN architecture is especially needed because with multilayer perceptron (MLP) [45] (used in [34]) the model's error does not plateau (Fig. 3)-with increasing e T the error is decreasing with log-linear rate (from certain e T ).…”

“…While in [34] MLP NN architecture was used for intelligent fBm identification, we will attach such architecture into plots.…”

“…First, 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. In the following section, the methods' evaluation and comparison occur.…”