This paper explores the capabilities of machine learning for the probabilistic forecasting of fractional Brownian motion (fBm). The focus is on predicting the probability of the value of an fBm time series exceeding a certain threshold after a specific number of time steps, given only the knowledge of its Hurst exponent. The study aims to determine if the self-similarity property is preserved in a forecasting time series and which machine learning algorithms are the most effective. Two types of forecasting methods are investigated: methods with a predefined distribution shape and those without. The results show that the self-similar properties of the fBm time series can be reliably reproduced in the continuations of the time series predicted by machine learning methods. The study also provides an experimental comparison of various probabilistic forecasting methods and their potential applications in the analysis and modeling of fractal time series.

This study investigates the robustness of mean-based and median-based methods for estimating the anomalous diffusion (AnDi) exponent in particle trajectories. AnDi, observed in heterogeneous environments, presents challenges due to the presence of outliers that inevitably arise during the segmentation of trajectories into homogeneous sections. We focus on short trajectories modeled by two-dimensional fractional Brownian motion and explore alternative strategies for robust estimation. Specifically, we propose an adaptation of the time-averaged mean square displacement method that uses the median (or quantile) instead of the mean, as well as a mean-based approach that trims extreme squared displacements. Experiments conducted on synthetic datasets with controlled outlier characteristics, as well as on segmented trajectories from a benchmark dataset of the 2nd AnDi challenge, reveal that although median- and quantile-based methods effectively suppress high-variance outliers, they incur larger estimation errors compared to mean-based methods. In contrast, a mean-based method that trims a fixed percentage of the largest squared displacements achieves an improved balance between robustness and accuracy. Furthermore, our results suggest a critical strategy for segmenting heterogeneous trajectories: under conditions of uncertainty, ambiguous sections should be associated with segments exhibiting higher variance. This approach minimizes the influence of outliers on diffusion parameter estimation.

The analysis of anomalous diffusion characteristics within single-particle tracking data is a key problem in several applied-science domains, including biosignal processing, bioinformatics, and biotechnology. This task becomes particularly challenging in the presence of short trajectories, localization errors, and non-ergodicity, features that are common in real experimental data. To address these limitations, this work proposes an approach that improves the robustness and accuracy of estimating the anomalous diffusion exponent α, even for very short trajectories of up to 10 points. The approach includes an ensemble-based variance estimation of the exponent α, along with a bias correction based on time–ensemble averaged mean squared displacement, which reduces the systematic bias. These components integrate well into neural network architectures and are suitable for analyzing experimental trajectories in biotechnology and bioprocess engineering applications.

Neural networks have shown excellent performance in the task of estimating the Hurst exponent, but their primary drawback is a lack of interpretability. We introduce a specialized neural network layer to address this. This layer is an implementation of the second-order generalized Hurst exponent method, designed as a non-trainable, differentiable layer compatible with any deep learning framework. We have implemented it for both TensorFlow and PyTorch. The layer also seamlessly handles missing values, which facilitates its integration into complex neural network architectures designed for analyzing heterogeneous trajectories. Our differentiable Hurst exponent estimation layer offers simplicity in deployment, as it eliminates the need for training and is ready to process time series of any length. While the convenience of not requiring training and its flexibility with series length are clear advantages, the key novelty of our work is the ability to provide interpretability by successfully translating a classical statistical method into a core deep learning component.