# Ensemble-Based Correction for Anomalous Diffusion Exponent Estimation in Single-Particle Tracking
## 1 Introduction  
Optical microscopy has served as a cornerstone tool for studying living cells with minimal invasiveness for centuries, and the emergence of single-molecule techniques over the past two decades has fundamentally transformed cell biology by enabling quantitative characterization of complex cellular dynamics. Among these advances, single-particle tracking (SPT) has established itself as a powerful methodology for the investigation of dynamic processes in life sciences, providing unprecedented access to single-molecule behavior within the natural contexts of living cells and allowing complete statistical characterization of biological systems [1, 2, 3]. The integration of artificial intelligence approaches in microscopy has further enhanced our ability to analyze the vast datasets generated by SPT experiments, particularly in the characterization of the anomalous diffusion behaviors commonly observed in cellular membranes [4, 5]. Anomalous diffusion, characterized by mean squared displacement (MSD) scaling relationships of the form MSD \(\propto\) t\({}^{\alpha}\) where \(\alpha\
eq 1\), represents a departure from classical Brownian motion and is quantified through ensemble-averaged MSD or time-averaged MSD (TA-MSD) measurements [6, 7, 8]. However, accurate determination of the anomalous diffusion exponent \(\alpha\) presents significantchallenges when analyzing the short trajectories typical of SPT experiments, as limited temporal sampling can lead to substantial uncertainties in parameter estimation and potential misclassification of the underlying diffusion mechanism governing molecular transport in living systems. Research on the statistical properties of the TA-MSD method for estimating anomalous diffusion exponents has primarily focused on analyzing the accuracy, systematic bias, and variance of estimates as functions of observed trajectory length. Studies such as [9; 10; 11; 12] have demonstrated that the variance of anomalous diffusion exponent estimates decreases inversely with increasing trajectory length, yet remains substantial for short time series. Additionally, the properties of time-averaged MSD and its logarithm have emerged as important subjects of analysis, with their characteristics proving critically important for accurate diffusion exponent determination [9; 12]. Particular attention has been devoted to systematic bias, which increases for short trajectory lengths and can lead to both overestimation and underestimation of the diffusion exponent [10; 11; 13]. Beyond theoretical considerations, most studies also provide practical recommendations for applying the TA-MSD method in biophysics, including analysis of the intracellular motion of telomeres, proteins, and other macromolecules [13; 14; 15]. The Anomalous Diffusion (AnDi) Challenge was organized in 2020 to objectively compare methods for the characterization of anomalous diffusion from particle trajectories [16; 17]. An open-source framework was developed to simulate trajectories for training and testing datasets modeled by five distinct diffusion processes [18]. The challenge revealed that for such diverse and complex datasets, machine learning-based methods for estimating anomalous diffusion exponents significantly outperform classical statistical approaches [19]. One of the best, easily applicable neural network approaches identified in the competition was RANDI, which is based on a two-layer long short-term memory structure [20]. The second AnDi Challenge (AnDi-2024) [21] aimed to evaluate methods for detecting and quantifying changes in single-particle motion. A key complication of this challenge was the strong heterogeneity of particle trajectories. In addition, the competition emphasized ensemble-based analyses. Here, \"ensemble\" refers to a collection of multiple particle trajectories. By analyzing multiple trajectories collectively, researchers were able to significantly improve estimation accuracy.