# Ensemble-Based Correction for Anomalous Diffusion Exponent Estimation in Single-Particle Tracking
## 3 Results  
### TEA-MSD Estimation Test  
In this subsection, we evaluate the effectiveness of the TEA-MSD method (Section 2.3) compared to the averaging estimates obtained from other methods. For this comparison, besides the TA-MSD method (Section 2.1), we selected the Whittle method [23] (implemented to compute a Hurst exponent estimator using maximum likelihood on the periodogram, which is effective for fBM). From among the neural network-based approaches, we selected the RANDI deep learning model [20], recognized as one of the top-performing methods in the challenge. To ensure compatibility with the published trained models of RANDI, we generated a synthetic two-dimensional AnDi-2024 dataset with trajectory lengths of 25 points.  
Ensemble trajectories were simulated from fractional Brownian motion processes, with \(\alpha\) following a normal distribution N(\(\mu\), \(\sigma^{2}\)) bounded between 0 and 2. The truncation was performed using the scipy.stats.truncnorm.rvs function in SciPy, which generates random variates only from the portion of the normal distribution within the specified bounds, using an efficient sampling algorithm that avoids simple rejection sampling. Conventional methods, including the machine learning-based approaches used in AnDi, estimate each trajectory individually. The parameter \(\mu\) can be estimated by taking the mean value of the individual estimates. By applying the TEA-MSD approach, we can directly obtain a single estimate for the entire ensemble. We experimentally verify how the effectiveness of \(\mu\) estimation methods differs for various values of \(\mu\) and \(\sigma\). The mean absolute error (MAE) metric, as adopted in the AnDi competition, is used to evaluate performance.  
For all our experiments, we adopted K = 4 (the number of lags \(\tau\)). This choice was primarily dictated by the conditions of the AnDi-2024 Challenge, on the benchmark dataset of which we evaluated our method; in this dataset, trajectories frequently consist of only several dozen points, and can be as short as four points.  
The results (Figure 1) show that for small and moderate values of \(\sigma\), TEA-MSD (red line) outperforms the other methods. In cases where \(\sigma\) is high, transforming the truncated normal distribution over the interval [0,2] into an almost uniform distribution, the effectiveness of all methods significantly decreases, making it difficult to select the best among them. Thus, the application of TEA-MSD instead of averaging individual trajectory \(\alpha\) estimates demonstrates its effectiveness in this experiment. The experiments were carried out using two distinct configurations. In the first, the diffusion coefficient D was kept constant across all trajectories in the ensemble, while in the second, D was independently assigned to each trajectory as a random value drawn from a normal distribution with high variance. Despite these differences, the results showed no significant variation between the two setups. The TEA-MSD method demonstrated a similar level of accuracy in estimating \(\mu\) in both cases, suggesting that it is not sensitive to variations in D, even when those variations are substantial.  
These two configurations were also applied in the experiments detailed in Sections 3.2 and 3.3, the consistency of the results of which further supported the conclusion that the performance of our correction framework remains stable regardless of the specific distribution or variability of the diffusion coefficient D.  
All code and experimental results are publicly available in the following repository: github.com/SophiaLavr/Ensemble-Based-Correction-for-Anomalous-Diffusion-Exponent-Estimation.