# Ensemble-Based Correction for Anomalous Diffusion Exponent Estimation in Single-Particle Tracking
## 4 Discussion  
Ensemble-based correction methods have clear limitations. If the variance of the diffusion exponent across an ensemble rises above 0.2-0.3, the underlying distribution over [0,2] becomes progressively flattened, leading to a less distinct peak and a shape that deviates increasingly from a typical Gaussian profile. When this happens, the estimation of the true mean becomes unreliable, and using ensemble-based correction can actually make results worse instead of better. Despite this, the method is still practical, especially in ensemble analysis, in which researchers look at multiple trajectories together and can apply these correction techniques. To avoid performance issues, researchers should only apply the correction when the diffusion exponent's variance is below the 0.2-0.3 threshold. This targeted approach improves the method's overall effectiveness by making sure that it is not used in situations where it would be harmful. We observed that the correction method detailed in Sections 2.1-2.3 does not perform optimally when K (the number of lags) exceeds approximately T/4, which may limit the applicability of this method in certain scenarios. However, it is important to note that in such cases, the generalized method described in Section 2.4 remains a suitable approach. The effectiveness of the proposed method, particularly in accurately estimating the ensemble variance (\(\sigma^{2}\) total) and applying the subsequent rescaling correction, depends inherently on the number of trajectories in the ensemble (M). While the TEA-MSD component demonstrates robust performance in estimating the mean diffusion exponent even for relatively small values of M, the reliability of the variance-based correction deteriorates when M falls below a certain threshold. Although we have not explicitly determined this critical minimum value of M within the scope of this study, our results were obtained using ensemble sizes consistent with those found in benchmark datasets such as AnDi-2024, in which M is typically sufficient to ensure stable variance estimation. In experimental settings where only a limited number of trajectories is available, the advantages of our correction approach may be reduced, and its applicability could become constrained. It is also worth noting that our method demonstrates robustness with respect to variations in the diffusion coefficient D across trajectories within the ensemble. As assumed in the AnDi-2024 challenge framework, the diffusion coefficient D may follow a normal distribution across the ensemble, yet this does not hinder the accurate estimation of the anomalous diffusion exponent \(\alpha\). Our experimental results confirm that the performance of the correction framework remains consistent regardless of whether D is fixed or varies significantly across trajectories. However, we emphasize that this property is not a novel finding of our study. It has been previously discussed in the literature, specifically in [13], where it was shown that the distribution of the diffusion coefficient does not affect the accurate extraction of the anomalous exponent. We refer interested readers to that work for a more detailed theoretical analysis of this invariance. Currently, our method has been rigorously validated only on synthetic fBM data. We acknowledge that the direct application of our current method to real SPT data is hindered by the absence of ground truth anomalous diffusion parameters in real-world datasets. This is a common challenge for solutions developed within the AnDi-2024 Challenge, as they are developed and tested on simulated data. Another significant challenge with respect to real data arises from highly heterogeneous systems (e.g., motion within living cells, tissues, or organisms, where motion types can vary both between and within tracks). Heterogeneity and its detection are, in fact, the central problems that the AnDi-2024 Challenge was organized to address.