# Ensemble-Based Correction for Anomalous Diffusion Exponent Estimation in Single-Particle Tracking
## 3 Results
### Generalization of the Correction Approach to Other a Estimation Methods  
Here we present the experimental results demonstrating the generalization of the correction approach outlined in Section 2.4. The correction is performed using Formulas (17) and (18).  
Appendix B shows how linear dependencies of Var(\(\alpha\)) on 1/T were established for several \(\mu\) values for three methods: TA-MSD (baseline), the Whittle method (based on periodograms, effective for fBm), and Cinnamon (based on machine learning but, unlike RANDI, independent of trajectory length). The investigated methods differ in their underlying approaches. The trajectory length range is selected according to the working range of the task. For subsequent use on the AnDi-2024 dataset (Section 3.4), in which the typical trajectory lengths are composed of several dozen points (with a minimum of 4 points and a maximum of 200), we selected a trajectory length range of T = 16 to T = 50 points for our detailed analysis. Given our choice of K = 4 (number of lags \(\tau\)), we specifically limited our experimental analysis to trajectory lengths starting from T = 16. This ensures that K \(\leq\) T/4, a condition under which the estimation remains stable and consistent. We observed that computational instability and increased estimation variance occur when K exceeds approximately T/4. This behavior is attributed to the decreasing number of independent segments available for reliable MSD averaging at larger lag times \(\tau\), which leads to TA-MSD estimates that are less reliable. It is important to note that the established dependency between T and Var(\(\alpha\)) within this range is expected to hold true for adjacent ranges as well, extending its applicability to both very short trajectories (e.g., from four points) and longer ones (up to several hundreds of points).  
For each method (Figure 3), we created a function for estimating Var(\(\alpha\)) from \(\mu\) and T using parametric interpolation. As a result, it is possible to correct the \(\alpha\) values of trajectories based on the method-specific variance estimates provided by the function and \(\mu\) estimates obtained using TEA-MSD.  
The consistent linear relationships across all three methods validate the generalizability of the proposed ensemble-based correction framework, demonstrating its applicability to both classical statistical methods and modern machine learning approaches for anomalous diffusion exponent estimation  
Figure 3 demonstrates the correction results across different \(\mu\) and \(\sigma^{2}\) values for trajectories with a length of 25, simulated from fractional Brownian motion processes with \(\alpha\) following a truncated normal distribution N(\(\mu\), \(\sigma^{2}\)) bounded between 0 and 2. The performance comparison includes five methods: TA-MSD (blue line), TA-MSD with correction (orange line), the Whittle method with correction (green line), uncorrected CINNAMON (red line), and CINNAMON with correction (purple line). The inclusion of the uncorrected methods allows for direct assessment of the correction effectiveness.