# Ensemble-Based Correction for Anomalous Diffusion Exponent Estimation in Single-Particle Tracking
## 2 Materials and Methods  
### The TA-MSD Method and Its Statistical Characteristics  
The mean squared displacement over a time interval \(\Delta\) is defined as the expected value of the squared particle displacement. In the two-dimensional case in the x- and y-coordinate planes, the MSD is defined as  
\[MSD(\Delta)=\mathrm{E}[(X(t+\Delta)-X(t))^{2}]+\mathrm{E}[(Y(t+\Delta)-Y(t))^{ 2}], \tag{1}\]  
where \(t\) is the time, and \(\Delta\) is the time interval over which the displacement is measured.  
For anomalous diffusion, the relationship between \(MSD\) and time \(\Delta\) is expressed as  
\[MSD(\Delta)\;\propto\;\Delta^{\alpha}, \tag{2}\]  
where \(\alpha\) is the anomalous diffusion exponent. This relationship characterizes anomalous diffusion, where \(\alpha=1\) corresponds to normal diffusion, \(\alpha<1\) to subdiffusion, and \(\alpha>1\) to superdiffusion.  
The estimation of anomalous diffusion exponents from single-particle trajectories is commonly performed using time-averaged mean square displacement (_TA-MSD_) analysis. For a two-dimensional trajectory (\(X\), \(Y\)) with time interval between observations \(\delta\)t, the TA-MSD is given by  
\[TAMSD(\tau)=\frac{1}{T-\tau}\cdot\sum_{i=1}^{T-\tau}[\left(X(i+\tau)-X(i) \right)^{2}+Y(i+\tau)-Y(i))^{2}], \tag{3}\]  
where T is the length of the particle trajectory, defined as the total number of discrete data points (i.e., time steps) over which the particle was tracked, and \(\tau\) is an integer multiple of the basic time step \(\delta\)t and represents the discrete time lag. For the method to be applicable, the condition T \(\gg\tau\) must hold. Estimation of the anomalous diffusion exponent \(\alpha\) is achieved according to Formula (2), by logarithmic transformation of TAMSD(\(\tau\)) versus \(\tau\). This yields a linear relation in logarithmic coordinates, and the slope of the regression corresponds to the estimate \(\hat{\alpha}\) of the anomalous diffusion exponent \(\alpha\).  
Previous studies [9; 11; 12; 13] have investigated the dependence of the estimation of \(\hat{\alpha}\) on the trajectory length \(T\) and shown that when estimating exponent \(\alpha\) by the TA-MSD method, the variance of the estimation is inversely proportional to the value of \(T\):  
\[Var[\hat{\alpha}]\propto 1/T \tag{4}\]To estimate the variance of \(\hat{\alpha}\) for the TA-MSD method in Formula (3), the estimate \(\hat{\alpha}\) is obtained through linear regression in a logarithmic scale:  
\[\log(\text{TAMSD}(\tau))\ \approx\ \hat{\alpha}\log(\tau)+\text{const}, \tag{5}\]  
where \(\log\) refers to a logarithm with an arbitrary base. In the two-dimensional case, since the displacements along the x- and y-coordinates are assumed to be independent, the variance of the increments is given by the sum of the variances along each direction. To estimate the variance of the exponent \(\alpha\), we use an approximate formula for the variance of the logarithm of a random variable based on the error propagation formula [26]:  
\[Var[log(Z)]=Var[Z]/(E[Z])^{2}, \tag{6}\]  
where \(Z=\text{TAMSD}(\tau)\). Based on Formula (6) and considering that for fractional Brownian motion, \(E[\Delta X(t,\ \tau)^{2}+\Delta Y(t,\ \tau)^{2}]=4D\tau^{\alpha}\) and \(Var[\Delta X(t,\ \tau)^{2}+\Delta Y(t,\ \tau)^{2}]=16D^{2}\tau^{2\alpha}\), it can be shown that the approximate relation is satisfied when \(T\gg\tau\):  
\[\text{Var}[\log(\text{TAMSD}(\tau))]\approx 1/\text{T} \tag{7}\]  
In this case, the determination takes into account that the estimation of \(\alpha\) is performed using a linear regression of the TAMSD as a function of \(\tau\) in log-log scale:  
\[Var[\hat{\alpha}]=\frac{Var[log(TAMSD(\tau))]}{\sum_{\tau=1}^{K}\Bigl{(}log( \tau)-\overline{log\tau}\Bigr{)}^{2}}=\frac{1/T}{\sum_{\tau=1}^{K}\Bigl{(}log (\tau)-\overline{log\tau}\Bigr{)}^{2}}, \tag{8}\]  
where \(K\) is the number of lags \(\tau\), and the overbar notation represents the arithmetic average.