Calculating diffusion coefficient from mean squared displacement data

To extract the diffusion coefficient from a nanoparticle tracking analysis experiment, especially when looking at diffusive brownian motion, we can calculate $\left<\Delta x^2\right>$ from the tracks themselves as:[@catipovic2013Improving the quantification of Brownian motion]

$$⟨\mathrm{\Delta }{x}^2(\tau )⟩=\frac{\sum _{i=1}^{N_i}(x({\tau }_i+\tau )-x({\tau }_i{)}^2)}{N_i}$$

In this case, instead of looking at the distribution (see: calculating diffusion coefficient from jump statistics), we can directly fit the temporal correlation, assuming the following relation:

$$⟨\mathrm{\Delta }{x}^2⟩(\tau )=2D\tau$$