# Determining the best number of fitting points for MSD data

- source: [ @ernst2013 Measuring a diffusion coefficient by single-particle tracking: statistical analysis of experimental mean squared displacement curves ]
- tags: #errors-nanoparticle-tracking-analysis

To measure the diffusion coefficient, we can fit the mean squared displacement curve with the following equation:

Experimentally, the accuracy of the MSD is limited, as well as the maximum

Considering more points does not necessarily improve the quality of the fit: the first few $\tau$ values are the result of the average over a large number of measurements, while the larger time delays are averaged only a couple of times.

In a setting where a single-particle is tracked over $10^5$ time-points, it is possible to divide the trajectory into different sub-sets and analyze how much the analysis changes compared to the analysis of the entire trajectory (presumably the most accurate one).

The figure above shows that using tracks of either $N=100$ or $N=1000$ points, the error in the determination of $\tilde{\textrm{D}}$ goes down initially, but after 4 or 5 fitting points, it increases dramatically. This means that the determination of the diffusion coefficient is best when using 4 or 5 points of $\tau$ and not the entire dataset.

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