Figure 1

Real and measured trajectory. (A) The real position $\widetilde{\stackrel{⃗}{r}}\left(t\right)$ is only governed by diffusion and can be defined at any time $t$. The observed position ${\vec{r}}_i=\vec{r}\left(t_i\right)$ can be defined only at discrete times and is affected by two sources of uncertainties. (B) For slow diffusion or very short frame duration (left), the uncertainty is dominated by noise, while for large enough diffusion coefficient or integration time (right), it can be dominated by diffusion.Reuse & Permissions

Figure 2

(Color online) Example of MSD fits of a $N=1000$ points simulated trajectory with $x=1000$ $\left(D={10}^{-4}\mu {\text{m}}^2/\text{s},\Delta t=100\text{ms},\sigma =100\text{nm}\right)$. Graph (B) is a zoom of graph (A) at short time lags. The simulated MSD is represented in thin red, the theoretical MSD in plain gray, and the $\text{MSD}\pm \text{SDV}$ in dashed gray. Unweighted least-squares fits obtained with different numbers of points $p$ of the MSD curve are represented with different styles in thick blue. In addition to introducing a positive offset, the presence of large relative localization error adds a clearly visible noise to the MSD at short time lags. Compared to the linear part of the MSD $\left(4Dt\right)$, the amplitude of this noise is very large [see Fig. 3B]. This results in erroneous fits when a suboptimal number of MSD points are used for the fit. For instance, the fit using $p=2$ points (dashed) shoots up, while the fit with $p=10$ points (dashed-dotted) results in a negative $D$. Fits using too many points ($p=500$, small dashed or $p=999$, dotted line) are biased for another reason: the large relative standard deviation of the MSD at large time lags [Fig. 3A]. The optimal number of fitting points for $x=1000$ given by Eq. (30) is $p=87$, which is close to the $p=100$ value chosen for the plain blue curve.Reuse & Permissions

Figure 3

(Color online) (A) Relative standard deviation of the mean-square displacement: comparison of theory (curves) and simulations (open circles). The curves show the effect of the ratio $x={\sigma }^2/\tilde{D}\Delta t$ on the relative MSD SDV. $x$ increases from left to right in the absence of localization error; the relative SDV is minimal for small time lag and increases to reach 100% at the last time lag. In the presence of significant localization error, the SDV is larger, but the MSD is also larger (dominated by the offset $4{\sigma }^2$), effectively resulting in a smaller ratio. The dashed-dotted and small dashed curves represent the asymptotic behavior at small time lag in the absence of localization error and at large time lag in the presence of large localization error, respectively $\left(K=N-n\right)$. Note that curves for $x=0$ and 0.1 are indistinguishable. The simulation partially overlapping the $K^{-1/2}$ asymptotic behavior at large time lags corresponds to $x={10}^5$. (B) Ratio of the MSD SDV over the linear component of the MSD $\left(n\alpha \right)$ for different $x$ values ($x$ decreases from top to bottom). For large $x$ values, the MSD SDV is much larger than the linear component of the MSD, which explains why a fit using few MSD points will fail to give a good estimate of the diffusion coefficient.Reuse & Permissions

Figure 4

(Color online) Relative error on fitted parameters (weighted fit, $N=1000$ points) [Eqs. (F17) and (F19)]. Evolution of the relative errors on fitted parameters [(A) intercept $a$; (B) slope $b$] as a function of the number of MSD points used for the fit. The curves correspond to different values of the reduced localization error $x$ [$x$ increases from top to bottom in (A) and from bottom to top in (B)]. (C) and (D) Comparison between theory and simulations. Plain curves: expected relative error on fit parameters [(C) intercept $a$; (D) slope $b$]. Dashed curves: observed relative standard deviation obtained from $N_S=1000$ simulations for each value of $x=10$, 100, and 1000 [$x$ increases from top to bottom in (C) and from bottom to top in (D)].Reuse & Permissions

Figure 5

(Color online) (A) Optimal number of MSD point $p_{min}$ to obtain the minimum relative error on the weighted fit parameters $a$ (open red symbols) and $b$ (plain black symbols) for different trajectory sizes $N=10$ (triangle), 100 (circle), or 1000 (square). These values are well fitted by a power law dependence (dashed curve) [Eq. (30)]. (B) and (C) Minimum values of the relative errors on (B) the intercept $a$ and (C) slope $b$. The minimum values for a weighted fit (plain symbols and curves) or unweighted fit (empty symbols and dashed curves) are identical for a given number of trajectory points ($N=10$: triangle, $N=100$: circle, $N=1000$: square). The minimum relative error increases when the number of trajectory points $N$ decreases. Note that plain and empty symbols are practically indistinguishable.Reuse & Permissions

Figure 6

(Color online) (A) and (B) Relative error on fitted parameters (unweighted fit, $N=1000$ points) [Eq. (F23)]. Evolution of the relative errors on fitted parameters [(A) intercept $a$; (B) slope $b$] as a function of the number of MSD points used for the fit. The curves correspond to different values of reduced localization error [$x$ increases from top to bottom in (A) and from bottom to top in (B)]. (C) and (D) Comparison between theory and simulations. Plain curves: expected relative error on fit parameters [(C) intercept $a$; (D) slope $b$]. Dashed curves: observed relative standard deviation obtained from $N_S=1000$ simulations for each value of $x$ [$x$ increases from top to bottom in (C) and from bottom to top in (D)].Reuse & Permissions

Figure 7

(Color online) Comparison between weighted (dashed-dotted curves) and unweighted (plain curves) fit parameters relative errors for trajectories with $N=1000$ points [(A) intercept $a$; (B) slope $b$). Although the weighted fits yield better results at large number of fitting points $p$, both types of fits have the same minimum error on the fitted parameters, obtained for similar number of fitting points [$x$ increases from top to bottom in (A) and from bottom to top in (B)].Reuse & Permissions

Figure 8

(Color online) Relative error (unweighted fit) on the fitted diffusion coefficient $D$ [Eq. (F23)] as a function of reduced localization uncertainty $x$ for different values of the number of fitting points $p$. (A) $N=1000$; (B) $N=100$ points per trajectory. $p$ increases from bottom to top for small $x$.Reuse & Permissions