due to CCV. Shades of gray indicate measured signal $M$; a cell with strong response (marked with $X$) moves through the cluster, leading to biases in gradient estimate (blue arrow). The characteristic relaxation time for this bias is $\tau_r$ (see text). b) This leads to a link between the timescale $\tau_r$, which is a measure of the cluster’s rheology, and chemotactic accuracy ${\ensuremath{\sigma_{\mathbf{g},T}}}$ (box). c) Different re-arrangement mechanisms will depend on cluster size in different ways (see text and Appendix \[app:mechanisms\]). []{data-label="fig:timeaverage"}](rearrangement_variation_drawing){width="90mm"}

How much does time-averaging reduce error? If we average the MLE estimate $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ over a time $T$ by applying a kernel $K_T(t)$, i.e., we define ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}(t) \equiv \int_{-\infty}^{\infty} {\ensuremath{\hat{{ {\mathbf g}}}}\xspace}(t') K_T(t-t') dt'$ and ${\ensuremath{\sigma_{\mathbf{g},T}}}^2 \equiv {\ensuremath{\left\langle |{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}-{ {\mathbf g}}|^2 \right\rangle}}$, we can derive (Appendix \[app:timeaverage\]) $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times \int_{-\infty}^\infty \frac{d\omega}{2\pi} |K_T(\omega)|^2 C_{rr}(\omega) \label{eq:snrt}$$ where $C_{rr}(t'-t'') \equiv \langle \boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t')\cdot\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t'') \rangle / \langle |\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}|^2 \rangle$ is the normalized cell position-position correlation function, $C_{rr}(\omega)$ its Fourier transform, and ${\ensuremath{\sigma_{\mathbf{g},0}}}^2 = 2 {\ensuremath{\sigma_{\Delta}}}^2/\chi$ is the error in the absence of time-averaging. To derive [Eq. ]{}\[eq:snrt\], we make two approximations: 1) the cluster has a constant and isotropic shape, 