# Ripley’s *L*(*r*) − *r* function is invariant to random subsampling.

(a) Probability map representation of a PALM image of *β*2–adrenergic receptor molecules labeled with mEos2 on the plasma membrane of HeLa cells, post agonist addition. Density: 650 molecules/*μm*^{−2}. (b) *L*(*r*) − *r* functions for the true and subsampled points, estimates for the latter obtained from both simulations and the analytical method presented. Continuous green: Ripley *L*(*r*) − *r* function *L*_{true}(*r*) − *r* corresponding to the points in (a). Orange: mean and 2*σ* bounds of *L*(*r*) − *r* functions corresponding to 10000 realizations of random sampling 50% of the points in (a). Broken lines: the mean and 2*σ* bounds corresponding to 50% subsampling, predicted by the analytical method presented. It can be seen that the mean values obtained from both simulations and analytical method coincide with *L*_{true}(*r*) − *r*, and that the 2*σ* curves obtained from the simulations and the analytical method coincide. (c) Histogram of *L*(*r*) − *r* of the subsampled realizations at *r* = *r*_{true}, where *r* = *r*_{true} is the cluster radius corresponding to the maxima of the *L*_{true}(*r*) − *r* function. It can be seen that it follows a normal distribution, with the fit parameters similar to that obtained from the analytical method. *r*_{true} is also plotted (dark green). The relative standard deviation (*σ*/*μ*, i.e,

σ

subsampled

L

true

(r)−r) at*r*=

*r*

_{true}is 2.6% for 50% subsampling.