The motivation for this study came from the task of analysing the kinetic behavior of single molecules in a living cell based on Single Molecule Localization Microscopy. Given measurements of both the motion of clusters and molecules, the main task consists in detecting if a molecule belongs to a cluster. While the exact size of the clusters is usually unknown, upper bounds are available. In this study, we simulate the cluster movement by a Brownian motion and those of the particles by a Gaussian mixture model with two modes depending on the position of the particle within or outside a cluster. We propose various variational models to detect if a particle lies within a cluster based on the Wasserstein and maximum mean discrepancy distances between measures. We compare the performance of the proposed models for simulated data.

翻译：本研究的动机源于基于单分子定位显微镜分析活细胞内单分子动力学行为的任务。给定簇与分子运动的观测数据，主要任务在于检测分子是否属于某个簇。尽管簇的精确尺寸通常未知，但可获得其上限。在本研究中，我们采用布朗运动模拟簇的运动，并通过高斯混合模型（包含两种模式，分别对应粒子位于簇内或簇外的位置）模拟粒子的运动。基于测度之间的Wasserstein距离与最大均值差异，我们提出多种变分模型以检测粒子是否位于簇内。最后，我们比较了所提模型在模拟数据上的性能。