Functional summary statistics and testing for independence in multi-type point processes on the surface of 3D shapes

Music City Center 

Recent advances in 3D biological imaging technologies now allow spatial point patterns of protein molecules to be observed directly on a cell's outer membrane, where the underlying geometry is complex and must be respected for principled statistical inference. In prior work, we developed functional summary statistics for point patterns on the surfaces of 3D convex shapes, leading to new insights into E. coli outer membrane assembly. In this work, we extend these methods to the multi-
type setting. We begin by developing functional summary statistics for multi-type homogeneous and inhomogeneous point processes on the sphere, and then generalise them to convex 3D surfaces, assuming a known bijective mapping from the shape to the sphere. To support inference in the inhomogeneous case, we employ a plug-in estimator for the intensity function of a spatial point process on a manifold. We demonstrate how these statistics can be used to test for independence between the component processes, with particular focus on methods for generating samples from the null distribution. We conclude with a discussion on extending the framework to a class of non-convex shapes.