Intrinsic geometry-inspired dependent toroidal distribution: application to regression model for astigmatism data

This paper introduces a dependent toroidal distribution to analyze astigmatism data following cataract surgery. Rather than utilizing the flat torus, we opt to represent the bivariate angular data on the surface of a curved torus, which naturally offers smooth edge identifiability and accommodates a variety of curvatures—positive, negative, and zero. Beginning with the area-uniform toroidal distribution on this curved surface, we develop a five-parameter-dependent toroidal distribution that harnesses its intrinsic geometry via the area element to model the distribution of two dependent circular random variables. We show that both marginal distributions are cardioid, with one of the conditional variables also following a cardioid distribution. This key feature enables us to propose a circular-circular regression model based on conditional expectations derived from circular moments. To address the high rejection rate (approximately 50%) in existing acceptance-rejection sampling methods for cardioid distributions, we introduce an exact sampling method based on a probabilistic transformation. Additionally, we generate random samples from the proposed dependent toroidal distribution through suitable conditioning. This bivariate distribution and the regression model are applied to analyze astigmatism data arising in the follow-up of one and three months due to cataract surgery.