An Efficient Sampling from Circular Distributions and its Extension to Toroidal Distributions

TL;DR

This paper introduces an efficient sampling framework for circular and toroidal distributions using upper Riemann sums, improving sampling efficiency and extending to complex curved surfaces, with practical applications demonstrated on wind direction data.

Contribution

It presents a novel sampling method based on upper Riemann sums for circular distributions and extends it to toroidal distributions, enhancing efficiency and flexibility.

Findings

Improved acceptance rates for von Mises distribution sampling

Successful extension to toroidal distributions using area elements

Practical application to wind direction data

Abstract

Sampling from circular distributions is a fundamental task in directional statistics. A key challenge in acceptance-rejection methods lies in selecting an efficient envelope density, as poor choices can lead to low acceptance rates and increased computational cost, especially in large-scale simulations. To address this, we propose a new sampling framework that utilizes the idea of upper Riemann sums to construct a piecewise envelope. This method ensures validity for any Riemann-integrable target density on a bounded interval. This method exhibits enhanced efficacy relative to the present sampling method for the von Mises distribution. Additionally, we introduce a flexible family of distributions defined on the surface of a curved torus, using its area element. The proposed sampling method is then employed to generate samples from the toroidal model. We explore the maximum entropy characterization and other theoretical properties of one of the marginal distributions arising from this construction for the von Mises distribution. To illustrate the practical utility of our framework, we apply the model to a real dataset on wind direction.