Statistical Inference in Circular Structural Model and Fitting Circles to Noisy Data

It is well known that commonly used algorithms for circle fitting perform poorly when sampling distribution of the points is not symmetric with respect to the circle center, for example, when the points are sampled from a circle arc. To overcome this difficulty we introduce and study a parametric circular structural model. In this model the points on the circumference are assumed to be sampled according to the von Mises distribution with unknown concentration and mean direction parameters. We develop maximum likelihood and method of moments estimators of the circle center and radius, and study their statistical properties. In particular, we show that the proposed maximum likelihood estimator is asymptotically normal and efficient. We also develop a test of uniformity for the sampling distribution along the circle. Based on the derived theoretical results we propose a numerically stable circle fitting algorithm, investigate its accuracy in a simulation study, and illustrate its behavior in a real data example. Supplementary materials for this article are available online.