The maximum likelihood based intensity estimate for circular point processes

Abstract

Identifying the geometrical nature of spatial point patterns plays an important role in many areas of scientific research. Common types of spatial point processes involve random, regular, and cluster patterns. However, some point patterns suggest identifiable geometrical shapes such as a circular or other conic patterns. These patterns may be recognized as either a specific clustered shape or an inhomogeneous point pattern. Less noisy conic shapes, including circular patterns, are heavily discussed in the pattern recognition literature, but the goodness-of-fit of conic-fitting algorithms is rarely discussed for very noisy data. This study addresses a parameter estimation technique for noisy circular point patterns using the maximum likelihood principle. Additionally, a spatial statistical tool known as the L-function is used to investigate whether the fitted location pattern is reasonably attributable to a circular shape. A novel quantity named ‘relative log-error’ ($\gamma$) is introduced to quantify the goodness-of-fit for circular model fits. An iteratively re-weighted least squares procedure is introduced and robustness is evaluated under several error structures. Computational efficiency of the current and novel circle-fitting methods is also discussed. The findings are applied to two environmental science data sets.