Year: 2025
Type: article
Abstract: Suppose x→ and y→ are two arbitrary fixed points in d-dimensional space and Z→ is a random vector with a known probability density. It is desired in some applications to find the joint probability distribution function for the distance between x→ and Z→ and the distance between y→ and Z→. This calculation has applications in signal processing, goodness-of-fit testing and two-sample testing. In this article, the efficient numerical calculation of the probability distribution is illustrated. The calculation reduces to the sum of two separate integrals where each integral is over a spherical cap. This is achieved by a transformation of the complex spherical intersection region into a sum of integrals over hypercubes via a carefully constructed variable change. The general approach applies to any application where integrals over a region defined by a spherical cap need to be evaluated.
Source: Mathematics
DOI: 10.3390/math14010011
Author: John Lawrence
Institution: United States Food and Drug Administration
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7116747665