Models and Measures for Measuring Random Angular Quantities

The models and measures for measuring random angular quantities (plane angles and phase shifts of cyclic signals) are considered, the basic concepts, terms, definitions and characteristics that are used in the statistical analysis of angular data are given. The definitions and characteristics of the integral probability distribution function (PDF) of random angles and the density of their distribution are given. A probabilistic model of a random angle on a unit circle is developed. Probabilistic measures of angular quantities are also determined. The most characteristic probability distributions of random angles are analyzed—von Mises, wrapped normal distribution, cardioid, triangular, wrapped Cauchy. The numerical characteristics of random angles are given—the characteristic function, the circular mean, circular dispersion, standard deviation and median. A comparative analysis of the numerical characteristics of random variables and random angles is performed. Definitions and analytical expressions are given for selective circular characteristics—trigonometric moments, mean, resulting vector length, dispersion, etc. The basic information about the models developed by the authors and measures of random phase shifts of cyclic signals for phase measurements are presented