ОБ ИННОВАЦИОННЫХ МЕТОДАХ УСРЕДНЕНИЯ ЧИСЛОВЫХ ДАННЫХ

Abstract

Numerical data refers to any finite set of data in the form of numbers, vectors, functions, matrices representing the results of an experiment or field observations. Averaging of deterministic, random variables and matrices is considered from a single point of view as a minimization of a function in the form of a generalized least squares problem. A new definition of the mean is given. Three generalizations of averages are obtained as solutions to the minimization problem. If the known averages are harmonic, geometric, arithmetic and quadratic averages and, perhaps, some other averages, then the first generalization of averages has already given an uncountable set of averages. Two new averages are derived from the first generalization. For particular types of averages arising from the first generalization, their interpretations are given in terms of absolute and relative deviations (errors). A sufficient condition of the mean is proved for all averages. Inequalities for six averages are proved. The law of nine numbers has been discovered. The concept of a complex average is given. The concept of optimal mean is introduced. New definitions of mathematical expectation and variance and their generalizations are proposed. In the family of mathematical expectations obtained, only the classical mathematical expectation turned out to be linear. The application of generalized mathematical expectation has led to the discovery of two new distributions in probability theory, namely, the harmonic and relative distributions of a continuous random variable are determined and analytically presented.