From a set of parts to an indivisible whole. Part I: Operations in a   closed mode
  This paper provides a description of a new method for information processing based on holistic approach wherein analysis is a direct product of synthesis. The core of the method is iterative averaging of all the elements of a system according to all the parameters describing the elements. Contrary to common logic, the iterative averaging of a system's elements does not result in homogenization of the system; instead, it causes an obligatory subdivision of the system into two alternative subgroups, leaving no outliers. Within each of the formed subgroups, similarity coefficients between the elements reach the value of 1, whereas similarity coefficients between the elements of different subgroups equal a certain constant value greater than 0 but lower than 1. When subjected to iterative averaging, any system consisting of three or more elements of which at least two elements are not completely identical undergo such a process of bifurcation that occurs non-linearly. Successive iterative averaging of each of the forming subgroups eventually provides a hierarchical system that reflects relationships between the elements of an input system under analysis. We propose a definition of a natural hierarchy that can exist only in conditions of closeness of a system and can be discovered upon providing such an effect onto a system which allows its elements interact with each other based on the principle of self-organization. Self-organization can be achieved through an overall and total cross-averaging of a system's elements. We demonstrate the application potentials of the proposed technology on a number of examples, including a system of scattered points, randomized datasets, as well as meteorological and demographical datasets. 
Leonid Andreev