The symptom-syndrome analysis of multivariate categorical data based on Zhegalkin polynomials

Abstract

In this article, we study the distribution, entropy and other informational properties of finite projective subspaces (syndromes) parameterized by impulse sequences with basic elements in the form of symptoms – polynomials over the field F2 which are known as Zhegalkin polynomials. It has been proven that the super syndrome, which is a linear syndrome with basic elements in the form of a multiplicative syndrome, is closed. If in the multiplication of two symptoms one is neutral, then we are talking about its majorization. The ordered by majorization symptoms form a majorized syndrome. Is proved that the majorized syndrome is closed and coincides with the super syndrome. The statements formulated in the first part of the paper are used to justify the convergence of the iterative procedure (PI), in which the most informative symptoms selected from partial super syndromes are again used in the next step. The stationary state of PI is obtained if all elements of the input set belong to either the same partial super syndrome or to the majorized syndrome. Thanks IP it is possible to quickly find the optimal syndrome from a large set of variables. An example from phthisiology shows how the specificity of classification can be improved using symptom analysis.