On Several New Generalized Entropy Optimization Methods

Abstract

Objective: In this study we have suggested new Generalized Entropy Optimization Methods (GEOM) for solving Entropy Optimization Problems (EOP) consisting of optimizing a given entropy optimization measure subject to constraints generated by given moment vector functions. These problems acquire in different scientific fields as statistics, information theory, biostatistics especially in survival data analysis and etc. Material and Methods: Mentioned problems in the form of GEOP2, GEOP3 based on GEOP1 have Generalized Entropy Optimization Distributions: GEOD2 in the form of Min MaxEnt , Max MaxEnt; GEOD3 in the form of Min MinxEnt , Max MinxEnt, where is the Jaynes optimization measure, is Kullback-Leibler optimization measure. It should be noted that formulation of GEOP1 uses only one optimization measure ( or ), however each of formulations of GEOP2, GEOP3 uses two measures , together. Results: GEOP 1,2,3 are conditional optimization problems which can be solved by Lagrange multipliers method. It must be noted that calculating Lagrange multipliers can be fulfilled by starting from arbitrary initial point for Newton approximations of constructed auxiliary equation. Conclusion: There are situations, for example in survival data analysis, when both MaxEnt and MinxEnt distributions are accepted to given statistical data (or distribution) in the sense of same goodness of fit test. For this reason, developed our methods to obtain distributions are fundamental in statistical analysis. Analogous generalized problems can be also considered by the virtue of other measures different from , in dependency of requirements of experimental situation.

Bulanık Üyelik Fonksiyonunun Yaklaşımı İçin Bir Yeni Metot ve Uygulama

“…The formula (14) represents maximum value of Max(F)Ent measure (1). It is seen that depends on Lagrange multipliers 0 , 1 , … , , moment functions 0 ( ), 1 ( ), … , ( ) and moment fuzzy values 0 , 1 , … , .…”

“…Before it is showed that maximum value of Max(F)Ent measure is represented by formula (14) in the dependency of moment functions 0 ( ), 1 ( ), … , ( ), Lagrange multipliers 0 , 1 , … , , and moment fuzzy values 0 , 1 , … , . Let us = ( 0 , 1 , … , ) be vector moment functions with components 0 , 1 , … , .…”

“…The aim of this study consists of developing a new Generalized Maximum Fuzzy Entropy Methods in the form of MinMax(F)EntM and MaxMax(F)EntM on the basis of entropy optimization theory [13][14][15] and present an application.…”

Minimum Cross Fuzzy Entropy Problem, The Existence of Its Solution and Generalized Minimum Cross Fuzzy Entropy Problems

“…According to [14][15][16][17][18], let be the compact set of moment vector functions ( ). ( ) reaches its least and greatest values in this compact set, because of its continuity property.…”

An Application of Generalized Entropy Optimization Methods in Survival Data Analysis

“…There are several examples in the literature that known statistical distributions do not conform to statistical data; however, the entropy optimization distributions conform well. Generalized Entropy Optimization Methods (GEOM) have suggested distributions in the form of MinMaxEnt which is the closest to statistical data, and MaxMaxEnt which is the furthest from mentioned data in the sense of information theory [1] [2], respectively. For this reason, GEOM can be more successfully applied in Survival Data Analysis.…”