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.

In this study, a new method to obtain approximate probability density function (pdf) of random variable of solution of stochastic differential equations (SDEs) by using generalized entropy optimization methods (GEOM) is developed. By starting given statistical data and Euler–Maruyama (EM) method approximating SDE are constructed several trajectories of SDEs. The constructed trajectories allow to obtain random variable according to the fixed time. An application of the newly developed method includes SDE model fitting on weekly closing prices of Honda Motor Company stock data between 02 July 2018 and 25 March 2019.

Every day, the number of newly confirmed cases of coronavirus (COVID-19) rises in many countries. It is critical to adjust policies and plans in order to investigate the relationships between the distributions of the spread of this virus in other countries. During this study, the intuitionistic fuzzy c-means (IFCM) clustering method is used to compare and cluster the distributions of COVID-19 spread in 62 countries. Using the IFCM clustering algorithm, the study aims to cluster the countries that use environmental, economic, social, health, and related measurements that affect disease spread to implement policies that regulate disease spread. As a result, countries that have similar factors can take proactive measures to address the pandemic. The data are obtained for 62 countries, and six different feature variables (factors associated with the spread of COVID-19) are determined. The data are obtained for 62 countries, and six variables with different characteristics (linked to the spread of COVID-19) are identified. In this study, the IFCM clustering algorithm is used to determine the dynamic behavior of COVID-19 based on real-world data for multiple countries and Turkey around the world. Data analysis is performed through MATLAB 2018a and R programs. The clustering results revealed that the distribution of dissemination in Brazil, India, and the United States was nearly identical and distinct from that of the 59 other countries.

In this study, we have developed one new approximate method to obtain a probability density function of a solution of a given stochastic differential equation (SDE) at a fixed time. The mentioned method is based on the estimation SDE fitting to given statistical data and approximate methods solving SDE. For this purpose, by approximate methods solving SDE trajectories of this equation are constructed. For example, it is possible to use the Euler-Maruyama (EM) method. By using trajectories at a fixed time are obtained reasonable random variables of the solution of SDE. The probability density function of the mentioned random variables is obtained. It is possible to use different statistical methods. These results are acquired by using the theorem. In our investigation, it is used Generalized Entropy Optimization Methods (GEOM). The reason using GEOM's is explained oneself by the fact that these methods represent distributions that are more flexible distributions. We illustrated the use of this new method to apply the SDE model fitting on S&P 500 stock data.

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Abstract:In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem (Minx(F)EntP) and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method.

In the present study we have formulated a new Maximum Fuzzy Entropy Problem (Max(F)EntP) for fuzzy membership function and proposed sufficient conditions for existence of its solution. Mentioned problem consists of approximation fuzzy membership function by maximizing Maximum Fuzzy Entropy (Max(F)Ent) measure with respect to membership functions with finite number of the fuzzy values subject to constraints generated by given moment functions. The existence of solution of mentioned problem is proved by virtue of convexity property of Max(F)Ent measure, the implicit function theorem and Lagrange multipliers method. Moreover, by using MATLAB programme one application of suggested method on fuzzy data analysis is given.

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