FUZZY INTEGRALS

Gupta, Megha and Jayaram, Balasubramaniam (2018) FUZZY INTEGRALS. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

The aim of this Project is to present the concept of Integrals that are based on Fuzzy Measures( known as Fuzzy Integrals). We bring an overview of Fuzzy Integrals, including historical remarks. After a short history of classical integration, several types of integrals (in particular, Choquet, Sugeno and Shilkret Integrals) with respect to monotone measures are discussed and exempli�ed. This thesis presents a synthesis on the use of fuzzy integrals as an aggregation operator in multicriteria decision making. An application of Choquet Integral based on a generalized � - fuzzy measure in image enhancement is given, and the corresponding experiment shows that di�erent values of � have di�erent in uences on an image.Then, we have implemented some Image Processing features using Sugeno Integral based on Counting Measure and Sugeno Measure in MATLAB. We study a framework for functionals based on monotone measures, which are referred as Universal Integrals and Choquet, Sugeno, and Shilkret integrals are particular cases of the Universal Integral. Next, we look at Fuzzy Integrals on di�erent structures. We consider fuzzy measures valued in complete lattices and their associated upper and lower fuzzy integrals and we see that in the setting of the complete lattices which are both Brouwerian and dually Brouwerian, the upper and lower fuzzy integrals are fuzzy integrals; while in that of the completely distributive complete lattices every fuzzy integral is both lower and upper fuzzy integral with respect to a fuzzy measure. Two new generalizations of the relation of comonotonicity of lattice-valued vectors are introduced and discussed. These new relations coincide on distributive lattices and they share several properties with the comonotonicity for the real-valued vectors (which need not hold for Lvalued vectors comonotonicity, in general). Based on these newly introduced generalized types of comonotonicity of L-valued vectors, several new axiomatizations of L-valued Sugeno integrals are introduced. One of them brings a substantial decrease of computational complexity when checking an aggregation function to be a Sugeno integral. Finally, we study characterisation theorems for four variants of Sugeno Integrals based on Heyting algebra

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