%0 Journal Article
%T Estimation of Fuzzy Measures Using Covariance Matrices in Gaussian Mixtures
%A Nishchal K. Verma
%J Applied Computational Intelligence and Soft Computing
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/402420
%X This paper presents a novel computational approach for estimating fuzzy measures directly from Gaussian mixtures model (GMM). The mixture components of GMM provide the membership functions for the input-output fuzzy sets. By treating consequent part as a function of fuzzy measures, we derived its coefficients from the covariance matrices found directly from GMM and the defuzzified output constructed from both the premise and consequent parts of the nonadditive fuzzy rules that takes the form of Choquet integral. The computational burden involved with the solution of 竹-measure is minimized using Q-measure. The fuzzy model whose fuzzy measures were computed using covariance matrices found in GMM has been successfully applied on two benchmark problems and one real-time electric load data of Indian utility. The performance of the resulting model for many experimental studies including the above-mentioned application is found to be better and comparable to recent available fuzzy models. The main contribution of this paper is the estimation of fuzzy measures efficiently and directly from covariance matrices found in GMM, avoiding the computational burden greatly while learning them iteratively and solving polynomial equations of order of the number of input-output variables. 1. Introduction Generalized fuzzy model (GFM) [1每3] is the backbone of this work that employs two norms for computing the strength of a rule: the multiplicative T-norm operator for determining the strength of a rule [4每6] and the additive S-norm operator for combining the outputs of all the rules. The effect of input fuzzy sets is taken into the defuzzified output in the form of rule strengths or weights. Gan et al. [7] have simplified the formulation of GFM by setting both the inputs and the output to be jointly Gaussian and proved that the input-output relation is as an expectation using the Bayesian framework. On simplification, this turns out to be the coveted Gaussian mixture model (GMM), which is linked by an additive function between the inputs and the output. The GMM, also known as cluster-weighted modeling (CWM) [8每12], is advocated by many researchers as means of statistical modeling of input-output systems [10, 13每15]. After the establishment of equivalence between GMM and GFM in [7], the difficult-to-compute index of fuzziness of GFM has become easy-to-compute prior probability [16每23] in GMM. Further simplification of GFM will be explored when it is converted into nonadditive case. The use of GMM in GFM has provided a generalized framework for additive fuzzy systems. A few
%U http://www.hindawi.com/journals/acisc/2012/402420/