Abstract
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 ofλ-measure is minimized usingQ-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.
Highlights
Generalized fuzzy model (GFM) [1,2,3] 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 [4,5,6] and the additive Snorm operator for combining the outputs of all the rules
We have proved that the defuzzified output of the non-additive GFM fuzzy rules is in the form of Choquet fuzzy integral
As we have already proved that the Choquet integral is the functionality of non-additive GFM, It is easy to extend to this functionality the Gaussian mixture model (GMM) case from the fact that the output is Gaussian in the nonadditive case too as per the expression p y | x, cm =
Summary
Generalized fuzzy model (GFM) [1,2,3] 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 [4,5,6] and the additive Snorm operator for combining the outputs of all the rules. The defuzzified output of the resultant non-additive GFM is shown to be in the form of Choquet integral [41]. This formulation is intended for real-life applications in which the information from different sources needs not to be additive; our efforts in this work will go a long way in evolving different types of non-additive fuzzy systems, like dynamic, adaptive, and so forth, but here our attempt is only on a simple non-additive fuzzy system. As the large number of input variables is increased for the purpose of fusion, the computational complexity grows exponentially [31, 51] with the λ-measure To overcome this problem a new fuzzy measure known as Q-Measure is introduced in [52].
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