The Sugeno fuzzy integral of log-convex functions

Abbaszadeh, Sadegh; Eshaghi, Madjid; Sen, Manuel De la

doi:10.1186/s13660-015-0862-6

Cited by 12 publications

(5 citation statements)

References 24 publications

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“…(1) If n i=1 g i = 1, then λ = 0; (2) If n i=1 g i < 1, then λ >0; (3) If n i=1 g i < 1, then − 1 ≤ λ > 0. There are many methods for calculating the fuzzy integral, such as Sugeno [20], Choquet [21], ordered weighted averaging AND operator (OWA-AND) [22], ordered weighted averaging OR operator (OWA-OR) [23], and Fuzzy min-max [24]. The Sugeno and Choquet methods were chosen in this study.…”

Section: The Proposed Efi-cnnsmentioning

confidence: 99%

Evolutionary-Fuzzy-Integral-Based Convolutional Neural Networks for Facial Image Classification

et al. 2019

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Various optimization methods and network architectures are used by convolutional neural networks (CNNs). Each optimization method and network architecture style have their own advantages and representation abilities. To make the most of these advantages, evolutionary-fuzzy-integral-based convolutional neural networks (EFI-CNNs) are proposed in this paper. The proposed EFI-CNNs were verified by way of face classification of age and gender. The trained CNNs’ outputs were set as inputs of a fuzzy integral. The classification results were operated using either Sugeno or Choquet output rules. The conventional fuzzy density values of the fuzzy integral were decided by heuristic experiments. In this paper, particle swarm optimization (PSO) was used to adaptively find optimal fuzzy density values. To combine the advantages of each CNN type, the evaluation of each CNN type in EFI-CNNs is necessary. Three CNN structures, AlexNet, very deep convolutional neural network (VGG16), and GoogLeNet, and three databases, computational intelligence application laboratory (CIA), Morph, and cross-age celebrity dataset (CACD2000), were used in experiments to classify age and gender. The experimental results show that the proposed method achieved 5.95% and 3.1% higher accuracy, respectively, in classifying age and gender.

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Section: The Proposed Efi-cnnsmentioning

confidence: 99%

Evolutionary-Fuzzy-Integral-Based Convolutional Neural Networks for Facial Image Classification

et al. 2019

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“…Recently, the integral inequalities for Sugeno integrals using different kinds of convexities and other results on several other types of inequalities based on Sugeno integrals are a thought-provoking topic to many authors in the field of fuzzy integrals, see for instance [20,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Hermite–Hadamard-Type Inequalities for Coordinated Convex Functions Using Fuzzy Integrals

Latif

2023

Mathematics

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In this paper, some estimates of third and fourth inequalities in Hermite–Hadamard-type inequalities for coordinated convex functions are proved using the non-additivity of the integrals and Fubini’s theorem for fuzzy integrals. That is, the results are obtained in the fuzzy context and using the Lebesgue measure. Several examples are provided on how to evaluate these estimates in order to illustrate the obtained results.

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“…It is well known that the Hadamard inequality is a famous and important result, which provides the upper (lower) bound for the mean value of a log-convex (log-concave) function. Abbaszadeh et al [ 19 ] studied the Sugeno fuzzy integral of log-convex functions and showed that the Hadamard inequality is not valid for this kind of Sugeno fuzzy integral. Motivated by [ 19 ], we naturally wonder whether the Hadamard inequality still holds for the Choquet integral.…”

Section: Introductionmentioning

confidence: 99%

“…Abbaszadeh et al [ 19 ] studied the Sugeno fuzzy integral of log-convex functions and showed that the Hadamard inequality is not valid for this kind of Sugeno fuzzy integral. Motivated by [ 19 ], we naturally wonder whether the Hadamard inequality still holds for the Choquet integral. If the Hadamard inequality is not valid, then it is necessary to estimate the upper bound and the lower bound of the Choquet integral for log-convex functions.…”

Section: Introductionmentioning

confidence: 99%

The Choquet integral of log-convex functions

Wang

2018

J Inequal Appl

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In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the upper bound of the Choquet integral for a general log-convex function, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present Jensen’s inequality of the Choquet integral for log-convex functions, which can be used to estimate the lower bound of this kind when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.

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The Sugeno fuzzy integral of log-convex functions

Abstract: In this paper, we give an upper bound for the Sugeno fuzzy integral of log-convex functions using the classical Hadamard integral inequality. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.

Cited by 12 publications

References 24 publications

Evolutionary-Fuzzy-Integral-Based Convolutional Neural Networks for Facial Image Classification

Evolutionary-Fuzzy-Integral-Based Convolutional Neural Networks for Facial Image Classification

Hermite–Hadamard-Type Inequalities for Coordinated Convex Functions Using Fuzzy Integrals

The Choquet integral of log-convex functions

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