Weighted Fuzzy Averages in Fuzzy Environment: Part II. Generalized Weighted Fuzzy Expected Values in Fuzzy Environment

2003 ◽  
Vol 11 (02) ◽  
pp. 159-172 ◽  
Author(s):  
G. Sirbiladze ◽  
A. Sikharulidze
Keyword(s):  
Measure Space ◽  
Expected Value ◽  
Fuzzy Measure ◽  
Weighted Averages ◽  
Fuzzy Environment ◽  
Fuzzy Expected Value ◽  
Interval Extension ◽  
Finite Set ◽  
Expected Values ◽  
Iteration Processes

The weighted fuzzy expected value (WFEV) of the population for a sampling distribution was introduced in 1. In 2 the notion of WFEV is generalized for any fuzzy measure on a finite set (WFEVg). The latter paper also describes the notions of weighted fuzzy expected intervals WFEI and WFEIg which are an interval extension of WFEV and WFEVg, respectively, when due to ''scarce'' data the fuzzy expected value (FEV) 3 does not exist, but the fuzzy expected interval (FEI) 3 does. In this paper, The generalizations GWFEVg and GWFEIg of WFEVg and WFEIg, respectively, are introduced for any fuzzy measure space. Furthermore, the generalized weighted fuzzy expected value is expressed in terms of two monotone expectation (ME)4 values with respect to the Lebesgue measure on [0,1]. The convergence of iteration processes is provided by an appropriate choice of a ''weight'' function. In the interval extension (GWFEIg) the so-called combinatorial interval extension of a function 5 is successfully used, which is clearly illustrated by examples. Several examples of the use of the new weighted averages are discussed. In many cases these averages give better estimations than classical estimators of central tendencies such as mean, median or the fuzzy ''classical'' estimators FEV, FEI and ME.

Weighted Fuzzy Averages in Fuzzy Environment: Part I. Insufficient Expert Data and Fuzzy Averages

2003 ◽  
Vol 11 (02) ◽  
pp. 139-157 ◽  
Author(s):  
G. Sirbiladze ◽  
A. Sikharulidze
Keyword(s):  
Expected Value ◽  
Fuzzy Measure ◽  
Construction Process ◽  
Weighted Averages ◽  
The Third ◽  
Fuzzy Environment ◽  
Fuzzy Expected Value ◽  
Interval Extension ◽  
Finite Set ◽  
Sampling Probability

Three new versions of the most typical value (MTV)1,2 of the population (generalized weighted averages) are introduced. The first version, WFEVg, is a generalization of the weighted fuzzy expected value (WFEV)3 for any fuzzy measure g on a finite set and it coincides with the WFEV when a sampling probability distribution is used. The second and the third version are respectively the weighted fuzzy expected intervals WFEI and WFEIg which are generalizations of the WFEV, namely, MTV s of the population for a sampling distribution and for any fuzzy measure g on a finite set, respectively, when the fuzzy expected interval (FEI)4 exists but the fuzzy expected value (FEV)4 does not. The construction process is based on the Friedman-Schneider-Kandel (FSK)3 principle and results in the new MTV s called the WFEI and the WFEIg when the combinatorial interval extension of a function5 is used.

ON THE THEORY OF TYPICALITY

1995 ◽  
Vol 03 (02) ◽  
pp. 127-142 ◽  
Author(s):  
MENAHEM FRIEDMAN ◽  
MA MING ◽  
ABRAHAM KANDEL
Keyword(s):  
Standard Deviation ◽  
Fuzzy Set ◽  
Typical Feature ◽  
Expected Value ◽  
Central Tendency ◽  
Data Set ◽  
Fuzzy Expected Value ◽  
Finite Set ◽  
The Given

The classical definitions of ‘expected value’ and ‘standard deviation’ may sometimes lead to quantities which fail to represent a ‘typical’ feature of a given data set, whenever this set consists of more than one cluster. The use of the fuzzy expected value (FEV) and the clustering fuzzy expected value (CFEV) also yield central tendency and in general cannot represent a typical value of the given data. In this work a new quantity—a Most Typical Value (MTV) is defined and investigated. A given fuzzy set in Rn is first clustered and replaced by a finite set of clusters. This set is then represented by a single vector—the most typical value of the set.

scholarly journals Refined Expected Value Decision Rules under Orthopair Fuzzy Environment

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 442 ◽  
Author(s):  
Yige Xue ◽  
Yong Deng
Keyword(s):  
Decision Making ◽  
Choquet Integral ◽  
Decision Rules ◽  
Expected Value ◽  
Experimental Results ◽  
Numerical Examples ◽  
Fuzzy Environment ◽  
Proposed Model ◽  
Expected Values ◽  
Function Measure

Refined expected value decision rules can refine the calculation of the expected value and make decisions by estimating the expected values of different alternatives, which use many theories, such as Choquet integral, PM function, measure and so on. However, the refined expected value decision rules have not been applied to the orthopair fuzzy environment yet. To address this issue, in this paper we propose the refined expected value decision rules under the orthopair fuzzy environment, which can apply the refined expected value decision rules on the issues of decision making that is described in the orthopair fuzzy environment. Numerical examples were applied to verify the availability and flexibility of the new refined expected value decision rules model. The experimental results demonstrate that the proposed model can apply refined expected value decision rules in the orthopair fuzzy environment and solve the decision making issues with the orthopair fuzzy environment successfully.

scholarly journals Reward Value Coding Distinct From Risk Attitude-Related Uncertainty Coding in Human Reward Systems

2007 ◽  
Vol 97 (2) ◽  
pp. 1621-1632 ◽  
Author(s):  
Philippe N. Tobler ◽  
John P. O'Doherty ◽  
Raymond J. Dolan ◽  
Wolfram Schultz
Keyword(s):  
Risk Aversion ◽  
Risk Attitude ◽  
Mean Value ◽  
Expected Value ◽  
Individual Risk ◽  
Functional Magnetic Resonance ◽  
Coding Regions ◽  
Monetary Rewards ◽  
Expected Values ◽  
Value Coding

When deciding between different options, individuals are guided by the expected (mean) value of the different outcomes and by the associated degrees of uncertainty. We used functional magnetic resonance imaging to identify brain activations coding the key decision parameters of expected value (magnitude and probability) separately from uncertainty (statistical variance) of monetary rewards. Participants discriminated behaviorally between stimuli associated with different expected values and uncertainty. Stimuli associated with higher expected values elicited monotonically increasing activations in distinct regions of the striatum, irrespective of different combinations of magnitude and probability. Stimuli associated with higher uncertainty (variance) elicited increasing activations in the lateral orbitofrontal cortex. Uncertainty-related activations covaried with individual risk aversion in lateral orbitofrontal regions and risk-seeking in more medial areas. Furthermore, activations in expected value-coding regions in prefrontal cortex covaried differentially with uncertainty depending on risk attitudes of individual participants, suggesting that separate prefrontal regions are involved in risk aversion and seeking. These data demonstrate the distinct coding in key reward structures of the two basic and crucial decision parameters, expected value, and uncertainty.

scholarly journals THE INTERVAL-VALUED INTUITIONISTIC FUZZY GEOMETRIC CHOQUET AGGREGATION OPERATOR BASED ON THE GENERALIZED BANZHAF INDEX AND 2-ADDITIVE MEASURE

2015 ◽  
Vol 21 (2) ◽  
pp. 186-215 ◽  
Author(s):  
Fanyong MENG ◽  
Qiang ZHANG ◽  
Jiaquan ZHAN
Keyword(s):  
Geometric Mean ◽  
Fuzzy Measure ◽  
Entropy Measure ◽  
Additive Measure ◽  
Banzhaf Index ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Environment ◽  
Operational Laws ◽  
Additive Measures ◽  
Interval Valued

Based on the operational laws on interval-valued intuitionistic fuzzy sets, the generalized Banzhaf interval-valued intuitionistic fuzzy geometric Choquet (GBIVIFGC) operator is proposed, which is also an interval-valued intuitionistic fuzzy value. It is worth pointing out that the GBIVIFGC operator can be seen as an extension of some geometric mean operators. Since the fuzzy measure is defined on the power set, it makes the problem exponentially complex. In order to overall reflect the interaction among elements and reduce the complexity of solving a fuzzy measure, we further introduce the GBIVIFGC operator w.r.t. 2-additive measures. Furthermore, if the information about weights of experts and attributes is incompletely known, the models of obtaining the optimal 2-additive measures on criteria set and expert set are given by using the introduced cross entropy measure and the Banzhaf index. Finally, an approach to pattern recognition and multi-criteria group decision making under interval-valued intuitionistic fuzzy environment is developed, respectively.

Fuzzy Measures: A solution to deal with community detection problems for networks with additional information

2020 ◽  
Vol 39 (5) ◽  
pp. 6217-6230
Author(s):  
Inmaculada Gutiérrez ◽  
Daniel Gómez ◽  
Javier Castro ◽  
Rosa Espínola
Keyword(s):  
Community Detection ◽  
Weighted Graph ◽  
Fuzzy Relation ◽  
Fuzzy Measure ◽  
Fuzzy Measures ◽  
Additional Information ◽  
Fuzzy Graphs ◽  
Finite Set ◽  
Definition Of ◽  
Community Detection Problems

In this work we introduce the notion of the weighted graph associated with a fuzzy measure. Having a finite set of elements between which there exists an affinity fuzzy relation, we propose the definition of a group based on that affinity fuzzy relation between the individuals. Then, we propose an algorithm based on the Louvain’s method to deal with community detection problems with additional information independent of the graph. We also provide a particular method to solve community detection problems over extended fuzzy graphs. Finally, we test the performance of our proposal by means of some detailed computational tests calculated in several benchmark models.

Borel-Cantelli Lemma for Sugeno Measure

2014 ◽  
Vol 614 ◽  
pp. 367-370 ◽  
Author(s):  
Chun Qin Zhang ◽  
Hui Zhang
Keyword(s):  
Measure Space ◽  
Natural Extension ◽  
Fuzzy Measure ◽  
Cantelli Lemma ◽  
Sugeno Measure

Sugeno measure is a fuzzy measure. In this paper, we derive the Borel-Cantelli lemma for Sugeno measure. This result is a natural extension of the classical Borel-Cantelli lemma to the case where the measure tool is fuzzy. The properties of Sugeno measure are further discussed. Then the Borel-Cantelli lemma will be proven on Sugeno measure space. This work generalizes the research and applications of the Borel-Cantelli lemma.

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