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Data obtained in association with many real-life random experiments from different fields cannot be perfectly/exactly quantified.hspace{.1cm}Often the underlying imprecision can be suitably described in terms of fuzzy numbers/\values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting.This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.

Age replacement policy is concerned with finding an optional time tominimize the cost, at which time the unit is replaced even if itdoes not fail. So far, age replacement policy involving random agehas been proposed. This paper will assume the age of the unit is anuncertain variable, and find the optimal time to replace the unit.

Detection of brain tissues using magnetic resonance imaging (MRI) is an active and challenging research area in computational neuroscience. Brain MRI artifacts lead to an uncertainty in pixel values. Therefore, brain MRI segmentation is a complicated concern which is tackled by a novel data fusion approach. The proposed algorithm has two main steps. In the first step the brain MRI is divided to some main and ancillary cluster which is done using Fuzzy c-mean (FCM). In the second step, the considering ancillary clusters are merged with main clusters employing Dempster-Shafer Theory. The proposed method was validated on simulated brain images from the commonly used BrainWeb dataset. The results of the proposed method are evaluated by using Dice and Tanimoto coefficients which demonstrate well performance and robustness of this algorithm.

t is the purpose of this paper to contribute to the discussion initiated byWachter about the parallelism between principal component (PC) and atypological grade of membership (GoM) analysis. The author testedempirically the close relationship between both analysis in a lowdimensional framework comprising up to nine dichotomous variables and twotypologies. Our contribution to the subject is also empirical. It relies ona dataset from a survey which was especially designed to study the reward ofskills in the banking sector in Portugal. The statistical data comprisethirty polythomous variables and were decomposed in four typologies using anoptimality criterion. The empirical evidence shows a high correlationbetween the first PC scores and individual GoM scores. No correlation withthe remaining PCs was found, however. In addtion to that, the first PC alsoproved effective to rank individuals by skill following the particularity ofdata distribution meanwhile unveiled in GoM analysis.

We provide a framework for the study of statistical quantitiesrelated to the Hurst phenomenon when the data are non-precise with boundedsupport.

In this paper, an anomaly detection method in cluster-based mobile ad hoc networks with ad hoc on demand distance vector (AODV) routing protocol is proposed. In the method, the required features for describing the normal behavior of AODV are defined via step by step analysis of AODV and independent of any attack. In order to learn the normal behavior of AODV, a fuzzy averaging method is used for combining one-class support vector machine (OCSVM), mixture of Gaussians (MoG), and self-organizing maps (SOM) one-class classifiers and the combined model is utilized to partially detect the attacks in cluster members. The votes of cluster members are periodically transmitted to the cluster head and final decision on attack detection is carried out in the cluster head. In the proposed method, an adaptive ordered weighted averaging (OWA) operator is used for aggregating the votes of cluster members in the cluster head. Since the network topology, traffic, and environmental conditions of a MANET as well as the number of nodes in each cluster dynamically change, the mere use of a fixed quantifier-based weight generation approach for OWA operator is not efficient. We propose a condition-based weight generation method for OWA operator in which the number of cluster members that participate in decision making may be varying in time and OWA weights are calculated periodically and dynamically based on the conditions of the network. Simulation results demonstrate the effectiveness of the proposed method in detecting rushing, RouteError fabrication, and wormhole attacks.

A manufacturing process cannot be released to production until it has been proven to be stable. Also, we cannot begin to talk about process capability until we have demonstrated stability in our process. This means that the process variation is the result of random causes only and all assignable or special causes have been removed. In complicated manufacturing processes, such as drilling process, the natural instability of the process impedes the use of any control charts for the mean and standard deviation. However, a complicated manufacturing process can be capable in spite of this natural instability.In this paper we discuss the $widetilde{C}_{pk}$ process capability index. We find the membership function of $widetilde{C}_{pk}$ based on fuzzy data. Also, by using the definition of classical control charts and the method of V$ddot{a}$nnman and Castagliola, we propose new control charts that are constructed by the $alpha$-cut sets of $widetilde{C}_{pk}$ for the natural instable manufacturing processes with fuzzy normal distributions. The results are concluded for $alpha=0.6$, that is chosen arbitrarily.

Considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. This paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.We study interrelationships between this theory and other fuzzy theories suchas intuitionistic fuzzy metric spaces, Kramosil and Michalek's spaces, Kalevaand Seikkala's spaces, probabilistic metric spaces, probabilisticmetric co-spaces, Menger spaces and intuitionistic probabilistic metricspaces, determining their position in the framework of theses different theories.

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