algorithms for improving consistency or consensus of reciprocal 0 1 valued preference relations we investigate the consistency and consensus of reciprocal 0 1 valued preference relations also called fuzzy preference relations by many authors based on the multiplicative consistency property which is an important issue in fuzzy set theory an algorithm is first developed to improve the consistency level of a reciprocal 0 1 valued preference relation and the corresponding algorithm for the incomplete reciprocal 0 1 valued preference relation is also developed we further propose the consensus improving algorithms for individual reciprocal 0 1 valued preference relations or incomplete ones the convergence and robustness of the algorithms are proven and some important conclusions are obtained in addition the proposed algorithms can improve the consistency or consensus of reciprocal 0 1 valued preference relations with less interactions with the decision makers which can save a lot of time and obtain the results quickly

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

the owa based consensus operator under linguistic representation models using position indexes when using linguistic approaches to solve decision problems we need linguistic representation models the symbolic model the 2 tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes together with these three linguistic models the corresponding ordered weighted averaging operators such as the linguistic ordered weighted averaging operator the 2 tuple ordered weighted averaging operator and the extended ordered weighted averaging operator have been developed respectively in this paper we analyze the internal relationship among these operators and propose a consensus operator under the continuous linguistic model or the 2 tuple fuzzy linguistic representation model the proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures some desired properties of the consensus operator are also presented in particular the consensus operator provides an alternative consensus model for group decision making this consensus model preserves the original preference information given by the decision makers as much as possible and supports consensus process automatically without moderator

general transitivity conditions for fuzzy reciprocal preference matrices a reciprocal fuzzy matrix relation is a non negative matrix q qij such that qij qji 1 for all i j 1 2 n we define general transitivity conditions named fg transitivities for fuzzy reciprocal preference relations and show that they generalize some well known transitivities we also study relationships of these conditions with two models of rational preferences the so called utility model and the multidimensional model

cumulative voting consensus method for partitions with variable number of clusters over the past few years there has been a renewed interest in the consensus clustering problem several new methods have been proposed for finding a consensus partition for a set of n data objects that optimally summarizes an ensemble in this paper we propose new consensus clustering algorithms with linear computational complexity in n we consider clusterings generated with a random number of clusters which we describe by categorical random variables we introduce the idea of cumulative voting as a solution for the problem of cluster label alignment where unlike the common one to one voting scheme a probabilistic mapping is computed we seek a first summary of the ensemble that minimizes the average squared distance between the mapped partitions and the optimal representation of the ensemble where the selection criterion of the reference clustering is defined based on maximizing the information content as measured by the entropy we describe cumulative vote weighting schemes and corresponding algorithms to compute an empirical probability distribution summarizing the ensemble given the arbitrary number of clusters of the input partitions we formulate the problem of extracting the optimal consensus as that of finding a compressed summary of the estimated distribution that preserves the maximum relevant information an efficient solution is obtained using an agglomerative algorithm that minimizes the average generalized jensen shannon divergence within the cluster the empirical study demonstrates significant gains in accuracy and superior performance compared to several recent consensus clustering algorithms

fuzzy sets and decision analysis abstract this survey points out recent advances in multiple attribute decision making methods dealing with fuzzy or ill defined information fuzzy maut as well as fuzzy outranking methods are reviewed aggregation procedures choice problems and treatment of interactive attributes are covered trends in research and open problems are indicated

acyclicity of schur complexes and torsion freeness of schur modules abstract let r be a noetherian commutative ring and m a r module with pd r m 1 that has rank necessary and sufficient conditions were provided in 1 for an exterior power k m to be torsion free when m is an ideal of r similar necessary and sufficient conditions were provided in 2 for a symmetric power s k m to be torsion free we extend these results to a broad class of schur modules l m en route for any map of finite free r modules f g we also study the general structure of the schur complexes l and provide necessary and sufficient conditions for the acyclicity of any given l by computing explicitly the radicals of the ideals of maximal minors of all its differentials

aggregation functions based on penalties this article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs or equivalently minimizing a penalty imposed for deviations of individual inputs from the aggregated value we provide a systematization of various types of penalty based aggregation functions and show how many special cases arise as the result we show how new aggregation functions can be constructed either analytically or numerically and provide many examples we establish connection with the maximum likelihood principle and present tools for averaging experimental noisy data with distinct noise distributions

bagging for path based clustering a resampling scheme for clustering with similarity to bootstrap aggregation bagging is presented bagging is used to improve the quality of path based clustering a data clustering method that can extract elongated structures from data in a noise robust way the results of an agglomerative optimization method are influenced by small fluctuations of the input data to increase the reliability of clustering solutions a stochastic resampling method is developed to infer consensus clusters a related reliability measure allows us to estimate the number of clusters based on the stability of an optimized cluster solution under resampling the quality of path based clustering with resampling is evaluated on a large image data set of human segmentations

on the resolutions of the powers of the pfaffian ideal x is a generic skew symmetric matrix and we let s stand for the polynomial ring r x r z the pfaffians of the 2k order principal submatrices of x generate an ideal of s the pfalfian ideal denoted by i it is well known that i is a generically perfect gorenstein prime ideal of grade 3 in fact a prototype of all the ideals of this kind cf b e 31 hence i has a finite free s resolution of length 3 a resolution which looks the same regardless of the ring r

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

combining multiple clusterings using evidence accumulation we explore the idea of evidence accumulation eac for combining the results of multiple clusterings first a clustering ensemble a set of object partitions is produced given a data set n objects or patterns in d dimensions different ways of producing data partitions are 1 applying different clustering algorithms and 2 applying the same clustering algorithm with different values of parameters or initializations further combinations of different data representations feature spaces and clustering algorithms can also provide a multitude of significantly different data partitionings we propose a simple framework for extracting a consistent clustering given the various partitions in a clustering ensemble according to the eac concept each partition is viewed as an independent evidence of data organization individual data partitions being combined based on a voting mechanism to generate a new n times n similarity matrix between the n patterns the final data partition of the n patterns is obtained by applying a hierarchical agglomerative clustering algorithm on this matrix we have developed a theoretical framework for the analysis of the proposed clustering combination strategy and its evaluation based on the concept of mutual information between data partitions stability of the results is evaluated using bootstrapping techniques a detailed discussion of an evidence accumulation based clustering algorithm using a split and merge strategy based on the k means clustering algorithm is presented experimental results of the proposed method on several synthetic and real data sets are compared with other combination strategies and with individual clustering results produced by well known clustering algorithms

a rational consensus model in group decision making using linguistic assessments abstract a new consensus model for the consensus reaching process in a linguistic framework is presented in heterogeneous group decision making problems called rational consensus model it is guided by some linguistic consensus and linguistic consistency meaures all the measures are calculated from a set of linguistic preference relations used to provide experts opinions this consensus model allows more rational consensus solutions to be obtained and thus more human consistency to be incorporated in decision support systems

the self dual core and the anti self dual remainder of an aggregation operator in most decisional models based on pairwise comparison between alternatives the reciprocity of the individual preference representations expresses a natural assumption of rationality in those models self dual aggregation operators play a central role in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences in this paper we propose a simple method by which one can associate a self dual aggregation operator to any aggregation operator on the unit interval the resulting aggregation operator is said to be the self dual core of the original one and inherits most of its properties our method constitutes thus a new characterization of self duality with some technical advantages relatively to the traditional symmetric sums method due to silvert in our framework moreover every aggregation operator can be written as a sum of a self dual core and an anti self dual remainder which in some cases seems to give some indication on the dispersion of the variables in order to illustrate the method proposed we apply it to two important classes of continuous aggregation operators with the properties of idempotency symmetry and stability for translations the owa operators and the exponential quasiarithmetic means

reliability based fuzzy clustering ensemble abstract in the clustering ensemble the quality of base clusterings influences the consensus clustering although some researches have been devoted to weighting the base clustering fuzzy cluster level weighting has been ignored more specifically they did not pay attention to the role of cluster reliability in the fuzzy clustering ensemble in this paper we propose a new fuzzy clustering ensemble framework without access to the features of data objects based on fuzzy cluster level weighting the reliability of each fuzzy cluster is computed based on estimation of its unreliability and is considered as its weight in the ensemble the unreliability of fuzzy clusters is estimated by applying the similarity between fuzzy clusters in the ensemble based on an entropic criterion in our framework the final clustering is produced by two types of consensus functions 1 a reliability based weighted fuzzy co association matrix is constructed from the base clusterings and then a single traditional clustering such as hierarchical agglomerative clustering or k means is applied over the matrix to produce the final clustering 2 a new graph based fuzzy consensuses function the graph based consensus function has linear time complexity in the number of data objects experimental results on various standard datasets demonstrated the effectiveness of the proposed approach compared to the state of the art methods in terms of evaluation criteria and clustering robustness

the group consensus based evidential reasoning approach for multiple attributive group decision analysis abstract many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance fuzziness interval data and interval belief degrees an evidential reasoning er approach developed in the 1990s and in recent years can be used to model these problems in this paper the er approach is extended to group consensus gc situations for multiple attributive group decision analysis problems in order to construct and check the gc a compatibility measure between two belief structures is developed first considering two experts utilities the compatibility between their assessments is naturally constructed using the compatibility measure based on the compatibility between two experts assessments the gc at a specific level that may be the attribute level the alternative level or the global level can be constructed and reached after the group analysis and discussion within specified times under the condition of gc we conduct a study on the forming of group assessments for alternatives the achievement of the aggregated utilities of assessment grades and the properties and procedure of the extended er approach an engineering project management software selection problem is solved by the extended er approach to demonstrate its detailed implementation process and its validity and applicability

modeling rationality in a linguistic framework in classical decision theory there exists a large class of rationality models which try to capture different kinds of behavior when individuals compare by pairs a set of alternatives all these models assume that decision makers have dichotomous preferences however in real decisions individuals feel different degrees of preference in this paper we have checked the mentioned models in a real case where different kinds of linguistic preferences are allowed after the empirical analysis the main conclusion is that the fulfillment of rational conditions decreases when individuals have non extreme preferences based on the obtained empirical evidences we propose some classes of transitivity conditions in the framework of linguistic preferences

positional and confidence voting based consensus functions for fuzzy cluster ensembles consensus clustering i e the task of combining the outcomes of several clustering systems into a single partition has lately attracted the attention of researchers in the unsupervised classification field as it allows the creation of clustering committees that can be applied with multiple interesting purposes such as knowledge reuse or distributed clustering however little attention has been paid to the development of algorithms known as consensus functions especially designed for consolidating the outcomes of multiple fuzzy or soft clustering systems into a single fuzzy partition despite the fact that fuzzy clustering is far more informative than its crisp counterpart as it provides information regarding the degree of association between objects and clusters that can be helpful for deriving richer descriptive data models for this reason this paper presents a set of fuzzy consensus functions capable of creating soft consensus partitions by fusing a collection of fuzzy clusterings our proposals base clustering combination on a cluster disambiguation process followed by the application of positional and confidence voting techniques the modular design of these algorithms makes it possible to sequence their constituting steps in different manners which allows to derive versions of the proposed consensus functions optimized from a computational standpoint the proposed consensus functions have been evaluated in terms of the quality of the consensus partitions they deliver and in terms of their running time on multiple benchmark data sets a comparison against several representative state of the art consensus functions reveals that our proposals constitute an appealing alternative for conducting fuzzy consensus clustering as they are capable of yielding high quality consensus partitions at a low computational cost

a method for repairing the inconsistency of fuzzy preference relations this paper investigates the inconsistency problems of preference information about alternatives expressed as a fuzzy preference relation by a decision maker an analysis method is presented to identify the inconsistency and weak transitivity of a fuzzy preference relation and to repair its inconsistency to reach weak transitivity first several definitions are given on additive consistency inconsistency and weak transitivity of a fuzzy preference relation next two methods derived from graph theory are presented to judge whether a fuzzy preference relation has weak transitivity or not then an algorithm is developed to repair an inconsistent fuzzy preference relation and to make it become one with weak transitivity via a synthesis matrix which reflects the relationship between the fuzzy preference relation with additive consistency and the initial one given by a decision maker a convergence theorem is also given for the algorithm lastly two numerical examples are shown to illustrate the proposed method

fuzzy consensus clustering with applications on big data consensus clustering aims to find a single partition of data that agrees as much as possible with existing basic partitions given its robustness and generalizability consensus clustering has emerged as a promising solution to find cluster structures inside heterogeneous big data rising from various application domains in the area of fuzzy systems however research along this line is still in its initial stage with some unsystematic algorithmic studies finding a fuzzy consensus partition from multiple fuzzy basic partitions in an efficient flexible and robust way is still an exciting open problem calling for further investigation in light of this this paper provides a systematic study of fuzzy consensus clustering fcc from a utility perspective specifically we first define the objective function of fcc clearly using the novel fuzzified contingency matrix we then derive a family of fcc utility functions termed as fccu that can transform fcc to a weighted piecewise fuzzy c means clustering pifcm problem this helps us to establish an algorithmic framework for fcc with flexible choice of utility functions and speeds fcc significantly with a fcm like iterative process of pifcm to meet the big data challenge we further parallelize fcc on the spark platform with both vertical and horizontal segmentation schemes extensive experiments on various real world datasets demonstrate the excellent performance of fcc even with a majority of poor basic partitions in particular our method exhibits interesting potential for big data clustering in two real life applications concerned with online event detection and overlapping community detection respectively

fuzzy rule base ensemble generated from data by linguistic associations mining as there are many various methods for time series prediction developed but none of them generally outperforms all the others there always exists a danger of choosing a method that is inappropriate for a given time series to overcome such a problem distinct ensemble techniques that combine several individual forecasts are being proposed in this contribution we employ the so called fuzzy rule based ensemble this method is constructed as a linear combination of a small number of forecasting methods where the weights of the combination are determined by fuzzy rule bases based on time series features such as trend seasonality or stationarity for identification of fuzzy rule bases we use the linguistic association mining a huge experimental justification is provided

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

group consensus algorithms based on preference relations in many group decision making situations decision makers preferences for alternatives are expressed in preference relations including fuzzy preference relations and multiplicative preference relations an important step in the process of aggregating preference relations is to determine the importance weight of each preference relation in this paper we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations we further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations finally we include an illustrative example

a concise consensus support model for group decision making with reciprocal preference relations based on deviation measures supporting the process of consensus building has become a major research field in group decision making gdm the purpose of this paper is to present a consistency consensus based decision support model for gdm based on reciprocal preference relations inspired by the concept of a deviation measure between two matrices an individual consistency index and a group consensus index are presented in the support model a consistency control process is designed to make an inconsistent reciprocal preference relation one of acceptable consistency to achieve a predefined consensus level an interactive consensus reaching process is provided where the group reciprocal preference relation is seen as a reference when experts change their preferences theoretical foundations are then established for the proposed model the proposed model revises the original information in a simple and straightforward gradual way in the consensus process the individual rationality level in terms of consistency is maintained two examples are examined to demonstrate the model in practical problems

an extension on statistical comparisons of classifiers over multiple data sets for all pairwise comparisons in a recently published paper in jmlr dem sar 2006 recommends a set of non parametric statistical tests and procedures which can be safely used for comparing the performance of classifiers over multiple data sets after studying the paper we realize that the paper correctly introduces the basic procedures and some of the most advanced ones when comparing a control method however it does not deal with some advanced topics in depth regarding these topics we focus on more powerful proposals of statistical procedures for comparing n n classifiers moreover we illustrate an easy way of obtaining adjusted and comparable p values in multiple comparison procedures

multi attribute aggregation operators this paper deals with the idea of aggregation a new enlarged notion of aggregation operator is given along with a classification of classical properties into some main groups the concept of multi attribute aggregation operator which incorporates many classical aggregation methods is provided an extension of different properties to multi attribute aggregation operators is proposed

binary survival aggregation functions we introduce a transformation that acts on binary aggregation functions and that generalizes the transformation that maps copulas a well studied class of binary aggregation functions with a profound probabilistic interpretation to their associated survival copulas the new transformation called double flipping is the composition of two elementary flipping transformations introduced earlier each operating on one of the arguments of the aggregation function we lay bare the relationships between these elementary flipping operations and double flipping we study invariants under these transformations furthermore we characterize the transformations that preserve flippability most importantly we characterize different subclasses of flippable aggregation functions in particular aggregation functions that have an absorbing element or that have a neutral element in this investigation the key role played by quasi copulas and their dual operations is highlighted these findings support the introduction of the term survival aggregation function

cluster ensembles a knowledge reuse framework for combining multiple partitions this paper introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings we first identify several application scenarios for the resultant knowledge reuse framework that we call cluster ensembles the cluster ensemble problem is then formalized as a combinatorial optimization problem in terms of shared mutual information in addition to a direct maximization approach we propose three effective and efficient techniques for obtaining high quality combiners consensus functions the first combiner induces a similarity measure from the partitionings and then reclusters the objects the second combiner is based on hypergraph partitioning the third one collapses groups of clusters into meta clusters which then compete for each object to determine the combined clustering due to the low computational costs of our techniques it is quite feasible to use a supra consensus function that evaluates all three approaches against the objective function and picks the best solution for a given situation we evaluate the effectiveness of cluster ensembles in three qualitatively different application scenarios i where the original clusters were formed based on non identical sets of features ii where the original clustering algorithms worked on non identical sets of objects and iii where a common data set is used and the main purpose of combining multiple clusterings is to improve the quality and robustness of the solution promising results are obtained in all three situations for synthetic as well as real data sets

some issues on consistency of fuzzy preference relations in decision making in order to avoid misleading solutions the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions in decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property in this paper a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented using this new characterization a method for constructing consistent fuzzy preference relations from a set of n 1 preference data is proposed applying this method it is possible to assure better consistency of the fuzzy preference relations provided by the decision makers and in such a way to avoid the inconsistent solutions in the decision making processes additionally a similar study of consistency is developed for the case of multiplicative preference relations 2002 elsevier b v all rights reserved

transitivity frameworks for reciprocal relations cycle transitivity versus fg transitivity for a reciprocal relation q on a set of alternatives a two transitivity frameworks which generalize both t transitivity and stochastic transitivity are compared the framework of cycle transitivity introduced by the present authors soc choice welf to appear and which is based upon the ordering of the numbers q a b q b c and q c a for all a b c a 3 and the framework of fg transitivity introduced by switalski fuzzy sets and systems 137 2003 85 as an immediate generalization of stochastic transitivity the rules that enable to express fg transitivity in the form of cycle transitivity and cycle transitivity in the form of fg transitivity illustrate that for reciprocal relations the concept of cycle transitivity provides a framework that can cover more types of transitivity than does the concept of fg transitivity

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal