a fuzzy extension of the silhouette width criterion for cluster analysis the present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis this measure named fuzzy silhouette is a generalization to the fuzzy case of the average silhouette width criterion originally conceived to assess crisp non fuzzy data partitions the fuzzy silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm in addition it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters the performance of the fuzzy silhouette is evaluated and compared to that of five well known cluster validity measures six data sets are used to illustrate different scenarios in which the proposed fuzzy silhouette performs similar to or better than these other criteria thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis

segmentation and classification of range images the recognition of objects in three dimensional space is a desirable capability of a computer vision system range images which directly measure 3 d surface coordinates of a scene are well suited for this task in this paper we report a procedure to detect connected planar convex and concave surfaces of 3 d objects this is accomplished in three stages the first stage segments the range image into surface patches by a square error criterion clustering algorithm using surface points and associated surface normals the second stage classifies these patches as planar convex or concave based on a non parametric statistical test for trend curvature values and eigenvalue analysis in the final stage boundaries between adjacent surface patches are classified as crease or noncrease edges and this information is used to merge compatible patches to produce reasonable faces of the object s this procedure has been successfully applied to a large number of real and synthetic images four of which we present in this paper

image filtering edge detection and edge tracing using fuzzy reasoning we characterize the problem of detecting edges in images as a fuzzy reasoning problem the edge detection problem is divided into three stages filtering detection and tracing images are filtered by applying fuzzy reasoning based on local pixel characteristics to control the degree of gaussian smoothing filtered images are then subjected to a simple edge detection algorithm which evaluates the edge fuzzy membership value for each pixel based on local image characteristics finally pixels having high edge membership are traced and assembled into structures again using fuzzy reasoning to guide the tracing process the filtering detection and tracing algorithms are tested on several test images comparison is made with a standard edge detection technique

a comparison of discrete algorithms for fuzzy weighted average fuzzy weighted average fwa which can be applied to various fields such as engineering design decision analysis etc and as function of fuzzy numbers is suitable for the problem of multiple occurrences of fuzzy parameters additional fuzziness may be introduced in the alpha cut arithmetic this paper reviews and compares discrete algorithms for the fwas in both theoretical comparison and numerical comparisons as opposed to the linear programming algorithms that may be efficient but require the help of linear programming software an alternative efficient algorithm is also proposed the algorithm introduces an all candidate criteria ratings weights replaced benchmark adjusting procedure other than a binary dichotomy search in the existing methods the theoretical worst case comparison shows that the algorithms of guu and guh and our alternative algorithm require the same order of calculation complexity o n where n is the number of fwa terms yet the guh algorithm always requires 2 n 1 calculations to be performed the algorithm of guu requires the elemental comparison complexity o n and is superior to the other algorithms in the worst case on the other hand our alternative algorithm and lee and park algorithm have the same comparison complexity o n log n yet our alternative algorithm requires o n calculation complexity which is better than o n log n of the lee and park algorithm the numerical experiments show that the algorithm of guu and our alternative algorithm generally perform and converge faster by requiring fewer calculations than that of lee and park s algorithm the alternative algorithm requires a slightly smaller number of calculations than that of the guu algorithm due to the use of the convergent benchmark adjustment procedure rather than a somewhat fixed binary search conversely in the number of element comparisons lee and park s algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used the alternative algorithm also requires fewer comparisons than that of the guu algorithm furthermore in comparison of the average cpu time requirements the ranks of the results of these algorithms are consistent with those of the algorithms for the average numbers of calculations in general the alternative algorithm may provide an efficient alternative algorithm to the fwas in the worse case situation the guu algorithm still may be considered as an efficient alternative

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

calculating functions of fuzzy numbers abstract this paper discusses the calculation of functions of fuzzy numbers our general approach follows that of the fuzzy weighted algorithm fwa of wong and dong however by developing a classification scheme for functions and their arguments we show that there are many cases for which the fwa and all other published methods will give wrong results we suggest an alternative approach which will work in all cases for certain restricted classes of functions we develop methods which require less computation than the fwa these algorithms have been implemented in a constraint based logic programming language examples are given to illustrate the method and its relation to other discrete methods and the exact nonlinear programming approach

general c means clustering model partitional clustering is an important part of cluster analysis based on various theories numerous clustering algorithms have been developed and new clustering algorithms continue to appear in the literature it is known that occam s razor plays a pivotal role in data based models and partitional clustering is categorized as a data based model however no relation had previously been discovered between occam s razor and partitional clustering as we discuss in this paper the three main contributions of this paper can be summarized as follows 1 according to a novel definition of the mean a unifying generative framework for partitional clustering algorithms called a general c means clustering model gcm is presented and studied and 2 based on the local optimality test of the gcm the connection between occam s razor and partitional clustering is established for the first time as its application a comprehensive review of the existing objective function based clustering algorithms is presented based on gcm 3 under a common assumption about partitional clustering a theoretical guide for devising and implementing clustering algorithm is discovered these conclusions are verified by numerical experimental results

dynamic journeying under uncertainty abstract we introduce a journey planning problem in multi modal transportation networks under uncertainty the goal is to find a journey possibly involving transfers between different transport services from a given origin to a given destination within a specified time horizon due to uncertainty in travel times the arrival times of transport services at public transport stops are modeled as random variables if a transfer between two services is rendered unsuccessful the commuter has to reconsider the remaining path to the destination the problem is modeled as a markov decision process in which states are defined as paths in the transport network the main contribution is a backward induction method that generates an optimal policy for traversing the public transport network in terms of maximizing the probability of reaching the destination in time by assuming history independence and independence of successful transfers between services we obtain approximate methods for the same problem analysis and numerical experiments suggest that while solving the path dependent model requires the enumeration of all paths from the origin to the destination the proposed approximations may be useful for practical purposes due to their computational simplicity in addition to on time arrival probability we show how travel and overdue costs can be taken into account making the model applicable to freight transportation problems

computation of the output of a function with fuzzy inputs based on a low rank tensor approximation we apply a derivative free optimization method based on novel low rank tensor methods to the problem of propagating fuzzy uncertainty through a continuous real valued function adhering to zadeh s extension principle such a problem can be reformulated as a sequence of optimization problems over nested search spaces the optimization method we use is based on a low rank tensor approximation of the function sampled on a grid and a search for the minimal and maximal entries in this low rank tensor in contrast to classical fuzzy uncertainty propagation methods such as the vertex method and the transformation method the method we propose does not exhibit an inherent exponential scaling for increasing dimension of the search space obviously no derivative free optimization algorithm can exist which shows sub exponential scaling with the dimension for all possible continuous functions the algorithm that we present here however can exploit a specific type of structure and regularity beyond continuity that is often present in real world optimization problems we illustrate this with some high dimensional numerical examples where the presented method clearly outperforms some established derivative free optimization codes

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

closed form fuzzy interpolation fuzzy interpolation enhances the robustness of fuzzy systems and helps to reduce systems complexity although a number of important fuzzy rule interpolation approaches have been proposed in the literature most of these approaches cannot be expressed in a closed form this is usually caused by the effort to avoid possible invalid interpolated results this paper proposes a different fuzzy rule interpolation approach it not only can be represented in a closed form but also guarantees that the interpolated results are valid fuzzy sets this approach is based on a direct use of the extension principle which has been widely utilised for the development of a variety of fuzzy systems the mathematical properties of the proposed approach are analysed by taking the advantage of the closed form representation this approach has been applied to a practical problem of predicting diarrhoeal disease rates in remote villages the results demonstrate the potential of the proposed work in enhancing the robustness of fuzzy interpolation

a novel approach to fuzzy clustering based on a dissimilarity relation extracted from data using a ts system clustering refers to the process of unsupervised partitioning of a data set based on a dissimilarity measure which determines the cluster shape considering that cluster shapes may change from one cluster to another it would be of the utmost importance to extract the dissimilarity measure directly from the data by means of a data model on the other hand a model construction requires some kind of supervision of the data structure which is exactly what we look for during clustering so the lower the supervision degree used to build the data model the more it makes sense to resort to a data model for clustering purposes conscious of this we propose to exploit very few pairs of patterns with known dissimilarity to build a ts system which models the dissimilarity relation among other things the rules of the ts system provide an intuitive description of the dissimilarity relation itself then we use the ts system to build a dissimilarity matrix which is fed as input to an unsupervised fuzzy relational clustering algorithm denoted any relation clustering algorithm arca which partitions the data set based on the proximity of the vectors containing the dissimilarity values between each pattern and all the other patterns in the data set we show that combining the ts system and the arca algorithm allows us to achieve high classification performance on a synthetic data set and on two real data sets further we discuss how the rules of the ts system represent a sort of linguistic description of the dissimilarity relation

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

left ventricle automated detection method in gated isotopic ventriculography using fuzzy clustering a method that uses the fuzzy isodata clustering algorithm and fourier analysis is proposed for automated detection of heart left ventricle contours this operation is used for quantitative analysis of cardiac function the computation begins by finding the phase image the fuzzy isodata algorithm is first applied to this image to generate a number of clusters that correspond to the organ substructures ventricles atria second the ventricles cluster is isolated and the intensities of its points are replaced by the corresponding ones from the original end diastolic frame finally a reduced image representing the ventricular region is obtained and an additional clustering is performed to find the left ventricular boundary automatically this algorithm is tested by application of 105 sets of 16 images each these results are compared with the measurements obtained with two semi automatic methods used respectively on the philips and the sopha medical gamma cameras

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

note an improved conditional monte carlo technique for the stochastic shortest path problem this paper describes a simulation procedure for estimating the distribution function of the shortest path length in a network with random arc lengths the method extends the concept of conditional monte carlo utilizing special properties of the uniformly directed cutsets and the unique arcs the objective here is to reduce the sampling effort and utilize known probability information to derive multivariate integrals of lower dimension the experimental results show that the proposed method is substantially cost effective and performs better than traditional monte carlo and conditional methods

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

kernel induced fuzzy clustering of image pixels with an improved differential evolution algorithm a modified differential evolution de algorithm is presented for clustering the pixels of an image in the gray scale intensity space the algorithm requires no prior information about the number of naturally occurring clusters in the image it uses a kernel induced similarity measure instead of the conventional sum of squares distance use of the kernel function makes it possible to partition data that is linearly non separable and non hyper spherical in the original input space into homogeneous groups in a transformed high dimensional feature space a novel search variable representation scheme is adopted for selecting the optimal number of clusters from several possible choices extensive performance comparison over a test suite of 10 gray scale images and objective comparison with manually segmented ground truth indicates that the proposed algorithm has an edge over a few state of the art algorithms for automatic multi class image segmentation

bootstrap technique in cluster analysis abstract we define a method to estimate the number of clusters in a data set e using the bootstrap technique this approach involves the generation of several fake data sets by sampling patterns with replacement in e bootstrapping for each number k of clusters a measure of stability of the k cluster partitions over the bootstrap samples is used to characterize the significance of the k cluster partition for the original data set the value of k which provides the most stable partitions is the estimate of the number of clusters in e the performance of this new technique is demonstrated on both synthetic and real data and is applied to the segmentation of range images

calculating functions of interval type 2 fuzzy numbers for fault current analysis in this work functions of type 2 fuzzy numbers are analyzed for the special case of interval type 2 fuzzy numbers the type 2 membership function of the output variable is calculated using the lower and upper membership functions of the input variables and the vertex method this procedure is used in an application where the type 2 fuzzy fault currents of an electric distribution system are calculated the results are shown and the advantages of the approach are discussed

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

stochastic shortest path problems with piecewise linear concave utility functions this paper considers a stochastic shortest path problem where the arc lengths are independent random variables following a normal distribution in this problem the optimal path is one that maximizes the expected utility with the utility function being piecewise linear and concave such a utility function can be used to approximate nonlinear utility functions that capture risk averse behaviour for a wide class of problems the principal contribution of this paper is the development of exact algorithms to solve large problem instances two algorithms are developed and incorporated in labelling procedures computational testing is done to evaluate the performance of the algorithms overall both algorithms are very effective in solving large problems quickly the relative performance of the two algorithms is found to depend on the curvature of the piecewise linear utility function

enhancement of the classification and reconstruction performance of fuzzy c means with refinements of prototypes abstract owning to their abilities to reveal structural relationships in data fuzzy clustering plays a pivotal role in fuzzy modeling pattern recognition and data analysis as supporting an unsupervised mode of learning fuzzy clustering brings about unique opportunities to build a structural backbone of numerous constructs in the areas identified above a follow up phase is required when the structural findings developed in the form of fuzzy clusters need some refinements usually when the clustering results are used afterwards in the model being developed in a supervised mode following this general line of thought in this study we propose a novel approach to optimize the clustering and classification performance of the fuzzy c means fcm algorithm proceeding with a collection of clusters information granules produced by the fcm we carry out the refinements of the results in order to improve the representation or classification capabilities of fuzzy clusters by adjusting a location of the prototypes so that a certain performance index becomes optimized at this phase the optimization is carried out in a supervised mode with the aid of differential evolution de we propose five different strategies to adjust a location of the prototypes experimental studies completed on synthetic data and publicly available real world data quantify the improvement of the representation and classification abilities of the clustering method

a novel cluster validity index for fuzzy clustering based on bipartite modularity a novel cluster validity index whose implementation is based on the membership degrees and improved bipartite modularity of bipartite network is proposed for the validation of partitions produced by the fuzzy c means fcm algorithm fcm algorithm is employed to group the dataset in order to obtain the membership degree of samples then a weighted bipartite network is constructed by samples and centroids of each cluster this allows the introduction of a new measurement for optimizing the numbers of clusters for fuzzy partitions the proposed index utilizes the optimum membership as its global property and the modularity of bipartite network as its local independent property the proposed index is compared with a number of popular validation indices on fifteen datasets the experimental results show that the effectiveness and reliability of the proposal is superior to other indices

a clustering performance measure based on fuzzy set decomposition clustering is primarily used to uncover the true underlying structure of a given data set and for this purpose it is desirable to subject the same data to several different clustering algorithms this paper attempts to put an order on the various partitions of a data set obtained from different clustering algorithms the goodness of each partition is expressed by means of a performance measure based on a fuzzy set decomposition of the data set under consideration several experiments reported in here show that the proposed performance measure puts an order on different partitions of the same data which is consistent with the error rate of a classifier designed on the basis of the obtained cluster labelings

performance evaluation of some clustering algorithms and validity indices in this article we evaluate the performance of three clustering algorithms hard k means single linkage and a simulated annealing sa based technique in conjunction with four cluster validity indices namely davies bouldin index dunn s index calinski harabasz index and a recently developed index i based on a relation between the index i and the dunn s index a lower bound of the value of the former is theoretically estimated in order to get unique hard k partition when the data set has distinct substructures the effectiveness of the different validity indices and clustering methods in automatically evolving the appropriate number of clusters is demonstrated experimentally for both artificial and real life data sets with the number of clusters varying from two to ten once the appropriate number of clusters is determined the sa based clustering technique is used for proper partitioning of the data into the said number of clusters

the extension principle and a decomposition of fuzzy sets we give an algorithm to decompose a fuzzy interval u using this decomposition and the multilinearization of a univariate function f we obtain an approximation of the fuzzy interval where is obtained from f by applying the extension principle we provide approximation bounds some numeric illustration is provided

a symmetry based multiobjective clustering technique for automatic evolution of clusters in this paper the problem of automatic clustering a data set is posed as solving a multiobjective optimization moo problem optimizing a set of cluster validity indices simultaneously the proposed multiobjective clustering technique utilizes a recently developed simulated annealing based multiobjective optimization method as the underlying optimization strategy here variable number of cluster centers is encoded in the string the number of clusters present in different strings varies over a range the points are assigned to different clusters based on the newly developed point symmetry based distance rather than the existing euclidean distance two cluster validity indices one based on the euclidean distance xb index and another recently developed point symmetry distance based cluster validity index sym index are optimized simultaneously in order to determine the appropriate number of clusters present in a data set thus the proposed clustering technique is able to detect both the proper number of clusters and the appropriate partitioning from data sets either having hyperspherical clusters or having point symmetric clusters a new semi supervised method is also proposed in the present paper to select a single solution from the final pareto optimal front of the proposed multiobjective clustering technique the efficacy of the proposed algorithm is shown for seven artificial data sets and six real life data sets of varying complexities results are also compared with those obtained by another multiobjective clustering technique mock two single objective genetic algorithm based automatic clustering techniques vgaps clustering and gcuk clustering

an efficient approach for unsupervised fuzzy clustering based on grouping evolution strategies fuzzy data clustering plays an important role in practical use and has become a prerequisite step for decision making in fuzzy environment in this paper we propose a new algorithm called fuzzyges for unsupervised fuzzy clustering based on adaptation of the recently proposed grouping evolution strategy ges to adapt ges for fuzzy clustering we devise a fuzzy counterpart of the grouping mutation operator typically used in ges and employ it in a two phase procedure to generate a new clustering solution unlike conventional clustering algorithms which should get the number of clusters as an input fuzzyges tries to determine the true number of clusters as well as providing the optimal cluster centroids after several iterations the proposed approach is validated through using several data sets and results are compared with those of fuzzy c means algorithm particle swarm optimization algorithm pso differential evolution de and league championship algorithm lca we also investigate the performance of fuzzyges through using different cluster validity indices our results indicate that fuzzyges is fast and provides acceptable results in terms of both determining the correct number of clusters and the accurate cluster centroids

technical note a note on the stochastic shortest route problem an optimality index for choosing the shortest route in an acyclic stochastic network has recently been proposed the present note identifies certain anomalies in using that index and proposes another optimality index the results of using these indices are compared in a simple example

arc reduction and path preference in stochastic acyclic networks the paper presents a heuristic for determining the path that maximizes the expected utility of a stochastic acyclic network the focus is on shortest route problems where a general nonlinear utility function is used to measure outcomes for such problems enumerating all feasible paths is the only way to guarantee that the global optimum has been found alternatively we develop a reduction algorithm based on stochastic dominance to speed up the computations monte carlo simulation is used to evaluate the approach in all 70 test problems comprising 20 to 60 nodes are randomly generated and analyzed the results indicate that the heuristic produces significant computational saving as the size of the network grows and that the quality of the reduced network solutions are better than those obtained from the original formulation

fuzzy system modeling by fuzzy partition and ga hybrid schemes abstract this paper presents an approach to building multi input and single output fuzzy models such a model is composed of fuzzy implications and its output is inferred by simplified reasoning the implications are automatically generated by the structure and parameter identification in structure identification the optimal or near optimal number of fuzzy implications is determined in view of valid partition of data set the parameters defining the fuzzy implications are identified by a ga genetic algorithm hybrid scheme to minimize mean square errors globally numerical examples are provided to evaluate the feasibility of the proposed approach comparison shows that the suggested approach can produce a fuzzy model with higher accuracy and a smaller number of fuzzy implications than the ones achieved previously in other methods the proposed approach has also been applied to construct a fuzzy model for the navigation control of a mobile robot the validity of the resultant model is demonstrated by experimentation

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

some new qualitative insights into quality of fuzzy rule based models rules in fuzzy rule based models convey essential knowledge about the system under discussion as such they capture the essence of relationships occurring among input and output variables while the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules although to a limited extent the quality of rules being regarded as generic pieces of knowledge has not been studied so far in this study we formulate and investigate this problem by looking at the quality of rules including aspects of a stability b generalizability and c conflict we identify a concept of rule multiplicity conflict and study an emergence of rule generalization a number of pertinent performance indices are developed and their usage is presented through a series of experimental studies

computing expensive multivariate functions of fuzzy numbers using sparse grids fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models with zadeh s extension principle one can obtain a fuzzy extension of any objective function computing expensive multivariate functions of fuzzy numbers however often poses a difficult problem due to non applicability of common fuzzy arithmetic algorithms severe overestimation or very high computational complexity this paper proposes a new approach based on sparse grids consisting of two parts first we compute a surrogate function using sparse grid interpolation second we perform the fuzzy valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic the new approach gives accurate results and requires only few function evaluations

iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data to deal with data patterns with linguistic ambiguity and with probabilistic uncertainty in a single framework we construct an interpretable probabilistic fuzzy rule based system that requires less human intervention and less prior knowledge than other state of the art methods specifically we present a new iterative fuzzy clustering algorithm that incorporates a supervisory scheme into an unsupervised fuzzy clustering process the learning process starts in a fully unsupervised manner using fuzzy c means fcm clustering algorithm and a cluster validity criterion and then gradually constructs meaningful fuzzy partitions over the input space the corresponding fuzzy rules with probabilities are obtained through an iterative learning process of selecting clusters with supervisory guidance based on the notions of cluster pureness and class separability the proposed algorithm is tested first with synthetic data sets and benchmark data sets from the uci repository of machine learning database and then with real facial expression data and tv viewing data

lda based clustering algorithm and its application to an unsupervised feature extraction research has shown fuzzy c means fcm clustering to be a powerful tool to partition samples into different categories however the objective function of fcm is based only on the sum of distances of samples to their cluster centers which is equal to the trace of the within cluster scatter matrix in this study we propose a clustering algorithm based on both within and between cluster scatter matrices extended from linear discriminant analysis lda and its application to an unsupervised feature extraction fe our proposed methods comprise between and within cluster scatter matrices modified from the between and within class scatter matrices of lda the scatter matrices of lda are special cases of our proposed unsupervised scatter matrices the results of experiments on both synthetic and real data show that the proposed clustering algorithm can generate similar or better clustering results than 11 popular clustering algorithms k means k medoid fcm the gustafson kessel gath geva possibilistic c means pcm fuzzy pcm possibilistic fcm fuzzy compactness and separation a fuzzy clustering algorithm based on a fuzzy treatment of finite mixtures of multivariate student s t distributions algorithms and a fuzzy mixture of the student s t factor analyzers model the results also show that the proposed fe outperforms principal component analysis and independent component analysis

bounding the project completion time distribution in pert networks we consider the pert model of a project composed of activities whose durations are random variables with known distributions for the situations in which the activity durations are completely independent we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below the bounding distribution can be used to obtain an upper bound on the mean completion time of the project we also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds

a multiresolution image segmentation technique based on pyramidal segmentation and fuzzy clustering in this paper an unsupervised image segmentation technique is presented which combines pyramidal image segmentation with the fuzzy c means clustering algorithm each layer of the pyramid is split into a number of regions by a root labeling technique and then fuzzy c means is used to merge the regions of the layer with the highest image resolution a cluster validity functional is used to find the optimal number of objects automatically segmentation of a number of synthetic as well as clinical images is illustrated and two fully automatic segmentation approaches are evaluated which determine the left ventricular volume lv in 140 cardiovascular magnetic resonance mr images first fuzzy c means is applied without pyramids in the second approach the regions generated by pyramidal segmentation are merged by fuzzy c means the correlation coefficients of manually and automatically defined lv lumen of all 140 and 20 end diastolic images were equal to 0 86 and 0 79 respectively when images were segmented with fuzzy c means alone these coefficients increased to 0 90 and 0 93 when the pyramidal segmentation was combined with fuzzy c means this method can be applied to any dimensional representation and at any resolution level of an image series the evaluation study shows good performance in detecting lv lumen in mr images

analytical and numerical solutions of fuzzy differential equations in this paper we study analytical and numerical solutions of fuzzy differential equations based on the extension principle for linear fuzzy differential equations we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative in order to approximate the solutions of linear and non linear fuzzy differential equations we propose a new fuzzification of the classical euler method and then incorporate an unconstrained optimisation technique this combination offers a powerful tool to tackle uncertainty in any numerical method an efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain several illustrative examples are given

design of robust fuzzy model based controller with sliding mode control for siso nonlinear systems in this paper we present the design of a new type of fuzzy controllers for controlling complex single input single output systems by incorporating sliding mode control theory with fuzzy control technology first a fuzzy model of the given nonlinear system is constructed to represent the local dynamic behaviors of the given nonlinear system a global controller is then constructed by combining all local state feedback controllers and a global supervisory sliding mode controller improved robustness and tracking performance are obtained by a seamless integration of fuzzy model based control and sliding mode control in addition stability of controlling the original underlying nonlinear systems is guaranteed owing to the robustness of the sliding mode control method to system parameter uncertainties two simulation examples are included to show the effectiveness and feasibility of the developed integrated controller

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

the fuzzy kalman filter state estimation using possibilistic techniques a new method to implement fuzzy kalman filters is introduced the combination of possibilistic techniques and the extended kalman filter has special application in fields where inaccurate information is involved the novelty of this article comes from the fact that by using possibility distributions instead of gaussian distributions a fuzzy description of the expected state and observation is sufficient to obtain a good estimation some characteristics of this approach are that uncertainty does not need to be symmetric and that a wide region of possible values for the expectations is allowed to implement the algorithm this approach also contributes a method to propagate uncertainty through the process model and the observation model based on trapezoidal possibility distributions finally several examples of a real mobile robot moving through a localization process while using qualitative landmarks are shown

detecting change points for shifts in mean and variance using fuzzy classification maximum likelihood change point algorithms knowing the time of changes called change point cp in a process is crucial for engineers to recognize the root cause fast and accurately since special causes may induce simultaneous changes in mean and variance detecting changes in both at once is required many methodologies in quality control were developed for detecting changes in either mean or variance only and process parameters were assumed known often however they are rarely known exactly and a small estimation error may lead to unfavorable cp estimates fuzzy partitioning is better suited to cases of vague boundaries between two segments which appear very often in reality a new mechanism called fuzzy classification maximum likelihood change point fcml cp algorithm is proposed to detect shifts in mean and variance simultaneously a cp framework is transferred into a mixture model and then a fcml cp algorithm is created through fuzzy classification maximum likelihood procedures the proposed fcml cp can be applied to phase i and ii processes without knowledge of in control process parameters it can estimate multiple cps of process mean or and variance simultaneously the effectiveness and superiority of fcml cp are shown by extensive experiments with numerical and real data sets specifically the proposed fcml cp is superior to the commonly used statistical mixture likelihood approach using expectation maximization em algorithm it is much more time saving especially the remarkable performance of fcml cp in detecting cps for small changes is particularly important and helpful for engineers to recognize the special cause fast and correctly since an out of control signal resulted from small changes is usually delayed long we propose a new algorithm fcml cp to detect change points in a statistical process the mixture likelihood embedded fuzzy c partition is utilized to estimate the change points the method outperforms statistical likelihood approaches various experiments show the effectiveness and practicability of the proposed method excellent performance in detecting small changes is helpful for root cause analyses

fuzzy system modeling in pharmacology an improved algorithm in this paper we propose an improved fuzzy system modeling algorithm to address some of the limitations of the existing approaches identified during our modeling with pharmacological data this algorithm differs from the existing ones in its approach to the cluster validity problem i e number of clusters the projection schema i e input membership assignment and rule determination and significant input determination the new algorithm is compared with the bazoon turksen model which is based on the well known sugeno yasukawa approach the comparison was made in terms of predictive performance using two different data sets the first comparison was with a two variable nonlinear function prediction problem and the second comparison was with a clinical pharmacokinetic modeling problem it is shown that the proposed algorithm provides more precise predictions determining the degree of significance for each input variable allows the user to distinguish their relative importance

a novel binary artificial bee colony algorithm based on genetic operators this study proposes a novel binary version of the artificial bee colony algorithm based on genetic operators gb abc such as crossover and swap to solve binary optimization problems integrated to the neighbourhood searching mechanism of the basic abc algorithm the modification comprises four stages 1 in neighbourhood of a current food source randomly select two food sources from population and generate a solution including zeros zero outside the population 2 apply two point crossover operator between the current two neighbourhood global best and zero food sources to create children food sources 3 apply swap operator to the children food sources to generate grandchildren food sources and 4 select the best food source as a neighbourhood food source of the current solution among the children and grandchildren food sources in this way the global local search ability of the basic abc algorithm is improved in binary domain the effectiveness of the proposed algorithm gb abc is tested on two well known binary optimization problems dynamic image clustering and 0 1 knapsack problems the obtained results clearly indicate that gb abc is the most suitable algorithm in binary optimization when compared with the other well known existing binary optimization algorithms in addition the achievement of the proposed algorithm is supported by applying it to the cec2005 benchmark numerical problems

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

cluster validity for fuzzy clustering algorithms abstract the proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm it is assumed that the output of an algorithm includes a fuzzy nembership function for each data point we show how to compute the proportion of possible memberships whose maximum entry exceeds the maximum entry of a given membership function and use these proportions to define the proportion exponent its use as a validity functional is illustrated with four numerical examples and its effectiveness compared to other validity functionals namely classification entropy and partition coefficient

clustering of the self organizing map using a clustering validity index based on inter cluster and intra cluster density the self organizing map som has been widely used in many industrial applications classical clustering methods based on the som often fail to deliver satisfactory results specially when clusters have arbitrary shapes in this paper through some preprocessing techniques for filtering out noises and outliers we propose a new two level som based clustering algorithm using a clustering validity index based on inter cluster and intra cluster density experimental results on synthetic and real data sets demonstrate that the proposed clustering algorithm is able to cluster data better than the classical clustering algorithms based on the som and find an optimal number of clusters

use of a fuzzy granulation degranulation criterion for assessing cluster validity the identification of a suitable clustering algorithm to partition data and assessment of the validity of the resultant partitioning are ongoing quests in unsupervised learning in this study a fuzzy granulation degranulation criterion is proposed to evaluate the goodness of a fuzzy partitioning of the data this in turn is used to determine the appropriate clustering algorithm suitable for a particular data set in general the quality of a partitioning is measured by computing the variance within it which is a measure of compactness of the obtained partitioning here a new error function which reflects how well the computed cluster centers represent the whole data set is used as the goodness measure of the obtained partitioning thus a clustering algorithm providing a good set of cluster centers which approximate well the whole data set is considered to be the most suited thereafter this new fuzzy granulation degranulation criterion is used to develop six new cluster validity indices these indices mimic the definitions of the existing and well known cluster validity indices such as pbm index xb index ps index fs index k index and sv index but use the new fuzzy granulation degranulation based error function instead of cluster compactness in order to evaluate the effectiveness of the proposed error function in correctly identifying the appropriate clustering algorithm for a particular data set eight well known clustering algorithms k means fuzzy c means gak means genetic algorithm based k means algorithm a newly developed genetic point symmetry based clustering technique gaps clustering average linkage clustering algorithm expectation maximization em clustering algorithm self organizing map som and spectral clustering technique are evaluated on a set of six artificially generated and six real life data sets results show that gak means is the most appropriate for most of the data sets used for the experiments thereafter the effectiveness of the proposed cluster validity indices in identifying the appropriate number of clusters automatically from different data sets are shown for above mentioned 12 data sets for the purpose of comparison results obtained with the original versions of the proposed cluster validity indices and results obtained by a density based clustering technique are also presented

cluster validity for the fuzzy c means clustering algorithrm the uniform data function is a function which assigns to the output of the fuzzy c means fc m or fuzzy isodata algorithm a number which measures the quality or validity of the clustering produced by the algorithm for the preselected number of cluster c the fc m algorithm produces c vectors in the space in which the data lie called cluster centers which represent points about which the data are concentrated it also produces for each data point c membership values numbers between zero and one which measure the similarity of the data points to each of the cluster centers it is these membership values which indicate how the point is classified they also indicate how well the point has been classified in that values close to one indicate that the point is close to a particular center but uniformly low memberships indicate that the point has not been classified clearly the uniform data functional udf combines the memberships in such a way as to indicate how well the data have been classified and is computed as follows for each data point compute the ratio of its smallest membership to its largest and then compute the probability that one could obtain a smaller ratio indicating better classification from a clustering of a standard data set in which there is no cluster structure these probabilities are then averaged over the data set to obtain the values of the udf

an approach to theory of fuzzy signals basic definitions the paper presents a new approach to the description of fourier transform and correlation function taking into consideration the uncertainty of the transformed function a conventional approach to uncertainty employs probabilistic description here fuzzy logic theory is applied to describe this uncertainty new definitions namely fuzzy fourier transform fuzzy correlation and possibility correlation are proposed the definition of correlation function is based on the concept of fuzzy functional spaces some examples of applications and a comparison with the conventional approach are shown

fuzzy radial basis function neural networks with information granulation and its parallel genetic optimization fuzzy modeling of complex systems is a challenging task which involves important problems of dimensionality reduction and calls for various ways of improving the accuracy of modeling the ig frbfnn a hybrid architecture of the ig fis fuzzy inference system and frbfnn fuzzy radial basis function neural networks is proposed to address these problems the paper is concerned with the analysis and design of ig frbfnns and their optimization by means of the hierarchical fair competition based parallel genetic algorithm hfc pga in the proposed network the membership functions of the premise part of the fuzzy rules of the ig based frbfnn model directly rely on the computation of the relevant distance between data points and the use of four types of polynomials such as constant linear quadratic and modified quadratic are considered for the consequent part of fuzzy rules moreover the weighted least square wls learning is exploited to estimate the coefficients of the polynomial forming the conclusion part of the rules since the performance of the ig rbfnn model is affected by some key design parameters such as a specific subset of input variables the fuzzification coefficient of the fcm the number of rules and the order of polynomial of the consequent part of fuzzy rules it becomes beneficial to carry out both structural as well as parametric optimization of the network in this study the hfc pga is used as a comprehensive optimization vehicle the performance of the proposed model is illustrated by means of several representative numerical examples

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

hierarchical unsupervised fuzzy clustering a recursive algorithm for hierarchical fuzzy partitioning is presented the algorithm has the advantages of hierarchical clustering while maintaining fuzzy clustering rules each pattern can have a nonzero membership in more than one subset of the data in the hierarchy optimal feature extraction and reduction is optionally reapplied for each subset combining hierarchical and fuzzy concepts is suggested as a natural feasible solution to the cluster validity problem of real data the convergence and membership conservation of the algorithm are proven the algorithm is shown to be effective for a variety of data sets with a wide dynamic range of both covariance matrices and number of members in each class

a clustering procedure for exploratory mining of vector time series a two step procedure is developed for the exploratory mining of real valued vector multivariate time series using partition based clustering methods the proposed procedure was tested with model generated data multiple sensor based process data as well as simulation data the test results indicate that the proposed procedure is quite effective in producing better clustering results than a hidden markov model hmm based clustering method if there is a priori knowledge about the number of clusters in the data two existing validity indices were tested and found ineffective in determining the actual number of clusters determining the appropriate number of clusters in the case that there is no a priori knowledge is a known unresolved research issue not only for our proposed procedure but also for the hmm based clustering method and further development is necessary

estimating network characteristics in stochastic activity networks this paper describes a monte carlo method based on the theory of quasirandom points for estimating the distribution functions and means of network completion time and shortest path time in a stochastic activity network in particular the method leads to estimators whose absolute errors converge as log kn k where k denotes the number of replications collected in the experiment and n is the number of dimensions for sampling this rate compares favorably with the standard error of estimate o1 k1 2 which obtains for experiments that use random sampling moreover since quasirandom points are nonrandom the upper bound log kn k is deterministic in contrast to the random sampling rate o1 k1 2 which is probabilistic the paper demonstrates how the use of a cutset of the network reduces n in the bound when estimating the distribution functions two examples illustrate the benefits and costs of using quasirandom points

a new fuzzy clustering validity index with a median factor for centroid based clustering determining the number of clusters which is usually approved by domain experts or evaluated by clustering validity indexes is an important issue in clustering analysis this study discusses the effectiveness of clustering validity indexes for centroid based partitional clustering algorithms most general purpose clustering validity indexes take the minimum maximum distance between a pair of data objects a pair of cluster centroids or an object and a centroid as an important evaluation factor however they may present unstable results especially when two centroids are allocated closely to alleviate this problem a new clustering validity index which is termed the wu and li index wli is proposed in this paper our proposed wli partially allows to some extent the existence of closely allocated centroids in the clustering results by considering not only the minimum but the median distances between a pair of centroids as well therefore possessing better stability the performances of wli and some existing clustering validity indexes are evaluated and compared by running the fuzzy c means algorithm for clustering various types of datasets including artificial datasets uci datasets and images experimental results have shown that wli has the more accurate and satisfactory performance than other indexes

exact algorithms for the stochastic shortest path problem with a decreasing deadline utility function in this paper the stochastic shortest path problem of determining a path that maximizes the expected utility is considered the nature of the utility function used to evaluate paths is of a decreasing deadline type the principal contribution of this paper is the development of exact algorithms that use two types of pruning techniques that are incorporated in labeling procedures one type of pruning makes use of the concept of local preference relations while the other type makes use of relaxations specifically two algorithms are developed each containing the same preference relation but two different relaxations our extensive computational testing indicate that both these algorithms are able to solve even large size problems quickly more importantly even for large problems both the algorithms solved them by enumerating a very small percentage of all paths

single point iterative weighted fuzzy c means clustering algorithm for remote sensing image segmentation in this paper a remote sensing image segmentation procedure that utilizes a single point iterative weighted fuzzy c means clustering algorithm is proposed based upon the prior information this method can solve the fuzzy c means algorithm s problem that the clustering quality is greatly affected by the data distributing and the stochastic initializing the centrals of clustering after the probability statistics of original data the weights of data attribute are designed to adjust original samples to the uniform distribution and added in the process of cyclic iteration which could be suitable for the character of fuzzy c means algorithm so as to improve the precision furthermore appropriate initial clustering centers adjacent to the actual final clustering centers can be found by the proposed single point adjustment method which could promote the convergence speed of the overall iterative process and drastically reduce the calculation time otherwise the modified algorithm is updated from multidimensional data analysis to color images clustering moreover with the comparison experiments of the uci data sets public berkeley segmentation dataset and the actual remote sensing data the real validity of proposed algorithm is proved

a set of practical experiments to validate the fuzzy kalman filter abstract the fuzzy kalman filter fkf was recently reformulated with a number of basic examples included in order to demonstrate its applicability we now look ahead to its practical implementation introducing deep analysis which includes very detailed simulations and complete real applications this will allow us to not only validate the fkf but also to draw important conclusions which will help users in the adjustment of the fuzzy filter parameters and the choosing of possibilistic distributions

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

gradual numbers and their application to fuzzy interval analysis in this paper we introduce a new way of looking at fuzzy intervals instead of considering them as fuzzy sets we see them as crisp sets of entities we call gradual real numbers they are a gradual extension of real numbers not of intervals such a concept is apparently missing in fuzzy set theory gradual numbers basically have the same algebraic properties as real numbers but they are functions a fuzzy interval is then viewed as a pair of fuzzy thresholds which are monotonic gradual real numbers this view enables interval analysis to be directly extended to fuzzy intervals without resorting to alpha cuts in agreement with zadeh s extension principle several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds our approach is illustrated on two known problems computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling

spl alpha cut fuzzy arithmetic simplifying rules and a fuzzy function optimization with a decision variable the problems of alpha cut fuzzy arithmetic have been shown like in interval arithmetic that distinct states of fuzzy parameters or fuzzy variable values may be chosen and produce an overestimated fuzziness meanwhile local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number s result previous approaches to overcoming these problems have appeared in literature yet the computational burden of these approaches became even heavier thus this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method these rules are devised through a function partitioned into subfunctions distinguishing the types of fuzzy parameter variable occurrences and types of subfunctions or functions with the various observations the improved efficiency has been found able to significantly reduce the combination vertex test of the vertex method for the fuzzy parameters alpha cut endpoints possibly to only a few fuzzy parameters endpoint combinations also as related a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument a single variable is examined with the vertex method as well a proper and useful preliminary algorithm is proposed numerical examples with results are provided