granular prototyping in fuzzy clustering we introduce a logic driven clustering in which prototypes are formed and evaluated in a sequential manner the way of revealing a structure in data is realized by maximizing a certain performance index objective function that takes into consideration an overall level of matching to be maximized and a similarity level between the prototypes the component to be minimized the prototypes identified in the process come with the optimal weight vector that serves to indicate the significance of the individual features coordinates in the data grouping represented by the prototype since the topologies of these groupings are in general quite diverse the optimal weight vectors are reflecting the anisotropy of the feature space i e they show some local ranking of features in the data space having found the prototypes we consider an inverse similarity problem and show how the relevance of the prototypes translates into their granularity

algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

a bernstein von mises theorem for smooth functionals in semiparametric models a bernstein von mises theorem is derived for general semiparametric functionals the result is applied to a variety of semiparametric problems in i i d and non i i d situations in particular new tools are developed to handle semiparametric bias in particular for nonlinear functionals and in cases where regularity is possibly low examples include the squared l 2 norm in gaussian white noise nonlinear functionals in density estimation as well as functionals in autoregressive models for density estimation a systematic study of bvm results for two important classes of priors is provided namely random histograms and gaussian process priors

bayesian factorizations of big sparse tensors it has become routine to collect data that are structured as multiway arrays tensors there is an enormous literature on low rank and sparse matrix factorizations but limited consideration of extensions to the tensor case in statistics the most common low rank tensor factorization relies on parallel factor analysis parafac which expresses a rank k tensor as a sum of rank one tensors in contingency table applications in which the sample size is massively less than the number of cells in the table the low rank assumption is not sufficient and parafac has poor performance we induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table taking a bayesian approach we place priors on terms in the factorization and develop an efficient gibbs sampler for posterior computation theory is provided showing posterior concentration rates in high dimensional settings and the methods are shown to have excellent performance in simulations and several

optimal bayesian estimation in random covariate design with a rescaled gaussian process prior in bayesian nonparametric models gaussian processes provide a popular prior choice for regression function estimation existing literature on the theoretical investigation of the resulting posterior distribution almost exclusively assume a fixed design for covariates the only random design result we are aware of van der vaart and van zanten 2011 assumes the assigned gaussian process to be supported on the smoothness class specified by the true function with probability one this is a fairly restrictive assumption as it essentially rules out the gaussian process prior with a squared exponential kernel when modeling rougher functions in this article we show that an appropriate rescaling of the above gaussian process leads to a rate optimal posterior distribution even when the covariates are independently realized from a known density on a compact set the proofs are based on deriving sharp concentration inequalities for frequentist kernel estimators the results might be of independent interest

dirichlet laplace priors for optimal shrinkage penalized regression methods such as l1 regularization are routinely used in high dimensional applications and there is a rich literature on optimality properties under sparsity assumptions in the bayesian paradigm sparsity is routinely induced through two component mixture priors having a probability mass at zero but such priors encounter daunting computational problems in high dimensions this has motivated continuous shrinkage priors which can be expressed as global local scale mixtures of gaussians facilitating computation in contrast to the frequentist literature little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution in this article we propose a new class of dirichlet laplace priors which possess optimal posterior concentration and lead to efficient posterior computation finite sample performance of dirichlet laplace priors relative to alternatives is assessed in simulated and real data examples

a semi supervised fuzzy clustering algorithm applied to gene expression data over the last decade there has been an increasing interest in semi supervised clustering several studies have suggested that even a small amount of supervised information can significantly improve the results of unsupervised learning one popular method of incorporating partial supervised information is through pair wise constraints indicating whether a certain pair of patterns should belong to the same must link or different dont link clusters in this study we propose a novel semi supervised fuzzy clustering algorithm ssfca the supervised information is incorporated via a method quantifying must link and or dont link constraints additionally we present an extension of ssfca that allows the algorithm to automatically detect the number of clusters in the data we apply ssfca to the intrinsic problem of gene expression profiles clustering the advantageous properties of fuzzy logic inherited to ssfca allow genes to belong to more than one group revealing this way more profound information concerning their multiple functioning roles finally we investigate the incorporation of prior biological knowledge arriving from gene ontology in the process of selecting pair wise constraints simulations on artificial and real life datasets proved that the proposed ssfca significantly outperformed other standard and semi supervised clustering methods

constructing a fuzzy controller from data abstract fuzzy control at the executive level can be interpreted as an approximation technique for a control function based on typical imprecisely specified input output tuples that are represented by fuzzy sets the imprecision is characterized by similarity relations that are induced by transformations of the canonical distance function between real numbers taking this interpretation of fuzzy controllers into account in order to derive a fuzzy controller from observed data typical input output tuples have to be identified in addition a concept of similarity based on a transformations of the canonical distance is needed in order to characterize the typical input output tuples by suitable fuzzy sets a variety of fuzzy clustering algorithms that are exactly working in this spirit exists they identify prototypes and assign fuzzy sets to the prototypes on the basis of a suitable transformed distance in this paper we discuss how such fuzzy clustering techniques can be applied to construct a fuzzy controller from data and introduce special clustering algorithms that are tailored for this problem

a bernstein von mises theorem in the nonparametric right censoring model in the recent bayesian nonparametric literature many examples have been reported in which bayesian estimators and posterior distributions do not achieve the optimal convergence rate indicating that the bernstein von mises theorem does not hold in this article we give a positive result in this direction by showing that the bernstein von mises theorem holds in survival models for a large class of prior processes neutral to the right we also show that for an arbitrarily given convergence rate n with 0 1 2 a prior process neutral to the right can be chosen so that its posterior distribution achieves the convergence rate n

clustering spatiotemporal data an augmented fuzzy c means in spatiotemporal data commonly encountered in geographical systems biomedical signals and the like each datum is composed of features comprising a spatial component and a temporal part clustering of data of this nature poses challenges especially in terms of a suitable treatment of the spatial and temporal components of the data in this study proceeding with the objective function based clustering such as e g fuzzy c means we revisit and augment the algorithm to make it applicable to spatiotemporal data an augmented distance function is discussed and the resulting clustering algorithm is provided two optimization criteria i e a reconstruction error and a prediction error are introduced and used as a vehicle to optimize the performance of the clustering method experimental results obtained for synthetic and real world data are reported

nonexistence results for a class of nonlinear elliptic equations involving critical sobolev exponents abstract in this paper we prove nonexistence results for some classes of nonlinear elliptic equations with critical growth of the form u g x u u 2 2 u x u 0 x u 0 x where 2 2 n n 2 g x u is a lower order perturbation of u 2 1 and is a bounded strictly star shaped domain in r n n 3 combining pohozaev s identity with classical and weak interpolation inequality we are able to exhibit examples of nonlinear problems without any nontrivial solution bifurcating from infinity in 0 for n 3 and n 4

a hierarchical neural model of matching abstract the issue of matching two fuzzy sets emerges as an essential design aspect in many algorithms of processing fuzzy information including fuzzy controllers pattern classifiers knowledge based systems etc this paper introduces a new model of matching the primordial features of the proposed model involve 1 i matching being carried out with respect to the grades of membership of the discussed fuzzy sets as well as some of their functionals like energy entropy transom 2 ii the concepts of hierarchies of matching in the model leading directly to a straightforward distinction between local and global levels of matching and 3 iii a distributed character of the model that is realized as a logic based neural network with enhanced learning capabilities

prototype construction and evaluation as inverse problems in pattern classification abstract a broad class of pattern recognition problems deals with a direct classification task it is based on measuring a distance between a pattern to be classified and some prototypes of classes studied in the problem quite frequently these prototypes are not provided and have to be computed a way of building prototypes of classes one is interested in not necessarily only those directly available in the training learning set is studied the algorithm is aided by a referential neural network structure also introduced are some self flagging mechanisms determining the feasibility of calculated prototypes the computational framework is constructed in terms of neural networks the conceptual knowledge representation counterpart of the classification problem is developed making use of selected concepts of fuzzy sets

a fuzzy approach for multitype relational data clustering mining interrelated data among multiple types of objects or entities is important in many real world applications despite extensive study on fuzzy clustering of vector space data very limited exploration has been made on fuzzy clustering of relational data that involve several object types in this paper we propose a new fuzzy clustering approach for multitype relational data fc mr in fc mr different types of objects are clustered simultaneously an object is assigned a large membership with respect to a cluster if its related objects in this cluster have high rankings in each cluster an object tends to have a high ranking if its related objects have large memberships in this cluster the fc mr approach is formulated to deal with multitype relational data with various structures the objective function of fc mr is locally optimized by an efficient iterative algorithm which updates the fuzzy membership matrix and the ranking matrix of one type at once while keeping those of other types constant we also discuss the simplified fc mr for multitype relational data with two special structures namely star structure and extended star structure experimental studies are conducted on benchmark document datasets to illustrate how the proposed approach can be applied flexibly under different scenarios in real world applications the experimental results demonstrate the feasibility and effectiveness of the new approach compared with existing ones

successive identification of a fuzzy model and its applications to prediction of a complex system abstract a successive identification method of a fuzzy model is suggested the identification mechanism consists of two levels one is the supervisor level and the other is the adjustment level the supervisor level determines a policy of parameter adjustment using a set of fuzzy adjustment rules the adjustment rules are derived from the fuzzy implications of a fuzzy model and are extended to fuzzy adjustment rules by using an extended concept of zadeh s contrast intensification the adjustment level executes the policy of parameter adjustment determined with the fuzzy adjustment rules the parameter adjustment consists of premise parameter adjustment and consequent parameter adjustment both of them are realized by the weighted recursive least square algorithm finally it is shown from two examples that the method is very useful for modeling complex systems

hierarchical neuro fuzzy quadtree models hybrid neuro fuzzy systems have been in evidence during the past few years due to its attractive combination of the learning capacity of artificial neural networks with the interpretability of the fuzzy systems this article proposes a new hybrid neuro fuzzy model named hierarchical neuro fuzzy quadtree hnfq which is based on a recursive partitioning method of the input space named quadtree the article describes the architecture of this new model presenting its basic cell and its learning algorithm the hnfq system is evaluated in three well known benchmark applications the sinc x y function approximation the mackey glass chaotic series forecast and the two spirals problem when compared to other neurofuzzy systems the hnfq exhibits competing results with two major advantages it automatically creates its own structure and it is not limited to few input variables

bayesian inverse problems with gaussian priors the posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter and the smoothness and scale of the prior correct combinations of these characteristics lead to the minimax rate the frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter with smoother priors leading to zero coverage and rougher priors to conservative coverage in the latter case credible sets are of the correct order of magnitude the results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive

fuzzy set based models of neurons and knowledge based networks we will introduce and study different fuzzy set oriented computational models of neurons the generic topologies of the neurons emerging there are significantly influenced by basic logic operators and or not encountered in the theory of fuzzy sets the logical flavor of the proposed constructs is expressed in terms of operators used in their formalization and a way of their superposition in the neurons the two broad categories of neurons embrace basic aggregation neurons named and and or neurons and referential processing units such as matching dominance inclusion neurons the specific features of the neurons are flexibly modeled with the aid of triangular norms the inhibitory and excitatory characteristics are captured by embodying direct and complemented negated input signals we will propose various topologies of neural networks put together with the use of these neurons and demonstrate straightforward relationships coming off between the problem specificity and the resulting architecture of the network this limpid way of mapping the domain knowledge onto the structure of the network contributes significantly toward enhancements in learning processes in the network and substantially facilitates its interpretation

alternating cluster estimation a new tool for clustering and function approximation many clustering models define good clusters as extrema of objective functions optimization of these models is often done using an alternating optimization ao algorithm driven by necessary conditions for local extrema we abandon the objective function model in favor of a generalized model called alternating cluster estimation ace ace uses an alternating iteration architecture but membership and prototype functions are selected directly by the user virtually every clustering model can be realized as an instance of ace out of a large variety of possible instances of non ao models we present two examples 1 an algorithm with a dynamically changing prototype function that extracts representative data and 2 a computationally efficient algorithm with hyperconic membership functions that allows easy extraction of membership functions we illustrate these non ao instances on three problems a simple clustering of plane data where we show that creating an unmatched ace algorithm overcomes some problems of fuzzy c means fcm ao and possibilistic c means pcm ao b functional approximation by clustering on a simple artificial data set and c functional approximation on a 12 input 1 output real world data set ace models work pretty well in all three cases

fuzzy shell clustering algorithms in image processing fuzzy c rectangular and 2 rectangular shells objective function based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors although there are special algorithms to discover lines the detection of rectangles needs further treatment a simple line detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary in this paper a special fuzzy shell clustering algorithm for rectangular contours is developed the principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles ellipses and lines

an output constrained clustering approach for the identification of fuzzy systems and fuzzy granular systems this paper presents an output constrained clustering approach for fuzzy system identification and fuzzy granular system identification the approach is unlike most existing clustering algorithms for structure identification of fuzzy systems where the focus is on input or combined input output clustering the output constrained clustering algorithm divides the output space into several partitions and each output partition is considered to be a constraint then input data are projected into clusters that are based on the input distribution constrained by the output partitions by introducing the key concept of separability of a set of clusters within each output constraint the proposed approach automatically finds an appropriate small and efficient number of clusters for each output constraint to have an appropriate small and efficient number of clusters in each output constraint results in a more compact final system structure and better accuracy this better performance is illustrated by experiments using benchmark function approximation and dynamic system identification

a generalization of distance functions for fuzzy c means clustering with centroids of arithmetic means fuzzy c means fcm is a widely used fuzzy clustering method which allows an object to belong to two or more clusters with a membership grade between zero and one despite the considerable efforts made by the clustering community the common characteristics of distance functions suitable for fcm remain unclear to fill this crucial void in this paper we first provide a generalized definition of distance functions that fit fcm directly the goal is to provide more flexibility to fcm in the choice of distance functions while preserving the simplicity of fcm by using the centroids of arithmetic means indeed we show that any distance function that fits fcm directly can be derived by a continuously differentiable convex function and thus is an instance of the generalized point to centroid distance p2c d by definition in addition we prove that if the membership grade matrix is nondegenerate any instance of the p2c d fits fcm directly finally extensive experiments have been conducted to demonstrate that the p2c d leads to the global convergence of fcm and that the clustering performances are significantly affected by the choices of distance functions

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

about the use of fuzzy clustering techniques for fuzzy model identification abstract in this work we present an alternative approach to generate fuzzy rules with a functional consequent associated to the tsk fuzzy model in our case using fuzzy clustering algorithms that look for linear behaviours in the product space of the input output data we analyse different methods to generate the associated fuzzy rules using in some cases multidimensional reference fuzzy sets in the product space of the input variables and in other cases fuzzy sets in each of the different dimensions in any case the rules being generated correspond to a tsk fuzzy model

the bernstein von mises theorem for the proportional hazard model we study large sample properties of bayesian analysis of the proportional hazard model with neutral to the right process priors on the baseline hazard function we show that the posterior distribution of the baseline cumulative hazard function and regression coefficients centered at the maximum likelihood estimator is jointly asymptotically equivalent to the sampling distribution of the maximum likelihood estimator

a fuzzy inverse model construction method for general monotonic multi input single output miso systems this paper presents a novel method of systematically constructing a fuzzy inverse model for general multi input single output miso systems represented with triangular input membership functions singleton output membership function and fuzzy mean defuzzification the fuzzy inverse model construction method has the ability of uniquely determining the inverse relationship for each input output pair it is derived in a straightforward way and the required input variables can be simultaneously obtained by the fuzzy inferencing calculation to realize the desired output value simulation examples are provided to demonstrate the effectiveness of the proposed method to find the inverse kinematics solutions for complex multiple degree of freedom industrial robot manipulators

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

genetic algorithms for learning in fuzzy relational structures abstract we propose an approach of integrated genetic learning in construction of fuzzy relational architectures described by fuzzy relational equations these equations have been widely utilized e g in fuzzy model identification fuzzy control and fuzzy controllers while the theoretical foundations of the equations are well developed they still call for more efficient and diversified schemes of learning the paper addresses this issue by synergistically combining some fundamental concepts of genetic algorithms ga and standard gradient based techniques into a unified scheme of a stratified learning especially we reveal how the ideas of gas can be effectively used at the level of the initialization of the gradient based learning schemes it is also elucidated how an optimal subpopulation of the strings can be helpful in forming feasibility regions guarding zones indispensable for supervising the subsequent phase of the parametric learning

direct and inverse problem in comparison of fuzzy data abstract discussed is the direct and inverse problem of comparison of two linguistic quantities specified as fuzzy sets relations in the same universe of discourse the proposed way of comparison makes use of some logical properties of connectives and implication operators stemming from the foundations of fuzzy sets a list of plausible equality indices is provided the inverse problem is formulated a set of its solutions is provided direct impacts of the discussed approach on development of fuzzy modelling are clearly indicated also it is pointed out how fuzzy sets of higher type or interval valued sets can be generated as a result of analysis of performance of any fuzzy model

partially supervised clustering for image segmentation all clustering algorithms process unlabeled data and consequently suffer from two problems p1 choosing and validating the correct number of clusters and p2 insuring that algorithmic labels correspond to meaningful physical labels clustering algorithms such as hard and fuzzy c means based on optimizing sums of squared errors objective functions suffer from a third problem p3 a tendency to recommend solutions that equalize cluster populations the semi supervised c means algorithms introduced in this paper attempt to overcome these problems domains where a few data from each clas can be labeled segmentation of magnetic resonance images is a problem of this type and we use it to illustrate the new algorithm our examples show that the semi supervised approach provides mri segmentations that are superior to ordinary fuzzy c means and to the crisp k nearest neighbor rule and further that the new method ameliorates p1 p3

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

knowledge based networks in classification problems abstract the paper proposes a distributed computational structure called knowledge based network as a classification scheme in pattern recognition unlike the existing architectures and algorithms of pattern recognition the network allows for an explicit representation of domain classification knowledge while maintaining its learning capabilities the knowledge based network blends useful properties of knowledge based systems namely explicit knowledge representation with those advantageous for neural networks viz learning the network is composed of basic and and or neurons fuzzy clustering constitutes a preprocessing phase leading towards developing geometric constructs they contribute to a conceptual level around which numerical processing of the classifier is centred

fuzzy relation equations theory as a basis of fuzzy modelling an overview abstract we give a wide overview on the applications of fuzzy relation equations theory to decision making processes to the construction of preference relations and to knowledge engineering mainly fuzzy control and fuzzy pattern recognition general theoretical results and methodological aspects are pointed out

functional horseshoe priors for subspace shrinkage we introduce a new shrinkage prior on function spaces called the functional horseshoe fhs prior that encourages shrinkage toward parametric classes of functions unlike other shrinkage priors f

multiple kernel fuzzy clustering while fuzzy c means is a popular soft clustering method its effectiveness is largely limited to spherical clusters by applying kernel tricks the kernel fuzzy c means algorithm attempts to address this problem by mapping data with nonlinear relationships to appropriate feature spaces kernel combination or selection is crucial for effective kernel clustering unfortunately for most applications it is uneasy to find the right combination we propose a multiple kernel fuzzy c means mkfc algorithm that extends the fuzzy c means algorithm with a multiple kernel learning setting by incorporating multiple kernels and automatically adjusting the kernel weights mkfc is more immune to ineffective kernels and irrelevant features this makes the choice of kernels less crucial in addition we show multiple kernel k means to be a special case of mkfc experiments on both synthetic and real world data demonstrate the effectiveness of the proposed mkfc algorithm

asymptotic normality of semiparametric and nonparametric posterior distributions in this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution this in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions in particular the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates mles or posterior means with covariance matrix the inverse of the fisher information additionally the semiparametric and nonparametric mles for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter with the same covariance matrix the results are a semiparametric version and a nonparametric version of the parametric bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric bayesian inference and their frequentis

identification of functional fuzzy models using multidimensional reference fuzzy sets abstract most commonly fuzzy systems for modelling and control use one dimensional orthogonal reference ffuzzy sets such as triangular or trapezoidal ones in this article a new description of functional fuzzy models by fuzzy rules with premises evaluating point affinities is presented this results in multidimensional reference fuzzy sets an algorithm to identify such systems using cluster algorithms is proposed two algorithms the fuzzy c means and the gustafson and kessel algorithm with locally varying distance measures are applied the performance of the identification algorithm is demonstrated by applying it to the identification of two nonlinear systems one of them is a gas furnace described by the well known box jenkins data

quasi bayesian analysis of nonparametric instrumental variables models this paper aims at developing a quasi bayesian analysis of the nonparametric instrumental variables model with a focus on the asymptotic properties of quasi posterior distributions in this paper instead of assuming a distributional assumption on the data generating process we consider a quasi likelihood induced from the conditional moment restriction and put priors on the function valued parameter we call the resulting posterior quasi posterior which corresponds to gibbs posterior in the literature here we focus on priors constructed on slowly growing finite dimensional sieves we derive rates of contraction and a nonparametric bernstein von mises type result for the quasi posterior distribution and rates of convergence for the quasi bayes estimator defined by the posterior expectation we show that with priors suitably chosen the quasi posterior distribution the quasi bayes estimator attains the minimax optimal rate of contraction convergence resp these results greatly sharpen the previous related work

unsupervised clustering and feature weighting based on generalized dirichlet mixture modeling abstract we propose a possibilistic approach for generalized dirichlet mixture parameter estimation data clustering and feature weighting the proposed algorithm called robust and unsupervised learning of finite generalized dirichlet mixture models rule gdm exploits a property of the generalized dirichlet distributions that transforms the data to make the features independent and follow beta distributions then it learns optimal relevance weights for each feature within each cluster this property makes rule gdm suitable for noisy and high dimensional feature spaces in addition rule gdm associates two types of memberships with each data sample the first one is the posterior probability and indicates how well a sample fits each estimated distribution the second membership represents the degree of typicality and is used to identify and discard noise points and outliers rule gdm minimizes one objective function which combines learning the two membership functions distribution parameters and the relevance weights for each feature within each distribution we also extend our algorithm to find the optimal number of clusters in an unsupervised and efficient way by exploiting some properties of the possibilistic membership function the performance of rule gdm is illustrated and compared to similar algorithms we use synthetic data to illustrate its robustness to noisy and high dimensional features we also compare our approach to other relevant algorithms using several standard data sets

relevancy of fuzzy model abstract a problem of relevancy of fuzzy models viz models with ability to cope with fuzziness is stated its importance in any identification scheme is underlined and discussed in light of some shortcomings existing in procedures of fuzzy model building two novel concepts viz equality index and confidence interval are introduced and utilized for fuzzy model evaluation a flexible way of interpreting results with the aid of fuzzy models is formulated by means of interval valued fuzzy sets fuzzy sets the clear importance of studies on the relevancy of fuzzy models for problem formulation and solution is pointed out

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

bernstein von mises theorems for gaussian regression with increasing number of regressors this paper brings a contribution to the bayesian theory of nonparametric and semiparametric estimation we are interested in the asymptotic normality of the posterior distribution in gaussian linear regression models when the number of regressors increases with the sample size two kinds of bernstein von mises theorems are obtained in this framework nonparametric theorems for the parameter itself and semiparametric theorems for functionals of the parameter we apply them to the gaussian sequence model and to the regression of functions in sobolev and c classes in which we get the minimax convergence rates adaptivity is reached for the bayesian estimators of functionals in our applications

frbc a fuzzy rule based clustering algorithm fuzzy clustering is superior to crisp clustering when the boundaries among the clusters are vague and ambiguous however the main limitation of both fuzzy and crisp clustering algorithms is their sensitivity to the number of potential clusters and or their initial positions moreover the comprehensibility of obtained clusters is not expertized whereupon in data mining applications the discovered knowledge is not understandable for human users to overcome these restrictions a novel fuzzy rule based clustering algorithm frbc is proposed in this paper like fuzzy rule based classifiers the frbc employs a supervised classification approach to do the unsupervised cluster analysis it tries to automatically explore the potential clusters in the data patterns and identify them with some interpretable fuzzy rules simultaneous classification of data patterns with these fuzzy rules can reveal the actual boundaries of the clusters to illustrate the capability of frbc to explore the clusters in data the experimental results on some benchmark datasets are obtained and compared with other fuzzy clustering algorithms the clusters specified by fuzzy rules are human understandable with acceptable accuracy

a clustering algorithm for fuzzy model identification the fuzzy model proposed by takagi and sugeno can represent highly nonlinear systems and is widely used for the representation of fuzzy rules in this paper the model is firstly modified to make its identification easier based on the fuzzy c partition space four criteria are proposed for optimization of the model parameters following that a clustering algorithm composed of fuzzy c linear functions clustering and like fuzzy c means clustering is developed for minimizing the four criteria an identification scheme for rule s premise and consequence parameters is deduced from the clustering algorithm in succession finally four examples are demonstrated to verify the effectiveness of the proposed algorithm

proximity based clustering a search for structural consistency in data with semantic blocks of features a class of clustering problems that is studied here is concerned with the development of a structure of a global nature given a collection of structures clusters constructed locally for data that are represented by several collections blocks of features these blocks of features come with a well defined semantics for instance in spatiotemporal data a certain block of features concerns a spatial component of the data say x y or x y z coordinates while another one deals with the features that describe time series associated with the corresponding locations the results of clustering that are being produced locally are reconciled by minimizing a distance between the proximity matrices that are formed at the higher conceptual level and induced by the individual partition matrices the optimization problem is formulated and presented along with its iterative scheme

fuzzy c p means clustering and its application to a fuzzy rule based classifier toward good generalization and good interpretability this paper introduces a new classifier design method based on a modification of the classical fuzzy c means clustering first a new fuzzy c means clustering with p constant prototypes is proposed this method can be considered a generalization of the concept of the conditional fuzzy clustering with some prototypes a priori known a special initialization of the prototypes is introduced next the proposed clustering method is used to construct the premises of an if then rule based classifier the conclusions of these rules are obtained by minimization of a criterion function with various approximations of a misclassification error e g based on the quadratic the linear the sigmoidal or the huber s loss function the conjugate gradient algorithm is used to minimize the proposed criterion function each if then rule is represented in the mamdani assilan form which has good interpretability finally an extensive experimental analysis on 14 benchmark datasets is performed to demonstrate the validity of the classifier introduced its competitiveness to the state of the art classifiers with respect to both performance and interpretability is also shown

adaptive bernstein von mises theorems in gaussian white noise we investigate bernstein von mises theorems for adaptive nonparametric bayesian procedures in the canonical gaussian white noise model we consider both a hilbert space and multiscale setting with applications in l 2 and l infty respectively this provides a theoretical justification for plug in procedures for example the use of certain credible sets for sufficiently smooth linear functionals we use this general approach to construct optimal frequentist confidence sets based on the posterior distribution we also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved

uncertain fuzzy self organization based clustering conventional unsupervised learning algorithms require knowledge of the desired number of clusters beforehand even if such knowledge is not required in advance empirical selection of the parameter values may limit the adaptive capability of the algorithm thereby restricting the clustering performance an inherent uncertainty in the number and size of clusters requires integration of fuzzy sets into a clustering algorithm in this paper we propose a type 1 t1 fuzzy art method that adaptively selects the vigilance parameter value which is critical in determining the network dynamics this results in improved clustering performance due to the added flexibility in dynamic selection of the number of clusters with the use of fuzzy sets to further manage the uncertainty associated with memberships we extend the proposed t1 fuzzy art with adaptive vigilance to an interval type 2 it2 fuzzy art method type reduction and defuzzification are then performed using the km algorithm to obtain a defuzzified vigilance parameter value we evaluate our proposed methods on several data sets to validate their effectiveness

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

fuzzy clustering with supervision this study is concerned with clustering carried out in presence of labeled patterns an objective of this optimization is to reconcile between the structure residing in data and being primarily discovered by the underlying clustering mechanism and the labels of the patterns forming such structure in this sense one can consider the supervised fuzzy clustering to be a framework of preliminary data analysis providing with a thorough insight into the structure of the data and supporting the ensuing design of detailed classifiers the proposed method augments the standard fuzzy c means algorithm by extending the original objective function by the supervision component labeled patterns experimental results illustrate the approach and discuss the use of this type of clustering in vector quantization

fuzzy neural networks with reference neurons as pattern classifiers a heterogeneous neural network consisting of logic neurons and realizing mappings in 0 1 hypercubes is presented the two kinds of neurons studied are utilized to perform matching functions equality or reference neurons and aggregation operations aggregation neurons all computations are driven by logic operations widely used in fuzzy set theory the network is heterogeneous in its nature and includes two types of neurons organized into a structure detecting individual regions of patterns using reference neurons and combining them to yield a final classification decision

directional fuzzy clustering and its application to fuzzy modelling abstract the paradigm of clustering unsupervised learning viewed as a fundamental tool for data analysis has been found useful in fuzzy modelling while the objective functions guiding the clustering mechanisms are by and large direction free namely they do not distinguish between independent input and dependent output variables for most of the models this discrimination becomes of vital importance the method of directional clustering takes the directionality requirement into account by incorporating the nature of the functional relationships into the objective function guiding the formation of the clusters the complete clustering algorithm is presented the role of this method in a two phase fuzzy identification scheme is also revealed in detail

fuzzy relational structures the state of art fuzzy relational structures built with the aid of fuzzy sets and fuzzy relations constitute a fundamental class of constructs arising in fuzzy set theory they give rise to fuzzy relation equations in this paper we will develop the concepts of single and multilevel relational structures different logic operations will be discussed in detail aspects of the knowledge representation will be studied and illustrated with several schemes of applications

a simultaneous learning framework for clustering and classification traditional pattern recognition generally involves two tasks unsupervised clustering and supervised classification when class information is available fusing the advantages of both clustering learning and classification learning into a single framework is an important problem worthy of study to date most algorithms generally treat clustering learning and classification learning in a sequential or two step manner i e first execute clustering learning to explore structures in data and then perform classification learning on top of the obtained structural information however such sequential algorithms cannot always guarantee the simultaneous optimality for both clustering and classification learning in fact the clustering learning in these algorithms just aids the subsequent classification learning and does not benefit from the latter to overcome this problem a simultaneous learning framework for clustering and classification scc is presented in this paper scc aims to achieve three goals 1 acquiring the robust classification and clustering simultaneously 2 designing an effective and transparent classification mechanism 3 revealing the underlying relationship between clusters and classes to this end with the bayesian theory and the cluster posterior probabilities of classes we define a single objective function to which the clustering process is directly embedded by optimizing this objective function the effective and robust clustering and classification results are achieved simultaneously experimental results on both synthetic and real life datasets show that scc achieves promising classification and clustering results at one time

processing in relational structures fuzzy relational equations abstract the paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations the origin and the central role of relational calculus in general and fuzzy sets in particular is explained with special emphasis paid to its role and representation capabilities it is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid well developed formalisms techniques and clarity of exposition we will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality such as for instance type 2 sets interval valued sets can be algorithmically determined finally we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory

fuzzy modelling fundamentals construction and evaluation abstract the paper is devoted to fundamental aspects of fuzzy modelling namely modelling articulated with the use of methodology of fuzzy sets open issues are addressed which summarize advantages of fuzzy models and point out some methodological deficiencies blurring existing approaches the problem of relevancy of fuzzy models and its direct impact on a suitable interpretation of derived results is studied in detail finally we analyze some particular fuzzy models developed in terms of relational equations

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

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

fuzzy set framework for development of a perception perspective abstract the question of perception attracts a lot of investigation concerning fundamental issues of knowledge representation and particularly application oriented techniques we clearly indicate the context dependent character of information processed in cognitive processes as opposed to statistical analysis in which no context is taken into account the problem being addressed here is translated into an optimization scheme under which algorithms of data analysis are worked out a general class of techniques dedicated to linguistic analysis of records of numerical data is introduced its essence relies on determining the level of data and fuzzy labels

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

a self generating method for fuzzy system design abstract in this paper a self generating method based on genetic algorithms is proposed to automatically construct a fuzzy system a parameter structure of fuzzy system is presented along with a fuzzy set elimination method which enables the self generating method not only to determine an appropriate fuzzy system with a high performance but also to favor the selected system with fewer fuzzy rules an inverted pendulum control and a system modeling problem are utilized respective to illustrate the efficiency of the proposed method in the fuzzy system design

identification of fuzzy systems by means of an auto tuning algorithm and its application to nonlinear systems abstract the study concerns a design procedure of rule based systems the proposed rule based fuzzy modeling implements system structure and parameter identification in the efficient form of if then statements and exploits the theory of system optimization and fuzzy implication rules two types of methods for rule based fuzzy modeling are studied this classification concerns the form of the conclusion part of the rules that can be either constant or formed by some linear functions both triangular and gaussian like membership function are studied the optimization hinges on an auto tuning algorithm that covers a modified constrained optimization method known as a complex method the study introduces a weighted performance index objective function that helps achieve a sound balance between the quality of results produced for the training and testing set this methodology sheds light on the role and impact of different parameters of the model on its performance in particular the mapping and predicting capabilities of the rule based computing the study is illustrated with the aid of several representative numerical examples

collaborative clustering with the use of fuzzy c means and its quantification in this study we introduce the concept of collaborative fuzzy clustering a conceptual and algorithmic machinery for the collective discovery of a common structure relationships within a finite family of data residing at individual data sites there are two fundamental features of the proposed optimization environment first given existing constraints which prevent individual sites from exchanging detailed numeric data any communication has to be realized at the level of information granules the specificity of these granules impacts the effectiveness of ensuing collaborative activities second the fuzzy clustering realized at the level of the individual data site has to constructively consider the findings communicated by other sites and act upon them while running the optimization confined to the particular data site adhering to these two general guidelines we develop a comprehensive optimization scheme and discuss its two phase character in which the communication phase of the granular findings intertwines with the local optimization being realized at the level of the individual site and exploits the evidence collected from other sites the proposed augmented form of the objective function is essential in the navigation of the overall optimization that has to be completed on a basis of the data and available information granules the intensity of collaboration is optimized by choosing a suitable tradeoff between the two components of the objective function the objective function based clustering used here concerns the well known fuzzy c means fcm algorithm experimental studies presented include some synthetic data selected data sets coming from the machine learning repository and the weather data coming from environment canada

a simple but powerful heuristic method for generating fuzzy rules from numerical data abstract in this paper we propose a simple but powerful heuristic method for automatically generating fuzzy if then rules from numerical data fuzzy if then rules with nonfuzzy singletons i e real numbers in the consequent parts are generated by the proposed heuristic method the main advantage of the proposed heuristic method is its simplicity i e it involves neither time consuming iterative learning procedures nor complicated rule generation mechanisms we also suggest a linguistic representation method for deriving linguistic rules from fuzzy if then rules with consequent real numbers the proposed linguistic approximation method consists of two linguistic rule tables which can realize exactly the same nonlinear mapping as an original system based on fuzzy if then rules with consequent real numbers using computer simulations on rice taste data we demonstrate the high performance of the proposed heuristic method and illustrate the proposed linguistic representation method

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model

rates of convergence of posterior distributions we compute the rate at which the posterior distribution concentrates around the true parameter value the spaces we work in are quite general and include infinite dimensional cases the rates are driven by two quantities the size of the space as measured by bracketing entropy and the degree to which the prior concentrates in a small ball around the true parameter we consider two examples

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection