are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

optimization of type 2 fuzzy systems based on bio inspired methods a concise review a review of the optimization methods used in the design of type 2 fuzzy systems which are relatively novel models of imprecision has been considered in this work the fundamental focus of the work has been based on the basic reasons of the need for optimizing type 2 fuzzy systems for different areas of application recently bio inspired methods have emerged as powerful optimization algorithms for solving complex problems in the case of designing type 2 fuzzy systems for particular applications the use of bio inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems in this review we consider the application of genetic algorithms particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type 2 fuzzy systems we also provide a comparison of the different optimization methods for the case of designing type 2 fuzzy systems

bootstrapping boosted random ferns for discriminative and efficient object classification in this paper we show that the performance of binary classifiers based on boosted random ferns can be significantly improved by appropriately bootstrapping the training step this results in a classifier which is both highly discriminative and computationally efficient and is particularly suitable when only small sets of training images are available during the learning process a small set of labeled images is used to train the boosting binary classifier the classifier is then evaluated over the training set and warped versions of the classified and misclassified patches are progressively added into the positive and negative sample sets for a new re training step in this paper we thoroughly study the conditions under which this bootstrapping scheme improves the detection rates in particular we assess the quality of detection both as a function of the number of bootstrapping iterations and the size of the training set we compare our algorithm against state of the art approaches for several databases including faces cars motorbikes and horses and show remarkable improvements in detection rates with just a few bootstrapping steps

stationary features and cat detection most discriminative techniques for detecting instances from object categories in still images consist of looping over a partition of a pose space with dedicated binary classifiers the efficiency of this strategy for a complex pose that is for fine grained descriptions can be assessed by measuring the effect of sample size and pose resolution on accuracy and computation two conclusions emerge 1 fragmenting the training data which is inevitable in dealing with high in class variation severely reduces accuracy 2 the computational cost at high resolution is prohibitive due to visiting a massive pose partition to overcome data fragmentation we propose a novel framework centered on pose indexed features which assign a response to a pair consisting of an image and a pose and are designed to be stationary the probability distribution of the response is always the same if an object is actually present such features allow for efficient one shot learning of pose specific classifiers to avoid expensive scene processing we arrange these classifiers in a hierarchy based on nested partitions of the pose as in previous work on coarse to fine search this allows for efficient processing the hierarchy is then folded for training all the classifiers at each level are derived from one base predictor learned from all the data the hierarchy is unfolded for testing parsing a scene amounts to examining increasingly finer object descriptions only when there is sufficient evidence for coarser ones in this way the detection results are equivalent to an exhaustive search at high resolution we illustrate these ideas by detecting and localizing cats in highly cluttered greyscale scenes

a lagrangean heuristic for the capacitated concave minimum cost network flow problem we propose a heuristic solution technique for the capacitated concave minimum cost network flow problem based on a lagrangean dualization of the problem despite its dual character the algorithm guarantees the generation of primal feasible solutions which are local optima and therefore candidates of being the global optimum the lagrangean dual is solved by a subgradient search procedure and provides a lower bound to the optimal value the lower bound is in general stronger than the one obtained by a linear approximation of the original problem it can be used as a judgement of the quality of the solution or in a branch and bound procedure computational results from randomly generated problems are presented

fuzzy modeling with tolerances abstract we consider the problem of the resolution of fuzzy equation systems a certain number of fuzzy inputs and outputs is given how can we determine a fuzzy relation which describes the behaviour of the object under consideration when attributing this problem to fuzzy equations in the classical sense it may happen that a solution does not exist therefore we introduce fuzzy tolerances as an extension of the solvability conception for this problem we get the whole solution set by means of posets moreover we consider the inverse problem how to determine tolerances to get consistency i e at least one solution for the arising equation system

multi class boosting with asymmetric binary weak learners we introduce a multi class generalization of adaboost with binary weak learners we use a vectorial codification to represent class labels and a multi class exponential loss function to evaluate classifier responses this representation produces a set of margin values that provide a range of punishments for failures and rewards for successes moreover the stage wise optimization of this model introduces an asymmetric boosting procedure whose costs depend on the number of classes separated by each weak learner in this way the boosting algorithm takes into account class imbalances when building the ensemble the experiments performed compare this new approach favorably to adaboost mh gentleboost and the samme algorithms

fuzzy linear regression and its applications to forecasting in uncertain environment abstract linear regression has been used for many years in developing mathematical models for application in marketing management and sales forecasting in this paper two different sales forecasting techniques are discussed the first technique involves non fuzzy abstract methods of linear regression and econometrics a study of the international market sales of cameras done in 1968 by john scott armstrong utilized these non fuzzy forecasting techniques the second sales forecasting technique uses fuzzy linear regression introduced by h tanaka s uejima and k asai in 1980 in this paper a study of the computer and peripheral equipment sales in the united states is discussed using fuzzy linear regression moreover fuzzy linear regression is applied to forecasting in an uncertain environment finally some possible improvements and suggestions for further study are mentioned

probabilistic local search algorithms for concave cost transportation network problems in practice concave cost transportation problems are characterized as np hard therefore cost functions are usually simplified as linear in order to facilitate problem solving however linear cost functions may not reflect actual operations which generally results in decreased operational performance this research employs the techniques of simulated annealing and threshold accepting to develop several heuristics that would efficiently solve these concave cost transportation network problems a network generator has also been designed to generate many instances on an hp workstation to test the heuristics the preliminary results show that these heuristics are potentially useful

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

a fuzzy neural network for pattern classification and feature selection a fuzzy neural network with memory connections for classification and weight connections for selection is introduced thereby solving simultaneously two major problems in pattern recognition pattern classification and feature selection the proposed network attempts to select important features from among the originally given plausible features while maintaining the maximum recognition rate the resulting value of weight connection represents the degree of importance of feature moreover the knowledge acquired by the network can be described as a set of interpretable rules the effectiveness of this new method has been validated by using anderson s iris data the results are first the use of two features selected by our method from among the original four in the proposed network results in virtually identical classifier performance and second the constructed classifier is described by three simple rules that are of if then form

fuzzy normal forms abstract linguistic expressions of a natural language are transformed at least three times before they are expressed in computational formulas between the linguistic and the computational expressions there are the meta linguistic and propositional expressions under various interpretations and assumptions fuzzy normal forms are canonical expressions derived at the propositional level with particular assumptions the fuzzy normal forms can be generated from fuzzy truth tables with a normal form derivation algorithm and an orthodox interpretation of conjunction disjunction and complementation operations it is shown that these fuzzy normal forms display a unique structure for certain general classes of crisp operators over fuzzy sets the fuzzy normal forms generated by crisp operators over fuzzy sets that correspond to max min standard complement de morgan triple are shown to be equivalent to the classical normal forms of two valued logic i e disjunctive and conjunctive normal forms known as dnf and cnf with a particular property i e the crisp containment fdnf 3 fcnf 3 whereas the fuzzy normal forms generated by crisp operators that correspond to non idempotent t norms t conorms and standard complement de morgan triples do not have in general a similar crisp containment property i e fdnf 2 fcnf 2 neither do the fuzzy normal forms generated by crisp operators over fuzzy sets that correspond to non idempotent and non commutative and non associative conjunction disjunction and standard complement de morgan triples have a crisp containment property i e fdnf 1 fcnf 1 however they still provide bounds on the combination of concepts

a review and comparison of six reasoning methods abstract five fuzzy reasoning methods are reviewed and their performance is compared with respect to a fuzzy control system model developed by an objective method based on three sets of real life data it is found from the investigation that the reasoning precision the calculation time and the number of possible input states to which a given reasoning method responds differ according to each reasoning method generally the point valued reasoning methods which are based on the assumption that connectives are crisp give better precision and shorter calculation time when the connectives are assumed to be linguistic reasoning with interval valued fuzzy sets are more appropriate to represent linguistic uncertainty for this reason type ii fuzziness generated with interval valued fuzzy sets are better handled with interval valued reasoning methods

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

groups of adjacent contour segments for object detection we present a family of scale invariant local shape features formed by chains of k connected roughly straight contour segments kas and their use for object class detection kas are able to cleanly encode pure fragments of an object boundary without including nearby clutter moreover they offer an attractive compromise between information content and repeatability and encompass a wide variety of local shape structures we also define a translation and scale invariant descriptor encoding the geometric configuration of the segments within a kas making kas easy to reuse in other frameworks for example as a replacement or addition to interest points ips software for detecting and describing kas is released at http lear inrialpes fr software we demonstrate the high performance of kas within a simple but powerful sliding window object detection scheme through extensive evaluations involving eight diverse object classes and more than 1 400 images we 1 study the evolution of performance as the degree of feature complexity k varies and determine the best degree 2 show that kas substantially outperform ips for detecting shape based classes and 3 compare our object detector to the recent state of the art system by dalal and triggs 2005

dependent multiple cue integration for robust tracking we propose a new technique for fusing multiple cues to robustly segment an object from its background in video sequences that suffer from abrupt changes of both illumination and position of the target robustness is achieved by the integration of appearance and geometric object features and by their estimation using bayesian filters such as kalman or particle filters in particular each filter estimates the state of a specific object feature conditionally dependent on another feature estimated by a distinct filter this dependence provides improved target representations permitting us to segment it out from the background even in nonstationary sequences considering that the procedure of the bayesian filters may be described by a hypotheses generation hypotheses correction strategy the major novelty of our methodology compared to previous approaches is that the mutual dependence between filters is considered during the feature observation that is into the hypotheses correction stage instead of considering it when generating the hypotheses this proves to be much more effective in terms of accuracy and reliability the proposed method is analytically justified and applied to develop a robust tracking system that adapts online and simultaneously the color space where the image points are represented the color distributions the contour of the object and its bounding box results with synthetic data and real video sequences demonstrate the robustness and versatility of our method

robust object detection with interleaved categorization and segmentation this paper presents a novel method for detecting and localizing objects of a visual category in cluttered real world scenes our approach considers object categorization and figure ground segmentation as two interleaved processes that closely collaborate towards a common goal as shown in our work the tight coupling between those two processes allows them to benefit from each other and improve the combined performance the core part of our approach is a highly flexible learned representation for object shape that can combine the information observed on different training examples in a probabilistic extension of the generalized hough transform the resulting approach can detect categorical objects in novel images and automatically infer a probabilistic segmentation from the recognition result this segmentation is then in turn used to again improve recognition by allowing the system to focus its efforts on object pixels and to discard misleading influences from the background moreover the information from where in the image a hypothesis draws its support is employed in an mdl based hypothesis verification stage to resolve ambiguities between overlapping hypotheses and factor out the effects of partial occlusion an extensive evaluation on several large data sets shows that the proposed system is applicable to a range of different object categories including both rigid and articulated objects in addition its flexible representation allows it to achieve competitive object detection performance already from training sets that are between one and two orders of magnitude smaller than those used in comparable systems

on characterization of generalized interval type 2 fuzzy rough sets this paper presents a systematic study of interval type 2 fuzzy rough sets integrating rough set theory with interval type 2 fuzzy set theory using constructive and axiomatic approaches from the perspective of a constructive approach a pair of lower and upper interval type 2 fuzzy rough approximation operators with respect to an interval type 2 fuzzy relation is defined the basic properties of the interval type 2 fuzzy rough approximation operators are studied using cut sets of interval type 2 fuzzy sets classical representations of interval type 2 fuzzy rough approximation operators are then presented and the connections between special interval type 2 fuzzy relations and interval type 2 fuzzy rough approximation operators are investigated adopting an axiomatic approach an operator oriented characterization of interval type 2 fuzzy rough sets is proposed in other words interval type 2 fuzzy rough approximation operators are characterized by axioms different axiom sets of interval type 2 fuzzy set theoretic operators guarantee the existence of different types of interval type 2 fuzzy relations that produce the same operators finally the relationship between interval type 2 fuzzy rough sets and interval type 2 fuzzy topological spaces is examined we obtain sufficient and necessary conditions for the conjecture that an interval type 2 fuzzy interior closure operator derived from an interval type 2 fuzzy topological space can associate with an interval type 2 fuzzy reflexive and transitive relation such that the corresponding lower upper interval type 2 fuzzy rough approximation operator is the interval type 2 fuzzy interior closure operator

experimental analysis of simulated annealing based algorithms for the layout problem abstract in this paper results from an experimental analysis involving the 2 way and 3 way exchange algorithms a modified penalty algorithm the simulated annealing algorithm and a hybrid simulated annealing algorithm are presented the hybrid simulated annealing algorithm uses the modified penalty algorithm to generate an initial solution and then improves it using simulated annealing it is tested on single row layout problems with facilities of unequal area and multi row layout problems with facilities of equal area the algorithm produced solutions better than those previously reported in the literature for two of the larger single row problems and five multi row layout problems for all the other single row problems and eight of ten multi row layout test problems it produced the best known solutions the hybrid simulated annealing algorithm appears to produce better quality solutions than the simulated annealing and tabu navigation algorithms but requires more computation time than simulated annealing

the symbol grounding problem there has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling this paper describes the symbol grounding problem how can the semantic interpretation of a formal symbol system be made intrinsic to the system rather than just parasitic on the meanings in our heads how can the meanings of the meaningless symbol tokens manipulated solely on the basis of their arbitrary shapes be grounded in anything but other meaningless symbols the problem is analogous to trying to learn chinese from a chinese chinese dictionary alone a candidate solution is sketched symbolic representations must be grounded bottom up in nonsymbolic representations of two kinds 1 iconic representations which are analogs of the proximal sensory projections of distal objects and events and 2 categorical representations which are learned and innate feature detectors that pick out the invariant features of object and event categories from their sensory projections elementary symbols are the names of these object and event categories assigned on the basis of their nonsymbolic categorical representations higher order 3 symbolic representations grounded in these elementary symbols consist of symbol strings describing category membership relations e g an x is a y that is z connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations thereby connecting names to the proximal projections of the distal objects they stand for in this way connectionism can be seen as a complementary component in a hybrid nonsymbolic symbolic model of the mind rather than a rival to purely symbolic modeling such a hybrid model would not have an autonomous symbolic module however the symbolic functions would emerge as an intrinsically dedicated symbol system as a consequence of the bottom up grounding of categories names in their sensory representations symbol manipulation would be governed not just by the arbitrary shapes of the symbol tokens but by the nonarbitrary shapes of the icons and category invariants in which they are grounded

classification of type 2 fuzzy sets represented as sequences of vertical slices granulation of information by using type 2 fuzzy sets is receiving more attention nowadays this is due to the superior capability of type 2 fuzzy sets in handling the data uncertainty from a theoretical perspective a set containing type 2 fuzzy sets has no trivial geometric structure therefore a proper metric cannot be easily defined as a consequence common pattern recognition systems which in one way or another rely on some geo metric structure of the input space are not easily applicable to a space of type 2 fuzzy sets in this paper we study the problem of designing a classifier in the input space of type 2 fuzzy sets type 2 fuzzy sets are hence interpreted as granular patterns forming a given input dataset by decomposing a type 2 fuzzy set into a sequence of simpler lower type fuzzy sets we explore the possibility of defining and building dissimilarity and kernel based classification systems on input spaces of type 2 fuzzy sets such an interpretation provided in terms of sequences allows us to conceive an effective sequence matching strategy which can be suitably embedded into well established pattern recognition systems we support the methodological developments by performing experiments on synthetically generated classification problems for datasets composed of type 2 fuzzy sets with adjustable and controlled level of difficulty results are promising and suggest to further investigate on the possibility of interpreting type 2 fuzzy sets as input patterns of a given data driven inference system

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

least squares model fitting to fuzzy vector data abstract we consider model fitting problems in which the mathematical model of an observed event is a system of equations involving model parameters and observables and in which the data are fuzzy vectors such problems naturally arise in applications when data are scarce and information is vague about distributions of variances that are contained in the observations then it may only be possible to obtain estimates of membership functions of the data a model can be fitted to such data by maximizing the membership values of the adjusted observations we achieve this by minimizing the sum of squares of the deviations of the membership values from one the optimization problem simplifies significantly if the membership functions of the fuzzy vectors belong to a class of conical functions which are defined in terms of an elliptic norm properties of these functions are discussed and a membership propagation formula derived which component wise obeys the extension principle the least squares model fitting to the described fuzzy vector data deviates in important aspects from ordinary least squares one obtains as a solution a fuzzy model parameter vector and a number of indicators which describe the quality of the fit viz the grade of compatibility between data and model their discord and dilators of model and data membership functions these new concepts are illustrated and discussed in the context of an example from a terminal ballistics problem

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

approximate reasoning for production planning abstract approximate reasoning for production planning could effectively be based on an interval valued representation of linguistic variables an interval valued representation provides a framework that i incorporates a dispersion about membership values and ii produces a robust result compared to classical results the proposed representation and inference is applied to a set of production problems in prototype studies the result of one such prototype study is summarized

an enhanced type reduction algorithm for type 2 fuzzy sets karnik and mendel proposed an algorithm to compute the centroid of an interval type 2 fuzzy set efficiently based on this algorithm liu developed a centroid type reduction strategy to carry out type reduction for type 2 fuzzy sets a type 2 fuzzy set is decomposed into a collection of interval type 2 fuzzy sets by cuts then the karnik mendel algorithm is called for each interval type 2 fuzzy set iteratively however the initialization of the switch point in each application of the karnik mendel algorithm is not a good one in this paper we present an improvement to liu s algorithm we employ the previously obtained result to construct the starting values in the current application of the karnik mendel algorithm convergence in each iteration except the first one can then speed up and type reduction for type 2 fuzzy sets can be carried out faster the efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results

a type 2 fuzzy time series model for stock index forecasting abstract most conventional fuzzy time series models type 1 models utilize only one variable in forecasting furthermore only part of the observations in relation to that variable are used to utilize more of that variable s observations in forecasting this study proposes the use of a type 2 fuzzy time series model in such a type 2 model extra observations are used to enrich or to refine the fuzzy relationships obtained from type 1 models and then to improve forecasting performance the taiwan stock index the taiex is used as the forecasting target the study period extends over the 2000 2003 period the taiex from january to october in each year is used for the estimation while that covering november and december is used for forecasting the empirical analyses show that type 2 model outperforms type 1 model

analysis of data generated from multidimensional type 1 and type 2 fuzzy membership functions due to the numerous applications that utilize different types of fuzzy membership functions mfs it may sometimes be difficult to choose an appropriate mf for a particular application in this paper we establish preliminary guidelines to direct this selection by proposing a three stage method in the forward stage different mfs such as crisp mfs type 1 t1 fuzzy mfs and type 2 t2 fuzzy mfs are generated from multidimensional data sets next in the reverse stage data is generated back from these mfs by considering different bin sizes in doing so various data sets may be generated for different applications which require fuzzy data finally for the similarity analysis stage we propose an iterative algorithm that makes use of the results of wilcoxon signed rank wsr and wilcoxon rank sum wrs tests to compare the original data and the generated data from the results of these tests recommendations concerning the suitability of mfs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application with this analysis the objective is to gain insight on when t2 fuzzy sets may be considered to outperform t1 fuzzy sets and vice versa several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions

a context model for fuzzy concept analysis based upon modal logic in this paper we present interesting relationships between the context model modal logic and fuzzy concept analysis it has been shown that the context model proposed by gebhardt and kruse int j approx reason 9 1993 283 can be semantically extended and considered as a data model for fuzzy concept analysis within the framework of the meta theory developed by resconi el al in 1990s consequently the context model provides a practical framework for constructing membership functions of fuzzy concepts and gives the basis for a theoretical justification of suitably use of well known t norm based connectives such as min max and product sum rules in applications furthermore an interpretation of mass assignments of fuzzy concepts within the context model is also established

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

type 2 fuzzy markov random fields and their application to handwritten chinese character recognition in this paper we integrate type 2 t2 fuzzy sets with markov random fields mrfs referred to as t2 fmrfs which may handle both fuzziness and randomness in the structural pattern representation on the one hand the t2 membership function mf has a 3 d structure in which the primary mf describes randomness and the secondary mf evaluates the fuzziness of the primary mf on the other hand mrfs can represent patterns statistical structurally in terms of neighborhood system and clique potentials and thus have been widely applied to image analysis and computer vision in the proposed t2 fmrfs we define the same neighborhood system as that in classical mrfs to describe uncertain structural information in patterns we derive the fuzzy likelihood clique potentials from t2 fuzzy gaussian mixture models the fuzzy prior clique potentials are penalties for the mismatched structures based on prior knowledge because chinese characters have hierarchical structures we use t2 fmrfs to model character structures in the handwritten chinese character recognition system the overall recognition rate is 99 07 which confirms the effectiveness of the proposed method

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

a self evolving interval type 2 fuzzy neural network with online structure and parameter learning this paper proposes a self evolving interval type 2 fuzzy neural network seit2fnn with online structure and parameter learning the antecedent parts in each fuzzy rule of the seit2fnn are interval type 2 fuzzy sets and the fuzzy rules are of the takagi sugeno kang tsk type the initial rule base in the seit2fnn is empty and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space to avoid generating highly overlapping fuzzy sets in each input variable an efficient fuzzy set reduction method is also proposed this method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated for parameter learning the consequent part parameters are tuned by the rule ordered kalman filter algorithm for high accuracy learning performance detailed learning equations on applying the rule ordered kalman filter algorithm to the seit2fnn consequent part learning with rules being generated online are derived the antecedent part parameters are learned by gradient descent algorithms the seit2fnn is applied to simulations on nonlinear plant modeling adaptive noise cancellation and chaotic signal prediction comparisons with other type 1 and type 2 fuzzy systems in these examples verify the performance of the seit2fnn

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

a decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure in this paper we develop a decomposition method using a pricing mechanism which has been widely applied to linear and convex programs for a class of nonconvex optimization problems that are min concave cost flow problems under directed uncapacitated networks with a hierarchical structure

designing of a type 2 fuzzy logic filter for improving edge preserving restoration of interlaced to progressive conversion this paper focuses on recently advanced fuzzy models and the application of type 2 fuzzy sets in video deinterlacing the final goal of the proposed deinterlacing algorithm is to exactly determine an unknown pixel value while preserving the edges and details of the image to begin we will discuss some artefacts of spatial temporal and spatio temporal domain deinterlacing methods in order to address the aforementioned issues we adopted type 2 fuzzy sets concepts to design a weight evaluating approach in the proposed method the upper and lower fuzzy membership functions of the type 2 fuzzy logic filters are derived from the type 1 or primary fuzzy membership function the weights from upper and lower membership functions are considered to be multiplied with the candidate deinterlaced pixels experimental results proved that the performance of the proposed method was superior both objectively and subjectively to other different conventional deinterlacing methods moreover the proposed method preserved the smoothness of the original image edges and produced a high quality progressive image

fuzzy artmap a neural network architecture for incremental supervised learning of analog multidimensional maps a neural network architecture is introduced for incremental supervised learning of recognition categories and multidimensional maps in response to arbitrary sequences of analog or binary input vectors which may represent fuzzy or crisp sets of features the architecture called fuzzy artmap achieves a synthesis of fuzzy logic and adaptive resonance theory art neural networks by exploiting a close formal similarity between the computations of fuzzy subsethood and art category choice resonance and learning four classes of simulation illustrated fuzzy artmap performance in relation to benchmark backpropagation and generic algorithm systems these simulations include finding points inside versus outside a circle learning to tell two spirals apart incremental approximation of a piecewise continuous function and a letter recognition database the fuzzy artmap system is also compared with salzberg s nge systems and with simpson s fmmc system

fuzzy sets spl minus a convenient fiction for modeling vagueness and possibility this paper is a reply to laviolette and seaman s critical discussion of fuzzy set theory rather than questioning the interest of the bayesian approach to uncertainty some reasons why bayesian find the idea of a fuzzy set not palatable are laid bare some links between fuzzy sets and probability that laviolette and seaman seem not to be aware of are pointed out these links suggest that contrary to the claim sometimes found in the literature probability theory is not a special case of fuzzy set theory the major objection to laviolette and seaman is that they found their critique on as very limited view of fuzzy sets including debatable papers while they fail to account for significant works pertaining to axiomatic derivation of fuzzy set connectives possibility theory fuzzy random variables among others

higher level fuzzy numbers arising from fuzzy regression consider a fuzzy random variable y with expectation symbol b x where b is an unknown fuzzy number and an unknown real number for n observations yi xi there is a model yi b xi ei i 1 2 n where ei are fuzzy valued errors independently and identically distributed in some sense the aim is to obtain estimates of b when all fuzzy numbers are triangular and the ei are uniformly distributed maximum likelihood estimators emerge naturally as a special form of higher level fuzzy number thus mle can be interpreted as an estimation procedure which transforms randomness into higher levels of fuzziness

a random activity network generator exact and heuristic procedures are often developed to obtain optimal and near optimal solutions to decision problems modeled as activity networks testing the accuracy and efficiency of these procedures requires the use of activity networks with various sizes structures and parameters the size of the network is determined by its number of nodes and arcs where the structure is chosen from the set of all structures for the specified network size the network parameters depend on the nature of the decision problem often it is desirable for test problems to be generated at random from the space of all feasible networks this paper deals with the problem of generating the size and structure of the network at random from the space of all feasible networks it develops a theory which guarantees the randomness of the network structure the theory is the basis for two methods one can be used to generate dense networks where the other is used to generate nondense networks the methods which are practical a

analysis of critical paths in a project network with fuzzy activity times abstract this paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers in that the membership function of the fuzzy total duration time is constructed the basic idea is based on the extension principle and linear programming formulation a pair of linear programs parameterized by possibility level is formulated to calculate the lower and upper bounds of the fuzzy total duration time at by enumerating different values of the membership function of the fuzzy total duration time is constructed and the fuzzy critical paths are identified at the same time moreover by applying the yager ranking method definitions of the most critical path and the relative degree of criticality of paths are developed and these definitions are theoretically sound and easy to use in practice two examples with activity times being fuzzy numbers of l r and l l types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach since the total duration time is completely expressed by a membership function rather than by a crisp value the fuzziness of activity times is conserved completely and more information is provided for critical path analysis

learning discriminative localization from weakly labeled data visual categorization problems such as object classification or action recognition are increasingly often approached using a detection strategy a classifier function is first applied to candidate subwindows of the image or the video and then the maximum classifier score is used for class decision traditionally the subwindow classifiers are trained on a large collection of examples manually annotated with masks or bounding boxes the reliance on time consuming human labeling effectively limits the application of these methods to problems involving very few categories furthermore the human selection of the masks introduces arbitrary biases e g in terms of window size and location which may be suboptimal for classification we propose a novel method for learning a discriminative subwindow classifier from examples annotated with binary labels indicating the presence of an object or action of interest but not its location during training our approach simultaneously localizes the instances of the positive class and learns a subwindow svm to recognize them we extend our method to classification of time series by presenting an algorithm that localizes the most discriminative set of temporal segments in the signal we evaluate our approach on several datasets for object and action recognition and show that it achieves results similar and in many cases superior to those obtained with full supervision highlightsa framework for discriminative localization and classification is proposed it works for both images and time series it only requires weakly annotated data it achieves results similar to those obtained with full supervision

impulse noise removal from digital images by a detail preserving filter based on type 2 fuzzy logic a novel image filter based on type 2 fuzzy logic techniques is proposed for detail preserving restoration of digital images corrupted by impulse noise the performance of the proposed filter is evaluated for different test images corrupted at various noise densities and also compared with representative conventional as well as state of the art impulse noise filters from the literature experimental results show that the proposed filter exhibits superior performance over the competing operators and is capable of efficiently suppressing the noise in the image while at the same time effectively preserving thin lines edges texture and other useful information within the image

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

a comparison of fuzzy strategies for corporate acquisition analysis analysing all prospective companies for acquisition in large market sectors is an onerous task a strategy that results in a shortlist of companies that meet certain basic criteria is required the short listed companies can then be further investigated in more detail later if desired fuzzy logic systems flss imbued with the expertise of a focal organisation s financial experts can be of great assistance in this process in this paper an investigation into the suitability of flss for acquisition analysis is presented the nuances of training and tuning are discussed in particular the difficulty of obtaining suitable amounts of expert data is a recurring theme throughout the paper a strategy for circumventing this issue is presented that relies on the design of a conventional fuzzy logic rule base with the assistance of a financial expert with the rule base created various scenarios such as the simulation of multiple experts and the creation of expert training data are investigated in particular two scenarios for the creation of simulated expert data are presented in the first the responses from the different experts are averaged and in the second scenario the responses from all the different experts are preserved in the training data this paper builds on previous work with scalable membership functions however the use of fuzzy c means clustering and backpropagation training are new developments additionally a type 2 fls is developed and its potential advantages are discussed for this application the type 2 system facilitates the inclusion of the opinions of multiple experts both the type 1 and type 2 flss were trained using the backpropagation algorithm with early stopping and verified with five fold cross validation multiple runs of the five fold method were conducted with different random orderings of the data for this particular application the type 1 system performed comparably with the type 2 system despite the considerable amount of variation in the expert training data the training results have proven the methods to be capable of efficient tuning of parameters and of reliable ranking of prospective companies

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

priority sampling for estimation of arbitrary subset sums from a high volume stream of weighted items we want to create a generic sample of a certain limited size that we can later use to estimate the total weight of arbitrary subsets applied to internet traffic analysis the items could be records summarizing the flows of packets streaming by a router subsets could be flow records from different time intervals of a worm attack whose signature is later determined the samples taken in the past thus allow us to trace the history of the attack even though the worm was unknown at the time of sampling estimation from the samples must be accurate even with heavy tailed distributions where most of the weight is concentrated on a few heavy items we want the sample to be weight sensitive giving priority to heavy items at the same time we want sampling without replacement in order to avoid selecting heavy items multiple times to fulfill these requirements we introduce priority sampling which is the first weight sensitive sampling scheme without replacement that works in a streaming context and is suitable for estimating subset sums testing priority sampling on internet traffic analysis we found it to perform an order of magnitude better than previous schemes priority sampling is simple to define and implement we consider a steam of items i e 0 n 1 with weights wi for each item i we generate a random number i 0 1 and create a priority qi e wi i the sample s consists of the k highest priority items let be the k p 1 th highest priority each sampled item i in s gets a weight estimate wi e max wi while nonsampled items get weight estimate wi e 0 magically it turns out that the weight estimates are unbiased that is e wi e wi and by linearity of expectation we get unbiased estimators over any subset sum simply by adding the sampled weight estimates from the subset also we can estimate the variance of the estimates and find surprisingly that the covariance between estimates wi and wj of different weights is zero finally we conjecture an extremely strong near optimality namely that for any weight sequence there exists no specialized scheme for sampling k items with unbiased weight estimators that gets smaller variance sum than priority sampling with k p 1 items szegedy settled this conjecture at stoc 06

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

experimental study of intelligent controllers under uncertainty using type 1 and type 2 fuzzy logic uncertainty is an inherent part in control systems used in real world applications the use of new methods for handling incomplete information is of fundamental importance type 1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems type 2 fuzzy sets that are used in type 2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom this paper deals with the design of control systems using type 2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise etc the experimental results are divided in two classes in the first class simulations of a feedback control system for a non linear plant using type 1 and type 2 fuzzy logic controllers are presented a comparative analysis of the systems response in both cases was performed with and without the presence of uncertainty for the second class a non linear identification problem for time series prediction is presented based on the experimental results the conclusion is that the best results are obtained using type 2 fuzzy systems

design of interval type 2 fuzzy sliding mode controller in this paper an interval type 2 fuzzy sliding mode controller it2fsmc is proposed for linear and nonlinear systems the proposed it2fsmc is a combination of the interval type 2 fuzzy logic control it2flc and the sliding mode control smc which inherits the benefits of these two methods the objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed loop system the lyapunov stability method is adopted to verify the stability of the interval type 2 fuzzy sliding mode controller system the design procedure of the it2fsmc is explored in detail a typical second order linear interval system with 50 parameter variations an inverted pendulum with variation of pole characteristics and a duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method the simulation results show that the it2fsmc achieves the best tracking performance in comparison with the type 1 fuzzy logic controller t1flc the it2flc and the type 1 fuzzy sliding mode controller t1fsmc

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

send and split method for minimum concave cost network flows many problems from inventory production and capacity planning and from network design exhibit scale economies and can be formulated in terms of finding minimum additive concave cost nonnegative network flows we reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic programming method called the send and split method to solve it the main work of the method entails repeatedly solving set splitting and minimum cost chain problems in uncapacitated networks with n nodes a arcs and d 1 demand nodes i e nodes with nonzero exogenous demand the algorithm requires up to n2 13d s2d operations additions and comparisons where s n log2n 3a is the number of operations required to solve a minimum cost chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph if also the network is k planar i e the graph is planar with all demand nodes lying on the boundary of k faces the method requires at most n2 kd3k sd2k operations the algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones these results unify significantly generalize e g to cyclic problems and sometimes improve upon e g for tandem facilities known polynomial time dynamic programming algorithms for wagner and whitin s wagner h m whitin t m 1958 dynamic version of the economic lot size model management sci 5 89 96 dynamic economic order quantity problem zangwill s zangwill w i 1969 a backlogging model and a multi echelon model of a dynamic economic lot size production system a network approach management sci 15 506 527 generalization to tandem facilities and veinott s veinott jr a f 1969 minimum concave cost solution of leontief substitution models of multi facility inventory systems oper res 17 262 291 increasing capacity warehousing problem the networks for the finite period versions of these problems are each 1 planar the method improves upon zangwill s zangwill w i 1968 minimum concave cost flows in certain networks management sci 14 429 450 related oand running time dynamic programming method for finding minimum additive concave cost non negative flows in circuitless single source networks we also implement the method to solve in polynomial time the d 1 demand node and k planar versions of the minimum cost forest and steiner problems in graphs the running time for the d 1 demand node steiner problem in graphs is comparable to that of dreyfus and wagner s dreyfus s e wagner r a 1971 the steiner problem in graphs networks1 195 207 method

data based system modeling using a type 2 fuzzy neural network with a hybrid learning algorithm we propose a novel approach for building a type 2 neural fuzzy system from a given set of input output training data a self constructing fuzzy clustering method is used to partition the training dataset into clusters through input similarity and output similarity tests the membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster then a type 2 fuzzy takagi sugeno kang if then rule is derived from each cluster to form a fuzzy rule base a fuzzy neural network is constructed accordingly and the associated parameters are refined by a hybrid learning algorithm which incorporates particle swarm optimization and a least squares estimation for a new input a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type 2 fuzzy set which is then defuzzified by applying a refined type reduction algorithm experimental results are presented to demonstrate the effectiveness of our proposed approach

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

interval valued fuzzy sets representation on multiple antecedent fuzzy s implications and reasoning abstract interval valued fuzzy sets are used to represent multiple antecedent s implications based on normal forms where linguistic connectives antecedents and consequents are assumed to be fuzzy these intervals are represented equivalently with their lower and upper bounds general forms are given for the bounds of multiple antecedent fuzzy s implications bounds for multiple antecedent reasoning are also investigated based on the bounds of multiple antecedent s implications it is suggested that modification functions utilized in approximation analogical reasoning aar be determined in such a way that consequents deduced by aar fall within the bounds of the reasoning an example is given to show how the bounds might be used to choose a modification scheme in aar

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

a fuzzy approach for multiresponse optimization an off line quality engineering problem abstract multiresponse optimization techniques are used to identify settings of process parameters that make the product s performance close to target values in the presence of multiple quality characteristics in this study a fuzzy multiresponse optimization procedure is proposed to search an appropriate combination of process parameter settings based on multiple quality characteristics or responses fuzzy regression models are first used to model the relations between process parameters and responses the possibility distributions of the prediction responses are then obtained the problem is reduced to a multiobjective optimization problem with fuzzy objectives a strategy of optimizing the most possible response values and minimizing the deviations from the most possible values is proposed we consider not only the most possible value but also the imprecision of the prediction responses through a die casting example we show how to use our approach to reach an appropriate machine setting which simultaneously optimize both casting quality and die life

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

neuro fuzzy clustering of radiographic tibia image data using type 2 fuzzy sets abstract this paper presents the results of using type 2 fuzzy sets to assist in the pre processing of data for use with neuro fuzzy clustering for classification of sports injuries in the lower leg this research is concerned with the analysis of bone scans from stress related injuries to the tibia of particular interest is whether neural network based clustering techniques can help the consultant in classifying the images the work was motivated by the situation where there is a relatively small amount of relevant data and difficulties are faced by consultants in classifying the various types of injuries for this particular problem the consultant s interpretation of the image lends itself to representation using type 2 fuzzy sets this research sets out to address whether with fuzzy neuro clustering techniques some insights may be provided to the consultant that they can use along with their experience and knowledge the results of this approach indicate that the use of neural clustering using a type 2 representation can improve the classification of shin images

an efficient and flexible mechanism for constructing membership functions abstract this paper introduces a bezier curve based mechanism for constructing membership functions of convex normal fuzzy sets the mechanism can fit any given data set with a minimum level of discrepancy in the absence of data the mechanism can be intuitively manipulated by the user to construct membership functions with the desired shape some numerical experiments are included to compare the performance of the proposed mechanism with conventional methods

on direct and inverse problems for fuzzy equation systems with tolerances abstract we investigate the resolution of fuzzy relational equation systems with tolerances which are a certain extension of fuzzy equations considered f i in 3 5 the extension of the concept of higashi and klir 3 enables us to describe the set of solutions to our problem for given tolerances by means of posets in a second part we investigate an inverse problem given upper lower tolerances how to determine lower upper tolerances such that the arising problem becomes consistent numerical examples are given

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

the measure of the degree of truth and the grade of membership abstract a method is presented to define the meaning of a fuzzy proposition and of a fuzzy set through the construction of fuzzy propositions with non ambiguous intermediate truth values

treating fuzziness in subjective evaluation data this paper proposes a technique to deal with fuzziness in subjective evaluation data and applies it to principal component analysis and correspondence analysis in the existing method or techniques developed directly from it fuzzy sets are defined from some standpoint on a data space and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares in this paper we try to map the variation in evaluation data into the parameter space while preserving information as much as possible and thereby define fuzzy sets in the parameter space clearly it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle however with a fuzzy model which uses the extension principle the possibility distribution spreads out as the explanatory variable values increase this does not necessarily make sense for subjective evaluations such as a 5 level evaluation for instance instead of doing so we propose a method for explicitly expressing the vagueness of evaluation using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread as a numerical example we present an analysis of subjective evaluation data on local environments

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

a fuzzy set model for market share and preference prediction abstract this paper presents a new method of modelling preferences using fuzzy sets a fuzzy set conjoint model is developed by incorporating fuzzy measurement of evaluations into the vector preference model a comparison test of rank prediction is conducted on data gathered from consumer subjects by an adaptive software program a methodology is developed to generate aggregate crisp market share estimates from individual fuzzy set preference predictions the result is a substantial improvement in individual predictive validity using the fuzzy set model and more accurate aggregate market share prediction for use in product design demand forecasting and other management functions

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

type i and type ii fuzzy system modeling abstract two system modeling approaches are proposed with a unification of fuzzy methodologies type i and type ii knowledge representation are reviewed together with their proposed approximate reasoning schemes type i representation and inference are based on boolean normal forms type i models are myopic because in these models either the disjunctive or the conjunctive boolean normal forms are selected rather arbitrarily for knowledge representation and inference and then they are fuzzified whereas in type ii models representation and inference are based on fuzzy disjunctive and conjunctive normal forms which are derived from the fuzzy truth tables

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space