comprehensive analysis of a new fuzzy rule interpolation method the first published result in fuzzy rule interpolation was the spl alpha cut based fuzzy rule interpolation termed as kh fuzzy rule interpolation originally devoted for complexity reduction a modified version of the kh approach has been presented by yam et al 1999 which eliminates the subnormality problem while at the same time intending to maintain the advantageous computational properties of the original method this paper presents a comprehensive analysis of the new method which includes detailed comparison with the original kh fuzzy rule interpolation method concerning the explicit functions of the methods preservation of piecewise linearity and stability the fuzziness of the conclusion with respect to the fuzziness of the observation is also investigated in comparison with several interpolation techniques all these comparisons shows that the new method preserves the advantageous properties of the kh method and alleviates its most significant disadvantage the problem of subnormality

granular counting of uncertain data abstract we propose a definition of granular count realized in the presence of uncertain data modeled through possibility distributions we show that the resulting counts are fuzzy intervals in the domain of natural numbers based on this result we devise two algorithms for granular counting an exact counting algorithm with quadratic time complexity and an approximate counting algorithm with linear time complexity we compare the two algorithms on synthetic data and show their application to a bioinformatics scenario concerning the assessment of gene expressions in cells

estimating and understanding exponential random graph models we introduce a method for the theoretical analysis of exponential random graph models the method is based on a large deviations approximation to the normalizing constant shown to be consistent using theory developed by chatterjee and varadhan european j combin 32 2011 1000 1017 the theory explains a host of difficulties encountered by applied workers many distinct models have essentially the same mle rendering the problems practically ill posed we give the first rigorous proofs of degeneracy observed in these models here almost all graphs have essentially no edges or are essentially complete we supplement recent work of bhamidi bresler and sly 2008 ieee 49th annual ieee symposium on foundations of computer science focs 2008 803 812 ieee showing that for many models the extra sufficient statistics are useless most realizations look like the results of a simple erd s renyi model we also find classes of models where the limiting graphs differ from erd s renyi graphs a limitation of our approach inherited from the limitation of graph limit theory is that it works only for dense graphs

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

probability and fuzziness we discuss some relationships between probability theory and statistics on one hand and the theory of fuzzy sets on the other hand we develop various statistical techniques for the analysis of imprecise data and for inference based on imprecise data we study the estimation of fuzzy parameters testing problems with fuzzy probabilities and confidence estimation under uncertainty finally we indicate some possible applications to the assessment of credibility in expert systems

practical inference with systems of gradual implicative rules a general approach to practical inference with gradual implicative rules and fuzzy inputs is presented gradual rules represent constraints restricting outputs of a fuzzy system for each input they are tailored for interpolative reasoning our approach to inference relies on the use of inferential independence it is based on fuzzy output computation under an interval valued input a double decomposition of fuzzy inputs is done in terms of alpha cuts and in terms of a partitioning of these cuts according to areas where only a few rules apply the case of 1 d and 2 d inputs is considered as well as higher dimensional cases an application to a cheese making process illustrates the approach

a similarity based bidirectional approximate reasoning method for decision making systems this paper presents a similarity based bidirectional approximate reasoning method which can express the decision maker s disposition for this we shall propose a new type of similarity measure based on the ordered weighted aggregation owa operator the proposed similarity measure has a structure to involve the disposition of a decision maker by choosing a suitable weighting vector of the owa operator from this property we derive a simple and flexible type of similarity based bidirectional approximate reasoning method which can support the making of a decision under more complex situations such as industrial inspection systems for example

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

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

self organizing maps for imprecise data self organizing maps soms consist of a set of neurons arranged in such a way that there are neighbourhood relationships among neurons following an unsupervised learning procedure the input space is divided into regions with common nearest neuron vector quantization allowing clustering of the input vectors in this paper we propose an extension of the soms for data imprecisely observed self organizing maps for imprecise data soms id the learning algorithm is based on two distances for imprecise data in order to illustrate the main features and to compare the performances of the proposed method we provide a simulation study and different substantive applications

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

a fuzzy measure similarity between sets of linguistic summaries in this paper we consider the problem of evaluating the similarity of two sets of linguistic summaries of sensor data huge amounts of available data cause a dramatic need for summarization in continuous monitoring it is useful to compare one time interval of data with another for example to detect anomalies or to predict the onset of a change from a normal state assuming that summaries capture the essence of the data it is sufficient to compare only those summaries i e they are descriptive features for recognition in previous work we developed a similarity measure between two individual summaries and proved that the associated dissimilarity is a metric additionally we proposed some basic methods to combine these similarities into an aggregate value here we develop a novel parameter free method which is based on fuzzy measures and integrals to fuse individual similarities that will produce a closeness measurement between sets of summaries we provide a case study from the eldercare domain where the goal is to compare different nighttime patterns for change detection the reasons for studying linguistic summaries for eldercare are twofold first linguistic summaries are the natural communication tool for health care providers in a decision support system and second due to the extremely large volume of raw data these summaries create compact features for an automated reasoning for detection and prediction of health changes as part of the decision support system

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

checking the coherence and redundancy of fuzzy knowledge bases checking the coherence of a set of rules is an important step in knowledge base validation coherence is also needed in the field of fuzzy systems indeed rules are often used regardless of their semantics and it sometimes leads to sets of rules that make no sense avoiding redundancy is also of interest in real time systems for which the inference engine is time consuming a knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base we more particularly consider knowledge bases composed of parallel fuzzy rules then coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted i e any value for these variables is possible or at least not more restrictive than integrity constraints fuzzy rule representations can be implication based or conjunction based we show that only implication based models may lead to coherence problems however unlike conjunction based models they allow to design coherence checking processes some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules the problem of redundancy which is also of interest for fuzzy knowledge bases validation is addressed for these two kinds of rules

weak regularity and finitely forcible graph limits graphons are analytic objects representing limits of convergent sequences of graphs lovasz and szegedy conjectured that every finitely forcible graphon i e any graphon determined by finitely many graph densities has a simple structure in particular one of their conjectures would imply that every finitely forcible graphon has a weak varepsilon regular partition with the number of parts bounded by a polynomial in varepsilon 1 we construct a finitely forcible graphon w such that the number of parts in any weak varepsilon regular partition of w is at least exponential in varepsilon 2 2 5 log varepsilon 2 this bound almost matches the known upper bound for graphs and in a certain sense is the best possible for graphons

fuzzy control of model car this paper discusses fuzzy control of a model car fuzzy control rules are derived by modelling an expert s driving actions experiments are performed using a model car with a sensing unit and a micro computer

compression quality prediction model for jpeg2000 a compression quality prediction model is proposed for grey images coding with jpeg2000 with this model the compression quality psnr could be estimated according to the given compression ratio cr and the image activity measures iam without coding images the image activity measure is the weighted sum of the iam values based on the 1 pixel distance and 2 pixel distance gradients along horizontal and vertical directions we have shown that iam is a function of the image variance and autocorrelation coefficients based on shannon s rate distortion theorem a theoretical justification is provided for the correlation of iam with psnr experimental results show that the prediction error is lower than 1 db for more than 70 sample images when cr is higher than 15 the prediction error is less than 2 db for over 90 images this prediction performance is acceptable for general applications

linguistic summarization using if then rules and interval type 2 fuzzy sets linguistic summarization ls is a data mining or knowledge discovery approach to extract patterns from databases many authors have used this technique to generate summaries like most senior workers have high salary which can be used to better understand and communicate about data however few of them have used it to generate if then rules like if x is large and y is medium then z is small which not only facilitate understanding and communication of data but can also be used in decision making in this paper an ls approach to generate if then rules for causal databases is proposed both type 1 and interval type 2 fuzzy sets are considered five quality measures the degrees of truth sufficient coverage reliability outlier and simplicity are defined among them the degree of reliability is especially valuable for finding the most reliable and representative rules and the degree of outlier can be used to identify outlier rules and data for close up investigation an improved parallel coordinates approach for visualizing the if then rules is also proposed experiments on two datasets demonstrate our ls and rule visualization approaches finally the relationships between our ls approach and the wang mendel wm method perceptual reasoning and granular computing are pointed out

classification and coding approaches to part family formation under a fuzzy environment classification and coding is a major approach to solve the part family formation problem in group technology applications traditionally all features in a classification and coding system are considered crisp it has been recognized that more realistically some features should be fuzzy this paper addresses issues involved in using the classification and coding approach for part family formation under a fuzzy environment methods for dealing with major issues such as part coding computation of similarity between parts and part family formation are proposed and discussed in detail examples were given to show how these methods could be applied

on the formulation of optimization under elastic constraints with control in mind abstract we give a basic survey of various approaches to defining the maximum point of a crisp numerical function over a fuzzy set this survey is based on several unifying ideas and includes original comparison results motivations and applications will be drawn mainly from control

geometric compatibility modification abstract compatibility modification is a rule based inference strategy that uses the similarity of the input with the antecedent of a rule to modify the consequent existing compatibility modification inference techniques have employed a set theoretic assessment of compatibility in this paper a distance based compatibility measure is derived from a generalization of the dissemblance index for fuzzy sets this measure is then used to develop an inference technique based on geometric compatibility this geometric approach is compared with two other distance based inference techniques linear rule interpolation and bound dependent linear revision

a generalized concept for fuzzy rule interpolation the concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993 it has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction the first implemented technique of fuzzy rule interpolation was termed as spl alpha cut distance based fuzzy rule base interpolation despite its advantageous properties in various approximation aspects and in complexity reduction it was shown that it has some essential deficiencies for instance it does not always result in immediately interpretable fuzzy membership functions this fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies this paper is an attempt into this direction it proposes an interpolation methodology whose key idea is based on the interpolation of relations instead of interpolating spl alpha cut distances and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques the proposed concept of interpolating relations is elaborated here using fuzzy and semantic relations this paper presents numerical examples in comparison with former approaches to show the effectiveness of the proposed interpolation methodology

a bit allocation method based on picture activity for still image coding bit allocation or the quantizer assignment problem is a basic and essential issue in lossy picture coding the optimal solution for the bit allocation problem can be found by the lagrangian method however it requires much computational time and memory to reduce complexity overhead we propose a fast scheme using a picture activity measure comparison among the existing activity measurement methods is presented to select the most reliable activity measure and the mapping relation between the activity value and a quantization parameter is proposed

from classic statistical characterization to fuzzy windowing based characterization for the exploratory analysis of miscellaneous time variables example in the field of car driving studies the problem of data characterization of quantitative and qualitative measurement scales is stated in the context of an exploratory multivariate statistical analysis an example from a car driving study is considered where the quantitative data correspond to the car and head movements while the qualitative data correspond to objects being viewed road bridge sign post etc for each of these two sets the literature is analyzed first in terms of data characterizing methods and relationship obtaining methods then we propose to evaluate and compare nine quantitative data characteriing methods five corresponding to classic statistical indicators two to crisp space windowing with either two or three windows and two on fuzzy windowing with either two or three windows logically the last method appears as the best according to our evaluation procedure then we propose a bidimensional fuzzy windowing instead of a crisp one to characterize the gaze positions finally the multiple correspondence analysis is used to investigate the membership value averages obtained from the characterization stage

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

generalized fuzzy indices for similarity matching the purpose of this paper is to introduce a new family of fuzzy similarity indices referred to as the generalised tversky index gti the development of gti is based on the theoretical findings by amos tversky regarding the human perception of similarity between different objects as formulated by the features contrast model fc although gti was developed starting from tversky s fc it is shown that it provides a fuzzy extension and generalization of several widely used similarity indices like the jaccard and dice coefficients the mathematical properties of two members of the gti family namely tim and tip are studied and their interpretation of similarity is explained by comparison to other conventional indices

optimal trellis based buffered compression and fast approximations the authors formalize the description of the buffer constrained adaptive quantization problem for a given set of admissible quantizers used to code a discrete nonstationary signal sequence in a buffer constrained environment they formulate the optimal solution they also develop slightly suboptimal but much faster approximations these solutions are valid for any globally minimum distortion criterion which is additive over the individual elements of the sequence as a first step they define the problem as one of constrained discrete optimization and establish its equivalence to some of the problems studied in the field of integer programming forward dynamic programming using the viterbi algorithm is shown to provide a way of computing the optimal solution then they provide a heuristic algorithm based on lagrangian optimization using an operational rate distortion framework that with computing complexity reduced by an order of magnitude approaches the optimally achievable performance the algorithms can serve as a benchmark for assessing the performance of buffer control strategies and are useful for applications such as multimedia workstation displays video encoding for cd roms and buffered jpeg coding environments where processing delay is not a concern but decoding buffer size has to be minimized

construction of a neurofuzzy network capable of extrapolating and interpolating with respect to the convex hull of a set of input samples in bb r n the problem of regression estimation is considered with a specific regard for the distinction between interpolation and extrapolation a neurofuzzy network named nfech is proposed that is capable of extrapolating and interpolating with respect to the convex hull of a finite set of input samples x sub ropfn the geometrical construction of the proposed network is explained both mathematically and graphically the illustrations explain how the particular parts of the construction work and also show the final surfaces of the obtained models the method is tested on artificial datasets generated from mathematical functions according to various statistical distributions also comparisons to the commonly used radial basis function rbf multilayered perceptron mlp neural networks and to fuzzy rule interpretation fri fuzzy rule extrapolation fre approach are presented

a fuzzy set based approach for modeling dependence among human errors the assessment of dependence among human errors is an important aspect of human reliability analysis when dependence between two tasks exists the probability of the operators failure on one task is higher if they have failed on the preceding task compared to when they have succeeded in current practice the task of assessing dependence among successive operator actions relies to a great extent to expert judgment often with lack of traceability and repeatability to overcome these limitations this work presents a systematic framework for the elicitation of expert knowledge on the factors influencing the dependence between two successive tasks the framework is based on a fuzzy expert system in which a set of transparent fuzzy logic rules is used to represent the relationship between the input factors and the conditional human error probability the proposed modeling approach is applied to two tasks required in response to an accident scenario at a nuclear power plant given the methodological scope of the work the fuzzy expert system is tested directly on a working model of dependence whereas no actual exercise of expert judgment elicitation is carried out

similarity and dissimilarity measures between fuzzy sets a formal relational study the paper deals with the well known notion of dis similarity measures between fuzzy sets we provide three separate lists of axioms that fit with the respective notions of general comparison measure similarity measure and dissimilarity measure then we review some of the most important axiomatic definitions of dis similarity measures in the literature by referring to the axioms in those lists satisfied by each specific definition this common framework will make our study about the formal relationships among different axiomatic definitions much easier some of them which are apparently different do in fact share many commonalities we provide a self contained picture of these relationships by providing formal results and counterexamples that reflect which of the dis similarity definitions in the literature are connected by implication relations and which of them are not we finalize the paper with an in depth study about the notion of duality between similarity and dissimilarity measures as well as with some concluding remarks

a distance metric for a space of linguistic summaries producing linguistic summaries of large databases or temporal sequences of measurements is an endeavor that is receiving increasing attention these summaries can be used in a continuous monitoring situation like eldercare where it is important to ascertain if the current summaries represent an abnormal condition it is therefore necessary to compute the distance between summaries as a basis for such a determination in this paper we propose a dissimilarity measure between summaries based on fuzzy protoforms and prove that this measure is a metric we take into account not only the linguistic meaning of the summaries but also two quality evaluations namely the truth values and the degrees of focus we present examples of how the distance metric behaves and show that it corresponds with intuition

ranking type 2 fuzzy numbers type 2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type 2 fuzzy set is characterized by a fuzzy membership function in this paper we consider the problem of ranking a set of type 2 fuzzy numbers we adopt a statistical viewpoint and interpret each type 2 fuzzy number as an ensemble of ordinary fuzzy numbers this enables us to define a type 2 fuzzy rank and a type 2 rank uncertainty for each intuitionistic fuzzy number we show the reasonableness of the results obtained by examining several test cases

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

interpolative reasoning with insufficient evidence in sparse fuzzy rule bases abstract rule based fuzzy approximate reasoning uses various techniques of modified modus ponens the observation is in most cases not identical with any of the antecedents in the rules however a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present if the rule base is sparse i e it contains insufficient information on the total state space it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired i e no conclusion can be computed on the basis of modus ponens in such a case interpolative reasoning in the strict sense can be applied some kind of weighted average of the flanking rules is calculated this technique can be extended to a form of extrapolation when the observation is not flanked from both sides linear interpolation and extrapolation is presented and then the idea is extended to arbitrary approximation

a statistical comparative study of different similarity measures of consensus in group decision making an essential aim in group decision making gdm problems is to achieve a high level of consensus among experts consensus is defined as general or widespread agreement and it is usually modelled mathematically via a similarity function measuring how close experts opinions or preferences are similarity functions are defined based on the use of a metric describing the distance between experts opinions or preferences in the literature different metrics or distance functions have been proposed to implement in consensus models but no study has been carried out to analyse the influence the use of different distance functions can have in the gdm process this paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in gdm by using the nonparametric wilcoxon matched pairs signed ranks test it is concluded that different distance functions can produce significantly different results moreover it is also shown that their application also has a significant effect on the speed of achieving consensus finally these results are analysed and used to derive decision support rules based on a convergent criterion that can be used to control the convergence speed of the consensus process using the compared distance functions

a similarity measure for fuzzy rulebases based on linguistic gradients fuzzy rulebases are an approximate representation of some interesting system as such they could potentially be used for indexing and searching candidate solutions in case based reasoning cbr systems in a variety of application areas however there is currently no method for directly and efficiently determining the similarity between two rulebases which is a necessary element of their use in any cbr system we propose a method for measuring similarity between two linguistic fuzzy rulebases based on the granular computing technique of linguistic gradients the proposed similarity measure is based on comparing the linguistic structure of two rulebases using the linguistic gradient operator to reveal that structure our algorithm operates at the level of linguistic rulebases rather than a defuzzified reasoning surface and thus belongs to the computing with words paradigm in a validation experiment we compare our new method with the root mean square difference between reasoning surfaces for 603 pairs of fuzzy rulebases drawn from the literature a spearman correlation analysis shows that our new linguistic method is consistent with the numerical rms results while being theoretically and empirically faster than computing an accurate rms difference

infinite dimensional finitely forcible graphon graphons are analytic objects associated with convergent sequences of dense graphs finitely forcible graphons that is those determined by finitely many subgraph densities are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science lovasz and szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension which would have implications on the minimum number of parts in its weak e regular partition we disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension

adaptive fuzzy interpolation fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems complexity however after a series of interpolations it is possible that multiple object values for a common variable are inferred leading to inconsistency in interpolated results such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations this paper presents a novel approach for identification and correction of defective rules in interpolative transformations thereby removing the inconsistencies in particular an assumption based truth maintenance system atms is used to record dependences between interpolations and the underlying technique that the classical general diagnostic engine gde employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations from this an algorithm is introduced to allow for the modification of the original linear interpolation to become first order piecewise linear the approach is applied to a realistic problem which predicates the diarrheal disease rates in remote villages to demonstrate the potential of this study

handling interpretability issues in anfis using rule base simplification and constrained learning abstract adaptive neuro fuzzy inference system anfis is a well known neuro fuzzy model for approximating highly complex non linear systems anfis uses precise fuzzy modelling concept that aims at accuracy of a fuzzy model being designed than on interpretability but interpretability of a fuzzy system is an equally important aspect of fuzzy modelling as accuracy so far the research based on anfis has been mostly application based due to which the various issues related to the interpretability of anfis have not been dealt with the rule base of anfis is typically obtained using data driven clustering algorithms but this process introduces redundancy in the system in terms of similar fuzzy sets and redundant fuzzy rules which unnecessarily increases system complexity this in turn reduces both the interpretability as well as generalization capability of anfis additionally in case of anfis unconstrained gradient descent based learning algorithms are used to fine tune the membership function parameters which usually result in a rule base with inconsistent excessively overlapping and indistinguishable membership functions for input variables and thus the interpretability of the final optimized system is not guaranteed this paper is based on addressing the issue of rule base redundancy in anfis to reduce complexity and enforcing constraints during learning phase to ensure interpretability of the final optimized system rule base redundancy is removed using similarity analysis based rule base simplification in which similar fuzzy sets are merged and subsequently resulting fuzzy rules with equal premises are combined hybrid learning technique which is an efficient parameter tuning method for anfis is constrained to prevent inconsistency excessive overlapping and inclusion of membership functions so that the final fuzzy partitions of inputs stay interpretable the empirical analysis of the impact of rule base simplification and constrained learning on anfis is done by application to two well known benchmark problems and a real world stock price prediction problem the introduction of rule base simplification and constrained learning in anfis modeling has shown better results in terms of obtaining a desired accuracy interpretability tradeoff than conventional anfis

a methodology to model fuzzy systems using fuzzy clustering in a rapid prototyping approach in this paper we present different approaches to the problem of fuzzy rules extraction by using fuzzy clustering they try to investigate the use of fuzzy clustering as a technique that let us obtain a first approximation of the fuzzy model in a rapid prototyping approach the key ideas here are to generate in a first step a collection of fuzzy rules from numerical data pairs subsequently these rules can be converted either in a collection of linguistic rules in order to obtain a descriptive approach or in an inference machine so that we obtain an approximative approach this can be considered as a quasi descriptive approach because it assigns a weight to the fuzzy rules like in the descriptive approaches but also it generates the fuzzy sets directly in the product space of the input variables like in approximative approaches both by using the information disclosed by means of a fuzzy clustering process

perceptual reasoning for perceptual computing a similarity based approach perceptual reasoning pr is an approximate reasoning method that can be used as a computing with words cww engine in perceptual computing there can be different approaches to implement pr e g firing interval based pr fi pr which has been proposed in j m mendel and d wu ieee trans fuzzy syst vol 16 no 6 pp 1550 1564 dec 2008 and similarity based pr s pr which is proposed in this paper both approaches satisfy the requirement on a cww engine that the result of combining fired rules should lead to a footprint of uncertainty fou that resembles the three kinds of fous in a cww codebook a comparative study shows that s pr leads to output fous that resemble word fous which are obtained from subject data much more closely than fi pr hence s pr is a better choice for a cww engine than fi pr

what are fuzzy rules and how to use them abstract fuzzy rules have been advocated as a key tool for expressing pieces of knowledge in fuzzy logic however there does not exist a unique kind of fuzzy rules nor is there only one type of fuzzy logic this diversity has caused many a misunderstanding in the literature of fuzzy control the paper is a survey of different possible semantics for a fuzzy rule and shows how they can be captured in the framework of fuzzy set and possibility theory it is pointed out that the interpretation of fuzzy rules dictates the way the fuzzy rules should be combined the various kinds of fuzzy rules considered in the paper gradual rules certainty rules possibility rules and others have different inference behaviors and correspond to various intended uses and applications the representation of fuzzy unless rules is briefly investigated on the basis of their intended meaning the problem of defining and checking the coherence of a block of parallel fuzzy rules is also briefly addressed this issue has been neglected in the fuzzy control literature although it looks important for validation purposes

improvements and critique on sugeno s and yasukawa s qualitative modeling investigates sugeno s and yasukawa s 1993 qualitative fuzzy modeling approach we propose some easily implementable solutions for the unclear details of the original paper such as trapezoid approximation of membership functions rule creation from sample data points and selection of important variables we further suggest an improved parameter identification algorithm to be applied instead of the original one these details are crucial concerning the method s performance as it is shown in a comparative analysis and helps to improve the accuracy of the built up model finally we propose a possible further rule base reduction which can be applied successfully in certain cases this improvement reduces the time requirement of the method by up to 16 in our experiments

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

gradual inference rules in approximate reasoning abstract gradual inference rules of the form the more x is f the more y is g or of similar forms with the less instead of the more which express a progressive change of the degree to which the entity y satisfies the gradual property g when the degree to which the entity x satisfies the gradual property f is modified are often encountered in commonsense reasoning a representation of such rules by means of fuzzy sets is proposed and discussed this representation turns out to be based on a special implication function already considered in multiple valued logic patterns of reasoning involving gradual inference rules are formalized their links with interpolation mechanisms are pointed out

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

a fuzzy clustering based rapid prototyping for fuzzy rule based modeling this paper presents different approaches to the problem of fuzzy rules extraction by using fuzzy clustering as the main tool within these approaches we describe six methods that represent different alternatives in the fuzzy modeling process and how they can be integrated with a genetic algorithms these approaches attempt to obtain a first approximation to the fuzzy rules without any assumption about the structure of the data because the main objective is to obtain an approximation the methods we propose must be as simple as possible but also they must have a great approximative capacity and in that way we work directly with fuzzy sets induced in the variables input space the methods are applied to four examples and the errors obtained are specified in the different cases

the asymptotics of large constrained graphs we show through local estimates and simulation that if one constrains simple graphs by their densitiesof edges and of triangles then asymptotically in the number of vertices for over 95 of the possible range of those densities there is a well defined typical graph and it has a very simple structure the vertices are decomposed into two subsets v1 and v2 of fixed relative size c and 1 c and there are well defined probabilities of edges g jk between v j vj and vk vk furthermore the four parameters c g11 g22 and g12 are smooth functions of except at two smooth phase transition curves

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

a collaborative recommender system based on probabilistic inference from fuzzy observations the problem of building recommender systems has attracted considerable attention in recent years the objective of this paper is to automatically suggest and rank a list of new items to a user based on the past voting patterns of other users with similar tastes the proposed model can be considered as a soft computing based collaborative recommender system the combination of bayesian networks which enables an intuitive representation of the mechanisms that govern the relationships between the users and the fuzzy set theory enabling us to represent ambiguity or vagueness in the description of the ratings improves the accuracy of the system

weighted fuzzy interpolative reasoning systems based on interval type 2 fuzzy sets abstract in this paper we present a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on interval type 2 fuzzy sets we also apply the proposed weighted fuzzy interpolative reasoning method to deal with the truck backer upper control problem the proposed method satisfies the seven evaluation indices for fuzzy interpolative reasoning the experimental results show that the proposed method outperforms the existing methods it provides us with a useful way for dealing with fuzzy interpolative reasoning in sparse fuzzy rule based systems

multiobjective linguistic optimization generalizing our earlier results on optimization with linguistic variables we introduce a novel statement of fuzzy multiobjective mathematical programming problems and provide a method for finding a fair solution to these problems suppose we are given a multiobjective mathematical programming problem in which the functional relationship between the decision variables and the objective functions is not completely known our knowledge base consists of a block of fuzzy if then rules where the antecedent part of the rules contains some linguistic values of the decision variables and the consequence part consists of a linguistic value of the objective functions we suggest the use of tsukamoto s fuzzy reasoning method to determine the crisp functional relationship between the decision variables and objective functions we model the anding of the objective functions by a t norm and solve the resulting usually nonlinear programming problem to find a fair optimal solution to the original fuzzy multiobjective problem

singularities in the entropy of asymptotically large simple graphs we prove that the asymptotic entropy of large simple graphs as a function of fixed edge and triangle densities is nondifferentiable along a certain curve we also determine the precise modified bipartite structure of asymptotic graphs with edge density 1 2 and triangle density in the interval displaystyle 0 1 8

stability of interpolative fuzzy kh controllers the classical approaches in fuzzy control zadeh and mamdani deal with dense rule bases when this is not the case i e in sparse rule bases one has to choose another method fuzzy rule interpolation proposed first by koczy and hirota internat j approx reason 9 1993 197 225 offers a possibility to construct fuzzy controllers kh controllers under such conditions the main result of this paper shows that the kh interpolation method is stable it also contributes to the application oriented use of balazs shepard interpolation operators investigated extensively by researchers in approximation theory the numerical analysis aspect of the result contributes to the well known problem of finding a stable interpolation method in the following sense we would like to approximate a function that is only known at distinct points e g where it can be measured since measurement errors cannot be eliminated in practice it is a natural requirement that the interpolation method should be stable independently from the selection of measurement points the classical interpolation methods generally do not fulfil this condition only with certain strong restrictions concerning the measurement points if we neglect the classical form of the approximating function polynomial and trigonometrical we can produce better behaving approximation the kh controller approximating function being a simple fractional function fulfils the stability condition this can also be interpreted as kh controllers are universal approximators in the space of continuous functions with respect to e g the supremum norm

finitely forcible graph limits are universal abstract the theory of graph limits represents large graphs by analytic objects called graphons graph limits determined by finitely many graph densities which are represented by finitely forcible graphons arise in various scenarios particularly within extremal combinatorics lovasz and szegedy conjectured that all such graphons possess a simple structure e g the space of their typical vertices is always finite dimensional this was disproved by several ad hoc constructions of complex finitely forcible graphons we prove that any graphon is a subgraphon of a finitely forcible graphon this dismisses any hope for a result showing that finitely forcible graphons possess a simple structure and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons in addition since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs our result also shows that such minimization problems which conceptually are among the simplest kind within extremal graph theory may in fact have unique optimal solutions with arbitrarily complex structure

review measures of divergence on credal sets when there is uncertainty about the values of a variable it is possible to have different pieces of information about it this can be due to the fact that they represent the points of view of different agents and these agents have followed different criteria when codify the pieces of information by means of numbers the aim of this paper is to provide tools for the comparison of these representations this is done in the framework of imprecise probabilities more concretely the credal set theory is used the reason is that credal sets are very general including as particular cases possibility theory probability theory and belief functions so it can serve as basis to compare the representations of information obtained by following these formalisms the measures used for comparison are inconsistency measure inclusion index and informative distance

representing membership functions as points in high dimensional spaces for fuzzy interpolation and extrapolation this paper introduces a new approach for fuzzy interpolation and extrapolation of sparse rule base comprising of membership functions with finite number of characteristic points the approach calls for representing membership functions as points in high dimensional cartesian spaces using the locations of their characteristic points as coordinates hence a fuzzy rule base can be viewed as a set of mappings between the antecedent and consequent spaces and the interpolation and extrapolation problem becomes searching for an image in the consequent space upon given an antecedent observation the present approach divides observations into two groups 1 observations within the antecedent spanning set contain the same geometric properties as the given antecedents and 2 observations lying outside the antecedent spanning set contain new geometric properties beyond those of the given rules heuristic reasoning must therefore be applied in this case a two step approach with certain flexibility to accommodate additional criteria and design objectives is formulated

fuzzy systems and approximation abstract the basic motivation of using fuzzy rule based systems especially for control purposes is to deduce simple and fast approximations of the unknown or too complicated models fuzzy rule based systems have become very popuar because of their transparency and easiness of tuning and modification recently some results concerning the explicit functions implemented by realistic fuzzy controllers presented the class of functions that could be implemented in this way some parallel results on the other hand attempted to prove that the main advantage of using fuzzy systems was the suitability for approximation with arbitrary accuracy in their universality the explicit formulas and some very recent theoretical results made it clear however that fuzzy systems were not really good approximators as realistic fuzzy controllers could generate only very rough approximations of given transference functions in connection with approximation the question can be asked whether there is an optimal fineness roughness of a fuzzy rule base that controls a certain action with roughness gives minimal time complexity as an example a target tracking problem was chosen cat and mouse or hawk and sparrow problem where the antagonistic criteria of minimizing inference time by the given rule base and minimizing action time search for the target with given uncertainty provided by the rule model were examined under certain assumptions the solution of this optimization problem leads to nontrivial rule base sizes these results have also practical applicability since if a fine enough model of the system is known it is always possible to generate a rougher version of the same by applying the model transformation technique offered by rule interpolation with levels

weighted fuzzy interpolative reasoning based on weighted increment transformation and weighted ratio transformation techniques in this paper we present a new weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems the proposed method uses weighted increment transformation and weighted ratio transformation techniques to handle weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems it allows each variable that appears in the antecedent parts of fuzzy rules to associate with a weight between zero and one moreover we also propose an algorithm that automatically tunes the optimal weights of the antecedent variables appearing in the antecedent parts of fuzzy rules we also apply the proposed weighted fuzzy interpolative reasoning method to handle the truck backer upper control problem the proposed weighted fuzzy interpolative reasoning method performs better than the ones obtained by the traditional fuzzy inference system 2000 huang and shen s method 2008 and chen and ko s method 2008 the proposed method provides us with a useful way to deal with weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems

fuzzy rule interpolation for multidimensional input spaces with applications a case study fuzzy rule based systems have been very popular in many engineering applications however when generating fuzzy rules from the available information this may result in a sparse fuzzy rule base fuzzy rule interpolation techniques have been established to solve the problems encountered in processing sparse fuzzy rule bases in most engineering applications the use of more than one input variable is common however the majority of the fuzzy rule interpolation techniques only present detailed analysis to one input variable case this paper investigates characteristics of two selected fuzzy rule interpolation techniques for multidimensional input spaces and proposes an improved fuzzy rule interpolation technique to handle multidimensional input spaces the three methods are compared by means of application examples in the field of petroleum engineering and mineral processing the results show that the proposed fuzzy rule interpolation technique for multidimensional input spaces can be used in engineering applications

weighted fuzzy rule interpolation based on ga based weight learning techniques in this paper we propose a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems it is based on a genetic algorithm ga based weight learning technique the proposed method can deal with fuzzy rule interpolation with weighted antecedent variables it also can deal with fuzzy rule interpolation based on polygonal membership functions and bell shaped membership functions we also propose a ga based weight learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules furthermore we apply the proposed weighted fuzzy interpolative reasoning method and the proposed ga based weight learning algorithm to deal with the truck backer upper control problem the computer activity prediction problem multivariate regression problems and time series prediction problems based on statistical analysis techniques the experimental results show that the proposed weighted fuzzy interpolative reasoning method by the use of the optimally learned weights that were obtained by the proposed ga based weight learning algorithm has statistically significantly smaller error rates than the existing methods

on the representation of fuzzy rules in terms of crisp rules there are two classical ways of viewing a fuzzy set in terms of a combination of crisp sets either as a weighted max aggregation of its level cuts or as a convex combination of the latter these views are applied to the particular case of fuzzy relations which are induced by fuzzy rules where the rules are modeled by implication connectives defined on a finite subset of the unit interval the paper studies more closely the representation of such fuzzy rules in terms of a convex combination of gradual rules a special type of implication based fuzzy rule inducing a crisp relation this representation which might be interpreted in a probabilistic way is shown to be unique on the assumption that the implication operator used for modeling the fuzzy rule does not satisfy a special kind of strict monotonicity condition in this case the crisp relations induced by the involved gradual rules correspond to level cuts of the fuzzy relation associated with the fuzzy rule however other representations might exist if the aforementioned property is not satisfied under a slightly stronger strict monotonicity condition the existence of further non consonant representations is even guaranteed then the crisp relations induced by gradual rules do not necessarily correspond to level cuts of the underlying fuzzy relation in this sense the decomposition of a fuzzy rule into a class of crisp implication based rules is more general than the classical decomposition of fuzzy sets

ordering distance and closeness of fuzzy sets abstract in dense rule bases where the observation usually overlaps with several antecedents in the rule base various algorithms are used for approximate reasoning and control if the antecedents are located sparsely and the observation does not overlap as a rule with any of the antecedents function approximation techniques combined with the resolution principle lead to applicable conclusions this kind of approximation is possible only if a new concept of ordering and distance i e a metric in the state space and a partial ordering among convex and normal fuzzy sets cnf sets is introduced so the fuzzy distance of two cnf sets can be defined and by this distance closeness and similarity of cnf sets as well

new methods for students evaluation using fuzzy sets in an earlier work biswas 1995 presented two methods for the application of fuzzy sets in students answerscripts evaluation in this paper we extend his work to propose two new methods for evaluating students answerscripts using fuzzy sets the proposed methods can overcome the drawbacks in biswas 1995 due to the fact that they do not need to perform the complicated matching operations and they can evaluate students answerscripts in a more fair manner

approximation of extensional fuzzy relations over a residuated lattice we investigate a relationship between extensionality of fuzzy relations and their lipschitz continuity on generalized metric spaces the duality of these notions is shown and moreover two particular applications of the extensionality property in the field of approximation are given

a comparison of similarity measures of fuzzy values abstract this paper extends the work of pappis and karacapilidis 1993 to present and compare the properities of several measures of similarity of fuzzy values the measures examined in this paper are based on the geometric model the set theoretic approach and the matching function s we presented in chen 1988 it is shown that several properties are common to all measures and some properties do not hold for all of them

fuzzy clustering of fuzzy data based on robust loss functions and ordered weighted averaging abstract in many real cases the data are not expressed in term of single values but are imprecise in all these cases standard clustering methods for single valued data are unable to properly take into account the imprecise nature of the data in this paper by considering the partitioning around medoids pam approach in a fuzzy framework we propose a fuzzy clustering method for imprecise data formalized in a fuzzy manner in particular in order to neutralize the negative effects of possible outlier fuzzy data in the clustering process we proposed a robust fuzzy c medoids clustering method for fuzzy data based on the combination of huber s m estimators and yager s owa ordered weighted averaging operators the proposed method is able to smooth the influence of anomalous data by means of a suitable parameter the so called typicality parameter capable to tune the influence of the outliers the performance of the proposed method has been shown by means of a simulation study composed of experiments on i simple two dimensional dataset ii benchmark datasets and iii the fuzzy art outliers dataset the comparison made with the robust clustering methods known from the literature indicates the competitiveness of the introduced method to others an application of the suggested method to a real dataset is also provided and the results of the method has been compared with other clustering methods suggested in the literature in the application the comparative assessment has shown the informational gain in term of additional information of the proposed method vs the other robust methods