induction of fuzzy rules and membership functions from training examples abstract most fuzzy controllers and fuzzy expert systems must predefine membership functions and fuzzy inference rules to map numeric data into linguistic variable terms and to make fuzzy reasoning work in this paper we propose a general learning method as a framework for automatically deriving membership functions and fuzzy if then rules from a set of given training examples to rapidly build a prototype fuzzy expert system based on the membership functions and the fuzzy rules derived a corresponding fuzzy inference procedure to process inputs is also developed

learning rules for a fuzzy inference model abstract the problem of learning rules for a fuzzy inference model directly from empirical observations without resorting to assessments from experts is considered we develop a method that builds uncertain rules from a set of examples these rules match the following pattern if x then y is b is where a and b are fuzzy sets representing fuzzy restrictions on the variables x and y and are real numbers expressing lower and upper degrees of certainty in the truth of the rule the method is based on the minimization of a distance measure between the real output associated to a given input and the output predicted by the inference model using a parameterized version of the same rule to be learnt our approach is computationally efficient in running time as well as in storage requirements moreover it can be used in both training batch processing and adaptation iterative processing modes of learning

fuzzy transform and least squares approximation analogies differences and generalizations investigating the relations between the least squares approximation techniques and the fuzzy transform in this paper we show that the discrete fuzzy transform is invariant with respect to the interpolating and least squares approximation additionally the fuzzy transform is evaluated at any point by simply resampling the continuous approximation underlying the input data using numerical linear algebra we also derive new properties e g stability to noise additivity with respect to the input data and characterizations e g radial and dual membership maps of the discrete fuzzy transform finally we define the geometry and confidence driven discrete fuzzy transforms which take into account the intrinsic geometry and the confidence weights associated to the data

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

data mining by attribute generalization with fuzzy hierarchies in fuzzy databases in this paper we examine the soft computing issues involving data generalization on fuzzy information the attribute induction process is formalized for these three cases crisp hierarchy fuzzy data fuzzy hierarchy crisp data fuzzy hierarchy fuzzy data each case is exemplified by applications of the algorithm to a prototype database

structure of efficient sets for convex objectives this paper considers multiple objective problems in which each objective is the minimization of a convex function defined over a convex solution space and for which the number of objectives is greater than the dimension of the solution space such problems arise in location analysis where the solution space of possible facility sites may be represented by a plane but many location objectives may exist we show that the set of efficient solutions to such problems exhibit structures similar to convex hulls that is the solutions on the boundary of the efficient set are efficient for subproblems obtained by considering only small subsets of objectives the number of objectives in these subproblems is at most equal to the dimension of the solution space when the objectives are continuous and attain their minimum values in a bounded region over the solution space the efficient set of the subproblems completely determine an efficient set for the larger problem

on efficient solutions to multiple objective mathematical programs this note develops properties of quasi efficient solutions and explores interrelationships to the classical concept of efficiency in particular a point is a quasi efficient solution to a multiple objective mathematical program if and only if it is an optimal solution to a scalar maximum problem for some set of nonnegative weights on the objectives this result is then used to characterize the set of quasi efficient solutions as the union of efficient solutions to a multiple objective problem over all nonempty subsets of the objectives

a novel group ranking model for revealing sequence and quantity knowledge abstract the aggregation of individuals preferences into a consensus ranking is a group ranking problem which has been widely utilized in various applications such as decision support systems recommendation systems and voting systems gathering the comparison of preferences and aggregate them to gain consensuses is a conventional issue for example b c d a indicates that b is favorable to c and c d is somewhat favorable but not fully favorable to d a where and are comparators and a b c and d are items recently a new type of ranking model was proposed to provide temporal orders of items the order b c a means that b and c can occur simultaneously and are also before a although this model can derive the order ranking of items the knowledge about quantity related items is also of importance to approach more real life circumstances for example when enterprises or individuals handle their portfolios in financial management two considerations the sequences and the amount of money for investment objects should be raised initially in this study we propose a model for discovering consensus sequential patterns with quantitative linguistic terms experiments using synthetic and real datasets showed the model s computational efficiency scalability and effectiveness

a new approach for discovering fuzzy quantitative sequential patterns in sequence databases given a sequence database and minimum support threshold the goal of mining quantitative sequential patterns is to discover the complete set of sequential patterns with purchased quantities in databases although this type of pattern can provide more information than the traditional sequential pattern it also causes a sharp boundary problem this means that when an item s quantity is close to the boundary of two adjacent quantity intervals it is either ignored or overemphasized in view of this weakness a recent paper from hong kuo and chi proposed a new kind of extended patterns called fuzzy quantitative sequential patterns fqsp where an item s quantity in the pattern is represented by a fuzzy term rather than a quantity interval in their work an apriori like algorithm was developed to mine all fqsp in this paper we propose a new and novel algorithm to mine fqsp based on the divide and conquer strategy since the proposed algorithm greatly reduces the candidate subsequence generation efforts the performance is improved significantly experiments show that the proposed algorithm is much more efficient and scalable than the previous algorithm

fuzzy association rules and the extended mining algorithms this paper focuses on the notion of fuzzy association rules that are of the form x y where either x or y is a collection of fuzzy sets the work stems from two observations first in generalized association rule mining the taxonomies concerned may not be crisp but fuzzy e g tomato could be regarded as both fruit and vegetable each at a different degree second managers often refer to decision rules in terms of linguistic expressions that may or may not be the nodes of the taxonomies e g very expensive cloth tropical fruit the paper deals with the fuzziness based upon fuzzy taxonomies that reflect partial belongings among itemsets as well as upon the extended settings for the degree of support and the degree of confidence apriori algorithms are extended accordingly to discover association rules across the higher level taxonomic nodes which are fuzzy sets in general as a result the discovered rules are fuzzy rules furthermore linguistic hedges are also incorporated in mining fuzzy rules to express more meaningful knowledge moreover the extended algorithms are tested with the synthetic data revealing similar computational complexities to that of the classical algorithm finally the extended algorithms are applied to a real data set with an explanation of the semantics of discovered fuzzy rules

a cardinal function algorithm for computing multivariate quadrature points we present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space an effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights symmetry conditions are often used to reduce the number of equations and unknowns our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points n be equal to the dimension of a prescribed lower dimensional polynomial space the cardinal functions allow us to treat the quadrature weights as dependent variables and remove them as well as an equivalent number of equations from the numerical optimization procedure we give results for the triangle where for all degrees d le 25 we find quadrature formulas of this form which have positive weights and contain no points outside the triangle seven of these quadrature formulas improve on previously known results

symmetric quadrature rules on a triangle abstract we present a class of quadrature rules on triangles in r 2 which somewhat similar to gaussian rules on intervals in r 1 have rapid convergence positive weights and symmetry by a scheme combining simple group theory and numerical optimization we obtain quadrature rules of this kind up to the order 30 on triangles this scheme essentially a formalization and generalization of the approach used by lyness and jespersen over 25 years ago can be easily extended to other regions in r 2 and surfaces in higher dimensions such as squares spheres we present example formulae and relevant numerical results

compressed algebraic cubature over polygons with applications to optical design abstract in this paper we propose an algorithm to determine cubature rules of algebraic degree of exactness on general polygons p by means of matlab polyshape objects and near minimal rules on triangles obtaining by caratheodory tchakaloff subsampling a pi positive interior final formula with cardinality at most n 1 2 2 we test our algorithm on polygons with different shape and we also discuss an application to the computation of the rmswe root mean square wavefront error on obscured and vignetted pupils in the framework of optical design by numerical ray tracing for the lsst large synoptic survey telescope

a neural fuzzy modelling framework based on granular computing concepts and applications fuzzy and neural fuzzy systems have successfully and extensively applied to solve problems in many research areas such as those associated with industrial medical and academic applications however recent trends reveal a demand for a workflow with a particular emphasis on transparency simplicity system interpretability as well as on a strong link with human cognition such requirement is mainly driven by research areas where expert knowledge is of very high importance and any new proposed modelling system falls under the interpretability scrutiny of experts in order to confirm the system s validity the relatively recent paradigm of granular computing grc offers an ideal opportunity for a transparent knowledge discovery methodology to be combined with fuzzy logic thereby towards a systematic modelling framework with a focus on the overall transparency of the system such transparency in the workflow allows for better interaction between the expert process knowledge and the system design which translates into a better performing system in this paper a systematic modelling approach using granular computing grc and neural fuzzy modelling is presented in this research study a grc algorithm is used to extract relational information and data characteristics out of an initial database the extracted knowledge and granular features are then translated into a linguistic rule base of a fuzzy system this rule base is finally elicited and optimised via a neural fuzzy modelling structure during the various steps of this methodology the transparency features are highlighted and it is shown here how the system designer can take advantage of such features to enhance the system the proposed modelling framework is applied to a multi dimensional and complex data set consisting of measurements of mechanical properties of heat treated steel the data set is collected from a real industrial process and the measurements are dictated by customer production demands and the data set is very sparse with many discontinuities the proposed framework successfully models the mechanical properties of heat treated steel and it further improves upon the performance of previously established modelling structures

a numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions we present a numerical algorithm for the construction of efficient high order quadratures in two and higher dimensions quadrature rules constructed via this algorithm possess positive weights and interior nodes resembling the gaussian quadratures in one dimension in addition rules can be generated with varying degrees of symmetry adaptable to individual domains we illustrate the performance of our method with numerical examples and report quadrature rules for polynomials on triangles squares and cubes up to degree 50 these formulae are near optimal in the number of nodes used and many of them appear to be new

extracting compact and information lossless sets of fuzzy association rules applying classical association rule extraction framework on fuzzy datasets leads to an unmanageably highly sized association rule sets moreover the discretization operation leads to information loss and constitutes a hamper towards an efficient exploitation of the mined knowledge to overcome such a drawback this paper proposes the extraction and the exploitation of compact and informative generic basis of fuzzy association rules the presented approach relies on the extension within the fuzzy context of the notion of closure and galois connection that we introduce in this paper in order to select without loss of information a generic subset of all fuzzy association rules we define three fuzzy generic basis from which remaining redundant fars are generated this generic basis constitutes a compact nucleus of fuzzy association rules from which it is possible to informatively derive all the remaining rules in order to ensure a sound and complete derivation process we introduce an axiomatic system allowing the complete derivation of all the redundant rules the results obtained from experiments carried out on benchmark datasets are very encouraging they highlight a very important reduction of the number of the extracted fuzzy association rules without information loss

mining association rules from imprecise ordinal data categorical data can generally be classified into ordinal data and nominal data although there have been numerous studies on finding association rules from nominal data few have tried to do so from ordinal data additionally previous mining algorithms usually assume that the input data is precise and clean which is unrealistic in practical situations real world data tends to be imprecise due to human errors instrument errors recording errors and so on therefore this paper proposes a new approach to discovering association rules from imprecise ordinal data experimental results from the survey data show the feasibility of the proposed mining algorithm performance analyses of the algorithms also show that the proposed approach can discover interesting and valuable rules that could never be found using the conventional approach the apriori algorithm

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

finding relevant attributes and membership functions fuzzy systems that automatically derive fuzzy if then rules from numeric data have been developed most have to predefine membership functions in order to learn hong and lee proposed a general learning method that automatically derives fuzzy if then rules and membership functions from a set of given training examples using a decision table all available attributes were included in the decision table and the initial membership functions for each attribute were built according to the predefined smallest unit although hong and lee s method accurately derives the fuzzy if then rules and final membership functions the decision table and the initial membership functions are complex if there are many attributes or if the predefined unit is small we improve hong and lee s method by first selecting relevant attributes and building appropriate initial membership functions these attributes and membership functions are then used in a decision table to derive final fuzzy if then rules and membership functions experimental results on iris data show that the proposed method effectively induces membership functions and fuzzy if then rules

a fuzzy inductive learning strategy for modular rules abstract in real applications data provided to a learning system usually contain linguistic information which greatly influences concept descriptions derived by conventional inductive learning methods the design of learning methods to learn concept descriptions in working with vague data is thus very important in this paper we apply fuzzy set concepts to machine learning to solve this problem a fuzzy learning algorithm based on the maximum information gain is proposed to manage linguistic information the proposed learning algorithm generates fuzzy rules from soft instances which differ from conventional instances in that they have class membership values experiments on the sports and the iris flower classification problems are presented to compare the accuracy of the proposed algorithm with those of some other learning algorithms experimental results show that the rules derived from our approach are simpler and yield higher accuracy than those from some other learning algorithms

an algebraic cubature formula on curvilinear polygons abstract we implement in matlab a gauss like cubature formula on bivariate domains whose boundary is a piecewise smooth jordan curve curvilinear polygons the key tools are green s integral formula together with the recent software package chebfun to approximate the boundary curve close to machine precision by piecewise chebyshev interpolation several tests are presented including some comparisons of this new routine chebfungauss with the recent splinegauss that approximates the boundary by splines

compensatory neurofuzzy systems with fast learning algorithms in this paper a new adaptive fuzzy reasoning method using compensatory fuzzy operators is proposed to make a fuzzy logic system more adaptive and more effective such a compensatory fuzzy logic system is proved to be a universal approximator the compensatory neural fuzzy networks built by both control oriented fuzzy neurons and decision oriented fuzzy neurons cannot only adaptively adjust fuzzy membership functions but also dynamically optimize the adaptive fuzzy reasoning by using a compensatory learning algorithm the simulation results of a cart pole balancing system and nonlinear system modeling have shown that 1 the compensatory neurofuzzy system can effectively learn commonly used fuzzy if then rules from either well defined initial data or ill defined data 2 the convergence speed of the compensatory learning algorithm is faster than that of the conventional backpropagation algorithm and 3 the efficiency of the compensatory learning algorithm can be improved by choosing an appropriate compensatory degree

new fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases cubature rules on the triangle have been extensively studied as they are of great practical interest in numerical analysis in most cases the process by which new rules are obtained does not preclude the existence of similar rules with better characteristics there is therefore clear interest in searching for better cubature rules here we present a number of new cubature rules on the triangle exhibiting full or rotational symmetry that improve on those available in the literature either in terms of number of points or in terms of quality these rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules as shown in specific benchmark examples this results in significantly better performance of the employed algorithm

an inductive learning procedure to identify fuzzy systems abstract in many real cases a system may not be described by a functional input output description and the only way to describe it is through a finite set of fuzzy rules even in some cases where exact identification is possible looking for that rule description may be useful for the sake of ease and effectiveness the automatic identification of a system through the analysis of an input output data set is the aim of this work in order to identify a system from raw data obtained in repeated instances of its own system the frequency of appearance of some kind of patterns may be particularly interesting nevertheless when the base domain is fuzzy and the data can be crisp statistical methods are not very appropriate and it is necessary to study a new model for frequencies on fuzzy domains the proposed model for identifying systems is based on this definition

fuzzy data mining for interesting generalized association rules due to the increasing use of very large databases and data warehouses mining useful information and helpful knowledge from transactions is evolving into an important research area most conventional data mining algorithms identify the relationships among transactions using binary values and find rules at a single concept level transactions with quantitative values and items with hierarchy relation are however commonly seen in real world applications in this paper we thus introduce the problem of mining fuzzy generalized association rules from quantitative data a fuzzy mining algorithm based on srikant and agrawal s method is proposed for extracting implicit generalized knowledge from transactions stored as quantitative values it integrates fuzzy set concepts and generalized data mining technologies to achieve this purpose items in rules may be from any level of the given taxonomy the effect of numbers of fuzzy regions on the performance of the proposed algorithm is also discussed

processing individual fuzzy attributes for fuzzy rule induction fuzzy systems that can automatically derive fuzzy if then rules and membership functions from numeric data have recently been developed in this paper we propose two new fuzzy learning methods for automatically deriving membership functions and fuzzy if then rules from a set of given training examples the proposed methods first select relevant attributes and build appropriate initial membership functions they then simplify the intervals and the membership functions of each attribute before forming a decision table these attributes and membership functions are then used in a decision table to derive the final fuzzy if then rules and membership functions experimental results for the iris data show that our methods can achieve a high degree of accuracy the proposed methods are thus useful in constructing membership functions and in managing uncertainty and vagueness they can also reduce the time and effort needed to develop a fuzzy knowledge base

on the representation measurement and discovery of fuzzy associations the use of fuzzy sets to describe associations between data extends the types of relationships that may be represented facilitates the interpretation of rules in linguistic terms and avoids unnatural boundaries in the partitioning of the attribute domains in addition the partial membership values provide a method for incorporating the distribution of the data into the assessment of a rule this paper investigates techniques to identify and evaluate associations in a relational database that are expressible by fuzzy if then rules extensions of the classical confidence measure based on the spl alpha cut decompositions of the fuzzy sets are proposed to incorporate the distribution of the data into the assessment of a relationship and identify robustness in an association a rule learning strategy that discovers both the presence and the type of an association is presented

from crispness to fuzziness three algorithms for soft sequential pattern mining most real world databases consist of historical and numerical data such as sensor scientific or even demographic data in this context classical algorithms extracting sequential patterns which are well adapted to the temporal aspect of data do not allow numerical information processing therefore the data are pre processed to be transformed into a binary representation which leads to a loss of information fuzzy algorithms have been proposed to process numerical data using intervals particularly fuzzy intervals but none of these methods is satisfactory therefore this paper completely defines the concepts linked to fuzzy sequential pattern mining using different fuzzification levels we propose three methods to mine fuzzy sequential patterns and detail the resulting algorithms speedyfuzzy minifuzzy and totallyfuzzy finally we assess them through different experiments thus revealing the robustness and the relevancy of this work

discovery of fuzzy quantitative sequential patterns with multiple minimum supports and adjustable membership functions the objective of mining quantitative sequential patterns is to discover complete sets of sequential patterns with quantities in databases although this novel type of pattern considers information compared to traditional sequential patterns it contains a sharp boundary problem that is when the quantity of an item is near the boundary of two predetermined quantity intervals it can be either ignored or overemphasized hong et al proposed a new type of pattern called a fuzzy quantitative sequential pattern fqsp to solve this problem although this type of pattern provides summary information to expedite decision making fqsps confront unrealistic circumstances that is they offer only a unique minimum support min sup to all items for the whole database a higher min sup fails to find rare items with higher profit whereas a lower min sup leads to a combinatorial explosion problem instead of this type of mechanism we used the idea of multiple minimum supports to mine an fqsp furthermore the utility of fqsp is also disturbed because this approach uses only a single membership function without considering the price quantity relation the idea of adjustable membership functions was also provided to address this problem this study proposes a new model to discover an fqsp with both multiple minimum supports and adjustable membership functions experiments using synthetic and real datasets demonstrated the computational efficiency scalability and effectiveness of the model

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

identification of fuzzy models using a successive tuning method with a variant identification ratio the paper is concerned with the identification of fuzzy inference systems fuzzy models realized with the aid of the successive tuning method using a variant identification ratio that is based on hierarchical fair competition based parallel genetic algorithms hfc pga and information granulation the hfc pga is a certain multi population version of parallel genetic algorithms pga which is suitable for a simultaneous optimization of both the structure of the fuzzy model as well as its parameters the granulation of information is realized with the aid of the c means clustering algorithm information granules formed in this way become essential at further stages of the construction of the fuzzy models by forming the centers modal values of the fuzzy sets constituting individual rules of the inference schemes further optimization of the fuzzy model deals with an adjustment of a suite of parameters such as the number of input variables to be used in the model a collection of specific subsets of the input variables and the number of membership functions being used by these variables and the order and parameters of the polynomial occurring in the conclusions of the corresponding rules an iterative development of the fuzzy model deals with its structural as well as parametric optimization via hfc pga the c means algorithm and a standard least square estimation method to evaluate the performance of the proposed model we exploit well known and commonly used data sets such as gas furnace mackey glass time series as well as medical imaging system a comparative analysis demonstrates that the proposed model leads to the superior performance when compared with other fuzzy models reported in the literature

a fuzzy rule based approach for screening international distribution centres in practical situations a decision maker often has to deal with numerous alternatives in evaluating and selecting the most suitable international distribution centre idc location for enhancing the organizational competitiveness to improve the effectiveness of the decision making process it is critical to reduce the number of alternatives through screening so that the decision maker can focus on the most attractive alternatives for further consideration this paper presents a fuzzy rule based approach to effectively screen decision alternatives in adequately solving the problem of evaluating and selecting idc locations case based reasoning with the use of fuzzy production rules is adopted to screen out unattractive alternatives based on past experience in a specific decision making situation a decision support system based on the proposed approach is presented to improve the effectiveness of the screening process an example is given to demonstrate the applicability of the proposed approach in improving the effectiveness of the decision making process in multicriteria decision making

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

a new algorithm for solving planar multiobjective location problems involving the manhattan norm abstract this paper is devoted to the study of unconstrained planar multiobjective location problems where distances between points are defined by means of the manhattan norm we characterize the nonessential objectives and by eliminating them we develop an effective algorithm for generating the whole set of efficient solutions as the union of a special family of rectangles and line segments we prove the correctness of this algorithm analyze its complexity and present illustrative computational results obtained by a matlab based implementation

developing an efficient knowledge discovering model for mining fuzzy multi level sequential patterns in sequence databases sequential pattern mining from sequence databases has been recognized as an important data mining problem with various applications items in a sequence database can be organized into a concept hierarchy according to taxonomy based on the hierarchy sequential patterns can be found not only at the leaf nodes individual items of the hierarchy but also at higher levels of the hierarchy this is called multiple level sequential pattern mining in previous research taxonomies based on crisp relationships between any two disjointed levels however cannot handle the uncertainties and fuzziness in real life for example tomatoes could be classified into the fruit category but could be also regarded as the vegetable category to deal with the fuzzy nature of taxonomy chen and huang developed a novel knowledge discovering model to mine fuzzy multi level sequential patterns where the relationships from one level to another can be represented by a value between 0 and 1 in their work a generalized sequential patterns gsp like algorithm was developed to find fuzzy multi level sequential patterns this algorithm however faces a difficult problem since the mining process may have to generate and examine a huge set of combinatorial subsequences and requires multiple scans of the database in this paper we propose a new efficient algorithm to mine this type of pattern based on the divide and conquer strategy in addition another efficient algorithm is developed to discover fuzzy cross level sequential patterns since the proposed algorithm greatly reduces the candidate subsequence generation efforts the performance is improved significantly experiments show that the proposed algorithm is much more efficient and scalable than the previous one in mining real life databases our works enhance the model s practicability and could promote more applications in business

knowledge gathering of fuzzy multi time interval sequential patterns mining sequential patterns to find ordered events or subsequence patterns is essential in many applications such as analysis of consumer shopping data web clickstreams and biological sequences traditional patterns reveal which items are frequently purchased together and in what order however information about the time intervals between purchases is missing therefore yang proposed using multi time interval sequential patterns to consider the time intervals between each pair of items in a pattern for example means that bread is bought before milk within an interval of ti 1 and jam is bought after bread and milk within intervals of ti 2 and ti 1 respectively where ti 1 and ti 2 are predefined time intervals although this new type of pattern considers the intervals between all pairs of items it contains a sharp boundary problem that is when the time interval between two purchases is near the boundary of two predetermined time ranges we either ignore or overemphasize it in this study we applied the concept of fuzzy sets to solve the sharp boundary problem the discovered patterns called fuzzy multi time interval sequential patterns describe time intervals in linguistic terms for better understanding two algorithms fuzzmi apriori and fuzzmi prefixspan were developed for mining fuzzy multi time interval patterns experiments using synthetic and real datasets showed the algorithms computational efficiency scalability and effectiveness

input selection and partition validation for fuzzy modelling using neural network a simple and effective method for selecting significant input variables and determining optimal number of fuzzy rules when building a fuzzy model from data is proposed in contrast to the existing clustering based methods in this approach both input selecting and partition validating are determined on the basis of a class of sub clusters created by a self organising network instead of on the data the important input variables which independently and significantly influence the system output can be extracted by a fuzzy neural network on the other hand the optimal number of fuzzy rules can be determined separately via the fuzzy c means algorithm with a modified fuzzy entropy as the criterion of cluster validation the simulation results show that the proposed method can provide good model structures for fuzzy modelling and has high computing efficiency

weighted fuzzy production rules abstract most of the research works on fuzzy production rules fprs show that its knowledge representation power suffers from a serious shortcoming in that all propositions in the antecedent part are assumed to have equal importance the lack of local weight concept and that a number of rules executed in an inference path leading to a specified goal or the same rule employed in various inference paths leading to distinct final goals may have relative degrees of importance has not been explored the lack of global weight concept this paper presents a weighted fpr wfpr incorporating the concepts of local and global weights an improved method to compute the fuzzy value and the certainty factor of the consequent assertion and a better way to interpret the linguistic meaning of the consequent are proposed for this wfpr our approach offers the advantages of enhancing the knowledge representation power of a fpr reducing the undesirable effects when computing the certainty factor of the consequent part by generalizing the traditional method of computation and overcoming the tedious steps involved in zadeh s compositional rule of inference cri method as far as interpretation of the linguistic meaning of the consequent part is concerned our method is found to be better than the ad hoc approach of cri based methods

finding efficient solutions for rectilinear distance location problems efficiently abstract given n planar existing facility locations a planar new facility location x is called efficient if there is no other location y for which the rectilinear distance between y and each existing facility is at least as small as between x and each existing facility and strictly less for at least one existing facility rectilinear distances are typically used to measure travel distances between points via rectilinear aisles or street networks we first present a simple arrow algorithm based entirely on geometrical analysis that constructs all efficient locations we then present a row algorithm which is of order n log n that constructs all efficient locations and establish that no alternative algorithm can be of a lower order

evolutionary fuzzy rule induction process for subgroup discovery a case study in marketing this paper presents a genetic fuzzy system for the data mining task of subgroup discovery the subgroup discovery iterative genetic algorithm sdiga which obtains fuzzy rules for subgroup discovery in disjunctive normal form this kind of fuzzy rule allows us to represent knowledge about patterns of interest in an explanatory and understandable form that can be used by the expert experimental evaluation of the algorithm and a comparison with other subgroup discovery algorithms show the validity of the proposal sdiga is applied to a market problem studied in the university of mondragon spain in which it is necessary to extract automatically relevant and interesting information that helps to improve fair planning policies the application of sdiga to this problem allows us to obtain novel and valuable knowledge for experts

nonessential objectives within network approaches for mcdm in gal and hanne eur j oper res 119 1999 373 the problem of using several methods to solve a multiple criteria decision making mcdm problem with linear objective functions after dropping nonessential objectives is analyzed it turned out that the solution does not need be the same when using various methods for solving the system containing the nonessential objectives or not in this paper we consider the application of network approaches for multicriteria decision making such as neural networks and an approach for combining mcdm methods called mcdm networks we discuss questions of comparing the results obtained with several methods as applied to the problem with or without nonessential objectives especially we argue for considering redundancies such as nonessential objectives as a native feature in complex information processing in contrast to previous results on nonessential objectives the current paper focuses on discrete mcdm problems which are also denoted as multiple attribute decision making madm