test cost sensitive multigranulation rough set model and minimal cost selection abstract multigranulation rough set is an expansion of the classical rough set by using multiple granular structures presently three important multigranulation rough sets have been proposed they are optimistic pessimistic and multigranulation approaches however such three multigranulation rough sets do not take the test cost into consideration which is an important issue in both data mining and machine learning to solve such problem we propose a test cost sensitive multigranulation rough set model in this paper we show that test cost sensitive multigranulation rough set is a generalization of optimistic pessimistic and multigranulation rough sets furthermore it is found that the traditional heuristic algorithm is not suitable for granular structure selection with lower test cost we then propose a backtracking algorithm for granular structure selection with minimal test cost the algorithms are tested on ten uci university of california irvine data sets experimental results show the effectiveness of backtracking algorithm by comparing with heuristic algorithm this study suggests potential application areas and new research trends concerning multigranulation rough set theory

an application of rough sets to graph theory this paper studies the testing bipartiteness of simple undirected graphs based on generalized rough sets a new binary relation induced from a simple undirected graph is first defined the concept of quasi outer definable sets is then introduced and some of its important properties are investigated finally based on the quasi outer definable set a new algorithm is designed for testing bipartiteness of a simple undirected graph experiments show that the proposed algorithm is more efficient than the traditional one when the graph is represented by an adjacency matrix

distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution summary a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry and to generate a whole set of non symmetric densities the approach is sufficiently general to encompass some recent proposals in the literature variously related to the skew normal distribution the special case of skew elliptical densities is examined in detail establishing connections with existing similar work the final part of the paper specializes further to a form of multivariate skew t density likelihood inference for this distribution is examined and it is illustrated with numerical examples

discernibility matrix simplification for constructing attribute reducts this paper proposes a reduct construction method based on discernibility matrix simplification the method works in a similar way to the classical gaussian elimination method for solving a system of linear equations elementary matrix simplification operations are introduced each operation transforms a matrix into a simpler form by applying these operations a finite number of times one can transform a discernibility matrix into one of its minimum i e the simplest forms elements of a minimum discernibility matrix are either the empty set or singleton subsets in which the union derives a reduct with respect to an ordering of attributes which is either computed based on a certain measure of attributes or directly given by a user two heuristic reduct construction algorithms are presented one algorithm attempts to exclude unimportant attributes from a reduct and the other attempts to include important attributes in a reduct

extended rough set based attribute reduction in inconsistent incomplete decision systems a systematic study of attribute reduction in inconsistent incomplete decision systems iidss has not yet been performed and no complete methodology of attribute reduction has been developed for iidss to date in an iids there are various ways to handle missing values in this paper a missing attribute value may be replaced with any known value of a corresponding attribute such a missing attribute value is called a do not care condition in this way this paper establishes reduction concepts specifically for iidss mainly by extending related reduction concepts from other types of decision systems into iidss and then derives their relationships and properties with these derived properties the extended reducts are divided into two distinct types heritable reducts and nonheritable reducts and algorithms for computing them are presented using the relationships derived here the eight types of extended reducts established for iidss can be converted to five equivalent types then five discernibility function based approaches are proposed each for a particular kind of reduct each approach can find all reducts of its associated type the theoretical analysis of the proposed approaches is described in detail finally numerical experiments have shown that the proposed approaches are effective and suitable for handling both numerical and categorical attributes but that they have different application conditions the proposed approaches can provide a solution to the reduction problem for iidss

on the combination and normalization of interval valued belief structures this paper investigates the issues of combination and normalization of interval valued belief structures within the framework of dempster shafer theory dst of evidence existing approaches are reviewed examined and critically analysed they either ignore the normalization or separate it from the combination process leading to irrational or suboptimal interval valued belief structures a new logically correct optimality approach is developed where the combination and the normalization are optimised together rather than separately numerical examples are provided throughout the paper

existence of homoclinic solutions for a fourth order differential equation with a parameter in this paper we study the existence of homoclinic solutions for a class of fourth order differential equations by using variational methods the existence and the non existence of nontrivial homoclinic solutions are obtained depending on a parameter

reducts within the variable precision rough sets model a further investigation abstract one fundamental aspect of the variable precision rough sets vprs model involves a search for subsets of condition attributes which provide the same information for classification purposes as the full set of available attributes such subsets are labelled approximate reducts or reducts being defined for a specified classification error denoted by this paper undertakes a further investigation of the criteria for a reduct within vprs certain anomalies and interesting implications are identified an additional condition is suggested for finding reducts which assures a more general level knowledge equivalent to that of the full set of attributes

a novel and better fitness evaluation for rough set based minimum attribute reduction problem the minimum attribute reduction mar problem in the context of rough set theory is known to be np hard one popular way of dealing with this problem is to first transform it into a fitness maximization problem over a multi dimensional boolean space and to then solve this problem using population based stochastic optimization algorithms it is therefore important to have an appropriate fitness function in this paper two examples are presented to show that existing fitness functions either do not guarantee optimality equivalence between the mar problem and the transformed fitness maximization problem or may produce the so called overemphasis phenomenon that affects the performance of population based stochastic optimization algorithms to overcome these drawbacks we propose a new fitness function that we prove both guarantees the optimality equivalence and reduces the overemphasis phenomenon experimental results show that the proposed fitness function is better than existing fitness functions in terms of solution quality

the relationship between attribute reducts in rough sets and minimal vertex covers of graphs the problems to find attribute reduction in rough sets and to obtain the minimal vertex cover for graphs are both np hard problems this paper studies the relationship between the two problems the vertex cover problem for graphs from the perspective of rough sets is first investigated the attribute reduction of an information system is then studied in the framework of graph theory the results in this paper show that finding the minimal vertex cover of a graph is equivalent to finding the attribute reduction of an information system induced from the graph conversely the attribute reduction computation can be translated into the calculation of the minimal vertex cover of a derivative graph finally a new algorithm for the vertex cover problem based on rough sets is presented furthermore experiments are conducted to verify the effectiveness of the proposed method

attribute reduction in ordered decision tables via evidence theory rough set theory and dempster shafer theory of evidence are two distinct but closely related approaches to modeling and manipulating uncertain information it is quite natural to set up a hybrid model based on these two theories in this paper we investigate the problem of attribute reduction for ordered decision tables based on evidence theory belief and plausibility functions which are strongly connected with lower and upper approximation operators in dominance based rough set approach are proposed to define relative belief and plausibility reducts of ordered decision tables relationships among various types of relative reducts are thoroughly studied in consistent and inconsistent ordered decision tables a pair of numeric measures the inner and outer significance measures of a criterion is presented to search for a relative belief plausibility reduct which is meaningful for practical problems some real world tasks taken from the uci repository are employed to verify the feasibility and effectiveness of the proposed technique

an efficient accelerator for attribute reduction from incomplete data in rough set framework feature selection attribute reduction from large scale incomplete data is a challenging problem in areas such as pattern recognition machine learning and data mining in rough set theory feature selection from incomplete data aims to retain the discriminatory power of original features to address this issue many feature selection algorithms have been proposed however these algorithms are often computationally time consuming to overcome this shortcoming we introduce in this paper a theoretic framework based on rough set theory which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data as an application of the proposed accelerator a general feature selection algorithm is designed by integrating the accelerator into a heuristic algorithm we obtain several modified representative heuristic feature selection algorithms in rough set theory experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

space structure and clustering of categorical data learning from categorical data plays a fundamental role in such areas as pattern recognition machine learning data mining and knowledge discovery to effectively discover the group structure inherent in a set of categorical objects many categorical clustering algorithms have been developed in the literature among which k modes type algorithms are very representative because of their good performance nevertheless there is still much room for improving their clustering performance in comparison with the clustering algorithms for the numeric data this may arise from the fact that the categorical data lack a clear space structure as that of the numeric data to address this issue we propose in this paper a novel data representation scheme for the categorical data which maps a set of categorical objects into a euclidean space based on the data representation scheme a general framework for space structure based categorical clustering algorithms sbc is designed this framework together with the applications of two kinds of dissimilarities leads two versions of the sbc type algorithms to verify the performance of the sbc type algorithms we employ as references four representative algorithms of the k modes type algorithms experiments show that the proposed sbc type algorithms significantly outperform the k modes type algorithms

a novel attribute reduction approach for multi label data based on rough set theory multi label classification is an active research field in machine learning because of the high dimensionality of multi label data attribute reduction also known as feature selection is often necessary to improve multi label classification performance rough set theory has been widely used for attribute reduction with much success however little work has been done on applying rough set theory to attribute reduction in multi label classification in this paper a novel attribute reduction method based on rough set theory is proposed for multi label data first the uncertainties conveyed by labels are analyzed and a new type of attribute reduct is introduced called complementary decision reduct the relationships between complementary decision reduct and two representative types of attribute reducts are also investigated showing significant advantages of complementary decision reduct in revealing the uncertainties implied in multi label data second a discernibility matrix based approach is introduced for computing all complementary decision reducts and a heuristic algorithm is proposed for effectively computing a single complementary decision reduct experiments on real life data demonstrate that the proposed approach can effectively reduce unnecessary attributes and improve multi label classification accuracy

quick attribute reduction in inconsistent decision tables this paper focuses on three types of attribute reducts in inconsistent decision tables assignment reduct distribution reduct and maximum distribution reduct it is quite inconvenient to judge these three types of reduct directly according to their definitions this paper proposes judgment theorems for the assignment reduct the distribution reduct and the maximum distribution reduct which are expected to greatly simplify the judging of these three types of reducts on this basis we derive three new types of attribute significance measures and construct the q ara quick assignment reduction algorithm the q dra quick distribution reduction algorithm and the q mdra quick maximum distribution reduction algorithm these three algorithms correspond to the three types of reducts we conduct a series of comparative experiments with twelve uci machine learning data repository university of california at irvine data sets including consistent and inconsistent decision tables to evaluate the performance of the three reduction algorithms proposed with the relevant algorithm quickreduct 9 34 the experimental results show that quickreduct possesses weak robustness because it cannot find the reduct even for consistent data sets whereas our proposed three algorithms show strong robustness because they can find the reduct for each data set in addition we compare the q dra quick distribution reduction algorithm with the cebarknc conditional entropy based algorithm for reduction of knowledge without a computing core 43 because both find the distribution reduct by using a heuristic search the experimental results demonstrate that q dra runs faster than cebarknc does because the distribution function of q dra has a lower calculation cost instructive conclusions for these reduction algorithms are drawn from the perspective of classification performance for the c4 5 and rbf svm classifiers last we make a comparison between discernibility matrix based methods and our algorithms the experimental results indicate that our algorithms are efficient and feasible

relative reducts in consistent and inconsistent decision tables of the pawlak rough set model a relative reduct can be considered as a minimum set of attributes that preserves a certain classification property this paper investigates three different classification properties and suggests three distinct definitions accordingly in the pawlak rough set model while the three definitions yield the same set of relative reducts in consistent decision tables they may result in different sets in inconsistent tables relative reduct construction can be carried out based on a discernibility matrix the study explicitly stresses a fact that the definition of a discernibility matrix should be tied to a certain property regarding the three classification properties we can define three distinct definitions accordingly based on the common structure of the specific definitions of relative reducts and discernibility matrices general definitions of relative reducts and discernibility matrices are suggested

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs

rough set based conflict analysis model and method over two universes rough set theory as a new mathematical tool to handle uncertainty decision making problems was used to study conflict analysis decision making in late90 s and then the pawlak conflict analysis model was established the approach of rough set provides a new perspective and also gives an effective tool to deal with conflict analysis decision making both theoretically and practically in this paper we propose a new rough set model of conflict analysis a conflict analysis decision model based on rough set theory over two universes it is a natural extension of the pawlak conflict analysis model subsequently we define the conflict matrix and a consistent disagreement matrix on the basis of the model proposed in this paper then we can find the intrinsic reasons for conflict and attain a feasible strategy for a conflict situation furthermore we define a positive alliance matrix and a negative alliance matrix for a conflict situation these can help us analyze the opinions of different agents in the conflict situation moreover we develop a matrix approach for a conflict analysis model based on rough set theory over two universes which provides a convenient way to analyze and solve the conflict situation comparing to the pawlak conflict analysis model our proposed model not only could reveal the core causes for a conflict situation but also can find a possible optimal feasible consensus strategy to solve the conflict situation which satisfies the agents as much as possible finally we illustrate the idea and the basic principles established in this paper by analyzing a conflict decision making scenario

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

two stage selection and testing designs for comparative clinical trials summary a two stage design which selects the best of several experimental treatments and compares it to a standard control is proposed the design allows early termination with acceptance of the global null hypothesis optimal sample size and cut off parameters are obtained by minimizing expected total sample size for fixed significance level and power with many other factors moreover the usual error rate computations associated with comparative testing do not account for the preliminary selection process consequently the overall procedure may be neither effective nor efficient for identifying an experimental treatment which is an improvement over c in this paper we propose a new approach to the problem of identifying the best of k experimental treatments and determining whether it is superior to a control we deal with the binomial setting where patient response may be characterized as either success or failure with ok the success probability for ek k 0 corresponds to c for ease of notation assume 01 ok we propose a two stage procedure which allows early termination with acceptance of h0 o 01 ok with design parameters chosen to minimize expected total sample size the design and accompanying generalized definitions of size a and power 1 3 are given in 2 the algorithm used for optimization is described in 3 numerical results are presented in 4 followed by a discussion of the relative merits of the proposed design

boundary region based rough sets and uncertainty measures in the approximation space the approximation operators play an important role in rough set theory which are mainly defined by means of neighborhood systems in this paper firstly we try to propose a class of novel definitions of the approximation operators via a predefined boundary region based on a binary relation then we compare the proposed concepts with the originals the necessary and sufficient conditions of their equipollence are investigated secondly we give the definitions of boundary region based on a covering by employing the boundary a class of novel definitions of the approximation operators based on a covering are proposed it is shown that the proposed operators are equivalent to a class of covering approximation operators introduced by zakowski thirdly the relationship between general binary relations and coverings based approximation operators is investigated with the aid of the novel boundary region finally the more reasonable characterizations of the accuracy and the roughness are proposed by employing the boundary operators meanwhile we study uncertainty measures of approximation spaces based on a partition and a covering

statistical applications of the multivariate skew normal distribution azzalini and dalla valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter the first part of the present paper examines further probabilistic properties of the distribution with special emphasis on aspects of statistical relevance inferential and other statistical issues are discussed in the following part with applications to some multivariate statistics problems illustrated by numerical examples finally a further extension is described which introduces a skewing factor of an elliptical density

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

communication between information systems with covering based rough sets communication between information systems is considered as an important issue in granular computing the concept of homomorphism is an effective mathematical tool to study information exchange between information systems this paper provides a study on some basic properties of covering information systems and decision systems under homomorphisms first we define consistent functions related to coverings and covering mappings between two universes and study their properties then we introduce the notions of homomorphisms of covering information systems and point out that a homomorphism is a special covering mapping between information systems furthermore we investigate some important properties of homomorphisms in covering information systems and decision systems it is proved that some basic properties of original systems such as set approximations attribute reductions can be reserved under the condition of homomorphisms in both covering information systems and covering decision systems

attribute reduction based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

web based medical decision support systems for three way medical decision making with game theoretic rough sets the realization of the web as a common platform medium and interface for supporting human activities has attracted many researchers to the study of web based support systems wss an important branch of wss is web based decision support systems that provide intelligent support for making effective decisions in different domains we focus on decision making in web based medical decision support systems wmdss uncertainty is a critical factor that affects decision making and reasoning in the medical field a three way decision making approach is an effective and better choice to lessen the effects of uncertainty it provides the provision for delaying certain or definite decisions in situations that lack sufficient evidence or accurate information in reaching certain conclusions particularly the option of deferment decisions is added in this approach that provides the flexibility to further examine and investigate the uncertain and doubtful cases the game theoretic rough set gtrs model is a recent development in rough sets that can be used to determine the three rough set regions in the probabilistic rough sets framework by determining a pair of thresholds the three regions are used to obtain three way decision rules in the form of acceptance rejection and deferment rules in this paper we extend the gtrs model to analyze uncertainty involved in medical decision making experimental results with a gtrs based approach on different health care datasets suggest that the approach may improve the overall quality of decision making in the medical field as well as other fields it is hoped that the incorporation of a gtrs component in wmdss will enrich and enhance its decision making capabilities

attribute reduction and optimal decision rules acquisition for continuous valued information systems for continuous valued information systems the attribute values of objects for the same attribute represent not only their ordinal relationship but also their relative distances therefore the classical rough set model is not suitable for deducing attribute reductions and optimal decision rules for continuous valued information systems though some discretization methods are proposed to transform the continuous valued information systems into discrete ones but those methods are too categorical and may lead to loss of information in some cases to solve such information loss problem we propose a tolerance rough set model in this paper with a given level the proposed model can divide a universe into some maximal tolerance classes also two types of lower and upper approximations are defined accordingly then the reductions of the maximal tolerance class and optimal decision rules based on the proposed attribute descriptors are defined and the approximate discernibility function for the maximal tolerance class is constructed and used to compute all the corresponding optimal decision rules via using boolean reasoning techniques finally the general reductions and consistent reductions for continuous valued information systems are discussed

heteroclinic solutions for the extended fisher kolmogorov equation abstract we study the extended fisher kolmogorov efk equation and its variants by variational approach we show that for every global minimum of the potential function v there is a pair of heteroclinic solutions one emanating from and the other terminating at we require neither superquadratic growth of v nor the presence of saddle focus equilibria moreover v is allowed to approach its minimal level near infinity

uncertainly measures of rough set prediction abstract the main statistics used in rough set data analysis the approximation quality is of limited value when there is a choice of competing models for predicting a decision variable in keeping within the rough set philosophy of non invasive data analysis we present three model selection criteria using information theoretic entropy in the spirit of the minimum description length principle our main procedure is based on the principle of indifference combined with the maximum entropy principle thus keeping external model assumptions to a minimum the applicability of the proposed method is demonstrated by a comparison of its error rates with results of c4 5 using 14 published data sets

maximal confidence intervals of the interval valued belief structure and applications an unsolved problem in the interval valued belief structure is the existence of maximal confidence intervals referring to the credible and maximal interval valued beliefs assigned to focal elements in which every point can be reached by a combination rule in this study the existence of maximal confidence intervals of the interval valued belief structure was investigated and validated in addition some applications of the maximal confidence interval in constructing valid and normalized interval valued belief structures are discussed a numerical experiment was conducted to illustrate the existence of maximal confidence intervals several propositions and a theorem were further proposed to demonstrate the validity of the maximal confidence interval by using the concept of the maximal confidence interval a procedure was proposed to construct valid and normalized interval valued belief structures and some meaningful guidance for assigning interval valued beliefs was also suggested to the experts finally a series of well designed examples was presented as applications of the maximal confidence interval the proposed procedure and the suggestions they indicated that valid and normalized interval valued belief structures can be achieved accurately and efficiently in practical applications with the proposed procedure and guidance on the basis of the concept of the maximal confidence interval

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

consistency measure inclusion degree and fuzzy measure in decision tables classical consistency degree has some limitations for measuring the consistency of a decision table in which the lower approximation of a target decision is only taken into consideration in this paper we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory for three types of decision tables complete incomplete and maximal consistent blocks the membership functions of an object are defined through using the equivalence class tolerance class and maximal consistent blocks including itself respectively based on these membership functions we introduce consistency measures to assess the consistencies of a target set and a decision table and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables in addition the relationships among the consistency inclusion degree and fuzzy measure are established as well these results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues

a multicriteria approach for analysis of conflicts in evidence theory we propose a new approach to model the conflict on dst as multi criteria problem we introduce a new subjective notion of conflict through a pseudo criteria analysis we present a numerical application of the proposed model based on 2 and 3 conflict measures therefore it is showed that is possible to capture conflict in dst using more that to 2 criteria dempster shafer theory dst or evidence theory is regarded as one of the leading theories for modeling uncertainty in imprecise situations the main advantage of this theory arises from the possibility of combining different bodies of evidence originally developed by using dempster s combination rule however this rule leads to counter intuitive results when the bodies of evidence conflict with each other to a high degree thus different combinations of conflict management rules have been developed over the years where regardless of the method used what should be identified first of all is the level of conflict between the bodies of evidence therefore different metrics were used to classify or quantify the conflict but no single one of these was successful because it is impracticable to represent all situations of conflict in this theory by using only one metric therefore the contribution of this article is to analyze conflict within dst by using a multi criteria analysis for which the multi criteria decision making mcdm method considered was electre tri on modeling the problem three classes of conflict low medium and high were considered to validate the model a numerical analysis was conducted that included the use of a method to point up conflict so as to infer parameters

knowledge reduction based on the equivalence relations defined on attribute set and its power set one of the key problems of knowledge discovery is knowledge reduction this paper proposes a new method for knowledge reduction in information systems first two families of closed sets c r and c r are defined where r and r are equivalence relations defined on the attribute set and its power set respectively the properties of c r and c r are also discussed the necessary and sufficient condition for c r c r is then given and employed to construct an approach to attribute reduction in information systems it is also proved that under the condition c r c r the proposed approach to knowledge reduction is equivalent to the well accepted one in reference w x zhang y leung w z wu information systems and knowledge discovery science publishing company beijing 2003

decision region distribution preservation reduction in decision theoretic rough set model abstract in the pawlak rough set model the positive region the boundary region and the non negative region are monotonic with respect to the set inclusion of attributes however the monotonicity property of the decision regions positive region boundary region or non negative region with respect to the set inclusion of attributes does not hold in the decision theoretic rough set model therefore the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions this effect is observed partly because three decision regions are defined by introducing the probabilistic threshold values in addition heuristic reduction algorithms based on decision regions may find super reducts because of the non monotonicity of decision regions to address the above issues this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision theoretic rough set model first the positive region distribution preservation reduct the boundary region distribution preservation reduct and the negative region distribution preservation reduct are introduced into the decision theoretic rough set model second three new monotonic measures are constructed by considering variants of the conditional information entropy from which we can obtain the heuristic reduction algorithms the results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts

set valued ordered information systems set valued ordered information systems can be classified into two categories disjunctive and conjunctive systems through introducing two new dominance relations to set valued information systems we first introduce the conjunctive disjunctive set valued ordered information systems and develop an approach to queuing problems for objects in presence of multiple attributes and criteria then we present a dominance based rough set approach for these two types of set valued ordered information systems which is mainly based on substitution of the indiscernibility relation by a dominance relation through the lower upper approximation of a decision some certain possible decision rules from a so called set valued ordered decision table can be extracted finally we present attribute reduction also called criteria reduction in ordered information systems approaches to these two types of ordered information systems and ordered decision tables which can be used to simplify a set valued ordered information system and find decision rules directly from a set valued ordered decision table these criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules

attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems attribute reduction is viewed as an important issue in data mining and knowledge representation this paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory the definitions of several kinds of attribute reducts are introduced the relationships among these reducts are then investigated in a fuzzy decision system it is proved that the concepts of fuzzy positive region reduct lower approximation reduct and generalized fuzzy belief reduct are all equivalent the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set in a consistent fuzzy decision system an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct but in an inconsistent fuzzy decision system a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general

on generalized fuzzy belief functions in infinite spaces determined by a fuzzy implication operator a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined relationship between the belief structure based and the belief space based fuzzy dempster shafer models is then established it is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions for any fuzzy belief structure there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space respectively essential properties of the fuzzy belief and plausibility functions are also examined the fuzzy belief and plausibility functions are respectively a fuzzy monotone choquet capacity and a fuzzy alternating choquet capacity of infinite order

measures for evaluating the decision performance of a decision table in rough set theory as two classical measures approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table however these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero to overcome this shortcoming we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision rule set extracted from a decision table we then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision rule set and are much better than the two classical measures

a unified framework for characterizing rough sets with evidence theory in various approximation spaces abstract different rough set models may be numerically characterized with evidence theory by adopting different methodologies that is to say the evidence theory based characteristics of rough sets vary with different approximation spaces and different rough set approximations to address this issue this paper proposes a theoretic framework within which rough set theory can be unifiedly characterized by evidence theory no matter what types of rough set approximations and approximation spaces they are the main works are presented as follows first we declare a principle that in any given approximation space a belief structure can be constructed based on which the lower and upper approximation operators can be measured by the belief and plausibility functions if and only if a basic condition is satisfied second in a given decision approximation space a belief structure is also derived based on which the lower and upper approximation operators can be always measured by the belief and plausibility functions without any additional condition as theoretical applications we employ the new principles to examine the evidence theory based characteristics of four types of substantial rough sets including the covering rough sets in covering approximation spaces the decision theoretic rough sets in pawlak approximation spaces the pawlak rough sets in pawlak decision approximation spaces and the multigranulation rough sets in pawlak decision approximation spaces respectively the proposed framework is applicable to various approximation spaces and various rough set approximations and hence can make us clear the basic principles for relating rough set theory with evidence theory

reductions in a knowledge base one of the advantages of rough set theory is the fact that an unknown target concept can be approximately characterized based on existing knowledge structures in a knowledge base knowledge reductions are one of the key problems in the study of rough set theory in this paper reductions and their features in a knowledge base are explored moreover communication between knowledge bases is proposed based on the use of homomorphisms between relation information systems and several invariant characterizations under homomorphisms are obtained these results will be significant for establishing a framework for granular computing in knowledge bases

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

a new approach of optimal scale selection to multi scale decision tables as a special case of information table multi scale decision table can usually be observed in real life world in such table objects may take different values under the same attribute measured at different scales based on inclusion relation of subsets of attributes and coarse relation of scales of attributes multi layered granulations and stratified rough set approximations in multi scale decision tables are shown from the perspective of granular computing compared with a special case studied by wu and leung the multi scale decision tables of diverse attributes with different numbers of levels of scales are studied in this paper furthermore complement model and lattice model are proposed to analyze the optimal scale selection for multi scale decision tables correspondingly algorithms of the two models are designed and some experiments are performed to testify feasibilities of these proposed algorithms and to make comparisons of the models

converse approximation and rule extraction from decision tables in rough set theory in this paper the concept of a granulation order is proposed in an information system the converse approximation of a target concept under a granulation order is defined and some of its important properties are obtained which can be used to characterize the structure of a set approximation for a subset of the universe in an information system its converge degree is monotonously increasing under a granulation order this means that a proper family of granulations can be chosen for a target concept approximation according to user requirements as an application of the converse approximation an algorithm based on the converse approximation called rebca is designed for decision rule extraction from a decision table which has a time complexity of o m2 c 2 u log 2 u and its practical applications are illustrated by two examples

an approach to dimensionality reduction in time series many methods of dimensionality reduction of data series time series have been introduced over the past decades some of them rely on a symbolic representation of the original data however in this case the obtained dimensionality reduction is not substantial in this paper we introduce a new approach referred to as symbolic essential attributes approximation seaa to reduce the dimensionality of multidimensional time series in such a way we form a new nominal representation of the original data series the approach is based on the concept of data series envelopes and essential attributes generated by a multilayer neural network the real valued attributes are discretized and in this way symbolic data series representation is formed the seaa generates a vector of nominal values of new attributes which form the compressed representation of original data series the nominal attributes are synthetic and while not being directly interpretable they still retain important features of the original data series a validation of usefulness of the proposed dimensionality reduction is carried out for classification and clustering tasks the experiments have shown that even for a significant reduction of dimensionality the new representation retains information about the data series sufficient for classification and clustering of the time series

knowledge reduction of dynamic covering decision information systems when varying covering cardinalities in covering based rough set theory non incremental approaches are time consuming for performing knowledge reduction of dynamic covering decision information systems when the cardinalities of coverings change as a result of object immigration and emigration because computing approximations of sets is an important step for knowledge reduction of dynamic covering decision information systems efficient approaches to calculating the second and sixth lower and upper approximations of sets using the type 1 and type 2 characteristic matrices respectively are essential in this paper we provide incremental approaches to computing the type 1 and type 2 characteristic matrices of dynamic coverings whose cardinalities vary with the immigration and emigration of objects we also design incremental algorithms to compute the second and sixth lower and upper set approximations experimental results demonstrate that the incremental approaches effectively improve the efficiency of set approximation computation finally we employ several examples to illustrate the feasibility of the incremental approaches for knowledge reduction of dynamic covering decision information systems when increasing the cardinalities of coverings

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

infinitely many homoclinic solutions for a nonperiodic fourth order differential equation without ar condition in this paper we study the existence of infinitely homoclinic solutions for a nonperiodic fourth order differential equation under the superlinear condition but not assuming the well known ar condition the proof is based on variational methods

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

a graph approach for fuzzy rough feature selection abstract rough sets especially fuzzy rough sets have proven to be a powerful tool for dealing with vagueness and uncertainty in data analysis fuzzy rough feature selection has been shown to be highly useful in data dimensionality reduction however many fuzzy rough feature selection algorithms are still time consuming when dealing with the large scale data sets in this paper the problem of feature selection in fuzzy rough sets is studied in the framework of graph theory we propose a new mechanism for fuzzy rough feature selection it is shown that finding the attribute reduction of a fuzzy decision system can be translated into finding the transversal of a derivative hypergraph based on the graph representation model a novel graph theoretic algorithm for fuzzy rough feature selection is proposed the performance of the proposed method is compared with those of the state of the art methods on various classification tasks experimental results show that the proposed technique outperforms all other known feature selection methods in terms of the computation time especially for the large scale data sets it demonstrates promising performance moreover our proposed method can achieve better classification accuracies with the usage of small number of features

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

signed and sign changing solutions of bi nonlocal fourth order elliptic problem we study a class of nonlocal fourth order elliptic problem which is seldom studied because of the presence of the biharmonic operator and binonlocal terms we are interested in the existence of signed and sign changing solutions our approach is based on variational invariant sets of descending flow and cone theory our results extend and improve some recent work we study a class of nonlocal fourth order elliptic problem which is seldom studied because of the presence of the biharmonic operator and binonlocal terms we are interested in the existence of signed and sign changing solutions our approach is based on variational invariant sets of descending flow and cone theory our results extend and improve some recent work

homoclinic solutions for a nonperiodic fourth order differential equations without coercive conditions in this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equations u 4 wu a x u f x u fde wherew is a constant a c r r and f c r r r the novelty of this paper is that when fde is nonperiodic i e a and f are nonperiodic in x and assuming that a does not fulfil the coercive conditions and f satisfies some more general ar condition we establish one new criterion to guarantee that fde has at least one nontrivial homoclinic solution via using the mountain pass theorem recent results in the literature are generalized and significantly improved

three way decision approaches to conflict analysis using decision theoretic rough set theory social progress normally occurs through a sequence of struggles and conflicts and there has been relatively little progress in developing effective methods for conflict analysis decision theoretic rough set theory is a powerful mathematical tool for depicting ambiguous information and it can provide constructive advice for decision making in this paper we first present the concepts of probabilistic conflict neutral and allied sets of conflicts and then discuss the mechanism for computing the thresholds and for conflict analysis using decision theoretic rough set theory then we describe incremental algorithms for constructing the probabilistic conflict neutral and allied sets in dynamic information systems and their effectiveness is illustrated by experimental results in light of the relationship between maximal coalitions and allied sets we finally provide efficient approaches to help a government adjust various policies according to changes in the present international situation to calculate the maximal coalitions in dynamic information systems

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

interim treatment selection using the normal approximation approach in clinical trials we consider a study starting with two treatment groups and a control group with a planned interim analysis the inferior treatment group will be dropped after the interim analysis and only the winning treatment and the control will continue to the end of the study this two stage winner design is based on the concepts of multiple comparison adaptive design and winner selection in a study with such a design there is less multiplicity but more adaptability if the interim selection is performed at an early stage if the interim selection is performed close to the end of the study the situation becomes the conventional multiple comparison where dunnett s method may be applied the unconditional distribution of the final test statistic from the winner treatment is no longer normal the exact distribution of which is provided in this paper but numerical integration is needed for its calculation to avoid complex computations we propose a normal approximation approach to calculate the type i error the power the point estimate and the confidence intervals due to the well understood and attractive properties of the normal distribution the winner design can be easily planned and adequately executed which is demonstrated by an example we also provide detailed discussion on how the proposed design should be practically implemented by optimizing the timing of the interim look and the probability of winner selection

rough sets and matroids from a lattice theoretic viewpoint this paper studies rough sets via matroidal approaches from a lattice theoretic viewpoint we firstly give a new interpretation of definable sets of pawlak rough set model i e the set of definable sets defines uniquely a matroid in which it is the family of open and closed sets then we induce two equivalence relations on a given universe based on a matroid defined on this universe one of the equivalence relations actually is defined on the set of all atoms of a geometric lattice corresponding to the matroid another is based on the transitivity of circuits properties of these two equivalence relations are then studied besides we also investigate the connections between relation based rough sets and matroids finally we point out that a geometric lattice can induce a series of coverings of a universe on which the corresponding matroid is defined and further relations of approximations based on the induced coverings are studied

on measurements of covering rough sets based on granules and evidence theory covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings the existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems less efforts have been made in measurements of covering rough sets to enrich this field in this paper we investigate the measurements of covering rough sets based on granules and evidence theory which can be used to solve more complex practical problems firstly it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non dual covering approximation operators according to definitions based on element granule and subsystem secondly we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework the underlying uncertainty of covering rough sets results from the roughness of granulation of coverings finally we investigate the relationships between the covering approximation operators and evidence theory it shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions while others do not share this property the sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored and the corresponding counterexamples are given to illustrate those that cannot be measured based on above discussions we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory

positive approximation an accelerator for attribute reduction in rough set theory feature selection is a challenging problem in areas such as pattern recognition machine learning and data mining considering a consistency measure introduced in rough set theory the problem of feature selection also called attribute reduction aims to retain the discriminatory power of original features many heuristic attribute reduction algorithms have been proposed however quite often these methods are computationally time consuming to overcome this shortcoming we introduce a theoretic framework based on rough set theory called positive approximation which can be used to accelerate a heuristic process of attribute reduction based on the proposed accelerator a general attribute reduction algorithm is designed through the use of the accelerator several representative heuristic attribute reduction algorithms in rough set theory have been enhanced note that each of the modified algorithms can choose the same attribute reduct as its original version and hence possesses the same classification accuracy experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

communicating between information systems communication between information systems is a basic problem in granular computing the concept of homomorphism is a useful mathematical tool to study the communication between two information systems in this paper some properties of information systems under homomorphisms are investigated the concepts of consistent functions are first introduced and their properties are investigated the concepts of relation mappings between two universes are then proposed in order to construct a binary relation on one universe according to the given binary relation on the other universe the main properties of the mappings are studied finally the notions of homomorphisms of information systems based on arbitrary binary relations are proposed and it is proved that attribute reductions in the original system and image system are equivalent to each other under the condition of homomorphism

existence of homoclinic solutions for nonlinear second order coupled systems abstract this work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature the arguments apply the fixed point theory green s functions technique l 1 caratheodory functions lower and upper solutions and schauder s fixed point theorem

an information fusion approach by combining multigranulation rough sets and evidence theory multigranulation rough set mgrs theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions they are used to aggregate multiple granular structures from a set theoretic standpoint however the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems dempster s combination rule in the evidence theory has been employed to aggregate information coming from multiple sources however it fails to deal with conflict evidence to overcome these limitations we focus on the combination of granular structures with both reliability and conflict from multiple sources which has been a challenging task in the field of granular computing we first address the connection between multigranulation rough set theory and the evidence theory then a two grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed which is based on a well defined distance function among granulation structures finally an illustrative example is given to show the effectiveness of the proposed fusion method the results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing

two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation abstract in this paper we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation by establishing a compactness lemma and using variational methods the existence result of two homoclinic solutions is obtained under some suitable assumptions but not requiring the periodicity condition some recent results are improved and extended