an action learning evaluation procedure for multiple criteria decision making problems abstract in this paper we focus on an extension of the analytic hierarchy process ahp that accommodates ambiguity on the part of the decision maker dm and facilitates the exploration of the decision domain we propose a systematic action learning process that builds confidence as it converges from numeric interval estimates to numeric point estimates our multiple criteria decision making mcdm problem procedure structures the problem as a hierarchy evaluates all objects using pairwise comparisons that accommodate vagueness and ambiguity uses interval prioritization techniques and does synthesis using the linear additive value function this action learning process facilitates the understanding of key stakeholders which is imperative for the successful implementation of the subsequent decision

variable threshold concept lattices in this paper the definition of a variable threshold concept lattice is introduced based on a galois connection three kinds of variable threshold concept lattices in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold are defined the number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice the three kinds of variable threshold concept lattices are constructed between two crisp sets between a crisp set and a fuzzy set and between a fuzzy set and a crisp set their properties are analogous to that of the classical concept lattices and can be induced by the fuzzy concept lattice

interval ordered information systems interval information systems are generalized models of single valued information systems by introducing a dominance relation to interval information systems we propose a ranking approach for all objects based on dominance classes and establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation furthermore we discuss interval ordered decision tables and dominance rules to simplify knowledge representation and extract much simpler dominance rules we propose attribute reductions of interval ordered information systems and decision tables that eliminate only the information that are not essential from the viewpoint of the ordering of objects or dominance rules the approaches show how to simplify an interval ordered information system and find dominance rules directly from an interval ordered decision table these results will be helpful for decision making analysis in interval information systems

fast non searching method for maneuvering target detection and motion parameters estimation this paper considers the coherent integration problem for maneuvering target detection and motion parameters estimation involving range migration rm and doppler frequency migration dfm within the coherent pulse interval a fast non searching method based on adjacent cross correlation function accf and lv s distribution lvd is proposed where the adjacent correlation operation is first employed to remove the rm and reduce the order of dfm after that lvd is applied to realize the coherent integration target detection and parameters estimation in addition at the cost of some performance loss another fast method via accf iteratively is also introduced to further reduce the computational complexity and obtain the motion parameters estimation the proposed two methods are fast in that they can be easily implemented by using complex multiplications the fast fourier transform fft and inverse fft ifft compared with the existing methods the presented algorithms can obtain the motion parameters estimation without any searching procedure and can achieve a good balance between the computational cost and the detection ability as well as parameters estimation performance finally several simulation experiments are provided to demonstrate the effectiveness

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

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 consistent block technique for rule acquisition in incomplete information systems in this paper the concept of a maximal consistent block is applied to formulate a new approximation to an object set in incomplete information systems with higher level of accuracy similar to the method in inform sci 112 1998 39 113 1999 271 the proposed rough set based rule acquisition method does not require change in the size of the original incomplete system it however has the additional advantage of using a set of simpler discernibility functions of an incomplete system this means that it can provide a more efficient computation for knowledge acquisition especially in large incomplete systems

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

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

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

set valued information systems set valued information systems are generalized models of single valued information systems incomplete information systems can be viewed as disjunctively interpreted set valued information systems since some objects in set valued information systems may have more than one value for an attribute so we define tolerance relation and use the maximal tolerance classes to classify the universe of discourse in order to derive optimal decision rules from set valued decision information systems we propose the concept of relative reduct of maximal tolerance classes and define a kind of discernibility function to compute the relative reduct by boolean reasoning techniques finally we define three kinds of relative reducts for set valued information systems and used them to evaluate the significance of attributes

rough approximation quality revisited in rough set theory the approximation quality is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes in this paper we re interpret the classical in terms of a classic measure based on sets the marczewski steinhaus metric and also in terms of proportional reduction of errors pre measures we also exhibit infinitely many possibilities to define like statistics which are meaningful in situations different from the classical one and provide tools to ascertain the statistical significance of the proposed measures which are valid for any kind of sample copyright 2001 elsevier science b v

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

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

neighborhood operator systems and approximations this paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k step neighborhood systems based on a binary relation on a finite universe six families of binary relations are obtained and the corresponding six classes of k step neighborhood systems are derived extensions of pawlak s rough set approximation operators based on such neighborhood systems are proposed properties of neighborhood operator systems and rough set approximation operators are investigated and their connections are examined

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

granular computing and dual galois connection a covering model for granular computing in a set theoretic setting is studied in this paper under this model a zooming in operator is redefined combinations of the zooming in and zooming out operators form two pairs of approximation operators of the original and the granulated universe of discourse their properties are examined in detail for the two pairs of lower and upper approximation operators it is proved that the duality is always true for a generalized approximation space the approximation representations are just the combination operators formed on the basis of the zooming out and zooming in operators relationships between these combination operators and the dual galois connection are also analyzed

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined