quantitative analysis for covering based rough sets through the upper approximation number coverings are a useful form of data while covering based rough sets provide an effective tool for dealing with this data covering based rough sets have been widely used in attribute reduction and rule extraction however few quantitative analyses for covering based rough sets have been conducted while many advances for classical rough sets have been obtained through quantitative tools in this paper the upper approximation number is defined as a measurement to quantify covering based rough sets and a pair of upper and lower approximation operators are constructed using the approximation number the operators not only inherit some important properties of existing approximation operators but also exhibit some new quantitative characteristics it is interesting to note that the upper approximation number of a covering approximation space is similar to the dimension of a vector space or the rank of a matrix

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

rough 3 valued algebras rough set theory is an important tool for dealing with granularity and vagueness in information systems this paper studies a kind of rough set algebra the collection of all the rough sets of an approximation space can be made into a 3 valued lukasiewicz algebra we call the algebra a rough 3 valued lukasiewicz algebra in this paper we focus on the rough 3 valued lukasiewicz algebras which are a special kind of 3 valued lukasiewicz algebras firstly we examine whether the rough 3 valued lukasiewicz algebra is an axled 3 valued lukasiewicz algebra secondly we present the condition under which the rough 3 valued lukasiewicz algebra is also a 3 valued post algebra then we investigate the 3 valued post subalgebra problem of the rough 3 valued lukasiewicz algebra finally this paper studies the relationship between the rough 3 valued lukasiewicz algebra and the boolean algebra constructed by all the exact sets of the corresponding approximation space

optimum accelerated life tests wth a nonconstant scale parameter previous work on planning accelerated life tests has assumed that the scale parameter for a location scale distribution of log lifetime remains constant over all stress levels this assumption is however inappropriate for many applications including accelerated tests for metal fatigue and certain electronic components this article extends the existing maximum likelihood theory for test planning to the nonconstant model and presents test plans for a large range of practical testing situations the test plans are optimum in that they minimize the asymptotic variance of the maximum likelihood estimator of a specified quantile at the design stress the development and discussion in the article as well as the theory given in the appendix applies to accelerated life test models in which the log time to failure can be modeled as a location scale distribution the test setup assumes simultaneous testing of units with time censoring we give particular numerical results for the weibull failure time distr

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

the ϑ lower and t upper fuzzy rough approximation operators on a semigroup in this paper we consider a semigroup as a universal set and investigate the properties of q lower and t upper fuzzy rough approximation operators a new fuzzy algebraic structure called a tl fuzzy rough semigroup is introduced and some related properties are studied finally the homomorphism properties of q lower and t upper fuzzy rough approximation operators are investigated

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

lower bounds in minimum rank problems abstract the minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph the minimum semidefinite rank of a graph is the minimum rank among hermitian positive semidefinite matrices with the given graph we explore connections between os sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large

a class of rough multiple objective programming and its application to solid transportation problem this paper establishes a rough multiple objective programming model for a solid transportation problem furthermore a general model for rough multiple objective programming problem is presented properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated in addition compromise solutions are obtained by using the interactive fuzzy satisfying method to solve the rough multiple objective solid transportation problem the rough simulation based genetic algorithm is proposed in which the rough simulation is embedded into the genetic algorithm finally an application of the solid transportation problem at xiluodu hydropower station is provided as an illustration

categorical properties of m indiscernibility spaces this paper discusses categorical aspect of the pawlak rough set theory it is proved that the category of all m indiscernibility spaces and m equivalence relation preserving mappings between them is both a topological construct and a topos as an application of these results the notions of product m indiscernibility space sum m indiscernibility space quotient m indiscernibility space m indiscernibility subspace quotient mapping and isomorphism mapping are defined and structures of these m indiscernibility spaces and mappings are also given

semiparametric analysis of general additive multiplicative hazard models for counting processes the additive multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process z w t x t t takes the form of t z g 0 t w t 0 t h 0 t x t where 0 0 t 0 t t is a vector of unknown regression parameters g and h are known link functions and 0 is an unspecified baseline hazard function in this paper we develop a class of simple estimating functions for 0 which contains the partial likelihood score function in the special case of proportional hazards models the resulting estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions weak convergence of the aalen breslow type estimators for the cumulative baseline hazard function 0 t 0 t 0 u du is also established furthermore we construct adaptive estimators for 0 and o that achieve the semiparametric information bounds finally a real example is provided along with some simulation results

an extension of the proportional hazards model for grouped data summary a family of transformations for probabilities is considered for the analysis of grouped survival data additive and multiplicative models for the hazard function are compared a method is suggested for estimating the scale on which an additive representation of the hazard in terms of explanatory variables is appropriate computations are performed using glim a test for departures from the grouped proportional hazards model is introduced an example is analyzed by the methods put forward in the analysis of survival data the effects of explanatory variables on failure time t are of great interest these effects are perhaps most conveniently investigated by establishing simple models for the hazard function in terms of explanatory variables we focus attention on additive and multiplicative models for the hazard the approach taken is to analyze sequences of contingency tables constructed from the data this follows mantel 1966 though the way in which we determine the sequences parallels work by holford 1976 certain similarities to problems involving binary data are exploited in particular a family of transformations for probabilities is defined which includes both additive and multiplicative models as special cases generalized linear models nelder and wedderburn 1972 associated with the transformations may be tried on the data and their fits compared in 3 a test for departures from the grouped multiplicative model in the direction of the grouped additive model is proposed and the forms of two simple estimates of the survivor function are given the methods are applied to an example in 4

a comparison of accelerated life test plans for weibull and lognormal distributions and type i censoring previous work on planning accelerated life test plans for the weibull and lognormal distributions has concentrated on optimum test plans that minimize the variance of some specified estimator however these test plans use tests at only two levels of stress and thus have serious practical limitations this article compares optimum test plans and some compromise test plans with respect to additional criteria including a the ability to detect departures from the assumed stress life relationship and b robustness to departures from the assumptions used in determining the plans the comparisons are based on the large sample properties of maximum likelihood estimators and the test plans are compared over a range of practical testing situations the comparisons suggest some general rules for planning accelerated life tests

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

an upper bound for the minimum rank of a graph abstract for a graph g of order n the minimum rank of g is defined to be the smallest possible rank over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise we prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs including all bipartite graphs and conjecture this bound is true over for all graphs and prove a related bound for all zero nonzero patterns of not necessarily symmetric matrices most of the results are valid for matrices over any infinite field but need not be true for matrices over finite fields

characteristic matrix of covering and its application to boolean matrix decomposition covering based rough sets provide an efficient means of dealing with covering data which occur widely in practical applications boolean matrix decomposition has frequently been applied to data mining and machine learning in this paper three types of existing covering approximation operators are represented by boolean matrices and then used in boolean matrix decomposition first we define two characteristic matrices of a covering through these boolean characteristic matrices three types of existing covering approximation operator are concisely and equivalently represented second these operator representations are applied to boolean matrix decomposition which has a close relationship with nonnegative matrix factorization a popular and efficient technique for machine learning we provide a sufficient and necessary condition for a square boolean matrix to decompose into the boolean product of another matrix and its transpose we then develop an algorithm for this boolean matrix decomposition finally these three covering approximation operators are axiomatized using boolean matrices this work presents an interesting viewpoint from which to investigate covering based rough set theory and its applications

the algebraic structures of generalized rough set theory rough set theory is an important technique for knowledge discovery in databases and its algebraic structure is part of the foundation of rough set theory in this paper we present the structures of the lower and upper approximations based on arbitrary binary relations some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended based on the concepts of definable sets in rough set theory two important boolean subalgebras in the generalized rough sets are investigated an algorithm to compute atoms for these two boolean algebras is presented

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

invertible approximation operators of generalized rough sets and fuzzy rough sets this paper studies the classes of rough sets and fuzzy rough sets we discuss the invertible lower and upper approximations and present the necessary and sufficient conditions for the lower approximation to coincide with the upper approximation in both rough sets and fuzzy rough sets we also study the mathematical properties of a fuzzy rough set induced by a cyclic fuzzy relation

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

rough set theory applied to lattice theory in this paper we intend to study a connection between rough sets and lattice theory we introduce the concepts of upper and lower rough ideals filters in a lattice then we offer some of their properties with regard to prime ideals filters the set of all fixed points compact elements and homomorphisms

rough operations on boolean algebras in this paper we introduce two pairs of rough operations on boolean algebras first we define a pair of rough approximations based on a partition of the unity of a boolean algebra we then propose a pair of generalized rough approximations on boolean algebras after defining a basic assignment function between two different boolean algebras finally some discussions on the relationship between rough operations and some uncertainty measures are given to provide a better understanding of both rough operations and uncertainty measures on boolean algebras

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

information theoretic measures associated with rough set approximations in rough set theory some information theoretic measures of uncertainty and granularity have been proposed a common feature of these measures is that they are only dependent on the partitions and the cardinality of a universe which means that they are independent of the lower and upper approximations of rough sets this seems somewhat unreasonable since the basic idea of rough set theory is to describe incomplete or inexact concepts by the lower and upper approximations in light of this we develop a new pair of information theoretic entropy and co entropy functions associated to partitions and approximations in this paper such functions are used to measure the uncertainty and granularity of an approximation space after introducing the novel notions of entropy and co entropy we then examine their properties in particular we disclose the relationship of co entropies between different universes the theoretical development is accompanied by illustrative numerical examples

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

transfer functionals and semiparametric regression models summary a general approach to semiparametric regression model construction is proposed using the notation of a transfer functional estimators of unknown regression parameters are proposed generalising the estimators of the parameters of the proportional hazards and the additive risk models

a comparative study of rough sets for hybrid data to discover knowledge from hybrid data using rough sets researchers have developed several fuzzy rough set models and a neighborhood rough set model these models have been applied to many hybrid data processing applications for a particular purpose thus neglecting the issue of selecting an appropriate model to address this issue this paper mainly concerns the relationships among these rough set models investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules analyzing the relationships among rough approximations of these models shows that hu s fuzzy rough approximations are special cases of neighborhood and wang s fuzzy rough approximations respectively furthermore one to one correspondence relationships exist between wang s fuzzy and neighborhood rough approximations this study also finds that wang s fuzzy and neighborhood rough approximations are cut sets of dubois fuzzy rough approximations and radzikowska and kerre s fuzzy rough approximations respectively

the approximation number function and the characterization of covering approximation space there is renewed interest in the topic of covering based rough sets it is important to understand the characterization of covering approximation spaces recently zhu et al presented the concept of an approximation number function which can be viewed as a quantitative tool for analyzing the covering approximation spaces we focus on this concept and show that an approximation number function can be characterized by its properties we give the axiomatic definition of an approximation number function and prove that there is a one to one correspondence between the set of all lower approximation number functions and the set of all covering approximation spaces this suggests that the concept is fundamental to the characterization of covering approximation spaces we apply approximation number functions to the existing studies on approximation spaces and a series of matroidal structures are induced in an approximation space