extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

partial auc estimation and regression accurate diagnosis of disease is a critical part of health care new diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased from nondiseased states the partial area under the receiver operating characteristic roc curve is a measure of diagnostic test accuracy we present an interpretation of the partial area under the curve auc which gives rise to a nonparametric estimator this estimator is more robust than existing estimators which make parametric assumptions we show that the robustness is gained with only a moderate loss in efficiency we describe a regression modeling framework for making inference about covariate effects on the partial auc such models can refine knowledge about test accuracy model parameters can be estimated using binary regression methods we use the regression framework to compare two prostate specific antigen biomarkers and to evaluate the dependence of biomarker accuracy on the time prior to clinical diagnosis of prostate cancer

rules in incomplete information systems abstract a new method of computing all optimal certain rules from an incomplete information system is presented and proved the method does not require changing the size of the original incomplete system additionally several existing rough set methods of computing decision rules from incomplete information systems are analyzed and compared we show which of these methods are capable of generating all optimal certain rules or a class of optimal certain rules and which methods may lead to generation of false rules

n dimensional fractional fokker planck equation and its solutions for anomalous radial two phase flow in porous media in this paper we present a time fractional fokker planck equation ffpe for radial two phase flow of liquid and gas in porous media the ffpe of order a is solved for both two and three dimensional flow patterns using the laplace transform method the general solutions of the ffpe for both two and three dimensional flows are given as a convolution integral of the input and a kernel in the laplace domain special solutions for a large value and a periodic boundary condition are also given in the time domain when the inverse laplace transform can be found analytically the ffpe for two phase flow in porous media presented in this paper is the first report of its kind

extension of the fujita solution to air and water movement in soils for specific soil properties the fundamental solution of fujita 1952b is extended to include the effect of air movement on horizontal absorption this exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by parlange et al 1982 and sander and parlange 1984 the optimal sorptivity of parlange et al has very good accuracy while the surface water content s is near saturation but decreases in accuracy as s decreases the approximate sorptivity of sander and parlange however proves to be extremely accurate for all s with a maximum error of only 0 5 sander and parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0 5 the aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above alternatively the solution may be used for checking the accuracy of numerical models

some minimal axiom sets of rough sets the axiomatic approach is an important one to the development of rough set theory this paper proposes some independent and minimal axiom sets for characterizing the classical rough set first we investigate the correlations among some arbitrary binary relations in rough sets then we provide several forms of equivalence relations which are different from the original ones finally to well characterize the classical rough set we propose fifteen new minimal axiom sets of rough sets based on the new equivalence relations by employing relative tables the independence of axiom sets for characterizing the approximation operators is also examined

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

new results on single axioms for l fuzzy rough approximation operators abstract this paper further studies axiomatic characterizations of l fuzzy rough sets where l denotes a residuated lattice single axioms for upper l fuzzy rough approximation operators are investigated in two different approaches which describe upper l fuzzy approximation operators with ordinary l fuzzy operations and l fuzzy product operation respectively we characterize upper l fuzzy rough approximation operators by only one axiom with ordinary l fuzzy operations when the l fuzzy relation is euclidean as well as the composition of euclidean with serial resp reflexive we also discuss the single axioms for upper l fuzzy rough approximation operators with l fuzzy product operation on arbitrary l fuzzy relations that are not limited as either general l fuzzy relations or symmetric l fuzzy relations the single axioms for lower l fuzzy rough approximation operators are discussed with ordinary l fuzzy operations regardless of the regularity of residuated lattice l

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

on the statistical analysis of roc curves we introduce a new accuracy index for receiver operating characteristic roc curves namely the partial area under the binormal roc graph over any specified region of interest we propose a simple but general procedure based on a conventional analysis of variance for comparing accuracy indices derived from two or more different modalities the proposed method is related to and compared with existing methodology and is illustrated by results from an experiment on optimization of density and contrast yielded by multiform photographic images used for scintigraphy

semiparametric receiver operating characteristic analysis to evaluate biomarkers for disease the receiver operating characteristic roc curve is a popular method for characterizing the accuracy of diagnostic tests when test results are not binary various methodologies for estimating and comparing roc curves have been developed one approach due to pepe uses a parametric regression model rocx u g h0 u 0x with the baseline function h0 u specified up to a finite dimensional parameter in this article we extend the regression models by allowing arbitrary nonparametric baseline functions we also provide asymptotic distribution theory and procedures for making statistical inference we illustrate our approach with dataset from a prostate cancer biomarker study simulation studies suggest that the extra flexibility inherent in the semiparametric method is gained with little loss in statistical efficiency

semiparametric regression for the area under the receiver operating characteristic curve medical advances continue to provide new and potentially better means for detecting disease such is true in cancer for example where biomarkers are sought for early detection and where improvements in imaging methods may pick up the initial functional and molecular changes associated with cancer development in other binary classification tasks computational algorithms such as neural networks support vector machines and evolutionary algorithms have been applied to areas as diverse as credit scoring object recognition and peptide binding prediction before a classifier becomes an accepted technology it must undergo rigorous evaluation to determine its ability to discriminate between states characterization of factors influencing classifier performance is an important step in this process analysis of covariates may reveal subpopulations in which classifier performance is greatest or identify features of the classifier that improve accuracy we develop regression methods for the nonparametric area

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

on characterizations of i t fuzzy rough approximation operators in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties various fuzzy generalizations of rough approximations have been made in the literature this paper proposes a general framework for the study of i t fuzzy rough approximation operators within which both constructive and axiomatic approaches are used in the constructive approach a pair of lower and upper generalized fuzzy rough approximation operators determined by an implicator i and a triangular norm t is first defined basic properties of i t fuzzy rough approximation operators are investigated the connections between fuzzy relations and fuzzy rough approximation operators are further established in the axiomatic approach an operator oriented characterization of rough sets is proposed that is i t fuzzy approximation operators are defined by axioms different axiom sets of t upper and i lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators finally an open problem proposed by radzikowska and kerre in fuzzy sets and systems 126 2002 137 is solved

using one axiom to characterize l fuzzy rough approximation operators based on residuated lattices abstract axiomatic characterization of approximation operators plays an important role in the study of rough set theory different axiom sets of abstract operators can illustrate different classes of rough set systems in this paper we are devoted to searching for a single axiom to characterize l fuzzy rough approximation operators based on residuated lattices axioms of l fuzzy set theoretic operators make sure of the existence of certain types of l fuzzy relations which produce the same operators we demonstrate that the lower upper l fuzzy rough approximation operators generated by a generalized l fuzzy relation can be characterized by only one axiom furthermore we also use one axiom to characterize l fuzzy rough approximation operators produced by the l fuzzy serial reflexive symmetric and t transitive relations as well as any of their compositions

single axioms for lower fuzzy rough approximation operators determined by fuzzy implications abstract this paper further investigates axiomatic characterizations of lower fuzzy rough approximation operators determined by a semicontinuous fuzzy implication i the i lower fuzzy rough approximation operators on a general fuzzy relation can be characterized by only one axiom the axiomatic characterizations of i lower fuzzy rough approximation operators are studied in two approaches which characterize abstract i lower fuzzy rough approximation operators with ordinary fuzzy operations on fuzzy sets and product operation respectively considering the properties of fuzzy implications different axioms are proposed for i lower fuzzy rough approximation operators which are determined by an s implication an r implication and a fuzzy implication satisfying left neutrality property and regular property respectively when the fuzzy relation is special such as serial reflexive symmetric and so on i lower fuzzy rough approximation operators can also be characterized by a single axiom

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

an interpretation for the roc curve and inference using glm procedures the accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic roc curve this paper puts forth an interpretation for each point on the roc curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject this interpretation gives rise to new methods for making inference about roc curves it is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results one component of the pair being from a diseased subject and the other from a nondiseased subject within the generalized linear model glm binary regression framework roc curves can be estimated and we highlight a new semiparametric estimator covariate effects can also be evaluated with the glm models the methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters

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

a proposed design and analysis for comparing digital and analog mammography because randomized trials have shown a reduction in breast cancer mortality analog mammography for the early detection of breast cancer has gained widespread use recently several manufacturers have developed digital mammography which promises great advantages in the storage and transmission of images we were asked to design a study to compare the two types of mammography in terms of their performance for the early detection of breast cancer a standard measure of mammography performance is the receiver operating characteristic roc curve which is a plot of false and true positive rates for each ordered classification of the mammography images methods for study design and data analysis based on roc curves have been well developed for diagnostic tests particularly in radiology but for comparing the performance of mammography for the early detection of breast cancer among asymptomatic women special considerations motivate new designs and methodology first digital mammography may cost substantia

maximum likelihood estimation of receiver operating characteristic roc curves from continuously distributed data we show that truth state runs in rank ordered data constitute a natural categorization of continuously distributed test results for maximum likelihood ml estimation of roc curves on this basis we develop two new algorithms for fitting binormal roc curves to continuously distributed data a true ml algorithm labroc4 and a quasi ml algorithm labroc5 that requires substantially less computation with large data sets simulation studies indicate that both algorithms produce reliable estimates of the binormal roc curve parameters a and b the roc area index az and the standard errors of those estimates

axiomatic characterizations of s t fuzzy rough approximation operators axiomatic characterizations of approximation operators are of importance in the study of rough set theory in this paper axiomatic characterizations of relation based s t fuzzy rough approximation operators are investigated by employing a triangular conorm s and a triangular norm t on 0 1 we first introduced the constructive definitions of s lower and t upper fuzzy rough approximation operators with their essential properties we then propose an operator oriented characterization of s t fuzzy rough sets that is fuzzy set theoretic operators defined by axioms guarantee the existence of different types of fuzzy relations which produce the same operators we show that the s lower and respectively t upper fuzzy rough approximation operators generated by a generalized fuzzy relation can be described by only one axiom we further show that s t fuzzy rough approximation operators corresponding to special types of fuzzy relations such as serial reflexive symmetric and t transitive ones as well as any of their compositions can also be characterized by single axioms

the minimization of axiom sets characterizing generalized approximation operators in the axiomatic approach of rough set theory rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory in this paper the independence of axioms of generalized crisp approximation operators is investigated and their minimal sets of axioms are presented

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

lie group symmetry analysis of transport in porous media with variable transmissivity abstract we determine the lie group symmetries of the coupled partial differential equations governing a novel problem for the transient flow of a fluid containing a solidifiable gel through a hydraulically isotropic porous medium assuming that the permeability k of the porous medium is a function of the gel concentration c we determine a number of exact solutions corresponding to the cases where the concentration dependent permeability is either arbitrary or has a power law variation or is a constant each case admits a number of distinct lie symmetries and the solutions corresponding to the optimal systems are determined some typical concentration and pressure profiles are illustrated and a specific moving boundary problem is solved and the concentration and pressure profiles are displayed