textural unit operations in rough set theory abstract in this study we introduce a textural counterpart of the unit operation of wybraniec skardowska a unit di operation has two parts which are called a unit co operation and a unit operation respectively in this respect we have two types of symmetry we show that the symmetricity of direlations is equivalent to the galois connectivity of unit di operations for discrete textures we determine the image co operation of a unit co operation we prove that the symmetricity and duality are equivalent concepts for unit di operations if one of the compounds is symmetric we consider definability in terms of unit operations and unit co operations further we present a categorical discussion defining a category un whose structures of objects are unit operations we show that the category rel whose objects are approximation spaces and morphisms are relation preserving functions can be embedded into un

ditopological texture spaces and fuzzy topology i basic concepts this is the first of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of l topological spaces the authors begin by defining the notion of q sets which together with the p sets considered earlier enable the formulation of a powerful concept of duality this plays an important role in the theory of direlations and difunctions which is described here in detail difunctions are then taken as the morphisms of a category dftex whose objects are texture spaces several important subcategories are defined and the closely related construct ftex defined some properties of the functors between these categories are obtained

textures and fuzzy unit operations in rough set theory an approach to fuzzy rough set models abstract in this paper an approach for the fuzzy rough set models is presented using textures and a fuzzy version of the unit operations of wybraniec skardowska first a fuzzy unit operation and fuzzy unit co operation on fuzzy lattices are defined it is proved that the well known fuzzy rough set upper approximations as the approximation of dubois and prade are fuzzy unit operation using fuzzy direlations an axiomatic system for fuzzy unit operations is studied and it is shown that fuzzy rough set systems obtained by different fuzzy logical connectives as kleene dienes and godel implicators can be generated by the same textural fuzzy direlation finally it is observed that the approximations of two different fuzzy rough set models together constitute two different galois connections

treediagonal matrices and their inverses abstract a generalization of tridiagonal matrices is considered namely treediagonal matrices which have nonzero off diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements some properties of treediagonal matrices are given and their inverses are characterized and shown to have an interesting structure

monad transformers as monoid transformers the incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a pre existing monad a complication of this approach is that the operations associated to the pre existing monad need to be lifted to the new monad in a companion paper by jaskelioff the lifting problem has been addressed in the setting of system email protected here we recast and extend those results in a category theoretic setting we abstract and generalize from monads to monoids in a monoidal category and from monad transformers to monoid transformers the generalization brings more simplicity and clarity and opens the way for lifting of operations with applicability beyond monads

extending sliced inverse regression sliced inverse regression sir and an associated chi squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal in this article the assumptions on the predictor distribution under which the chi squared test was proved to apply are relaxed and the result is extended a general weighted chi squared test that does not require normal regressors for the dimension of a regression is given simulations show that the weighted chi squared test is more reliable than the chi squared test when the regressor distribution digresses from normality significantly and that it compares well with the chi squared test when the regressors are normal

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

the spectrum of infinite regular line graphs let g be an infinite d regular graph and l g its line graph we consider discrete laplacians on g and l g and show the exact relation between the spectrum of ag and that of al g our method is also applicable to di d2 semiregular graphs subdivision graphs and para line graphs

sufficient dimension reduction with missing predictors in high dimensional data analysis sufficient dimension reduction sdr methods are effective in reducing the predictor dimension while retaining full regression information and imposing no parametric models however it is common in high dimensional data that a subset of predictors may have missing observations existing sdr methods resort to the complete case analysis by removing all the subjects with missingness in any of the predictors under inquiry such an approach does not make effective use of the data and is valid only when missingness is independent of both observed and unobserved quantities in this article we propose a new class of sdr estimators under a more general missingness mechanism that allows missingness to depend on the observed data we focus on a widely used sdr method sliced inverse regression and propose an augmented inverse probability weighted sliced inverse regression estimator aipw sir we show that aipw sir is doubly robust and asymptotically consistent and demonstrate

inverses of weighted graphs abstract we consider only connected bipartite graphs g with a unique perfect matching m let g w be the weighted graph obtained from g by giving weights to its edges using the positive weight function w e g 0 such that w e 1 for each e m an unweighted graph g may be viewed as a weighted graph with the weight function w 1 a weighted graph g w is nonsingular if its adjacency matrix a g w is nonsingular the inverse of a nonsingular weighted graph g w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g w via a diagonal matrix of 1s by g m we denote the graph obtained from g by contracting each edge of m to a single vertex it is known that if g m is bipartite then g w is invertible for each weight function w in this article we consider the converse of this result and show that if g w is invertible for each w then the graph g m is bipartite we also supply exact conditions under which g w is invertible for one w will force that the graph g m is bipartite

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

adjusting for nonignorable drop out using semiparametric nonresponse models abstract consider a study whose design calls for the study subjects to be followed from enrollment time t 0 to time t t at which point a primary endpoint of interest y is to be measured the design of the study also calls for measurements on a vector v t of covariates to be made at one or more times t during the interval 0 t we are interested in making inferences about the marginal mean 0 of y when some subjects drop out of the study at random times q prior to the common fixed end of follow up time t the purpose of this article is to show how to make inferences about 0 when the continuous drop out time q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables in particular we consider two models for the conditional hazard of drop out given v t y where v t denotes the history of the process v t through time t t 0 t in the first model we assume that q t v t y exp 0 y where 0 is a scalar paramet

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

regression with missing x s a review abstract the literature of regression analysis with missing values of the independent variables is reviewed six classes of procedures are distinguished complete case analysis available case methods least squares on imputed data maximum likelihood bayesian methods and multiple imputation methods are compared and illustrated when missing data are confined to one independent variable and extensions to more general patterns are indicated attention is paid to the performance of methods when the missing data are not missing completely at random least squares methods that fill in missing x s using only data on the x s are contrasted with likelihood based methods that use data on the x s and y the latter approach is preferred and provides methods for elaboration of the basic normal linear regression model it is suggested that more widely distributed software is needed that advances beyond complete case analysis available case analysis and naive imputation methods bayesian simulation methods and mu

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

on reciprocal eigenvalue property of weighted trees an undirected unweighted graph g is called nonsingular if its adjacency matrix a g is nonsingular a graph is said to have property r if with every eigenvalue of a g 1 is also an eigenvalue if further the multiplicity of and 1 as eigenvalues of a g are the same then g is said to have property sr in a previous paper the classes of unweighted trees with property r and sr where shown to be the same and that this common class coincides with the class of corona trees that is trees which have been obtained from smaller trees by adding a pendent edge to each vertex in this paper we continue the study of nonsingular trees in two ways first we characterize the set of all graphs whose inverses are nonsingular trees second we investigate the extent of property r for weighted trees and characterize the property for all trees with 8 vertices or more under some conditions on the weights

on some graphs which satisfy reciprocal eigenvalue properties abstract we consider only simple graphs consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite on the class h g of such graphs g the equivalence of the following statements are known i the reciprocal of the spectral radius of the adjacency matrix a g is the least positive eigenvalue of the adjacency matrix ii the graph is isomorphic to its inverse where the inverse of a graph g is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g via a diagonal matrix of 1s iii the graph has the reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g iv the graph has the strong reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g and they both have the same multiplicities v the graph is a corona graph that is it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex it is known that the statements iii and iv are not equivalent if we allow nonbipartite graphs however the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings where we relax the condition of bipartiteness of the contracted graph in this article we establish the equivalence of the first four statements for a class larger than h g we then supply a constructive structural description of those graphs

generalized textural rough sets rough set models over two universes abstract the aim of this paper is to reveal the useful aspects of textures for generalized approximation spaces in rough set theory to this end we present a counterpart of rough sets over two universes for textures we compare the textural results with the well known basic properties of rough sets given in the literature first we define a t seriality and a t inverse seriality in textures we show that almost all basic results with respect to rough set model over two universes can be formulated using textures on the way we give new results and observations due to lower and upper approximation operators which are not taken into consideration by the researchers moreover we also discuss the revised approximation operators and attribute oriented formal concept lattices in a textural framework

strongly self dual graphs we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs if the h graphs g and h with adjacency matrices m g and m h satisfy m g 1 sm h s where s is a signature matrix we refer to h as the dual of g the dual is a type of graph inverse if the h graph g is isomorphic to its dual via a particular isomorphism we refer to g as strongly self dual we investigate the structural and spectral properties of strongly self dual graphs with a particular emphasis on identifying when such a graph has 1 as an eigenvalue

graph invertibility and median eigenvalues abstract let g w be a weighted graph with a weight function w e g r 0 a weighted graph g w is invertible to a new weighted graph if its adjacency matrix is invertible graph inverses have combinatorial interests and can be applied to bound median eigenvalues of graphs such as have physical meanings in this paper we characterize the inverse of a weighted graph based on its sachs subgraphs that are spanning subgraphs with only k 2 or cycles or loops as components the characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix if a graph has its spectra split about the origin i e half of eigenvalues are positive and half of them are negative then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse we characterize graphs with a unique sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching as applications we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval 1 1

a reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from 1 i abstract a weighted directed graph is a directed graph g whose underlying undirected graph is simple and whose edges have nonzero directional complex weights that is the presence of an edge u v of weight w is as good as the presence of the edge v u with weight w the complex conjugate of w let g be a weighted directed graph on vertices 1 2 n denote by w u v the weight of an edge u v e g the adjacency matrix a g of g is an n n matrix with entries a i j w i j or w j i or 0 depending on whether i j e g or j i e g or otherwise respectively we supply a characterization of those unicyclic weighted directed graphs g whose edges have weights from the set 1 i and whose adjacency matrix a g satisfies the following property is an eigenvalue of a g with multiplicity k if and only if 1 is an eigenvalue of a g with the same multiplicity

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

textures and c spaces in this paper we point out that textures simple textures and plain textures are respectively isomorphic to c spaces sober c spaces and partially ordered sets using the isomorphism between textures and c spaces we show that subtextures quotient textures sums and products of textures can be identified as subspaces quotient spaces sums and products of c spaces furthermore it is shown that textures except discrete textures do not satisfy any separation axioms t 1 t 4 other than t 0 and so they are not pseudometrizable then we also give four useful characterizations of the complete distributivity condition of textures

fusion refinement procedure for dimension reduction with missing response at random dimension reduction methods are useful for handling high dimensional data it is a common situation that responses of some subjects are not observed in practice generally the missingness carries additional information about the central subspace here we propose a two stage procedure known as the fusion refinement fr procedure in the first stage we obtain a subspace including the central subspace by fusing information on regression and missingness in the second stage we refine the obtained subspace to recover the central subspace by imputation method we use sliced inverse regression to illustrate the fr procedure we conduct simulation studies and suggest a data driven procedure to choose from the complete case analysis and the fr procedure for a purpose of a real application a real data analysis is used to illustrate our methods

strong reciprocal eigenvalue property of a class of weighted graphs let h be the class of connected bipartite graphs g with a unique perfect matching m for g h let wg be the set of weight functions w on the edge set e g such that w e 1 for each matching edge and w e 0 for each nonmatching edge let gw denote the weighted graph with g h and w wg the graph gw is said to satisfy the reciprocal eigenvalue property property r if 1 is an eigenvalue of the adjacency matrix a gw whenever is an eigenvalue of a gw moreover if the multiplicities of the reciprocal eigenvalues are the same we say gw has the strong reciprocal eigenvalue property property sr let hg g h g m is bipartite where g m is the graph obtained from g by contracting each edge in m to a vertex recently in 12 it was shown that if g hg then gw has property sr for some w wg if and only if gw has property sr for each w wg if and only if g is a corona graph obtained from another graph h by adding a new pendant vertex to each vertex of h now we have the following questions is there a graph g h hg such that gw has property sr for each w wg are there graphs g h hg such that gw never has property sr not even for one w wg are there graphs g h such that gw has property sr for some w wg but not for all w wg in this article we supply answers to these three questions we also supply a graph class larger than hg where for any graph g if gw has property sr for one w wg then g is a corona graph

directed intervals and the dual of a graph abstract we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs for the h graph g with adjacency matrix m we consider the problem of identifying exactly when m 1 is signature similar to the adjacency matrix of another h graph g when this holds g is a type of graph inverse of g which is known as the dual of g we present necessary and sufficient conditions for the existence of g as an application we provide a characterization of all dual pairs g and g where both graphs are unicyclic this characterization allows us to identify those h graphs g that are unicyclic and self dual

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

categories of rough sets and textures abstract it is known that the theories of rough sets and fuzzy sets have successful applications in computing textures as a theoretical model provide a new perspective for both rough sets and fuzzy sets indeed recent papers have shown that there is a natural link between rough sets and textures while a texture is an alternative point set based setting for fuzzy sets relations are representatives of information systems and induce approximation operators therefore the first step for the categorical discussions on rough sets involves the category rel of sets and relations in this context we observe that power sets and pairs of rough set approximation operators form a category denoted by r apr in particular we prove that r apr is isomorphic to a full subcategory of the category cdrtex whose objects are complemented textures and morphisms are complemented direlations therefore cdrtex may be regarded as a suitable abstract model of rough set theory here we show that r apr and cdrtex are new examples of dagger symmetric monoidal categories

textures and covering based rough sets this paper introduces the relations between covering and dicovering in the framework of rough sets the natural counterpart of a covering in textures is given by a family of pairs of sets which is called a dicovering this leads us to two dual concepts which are called star and co star of a set respectively here we define dicovering approximation space and we discuss on covering and dicovering based rough sets in this context star corresponds to the upper approximation operator defined by zakowski and co star corresponds to the lower approximation operator studied by pomykala yao and zhu and wang

inverses of triangular matrices and bipartite graphs abstract to a given nonsingular triangular matrix a with entries from a ring we associate a weighted bipartite graph g a and give a combinatorial description of the inverse of a by employing paths in g a under a certain condition nonsingular triangular matrices a such that a and a 1 have the same zero nonzero pattern are characterized a combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree

textural approach to generalized rough sets based on relations this paper presents a discussion on rough set theory from the textural point of view a texturing is a family of subsets of a given universal set u satisfying certain conditions which are generally basic properties of the power set the suitable morphisms between texture spaces are given by direlations defined as pairs r r where r is a relation and r is a corelation it is observed that the presections are natural generalizations for rough sets more precisely if r r is a complemented direlation then the inverse of the relation r the corelation r is actually a lower approximation operator an upper approximation operator

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

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

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

doubly robust estimation in missing data and causal inference models the goal of this article is to construct doubly robust dr estimators in ignorable missing data and causal inference models in a missing data model an estimator is dr if it remains consistent when either but not necessarily both a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified because with observational data one can never be sure that either a missingness model or a complete data model is correct perhaps the best that can be hoped for is to find a dr estimator dr estimators in contrast to standard likelihood based or nonaugmented inverse probability weighted estimators give the analyst two chances instead of only one to make a valid inference in a causal inference model an estimator is dr if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct inference based on dr estimators should improve upon previous approaches indeed we present the results of simulation studies which demonstrate that the finite sample performance of dr estimators is as impressive as theory would predict the proposed method is applied to a cardiovascular clinical trial

on unimodular graphs abstract we study graphs whose adjacency matrices have determinant equal to 1 or 1 and characterize certain subclasses of these graphs graphs whose adjacency matrices are totally unimodular are also characterized for bipartite graphs having a unique perfect matching we provide a formula for the inverse of the corresponding adjacency matrix and address the problem of when that inverse is diagonally similar to a nonnegative matrix special attention is paid to the case that such a graph is unicyclic

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat