uncertainty measures for interval set information tables based on interval δ similarity relation abstract the notion of uncertainty measure is one of the most important topics in rough set theory and has been studied in different kinds of information tables however few studies have focused on the interval set information table which is regarded as one of the generalized models of single valued information tables this paper aims at studying the uncertainty measurements for interval set information tables firstly an interval similarity relation is induced based on the similarity degree the similarity relation induces the granules which form a covering in interval set information tables secondly four types of granularity measures are defined to measure the granularity of a covering thirdly the concepts of accuracy and roughness in rough set theory are respectively extended to accuracy and roughness for interval set information tables furthermore four new combinations ofuncertainty measures by considering proposed granularity measures and accuracy and roughness are defined and analyzed theoretical analyses and experimental results illustrate that the proposed measures are effective and accurate for interval set information tables

uncertainty measurement for interval valued information systems abstract interval valued information systems are generalized models of single valued information systems accuracy and roughness are employed to depict the uncertainty of a set under an attribute subset in a pawlak rough set model based on equivalence classes information theoretic measures of uncertainty for rough sets have also been proposed however there are few studies on uncertainty measurements for interval valued information systems this paper addresses the uncertainty measurement problem in interval valued information systems the concept of the similarity degree based on the possible degree is introduced consequently the similarity relation between two interval objects are constructed by a given similarity rate based on the similarity relation similarity classes are defined under this definition accuracy and roughness are given for interval valued information systems which are generalizations of the concepts accuracy and roughness for the equivalence relation based rough set model moreover an alternative uncertainty measure called the rough degree is proposed theoretical studies and numerical experiments show that the proposed measures are effective and suitable for interval valued information systems

implicit discrete time systems and accessibility an intrinsic description for dynamic systems whose evolution along discrete time is governed by nonlinear implicit difference equations in one independent variable and zero order algebraic equations is presented by means of differential geometrical methods where systems are associated with appropriate geometric objects reflecting their dynamics dynamic systems given in implicit form have the peculiarity that they may contain so called hidden restrictions a normal form is presented which is characterized by the circumstances that there are no further restrictions in addition it is illustrated that such a normal form allows for an equivalent system representation in explicit form based on the geometric picture of implicit discrete time systems the qualitative property of accessibility along a fixed trajectory is discussed by applying symmetry groups of discrete time systems and studying invariants of these groups a formal approach is provided that allows us to gather local accessibility criteria successively which can be tested by computer algebra several examples illustrate the results

a linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary dimensions a sequential algorithm is presented for computing the exact euclidean distance transform dt of a k dimensional binary image in time linear in the total number of voxels n the algorithm which is based on dimensionality reduction and partial voronoi diagram construction can be used for computing the dt for a wide class of distance functions including the l sub p and chamfer metrics at each dimension level the dt is computed by constructing the intersection of the voronoi diagram whose sites are the feature voxels with each row of the image this construction is performed efficiently by using the dt in the next lower dimension the correctness and linear time complexity are demonstrated analytically and verified experimentally the algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast not only does it run in o n time but the time constant is small a simple modification of the algorithm computes the weighted euclidean dt which is useful for images with anisotropic voxel dimensions a parallel version of the algorithm runs in o n p time with p processors

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

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

relationships among generalized rough sets in six coverings and pure reflexive neighborhood system rough set is a useful tool to deal with partition related uncertainty granularity and incompleteness of knowledge although the classical rough set is constructed on the basis of an indiscernibility relation it can also be generalized by using some weaker binary relations in this paper a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system after two steps relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained the first step is to study the generalized rough sets in six coverings and to get relationships between every two covering rough set models the second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system the inclusion relations or equivalence relations among the seven upper lower approximations could be acquired finally the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared the relationships among seven accuracy measures are also obtained some illustrative examples are employed to demonstrate our arguments

discrete time transitivity and accessibility analytic systems a basic open question for discrete time nonlinear systems is that of determining when in analogy with the classical continuous time positive form of chow s lemma accessibility follows from transitivity of a natural group action this paper studies the problem and establishes the desired implication for analytic systems in several cases i compact state space ii under a poisson stability condition and iii in a generic sense in addition the paper studies accessibility properties of the controllability sets recently introduced in the context of dynamical systems studies finally various examples and counterexamples are provided relating the various lie algebras introduced in past work

complex wavelet structural similarity a new image similarity index we introduce a new measure of image similarity called the complex wavelet structural similarity cw ssim index and show its applicability as a general purpose image similarity index the key idea behind cw ssim is that certain image distortions lead to consistent phase changes in the local wavelet coefficients and that a consistent phase shift of the coefficients does not change the structural content of the image by conducting four case studies we have demonstrated the superiority of the cw ssim index against other indices e g dice hausdorff distance commonly used for assessing the similarity of a given pair of images in addition we show that the cw ssim index has a number of advantages it is robust to small rotations and translations it provides useful comparisons even without a preprocessing image registration step which is essential for other indices moreover it is computationally less expensive

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

theory and applications of granular labelled partitions in multi scale decision tables abstract granular computing and acquisition of if then rules are two basic issues in knowledge representation and data mining a formal approach to granular computing with multi scale data measured at different levels of granulations is proposed in this paper the concept of labelled blocks determined by a surjective function is first introduced lower and upper label block approximations of sets are then defined multi scale granular labelled partitions and multi scale decision granular labelled partitions as well as their derived rough set approximations are further formulated to analyze hierarchically structured data finally the concept of multi scale information tables in the context of rough set is proposed and the unravelling of decision rules at different scales in multi scale decision tables is discussed

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

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

invariant distributions for discrete time nonlinear systems the purpose of this paper is to introduce some notions of invariant distributions for the class of linear analytic discrete time systems and to give an algebraic characterization of them a brief final discussion motivates the usefulness of the notions introduced in the study of the geometric properties of the system

consistency of incomplete data consistency is well known for completely specified data sets a specified data set is defined as consistent when any pair of cases with the same attribute values belongs to the same concept in this paper we generalize the definition of consistency for incomplete data sets using rough set theory we discuss two types of missing attribute values lost values and do not care conditions for incomplete data sets there exist three definitions of approximations singleton subset and concept any approximation is lower or upper so we may define six types of consistencies we show that two pairs of such consistencies are equivalent hence there are only four distinct consistencies of incomplete data additionally we discuss probabilistic approximations and study properties of corresponding consistencies we illustrate the idea of consistency for incomplete data sets using experiments on many incomplete data sets derived from eight benchmark data sets

incremental update of approximations in dominance based rough sets approach under the variation of attribute values dominance based rough sets approach drsa has received much attention since it is able to acquire knowledge from information with preference ordered attribute domains and decision classes in many real life applications the information systems may evolve over time dynamically in a dynamic information system the obtained knowledge e g approximations in drsa need to be updated for decision making and other related tasks as a useful technique the incremental update can be applied to process dynamic information with revising the obtained knowledge in this paper we propose an incremental approach for maintaining approximations of drsa when attribute values vary over time some numerical examples illustrate that the incremental approach can renew approximations of drsa without beginning from scratch experimental evaluations show that the incremental algorithm can effectively reduce the computational time in comparison with the non incremental one when the ratio of the attribute values varied is less than a threshold

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

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

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

comparing images using the hausdorff distance the hausdorff distance measures the extent to which each point of a model set lies near some point of an image set and vice versa thus this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another efficient algorithms for computing the hausdorff distance between all possible relative positions of a binary image and a model are presented the focus is primarily on the case in which the model is only allowed to translate with respect to the image the techniques are extended to rigid motion the hausdorff distance computation differs from many other shape comparison methods in that no correspondence between the model and the image is derived the method is quite tolerant of small position errors such as those that occur with edge detectors and other feature extraction methods it is shown that the method extends naturally to the problem of comparing a portion of a model against an image

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

characterization and reduction of concept lattices through matroid theory concept lattice theory is an efficient tool for data analysis reduction of attributes and objects for a context is an important issue in concept lattice theory we explore the relationships between matroidal spaces and concepts for a given context with the assistance of matroid theory we conduct the attribute reduction and object reduction for a context and formulate the concepts for a given context as a generalization of the linear independence in vector spaces matroids provide well established platforms for some algorithms such as greedy algorithm in this paper we mainly propose a notion of matroidal space by the family of circuits of a matroid these relationships characterize the constructions of not only concepts but also concept lattices meanwhile we demonstrate that reduction of attributes and objects proposed in this paper preserve the original hierarchy order in the concept lattice for a given context

on the uniform observability of discrete time nonlinear systems in constructing an observer for a discrete time nonlinear system the system is commonly required to satisfy a certain kind of uniform observability condition that is the state should always be reconstructible from observation windows of a specific length irrespective of the values of the state and inputs in this technical note it is proved that this uniform requirement is unnecessary in the sense that if the initial state and inputs are on a compact set then the uniform observability is derived from its non uniform counterpart

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

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

dominance based rough set approach and knowledge reductions in incomplete ordered information system many methods based on the rough set to deal with incomplete information systems have been proposed in recent years however they are only suitable for the incomplete systems with regular attributes whose domains are not preference ordered this paper thus attempts to present research focusing on a complex incomplete information system the incomplete ordered information system in such incomplete information systems all attributes are considered as criterions a criterion indicates an attribute with preference ordered domain to conduct classification analysis in the incomplete ordered information system the concept of similarity dominance relation is first proposed two types of knowledge reductions are then formed for preserving two different notions of similarity dominance relations with introduction of the approximate distribution reduct into the incomplete ordered decision system the judgment theorems and discernibility matrixes associated with four novel approximate distribution reducts are obtained a numerical example is employed to substantiate the conceptual arguments

3d brain tumor segmentation in mri using fuzzy classification symmetry analysis and spatially constrained deformable models we propose a new general method for segmenting brain tumors in 3d magnetic resonance images our method is applicable to different types of tumors first the brain is segmented using a new approach robust to the presence of tumors then a first tumor detection is performed based on selecting asymmetric areas with respect to the approximate brain symmetry plane and fuzzy classification its result constitutes the initialization of a segmentation method based on a combination of a deformable model and spatial relations leading to a precise segmentation of the tumors imprecision and variability are taken into account at all levels using appropriate fuzzy models the results obtained on different types of tumors have been evaluated by comparison with manual segmentations

point set registration coherent point drift point set registration is a key component in many computer vision tasks the goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other multiple factors including an unknown nonrigid spatial transformation large dimensionality of point set noise and outliers make the point set registration a challenging problem we introduce a probabilistic method called the coherent point drift cpd algorithm for both rigid and nonrigid point set registration we consider the alignment of two point sets as a probability density estimation problem we fit the gaussian mixture model gmm centroids representing the first point set to the data the second point set by maximizing the likelihood we force the gmm centroids to move coherently as a group to preserve the topological structure of the point sets in the rigid case we impose the coherence constraint by reparameterization of gmm centroid locations with rigid parameters and derive a closed form solution of the maximization step of the em algorithm in arbitrary dimensions in the nonrigid case we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation we also introduce a fast algorithm that reduces the method computation complexity to linear we test the cpd algorithm for both rigid and nonrigid transformations in the presence of noise outliers and missing points where cpd shows accurate results and outperforms current state of the art methods

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

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

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

α dominance relation and rough sets in interval valued information systems though rough set has been widely used to study systems characterized by insufficient and incomplete information its performance in dealing with initial interval valued data needs to be seriously considered for improving the suitability and scalability the aim of this paper is to present a parameterized dominance based rough set approach to interval valued information systems first by considering the degree that an interval valued data is dominating another one we propose the concept of dominance relation second we present the dominance based rough set model in interval valued decision systems finally we introduce lower and upper approximate reducts into dominance based rough set for simplifying decision rules we also present the judgement theorems and discernibility functions which describe how lower and upper approximate reducts can be calculated this study suggests potential application areas and new research trends concerning rough set approach to interval valued information systems

mining incomplete data with singleton subset and concept probabilistic approximations abstract rough set theory provides a very useful idea of lower and upper approximations for inconsistent data for incomplete data these approximations are not unique in this paper we investigate properties of three well known generalizations of approximations singleton subset and concept these approximations were recently further generalized as to include an additional parameter interpreted as a probability in this paper we report novel properties of singleton subset and concept probabilistic approximations additionally we validated such approximations experimentally our main objective was to test which of the singleton subset and concept probabilistic approximations are the most useful for data mining our conclusion is that for a given incomplete data set all three approaches should be applied and the best approach should be selected as a result of ten fold cross validation finally we conducted experiments on complexity of rule sets and the total number of singleton subset and concept approximations

fuzzy rough sets assisted attribute selection attribute selection as refers to the problem of selecting those input attributes or features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing unlike other dimensionality reduction methods attribute selectors preserve the original meaning of the attributes after reduction this has found application in tasks that involve datasets containing huge numbers of attributes in the order of tens of thousands which for some learning algorithms might be impossible to process further recent examples include text processing and web content classification as techniques have also been applied to small and medium sized datasets in order to locate the most informative attributes for later use one of the many successful applications of rough set theory has been to this area the rough set ideology of using only the supplied data and no other information has many benefits in as where most other methods require supplementary knowledge however the main limitation of rough set based attribute selection in the literature is the restrictive requirement that all data is discrete in classical rough set theory it is not possible to consider real valued or noisy data this paper investigates a novel approach based on fuzzy rough sets fuzzy rough feature selection frfs that addresses these problems and retains dataset semantics frfs is applied to two challenging domains where a feature reducing step is important namely web content classification and complex systems monitoring the utility of this approach is demonstrated and is compared empirically with several dimensionality reducers in the experimental studies frfs is shown to equal or improve classification accuracy when compared to the results from unreduced data classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method in addition it is shown that frfs is more powerful than the other as techniques in the comparative study

relationships between generalized rough sets based on covering and reflexive neighborhood system rough set theory is an important tool to deal with uncertainty granularity and incompleteness of knowledge the covering based rough sets is one of the most important extensions of the classical pawlak rough sets in covering based rough set theory many types of rough sets were already defined in the literature to find out the relationships among different types of covering based rough sets and give an axiomatic definition is a very important issue the generalized rough sets in neighborhood system is another important extension of the classical pawlak rough sets in this paper we investigate relationships between covering based rough sets and generalized rough sets in neighborhood system then twenty three types of the element based the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems moreover properties of the generalized approximation operators in neighborhood systems are presented

characteristic sets and generalized maximal consistent blocks in mining incomplete data abstract mining incomplete data using approximations based on characteristic sets is a well established technique it is applicable to incomplete data sets with a few interpretations of missing attribute values e g lost values and do not care conditions on the other hand maximal consistent blocks were introduced for incomplete data sets with only do not care conditions using only lower and upper approximations in this paper we introduce an extension of the maximal consistent blocks to incomplete data sets with any interpretation of missing attribute values and with probabilistic approximations we prove new results on probabilistic approximations based on generalized maximal consistent blocks additionally we present results of experiments on mining incomplete data using both characteristic sets and maximal consistent blocks and using two interpretations of missing attribute values lost values and do not care conditions we show that there is some evidence that the best approach is using middle probabilistic approximations based on characteristic sets or on maximal consistent blocks

controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

a measurement theory view on the granularity of partitions measurement of granularity is one of the foundational issues in granular computing this paper investigates a class of measures of granularity of partitions the granularity of a set is defined by a strictly monotonic increasing transformation of the cardinality of the set the granularity of a partition is defined as the expected granularity of all blocks of the partition with respect to the probability distribution defined by the partition many existing measures of granularity are instances of the proposed class new measures of granularity of partitions are also introduced

evidence theory based numerical characterization of multigranulation rough sets in incomplete information systems multigranulation rough sets are desirable features in the field of rough set where this concept is approximated by multiple granular structures in this study we employ belief and plausibility functions from evidence theory to characterize the set approximations and attribute reductions in multigranulation rough set theory first we show that in an incomplete information system the pessimistic multigranulation approximations can be measured by belief and plausibility functions whereas the optimistic multigranulation approximations do not possess this characteristic in general we also give a sufficient and necessary condition for the numerical measurement of optimistic multigranulation approximations by belief and plausibility functions second in an incomplete decision system the pessimistic multigranulation approximations are also measured by belief and plausibility functions in the end an attribute reduction algorithm for multigranulation rough sets is proposed based on evidence theory and its efficiency is examined by an example thus belief and plausibility functions can be employed to numerically characterize the attribute reductions and to construct an attribute reduction algorithm for multigranulation rough sets

systematic studies on three way decisions with interval valued decision theoretic rough sets decision theoretic rough sets dtrs are a representative rough set model the loss function is a pivotal ingredient of dtrs which is associated with the decision maker s evaluation considering the value of loss function with the imprecise evaluation interval valued dtrs ivdtrs and its mechanism in this paper are explored first we construct a basic model of ivdtrs the comparison between dtrs and ivdtrs is discussed in the frame of ivdtrs we then focus on deriving three way decisions with the aid of two conventional methods i e a certain ranking method and a degree of possibility ranking method respectively the certain ranking method converts an interval value into single and derives decision rules under a certain risk attitude of decision maker the degree of possibility ranking method assumes the flexibility of interval and utilizes the preference between interval values all the combinations and their prerequisites are summarized in which we obtain two types of decision rules based on the above analysis we further propose an optimization method for three way decisions with ivdtrs which is designed to minimize the overall uncertainty based on the shannon entropy we also compare these methods based on standard data sets finally the criteria for choosing a suitable method to three way decisions with ivdtrs are generated these results can support decision making in the uncertain environment

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

relaxing realizability conditions for discrete time nonlinear systems this paper addresses two related topics first a nonrealizable set of input output equations can be made realizable in the classical state space form by adding a postcompensator whereas this is impossible in the continuous time case construction of such postcompensator of minimal order is presented second a general subclass of discrete time systems is suggested for which the observable subspace is non integrable the construction is based on the nonrealizable system which is realized by adding a postcompensator and then combined with the output of the original system

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

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

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

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

two bayesian approaches to rough sets abstract bayesian inference and probabilistic rough sets prss provide two methods for data analysis both of them use probabilities to express uncertainties and knowledge in data and to make inference about data many proposals have been made to combine bayesian inference and rough sets the main objective of this paper is to present a unified framework that enables us a to review and classify bayesian approaches to rough sets b to give proper perspectives of existing studies and c to examine basic ingredients and fundamental issues of bayesian approaches to rough sets by reviewing existing studies we identify two classes of bayesian approaches to prssand three fundamental issues one class is interpreted as bayesian classification rough sets which is built from decision theoretic rough set dtrs models proposed by yao wong and lingras the other class is interpreted as bayesian confirmation rough sets which is built from parameterized rough set models proposed by greco matarazzo and slowinski although the two classes share many similarities in terms of making use of bayes theorem and a pair of thresholds to produce three regions their semantic interpretations and hence intended applications are different the three fundamental issues are the computation and interpretation of thresholds the estimation of required conditional probabilities and the application of derived three regions dtrs models provide an interpretation and a method for computing a pair of thresholds according to bayesian decision theory naive bayesian rough set models give a practical technique for estimating probability based on bayes theorem and inference finally a theory of three way decisions offers a tool for building ternary classifiers the main contribution of the paper lies in weaving together existing results into a coherent study of bayesian approaches to rough sets rather than introducing new specific results

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

a local start search algorithm to compute exact hausdorff distance for arbitrary point sets this paper discovers the spatial locality feature of point sets and proposes a novel search algorithm called local start search lss to compute the exact hausdorff distance the lss algorithm can greatly reduce the running time when dealing with large scale of point set in which the spatial continuity and distance continuity are very common lss maintains high performance in both overlap and non overlap situations of a pair of regular point sets while earlybreak experiences degraded performance in overlap situations furthermore for general point sets in overlap situations the preprocess of excluding the intersection in earlybreak will fail however lss can still process the general point sets with different point sizes after ordering the sets our algorithm outperforms the state of the art algorithm experiments demonstrate the efficiency and accuracy of the proposed method the hausdorff distance hd is a very important similarity measurement in pattern recognition shape matching and artificial intelligence because of its inherent computational complexity computing the hd using the naivehd brute force algorithm is difficult especially for comparing the similarity between large scale point sets in the time of big data to overcome this problem we propose a novel efficient and general algorithm for computing the exact hd for arbitrary point sets which takes advantage of the spatial locality of point sets namely local start search lss different from the state of the art algorithm earlybreak in pami 2015 our idea comes from the observation and fact that the neighbor points of a break position in the current loop have higher probability to break the next loop than other points therefore in our algorithm we add a new mechanism to record the current break position as a start position which is initialized as search center of the next loop then lss executes the next loop by scanning the neighbor points around the center in this way lss maintains high performance in both overlap and non overlap situations furthermore the lss algorithm can process arbitrary data by adopting the morton curve to establish the order of scattered point sets while the earlybreak is mainly applied to regular data which require the same grid size such as medical images or voxel data in the non overlapping situation when comparing pairs of arbitrary point sets lss achieves performance as good as earlybreak algorithm while in the overlapping situation when comparing pairs of arbitrary point sets lss is faster than earlybreak by three orders of magnitude thus as a whole lss outperforms earlybreak in addition lss compared against the increment hausdorff distance calculation algorithm inc and significantly outperforms it by an order of magnitude faster experiments demonstrate the efficiency and accuracy of the proposed method

the fourth type of covering based rough sets as a technique for granular computing rough sets deal with the vagueness and granularity in information systems covering based rough sets have been proposed to generalize this theory for wider application three types of covering based rough sets have been studied for different situations to make the theory more complete this paper proposes a fourth type of covering based rough sets compared with the existing ones the new type shows its special characteristic in the interdependency between its lower and upper approximations we carry out a systematical study of this new theory first we discuss basic properties such as normality contraction and monotone then we investigate the conditions for this type of covering based rough sets to satisfy the properties of pawlak s rough sets and study the interdependency between the lower and upper approximation operations in addition axiomatic systems for the lower and upper approximation operations are established lastly we address the relationships between this type of covering based rough sets and the three existing ones

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

dynamic maintenance of approximations in set valued ordered decision systems under the attribute generalization set valued information systems are important type of data tables and generalized models of single valued information systems approximations are the focal point of approaches to knowledge discovery based on rough set theory which can be used to extract and represent the hidden knowledge in the form of decision rules attribute generalization refers to dynamic change of the attribute set in an information system with respect to the requirements of real life applications in this paper we focus on maintaining approximations dynamically in set valued ordered decision systems under the attribute generalization firstly a matrix based approach for computing approximations of upward and downward unions of decision classes is constructed by introducing the dominant and dominated matrices with respect to the dominance relation then incremental approaches for updating approximations are proposed which involves several modifications to relevant matrices without having to retrain from the start on all accumulated training data finally comparative experiments on data sets from uci as well as synthetic data sets show the proposed incremental updating methods are efficient and effective for dynamic attribute generalization

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

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

mgrs a multi granulation rough set the original rough set model was developed by pawlak which is mainly concerned with the approximation of sets described by a single binary relation on the universe in the view of granular computing the classical rough set theory is established through a single granulation this paper extends pawlak s rough set model to a multi granulation rough set model mgrs where the set approximations are defined by using multi equivalence relations on the universe a number of important properties of mgrs are obtained it is shown that some of the properties of pawlak s rough set theory are special instances of those of mgrs moreover several important measures such as accuracy email protected quality of email protected and precision of email protected are presented which are re interpreted in terms of a classic measure based on sets the marczewski steinhaus metric and the inclusion degree measure a concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in mgrs as well finally we discuss how to extract decision rules using mgrs unlike the decision rules and rules from pawlak s rough set model the form of decision rules in mgrs is or several pivotal algorithms are also designed which are helpful for applying this theory to practical issues the multi granulation rough set model provides an effective approach for problem solving in the context of multi granulations

single experiment observability decomposition of discrete time analytic systems this paper addresses the single experiment observability decomposition of discrete time analytic systems unlike the continuous time case there exist systems which cannot be decomposed into observable and unobservable subsystems due to the fact that the observable space is not integrable in this paper a necessary and sufficient condition for integrability of observable space is given as a corollary of this condition it is proven that if the system is reversible the observability decomposition can be always achieved moreover integrability of observable space is also addressed for delta domain models of non uniformly sampled systems

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

linearization of discrete time systems the algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete time nonlinear systems the accessibility problem for nonlinear discrete time systems is shown to be easy to tackle by means of standard linear algebraic tools whereas this is not the case for nonlinear continuous time systems in which case the most suitable approach is provided by differential geometry the feedback linearization problem for discrete time systems is recasted through the language of differential forms in the event that a system is not feedback linearizable the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a pfaffian system a discrete time system may be linearizable by dynamic state feedback though it is not linearizable by static state feedback necessary and sufficient conditions are given for the existence of a so called linearizing output which in turn is a sufficient condition for dynamic state feedback linearizability

on the topological properties of generalized rough sets in this paper we consider some topological properties of generalized rough sets induced by binary relations and show that1 any serial binary relation can induce a topology 2 let r be a binary relation on a universe u t r and e r denote the transitive closure and the equivalence closure of r respectively if r is a reflexive relation on u then r and t r induce the same topology i e t r t t r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators t r and t r respectively moreover r t r t r where r t r is the relation induced by the topology t r 3 when r is a reflexive and symmetric relation r and e r induce the same topology i e t r t e r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators e r and e r respectively moreover r t r e r 4 based on the above conclusions the notion of topological reduction of incomplete information systems is proposed and characterizations of reduction of consistent incomplete decision tables are obtained

incremental updating of rough approximations in interval valued information systems under attribute generalization interval valued information system ivis is a generalized model of single valued information system in which the attribute values of objects are all interval values instead of single values the attribute set in ivis is not static but rather dynamically changing over time with the collection of new information which results in the continuous updating of rough approximations for rough set based data analysis in this paper on the basis of the similarity based rough set model in ivis we develop incremental approaches for updating rough approximations in ivis under attribute generalization which refers to the dynamic changing of attributes firstly increment relationships between the original rough approximations and the updated ones when adding or deleting an attribute set are analyzed respectively and the incremental mechanisms for updating rough approximations in ivis are introduced which carry out the computation using the previous results from the original data set along with new results then the corresponding incremental algorithms are designed based on the proposed mechanisms finally comparative experiments on data sets from uci as well as artificial data sets are conducted respectively experimental results show that the proposed incremental algorithms can effectively reduce the running time for the computation of rough approximations in comparison with the static algorithm

an application of covering approximation spaces on network security based on the pawlak rough set theory this paper investigates separations in covering approximation spaces give some characterizations of these separations and some relations among these separations as an application of these results investigations on network security are converted into investigations on separations in covering approximation spaces by taking covering approximation spaces as mathematical models of networks results of this paper give further applications of the pawlak rough set theory in pattern recognition and artificial intelligence which makes it possible to research network security by logical methods and mathematical methods in computer science this contributes to giving risk assessments of securities and to raise grades of securities for networks

brief paper on the observability of discrete time dynamic systems a geometric approach a coordinate free description for dynamic systems described by explicit nonlinear difference equations in one independent variable via a differential geometric framework is presented based on this covariant approach suitable geometric objects for discrete time dynamics are introduced especially the observability along a trajectory is discussed and transformations to normal forms are derived in addition the obtained local observability criteria can be checked by computer algebra algorithms some examples illustrate the proposed approach

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

positive approximation and converse approximation in interval valued fuzzy rough sets methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval valued fuzzy information systems thus this paper deals with such systems instead of obtaining rules by attribute reduction which may have a negative effect on inducting good rules the objective of this paper is to extract rules without computing attribute reducts the data completeness of missing attribute values is first presented positive and converse approximations in interval valued fuzzy rough sets are then defined and their important properties are discussed two algorithms based on positive and converse approximations namely mine rules based on the positive approximation mrbpa and mine rules based on the converse approximation mrbca are proposed for rule extraction the two algorithms are evaluated by several data sets from the uc irvine machine learning repository the experimental results show that mrbpa and mrbca achieve better classification performances than the method based on attribute reduction

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

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

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

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

robust point set registration using gaussian mixture models in this paper we present a unified framework for the rigid and nonrigid point set registration problem in the presence of significant amounts of noise and outliers the key idea of this registration framework is to represent the input point sets using gaussian mixture models then the problem of point set registration is reformulated as the problem of aligning two gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized we show that the popular iterative closest point icp method and several existing point set registration methods in the field are closely related and can be reinterpreted meaningfully in our general framework our instantiation of this general framework is based on the the l2 distance between two gaussian mixtures which has the closed form expression and in turn leads to a computationally efficient registration algorithm the resulting registration algorithm exhibits inherent statistical robustness has an intuitive interpretation and is simple to implement we also provide theoretical and experimental comparisons with other robust methods for point set registration

approximate distribution reducts in inconsistent interval valued ordered decision tables abstract many methods based on the rough set theory to deal with information systems have been proposed in recent decades in practice some information systems are based on dominance relations and may be inconsistent because of various factors moreover taking the imprecise evaluations and assignments in the description of objects into account single valued information systems have been generalized to interval valued information systems in this paper by introducing a dominance relation to interval valued ordered information systems we establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation to extract the minimal decision rules approximate distribution reducts are proposed in inconsistent interval valued ordered decision tables this paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct and two equivalent definitions of approximate distribution reducts are also introduced in addition numerical examples are employed to examine the validity of the approaches proposed in this paper

further results on the uniform observability of discrete time nonlinear systems in this technical note the relation between notions of uniform observability of discrete time nonlinear systems based on injectivity of an observation map window the full rankness of its jacobian and a k function is investigated it is proven that a system is uniformly observable in the sense of injectivity of the observation map together with the full rankness of its jacobian if and only if it is so in some k function senses

an efficient algorithm for calculating the exact hausdorff distance the hausdorff distance hd between two point sets is a commonly used dissimilarity measure for comparing point sets and image segmentations especially when very large point sets are compared using the hd for example when evaluating magnetic resonance volume segmentations or when the underlying applications are based on time critical tasks like motion detection then the computational complexity of hd algorithms becomes an important issue in this paper we propose a novel efficient algorithm for computing the exact hausdorff distance in a runtime analysis the proposed algorithm is demonstrated to have nearly linear complexity furthermore it has efficient performance for large point set sizes as well as for large grid size performs equally for sparse and dense point sets and finally it is general without restrictions on the characteristics of the point set the proposed algorithm is tested against the hd algorithm of the widely used national library of medicine insight segmentation and registration toolkit itk using magnetic resonance volumes with extremely large size the proposed algorithm outperforms the itk hd algorithm both in speed and memory required in an experiment using trajectories from a road network the proposed algorithm significantly outperforms an hd algorithm based on r trees

pruning method for optimal solutions of intm intn decision making scheme abstract this paper aims to find a faster method for optimal solutions of feng et al s int m int n decision making scheme we first give theoretical characterizations of optimal decision sets then we develop a pruning method which filters out those objects that cannot be elements of any optimal decision sets in the beginning experimental results have shown that our method has higher efficiency in computing the optimal solutions of this scheme particularly when we are processing soft sets with a great quantity of data

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

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