test cost sensitive multigranulation rough set model and minimal cost selection abstract multigranulation rough set is an expansion of the classical rough set by using multiple granular structures presently three important multigranulation rough sets have been proposed they are optimistic pessimistic and multigranulation approaches however such three multigranulation rough sets do not take the test cost into consideration which is an important issue in both data mining and machine learning to solve such problem we propose a test cost sensitive multigranulation rough set model in this paper we show that test cost sensitive multigranulation rough set is a generalization of optimistic pessimistic and multigranulation rough sets furthermore it is found that the traditional heuristic algorithm is not suitable for granular structure selection with lower test cost we then propose a backtracking algorithm for granular structure selection with minimal test cost the algorithms are tested on ten uci university of california irvine data sets experimental results show the effectiveness of backtracking algorithm by comparing with heuristic algorithm this study suggests potential application areas and new research trends concerning multigranulation rough set theory

parametric output feedback controller design abstract a parameterization of fixed gain output feedback controllers which will assign a complete set of distinct eigenvalues to a given state space system is presented the main result is a compact parametric expression for the output feedback controller gain matrix explicitly characterized by two sets of free parameters the set of n closed loop system eigenvalues and a set of mr n design parameters for n state r input m output systems the free design parameters are determined by a sequential design procedure which automatically satisfies the basic orthogonality constraints on the left and right closed loop eigenvectors under the condition m r n the principal benefits of the explicit characterization of a parametric class of output feedback controllers lie in the ability to directly accommodate various different design criteria an illustration of the parametric output feedback design procedure is given by way of approximating the optimal full state regulator response of a fifth order pilot plant evaporator model

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

automatic determination about precision parameter value based on inclusion degree with variable precision rough set model the rough set theory provides a powerful approach for attributes reduction and data analysis the variable precision rough set vprs model an extension of the original rough set approach tolerates misclassifications of the training data to some degree which promotes the applications of rough set theory in inconsistent information systems however in most existing algorithms of feature reduction based on vprs the precision parameter is introduced as prior knowledge which restricts their applications because it is not clear how to set the value by studying consistency in the measurement of a decision table and the threshold value of the consistent decision table this paper presents an algorithm for automatic determination of the precision parameter value from a decision table based on vprs at the same time the precision parameter value from our proposed method is compared with the thresholds from the decision theoretic rough set dtrs the influences of the precision parameter are also discussed on attribute reduction which shows the necessity of the estimated precision parameter from a decision table the simulation results including vprs and other classification methods in real data further indicate that different precision parameter values make a great difference on rules and setting a precise parameter near the threshold value of the consistent decision table can precisely reflect the decision distribution of the decision table

a decision theoretic rough set approach for dynamic data mining uncertainty and fuzziness generally exist in real life data approximations are employed to describe the uncertain information approximately in rough set theory certain and uncertain rules are induced directly from different regions partitioned by approximations approximation can further be applied to data mining related task hbox e g attribute reduction nowadays different types of data collected from different applications evolve with time especially new attributes may appear while new objects are added this paper presents an approach for dynamic maintenance of approximations hbox w r t objects and attributes added simultaneously under the framework of decision theoretic rough set dtrs equivalence feature vector and matrix are defined first to update approximations of dtrs in different levels of granularity then the information system is decomposed into subspaces and the equivalence feature matrix is updated in different subspaces incrementally finally the approximations of dtrs are renewed during the process of updating the equivalence feature matrix extensive experimental results verify the effectiveness of the proposed methods

three way group decisions with decision theoretic rough sets gdm based three way decisions as an extension model of dtrss is proposed gdm provides a new semantic interpretation for three way decisions our method explains the imprecise origin of the existing literatures on consideration of the effectiveness of group decision making gdm in practical complex decision problems we introduce gdm into three way decisions with decision theoretic rough sets dtrss and propose gdm based three way decisions gdm based three way decisions extend the range of applications of three way decisions with dtrss and provide a novel interpretation of the determination of loss functions based on dtrss we firstly focus on analyzing the determination for the loss functions under the gdm environment with the aid of the principle of justifiable granularity we adopt the important and majority suggestions of experts to measure each loss function which supports a coherent way of designing information granules in presence of numerics in this case the loss functions are determined in the form of interval valued information granule by using the interval comparison method we further deduce the three way decisions and design a corresponding decision procedure of gdm based three way decisions then an example of strategy supply selection is given to elaborate the gdm based three way decisions finally we validate the performance of our proposed method by experimental analysis

a novel risk decision making based on decision theoretic rough sets under hesitant fuzzy information decision theoretic rough sets dtrss play a crucial role in risk decision making problems with respect to the minimum expected risk dtrss deduce the rules of three way decisions considering the new expression of evaluation information with hesitant fuzzy sets hfss we introduce hfss into dtrss and explore their decision mechanisms more specifically we take into account the losses of dtrss with hesitant fuzzy elements and propose a new model of hesitant fuzzy decision theoretic rough sets hfdtrss some properties of the expected losses and their corresponding scores are carefully investigated under the hesitant fuzzy information three way decisions and the associated cost of each object are further derived with the above analysis a novel risk decision making method with the aid of hfdtrss is developed besides the three way decisions with dtrss the method investigates the ranking and resource allocation by utilizing the associated costs of alternatives and multiobjective 0 1 integer programming our study also offers a solution in the aspect of determining losses of dtrs and extends the range of applications

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

reduction target structure based hierarchical attribute reduction for two category decision theoretic rough sets abstract attribute reduction is an essential subject in rough set theory but because of quantitative extension it becomes a problem when considering probabilistic rough set prs approaches the decision theoretic rough set dtrs has a threshold semantics and decision feature and thus becomes a typical and fundamental prs based on reduction target structures this paper investigates hierarchical attribute reduction for a two category dtrs and is divided into five parts 1 the knowledge preservation property and reduct are explored by knowledge coarsening 2 the consistency preservation principle and reduct are constructed by a consistency mechanism 3 region preservation is analyzed and the separability between consistency preservation and region preservation is concluded thus the double preservation principle and reduct are studied 4 structure targets are proposed by knowledge structures and an attribute reduction is further described and simulated thus general reducts are defined to preserve the structure targets or optimal measures 5 the hierarchical relationships of the relevant four targets and reducts are analyzed and a decision table example is provided for illustration this study offers promotion rationality structure hierarchy and generalization and it establishes a fundamental reduction framework for two category dtrs the relevant results also provide some new insights into the attribute reduction problem for prs

attribute reduction of data with error ranges and test costs in data mining applications we have a number of measurement methods to obtain a data item with different test costs and different error ranges test costs refer to time money or other resources spent in obtaining data items related to some object observational errors correspond to differences in measured and true value of a data item in supervised learning we need to decide which data items to obtain and which measurement methods to employ so as to minimize the total test cost and help in constructing classifiers this paper studies this problem in four steps first data models are built to address error ranges and test costs second error range based covering rough set is constructed to define lower and upper approximations positive regions and relative reducts a closely related theory deals with neighborhood rough set which has been successfully applied to heterogeneous attribute reduction the major difference between the two theories is the definition of neighborhood third the minimal test cost attribute reduction problem is redefined in the new theory fourth both backtrack and heuristic algorithms are proposed to deal with the new problem the algorithms are tested on ten uci university of california irvine datasets experimental results show that the backtrack algorithm is efficient on rational sized datasets the weighting mechanism for the heuristic information is effective and the competition approach can improve the quality of the result significantly this study suggests new research trends concerning attribute reduction and covering rough set

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

a new class of short distance universal amplitude ratios we propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point we will describe the critical system with a conformal field theory uv fixed point perturbed by an appropriate relevant operator in two dimensions the exact knowledge of the uv fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly in three dimensional o n scalar systems feasible extensions of some existing results should allow perturbative expansions for the ratios to be obtained by construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one point functions are not accessible in experiments or simulations

probabilistic model criteria with decision theoretic rough sets in dealing with risk in real decision problems decision theoretic rough sets with loss functions aim to obtain optimization decisions by minimizing the overall risk with bayesian decision procedures two parameters generated by loss functions divide the universe into three regions as the decision of acceptance deferment and rejection in this paper we discuss the semantics of loss functions and utilize the differences of losses replace actual losses to construct a new four level approach of probabilistic rules choosing criteria ten types of probabilistic rough set models can be generated by the four level approach and form two groups of models two way probabilistic decision models and three way probabilistic decision models a reasonable decision with these criteria is demonstrated by an illustration of oil investment

an expanded double quantitative model regarding probabilities and grades and its hierarchical double quantitative attribute reduction a double quantitative model on probabilities and grades is proposed by integration the model with quantitative semantics expands dtrs model grs model pawlak model model regions are hierarchically and optimally calculated via two algorithms three reduct types srp reduct crp reduct app reduct are hierarchically studied the double quantitative reducts expand basic qualitative and quantitative reducts probabilities and grades serve as relative and absolute measures respectively they are used to establish the decision theoretic rough set dtrs and graded rough set grs two basic quantitative models the double quantification of probabilities and grades exhibits systematicness and completeness in view of the two dimensional feature of the approximate space however double quantitative construction becomes a problem and double quantitative reduction is rarely reported thus this paper mainly constructs an expanded double quantitative model by logically integrating probabilities and grades it further studies relevant double quantitative reduction by hierarchically preserving specific regions 1 first a novel model is established via the logic integration and expansion requirement and its regional system and granular hierarchy are studied via granular computing thus regional semantics is extracted via basic semantics granules regional calculation is realized by two algorithms and the algorithm regarding calculation granules exhibits optimization according to algorithm analyses 2 second three types of model regional preservation reducts and their hierarchy are discussed in the two category case thus srp reduct crp reduct and app reduct are studied by exploring four region preservation properties constructing two region classification regions and preserving four original approximations respectively furthermore a relevant reduction hierarchy is thoroughly achieved 3 moreover the model and its reduction are illustrated by two examples of decision tables the constructional model conducts double quantification regarding probabilities and grades thus it exhibits double quantitative semantics and benignly expands dtrs model grs model and pawlak model furthermore its hierarchical reduction reflects some double quantitative reduction essence thus its reduction expands qualitative pawlak reduction while guides quantitative dtrs reduction and grs reduction

comparison of the probabilistic approximate classification and the fuzzy set model abstract it is shown that the generalized notion probabilistic approximate classification of rough sets can be conveniently described by the concept of fuzzy sets a discussion of the proper choice of the definition for the membership function of the intersection union of fuzzy sets is also presented however from the point of view of the probabilistic approximation space it is argued that there does not exist a universal definition for the fuzzy intersection union operation

double quantitative rough fuzzy set based decisions as two important expanded quantification rough set models the probabilistic rough set prs model and the graded rough set grs model are used to measure relative quantitative information and absolute quantitative information between the equivalence classes and a basic concept respectively the decision theoretic rough set dtrs model is a special case of prs model which mainly utilizes the conditional probability to express relative quantification since the fuzzy concept is more general than classical concept in real life how to make decision for a fuzzy concept using relative and absolute quantitative information is becoming a hot topic in this paper a couple of double quantitative decision theoretic rough fuzzy set dq dtrfs models based on logical conjunction and logical disjunction operation are proposed furthermore we discuss decision rules and the inner relationship between these two models then an experiment in the medical diagnosis is studied to support the theories finally to apply our methods to solve a pattern recognition problem in big data experiments on data sets downloaded from uci are conducted to test the proposed models in addition we also offer a comparative analysis using two non rough set based methods from the results obtained one finds that the proposed method is efficient for dealing with practical issues

interpretation of equilibria in game theoretic rough sets the gtrs provides a mechanism for determining a pair of thresholds in the probabilistic rough set model the equilibria play a critical role in determining thresholds in the gtrs model we address two equilibria related issues with the gtrs model the first issue is the interpretation of equilibria the second issue in the establishment of equilibria intelligent decision making models aim at improving the quality of decision making under uncertainty the fundamental issues that are generally encountered in these models are too many options to choose from and the involvement of contradictory decision making criteria the game theoretic rough set gtrs model provides an intelligent decision making mechanism that exploits a game theoretic environment for analyzing strategic situations between cooperative or conflicting decision making criteria in the probabilistic rough set framework the concept of equilibria is of central importance in the gtrs model which has not been sufficiently addressed in the current literature two key issues in this regard are the interpretation of equilibria and the establishment of their existence by reviewing examining and defining the basic game constructs in the gtrs model we are able to interpret an equilibrium in terms of the decision thresholds that control the rough sets based decision regions in particular an equilibrium is defined in terms of a pair of thresholds such that no player has a unilateral incentive to change these thresholds within the game an example game is considered to demonstrate the use of the interpretation in determining the thresholds the issue of existence of equilibria is addressed by considering a couple of typical two player games in the gtrs model the results suggest that the existence of equilibria may be established under certain limited conditions

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

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

criteria for choosing a rough set model one of the challenges a decision maker faces in using rough sets is to choose a suitable rough set model for data analysis we investigate how two rough set models the pawlak model and the probabilistic model influence the decision goals of a user two approaches use probabilities to define regions in the probabilistic model these approaches use either user defined parameters or derive the probability thresholds from the cost associated with making a classification by determining the implications of the results obtained from these models and approaches we observe that the availability of information regarding the analysis data is crucial for selecting a suitable rough set approach we present a list of decision types corresponding to the available information and user needs these results may help a user match their decision requirements and expectations to the model which fulfills these needs

decision theoretic rough fuzzy set model and application this article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space we propose the probabilistic rough fuzzy set by defining the conditional probability of a fuzzy event then we establish the model of probabilistic rough fuzzy set and discuss several properties in detail furthermore three generalizations of probabilistic rough fuzzy set namely 0 5 probabilistic rough fuzzy set variable precision probabilistic rough fuzzy set and bayesian rough fuzzy set are reported in order to give a systematic method of selecting parameters for the probabilistic rough fuzzy set we propose a decision theoretic rough fuzzy set that is we formulate a non parametric definition of the probabilistic rough fuzzy set moreover we illustrate the motivation and verify the validity of the decision theoretic rough fuzzy set by using a credit card applicant decision making problem furthermore the interrelationship between the decision theoretic rough fuzzy set and the probabilistic rough fuzzy set is explained the main contribution of this paper is twofold one is to extend the probabilistic rough set to fuzzy environment i e the probabilistic rough fuzzy set model another is to present an approach to select parameters needed in probabilistic rough fuzzy set modeling by using the process of decision making under conditions of risk

three way decisions with probabilistic rough sets the rough set theory approximates a concept by three regions namely the positive boundary and negative regions rules constructed from the three regions are associated with different actions and decisions which immediately leads to the notion of three way decision rules a positive rule makes a decision of acceptance a negative rule makes a decision of rejection and a boundary rule makes a decision of abstaining this paper provides an analysis of three way decision rules in the classical rough set model and the decision theoretic rough set model the results enrich the rough set theory by ideas from bayesian decision theory and hypothesis testing in statistics the connections established between the levels of tolerance for errors and costs of incorrect decisions make the rough set theory practical in applications

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

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

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

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones