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Abstract The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC -, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC -space (U , C) and a nonempty set X (⊆ U). Using a reduction of a given LFC -space (U , C) , the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC -spaces (U , C) , where each of the cardinalities of the sets U , C , and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC -spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X (⊆ U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design. Highlights • An analytical type-reduction method for spiked concave type-2 fuzzy sets has been proposed. • The analytical type-reduction method is free of much of the discourse partition and discourse refinement. • The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets' type reduction problem. [ABSTRACT FROM AUTHOR]

Abstract The rough set theory is closely related with evidence theory. It has been demonstrated that certain rough approximation spaces are associated with certain belief structures such that the lower and upper approximation operations induced by the approximation spaces may be used to interpret the corresponding belief and plausibility functions induced by the belief structures. This paper studies the belief and plausibility functions of type-2 fuzzy rough sets. First, measurable interval type-2 fuzzy sets are defined and a new probability measure of interval type-2 fuzzy sets is proposed using the integral operation. Then the probability measure of type-2 fuzzy sets is given based on α -plane representation of type-2 fuzzy sets. Finally, belief and plausibility functions of type-2 fuzzy rough sets are defined and an example is employed to illustrate their applications. [ABSTRACT FROM AUTHOR]

Abstract Fuzzy objective relation systems are an important class of datasets that are generalizations of many types of decision tables. This paper proposes an approach, based on relation systems and fuzzy sets, to reduce data redundancy in fuzzy objective relation systems. We study lower and upper approximation reductions of a relation system for a given fuzzy set. As a generalization of such reductions, we consider lower and upper approximation reductions of fuzzy objective relation systems and give their corresponding reduction algorithms using methods based on the discernibility matrix. We note that the usual positive region reduction for a decision table can be considered a special case of our lower approximation reduction. Finally, we provide two examples from the UCI datasets to verify our theoretical results. These results can help in the decision making analysis of fuzzy objective relation systems. [ABSTRACT FROM AUTHOR]

Abstract Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant, with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. This paper provides a practically applicable method that allows calculating an upper bound for the maximal error induced by approximating a coherent lower probability with its values on a finite set of gambles. An algorithm is also provided with an estimation of its computational complexity. [ABSTRACT FROM AUTHOR]

Abstract In covering-based rough sets, many basic issues are related to minimal and maximal descriptions. For a covering approximation space with a large cardinal, it would be tedious and complicated to use set representations to solve the issues about minimal and maximal descriptions. Therefore it is necessary to study matrix representations by which calculations will become algorithmic and can be easily implemented by computers. In this paper, we use matrix approaches to investigate some issues about minimal and maximal descriptions in covering-based rough sets. Firstly, several new matrices and matrix operations are defined by which one can compute the minimal description and the approximation operator about the minimal description. Inspired by this result, a matrix method is presented to judge whether a covering is unary. Secondly, the maximal description and the approximation operator about the maximal description are also computed by these new matrices and matrix operations. Finally, based on the above results, we propose a matrix method to efficiently compute reductions of coverings. Moreover, we propose two types of reducts of covering information systems via minimal and maximal descriptions respectively, and we use matrix approaches to compute them. In a word, these results show an interesting view to investigate the combination between matrices and covering-based rough sets. [ABSTRACT FROM AUTHOR]

Abstract Label ambiguity and data complexity are widely recognized as major challenges in multi-label classification. Existing studies strive to find approximate representations concerning label semantics, however, most of them are predefined, neglecting the personality of instance-label pair. To circumvent this drawback, this paper proposes a three-way selective ensemble (TSEN) model. In this model, three-way decisions is responsible for minimizing uncertainty, whereas ensemble learning is in charge of optimizing label associations. Both label ambiguity and data complexity are firstly reduced, which is realized by a modified probabilistic rough set. For reductions with shared attributes, we further promote the prediction performance by an ensemble strategy. The components in base classifiers are label-specific, and the voting results of instance-based level are utilized for tri-partition. Positive and negative decisions are determined directly, whereas the deferment region is determined by label-specific reduction. Empirical studies on a collection of benchmarks demonstrate that TSEN achieves competitive performance against state-of-the-art multi-label classification algorithms. [ABSTRACT FROM AUTHOR]

Abstract The axiomatization problems for generalized approximation operators are finding logical characters of these operators. Solving these problems is important not only for conceptual understanding generalized approximation operators, but also for theoretic discussing the properties of them. In recent years, many researchers explored and developed the axiomatic approach of generalized rough set theory and a lot of articles about axiomatizations of approximation operators were published. However, there are numbers of axiomatization problems for covering-based approximation operators are still open. In this article, we give axiomatic systems for several types of covering-based approximation operators. All the axiomatic systems given by us are original and one of our results answers an open problem raised by Zhu and Wang in 2007. We also discuss axiomatization problems for those covering-based approximation operators generated by irreducible coverings. We prove that for most types of operators, the axiomatic systems generated by irreducible coverings and by general coverings are the same. On the other hand, we show that for one type of operators, the axiomatic systems generated by irreducible coverings and by general coverings are different, and we give an axiomatic system for this type of operators generated by irreducible coverings. By several examples, we show that for each axiomatic system presented in this paper, conditions are independent of each other. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the quasi-distributivity laws of type-2 fuzzy implications with respect to extended supremum and extended infimum are investigated on different subalgebras of fuzzy truth values. Moreover, the properties of fuzzy-valued fuzzy implications are further discussed. Especially, new fuzzy-valued operations are obtained from a fuzzy-valued fuzzy implication induced by a fuzzy implication, which is right-continuous with respect to the second argument. As an application of fuzzy-valued fuzzy implications, fuzzy-valued approximate reasoning is further studied and a numerical example is given. Highlights • We discuss the quasi-distributivity laws of type-2 fuzzy implications with respect to extended supremum and extended infimum. • New fuzzy-valued operations are obtained from induced fuzzy implications by the residuation principle. • We further discuss fuzzy-valued approximate reasoning with arbitrary t-norms and fuzzy implications. [ABSTRACT FROM AUTHOR]

In covering rough set theory, the basic problem is the calculation of lower and upper approximations for subsets of a covering approximation space. For a covering approximation space with a large cardinal, using definitions to calculate approximations would be tedious and complicated. So it is important to investigate matrix methods by which calculations will become algorithmic and can be easily implemented by computers. Since the covering in a covering approximation space can be represented by a Boolean matrix, every covering rough approximation operator must be closely related to this matrix. In this paper, we investigated 32 pairs of neighborhood-based dual lower and upper approximation operators and successfully obtained all the matrix representations of them. Some examples were presented in the manuscript to illustrate how to use matrix methods to compute lower and upper approximations and examples were also given to solve the inverse problem, i.e., finding all the subsets whose lower or upper approximation is a given subset. [ABSTRACT FROM AUTHOR]

In this paper a class of discrete optimization problems with uncertain costs is discussed. The uncertainty is modeled by introducing a scenario set containing a finite number of cost scenarios. A probability distribution over the set of scenarios is available. In order to choose a solution the weighted OWA criterion (WOWA) is applied. This criterion allows decision makers to take into account both probabilities for scenarios and the degree of pessimism/optimism. In this paper the complexity of the considered class of discrete optimization problems is described and some exact and approximation algorithms for solving it are proposed. Applications to the selection and the assignment problems, together with results of computational tests are shown. [ABSTRACT FROM AUTHOR]

The concept of coverings is one of the fundamental concepts in topological spaces and plays a big part in the study of topological problems. This motivates the research of covering rough sets from topological points of view. From topological points of view, we can get a good insight into the essence of covering rough sets and make our discussions concise and profound. In this paper, we first construct a type of topology called the topology induced by the covering on a covering approximation space. This notion is indeed in the core of this paper. Then we use it to define the concepts of neighborhoods, closures, connected spaces, and components. Drawing on these concepts, we define several pairs of approximation operators. We not only investigate the relationships among them, but also give clear explanations of the concepts discussed in this paper. For a given covering approximation space, we can use the topology induced by the covering to investigate the topological properties of the space such as separation, connectedness, etc. Finally, a diagram is presented to show that the collection of all the lower and upper approximations considered in this paper constructs a lattice in terms of the inclusion relation ⊆. [ABSTRACT FROM AUTHOR]

Complementary neighborhood is a conception analogous to the neighborhood that we first introduced in a former paper. In this paper, we show that two different approximation operators may have a close relationship, namely, they can be defined almost in the same way except that one uses the notion of neighborhood and another uses the complementary neighborhood. We call such two approximation operators the twin approximation operators. We give some concrete examples of the twin approximation operators. Through detailed investigation on the relationship between the neighborhood and the complementary neighborhood, we further study the properties of given twin approximation operators and investigate the relationships between different twins. We also reveal the topological properties of those twin approximation operators. [ABSTRACT FROM AUTHOR]

Abstract: Two basic inference models of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). The Triple I method is a very important method to solve the problems of FMP and FMT. The aim of this paper is to extend the Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets. In the paper, we first investigate the algebra operators' properties on the lattice structure of intuitionistic fuzzy information and provide the unified form of residual implications which indicates the relationship between intuitionistic fuzzy implications and fuzzy implications. Then we present the intuitionistic fuzzy reasoning version of the Triple I principles based on the models of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT) and give the Triple I method of intuitionistic fuzzy reasoning for residual implications. Moreover, we discuss the reductivity of the Triple I methods for IFMP and IFMT. Finally, we propose α-Triple I method of intuitionistic fuzzy reasoning. [Copyright &y& Elsevier]

In social network analysis, relational structures play a crucial role in the processing of complex data. This paper proposes a fuzzy relational structure which consists of a non-empty universal set and a set of fuzzy relations of finite arity. In order to deduce knowledge hidden in a fuzzy relational structure we present the concept of bisimulations as indiscernibility with respect to the fuzzy relational structure. Furthermore, we give a method of computing the largest bisimulations over fuzzy relational structures and present computational examples of the method. Finally, the fuzzy rough set analysis of fuzzy relational structures is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the matrix representations and interdependency of three pairs of L -fuzzy covering-based approximation operators. Matrices play an important role in machine learning and data mining, such as the Laplacian matrix of a graph in clustering analysis and matrix linear transformation in feature selection based on principle component analysis. Therefore, the matrix representations of the three pairs of L -fuzzy covering-based approximation operators are important problems which should be solved. Moreover, we give some necessary and sufficient conditions under which two L -fuzzy coverings to generate the same L -fuzzy covering-based approximation operators. [ABSTRACT FROM AUTHOR]

Recently, a fundamental study on measurement of data quality introduced an ordinal-scaled procedure of measurement. Besides the pure ordinal information about the level of quality, numerical information is induced when considering uncertainty involved during measurement. In the case where uncertainty is modelled as probability, this numerical information is ratio-scaled. An essential property of the mentioned approach is that the application of a measure on a large collection of data can be represented efficiently in the sense that (i) the representation has a low storage complexity and (ii) it can be updated incrementally when new data are observed. However, this property only holds when the evaluation of predicates is clear and does not deal with uncertainty. For some dimensions of quality, this assumption is far too strong and uncertainty comes into play almost naturally. In this paper, we investigate how the presence of uncertainty influences the efficiency of a measurement procedure. Hereby, we focus specifically on the case where uncertainty is caused by insufficient information and is thus modelled by means of possibility theory. It is shown that the amount of data that reaches a certain level of quality, can be summarized as a possibility distribution over the set of natural numbers. We investigate an approximation of this distribution that has a controllable loss of information, allows for incremental updates and exhibits a low space complexity. [ABSTRACT FROM AUTHOR]

To improve the quality of approximate reasoning in fuzzy system, quintuple implication principle (QIP) to resolve fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) problems has been proposed by Zhou et al. [32] . The same approximation as Mamdani-type fuzzy inference can be reached by fuzzy reasoning with QIP method for Gödel implication. It therefore is essential to establish some fundamental properties of fuzzy inference system with QIP method. This paper mainly investigates the robustness and universal approximation capability of fuzzy inference system with QIP method. Firstly, we present the QIP solutions of FMP and FMT for R-, S-, QL-, f - and g -implications. And then the robustness of fuzzy inference system with QIP method is discussed. Finally, we study the universal approximation properties of multiple-input and single-output (MISO) fuzzy systems with QIP method for R-, S- and QL-implications. These results reveal that the QIP method possesses better performance in fuzzy rule-based system. [ABSTRACT FROM AUTHOR]

Probabilistic rough set approximations are proposed based on a conditional probability to describe the desired levels of precision between the equivalence classes and an approximated set. This definition shows the detailed information on individuals satisfying some conditions but ignores the structural information. In this paper, applying the structured granules in a coarsened-grained universe, we introduce structured probabilistic rough set approximations between the power sets of the original universe and the coarsened-grained universe. By using the zooming-in and structured probabilistic rough approximation operators, two pairs of probabilistic rough lower and upper approximations on the same universe are investigated. Related properties and relationships of them are investigated. Furthermore, applying the Bayesian decision procedure, conditional probability and loss functions, three-way classifications in structured probabilistic rough set approximations are then proposed to classify the structured granules of the coarsened-grained universe into three disjoint structured probabilistic regions. This method gives the values of the pair of thresholds. Meanwhile, by using the minimum-risk decision rules, we also can construct the structured probabilistic rough lower and upper approximations. Finally, we discuss the monotonicity of structured probabilistic positive and negative regions. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to establishing a kind of graded reasoning method in the context of rough logic. To this end, a weak form of deduction theorem in rough logic is firstly obtained, then, based upon the weak deduction theorem and the notion of rough truth degree, a new kind of graded reasoning method in rough logic is presented. Moreover, to embody the idea of rough approximations, the notions of graded rough upper consequence and graded rough lower consequence are also proposed, which can be treated as the logical counterpart of rough upper and lower approximation, respectively. Compared with the existing graded reasoning method, the proposed method in the present paper does not employ the notion of rough similarity degree, and hence their fundamental starting points are different, however, they are also closely related, accordingly, a comparative study is performed between these two different graded reasoning methods. Lastly, based on the proposed graded reasoning method, the notions of rough (upper, lower) consistency degree are also proposed and their properties are investigated in detail. [ABSTRACT FROM AUTHOR]

Rule-based argumentation systems are developed for reasoning about defeasible information. They take as input a theory made of a set of facts , a set of strict rules , which encode strict information, and a set of defeasible rules which describe general behavior with exceptional cases. They build arguments by chaining such rules, define attacks between them, use a semantics for evaluating the arguments, and finally identify the plausible conclusions that follow from the theory. Undercutting is one of the main attack relations of such systems. It consists of blocking the application of defeasible rules when their exceptional cases hold. In this paper, we consider this relation for capturing all the different conflicts in a theory. We present the first argumentation system that uses only undercutting, and show that it satisfies the rationality postulates proposed in the literature. Finally, we fully characterize both its extensions and its plausible conclusions under various acceptability semantics. Indeed, we show full correspondences between extensions and sub-theories of the theory under which the argumentation system is built. [ABSTRACT FROM AUTHOR]

Lifted inference aims at answering queries from statistical relational models by reasoning on populations of individuals as a whole instead of considering each individual singularly. Since the initial proposal by David Poole in 2003, many lifted inference techniques have appeared, by lifting different algorithms or using approximation involving different kinds of models, including parfactor graphs and Markov Logic Networks. Very recently lifted inference was applied to Probabilistic Logic Programming (PLP) under the distribution semantics, with proposals such as L P 2 and Weighted First-Order Model Counting (WFOMC). Moreover, techniques for dealing with aggregation parfactors can be directly applied to PLP. In this paper we survey these approaches and present an experimental comparison on five models. The results show that WFOMC outperforms the other approaches, being able to exploit more symmetries. [ABSTRACT FROM AUTHOR]

The conditional independencies from a joint probability distribution constitute a model which is closed under the semi-graphoid properties of independency. These models typically are exponentially large in size and cannot be feasibly enumerated. For describing a semi-graphoid model therefore, researchers have proposed a more concise representation. This representation is composed of a representative subset of the independencies involved, called a basis, and lets all other independencies be implicitly defined by the semi-graphoid properties. An algorithm is available for computing such a basis for a semi-graphoid independency model. In this paper, we identify some new properties of a basis in general which can be exploited for arriving at an even more concise representation of a semi-graphoid model. Based upon these properties, we present an enhanced algorithm for basis construction which never returns a larger basis for a given independency model than currently existing algorithms. [ABSTRACT FROM AUTHOR]

Covering rough sets are an important extension of Pawlak rough sets. This paper studies the relations arising from coverings and their topological structures. Every covering induces a reflexive and transitive relation. We represent the approximate pairs proposed by Ma (2015) [20] with different combinations of a relation and its inverse. Based on this representation, we give the relationship among approximate pairs. We also consider the topological structures induced by these lower approximations and establish the relationship among these topologies. The results show that the approximate pairs can be precisely characterized by a relation and its inverse. [ABSTRACT FROM AUTHOR]

Learning Bayesian networks with bounded tree-width has attracted much attention recently, because low tree-width allows exact inference to be performed efficiently. Some existing methods [24,29] tackle the problem by using k -trees to learn the optimal Bayesian network with tree-width up to k . Finding the best k -tree, however, is computationally intractable. In this paper, we propose a sampling method to efficiently find representative k -trees by introducing an informative score function to characterize the quality of a k -tree. To further improve the quality of the k -trees, we propose a probabilistic hill climbing approach that locally refines the sampled k -trees. The proposed algorithm can efficiently learn a quality Bayesian network with tree-width at most k . Experimental results demonstrate that our approach is more computationally efficient than the exact methods with comparable accuracy, and outperforms most existing approximate methods. [ABSTRACT FROM AUTHOR]

Abstract: We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method. [Copyright &y& Elsevier]

Abstract: This paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems. [Copyright &y& Elsevier]

Abstract: The variable precision rough set model and graded rough set model are 2 important extended rough set models. This paper aims to make a comparative study of the 2 models. Rough set regions, primitive notions, are proposed first for the 2 models, which classify the universe more precisely. Then, both of their logical meanings related to quantitative indexes and their basic structure are investigated, and their precise descriptions are obtained as well. Furthermore, in the graded rough set model, macroscopic and microscopic algorithms are proposed and analyzed to calculate rough set regions; then, the conclusion is drawn that macroscopic and microscopic algorithms have advantages in time and space complexities, respectively, and a medical example is provided to illustrate the rough set regions and the 2 algorithms. In addition, 3 new properties of the 2 models are investigated, which are the results of extending the classical rough set model, i.e. the relationships between approximations and the basic set, the power actions of approximation operators, and the modifications of approximation operator actions on set operations. Finally, the classical rough set model is used to obtain many corresponding results, and moreover, the relationship and transformation between the 2 models is investigated. The study results of this paper have extended and enriched rough set theory from both operator-oriented and set-oriented points of view. [Copyright &y& Elsevier]

Abstract: One of the key computational problems in Bayesian networks is computing the maximal posterior probability of a set of variables in the network, given an observation of the values of another set of variables. In its most simple form, this problem is known as the MPE-problem. In this paper, we give an overview of the computational complexity of many problem variants, including enumeration variants, parameterized problems, and approximation strategies to the MPE-problem with and without additional (neither observed nor explained) variables. Many of these complexity results appear elsewhere in the literature; other results have not been published yet. The paper aims to provide a fairly exhaustive overview of both the known and new results. [Copyright &y& Elsevier]

Abstract: This paper derives a particle filter algorithm within the Dempster–Shafer framework. Particle filtering is a well-established Bayesian Monte Carlo technique for estimating the current state of a hidden Markov process using a fixed number of samples. When dealing with incomplete information or qualitative assessments of uncertainty, however, Dempster–Shafer models with their explicit representation of ignorance often turn out to be more appropriate than Bayesian models. The contribution of this paper is twofold. First, the Dempster–Shafer formalism is applied to the problem of maintaining a belief distribution over the state space of a hidden Markov process by deriving the corresponding recursive update equations, which turn out to be a strict generalization of Bayesian filtering. Second, it is shown how the solution of these equations can be efficiently approximated via particle filtering based on importance sampling, which makes the Dempster–Shafer approach tractable even for large state spaces. The performance of the resulting algorithm is compared to exact evidential as well as Bayesian inference. [Copyright &y& Elsevier]

Abstract: The main goal of this paper is to describe inference in hybrid Bayesian networks (BNs) using mixture of polynomials (MOP) approximations of probability density functions (PDFs). Hybrid BNs contain a mix of discrete, continuous, and conditionally deterministic random variables. The conditionals for continuous variables are typically described by conditional PDFs. A major hurdle in making inference in hybrid BNs is marginalization of continuous variables, which involves integrating combinations of conditional PDFs. In this paper, we suggest the use of MOP approximations of PDFs, which are similar in spirit to using mixtures of truncated exponentials (MTEs) approximations. MOP functions can be easily integrated, and are closed under combination and marginalization. This enables us to propagate MOP potentials in the extended Shenoy–Shafer architecture for inference in hybrid BNs that can include deterministic variables. MOP approximations have several advantages over MTE approximations of PDFs. They are easier to find, even for multi-dimensional conditional PDFs, and are applicable for a larger class of deterministic functions in hybrid BNs. [Copyright &y& Elsevier]

Abstract: The present paper introduces a new kind of representation for the potentials in a Bayesian network: Binary Probability Trees. They enable the representation of context-specific independences in more detail than probability trees. This enhanced capability leads to more efficient inference algorithms for some types of Bayesian networks. This paper explains the procedure for building a binary probability tree from a given potential, which is similar to the one employed for building standard probability trees. It also offers a way of pruning a binary tree in order to reduce its size. This allows us to obtain exact or approximate results in inference depending on an input threshold. This paper also provides detailed algorithms for performing the basic operations on potentials (restriction, combination and marginalization) directly to binary trees. Finally, some experiments are described where binary trees are used with the variable elimination algorithm to compare the performance with that obtained for standard probability trees. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a granularity-based framework of deduction, induction, and abduction using variable precision rough set models proposed by Ziarko and measure-based semantics for modal logic proposed by Murai et al. The proposed framework is based on -level fuzzy measure models on the basis of background knowledge, as described in the paper. In the proposed framework, deduction, induction, and abduction are characterized as reasoning processes based on typical situations about the facts and rules used in these processes. Using variable precision rough set models, we consider -lower approximation of truth sets of nonmodal sentences as typical situations of the given facts and rules, instead of the truth sets of the sentences as correct representations of the facts and rules. Moreover, we represent deduction, induction, and abduction as relationships between typical situations. [Copyright &y& Elsevier]

Abstract: In decision theory under imprecise probabilities, discretizations are a crucial topic because many applications involve infinite sets whereas most procedures in the theory of imprecise probabilities can only be calculated for finite sets so far. The present paper develops a method for discretizing sample spaces in data-based decision theory under imprecise probabilities. The proposed method turns an original decision problem into a discretized decision problem. It is shown that any solution of the discretized decision problem approximately solves the original problem. In doing so, it is pointed out that the commonly used method of natural extension can be most instable. A way to avoid this instability is presented which is sufficient for the purpose of the paper. [Copyright &y& Elsevier]

Abstract: A strong implication operator (S-implication) is a fuzzy material implication operator defined in terms of a t-norm and a strong negation as a generalization of the classical equivalence between “IF A then B” and “Not(A and not B)”. This paper is concerned with a family of improper S-implications derived from an extension of the Schweizer–Sklar family of parameterized improper t-norms defined over (−∞,+1]. Analysis of fuzzy and defuzzified interpolation outputs includes proper and improper fuzzy set outputs and exact defuzzified solutions for important special cases such as Kleene–Dienes, Reichenbach, and Łukasiewicz implications. The effect of the Schweizer–Sklar parameter on interpolation between fuzzy points is outlined. The final section of the paper illustrates the use of the method with two examples, one in political geography and the other in rule-based control. [Copyright &y& Elsevier]

Obtaining reliable estimates of the parameters of a probabilistic classification model is often a challenging problem because the amount of available training data is limited. In this paper, we present a classification approach based on belief functions that makes the uncertainty resulting from limited amounts of training data explicit and thereby improves classification performance. In addition, we model classification as an active information acquisition problem where features are sequentially selected by maximizing the expected information gain with respect to the current belief distribution, thus reducing uncertainty as quickly as possible. For this, we consider different measures of uncertainty for belief functions and provide efficient algorithms for computing them. As a result, only a small subset of features need to be extracted without negatively impacting the recognition rate. We evaluate our approach on an object recognition task where we compare different evidential and Bayesian methods for obtaining likelihoods from training data and we investigate the influence of different uncertainty measures on the feature selection process. [ABSTRACT FROM AUTHOR]

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. Presented here, is a new proposal for a ‘metric’ approach to Rough Sets. We assume some finite measure space is defined on a given universe, and then use it to define various similarity indexes . A set of axioms and the concept of consistency for similarity indexes are also proposed. The core of the paper is a definition of the ‘ optimal ’ or ‘ best ’ approximation with respect to any particular similarity index, and an algorithm to find this optimal approximation by using the Marczewski–Steinhaus Index. This algorithm is also shown to hold for a class of similarity indexes that are consistent with the Marczewski–Steinhaus Index. [ABSTRACT FROM AUTHOR]

The Logic of Approximate Entailment ( LAE ) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAE where, at the semantic level, the underlying set of worlds is moreover endowed with an order structure. The chosen framework is designed in view of possible applications. [ABSTRACT FROM AUTHOR]

Rough set theory provides a powerful tool for dealing with uncertainty in data. Application of variety of rough set models to mining data stored in a single table has been widely studied. However, analysis of data stored in a relational structure using rough sets is still an extensive research area. This paper proposes compound approximation spaces and their constrained versions that are intended for handling uncertainty in relational data. The proposed spaces are expansions of tolerance approximation ones to a relational case. Compared with compound approximation spaces, the constrained version enables to derive new knowledge from relational data. The proposed approach can improve mining relational data that is uncertain, incomplete, or inconsistent. [ABSTRACT FROM AUTHOR]

The paper presents results on approximation in residuated lattices given that closeness is assessed by means of biresiduum. We describe central points and optimal central points of subsets of residuated lattices and examine several of their properties. In addition, we present algorithms for two problems regarding optimal approximation. [ABSTRACT FROM AUTHOR]

The logic LAE q discussed in this paper is based on an approximate entailment relation. LAE q generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance. We adopt the conceptual framework defined by E. Ruspini; our work is towards a contribution to the investigation of suitable logical calculi. LAE q is based on the assumption that the distance function is a quasimetric. We provide a proof calculus for LAE q and we show its soundness and completeness for finite theories. As our main tool for showing completeness, we use a representation of proofs by means of weighted directed graphs. [ABSTRACT FROM AUTHOR]

The notion of preference is crucial in many fields. This justifies the development of many formalisms for preferences representation such as CP-nets, qualitative choice logic and its extensions. Preferences help to choose the best option in decision making, to compare between arguments in argumentation theory, etc. In this paper, we establish a link between a preference formalism, called Prioritized Qualitative Choice Logic ( PQCL ) and argumentation theory. We show that for any set of preferences expressed using PQCL (called PQCL theory), a Value-based Argumentation Framework ( VAF ) can be built. However, we point out some problems related to the evaluation of arguments which does not guarantee the correspondence between elements of PQCL theory and those of its associated VAF . We show that the major problem is due to the evaluation of arguments defined in existing argumentation frameworks, where an absolute status is assigned to each argument: objectively (or skeptically) accepted if it belongs to every extension, subjectively (or credulously) accepted if it is in some extensions and not in others and rejected if it does not belong to any extension. To deal with this problem, we propose to revise the evaluation of arguments in the corresponding VAF . As a result, there is a direct relationship between preferred extensions of the corresponding VAF and preferred models of a set of preferences expressed using PQCL . In addition, rank ordering the set of arguments is possible. The relationship between the two formalisms is interesting since on the one hand, it points out that one should be careful in using argumentation theory for decision making purposes or in formalizing a given problem as an argumentation framework and on the other hand, it makes it possible to use an argumentation framework for preference elicitation. [ABSTRACT FROM AUTHOR]

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension , provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension . In this paper we consider probabilistic assessments in the form of a coherent lower prevision P ̲ , which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F ̲ ≥ P ̲ that is conglomerable, in case it exists. Past work has shown that F ̲ can be approximated by an increasing sequence ( E ̲ n ) n ∈ N of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many distinct elements. Moreover, we give sufficient conditions, of quite broad applicability, to make sure that the point-wise limit of the sequence is F ̲ in case P ̲ is the lower envelope of finitely many linear previsions. In addition, we study the question of the existence of F ̲ and its relationship with the notion of marginal extension. [ABSTRACT FROM AUTHOR]

Abstract: Recently, much attention has been given to multi-granulation rough sets (MGRS) and different kinds of multi-granulation rough set models have been developed from various viewpoints. In this paper, we propose four types of multi-granulation covering rough set (MGCRS) models under covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects in a given universe of discourse U. And then, we investigate a number of basic properties of the four types of MGCRS models, and discuss the relationships and differences among the classical MGRS model and our MGCRS models. Moreover, the conditions for two distinct MGCRS models which produce identical lower and upper approximations of a target concept in a covering approximation space are also studied. Finally, the relationships among the four types of MGCRS models are explored. We find that for any subset , the lower approximations of X and the upper approximations of X under the four types of MGCRS models can construct a lattice, if we consider the binary relation of inclusion. [Copyright &y& Elsevier]

Abstract: Distances in evidence theory are useful tools for belief function approximation or clustering. Efficient approaches are found in the literature, especially full metrics taking focal element interactions into account. In this paper, another aspect is investigated: the ability to detect common evidence pertaining to two different states of belief. This requirement, as well as the previously mentioned ones, are formalized through mathematical properties. To find a belief function distance satisfying the desired properties, matrix norms based distances between Dempsterian specialization matrices are investigated. It is proved that the Dempsterian matrix distance succeeds to fulfill all requirements. Interesting and unprecedented ties between the conjunctive combination rule and this distance are demonstrated. Several basic belief assignment distance experiments are proposed and interpreted thereby allowing to understand advantages and limitations of the newly introduced distances as compared to the existing ones. [Copyright &y& Elsevier]

Abstract: In this paper, we propose two variations of rough approximable concepts and investigate the order-theoretic properties of the associated concept hierarchies. We first show that every rough pseudo-concept hierarchy is a completely distributive lattice and its completely compact elements are exactly the rough pseudo-concepts generated from individual attributes. Next, we propose the notions of hyper-contexts and hyper-concepts, and prove that they provide an approach to restructuring algebraic domains. Finally, we set hyper-contexts into a category in which hyper-mappings serve as the morphisms. It turns out that this category is precisely equivalent to that of algebraic domains. [Copyright &y& Elsevier]

Many different proposals exist for the definition of lower and upper approximation operators in covering-based rough sets. In this paper, we establish relationships between the most commonly used operators, using especially concepts of duality, conjugacy and adjointness (also referred to as Galois connection). We highlight the importance of the adjointness condition as a way to provide a meaningful link, aside from duality, between a pair of approximation operators. Moreover, we show that a pair of a lower and an upper approximation operator can be dual and adjoint at the same time if and only if the upper approximation is self-conjugate, and we relate this result to a similar characterization obtained for the generalized rough set model based on a binary relation. [ABSTRACT FROM AUTHOR]

This paper introduces a flexible extension of rough set theory: multi-adjoint fuzzy rough sets, in which a family of adjoint pairs are considered to compute the lower and upper approximations. This new setting increases the number of applications in which rough set theory can be used. An important feature of the presented framework is that the user may represent explicit preferences among the objects in a decision system, by associating a particular adjoint triple with any pair of objects. Moreover, we verify mathematical properties of the model, study its relationships to multi-adjoint property-oriented concept lattices and discuss attribute selection in this framework. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a general framework for the study of dual fuzzy rough approximation operators determined by a fuzzy implication operator in infinite universes of discourse is investigated. Lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space in infinite universes of discourse are first introduced. Properties of -fuzzy rough approximation operators are then examined. An operator-oriented characterization of fuzzy rough sets is further proposed, that is, -fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper -fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, a comparative study of -fuzzy rough sets with fuzzy topological spaces is presented. It is proved that there exists a one-to-one correspondence between the set of all reflexive and -transitive fuzzy approximation spaces and the set of all fuzzy Alexandrov spaces such that the lower and upper -fuzzy rough approximation operators in a fuzzy approximation space are, respectively, the fuzzy interior and closure operators in a fuzzy topological space. [Copyright &y& Elsevier]

Abstract: The classical probabilistic rough set model is established based on a crisp binary relation. As a generalization of crisp binary relation, fuzzy relation makes descriptions of the objective world more realistic, practical, and accurate in some cases. Thus probabilistic rough set model based on a crisp binary relation limits its application domain. In this paper, based on a fuzzy relation, we propose a fuzzy probabilistic rough set model on two universes. Meanwhile, the concepts of the inverse lower and upper approximation operators are presented. We also study some properties of these approximation operators. Finally, a numerical example of the clinical diagnosis systems is applied to illustrate the validity of the proposed model. And we compare the proposed model with other models to show the superiority of the proposed model. [Copyright &y& Elsevier]