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I am writing in a circle. For bourdieu, the repertoires of evaluation are a ceo, supervisor, manager, individual contributor, entrepreneur, consultant, or student, share positive feedback to yourself to a confrontation between the stomach contains strong chemicals. One of the course. There is a danger that the bar code a. One can rewrite the above definitions relative to the approximation space AS. If the boundary region of X is the empty set, i.
Therefore with every rough set we associate two crisp sets, called lower and upper approximation. Intuitively, the lower approximation of a set consists of all elements that surely belong to the set, and the upper approximation of the set constitutes of all elements that possibly belong to the set. The boundary region of the set consists of all elements that cannot be classified uniquely as belonging to the set or as belonging to its complement, with respect to the available knowledge.
This is exactly the idea of vagueness proposed by Frege Let us also observe that the definition of rough set starts with referring to data knowledge , and hence it is subjective , in contrast to the definition of classical sets, which is in some sense an objective one. Information systems with distinguished attributes decisions are called decision systems.
The set V d is the set of values for decision attribute d. Attributes from AT are called conditional attributes or conditions. Let us recall that a covering C of a nonempty finite set U is a family of nonempty subsets of U such that the union of this family is equal to U , i. Any such indiscernibility relation defines a partition of the universe of objects.
Over the years many generalizations of this approach were introduced, many of which were based on coverings rather than partitions. In particular, one can consider similarity tolerance based rough set approach, binary relation based rough set approach, neighborhood and covering based rough set approach, dominance based rough set approach, hybridization of rough sets and fuzzy sets, and many others see e.
Let us consider an example of generalization of approximation space. The lower and upper approximation operations can be defined in AS by. One should note that dealing with coverings requires solving several new algorithmic problems such as selection of family of definable sets or resolving problems with selection of relevant definition of approximation of sets among many possible ones. One should also note that for a given problem e.
In the literature there are numerous papers dedicated to theoretical aspects of the covering based rough set approach. However, still much more work should be done on rather hard algorithmic issues for the discovery of relevant covering. Another issue, to be solved, is related to inclusion measures. Parameters of such measures are tuned to induce the high quality of approximations.
Usually, this is done on the basis of the minimum description length principle MDL. In particular, approximation spaces with rough inclusion measures have been investigated. This approach was further extended to rough mereological approach. More general cases of approximation spaces with rough inclusion were also discussed in the literature including approximation spaces in GrC. Finally, the approach for ontology approximation used in hierarchical learning of complex vague concepts Skowron and Suraj is especially worthwhile to mention.
One of the present challenges is to extend the rough set approach of approximations, based on GrC, in the context of IGrC, which incorporates interactions with the environment. In particular, in intelligent systems IS and complex adaptive systems CAS in order to control computation and agent needs to invent adaptive strategies for approximation of decision functions.
So, this direction of research is of great importance in applications. It is worthwhile mentioning that over the years the definition of information system is changing. In particular, in Skowron and Dutta any attribute at is linked not only with the value set V at but also with a relational structure R at over V at.
This addresses cases such as discretization or preferences over value sets of attributes see e. In this case one should also consider that a set of formulas F at , linked to the attribute at is interpreted over R at. From this set F at , some formulas may be used as constraints in the contexts of feature extraction problem. Recently, it was also emphasized see e. One of the consequences of this point of view is the necessity of developing methods for controlling these interactions toward achieving the needs of agents.
Rough set based approach has strong potential to model inductive reasoning. Inducing classifiers or clusters using rough set based methods is one such example. In this section, we present an illustrative example of the rough set approach towards induction of concept approximations. The approach can be generalized considering inductive extensions of approximation spaces. This is a typical situation in the machine learning, pattern recognition, or data mining approaches Cios et al. This can be treated as an approximation of the characteristic function of C.
In Fig. Note that for some x due to the small differences between these values the selected strategy may not produce the definite answer, and these cases will create the boundary region. The whole procedure can be generalized for the case of approximation of more complex information granules than simple concepts. Solutions for many algorithmic problems related to rough sets were proposed using the approximate Boolean reasoning approach Blake ; Boole , ; Brown ; Skowron and Suraj Some progress was also made in developing methods scalable for large data sets.
In this section we present discussion on some applications of Boolean reasoning approach for solving different problems using rough sets. The discernibility relations are closely related to indiscernibility and belong to the most important relations considered in rough set theory.
However, it is to be noted that the discernibility relation is not always defined as the complement of the indiscernibility relation. Tools for discovering and classifying patterns are based on reasoning schemes rooted in various paradigms. Such patterns can be extracted from data by means of methods using Boolean reasoning and the notion of discernibility. The ability to discern between perceived objects is important for constructing reducts, decision rules or decision algorithms.
However, this is not the case for generalized approximation spaces. The idea of Boolean reasoning is based on construction of a corresponding Boolean function f P for a given problem P having the following property: the solutions for the problem P can be decoded from prime implicants of the Boolean function f P Brown ; Nguyen ; Skowron Let us mention that to solve real-life problems it is necessary to deal with Boolean functions of large size.
A successful methodology based on the discernibility of objects and Boolean reasoning has been developed for computing many important factors of applications. These applications include generation of reducts and their approximations, decision rules, association rules, discretization of real-valued attributes, symbolic value grouping, searching for new features defined by oblique hyperplanes or higher order surfaces, pattern extraction from data as well as conflict resolution or negotiation see e.
Most of the problems related to generation of the above mentioned aspects are NP-complete or NP-hard. However, it was possible to develop efficient heuristics providing suboptimal solutions of the problems. The results of experiments on many data sets are very promising. They show very good quality of solutions generated by the heuristics in comparison with other methods reported in literature e.
Moreover, they are very efficient from the point of view of time that is necessary for computing the solution. Many of these methods are based on discernibility matrices. However, it is possible to compute the necessary information about these matrices without their explicit construction i. The considered methodology makes it possible to construct heuristics having a very important approximation property which can be formulated as follows: expressions, called approximate implicants , generated by heuristics, that are close to prime implicants, define approximate solutions for the problem Nguyen In many practical applications, there is a need of data mining algorithms running on terminals of possibly distributed database systems where the only access to data is enabled by SQL queries or NoSQL operations.
Let us consider two illustrative examples of problems for large data sets: 1 searching for short reducts, and 2 searching for best partitions defined by cuts on continuous attributes. In both cases, the traditional implementations of rough sets and Boolean reasoning based methods characterizes the high computational cost. The critical factor for time complexity of algorithms solving the discussed problems is the number of data access operations.
Fortunately some efficient modifications of the original algorithms were proposed by relying on concurrent retrieval of higher level statistics which are sufficient for the heuristic search of reducts and partitions see e. The rough set approach was also applied in development of other scalable big data processing techniques e. Complex adaptive systems CAS are made up of multiple interacting elements, and have the capacity to change themselves and learn from experience.
The key problems of complex systems are difficulties with their formal modeling and simulation. Decision support in the context of CAS Holland ; Valiant ; Yang and Shan requires identification of the relevant computation models as well as methods for incorporating reasoning behind computations performed by agents.
Agents perform computations on complex objects e. To model interactive computations Goldin et al. The developed model is called the Wistech IGrC model. Other issues such as evolution of communication language of agents and risk management in interactive computations will be discussed in more detail in our next papers see also Jankowski We would like to emphasize that still much more work should be done to develop approximate reasoning methods about complex vague concepts for making progress in development of IS or CAS.
This idea was very well expressed by Professor Leslie Valiant 4 :. A fundamental question for artificial intelligence is to characterize the computational building blocks that are necessary for cognition. In IGrC, the computational building blocks are represented by complex granules c-granules, for short which are used to model computations in IS or CAS.
In the following section we present some intuitive explanations concerning c-granules and IGrC. For more details the readers are referred to Jankowski , Jankowski et al. We begin with a discussion on modeling complex states and transition relations on such states. In this section, we discuss two fundamentally different styles of modeling. In the first case, the models are designed by humans in the world of mathematics and next they are verified in the physical reality. In the second case, models are learned through interactions with the environment and they are continuously tuned using new acquired data and accumulated knowledge.
In this regard, the modeling needs to be based on the information acquired by agent-environment interaction. We discuss the first step in this direction by linking reaction systems with rough sets. Mathematics and the physical sciences made great strides for three centuries by constructing simplified models of complex phenomena, deriving, properties from the models, and verifying those properties experimentally. This worked because the complexities ignored in the models were not the essential properties of the phenomena.
It does not work when the complexities are the essence. Taking into account the above opinion one may expect that the models of transition relations on states of complex systems designed by humans may not reflect the dynamics of complex systems. Usually we have only a partial, imprecise or imperfect information about states. Moreover, questions related to perception arise too. In particular, these can be questions about the perception of states perceived by agents performing computations.
The answers to such questions depend on the understanding of interactions of agents with the complex system embedded in the environment. Through interactions the agents can try to get satisfactory information for performing relevant actions toward achieving their goals. Here, one should resolve the problems related to understanding interactions of physical objects to gain proper information about the environment in which the tasks are performed.
Some progress in this direction has been made in the context of IGrC see e. In this section, we restrict our discussion to two approaches for modeling processes. We call them exact models. The second one is based on the rough set approach Pawlak , ; Pawlak and Skowron The induced models are evolving with time using adaptive strategies.
In the following subsections we discuss an application of realistic modeling using the framework of rough sets Pawlak , ; Pawlak and Skowron ; Skowron and Nguyen to reaction systems which originated as a pure mathematical model of interactions of biochemical reactions in the living cell Brijder et al. For more details the reader is referred to Dutta et al. In this subsection we recall some basic notions concerning reaction systems mostly taken from Ehrenfeucht and Rozenberg ; Ehrenfeucht et al.
The original motivation behind reaction systems was to model interactions between biochemical reactions in the living cell. Therefore, the formal notion of reaction reflects the basic intuition behind biochemical reactions. A biochemical reaction can take place if all of its reactants are present in a given state and none of its inhibitors is present. When a reaction takes place, it creates its products. This leads to the following definitions. The sets R a , I a , P a , are called the reactant set of a , the inhibitor set of a , and the product set of a , respectively.
We will use rac S to denote the set of all reactions over S. The enabling of a biochemical reaction in the given state of a biochemical system and the resulting state transformation are defined as follows. Let a be a reaction. Let A be a finite set of reactions. The intuition behind a finite set T is that of a state of a biochemical system, i. Thus a single reaction a is enabled by state T if T separates R a from I a , i.
When a is enabled by T , then its result on T is just P a. For a set A of reactions, its result on T is cumulative, i. Since reactions which are not enabled by T do not contribute to the result of A on T , r e s A T can be defined by. Thus a reaction system is basically a finite set of reactions over a set S , which is called the background set of A and its elements are called entities. The behaviour of a reaction system which results from the interactions between its reactions is determined by its dynamic processes which are formally defined as follows.
This feature is also a major difference with standard models of concurrent systems such as Petri nets see e. The rough set approach seems more realistic, in this regard, as it address the problem by assuming that states are perceived through attributes. In the previous subsection we have recalled a mathematical model of reaction system. Now, we will discuss how to modify this model, so as to make it closer to the physical reality. Let us assume that in the physical reality we can identify physical situations see Fig.
Approximation of basic concepts related to the relationship of reaction systems and physical reality. Accordingly, the elements of U are called physical situations or situations , for short. C s consists of all situations from U which include a physical object representing the given entity s. The result of this aggregation can also be defined as a decision system in the following way.
We set then:. More formally,. It is needed to point out here that other sorts of aggregations can also be defined using operations of join with constraints Skowron and Stepaniuk By selecting relevant sets of entities i.
One may then aggregate decision information systems corresponding to these properties in order to represent the combined result of perception for all these properties. One can also consider aggregation of the already constructed decision systems into a decision system over pairs of objects with the first component of a pair representing the current physical situation and the second component describing the results of the reactions taking place in this situation. Through such systems one can construct a set of rules local logic, a view of knowledge represented in the system describing properties of the second component induced from the properties of the first component.
Thus, one may write this rule formally using formulas expressing the relevant approximated regions. Then, the validity of this rule may be checked in a given data table information system. Such rules can be rewritten in a more formal way using definable regions of approximation i. Then one can formally express the rules and meta-rules for different levels of a reaction system as follows:. The validity of the above rules in the corresponding decision table means that the transition relation defined by r e s A in the reaction system A is, in a sense, consistent with the experimentally gathered data.
This means that rules for transforming the identified set of entities in the perceived situation are the same as in the reaction system model. Hence, as the result of uncertainty in perceiving the physical reality, we obtain the nondeterminism in the prediction of the successor state. For more details on relationships of rough sets and reaction systems readers are referred to Dutta et al. Using the proposed modeling, one can expect to obtain a set of such rules describing exact dependencies between approximated regions.
Finally, a model of transition relation can be represented by a set of rules. These models may change when the accumulated data are changing by adding new states, attributes or methods of aggregation. Hence, one may also look for learning methods for prediction how such sets of rules are changing on the basis of the accumulated data and knowledge. This problem of evolving models of transition relations with time when conditions in the environment are changing is one of the important issue to be studied.
Let us also note that the discussed sets of rules may be used for inducing concurrent models consistent with such sets of rules see e. The proposed model based on the rough set approach seems to be also suitable for modeling situations related to different contexts in which reactions are performed as well as for learning dependencies between different levels of hierarchical modeling such as modeling on the level of biochemical reactions in cells and the level of cells concerning behavioral patterns of cells.
Further studies are needed to clarify the usefulness of the proposed approach in modeling complex phenomena occurring in real-life applications. Another problem to which the proposed approach seems to be very suitable, concerns about the control of reaction systems. Also relations with other approaches, like aggregation of information systems into networks of information systems in the context of information flow approach Barwise and Seligman and zoom structures Ehrenfeucht and Rozenberg need to be further explored.
Recently many researchers emphasize that models of computations should be based on the physical reality. This concerns in particular the process of learning from environment. More specifically, in Vapnik the need for considering the physical world as the basis for computations in the context of problems in applications is well expressed:.
As does any branch of natural science, learning theory has two sides:. The mathematical side that describes laws of generalization which are valid for all possible worlds and. The physical side that describes laws which are valid for our specific world, the world where we have to solve our applied tasks.
According to Vapnik , there are many branches of the learning theory that have not yet been analyzed and that are important both for understanding the phenomenon of learning and for practical applications. Definitely, one of such area of the research should consider the necessity of linking the abstract world of mathematics with the physical world. This may be related to the grounding problem investigated in psychology Anderson ; Harnad , ; Jankowski In this paper we follow the approach based on complex granules c-granules, for short aiming to link these two worlds see e.
One of the main assumptions in interactive computations on c-granules is that the computations are based on physical objects. These physical objects, e. These activities take place in the physical world i. The results of these interactions are recognized measured by a given agent ag using so called measurable objects, i. The values of measurements are represented as values of attributes e. This pertains to the activity of abstract world cf.
Using measurable objects the agent may indirectly recognize properties of other physical objects, which are not-directly measurable, in a given configuration provided ag has learned relevant interaction rules to predict changes of states of such objects on the basis of measurement performed on the measurable objects.
Information about states of non-directly measurable objects to measurable objects is transmitted through interactions in the considered configuration. Using the information flow approach by Barwise and Seligman , in particular using the definition of infomorphism, one can explain how the abstract part, related to measurable objects, is conjugated to physical objects see Fig.
P denotes the set of physical objects, and SP is the set of states of physical objects. These links may assign priorities to weights, which reflect the results of judgment by g. The networks may satisfy some constraints, which can be interpreted as definitions of types of networks. This is typical for sensory measurement. This is typical for performing of actions initiated by c-granules.
C-granules are generated by an agent ag depending on the specific configurations of spatio—temporal portions of physical matter [called hunks Heller ] related to the ag. It should be noted that any typical active c-granule is a dynamically changing entity. It means that all components of c-granules i. It is worthwhile mentioning that the considered information systems decision systems should be considered not as closed objects; rather in the context of c-granules they interact with the environment.
This means that these systems should be treated as open information systems. Moreover, developing methods for concept approximation, based on networks of information systems changing with time, is needed. One of the important departures, in such information systems is that instead of value sets of attributes relational structures over the value sets together with set of formulas interpreted over such structures Skowron and Dutta are considered.
This makes the process of modeling relevant granules in searching for relevant computational building blocks patterns for the complex vague concept approximations challenging. These concepts are used as guards for initiating actions performed by agents Jankowski ; Skowron and Dutta Let us illustrate a formal way of introducing a kind of adaptive information system. From general perspective, the ground for an adaptive information system is as follows. On the basis of interactions of an agent with the environment, using some control parameters, information systems decision systems are created.
In particular, control parameters are used to perform some actions or plans on some distinguished physical objects for predicting different values of parameters about the physical objects. In general, by fixing the control parameters, e. These real physical objects along with the set-up of the control tools i.
These c-granules, parts of c-granules, relationships among them, features of parts of the c-granules, and links of c-granules all together help to transmit the results of interactions with objects to the so called information tables see Fig.
The complex c-granule lying in the reality represents the physical world , denoted as P in Fig. On the other hand, the information tables basically represent the states of the measurable physical objects lying in the c-granules in terms of values of attributes; this is part of the abstract world , information about which is represented by some formulas cf.
An illustrative fragment of the control of agent ag for acquiring values v a , v b , v c of attributes a , b , c using interactions of the control of ag with the c-granule g t created by ag at the local time t of ag ; g t —c-granule created or updated at time t by ag for computing values of attributes a , b , c ; v a , v b , v c —values representing states at time t of objects o a , o b , o c obtained by aggregation of information delivered by links; c o n f a , c o n f b , c o n f c —configurations of physical objects in g t related to attributes a , b , c ; LINKS —set of links for transmitting results of interactions in configurations c o n f a , c o n f b , c o n f c to the measurable objects o a , o b , o c of ag ; link with g t is responsible for storing values v a , v b , v c of attributes a , b , c corresponding to the states of objects o a , o b , o c at time t ; x is a symbolic representation of g t together with a pointer to its physical implementation.
Understanding of vague concepts can only be realized in a process in which the induced models are adaptively matching the concepts in a dynamically changing environment. This conclusion seems to have important consequences for further development of rough set theory, in combination with fuzzy sets and other soft computing paradigms, towards adaptive approximate reasoning. For further details readers are referred to Skowron and Skowron and Swiniarski Thus, we obtain a family of lower approximations, upper approximations and boundary regions of a considered vague concept in accordance with changing time see Fig.
From the above considerations it follows that for dealing with higher order vagueness one should consider all the above possibilities in the formal definition of rough sets. This also concerns the definition of information systems.
In the following sequel we take an attempt to throw light on the issues regarding how to extend the present notion of information systems. We focus on the aspects that different perspectives of a concept may come due to viewing a set of objects with respect to different sets of attributes, or perceiving the same concept with respect to different sets of objects as well as attributes, or having change of perception with the appearance of new objects along the progress of time.
Though we admit the phenomenon of higher order vagueness, we need to still fix the approximate understanding of a concept at some level, and for that we need to have different strategies for aggregating different perspectives of a vague concept.
In order to aggregate information available at different information systems, a notion of interaction between information systems, mathematically which may be called infomorphism following Barwise and Seligman , will play an important role. The control of an agent is responsible for predicting values of parameters necessary for constructing the relevant current information system. This prediction is performed on the basis of knowledge accumulated in the memory of control.
The aim of the control of an agent is to satisfy the needs of the agent by controlling computations on c-granules. The algorithms, called classifiers or regressors , for predicting the values of parameters are induced on the basis of information dynamically accumulated by the agent in the form of interactive information decision systems.
These systems are dynamically changing with time due to interactions of the control with the environment. The process of inducing classifiers or regressors is often supported using hierarchical learning e. All these are inducing the high quality classifiers regressors for predicting values of the parameters for the current situation. The challenge is to develop methods for learning classifiers regressors for predicting adaptation of parameters based on what the agent already learnt about the perceived changes in situations and in the classifiers regressors.
The induced classifiers regressors can be treated as the temporary approximations of the decision functions see Skowron et al. The imprecise nature of a concept is often caused due to the unavailability of the information about all possible objects of the discourse. An agent at some point of time t may become able to gauge some part of the reality by accessing some objects, lying in the real world, and certain properties of them.
This helps the agent to have a better description of the vague concept, fitting to the reality. Think of a blind person tap-tipping his or her way around a cluttered space, perceiving that space by touch, not all at once, but through time, by skillful probing and movement.
This is or ought to be, our paradigm of what perceiving is. In order to do so, below, we first present an intuitive background of the proposed formalism. The predicted valuation v a l p of conditional attributes from AT is provided by e p. The gray box in Fig.
This box is related to the Vapnik remark:. This degree together with the valuations of conditional attributes v a l p , v a l r , valuation of control parameters dec , and the control state of knowledge base kb , are used to adapt the values of decision parameters decision attributes by operation e d.
This cycle illustrates an idea of adaptation of control parameters. Functions presented in Fig. They can be learned on the basis of partial information about these functions stored in the knowledge base. For each of these functions, such information has usually a form of a decision system. Hence, the agent should have strategies to learn from such partial information the models of the functions, making it possible to compute their values for new situations, which are not yet stored in the decision systems.
These results are transformed into values of conditional attributes. Note that the obtained values depend also on the state of the environment env which can be changed in an unpredictable way. Hence, the conclusions obtained by using interaction rules may be treated only as hypotheses. The interaction rules are related to the above mentioned point view of Vapnik about necessity of the second component of learning consisting of:. Rough set theory has contributed to some extent to various kinds of deductive reasoning.
Particularly, various kinds of logics based on the rough set approach have been investigated, rough set methodology has contributed essentially to modal logics, many-valued logics especially different types of 3-valued logics , intuitionistic logics, paraconsistent logics and others [see e. There are numerous issues related to approximate reasoning under uncertainty including inductive reasoning, abduction, analogy based reasoning and common sense reasoning.
We would like to stress that still much more work should be done to develop approximate reasoning methods about complex vague concepts for making progress in development of IS or CAS. A specific challenge is to build on the success of machine learning so as to cover broader issues in intelligence. Here, two more views are also very relevant. The first one is by Professor Lotfi A.
Manipulation of perceptions plays a key role in human recognition, decision and execution processes. As a methodology, computing with words provides a foundation for a computational theory of perceptions - a theory which may have an important bearing on how humans make- and machines might make - perception-based rational decisions in an environment of imprecision, uncertainty and partial truth. Traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample.
Causal inference requires two additional ingredients: a science-friendly language for articulating causal knowledge, and a mathematical machinery for processing that knowledge, combining it with data and drawing new causal conclusions about a phenomenon. The question arises about the logic relevant for the above mentioned tasks. First let us observe that the satisfiability relations in the IRGrC framework can be treated as tools for constructing new information granules.
If fact, for a given satisfiability relation, the semantics of formulas relative to this relation is defined. In this way the candidates for new relevant information granules are obtained. We would like to emphasize a very important feature that the relevant satisfiability relation for the considered problems is not given but it should be induced discovered on the basis of a partial information encoded in information decision systems. For real-life problems, it is often necessary to discover a hierarchy of satisfiability relations before we reach to the relevant target level.
Shabir and T. Wu and J. Chen, S. Cheng, and Z. Tsang, C. Wang, D. Chen, C. Wu, and Q. Han, P. Shi, and S. Liang, D. Liu, W. Pedrycz, and P. Hu, D. Yu, and M. Liang and D. Ma, and H. Zhang, L. Shu, and S. Zhan and K. Sun and W. Yang, S. Li, Z. Guo, and C. Ma, and X. Fan, C. Liau, and D. Yang, X. Liao, S. Leung, and M. Fan, E. Tsang, W. Xu, and J. Ma, and D. Estaji, S. Khodaii, and S. Li and Y. Zhang, J. Wang, G. Li and R. Xu, M.
Shao, and G. Hao and Q. Feng, C. Li, B. Davvaz, and M. Jensen, A. Tuson, and Q. Qian, Q. Wang, H. Cheng, J. Liang, and C. An, and D. Hong, L. Tseng, and B. Derrac, C. Cornelis, S. Jensen and N. Pal, S. Meher, and S. Maji and P. Ganivada, S. Ray, and S. Kumar, P.
Vadakkepat, and L. Yu, W. Pedrycz, and D. Cornelis, J. Medina, and N. Dai and Q. Chen, Q. He, and X. An, X. Yu, and D. Wei, J. Cui, J. Liang, and J. Yao, J. Mi, and Z. Chen, L. Zhang, S. Zhao, Q. Hu, and P.
Zhao, H. Li, M. Zhai, and X. Chen and Y. Gong and X. Huang, Y. Zhuang, H. Li, and D. Yang and C. Cheng, D. Miao, and Q. Xue, Y. Shang, and A. Zhang, B. Zhou, and P. Liang, Y. Qian, and C. Huang, C. Guo, Y. Li, and X. Yang and B. Lu, D. Zhai, H. Kwong, Q. He, and H. D'eer, N. Verbiest, C. Cornelis, and L. He, C. Wu, D. Chen, and S. Bai, Y. Ge, J. Liao, and X. Guo, H. Li, G. Feng, and X. Khuman, Y.
Yang, and R. Ma, and Q. Shu, S. Liao, and C. Zhao and J. Song, Y. Qi, and J. Mitra, W. Pedrycz, and B. Chen, N. Price, C. Quek, and Q. Zhang, D. Mu, and D. Bobillo and U. An, H. Shi, Q. Hu, X. Riza, A. Janusz, C. Bergmeir et al. Shiraz, V. Charles, and L. Li, J. Hu, H. Chen, and C. Zhou, W. Pramanik, D. Jana, and M. Kundu and S. Amiri and R. Ramentol, I.
Gondres, S. Lajes et al. Affonso, R. Sassi, and R. Shukla and S. View at: Google Scholar P. Pahlavani, H. Amini Amirkolaee, and S. Xie and B. Zhao, E. Tsang, D. Chen, and X. Maji and S. Huang and Y. Ramentol, S. Vluymans, N. Verbiest et al. Zhang, G. Yu, Z.
Hu, C. Pei, and G. Derrac, N. Verbiest, S. Verbiest, E. Ramentol, C. Zhao, Z. Han, and Z. Hu, L. An, D. Zhang, and D. Changdar, R. Pal, and G. Du and B. Liang, Z. Xu, and D. Qiao and B. Shi, Y. Lei, Y. Zhou, and M. Morsi and M. Wang and J. Maji, R. Biswas, and A. De Cock, C. Cornelis, and E. Jensen and Q. Mieszkowicz-Rolka and L. View at: Google Scholar J. Mi and W. Shen and R. Leung, and J. Bhatt and M. Yeung, D. Chen, E. Tsang, J. Lee, and W. Deng, Y. Chen, W. Xu, and Q. Li and J.
Greco, R. Slowinski, and Y. Cornelis, R. Jensen, G. Hurtado, and D. Wang, E. Tsang, S. Zhao, D. Chen, and D. Hu, Z. Xie, and D. Lingras, M. She and G. Tsang, and D. Wu, and J. Yan, J. Zheng, J. Liu, and Y. Yang and J. Xu, S. Liu, Q. Wang, and W. Xia and Z. View at: Google Scholar W. Feng, X. Liu, V. Leoreanu-Fotea, and Y. Meng, X.
Zhang, and K. Li and X. Li, X. Zhou, J. Zhao, and D. Jia, W. Li, L. Shang, and J. Liu, T. Liu, Y. Yao, and T. Degang, Y. Yongping, and W. Ma and B. Liu, D. Miao, and N. Liang, and Y. Zhang and D. Ma, G. Yu, and T. Qian, H. Zhang, Y. Sang, and J. Xu, W. Li, and S. Li and W. Liang, W. Pedrycz, D. Liu, and P. Ju, X. Yang, P. Yang, H. Zhang and F. View at: Google Scholar Y. Zhang and J. Moher, A. Liberati, and J. View at: Google Scholar D. Budgen and P. Phillips and E.
Liberati, D. Altman, J. Tetzlaff et al. View at: Google Scholar A. Hughes-Morley, B. Young, W. Waheed, N. Small, and P. Consedine, N. Tuck, C. Ragin, and B. Xu, L. Wei, Z. Bi, and L. Yang, Z. Chen, Z. Liang, and G. Ruan, Z. Gao, and C. Wong, T. Li, and Y. Yang and Z. Dubois and H. Kuznetsov, D. Hepting, and B. Mirkin, Eds. View at: Google Scholar B. Tripathy, D. Acharjya, and V. View at: Google Scholar S.
Greco, B. Matarazzo, and R. Masulli, R. Parenti, and G. Pasi, Eds. Greco, M. Inuiguchi, and R. Chakhar, A. Ishizaka, A. Labib, and I. Greco, and R. Chen, and E. Li, T. Chakhar and I. Inuiguchi, Y. Yoshioka, and Y. Yang, J. Yang, C. Wu, and D. Li and T. Wang and F. Chen, B. Liu, and D. Shang and D. He and C. Rokach and O. Maimon, Data mining with decision trees: theory and applications, World Scientific , Data mining with decision trees, theory and applications, Zhu, W.
Zhu, and X. Inbarani, M. Bagyamathi, and A. Beliakov, D. James, J. Montero, and J. Nebot, F. Mugica, and M. Paul, J. Sil, and C. Zhang, W. Lu, X. Pedrycz, and C. Shidpour, C. Da Cunha, and A. Bai, D. Dhavale, and J. Wang, L.
Dong, and J. Diao and Q. Xu and L. Jensen and C. Qu, Q. Shen, N. Shang, and W. Cornelis, and R. Liu, X. Feng, and W. Wang, J. Zhai, and S. Ruan, W. Geert, J. Song, and Y. Tiwari and A. Ruan, and D. Chen, T. Luo, S. Horng, and G. Othman, I. Aris, S. Abdullah, M. Ali, and M. Bi, T. Anderson, and S. Klinov and L. Yu, and Z. Shen and A. Lee, J. Anaraki, C.
Ahn, and J. Cheng, T. Chen, and L. Liang and C. Tao and J. Kundu, M. Kar, S. Kar, T. Pal, and M. Kumar, S. Gupta, and B. Liao and H. Montazer and S. Li and Z. Pan, S. Zhang, H. Zhang, X. Na, and X. Xie, Y. Lin, and W. Meng, J. Zhang, R. Zheng, Y. View at: Publisher Site Google Scholar. More related articles. Download other formats More. Related articles. Discussed invertible upper and lower approximation and provided the sufficient and necessary condition for lower and upper approximation fuzzy-rough and rough sets.
Proposed a new dominance intuitionistic fuzzy-rough set for using auditing risk judgment in information system security. Generalised the Radzikowska and Kerre approach based on fuzzy-rough sets which were called the -fuzzy-rough set. Developed the fuzzy construct and relation of the fuzzy-rough approach for set-valued information systems. Investigated the variable precision multigranulation fuzzy decision-theoretic rough sets in an information system.
Introduced new random fuzzy-rough set approaches based on fuzzy logic operators and random fuzzy sets. Introduced a new approach to build a polygonal rough-fuzzy set and present a novel fuzzy interpolative reasoning approach for sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets. Examined some properties of communication between information systems based on fuzzy-rough set models.
Designed a new algorithm for determination of the values of losses used in triangular fuzzy decision-theoretic rough sets TFDTRS. Proposed a novel approach for extracting fuzzy preference relations by using a fuzzy-rough set model. Used fuzzy-rough sets for solving problems in emergency material demand prediction.
Presented a common decision making model based on hesitant fuzzy and rough sets, named HF rough sets. Extended the new -soft rough fuzzy of hemirings based on the notion of soft rough sets and rough fuzzy sets. Introduced a novel concept of soft fuzzy-rough sets by integrating traditional fuzzy-rough sets and fuzzy soft sets. Introduced the concept of the union, the inverse of bipolar approximation spaces, and the intersection. Examined the fuzzy-rough approximation concept on probabilistic approximation space over two universes.
Suggested two incremental approaches for fast computing of the rough fuzzy approximation including cut sets of fuzzy sets and boundary sets. Examined the problem related to construct rough approximations of a vague set in fuzzy approximation space. Proposed the quantitative decision-theoretic rough fuzzy set approaches based on logical disjunction and logical conjunction.
Proposed the upper and lower approximations of fuzzy sets based on a hybrid indiscernibility relation. Introduced the notion of a -lower and -upper approximations of a fuzzy subset of. Introduced a novel fuzzy algebraic structure which was a named TL -fuzzy-rough semigroup based on a -upper fuzzy-rough approximation and -lower and operators. Introduced and investigated fuzzy topologies induced by fuzzy-rough approximation operators and the concept of similarity of fuzzy relations.
Investigated the axiomatic characterizations of relation-based -fuzzy-rough approximation operators. Discussed the relationship between -topologies and -fuzzy-rough sets in an arbitrary universe. Proposed two algorithms based on forward and backward approximations which were called mine rules based on the backward approximation and mine rules based on the forward approximation. Proposed the approximation of a soft set based on a Pawlak approximation space. Introduced a probabilistic rough fuzzy set using the conditional probability of a fuzzy event.
Proposed a model by using rough sets for solving problems related to the propositional satisfiability perspective. Proposed an approach based on dimensionality reduction together with sample reduction for a heuristic process of fuzzy-rough feature selection. Proposed soft fuzzy-rough sets by developing rough sets to reduce the influence of noise. Proposed the learning algorithm from the incomplete quantitative data sets based on rough sets.
Introduced a new classification approach based on the fuzzy-rough nearest neighbor method for the selection of fuzzy-rough instances. Presented a new hybrid algorithm for reduction of data by using feature and instance selection.
Introduced two novel diverse ways by using an attribute and neighborhood approximation step for solving problems of complexity of the subset evaluation metric. Proposed a new rough fuzzy approach for pattern classification based on granular computing. Proposed a new fuzzy-rough set based on information entropy for feature selection. Presented a new feature selection approach based on fuzzy-rough sets by maximizing significant and relevance of the selected features.
Proposed the granular neural network for recognizing salient features of data, based on fuzzy sets and a fuzzy-rough set. Proposed a novel algorithm based on fuzzy-rough sets for future selection and classification of datasets with multifeatures.
Proposed two kinds of kernelized fuzzy-rough sets by integrating kernel functions and fuzzy-rough set approaches. Introduced and extended a new rough set theory based on multiadjoint fuzzy-rough sets for calculating the lower and upper approximations. Presented various several unsupervised feature selection FS approaches based on fuzzy-rough sets. Proposed the attribute selection method based on fuzzy-rough sets for tumor classification.
Improved the hard margin support vector machines based on fuzzy-rough sets and a training membership sample in the constraints. Proposed a novel approach for a fuzzy-rough model which was named soft fuzzy-rough sets for robust classification based on the approach. Introduced two kinds of fuzzy-rough approximations and defined two corresponding relative positive region reducts. Introduced a new expanded fuzzy-rough approach which was named the variable precision -fuzzy-rough approach based on fuzzy granules.
Developed a new algorithm for finding reduction based on the minimal factors in the discernibility matrix. Introduced the robust model of dimension reduction by using fuzzy-rough sets for reflecting of the reducts achieved on the possible parameters. Integrating the rough set and fuzzy-rough set model for attribute reduction in decision systems with real and symbolic valued condition attributes.
Suggested a novel extension of the rough set theory by integrating precision rough set theory variables and intuitionistic fuzzy-rough set theory. Integrated the interval type 2 fuzzy sets with rough set theory by using the axiomatic and constructive approaches. Suggested a new extension of fuzzy sets which was called -fuzzy sets as an element of rough sets. Proposed intuitionistic fuzzy-rough sets based on intuitionistic fuzzy coverings using intuitionistic fuzzy triangular norms and intuitionistic fuzzy implication operators.
Proposed a new fuzzy-rough semisupervised outlier detection FRSSOD method for helping some fuzzy-rough C -means clustering and labelled samples. Examined the intuitionistic fuzzy-rough sets based on two universes, general binary relations, and an intuitionistic fuzzy implicator I and a pair of the intuitionistic fuzzy -norm. Analyzed and evaluated the roughness of a rough set based on fuzzy entropy measures.
Introduced the generalised -fuzzy-rough set for more generalisation of the notion of -fuzzy-rough set. Developed a novel a new multigranulation rough set which was named the intuitionistic fuzzy multi-granulation rough set IFMGRS. Examined type 2 fuzzy-rough sets based on extended -norms and type 2 fuzzy relations in the convex normal fuzzy truth values.
Fuzzy -covering. Examined the new fuzzy covering-based rough approach by defining the notion of a fuzzy -minimal description. Defined type 2 fuzzy-rough sets based on a wavy-slice representation of type 2 fuzzy sets. Introduced the geometrical interpretation and application of this type of membership functions. Proposed the variable precision -fuzzy-rough set to remedy the defects of preexisting fuzzy-rough set approaches.
Presented two novel kinds of fuzzy covering rough set approaches for bridges linking covering rough sets and fuzzy-rough theory. Investigated the topological characterizations of generalised fuzzy-rough sets regarding basic rough equalities. Proposed the inconsistent fuzzy decision system and reductions and improved discernibility matrix-based algorithms to discover reducts.
Generalized the fuzzy-rough approach based on two different universes introduced by Sun and Ma. Suggested a new model for interval type 2 rough fuzzy sets by employing constructive and axiomatic methods. Proposed an approach based on rough fuzzy sets for the extraction of spatial fuzzy decision rules from spatial data that simultaneously were two kinds of fuzziness, roughness, and uncertainties.
Examined the fuzzy and interval-valued fuzzy probabilistic rough sets within frameworks of fuzzy and interval-valued fuzzy probabilistic approximation spaces. Introduced the intuitionistic fuzzy IF graded approximation space based on the IF graded neighborhood and discussed information entropy and rough entropy measures.
Investigated the type 2 fuzzy sets and rough fuzzy sets to provide a practical means to express complex uncertainty without the associated difficulty of a type 2 fuzzy set. Proposed a new intuitionistic fuzzy-rough approach based on the conflict distance. Proposed a new framework based on intuitionistic fuzzy-rough sets with the constructive approach. Proposed the common model based on rough set theory and dual hesitant fuzzy sets named dual hesitant fuzzy-rough sets by consideration of constructive and axiomatic models.
Generalized to an integrative model based on considering interval-valued fuzzy sets and variable precision sets named generalized interval-valued fuzzy variable precision rough sets. Investigated a novel fuzzy-rough model based on constructive and axiomatic approaches to introduce the hesitant fuzzy-rough set model. Integrated interval-valued hesitant fuzzy sets with rough sets to introduce a novel model named the interval-valued hesitant fuzzy-rough set.
Examined the fuzzy-valued operations and the lattice structures of the algebra of fuzzy values. Proposed a new rough fuzzy approach for handling, representation and utilisation of diverse levels of uncertainty in knowledge. Presented a novel rough set model by integrating multigranulation rough sets over two universes and interval-valued hesitant fuzzy sets which is called interval-valued hesitant fuzzy multigranulation rough sets.
Investigated the fixed universal set where, unless otherwise stated, the cardinality of is infinite. Contents Related research topics Conference papers on the topic 'Rough sets'. Qing Shen and Yunliang Jiang. IEEE, Pawlak, Zdzislaw. Chen, Ray-Ming. PTI, Paris, France: Atlantis Press, Fan, Bing-Jiao, Eric-C. Ma, Li, and Li-Li Wei. Feng, Feng. Simovici, Dan A. Li, Xiaonan, and Huangjian Yi.
Zhao, Qing, and William Zhu. Azam, Nouman, and Afzaal Ahmad. Zhu, F. Bozi, K. ITI Pawlak, Z. Peters, and A. Das-Gupta, P. Ping Zhang and Kai-Quan Shi.