important in the decision model. The condition attribute c 3 could be omitted from the decision table, when
u ¼ 0:8 is assumed. 6. Conclusions
The presented variable precision **fuzzy** **rough** set (VPFRS) model with asymmetric bounds is a suitable tool for analyzing information systems with crisp or **fuzzy** attributes. We proposed to express the human operator’s decision model in the form of decision table with **fuzzy** attributes. The **fuzzy** character of attributes corre- sponds with the human ability to inference, using linguistic concepts rather than numbers. Particular steps of the VPFRS approach were illustrated by simple examples. It was shown that relaxation of strong inclusion requirements of one **fuzzy** set in another **fuzzy** set (admitting of a certain misclassiﬁcation level in the human operator’s control) leads to an increase of the approximation quality of the decision system. In this paper, we discussed an extended and improved version of our VPFRS model. It is compatible with the **fuzzy** **rough** set concept of Dubois and Prade. However, there is more than one possibility to deﬁne **fuzzy** **rough** sets. This is not only due to diﬀerent forms of **fuzzy** operators (intersection, union, implication) that can be used in def- initions of **fuzzy** **rough** **approximations**. We proposed a uniﬁed form of the crisp lower and upper approxima- tions that could also be applied to deﬁning new models of **fuzzy** **rough** set. Thus, our aim of future research is to introduce and investigate new classes of **fuzzy** **rough** sets and VPFRS models, which go beyond the widely used concept of Dubois and Prade.

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Auto Associative Neural Networks are a kind of neural network that are connected tightly with other neurons of their previous and next levels. In [16] the authors used an AANN to impute missing values. In the first step, an AANN is trained with instances which do not contain missing values. During the learning procedure, when instances containing missing values are given to the AANN, the weights are not updated but instead missing values are predicted and the **process** is repeated again. Another method based on AANN is introduced in [16]. In this model an AANN is used with inputs from both the dataset and the output of a genetic algorithm estimator. The GA estimates **approximations** for the missing values and gives them to the AANN. Then the outputs of the AANN are examined to see if the error is in a sensible range or not. If the error is minimized then the final values of the missing features are produced. Otherwise, the GA component generates another approximation for the missing values and this is given to the AANN again. This cycle is repeated until the error is minimized. Multi Task Learning (MTL) [20] is another type of neural network which solves more than one problem at the same time. These usually have more outputs than standard networks because they solve more than one problem. In the missing value imputation domain MTLs have one standard output which is the decision feature of the dataset and also outputs which try to predict missing values of features. Its major drawback is that it uses the quadratic error as the cost function to be minimized during the training phase.

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rules are more compact. A good feature selection step will remove unnecessary attributes which may affect both rule comprehension and rule prediction performance.
The work on **rough** set theory (RST) offers an alternative, and formal methodology that can be employed to reduce dimensionality of datasets, as a preprocessing step to assist any chosen modelling method for learning from **data**. It helps to select the most information-rich features in a dataset, without transforming the **data**, whilst at the same time attempting to minimise information loss during the selection **process**. Computationally, the approach is highly efficient, relying on simple set operations, which makes it suitable as a preprocessor for techniques that are much more complex. Unlike statistical correlation-reduction approaches [5], RST requires no human input or intervention. Most importantly however, it retains the underlying semantics of the **data**, which results in models that are more transparent to human scrutiny.

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Integration of **fuzzy** sets [ 6 , 7 ] and **rough** sets [ 14 – 17 ] under **fuzzy** **rough** computing or **rough** **fuzzy** computing has recently drawn the attention of researchers. Many relationships have been established to extend and integrate the underlying concepts of these two methodologies judiciously to deal with additional aspects of **data** imperfection, especially in the context of granular computing. The main purpose of such hybridization [ 19 ] is to provide a high degree of flexibility [ 20 ], robust solutions and advanced tools for **data** analysis [ 21 ], and a framework for efficient uncertainty handling [ 22 ]. This **rough** **fuzzy** paradigm may also be considered as a tool for modeling the aforesaid f-granular characteristics of perception-based computing. In **rough** set theory, one starts with crisp equivalence classes, in the same way that **fuzzy** equivalence classes are central to the **fuzzy** **rough** sets approach. Each equivalence class may be used as a granule. The concept of crisp equivalence classes can be extended to **fuzzy** equivalence classes by the inclusion of a **fuzzy** tolerance relation on the universe, which determines the extent that two elements are similar in a relation. **Fuzzy** **rough** sets [ 11 ], based on a **fuzzy** tolerance relation, provide a means by which discrete or real-valued noisy **data** can be effectively reduced without any additional information about the **data** (such as thresholds on a particular domain of universe) for its analysis. The granulation structure produced by an equivalence class provides a partition of the universe. The intension of it is to approximate an imprecise concept in the domain of universe by a pair of approximation concepts, called lower and upper **approximations**. These **approximations** are used to define the notion of positive degree of each object, and this is used to define the dependency factor of each conditional attribute, all of which are then used to extract the domain knowledge about the **data**.

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Proof. The properties depend on the following crucial fact, which can be derived easily from ( 18 ) and ( 19 ).
Cl ≤ t ( x ) = ¬ Cl ≥ t +1 ( x ) (22)
for x ∈ U and 1 ≤ t ≤ n − 1.
The precisiation of **data** means any new information about objects in U that is added to the **data** tables. It can be a new criterion or a piece of information that further refines the subsets of evaluations or assignments of an object. **Rough** set approaches usually require that more precise information about objects does not reduce the lower **approximations** of the decision classes. Thus, two types of precisiation properties are investigated in [ 1 ]. One is for the situation when new attributes or criteria are added to the decision table; and the other is for when more specific information about the evaluations and assignments of objects is added. The first type depends on the monotonicity of the t-norm, s-norm, and implication operations. Recall that the t-norm and s-norm operations are non-decreasing in their respective arguments, while the implication operation should be non-increasing in its left argument and non-decreasing in its right argument.

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pen as part of the same transaction. We benefit both from the robust recovery architecture of the storage subsystem and also from other features such as online backup mechanisms. The recovery operation for se- rial processing is a more challenging because the large amounts of runtime state managed in main memory structures by operators in executor and the decoupled nature in which durable state is written out originally by the archiver. The crash recovery therefore involves standard database style recovery of all durable state, making all serial processing archives self-consistent with each other as well as the latest committed **data** from their underlying archive and rebuilding the run- time state of the various operators in executor. The ability to have robust recovery implementation that is capable of quickly recovering from a failure is essen- tial. Furthermore, the longer it takes to recover from a failure the more the amount of pent-up **data** has gath- ered and the longer it takes to catch up to the live **data**. The recovery from failure in distributed PFRSVM is assessed in terms of delay, **process** and correct (DPC) protocol which handles crash failures of processing nodes and network failures [24]. Here, the choice is made explicit as the user specifies an availability bound and it attempts to minimize the resulting in- consistency between the node replicas while meeting the given delay threshold. The protocol tolerates oc- currence of multiple simultaneous failures that occur during recovery. In DPC each node replica manages its own availability and consistency by implementing state machine as shown in figure 9 that has three states viz. STABLE, UPSTREAM FAILURE (UP FAIL- URE), and STABILIZATION.

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The **rough** set theory and **fuzzy** set theory are extensions of classical set theory, and they are related but distinct and complementary theories. The **rough** set theory[1,2] is mainly focused on crisp information granulation, while its basic concept is indiscernibility, for example, an indiscernibility between different objects deduced by different attribute values of described objects in the information system; whereas the **fuzzy** set theory is regarded as a mathematical tool for imitating the fuzziness in the human classification mechanism, which mainly deals with **fuzzy** information granulation. Because of its simplicity and similarity with the human mind, its concept is always used to express quantity **data** expressed by language and membership functions in the intelligent system. In **fuzzy** sets, the attributes of elements may be between yes and no. For example, a beautiful scenery, we cannot simply classify the beautiful scenery into a category between yes and no. For the set of beautiful scenery, there does not exist good and definite border. The **fuzzy** sets cannot be described with any precise mathematical formula, but it is included in the physical and psychological **process** of human's way of thinking, because the physiology of human reasoning is never used any precise mathematical formula during the physical **process** of reasoning, and **fuzzy** sets is important in the pattern classification. Essentially, these two theories both study the problems of information granularity. The **rough** set theory[3,4] studies **rough** non-overlapping type and the **rough** concept; while the **fuzzy** set theory studies the fuzziness between overlapping sets, and these naturally lead to investigating the possibility of the "hybrid" between the **rough** sets and the **fuzzy** set. The hybrid of **rough** set and **fuzzy** set can be divided into three kinds of **approximations** that are the **approximations** of **fuzzy** sets in a crisp space, the **approximations** of crisp sets in **fuzzy** approximate space, and the **approximations** of **fuzzy** sets in **fuzzy** approximate space[8,9]. The paper mainly discusses the second case. In order to simulate the situation of this type, Dubios introduced the concept of **fuzzy** **rough** sets (Dubois and Prade, 1990)[6], which is an extension of **rough** set approximation deduced from a crisp set in **fuzzy** approximate space.

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2.5. A novel approach: A high-performance parallel and distributed computation using mapreduce In order to compute an optimal set of cut-points, most of discretization algorithms perform an iterative search in the space of candidate discretizations, using different types of scoring functions for evaluating a discretization, that take a lot of time. In this paper, we propose a parallel **process** of discretization based on MapReduce using sliding grid. A sliding grid is specified by defining its range M and slide S. The range M is an interval of discretization while the slide S specifies the portion of the grid that is moved forward. A sliding window is specified as a tuple (M,s). A smooth sliding specification is highly desired where the slide S issmall relative to the range M. where𝑆 < 𝑀. The proposed algorithm based on MapReduce computed for each node i(𝑃 𝑖 ⊆ 𝐴) is a parallel **process** that consists of three steps: map, shuffle, and reduce as shown in Figure 5.

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Covering is a common type of **data** structure and covering-based **rough** set theory is an efficient tool to **process** this type of **data**. Lattice is an important algebraic structure and used extensively in investigating some types of generalized **rough** sets. This paper presents the lattice based on covering **rough** **approximations** and lattice for covering numbers. An important result is investigated to illustrate the paper.

Keywords: **Rough** sets; **Fuzzy** sets; **Fuzzy** **rough** sets; Lower **approximations**; Upper **approximations**; Axioms
1. Introduction
**Rough** set theory [16,17] is a new approach for reasoning about **data**. It has achieved a large amount of real applications such as medicine, information analysis, **data** mining, control and linguistics. As a tool to handling imperfect **data**, it complements other theories that deal with uncertain **data**, such as probability theory, evi- dence theory and **fuzzy** set theory. The main idea of **rough** sets corresponds to the lower and upper approx- imations. Pawlak’s deﬁnitions for the lower and upper **approximations** were originally introduced with reference to an equivalence relation. Many interesting properties of the lower and upper **approximations** have been derived by Pawlak [16,17] based on the equivalence relations. In this paper, we study a reverse problem. That is, can we characterize the notion of the lower and upper **approximations** in terms of those properties? We answer the question aﬃrmatively.

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In (**fuzzy**) **rough** set theory, the **data** at hand is represented as an information system, consisting of objects described by a set of attributes. Examples include an information system of patients (objects) described by symptoms and lab results (attributes), or an information system of online customers (objects) described by their demographics and purchase his- tory (attributes). Inconsistencies arise when objects in the information system have the same attribute values but belong to a different concept (i.e. a subset of objects in the **data**), meaning that the available attributes are not able to discern the concept. For instance, two patients with very similar symptoms might still have been diagnosed with a different disease, and out of two customers with very similar demographics and purchase history, one might have clicked on an advertisement while the other one did not. An important question is how to take such inconsistencies that arise in training **data** into account when building models to predict medical diagnosis for patients, or click behavior of future customers. To tackle this problem, Pawlak’s **rough** set theory [11] relies heavily on the construction of the so-called lower and upper **approximations** of concepts. These **approximations** are constructed based on an indiscernibility relation induced by the attributes in the information system. The original **rough** set theory could only handle nominal (discrete) attributes and requires discretization of continous attributes. **Fuzzy** **rough** set theory extends the original **rough** set theory for information systems with con- tinuous valued attributes, and detects gradual inconsistencies within the **data**.

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Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
1. Introduction
**Fuzzy** set theory was ﬁrst proposed by Zadeh [1] . It is an important mathematical approach to uncertain and vague **data** analysis, and has been widely used in the area of **fuzzy** decision making, **fuzzy** control, **fuzzy** inference, and so on [2–5] . Thereafter, the theory of **rough** set was proposed by Pawlak [6] , which was though of as another powerful tool for managing uncertainty that arises from inexact, noisy, or incomplete information. In terms of method, it was turned out to be method- ologically signiﬁcant in the domains of artiﬁcial intelligence and cognitive science, especially when representating or reason- ing with imprecise knowledge, machine learning and knowledge discovery. In recent years, the combination of **fuzzy** set theory and **rough** set theory has been studied by many researchers [7–13,31–35] , hence, many new mathematical methods are generated for dealing with the uncertain and imprecise information, such as the **fuzzy** **rough** sets and **rough** **fuzzy** sets, etc. Meantime, many metric methods are presented and investigated by different authors, in order to measure the uncer- tainty and ambiguity of the different sets [14–20] .

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Pawlak [11, 12] proposed the theory of **rough** sets. Keyun Qin and Pei [7] successfully compared **fuzzy** **rough** set models and **fuzzy** topologies on a finite universe. Mathew and John [3] established and developed topological structures on **rough** sets. **Rough** topology in terms of **rough** sets was introduced by Lellis Thivagar et al. [8]. In [4, 5, 10], the concept of **fuzzy** **rough** sets were studied by replacing crisp binary relations with **fuzzy** relations on the universe.

Abstract. We define the concept of **rough** limit set of a sequence of **fuzzy** numbers and obtain the relation between the set of **rough** limit and the extreme limit points of a sequence of **fuzzy** numbers. Finally, we investigate some properties of the **rough** limit set.

[16] L.A. Zadeh, “Similarity Relations and **Fuzzy** Orderings”, Information Science, vol. 3, no. 2, 1970, pp. 177-200.
Rolly Intan received his B.Sc. in Computer Engineering from Sepuluh Nopember Institute of Technology, Surabaya Indonesia in 1991, and his M.A.Sc. in Compter Science from International Christian University, Tokyo, Japan in 2000. His Doctor of Enggineering in Computer Science and Engineering was obtained from Meiji University, Tokyo, Japan in 2003. He is currently a professor in Informatics Engineering, Petra Christian University, Surabaya, Indonesia. He is the former rector (president) of Petra Christian University, Surabaya. He serves as chair, co-chair and committee member in many international conferences, including IAENG International Conference on **Data** Mining and Applications since 2008. His primary research interests are **Fuzzy** Logic and Systems, **Rough** Sets, Granular Computing and **Data** Mining.

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We therefore consider here a Gaussian **process** approximation approach to speed-up real time analysis of disease **data** when other methods are too complex. This simultaneously accounts for stochasticity and avoids the problems of direct fitting of ODE models. This approximation is applied initially to the stochastic SIR (susceptible-infectious-removed) model. We also consider the SEIR (susceptible-exposed-infectious-removed) model, although it is easy to use this approximation scheme with a more complex compartmental model or even models used outside epidemiology. We stress that our approach involves an additional approximation step beyond the derivation of a stochastic differential equation (SDE) limit, that must be controlled and which is the main technical component of our work.

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L.A.Zadeh[12] introduced the notion of a **fuzzy** sets, and it is now a rigorous area of research with manifold applications ranging from engineering and computer science to medical diagnosis and social behaviour studies. Rosenfeld[8], applied the notion of **fuzzy** sets to algebra and introduced the notion of **fuzzy** subgroups. Jun et.al.,[4] defined **fuzzy** ideals in -near-rings. Meenakumari and Tamizhl Chelvan[5] discussed **fuzzy** bi-ideals in -near-ring.