Abstract
Multisource information systems and multigranulation intuitionistic fuzzy rough sets are important extended types of Pawlak’s classical rough set model. Multigranulation intuitionistic fuzzy rough sets have been investigated in depth in recent years. However, few studies have considered this combination of multisource information systems and intuitionistic fuzzy rough sets. In this paper, we give the uncertainty measure for multisource intuitionistic fuzzy information system. Against the background of multisource intuitionistic fuzzy information system, each information source is regarded as a granularity level. Considering the different importance of information sources, we assign different weights to them. Firstly, the paper proposes an optimal source selection method. Secondly, we study the weighted generalized, weighted optimistic, and weighted pessimistic multigranularity intuitionistic fuzzy rough set models and uncertainty measurement methods in the multisource intuitionistic fuzzy information system, and we further study the relationship between the three models and related properties. Finally, an example is given to verify the validity of the models and methods.
1. Introduction
Rough set theory [1] was proposed by Polish mathematician Pawlak in 1982. It is an effective mathematical tool to analyze and process inaccurate data and uncertain information. Rough set theory has received more and more attention in recent years. It is widely used in many fields such as natural science, social science, and engineering technology [2–11]. Uncertainty measurement [12, 13] is one of the important research contents in rough set theory. It can measure the dependency and similarity between attributes and provide an effective measurement tool for attribute reduction and cluster analysis [14]. Traditional uncertainty measurement considers single-source information system. With the advent of the era of big data, it is necessary to study uncertainty measurement methods in multisource intuitionistic fuzzy information system [15].
In current study, by extending the equivalence relationship to a general binary relationship, Qian et al. extended the single-granularity rough set model to multigranularity structure [16, 17]. By combining fuzzy set theory and rough set theory, a fuzzy rough set model [18] and rough fuzzy set model [19, 20] are obtained. Literature [21] extends the fuzzy set theory to the intuitionistic fuzzy set theory and extends the relationship between elements and sets from membership degree to nonmembership degree and hesitation degree. Therefore, the intuitionistic fuzzy rough set theory is a very effective mathematical tool when analyzing and processing inaccurate, incomplete, and other rough information, and the result accuracy is significantly improved. Literature [22, 23] combined rough set theory with intuitionistic fuzzy set theory and established an intuitionistic fuzzy rough set model. Literature [24] investigated the upper approximation reduction problem of intuitionistic fuzzy information system based on dominant relationship. Literature [25, 26] extended the single-granularity intuitionistic fuzzy rough set model to the multigranularity intuitionistic fuzzy rough set model and presented the optimistic multigranularity and pessimistic multigranularity intuitionistic fuzzy rough set models under the dominant relationship. Literature [27] proposes the intuitionistic fuzzy soft set, which is an effective tool for solving multiple attribute decision-making with intuitionistic fuzzy information. Literature [28] proposes the hesitant fuzzy sets and studies their relationship with intuitionistic fuzzy sets.
In the research of multigranularity intuitionistic fuzzy rough set model, if all granularity levels are considered equally important, optimistic multigranularity only requires one granularity knowledge and target concept to meet inclusion relation, while pessimistic multigranularity requires all granularity knowledge and target cocept to meet inclusion relation, ultimately leading to inaccurate decision-making results. The importance of granulation is usually different for decision-making in real life. Taking various factors into account, some granularity is very important, so we give it a larger weight, but some granularity is not important, so we give it a smaller weight. Considering the importance of different granularity levels, literature [29] proposed a weighted multigranularity intuitionistic fuzzy rough set model.
Now, in our real life, we no longer face a single-source information system, but a multisource information system. Multisource information systems [30] are used to represent information that comes from multiple sources. Literature [31] proposed a fuzzy multigranulation decision-theoretic rough set model in multisource fuzzy information systems. Literature [32] investigated the attribute reduction in multisource decision systems. Literature [33] combined rough set model and multisource decision systems and established a decision-theoretic rough set model of multisource decision systems. Literature [34] built the information source selection criteria and proposed some principles of information fusion. Literature [35–42] proposed information fusion methods of multisource information system. In this paper, we treat each single information system as a granular structure. As the number of information sources increases, a large amount of data is unreliable. Therefore, we propose a corresponding algorithm to select reliable information sources and one that can greatly improve the efficiency of information processing. Up to now, few scholars have combined the rough set model with the multisource intuitionistic fuzzy information system. Different weights are given to the granularity; the weighted multigranularity intuitionistic fuzzy rough set models and the uncertainty measurement methods for multisource intuitionistic fuzzy information system are proposed. This is the purpose of this article.
The rest of this paper is organized as follows. In Section 2, we mainly review the relevant concepts and properties of intuitionistic fuzzy rough sets, multigranulation intuitionistic fuzzy rough sets, and multisource intuitionistic fuzzy information system. In Section 3, we first study the optimal source selection of multisource intuitionistic fuzzy information system. Further, the weighted multigranulation intuitionistic fuzzy rough set model and its related properties of multisource intuitionistic fuzzy information system are researched. In Section 4, we give the uncertainty measurement methods for the weighted multigranulation intuitionistic fuzzy rough set model of multisource intuitionistic fuzzy information system. At the same time, we verify the effectiveness of the proposed models and methods through a specific example. Section 5 uses a numerical experiment to verify the effectiveness of the proposed methods. In Section 6, we present the conclusion and the future work.
2. Preliminaries
This section mainly reviews related concepts and properties of intuitionistic fuzzy rough set models, multigranulation intuitionistic fuzzy rough set models, and multisource intuitionistic fuzzy information system.
2.1. Intuitionistic Fuzzy Rough Set Model
Definition 1. (see [21]): Given the universe of discourse , an intuitionistic fuzzy rough set on is defined by , where the functions and satisfy for all . and are called the degrees of membership and nonmembership of element to , respectively. The family of all intuitionistic fuzzy sets in is denoted by . When , degenerates into a fuzzy set.
Definition 2. (see [21]): Let , ; then:(1)The supplementary set of .(2).(3).(4).(5).(6).(7).
Definition 3. (see [21]): Let be an intuitionistic fuzzy information system, where is a nonempty finite set of objects called the universe of discourse; is a nonempty finite attribute set; is the attribute value set; is the information function, ; a binary dominant relation is defined as follows:Obviously, this binary dominant relation satisfies reflexivity, antisymmetry, and transitivity, and . Based on the above dominant relation, the dominant class definition of object can be obtained as follows:
Definition 4. (see [22]): Let be an intuitionistic fuzzy information system; for any , the lower approximation and upper approximation of for are defined as follows:By definition, consists of all objects that are definitely contained in the set . consists of all objects that are possibly contained in the set . If , then is the exact set of ; otherwise, it is the rough set about .The positive, negative, and boundary regions of can be defined as follows:
Definition 5. (see [22]): Let be an intuitionistic fuzzy information system; for any , the approximate accuracy and the roughness of the target set are defined as follows:when , .
Definition 6. (see [23]): Let be a fuzzy sets on universe ; the nonmembership degree of is defined as follows:Then, is the correspondence intuitionistic fuzzy sets of the fuzzy set .
2.2. Multigranulation Intuitionistic Fuzzy Rough Set Model
Multigranulation rough set model was first proposed by Qian et al. It is an extension of the classical rough set theory. In this subsection, the single-granulation intuitionistic fuzzy rough set is extended to the multigranulation intuitionistic fuzzy rough set model under multiple dominant relations. Generalized multigranulation intuitionistic fuzzy rough set is the generalization of optimistic multigranulation intuitionistic fuzzy rough set and pessimistic multigranulation intuitionistic fuzzy rough set.
Given the definition of the support characteristic function, we use this function to complete the object selection.
Definition 7. (see [26]): Let be an intuitionistic fuzzy information system, , ; for any , the support characteristic function of for is denoted asThe support characteristic function is used to describe the inclusion relation between dominance class and concept , Which indicates whether object accurately supports by , or whether object has a positive description of by .
Definition 8. (see [26]): Let be an intuitionistic fuzzy information system, , ; is the characteristic function of for . Given information level , for any , the generalized lower and upper approximation of are defined as follows:If , then the target set is generalized and definable; otherwise, it is generalized and rough. is called the generalized multigranulation intuitionistic fuzzy rough set model .
Definition 9. (see [26]): Let be an intuitionistic fuzzy information system, , ; is the support characteristic function of for ; for any , the pessimistic lower and upper approximation of are defined as follows:where “” denotes “or” and “” denotes “and.”
If , then the target set is pessimistic and definable; otherwise, is pessimistic and rough. is called the pessimistic multigranulation intuitionistic fuzzy rough set model .
Definition 10. (see [26]): Let be an intuitionistic fuzzy information system, , ; is the support characteristic function of for ; for any , the optimistic lower and upper approximation of are defined as follows:where “” denotes “or” and “” denotes “and.”
If , then the target set is optimistic and definable; otherwise, is optimistic and rough. is called the optimistic multigranulation intuitionistic fuzzy rough set model .
2.3. Attribute Reduction of Intuitionistic Fuzzy Information System Based on Dominant Relation
In this subsection, we define the attribute importance of intuitionistic fuzzy information system based on the dominant relation. We also give the attribute reduction algorithm of intuitionistic fuzzy information system based on knowledge rough entropy.
Definition 11. (see [9]): Let be an intuitionistic fuzzy information system, , ; is a binary dominant relation; the rough entropy of is defined asIt is obvious that there are minimum and maximum values of the rough entropy. When is the finest classification, the rough entropy of has a minimum value of 0; if is the coarsest classification, the rough entropy of has a maximum value of .
Each information system has many attributes, but some of these are redundant. To measure the significance of a single attribute, [3, 4] give the concepts of relative importance and absolute importance of attribute.
Definition 12. Let be an intuitionistic fuzzy information system, ; for any , for any , the relative importance and absolute importance of attribute in attribute set is defined asAccording to this definition, the following properties are established:(1).(2)Attribute a is necessary .(3)The attribute core of A is .
Definition 13. Let be an intuitionistic fuzzy information system, ; if , for any and , is the attribute reduction of intuitionistic fuzzy information system based on dominance relation.
Since the attribute core is a subset of attribute reduction, in the heuristic reduction process starting from the attribute core, we often add attributes to attribute core through a measurement method to obtain attribute reduction. Next, an attribute reduction algorithm for intuitionistic fuzzy information system based on knowledge rough entropy is given.
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Example 1. Table 1 shows an intuitionistic fuzzy information system , where the universe and attribute set . We use Algorithm 1 to calculate the minimum attribute reduction of the intuitionistic fuzzy information system given in Table 1.
From this table, it is easy to calculate the granular structure:
.
According to Definition 11, the rough entropy of is as follows:
.
After calculating the core of attribute set ,
,
Hence, , .
, .
Accordingly, . Let .
, .
Therefore, , .
; for any , we calculate as follows:
,
,
.
Thus, .
Then, let or .
Similarly, by calculation, we know that , .
Therefore, the minimal attribute reduction of is or .
2.4. Multisource Intuitionistic Fuzzy Information System (MSIFIS)
In this subsection, we introduce multisource intuitionistic fuzzy information systems. When a person obtains information about a group of objects from different sources, each source can be regarded as a classical information system, which is an attribute with some intuitionistic fuzzy attribute values. The information system is called multisource intuitionistic fuzzy information system (MSIFIS). A is shown in Figure 1.
This paper mainly discusses the case which shares the same structure. That is, the same object, attributes, and the value of the object’s attributes have the same digital characteristics under different information sources.
Definition 14. (see [30]): Let be a multisource intuitionistic fuzzy information system, where:(1) is a set of nonempty finite objects called the universe.(2) is a finite nonempty set of the attributes of each subsystem.(3), is the value range of attribute .(4) represents the corresponding relation between the object and the feature under the information source .Let be a multisource intuitionistic fuzzy decision information system, where and is the decision attribute.
3. Weighted Multigranulation Intuitionistic Fuzzy Rough Set Model for MSIFIS
In the theory of multigranulation intuitionistic fuzzy rough set, optimistic multigranulation only requires the existence of the granular knowledge and the target concept to satisfy the inclusion relation; the requirements are too loose for approximate characterization. However, the pessimistic multigranularity requires that the knowledge granule and the target concept satisfy the inclusion relation at all granularity spaces; the requirements are too strict for approximate characterization. These multigranularity models treat each granularity space equally.
However, in practical applications, considering the application background and user preferences, the granularity spaces are not equally important. Therefore, in this section, we study multigranularity intuitionistic fuzzy rough set with weights in multisource intuitionistic fuzzy information system.
3.1. Optimal Source Selection for
MSIFISs are used to express information from multiple sources. As the number of information sources increases, the selection of reliable information sources is a key issue in the field of information technology research. To characterize the effectiveness of an information source, we define the two source quality metrics of the internal-confidence degree and external-confidence degree.
Definition 15. Let be a multisource intuitionistic fuzzy information system. For each single information source , let be the reduction of . The internal-confidence degree of can be defined as follows:From the above definition, the following is obvious:(1) is the ratio of the cardinalities of and , and .(2)If , the majority of the attributes are useful and the source is reliable. In practical applications, different thresholds can be defined in different fields according to the specific requirements.
Definition 16. Let be a multisource intuitionistic fuzzy information system; , the difference between them can be defined as follows:where is the dominant class of with respect to .
According to this definition, the following is obvious:(1).(2).(3)When the single information sources and have the same granular structure, the difference between and has a minimum of 0. If the single information source has the finest granular structure and has the coarsest structure, then the difference between and reaches a maximum of .
Definition 17. Let be a multisource intuitionistic fuzzy information system; , the external correlation between and can be defined aswhere is the difference between and .
From this definition, the following is obvious:
To clarify the relationship between the external-confidence degrees of and , we can construct an external-confidence degree matrix as follows:where .
Definition 18. Let be a multisource intuitionistic fuzzy information system; , the external-confidence degree between and can be defined asFrom this definition, the following is obvious:(1)For any , .(2)Similar to the internal-confidence degree, different thresholds may be applicable in different fields according to specific requirements.
Definition 19. Let be a multisource intuitionistic fuzzy information system; , we define a total score for information source as follows:where is the internal-confidence degree of a source , and is the external-confidence degree of a source .
Example 2. Table 2 shows a multisource intuitionistic fuzzy information system, which consists of three intuitionistic fuzzy information tables; represents six evaluated objects, and represents conditional attribute set.
From this table, it is easy to calculate the granular structure for each information source:
, , .
From the result of example 1, .
Similarly, , and .The calculation results of all sources are presented in Table 3.
Then, the internal-confidence degree of source is as follows:
,
,
.
Next, the external-confidence degree between and can be calculated as follows:
,
, , .
By Definition 18, the external correlation between and can be computed. The result is as follows:
,
, , .Thus, we can get the external-confidence degree matrix as follows:Therefore, the external-confidence degrees for each information source is .
Furthermore, the total score for each information source can be calculated using Definition 19: , , .Therefore, the quality ranking of information sources is , and is the optimal source of the multisource intuitionistic fuzzy information system.
3.2. Weighted Generalized Multigranularity Intuitionistic Fuzzy Rough Set Model for
Definition 20. Let be a multisource intuitionistic fuzzy information system; is the support characteristic function of for . If the weight corresponding to each granularity space is derived from each dominance relation as , parameter denotes the information level with respect to ; for any , the lower approximation and upper approximation of weighted generalized multigranularity intuitionistic fuzzy rough set are defined as follows:If , then the target set is definable for ; otherwise, is rough for . is called the weighted generalized multigranularity intuitionistic fuzzy rough set model for .
In the weighted generalized multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions of can be defined as follows:The basic properties of the weighted generalized multigranularity intuitionistic fuzzy rough set for are given by the following theorem.
Theorem 1. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , parameter denotes the information level with respect to ; , the following conclusions hold:(1), .(2).(3).(4).(5), .(6), .(7), .(8), .(9), .
Proof. (1)Because . Similarly, . Therefore, the property is clearly established.(2)For any , we can know . In addition, , so , s.t. . Namely, . We have ; thus, . . Hence, . Therefore, property (2) has been proved.(3)For any , we can easily get . Therefore, . Thus, , . Therefore, property (3) has been proved.(4)According to (3), , . Therefore, property (4) has been proved.(5)For any . Therefore, . Then, we can get that . Meanwhile, for any , , . Then, we obtain that . Therefore, the property is clearly established.(6) . Thus, . Meanwhile, for any . Therefore, the property is clearly established.(7)Because , we can know . According to property (5), . Therefore, . Thus, . Meanwhile, according to property (5), . Therefore, . Thus, . Therefore, the property is clearly established.(8)For any , we have . Therefore, . Then, implies that . Similarly, we can prove that . Therefore, property (8) has been proved.(9)According to properties (2) and (7), .The following is the proof: .
For any , we can know .
; thus, ; namely, .
Then, according to property (7), we have .
Therefore, ; we can get , and .
Therefore, , from which one can get that .
Meanwhile, we can prove that .
Property (1) shows that in multisource intuitionistic fuzzy information system, the lower approximation operator, and the upper approximation operator of weighted generalized multigranularity intuitionistic fuzzy rough sets satisfy duality. Property (2) illustrates the inclusion relationship between the lower approximation, the upper approximation, and the target concept. Properties (3) and (4) show the approximation of two special sets. The lower and upper approximation of the empty set and the universe are themselves. Properties state the monotonicity of and . Moreover, property (9) expresses the idempotency between the lower approximation operator and the upper approximation operator.
3.3. Weighted Optimistic Multigranularity Intuitionistic Fuzzy Rough Set Model for
Definition 21. Let be a multisource intuitionistic fuzzy information system; is the support characteristic function of for . If the weight corresponding to each granularity space is derived from each dominance relation as , , the lower approximation and upper approximation of weighted optimistic multigranularity intuitionistic fuzzy rough set are defined as follows:If , then the target set is optimistic and definable for ; otherwise, is optimistic and rough for . is called the weighted optimistic multigranularity intuitionistic fuzzy rough set model for .
In the weighted optimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions of can be defined as follows:The basic properties of the weighted optimistic multigranularity intuitionistic fuzzy rough set for are given by the following theorem.
Theorem 2. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , , the following conclusions hold:(1), .(2).(3).(4).(5), .(6), .(7), .(8), .
Proof. It is similar to that of Theorem 1.
3.4. Weighted Pessimistic Multigranularity Intuitionistic Fuzzy Rough Set Model for
Definition 22. Let be a multisource intuitionistic fuzzy information system; is the support characteristic function of for . If the weight corresponding to each granularity space is derived from each dominance relation as , , the lower approximation and upper approximation of weighted pessimistic multigranularity intuitionistic fuzzy rough set are defined as follows:If , then the target set is pessimistic and definable for ; otherwise, is pessimistic and rough for . is called the weighted pessimistic multigranularity intuitionistic fuzzy rough set model for .
In the weighted pessimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions of can be defined as follows:The basic properties of the weighted pessimistic multigranularity intuitionistic fuzzy rough set for are given by the following theorem.
Theorem 3. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , , the following conclusions hold:(1), .(2).(3).(4).(5), .(6), .(7), .(8), .
Proof. It is similar to that of Theorem 1.
3.5. The Relationship between the Three Models of
In this subsection, we investigate the relationship between the three models of .
Theorem 4. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , , the following equations can be obtained:(1)If , then(2)If , then
Proof. It is easily obtained from Definition 15, Definition 20, and Definition 21.
From the above theorems, if , weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted pessimistic multigranularity intuitionistic fuzzy rough set. If , weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted optimistic multigranularity intuitionistic fuzzy rough set. Hence, weighted generalized multigranularity intuitionistic fuzzy rough set is a generalization of weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set. On the other hand, weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set are the special cases of weighted generalized multigranularity intuitionistic fuzzy rough set.
Theorem 5. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , , the following properties are established:(1).(2).
Proof. From Definition 20–Definition 22, the theorem clearly holds.
The theorem shows that the lower and upper approximation of the weighted generalized multigranularity intuitionistic fuzzy rough set are between weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set.
4. Uncertainty Measurement of
Like Pawlak rough sets, the uncertainty of knowledge is caused by the boundary region. The larger the boundary area, the lower the accuracy and the higher roughness. For the weighted multigranularity intuitionistic fuzzy rough set model in the MSIFIS, this section gives definition of rough accuracy, roughness, and attribute dependence.
Definition 23. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , parameter denotes the information level with respect to ; , the approximation accuracy and roughness of the weighted generalized multigranularity intuitionistic fuzzy rough set model of the set with respect to are defined as follows:Similarly, the approximation accuracy and roughness of the weighted optimistic multigranularity intuitionistic fuzzy rough set model of the set with respect to are defined as follows:The approximation accuracy and roughness of the weighted pessimistic multigranularity intuitionistic fuzzy rough set model of the set with respect to are defined as follows:
Definition 24. Let be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as , parameter denotes the information level with respect to ; , the approximation accuracy and roughness of the weighted generalized multigranularity intuitionistic fuzzy rough set model of the set with respect to are defined as follows:Similarly, the dependence of the weighted optimistic multigranularity intuitionistic fuzzy rough set model of the set with respect to is defined as follows:The dependence of the weighted pessimistic multigranularity intuitionistic fuzzy rough set model of the set with respect to is defined as follows:
Theorem 6. Let be a multisource intuitionistic fuzzy information system, where ; is the support characteristic function of for . If the weight corresponding to each granularity space is derived from each dominance relation as , for any , under the three rough set models, the relationships between the approximate accuracy, roughness, and dependence of the set with respect to are as follows:(1).(2).(3).
Proof. It is easy to obtain by Definition 23 and Definition 24.
The theorem shows that the accuracy, roughness, and dependence of the weighted generalized multigranularity intuitionistic fuzzy rough set model are between weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set model.
Example 3. In example 1, the quality ranking of the three information sources in the multisource intuitionistic fuzzy information system is calculated. Let us suppose that the granularity weights corresponding to the three information systems are assigned as ; the distribution of granularity weights in specific applications can be given subjectively based on the experience of domain experts, with threshold .
According to Definition 2, we calculate the dominant classes of object under each source.
Dominant classes under source 1 are as follows:
, ,
, ,
, .
Dominant classes under source 2 are as follows:
, ,
, ,
, .
Dominant classes under source 3 are as follows:
, ,
, ,
, .
Given a concept set , the support characteristic function of and support characteristic function of under each source are computed.
, , ,
, , ,
, , ,
, , ,
, , ,
, , .
,
, , ,
, , ,
, , ,
, , ,
, , ,
, , .
According to Definition 20, the lower approximation and upper approximation of the weighted generalized multigranularity intuitionistic fuzzy rough set are obtained as follows:
,
.
The positive region, boundary region, and negative region of the weighted generalized multigranularity intuitionistic fuzzy rough set are as follows:
,
,
.
The approximation accuracy, roughness, and dependence of the weighted generalized multigranularity intuitionistic fuzzy rough set are as follows:
According to Definition 21, the lower approximation and upper approximation of the weighted optimistic multigranularity intuitionistic fuzzy rough set are obtained as follows:
The positive region, boundary region, and negative region of the weighted optimistic multigranularity intuitionistic fuzzy rough set are as follows:
The approximation accuracy, roughness, and dependence of the weighted optimistic multigranularity intuitionistic fuzzy rough set are as follows:
According to Definition 22, the lower approximation and upper approximation of the weighted pessimistic multigranularity intuitionistic fuzzy rough set are obtained as follows:
The positive region, boundary region, and negative region of the weighted pessimistic multigranularity intuitionistic fuzzy rough set are as follows:
,
,
.
The approximation accuracy, roughness, and dependence of the weighted pessimistic multigranularity intuitionistic fuzzy rough set are as follows:
,
,
.
Hence,
,
.
,
,
.
5. Experimental Evaluation and Analysis
In this paper, three weighted multigranulation intuitionistic fuzzy rough set models for are studied, and the uncertainty measurement methods of different models are discussed. In this section, we use an experiment to show the effectiveness of the models and methods proposed in this paper. We propose Algorithm 2 to calculate the uncertainty measure of the weighted generalized multigranulation intuitionistic fuzzy rough set in . Similarly, by changing the threshold , a calculated weighted optimistic and pessimistic uncertainty measurement algorithm can be obtained.
The time complexity analysis of Algorithm 2 is as folloxws. From Steps , we calculate the dominant class of each object under each information source, and its time complexity is (q is the number of sources). From Steps , we calculate the support feature function of and under each information source, and the time complexity is . From Steps , we compute the upper and lower approximations of in MSIFIS, and the time complexity is .
As we all know, we cannot get directly from UCI (http://archive.ics.uci.edu/ml/datasets.html), so we need to process the data. First, we need to download the “winequality-red” and “winequality-white” data sets from UCI and divide them by the maximum number of each column in the data table to make them fuzzy data tables. Then, we use the MATLAB software to randomly generate two fuzzy data sets. Finally, the method given in literature [23] is used to make them intuitionistic fuzzy data sets.
The details of the four data tables are shown in Table 4. We use the above four data sets as the original intuitionistic fuzzy information system. Finally, we randomly select of the original data to add white noise and then randomly select of the remaining data to add random noise, and the rest of the data remains unchanged to generate four with ten sources. The entire experiment was run on a private computer. The specific operating environment, including hardware and software, is shown in Table 5.
We add white noise as follows: the real numbers with a normal distribution were first generated by MATLAB.We add random noise as follows:
We conducted ten experiments on each data set, assuming that the granularity weights corresponding to the three information systems are assigned as , with threshold , and the concept set is randomly generated from the target set. The uncertainty measurement of the three rough set models corresponding to each data set is shown in Tables 6–8, and the results are illustrated inFigure 2–5.
From Figures 2–5, we can find that the weighted pessimistic approximation accuracy is less than or equal to the weighted generalized approximation accuracy, which is also less than or equal to the weighted optimistic approximation accuracy; the weighted optimistic approximation roughness is less than or equal to the weighted generalized approximation roughness, which is also less than or equal to the weighted pessimistic approximation roughness. Similarly, the weighted pessimistic approximation dependence is less than or equal to the weighted generalized approximation dependence, which is also less than or equal to the weighted optimistic approximation dependence.
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optimistic multigranularity intuitionistic fuzzy rough set model requirements are too loose in selecting objects, but the pessimistic multigranularity intuitionistic fuzzy rough set model requirements are too strict in selecting objects. Therefore, in practical applications, the object selection can be completed by the weighted generalized multigranular intuitionistic fuzzy rough set model.
From Figures 2–5, in the same data set, has different calculation results in different multigranularity intuitionistic fuzzy rough set models. Therefore, in practical applications, we can complete the object selection by changing the threshold .
6. Conclusions
In this paper, in order to solve the problem of knowledge discovery in the , the weighted multigranulation intuitionistic fuzzy rough set models, combined with the idea of multigranulation, are studied. We also studied the relationship between them. In order to further study the multigranulation intuitionistic fuzzy rough set model in the , the uncertainty measurement methods of different models are discussed. Finally, the effectiveness of the proposed models and methods is verified through an example. In the future, we need to continue to study the granularity weight distribution of multisource intuitionistic fuzzy information system and its application to decision-making.
Data Availability
The original data were obtained from UCI.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Nature Science Foundation of Shanxi Province (No. 201901D111280).