A data fusion method in wireless sensor network based on belief structure

Considering the issue with respect to the high data redundancy and high cost of information collection in wireless sensor nodes, this paper proposes a data fusion method based on belief structure to reduce attribution in multi-granulation rough set. By introducing belief structure, attribute reduction is carried out for multi-granulation rough sets. From the view of granular computing, this paper studies the evidential characteristics of incomplete multi-granulation ordered information systems. On this basis, the positive region reduction, belief reduction and plausibility reduction are put forward in incomplete multi-granulation ordered information system and analyze the consistency in the same level and transitivity in different levels. The positive region reduction and belief reduction are equivalent, and the positive region reduction and belief reduction are unnecessary and sufficient conditional plausibility reduction in the same level, if the cover structure order of different levels are the same the corresponding equivalent positive region reduction. The algorithm proposed in this paper not only performs three reductions, but also reduces the time complexity largely. The above study fuses the node data which reduces the amount of data that needs to be transmitted and effectively improves the information processing efficiency.

strong support data fusion technology in the real world of imprecise research [1][2][3][4][5][6] and proposed other two reductions based on the positive reduction. The propose of three reductions will provide novel insights and different angles to information processing in wireless communication and transmission.
The basic structure of rough set theory is an approximation space consisting of a universe of discourse, in which lower and upper approximations are defined to approximate a undefinable set by using equivalence relations [7][8][9]. Research on rough set mainly focuses on attribute reduction [10] to fuse information on sensor nodes. From the perspective of granular computing, three extensions of rough set model have been proposed in terms of the characters of data, respectively, multi-granulation rough set based on multi-scale, multi-level and multi-angle [11][12][13][14][15][16][17][18][19][20]. A general concept of multi-granulation rough set based on multi-scale describes that an attribute of an object can only take one value in a single-scale information system where the object information is reflected at a fixed scale. We call such a single-scale information system as the classic Pawlak's information system. However, in practical, an object could take on as many different hierarchical values under the same attribute with respect to different scales. And, there do exist special relationships among these hierarchical levels. One example is that the examination results of English for students can be recorded as natural numbers between 0 and 100, and it can also be graded as "Excellent, " "Good, " "Medium, " "Bad, " "Unacceptable. " Sometimes, if needed, it might be graded into two values, "Passed" and "Failed". A hierarchy of such obtained information granules can be organized to a system which is called multi-scale information system.
The evidence theory represents the uncertainty through the belief and plausibility function derived by the mass function which the core concept is belief structure and evidence structure [21][22][23]. Recently, the combination of evidence theory and rough set model become one of the research hotspots. As introduced in Yao et al. [24], the adequate condition for belief structure exactly exists in the classic rough set. On the above basis, this study was extended to covering rough set by Chen et al. [25,26], who successfully employ the belief function and the plausibility function to describe the upper and lower approximations of the covering rough set, which means the numerical features of the rough set can be characterized by evidence theory. In particular, from the perspective of information fusion, Lin et al. [27] explore the relationship between evidence theory and classical multi-granulation rough sets, which shows that, in general, the classic optimistic multi-granulation rough set does not have its corresponding belief structure.
By introducing belief structure, this paper firstly studies the evidential characteristics of multi-granulation rough set based on multi-scale. On this basis, the positive region reduction, belief reduction and plausibility reduction are put forward in incomplete multi-granulation ordered information system and then analyze the consistency in the same level and transitivity in different levels, which can reduce data redundancy and circuit complexity and save node limited resources through data fusion.

Method
This study puts forward the positive region reduction, belief reduction and plausibility reduction in terms of reducing data redundancy of WSN and proposes an algorithm to reduce the time complexity of attribute reduction. This section firstly introduces the basic preliminaries of WSN information processing, multi-granulation rough set and belief structure. On this basis, the above three reductions are proposed and will be conducted in Sect. 3.

WSN information processing
WSN is mainly composed of nodes, sensor network and users, which the core task of nodes is data perceiving and processing [28][29][30]. According to a certain standard, n nodes can form m clusters and the cluster header is selected in each cluster, which can also represent this cluster at a higher level. Meanwhile, the same mechanism is also applied between cluster headers to form a hierarchical structure [1]. In the above model, the real world is regarded as an information system according to the realistic data observed and measured by WSN. Every single node is considered as an object from the world, and the environment is descripted by a group of attributions which also called observation data. The attribute set can be divided into condition attributes and decision attributes for practical requirements, which are the input and output of the real world, respectively [1,2,30].
This hierarchical routing structure focuses on data which makes the node only interact with their neighbors within a certain range through localized principle, as Fig. 1 shown. And the cluster header will perform data fusion in the cluster so that the sensor node only automatically obtains and transmits effective information. This is also the key to WSN information fusion. In this case, rough set theory particularly suitable for intelligent information fusion at the global level while data from different cluster headers will be aggregate in the sink node, which means that the multi-granulation rough set based on multi-scale can fuse data in the cluster, which ensure that a small amount of effective information is transmitted between the cluster header and sink node with respect to effectively balance information processing, energy consumption and system performance.

Multi-granulation rough set based on multi-scale
Definition 1 [18] Let U = {x 1 , x 2 , . . . , x n } be a nonempty finite set of objects called the universe of discourse, AT k = a k 1 , a k 2 , . . . , a k m be a set of attributions and a j be the attribute of multi-granulation. For each object in U, the attribute a j can take different values on the different levels of granulations. If f is the attribute value surjective function of different levels (that is, for every k representing the number of levels with a value of positive integer, there exists x ∈ U such that f (x) = k ) and V k is the domain of the attribute a k , then the quaternary MGIS = (U , AT k , f k , V k ) is called a multi-granulation information system.
From the above definition, the multi-scale information system will degenerate into the classic Pawlak information system when the number of granular levels is k = 1 . For convenience of description, the following simplifies the multi-granulation information system based on multi-scale as a multi-granulation information system. Definition 2 [18] Let MGIS = (U , AT k , f k , V k ) be a multi-granulation information system which arbitrary attribute a j has I levels of granulations. We further define the attribute of a j on the k-th level of granulations a k j : U → V k j represents a surjective function and V k j is the domain of the k-th scale attribute a k j (that is, for any 1 ≤ k ≤ I , there exists x ∈ U such that a k j (x) = * , where ( * ) means variable quantity). And the surjective function g k,k+1 is called the granular transformation function with variable quantity ( * ) as defined as follows: On the basis of Definition 2, clearly, the value of an object between different levels of granulations is not arbitrary and depended on the value of the lower level in a multigranulation information system, which means the value of a k+1 j (x) is determined by a k j (x).
Definition 3 [18] Let MGIS = (U , AT k , f k , V k ) be a multi-granulation information system which arbitrary attribute a j has I levels of granulations. For any 1 ≤ k ≤ I , the multi-granulation information system MGOIS can be called multi-granulation ordered information system if the attribute value range of any levels of granulations is all partial ordering.
And MGOIS * ≥ , a multi-granulation ordered information system with variable values ( * ) and null values, is collectively referred to as an incomplete multi-granulation ordered information system.
Definition 4 [18] Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation information system which has I levels of granulations. For any 1 ≤ k ≤ I , if the attribute set A k ⊆ AT k and two arbitrary elements x, y ∈ U , then there exists can be called an incomplete multi-granulation ordered information system dominance relationship.
On the above basis, if the pair (x, y) ∈ R * ≥ AT k , then [X] * ≥ AT k means that y is finer than x or x is coarser than y. The relationship R * ≥ AT k can be considered as a kind of surjection from U to P(U) where P(U) is a power set. U /R = y ∈ P(U) | x ∈ U is a covering of universe of discourse.
Definition 5 [18] Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For every X ⊆ Uand any 1 ≤ k ≤ I , the lower and upper approximations of X in the k-th level of granulations are defined as From Definition 5, the relationship between the lower and upper approximations in the same levels of granulations has been clearly proved. R * ≥ AT k (X) and R * ≥ AT k (X) satisfy the following properties, which will be the theoretical foundation for the further discussions in this paper.
Proposition 1 [18] Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multigranulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I and two elements X, Y ⊆ U , we denote the complement of X in U as ∼ U , i.e., Theorem 1 Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I and every x ∈ U , denote attribute subset Proof On the basis of granular transformation function from Definition 2, it is easy to see that for any a k j , there exists g k,k+1 Theorem 1 represents the relationship of the attribute set AT in different levels of granulations, i.e., R * ≥ AT k+1 is subdivided by the relationship R * ≥ AT k defined on the attribute set AT, and thus obtains the relationship between the upper and lower approximations in different levels of granulations. □ be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I and every x ∈ U , denote attribute subset A k ⊆ AT k , then, From Proposition 2, clearly, the lower approximation in the (k + 1)-th level of granulations of X further subdivides the same in the k-th and the upper approximation in the (k + 1)-th level of granulation of X further subdivides the same in the k-th. Moreover, we have the corresponding hierarchical sequence of approximations as follows.
In the following, employ for understanding the above eqs more conveniently. Example 1 will provide a table of the quantity of rape pests detected by WSN. For facilitating reduction, in the first level of granulation, the quantity will be regarded as 90 when it is between 80 and 100 or 70 when it is between 60 and 80, and it can deduce the rest from this. However, missing values will exist due to perception errors and access limitations in WSN to a certain extent, and unclear pictures or imprecise targets will be regarded as missing values. □ Table 1 is an incomplete multi-granulation ordered information system table of Rape pests detected by WSN in a certain period of time, where granularity I = 3 , 12) represents different clusters which stand for Cabbage butterfly, Aphids, Cabbage bug and Cricket, respectively. And AT k is the attribute set where k = 1, 2, 3 , which is the different levels of granulations. And * is the missing value. From a multi-granular information system structure, we obtain the hierarchical sequence of attributes in different levels of granulations as follows.
(1) The sequence of the value in the first level of granulation of the quantity of pests is The sequence of the value in the second level of granulation of the grade of the quantity is r < f < m < l < a where r, f, m, l and a represent rarely, few medium, lot and abundance, respectively. (3) The sequence of the value in the third level of granulation of the grade of risk is {S < F } where S and F represent Seconds and Firsts, respectively.
For the above levels of granulations, the system is decomposed into three decision tables which are described as Tables 2, 3, 4, respectively.
From Table 2, we can derive that the value of the first level of granulation is as follows.  x 12 10 10 10 * Table 3 The incomplete ordered information table with the second level of granulations x 11 m l m l x 12 r r r * Table 4 The incomplete ordered information table with the third level of granulations  Table 3, we can derive that the value of the second level of granulation is as follows.
Suppose X = {x 1 , x 6 , x 8 , x 10 , x 11 } , then the reductions of the lower and upper approximation are R * ≥ AT 2 (X) = {x 1 , x 8 , x 10 } and R * ≥ AT 2 (X) = {U } , respectively. And for every Table 4, we can obtain that the value of the third level of granulations is as follows.
, then the reduction of the lower and upper approximation are R * ≥ AT 3 (X) = ∅ and R * ≥ AT 3 (X) = {U } , respectively. And for every x ∈ U , we can obtain [x] * ≥ AT 2 ⊆ [x] * ≥ AT 3 . Example 1 illustrates that it is not arbitrary for the value of the same attribute of the same object in different levels of granulations and proves that the value of the higher level of granulations is determined by the lower, i.e., the a from the attribute a 1 of x 1 in the second level of granulations is determined by 90 from the value of the first level.

Evidence structure
Definition 6 [20][21][22] Let be a finite and nonempty set which is called the frame of discernment, where A is the arbitrary subset of . If there exists a mapping function m : 2 → [0, 1] that satisfies m(∅) = 0 and X⊆U m(X) = 1 , then we define that the function m is the basic probability assignment function or the mass function on 2 .
The degree of evidence exactly to A is indicated by m(A) . If there exists m(A) > 0 , then we suppose that A is called the focal element of m and a family of all focal elements are viewed as the core. A pair of (F, m) is called a belief structure on the core. And we can obtain the other pair of the belief and plausibility functions can be derived as in terms of the mass function as Definition 7.
Definition 7 [20][21][22] Let be a finite and nonempty set which is called the frame of discernment, where A is the subset of and m is the basic probability assignment function of the frame of discernment . The belief function is a mapping Bel(X) that satisfies Bel(X) = A⊆U m(A) and the plausibility is a mapping Pl(X) that satisfies Pl(X) = A∩X� =∅ m(A) .
The Belief function Bel(X) represents the true degree of trust for X, while the plausibility function Pl(X) indicates that it is no doubt with trust is not true for X. These two functions are based on the same belief structure that are connected by the dual property, i.e., Bel(X) = 1 − Pl(∼ X) , where ∼ X is the complement of X. Also, the belief function can be defined by semi-additive measure as Definition 8.

an incomplete multi-granulation ordered information system which has I levels of granulations. For any
are the belief and plausibility function of the k-th level of granulations, respectively, and the corresponding mass function is Proof According to Definition 6, we can derive that m A k (X) is called mass function and then only need to demonstrate that Bel * ≥ A k satisfies three conditions of Definition 8. From the basic Definition 7, we take Bel * ≥ A k (∅) = 0 and Bel * ≥ A k (U ) = 1 , respectively. Next, prove the condition (3) of Definition 8. Considering a collection {x 1 , x 2 , . . . , x n } ⊆ U , then we have Hence, Bel * ≥ A k (X) is a belief function. And Pl * ≥ A k (X) is also a plausibility function due to the duality of the belief and plausibility functions.
With Theorem 3, there exactly exists the corresponding belief structure of multi-granulation rough set and the consistency of belief structure in different levels of granulations from Theorem 2 can be derived as follows. □

Proposition 3
Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I , denote the subset A k ⊆ AT k . For x ⊆ U , Bel * ≥ A k (X) and Pl * ≥ A k (X) are the belief and plausibility functions, respectively, and P(X) = |X| |U | . By the above analysis, we have the properties as follows.

Reduction in incomplete multi-granulation ordered information system
First, the positive region reduction, belief reduction and plausibility reduction are put forward in incomplete multi-granulation ordered information system.

Definition 9
Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I , denote the subset A k ⊆ AT k , and the positive region, belief and plausibility reduction are developed in the k-th level of granulations of information system as follows.
(1) If there exists U /R * ≥ A k = U /R * ≥ A k , then assume that A k is a consistent set. Furthermore, if any true subset of A k is not a consistent set, then A k can be defined as the positive region reduction.
, where for every X ∈ U /R * ≥ A k , then assume that A k is a belief consistent set. Furthermore, if any true subset of A k is not a belief consistent set, then A k can be defined as belief reduction.
, where for every X ∈ U /R * ≥ A k , then assume that A k is a plausibility consistent set. Furthermore, if any true subset of A k is not a plausibility consistent set, then A k can be defined as plausibility reduction.
Based on Definition 9, belief reduction and plausibility reduction are the minimal attribute set to keep the degree of belief and plausibility. Next, we analyze the consistency of three ways of reduction in the same level of granulations. Theorem 3 Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I , denote the subset A k ⊆ AT k , and we have some properties in the k-th level of granulations of information system as follows.

Proof
(1) Suppose that A k is the consistency set in the k-th level of granulations of information system. Then, we have U /R * ≥ A k = U /R * ≥ A k . Clearly, A k is also the belief consistency set in the same level with Definition 9. If A k is the belief reduction in the k-th level of granulations. Then, we have Bel * ≥ A k (X) = Bel * ≥ AT k (X),X ∈ U /R * ≥ A k (i). According to Eqs.(i), one obtains Bel * ≥ A k ([X] * ≥ AT k ) = Bel * ≥ AT k ([X] * ≥ AT k ) (ii), According to Eqs.(ii), one obtains According to the conjunction of Theorem 2 and Eqs(iii), we can obtain . By the definition of the lower approximation from Definition 7 and A k ⊆ AT k ,we can get the following relationship: Consequently, A k is the consistency set in the k-th level of granulations of information system.
(2) Similar to the proof of (1), it can be proved.
(3) Suppose that A k is the consistency set in the k-th level of granulations of information system. Then, we have U /R * ≥ A k = U /R * ≥ A k . Clearly, A k is also the plausibility consistency set in the same level with Definition 9. (4) Similar to the proof of (3), it can be proved. As demonstrated above, we obtain the consistency of several reduction in the same level, which means the positive region reduction is equivalent to belief reduction. And it is also proved that the positive region reduction and belief reduction are adequate condition for plausibility reduction. Next, analyze the transitivity of the above three reduction in different levels.
□ Theorem 4 Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system which has I levels of granulations. For any 1 ≤ k ≤ I , denote the subset A k ⊆ AT k and we have some properties as follows.
(1) For any x ∈ U , we can obtain where A k is the positive region reduction in the k-th level of granulations of information system. Then, we define that A k+1 is the positive region reduction in the k-th level of granulations.
(2) Reversely, if for x ∈ U,we can obtain [X] * ≥ A k = [X] * ≥ A k+1 , where A k+1 is the positive region reduction in the (k + 1)-th level of granulations of information system. Then, we define that A k is the positive region reduction in the (k + 1)-th level of granulations.
is not the positive region reduction in the (k + 1)-th level of granulations of information system. Then, we define that A k is not the positive region reduction in the k-th level of granulations.

Proof
(1) If A k is the positive region reduction in the k-th level of granulations of information system, then we get And then, [X] * ≥ A k+1 = [X] * ≥ AT k can be verified. Hence, we get the result that A k+1 is the positive region reduction in the k-th level of granulations.
(2) Similar to the proof of (1), it can be proved.
Hence, we can get the result that A k is not the positive region reduction in the k-th level of granulations.
It is demonstrated in Theorem 4 that, when the condition [X] * ≥ A k = [X] * ≥ A k+1 where for every x ∈ U is satisfied, reducibility has bidirectional transitivity between different levels of granulations and nonreducibility has only unidirectional transitivity between different levels of granulations, i.e., if the higher level of granulations has nonreducibility, then the lower level has irreducibility, too. Whereas it is impossible to judge whether the higher level can be reduced if the lower level has irreducibility. On the above analysis, the following inference can be put forward.
(1) Suppose that A i is the positive region reduction in the i-th level of granulations and for every x ∈ U,we have [X] * ≥ A k = [X] * ≥ A k+m ,where m > 0,k + m ≤ I and for every I, there exists k ≤ i ≤ k + m . Then, A k ,A k+1 , . . . ,A k+m are the positive region reductions in the k-th, (k + 1)-th, . . . , (k + m)-th levels of granulations of information system, respectively. Specially, when i = 1 , incomplete multi-granulation ordered information system completely can be positively reduced.
(2) Conversely, suppose that A i is not the positive region reduction in the i-th level of granulations and for every x ∈ U , we have [X] * ≥ A k = [X] * ≥ A k+m , where m > 0 , k + m ≤ I and for every I, there exists k ≤ i ≤ k + m . Then, A k ,A k+1 , . . . ,A k+m are not the positive region reductions in the k-th, (k + 1)-th, . . . ,(k + m)-th levels of granulations of information system, respectively. Specially, when i = 1 , incomplete multigranulation ordered information system completely can't be positive region reduced.

Proof
(1) First, prove that A k is the positive region reduction in the i-th level of granulations.
If A i is the positive region reduction in the i-th level of granulations of information system, then we get AT i can be verified. Thus, we get that A k is the positive region reduction in the i-th level of granulations. Second, similar to proof of (1), we can derive that A i is the positive region reduction in the k-th level of granulations,A i is the positive region reduction in the (k + m)-th level of granulations and A k+m is the positive region reduction in the i-th level of granulations Hence, we can get the result that A k ,A k+1 , . . . ,A k+m are the positive region reductions in the k-th, (k + 1)-th, . . . ,(k + m)-th levels of granulations of information system, respectively.
(2) First, prove that A k is not the positive region reduction in the k-th level of granulations.
If A i is the positive region reduction in the i-th level of granulations of information system, then we get And then, [X] * ≥ AT k ⊂ [X] * ≥ A k can be verified.
Thus, we can get that A k is not the positive region reduction in the k-th level of granulations.
Second, similar to proof of (1), we can derive that A k+m is not the positive region reduction in the (k + m)-th level of granulations.
Hence, we can get the result that A k ,A k+1 , . . . ,A k+m are not the positive region reductions in the k-th, (k + 1)-th, . . . ,(k + m)-th levels of granulations of information system, respectively.
In order to describe the relationship among the reductions above conveniently, given the relationship figures as follows, clearly, Fig. 2 is the consistency of several reductions in the same level of granulations. In this representation, each of nodes represents a kind of reduction and the unidirectional arrows are granted as a reduction of the end point from the starting point, i.e., the positive region reduction A must be the plausibility reduction at the mean time. The bidirectional arrows mean the equivalent of the ends of the arrow. Figure 3 represents the transitivity of several reductions in the different levels of granulations. When [X] * ≥ A k ⊂ [X] * ≥ A k+1 are satisfied, we can get that A is also the positive region reduction of the k-th and (k + 1)-th levels of granulations.

Results and discussion
In this section, we will simulate numerically the relationship of three reductions of an incomplete multi-granulation ordered information system based on belief structure that is defined in Sect. 3 and give the notions of the significance to explain whether the corresponding attribute is dispensable or not. Based on this fact, we proposed an algorithm to find out reductions of an incomplete multi-granulation ordered information system.

Algorithm design
Definition 10 Let MGOIS * ≥ = (U , AT k , f k , V k ) be an incomplete multi-granulation ordered information system. If AT k = a k 1 , a k 2 , . . . , a k m , then for every attribute a k i ∈ AT k , we define the significance degree of the attribute AT k as follows.
sig 1 (a k i , AT k ) > 0 explains that the attribute a k i is not dispensable for AT k and a k i should be included in the positive region reduction. If sig 1 (a k i , AT k ) ≤ 0 , it shows the attribute a k i is dispensable for AT k and a k i should not be included in positive region reduction. sig 2 (a k i , AT k ) > 0 explains that the belief function of the attribute a k AT − a k i is equal to AT k , which means a k i should be included in the belief reduction. If sig 2 (a k i , AT k ) ≤ 0 , it shows the belief function of the attribute a k AT − a k i is not equal to AT k , and a k i should not be included in belief reduction. sig 3 (a k i , AT k ) > 0 explains that the plausibility function of the attribute a k AT − a k i is equal to AT k and a k i should be included in the plausibility reduction. If sig 3 (a k i , AT k ) ≤ 0 , it shows the plausibility function of the attribute a k AT − a k i is equal to AT k , and a k i should not be included in the plausibility reduction. Fig. 3 The transitivity of several reductions between k and k+1 levels of granulations The specific steps of the reduction will be given in the following. Since that it is same for reduction steps in different levels of granulations, just given the reduction process of a certain level of granulations as Algorithm 1 shows.

Algorithm 1
Reduction in incomplete multi-granulation ordered information system based on belief structure Input: incomplete information system IS * ≥ = (U , AT , f , V ); Output: let the positive region reduction be Red, belief reduction be Red Bel , plausibility reduction be Red Pl .
Step 1: let Red = ∅ , Red Bel = ∅,Red Pl = ∅ and AT ′ = AT; Step 2: according to Definition 9, calculate the positive region consistent set U /R AT , belief consistent set Bel AT and plausibility consistent set Pl AT of the attribute AT; Step 3: let a i ∈ AT , according to Definition 9, calculate the positive region consistent set U /R AT −{a i } , belief consistent set Bel AT −{a i } and plausibility consistent set Pl AT −{a i } of the attribute AT − {a i }; Step 4: Let sig j (a i , AT ) be the significance and relative of the attribute a i where1 ≤ |j| ≤ 3 , Step 5: If sig 1 (a i , AT ) > 0 , then suppose the attribute a i is important, and add it into the positive region reduction set, and obtain a i ∈ Red , then go to Step 8, else go to Step 8 directly; Step 6: If sig 2 (a i , AT ) > 0 , then suppose the attribute a i is important, and add it into the belief reduction set, and obtain a i ∈ Red Bel , then go to Step 8, else go to Step 8 directly; Step 7: If sig 3 (a i , AT ) > 0 , then suppose the attribute a i is important, and add it into the plausibility reduction set, and obtain a i ∈ Red Pl , then go to Step 8, else go to Step 8 directly; Step 8: Let AT ′ = AT ′ − {a i } , if AT ′ = ∅ , then return step 8, else let a i = a j , and return Step 3, where i = j,; Step 9: output the positive region reduction Red, belief reduction Red Bel and plausibility reduction Red Pl as reduction.
Suppose that the size of U is n and the number of attributions is m, then the time complexity of Algorithm 1 is O(m * n) . Table 5 and Fig. 4 both show the comparison of time complexity among different rough set measures, like covering rough set [25], traditional rough set [7] and multi-granulation rough set [18], illustrating that the calculating time can be reduced largely. In Fig. 4, suppose m = 30 and n = 0, 500, 1000, . . . , 10000 . For clearly observing, the final result of our idea is divided by 1000, MGRS is divided by 10000, and the rest methods are divided by 100000. This figure shows the great superiority of MGRS based on belief structure with low time complexity clearly and intuitively. And the following analysis of Example 1, which is analyzed briefly in Sect. 3.2, is employed to illustrate our idea.

Algorithm implementation
Example 1.1. It is an incomplete multi-granulation ordered information system table of Rape pests detected by WSN in a certain period of time, where granularity I = 3 , U = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 1 0, x 1 1, x 1 2} and AT k = a k 1 , a k 2 , a k 3 , a k 4 . X i (i = 1, 2, . . . , 12) represents different clusters which stand for Cabbage butterfly, Aphids, Cabbage bug and Cricket, respectively. And AT k is the attribute set where k = 1, 2, 3 , which is the different levels of granulations. Since that, it is same for reduction steps in different levels of granulations, just given the reduction process of the first level of granulations as Algorithm 1 shows.
(1) Let Red = ∅ , Red Bel = ∅ , Red Pl = ∅ and AT ′ = AT; (2) According to Definition 9, calculate the positive region consistent set U /R AT 1 , belief consistent set Bel AT 1 and plausibility consistent set Pl AT 1 of the attribute AT 1 .