Dynamic and Updating Multigranulation Decision-Theoretic Rough Approximations When Adding or Deleting Objects

Along with the development of big data, knowledge updating occurs in various situations, some scholars had studied the dynamic method of updating knowledge in multigranulation decision-theoretic rough sets when adding or deleting granular structures. However, there is no study about the case of adding or deleting objects, which limits the development of multigranulation decision-theoretic rough sets. Based on the matrix method, the dynamic knowledge updating of optimistic multigranulation decision-theoretic rough set and pessimistic multigranulation decision-theoretic rough set were studied in this paper, then given the static algorithm and dynamic algorithms, and the time complexity of three algorithms was analyzed. The theory and experimental results show that two dynamic algorithms are both more effective than the static algorithm.


I. INTRODUCTION
As a data analysing and processing theory, rough set theory was put forward by Z. Pawlak from Poland [1], it has made great progress in both theory and application, and has been applied to data mining [2], machine learning [3]- [5], knowledge discovery [6] and other fields. As the basic calculation of rough set, the calculation of lower and upper approximations are necessary steps to obtain other significant achievements, many scholars have popularized rough set in various aspects.
To overcome the influence of noise data, Yu et al. [7] proposed the decision-theoretic rough set, which simulates the process of human decision-making under uncertainty and risk, we can obtain the calculation of thresholds through decision risk minimization based on bayesian decision theory [8], conditional probability was estimated by naive bayesian model [9], and then given the concepts of positive domain, negative domain and edge domain, they are the basis of threeway decision [10], [11]. Therefore, decision-theoretic rough set is a model with solid theoretical foundation and practical application.
Pawlak rough set model and decision-theoretic rough set model are based on a single equivalence relation, due to The associate editor coordinating the review of this manuscript and approving it for publication was Sathish Kumar . practical needs, multigranulation rough set based on multiple equivalence relations was put forward by Qian et al. [12]. With the development of multigranulation rough set, several generalized multigranulation rough set models had been proposed to solve real-world problems with complex environments [13]- [17]. Multigranulation decision-theoretic rough set was proposed by Qian et al. [18], and several of its extended models were studied [19]- [23].
With the arrival of the big data, many scholars have studied incremental knowledge updating, using existing knowledge to update new knowledge, rather than recalculating, which will save a lot of time and space. There are three main aspects of the current dynamic knowledge updating: the dynamic change of objects, the dynamic change of attributes and the dynamic change of attribute values.
For the decision rules for the decision table with increase of objects, Liu et al. [24] proposed the corresponding incremental algorithm. Liang et al. [25] proposed an incremental feature selection algorithm based on rough set while increasing objects. Considering the addition and deletion of single object in neighborhood fuzzy decision system, Zeng et al. [26] proposed an incremental updating method of approximations in fuzzy rough set theory. Hu et al. [27] presented a method to update multigranulation approximations with the variation of granulars. Li et al. [28] proposed a matrix-based method of approximates when attribute values were updated dynamically in ordered information system. Li et al. [29] put forward the rule extraction algorithm based on the characteristic relation when adding and deleting multiple attributes. Qian et al. [30] proposed a solution for the attribute reduction problem to avoid redundant steps of incremental calculation with increasing attributes. Zeng et al. [31] proposed an incremental feature extraction algorithm for fuzzy rough set aiming at dynamic changes of attributes in mixed information system with many data types.
The rest of the paper is organized as follows. Section 2 briefly introduces the basic concepts of rough sets, multigranulation decision-theoretic rough sets and relation matrix. In section 3, the method of computing the lower and upper approximations are proposed by the matrix-based, then updating approximations when adding or deleting objects. Several dynamic algorithms are given in Section 4. In section 5, we verify the effectiveness of proposed dynamic algorithms experimentally. Finally, summarized the full text in Section 6 and put forward further directions of research.

Give an information system
. , x n } is a non-empty finite set of objects called universe; AT is a non-empty finite set of attributes, a ∈ AT is called an attribute; V = a∈AT V a is a set of attribute values, V a is a non-empty set of values of attribute a ∈ AT , called the domain of a; f : U × AT → V is an information function that maps an object in U to exactly one value in V a such that For a subset of attributes B, an indiscernibility relation R B is defined as follows: where R B is an equivalence relation on U . The equivalence relation R B partitions the universe U into a family of disjoint subsets called equivalence classes, the equivalence class including x with respect to B is denoted as follows: For a set X ⊆ U , the lower and upper approximations of X with repect to R are defined as follows: (X ) > is called the pessimistic multi-granularity decision-theoretic rough sets of X .

C. RELATION MATRIX
Let IS = U , AT , V , f is an information system, where U = {x 1 , x 2 , . . . , x n } and ∀X ⊆ U , the characteristic function F(X ) = (f 1 , f 2 , . . . , f n ) T (T represents a transpose operation) of X is constructed as: Due to the need of practical application, this section discusses multigranulation decision-theoretic rough approximations based on the matrix and give some properties.
Definition 2: of A k is constructed as: , two cut matrices of the lower and upper approximations of A k are constructed as I where 'max' indicate a maximum value among the corresponding position values of multiple matrices with same size.
Proof: (1) By the definitions of characteristic function and cut matrix, we can obtain ). (2) By the definitions of characteristic function and cut matrix, we can obtain where 'min' indicate a minimum value among the corresponding position values of multiple matrices with same size.
Proof: The process of proof is similar to Theorem 1. The following example shows that there is no inclusion relationship between them.

B. UPDATING MULTIGRANULATION DECISION-THEORETIC ROUGH APPROXIMATIONS WHILE INCREASING OBJECTS
In the previous section, we introduced multigranulation decision-theoretic rough approximations based on the matrix. In this section, we will introduce dynamic multigranulation decision-theoretic rough approximations with increasing objects.
Definition 4: , f n+2 , . . . , f n+n + ) T of X + is constructed as: is the characteristic function of X and the following result hold: Proof: It's obvious the theorem holds by Definition 4.

C. UPDATING MULTIGRANULATION DECISION-THEORETIC ROUGH APPROXIMATIONS WHILE DECREASING OBJECTS
In the previous section, we introduced multigranulation decision-theoretic rough approximations based on the matrix. In this section, we will introduce dynamic multigranulation decision-theoretic rough approximations with decreasing objects.
A k ] n − ×n − are constructed as: A k ] (n−n − )×(n−n − ) is the new relation matrix and the following result hold: A k is the transport matrix of S A k . Proof: It's obvious the theorem holds by Definition 7.
then the following results hold: Proof: It's obvious the theorem holds by Definition 1. According to Theorem 6, we can get the characteristic function of X − :

IV. THE ALGORITHMS FOR UPDATING MULTIGRANULATION DECISION-THEORETIC APPROXIMATIONS WHILE ADDING OR DELETING OBJECTS
In this section, we give a static algorithm and two fast algorithms for updating multigranulation decision-theoretic rough approximations. For Algorithm 2, Step 1 update the universe and the set X , it's time complexity is O(1); Step 2 calculate the added char-

Algorithm 1 Static Algorithm for Computing Multigranulation Decision-Theoretic Rough Approximations
Input: An information system IS = U , AT , V , f , the set X , two thresholds α and β.

V. EXPERIMENTAL EVALUATION AND ANALYSIS
We conducted several experiments to evaluate the performance of the proposed incremental algorithms. From the UCI machine learning repository, the basic information of six data sets were wrote in Table 4, and experiments are implemented on a PC with Windows10, AMD Ryzen5 3550H CPU, 2.10 GHz and 16 GB memory, Algorithm 1 and Algorithm 2, Algorithm 3 were compared respectively. Each data set in Table 4 was divided into an average of 10 sub-data sets, and the first sub-data set was seen as the first basic data set, the combination of the first and second sub-data set was seen as the second basic data set, and so on.

A. EXPERIMENTS WITH DIFFERENT SIZED DATA SETS WHEN ADDING OBJECTS
In this subsection, for each basic data set, we randomly select 5% of the size of the basic data set from its complement set in the universe as the inserted new data set. By comparing the calculation time of Algorithm 1 and Algorithm 2, we show the efficiency of Algorithm 2 and the experimental results were listed in Table 5. With the increase of size for data set, the more detailed information of two algorithms were shown in Figure 1, it is easy to see from Figure 1 that the calculation time of two algorithms usually increase with the increase of the basic data set and Algorithm 2 is always faster than Algorithm 1, the larger the basic data set, the greater the difference in efficiency.

B. EXPERIMENTS WITH DIFFERENT SIZED DATA SETS WHEN DELETING OBJECTS
In this subsection, for each basic data set, we randomly select 5% of the size of the basic data set from its complement set in the universe as the inserted new data set. By comparing the  calculation time of Algorithm 1 and Algorithm 3, we show the efficiency of Algorithm 3 and the experimental results were listed in Table 6. With the increase of size for data set, the more detailed information of two algorithms were shown in Figure 2, it is easy to see from Figure 2 that the calculation time of two algorithms usually increase with the increase of the basic data set and Algorithm 3 is always faster than Algorithm 1, the larger the basic data set, the greater the difference in efficiency.

VI. CONCLUSION
In this paper, we propose the method of computing approximations based on the matrix of multigranulation decision-theoretic rough sets. On this basis, the method of dynamic updating approximations with objects increased or deleted of multigranulation decision-theoretic rough sets are proposed, and some related properties are studied. For each case, an example is given to verify its validity. Finally, experimental studies show that two proposed incremental algorithms can significantly reduce unnecessary computing time.