The construction of equivalence relations for numerical type data first requires the discretization of the original data, and this method will inevitably cause the loss of information. On the basis of neighborhood relations, a neighborhood rough set model was proposed by Hu et al. [11–13].

Assume that information system is expressed as S = U , A T = C ∪ D , f , V .Among them, U = {x ₁, x ₂, … , x _n } represents a collection of non-empty limited objects, AT stands for the set of attributes, containing conditional attribute set C and decision attribute set D.

Definition 1

Suppose the information system is S = U , A T = C ∪ D , f , V , ∀ x ∈ U , B ⊆ C , the δ -neighborhood of x in B is defined as: δ B x = y ∈ U | d i s B x , y ≤ δ , δ > 0 (1) where dis (•) represents the distance between any objects, using Euclidean distance commonly.

Definition 2

Suppose the information system is S = U , A T = C ∪ D , f , V , ∀ x ∈ U , X ⊆ U , B ⊆ C , the rough affiliation μ B x of x to X in B is defined as: μ B x = P X | δ B x = X ∩ δ B x δ B x (2) where P X | δ B x represents the conditional probability of classification, and • represents the number of elements in the combination.

Definition 3

Suppose the information system is S = U , A T = C ∪ D , f , V , X ⊆ U , B ⊆ C , the lower and upper approximations of the decision D in B are defined as: δ B ¯ X = x ∈ U | δ B x ∩ X ≠ ∅ (3) δ B ¯ X = x ∈ U | δ B x ⊆ X (4)

The following definitions apply to the positive, negative, and boundary regions of X in B: P O S B X = δ B ¯ X = x ∈ U P X | δ B X = 1 (5) N E G B X = U − δ B ¯ x = x ∈ U P X | δ B x = 0 (6) B N D B X = δ B ¯ x − δ B ¯ x = x ∈ U 0 < P ( X δ B x < 1 (7)

From the above definition, it can be found that the conditions on which the neighborhood rough set is based in taking both acceptance and rejection decisions are too severe and lack a certain degree of fault tolerance. Only elements that are completely correctly classified are grouped into the positive domain. Alternatively, only elements that are completely misclassified are classified in the negative domain. The result of such a definition makes the boundary domain too large.

The rough set model for decision-making put forth by Yao et al. [5] lacks the ability to directly process numerical data. In order to address this weakness, a rough set model of decision theory based on neighborhood was proposed by Li et al. [14] through the integration of the neighborhood rough set and the decision rough set.

The decision rough set has two important elements: Ω = X , ∼ X and A c t i o n = a P , a B , a N . When different decision-making actions are taken, different losses will occur. λ P P , λ B P , λ N P respectively represent the cost of a P , a B and a N when X owns the object, λ P N , λ B N , λ N N respectively represent the cost of a P , a B and a N when X is not the owner of the object. Through cost risk analysis, the solution formula of α , β is given [5] as follows: α = λ P N − λ B N λ P N − λ N N + λ B P − λ P P (8) β = λ B N − λ N N λ B N − λ N N + λ N P − λ B P (9)

In addition, Yao proposed three decision theories based on decision rough set model [5], including P rule, N rule and B rule.

Definition 4

Suppose the information system S = U , A T = C ∪ D , f , V , X ⊆ U , B ⊆ C , then the P, B, and N rules of X on δ -neighborhood under attribute set B are defined as:

P rule: if x ∈ U , P X | δ B X ≥ α , then x ∈ P O S B X ;

B rule: if x ∈ U , β < P X | δ B X < α , then x ∈ B N D B X ;

N rule: if x ∈ U , P X | δ B X ≤ β , then x ∈ N E G B X .

Definition 5

Suppose the information system S = U , A T = C ∪ D , f , V , X ⊆ U , B ⊆ C , then the X of the attribute set B is related to the upper and lower approximation sets of the δ -neighborhood based local rough set, which are defined as: δ B L ¯ X = x ∈ X | P X | δ B x > β (10) δ B L ¯ X = x ∈ X | P X | δ B x ≥ α (11)

The following definitions apply to the positive, negative, and boundary regions of X in B: P O S B X = δ B L ¯ X = x ∈ X P X | δ B X ≥ α ; (12) N E G B X = U − δ B L ¯ x = x ∈ X P X | δ B x ≤ β ; (13) B N D B X = δ B L ¯ x − δ B L ¯ x = x ∈ X β < P ( X δ B x < α . (14)

The most significant difference between the local neighborhood rough set model and the neighborhood rough set model is the different search scope when finding the upper and lower approximation sets. In the neighborhood rough set model, finding the approximation set for each decision category requires traversing all the data points in the data set. However, in the local neighborhood rough set model, the focus is on the data points of the same category, and only the data points of the same decision category need to be traversed. This greatly reduces the computational effort and increases the computational speed [14]. This model not only improves computational efficiency, but also eliminates the interference of noisy points.

In addition, the traditional method of calculating rough affiliation does not take into account the complexity of the data. In this paper, the calculation process of affiliation degree is improved for the affiliation degree, and the process is as follows:

Suppose S = U , A T = C ∪ D , f , V is a decision system, U / D = X 1 , X 2 , . . . , X d is the decision attribute of all objects U in the decision attribute set D, ∀ x ∈ U , the neighborhood of x is expressed as δ x , the decision value of the information system is L = 1,2 , . . . , d . Now suppose that the decision value of the sample x to be investigated is q.

(1) δ x ≤ N (N represents a small positive integer), this paper sets rough membership degree to P X | δ x = e − 5 .

Depending on which of the data points in the neighborhood information granularity are specifically situated, above rules is used to define the rough membership function for each category of data points.

Based on the above discussion, this paper designs the following Algorithm 1 to calculate the upper and lower approximation sets and identify the anomalous data. Different from the classical method that only considers the upper and lower approximation sets, Algorithm 1 not only identifies label noise data points and outlier data points based on the neighborhood information, making the upper and lower approximation sets more accurate. It also appends category information to the label noise data, which is referred to as pseudo-tagging in this paper.

Algorithm 1

The upper and lower approximation sets of local neighborhood rough set.

Input: S = U , A T = C ∪ D , neighborhood radius δ , cost matrix λ .

Algorithm 1 detects outliers and labeled noisy points, as well as enables the detection of data points for high-density areas. In fact, some samples are not always considered as outlier data or noise, and their decisions sometimes depend on the choice of neighborhood radius.

According to the above-mentioned rough set model, a smaller neighborhood radius contains very little information, while a larger radius may cause the next approximate set to be an empty set. This paper introduces the justifiable granularity criterion [15, 16]. There are generally two functions in the construction of information granules, namely, covering function and particularity function.

The coverage function describes how much data is in the constructed information granule. This paper designs the coverage index function as shown below: c o v δ = max 0 , F 1 + F 2 (15) where F 1 = 1 X q ∑ x ∈ X q δ q x − max j = 1 , … , d j ≠ q δ j x and F 2 = 1 δ x (|POS( X q )|-|BND( X q )|).

The coverage index function mentioned above is considered from two perspectives, namely, neighborhood information granularity and approximate set. In terms of specificity criteria, the smaller the neighborhood radius, the better. Therefore, the specificity function can be designed as: s p δ = 1 − δ .

Obviously, the two are contradictory. Therefore, the function for optimized performance can be written as the multiplication of specificity and coverage, which is: Q = c o v δ × s p δ .

In this way, the optimal neighborhood about X q can be obtained. To further elaborate, the cumulative behavior can be represented in terms of the decision partition set U / D = X 1 , X 2 , . . . , X d as follows: Q = Q 1 + Q 2 + . . . + Q d , where Q 1 , Q 2 ,…, Q d correspond to the optimized value of each decision class.

To achieve the optimal Q value and the corresponding optimal neighborhood radius. In this paper, PSO algorithm is used for optimization [17, 18], which is an evolutionary algorithm based on population intelligence, proposed by Drs. Kennedy and Eberhart in 1995. In this paper, we use the PSO algorithm to intelligently optimize the selection of neighborhoods and select the appropriate granularity as a way to improve the accuracy of decision making.

Moreover, to update the dataset, one can utilize the noise identification strategy along with the set of predicted pseudo-decision labels. The main steps are described in Algorithm 2.

Algorithm 2

Update of rough approximation set in label noise injection environment.

 1: Obtain the optimal neighborhood radius δ using PSO optimization algorithm;

In this paper the risky decision cost will be used to reduce the attributes. It comes from the Bayesian decision process, which is comparable to the classical rough set. Risky decision costs for P, N and B rule can be separately expressed as: COS T P O S = ∑ X j ∈ U / D ∑ x ∈ P O S X j ∑ k = 1 m λ P P k • P X j | x C k + λ P N k • P ∼ X j | x C k (16) COS T N E G = ∑ X j ∈ U / D ∑ x ∈ N E G X j ∑ k = 1 m λ N P k • P X j | x C k + λ N N k • P ∼ X j | x C k (17) COS T B N D = ∑ X j ∈ U / D ∑ x ∈ B N D X j ∑ k = 1 m λ B P k • P X j | x C k + λ B N k • P ∼ X j | x C k (18)

As discussed above, the cost of making a risky decision for all decision rules can be obtained as: COS T B = COS T P O S + COS T N E G + COS T B N D (19)

Obviously, the higher COS T B , the greater the significance of the attribute becomes evident.

Definition 6

Suppose the information system S = U , A T = C ∪ D , f , V , B ⊆ C , a ∉ B , the significance of an attribute is defined as: s i g a , B , D = COS T B ∪ a D − COS T B D (20)

A scheme based on neighborhood decision rough sets is designed for forward search to achieve the optimal reduction. Its specific steps are shown in Algorithm 3.

Algorithm 3

Attribute reduction based on neighborhood decision rough set model.

 1:  RED = ∅ .

To assess the effectiveness of the suggested approach, this article discusses the following two evaluation indicators: the lower approximation and information granularity.

Approximation quality (AQ): Given decision information system S = U , A T = C ∪ D , f , V , A ⊆ C , the approximate quality of A relative to D [19] is defined as: γ = ∪ δ A X q _ U , q = 1,2 , . . . , d (21)

The γ value is expressed as the ratio of the number of objects correctly classified by the conditional attribute set A to the number of all objects in the decision information system. The performance of the proposed granularity description is evaluated in terms of the lower approximation.

Neighborhood number(NN): x ∈ U , suppose x ∈ X q . δ q x is the set of data points with decision label q in the neighborhood of x. Therefore, the categories of similar decision label data and different data in the neighborhood can be described as: N N = ∑ x ∈ X q δ q x − ∑ j = 1 , j ≠ q d δ j x (22)

The larger value of N N indicates that the information granularity provides greater information value to the decision maker and more reasonable granularity.