Local fuzzy rough set model over two universes and its reduction

The fuzzy information system over two universes formalizes a data table corresponding to two universes as well as their relations. Although many analyses have focused on this topic, the existing models are mixed with some uncertain factors in the process of given relation. And it is unnecessary to analyze the fuzzy objects with small membership degree. This motivates us to develop a better method to handle complex data. This paper proposes a local fuzzy rough set model over two universes, then further analyzes its properties and decision rules. Moreover, the reduction method of the model is being studied. The experiment results show that the proposed model has better performance in classfication and decision-making.


Introduction
As a data processing tool, rough set (Pawlak 1982;Zhang and Leung 1996) has been widely promoted and applied in artification intelligence, machine learning and expert system. Among them, in order to avoid the result of overfitting, Qian proposed the local rough set model , which has attracted wide attention by virtue of the speed advantage of the algorithm Guo and Eric 2019;Xie and Lin 2021). In order to make the rough set model better handle continuous data, Dubois combines the fuzzy set theory with the rough set model. He proposed the fuzzy rough set model (Dubois and Prade 1990) in 1990, which is the first appearance of the fuzzy rough set model. Many scholars define the fuzzy rough sets by using triangular norms and triangular co-norms, and propose a more general fuzzy rough set model (Morsi and Yakout 1998;Menger 1942;Bogar 1960). However, different from the classical rough sets, the definition of fuzzy similarity classes of fuzzy rough sets is often complex. Many scholars use the relation to define the fuzzy similarity classes. This definition method does not consider the impact of the membership degree of fuzzy target concepts. Therefore, Chen et al. (2011); Chen (2013) discussed the general granular structure of fuzzy rough sets and gave another granular structure expression of fuzzy rough sets. Due to the existence complexity and uncertainty of many practical problems, several extended fuzzy rough set models have been proposed to meet various requirements (Zhang et al. 2020;Sun et al. 2019;An et al. 2021;Wang and Hu 2015;Wang and Wan 2020;Zhang and Zhu 2022;Fan et al. 2017;Li et al. 2018).
We can find that all rough set models discussed above are based on the classical rough set theory. These models limit the objects discussed to the same universe. In practice, however, two or more universes can describe the problem more effectively and rationally. To solve the problem of multiple features and decisions, Yao proposed a rough set model on two universes (Yao et al. 1995). It is widely used in applications such as recommendation systems and disease diagnosis systems. Similar to the classical rough set, more and more scholars have recognized it and made some efforts on two universes (Shen and Wang 2011;Ma and Sun 2012;Liu et al. 2012;Sun et al. 2016;Li et al. 2020;Xu et al. 2015;Yang 2016). In order to improve the use of the model in continuous data, Li extended the rough approximation operator on two universes to fuzzy cases (Li and Zhang 2008). And the fuzzy rough set models were considered on two universes (Xu et al. 2013) by Xu. Aysegui Altay also used norms to define upper and lower approximations (Aysegui and Murat 2020). Many scholars have applied this model to various fields, such as emergency material demand prediction and the analysis of the human body comfort Li et al. 2019;Zhou et al. 2021;Yang et al. 2013). Because it is impossible to avoid the artificial influence that may occur when a doctor decides the relation between disease and symptom. Therefore, in order to make full use of the original data, many scholars introduce multi-granulation into the fuzzy rough set over two universes Yan et al. 2022;Sun et al. 2017aSun et al. , b, 2015Tan et al. 2019;Hu et al. 2017;. These models can deal with multi-standard decision problems, but only special cases are considered.
The first motivation of this paper is that in order to avoid artificial influence in the relation calculation progress, should we calculate directly from the raw data? The main aim of this paper is to fill this gap. The second motivation of this paper is that some fuzzy rough set models cannot be applied in fuzzy decision. However, whether fuzzy decision can obtain more accurate result? For example, for doctors, the fuzzy decision result is more convenient for their dispensing. Although there are some models over two universes with fuzzy decision, the results are generated from fuzzy relations. The second motivation of this paper is that in order to avoid artificial influence in the relation calculation progress, should we calculate directly from the raw data? The main aim of this paper is to fill this gap. The third motivation is to consider improving the computational inefficiency of the model and avoiding the overfitting problem in reduction. In order to solve these problems, this paper proposed the local fuzzy rough sets over two universes. This new model is not only be viewed as an extension of classical rough sets but also the direct extension of the fuzzy rough sets over two universes.
As we all know, there are many objects and features in an information system, but some do not always need to be based on lower and upper approximations. Therefore, based on the classification ability of the local fuzzy rough set model on two universes unchanged, a new reduction algorithm based on the significance is proposed in this paper. The analysis and experiments show that this local reduction algorithm can improve the computational efficiency of the model to a greater extent than the global reduction algorithm.
Therefore, the contributions of this study are threefold: 1. we proposed a new method to discuss the data sets over two universes; 2. a novel reduction algorithm to improve speed and accuracy is proposed; 3. the advantages of this model in decision-making and reduction are confirmed by experimental comparison with other models.
The rest of the paper is briefly summarized as follows. Section 2, the basic concepts of the local fuzzy rough sets and fuzzy rough sets over two universes are introduced. Section 3, a local fuzzy rough set model over two universes is proposed, we give the corresponding properties and use an example to compare this model with other models. Section 4, a novel reduction method is introduced. The results of numerical experiments are analyzed in Sect. 5, and finally, a brief conclusion is given in Sect. 6.

Preliminaries
In order to facilitate a clear understanding of the content of this paper, we present some relevant basic concepts in this section. This part can provide a more complete description for the following.

Fuzzy rough set model over single universe
In this section, we briefly review the concept of fuzzy rough set over single universe (Zhang and Leung 1996;Menger 1942;Bogar 1960;Chen 2013;Shen and Wang 2011;Xie and Lin 2021). Menger (1942); Bogar (1960)  where the notation | · | denotes the cardinality of set. Especially, if the set is crisp, then it denotes that the set contains the number of elements.

Definition 2.1
Consider U is a nonempty finite set of objects, andR is the T -fuzzy similarity relation on U . The pair (U ,R) is called a fuzzy information system. Definition 2.4 Xie and Lin (2021) Let I = (U ,R) be a fuzzy information system, and x λ be a fuzzy point, x ∈ U , λ ∈ [0, 1] and 0 ≤ β < α ≤ 1, then for any target conceptX ∈ F(U ), its α-local lower approximation and β-local upper approximation are respectively defined as follows: where the inclusion degreeD is: .

Fuzzy rough set model over two universes
Next, we will list some basic concepts of the fuzzy rough set models on two universes. For more details, please refer to Wu and Zhang (2004); Li et al. (2019). Let U and V be two non-empty finite universes andR be a fuzzy relation from U to V ,R : is called a fuzzy approximation space on two universes. Definition 2.5 Wu and Zhang (2004) Let (U , V ,R) be a fuzzy approximation space on two universes. For anỹ Y ∈ F(V ), the θ -fuzzy lower approximation operator and the T -fuzzy upper approximation operator ofỸ are defined as follows: for any x ∈ U , The pair (R θ (Ỹ ),R T (Ỹ )) is called the fuzzy rough set ofỸ on the information system (U , V ,R). Particularly, if (R θ (Ỹ ) =R T (Ỹ )), thenỸ is called a definable set of V , otherwise,Ỹ is called fuzzy rough set of V .
Definition 2.6 Li et al. (2019) Let (U , V ,R) be a fuzzy approximation space on two universes,P is a probability measure defined on an σ algebra composed of a subset of V , Universes of all objects with decision result λ ≥ which (U , V ,R,P) is called a fuzzy probability approximation space on two universes, for any k ∈ (0, 1], 0 ≤ β < α ≤ 1,Ỹ ∈ F(V ). The lower approximation and upper approximation ofỸ about the parameters λ, α and β are defined as follows: The pair (PR (k,α) (Ỹ ), PR (k,β) (Ỹ )) is called the fuzzy probabilistic rough set on two universes.

Notation
Notation for this paper is described in Table 1.

Local fuzzy rough set model over two universes
In this section, the basic concepts, properties and theorems of the proposed model are introduced in Subsection 3.1. Then in 3.2, we give a practical example to compare the proposed model with three existing related models.

Local fuzzy rough set model over two universes
We know that in the classical rough set model, the upper approximation of a single point set is the equivalent class of this single point set as the representative element. Chen found that this property is also satisfied in fuzzy rough set theory, so he proposed the granular computing based on fuzzy similarity relation (Chen et al. 2011). Here, according to Definition 2.6, we also give the definition of fuzzy similarity classes corresponding to the fuzzy rough set over two universes as follows: Definition 3.1 Let (U , V ,R) be a fuzzy approximation space over two universes, x λ is a fuzzy single point set, for x ∈ U , y ∈ V and λ ∈ (0, 1], fuzzy set [y λ ] TR and [y λ ] SR are defined as follows: According to Definition 2.6, the relationR is symmetric, that is for any x ∈ U , y ∈ V and λ ∈ (0, 1], we havẽ Definition 3.2 Let (U , V ,R) be a fuzzy approximation space over two universes, for any λ ∈ (0, 1], 0 ≤ β < α ≤ 1, x λ is a fuzzy single point set, for anyỸ ∈ F(V ), the θ -local fuzzy lower approximation operator and the T -local fuzzy upper approximation operator ofỸ are defined as follows: for any x ∈ U , The pair (R αθ (Ỹ ),R βT (Ỹ )) is called the local fuzzy rough set ofỸ with respect to (U , V ,R), α and β. Particularly, ifR αθ (Ỹ ) =R βT (Ỹ ), thenỸ is called definable set of V , otherwise,Ỹ is called fuzzy rough set of V . It can also be written in another form as follows: And for convenience, local fuzzy rough set model over two universes is abbreviated as TLFRS in this paper. According to Definition 3.2, for any x λ ∈ F(U ), if λ ≤ , where ≥ 0 is a very small parameter, then it's unnecessary to calculate and discuss the fuzzy similarity class of these objects. There are two reasons for removing this part of objects. Firstly, according to the actual situation, when λ ≤ , it means that the patient's illness degree is not high, so the reference significance of this sample is not high. Secondly, if the target whose membership degree is λ ≤ , then according the [x λ ] TR (y) = T (R(x λ , y), λ), the fuzzy similarity class will tend to ∅, which will have no impact on the final decision result.
It can be seen that the θ -local fuzzy lower approximation and T -local fuzzy upper approximation of the fuzzy subsetỸ on the universe V belong to the fuzzy set on another universe U . This means that the feature of the model on two universes is still holds in this model. This is also a significant difference between a fuzzy rough set over two universes and fuzzy rough set over single universe.
Similarly, an important application of information system on two universes is to analyze and obtain the decision rules. Therefore, the local positive region, local negative region and local boundary region of the fuzzy setỸ on the fuzzy approximation space (U , V ,R) are defined as follows: A new approach for computing the three regions of local fuzzy rough set model over two universes in fuzzy information system is given in Algorithm 1.
In Algorithm 1, the |U | fuzzy similarity classes need to be compute through scanning the entire universe V , hence the time complexity of step 1 is O(|U | · |V |), where |U | refer to the number of x λ ∈ F(U ) and λ ≥ . In steps 6-14, we compare the |U | fuzzy similarity classes with the fuzzy target conceptỸ for obtaining its three regions, thus its time complexity is O(|U | · |Ỹ |). Hence, the time complexity of the Algorithm 1 is only linear O(|U | · (|Ỹ | + |V |)) in terms of O(|U |).
In the following, we will cite an example to interpret the Algorithm 1.
Through the introduction of the corresponding references, we can see that the model over two universes is often used in medical diagnosis problems. The general fuzzy rough set model can only discuss the situation of multiple patients in each symptom and their diagnosis in a certain disease. However, in practical application, a doctor should consider the possibility of several similar diseases according to multiple symptoms when visiting a new patient. So the fuzzy rough set model over two universes can discuss the relationship between some symptoms and some diseases. Then judge the illness of the new patient according to the degree of each symptom. The local fuzzy rough set model over two uni-Algorithm 1 The three regions of local fuzzy rough set model over two universes.
Input: Fuzzy information system (U , V ,R), parameters α, β and , fuzzy target conceptỸ . Output: Local positive region L P O S(Ỹ ), local negative region L N EG(Ỹ ) and local boundary region L B N D(Ỹ ) of the target con-ceptỸ . 1: Calculate the fuzzy similarity class [x iλ ] TR for each object Calculate the inclusion degreeD(Ỹ /[x iλ ] TR ) of each fuzzy similarity class [x iλ ] TR and the target conceptỸ .
verses proposed in this paper is also applied to this situation. The difference is that we only treat patients with different diseases as objects, not diseases. And refine the types of diseases with different diseases and different degrees. Then further analyzes and delete some objects with low disease degree to improve the performance of the model. Example 3.1 illustrates this more clearly.
Example 3.1 Let (U , V ,R) be a disease diagnosis information system on two universes. U is a group of disease, where U = {x 1 , x 2 , x 3 }, x 1 is cold, x 2 is fever and x 3 is pneumonia. The information system directly gives relations between patients with corresponding illness {x 1 0.7 , x 2 0.7 , x 3 0.5 , x 3 0.9 , x 1 0.3 } and symptoms of V . V = {y 1 , y 2 , y 3 , y 4 , y 5 } is a group of symptoms, where y 1 is low fever, y 2 is dizziness, y 3 is runny nose, y 4 is cough and y 5 is sweating. In real life, a disease is often accompanied by several symptoms.R refers to the fuzzy relations between patients (illnesses) of U and different symptoms of V . For example,R(x 1 0.7 , y 1 ) represents that the patient who has a cold and the degree of cold is 0.7 has a 0.6 fever.
First, the fuzzy similarity class [x i λ ] TR is calculated: x 3 . At this time, it can be concluded that when the patient's symptom set isỸ , it is very likely to suffer from disease y 1 , y 2 , and the suffer degree is 0.7, and may suffer from disease y 3 , and the suffer degree is 0.5.
Based on the above definitions, some of the properties of the θ -local fuzzy lower approximation operator and the T -local fuzzy upper approximation operator ofỸ will be obtained.

Proposition 3.1 Let U and V be two non-empty finite universes andR is a fuzzy relation from U to V , that isR
, the θ -local fuzzy lower approximation operator and the T -local fuzzy upper approximation operator ofỸ have the following properties, where fuzzy setỸ = ∅ represents for any object y ∈ V ,Ỹ (y) = 0; fuzzy setỸ = V represents for any object y ∈ V ,Ỹ (y) = 1.

Comparisons of the proposed model and other models
It is known that in the general fuzzy rough set model, the uncertainty of the model is due to the existence of a boundary region in the fuzzy rough set. Similar to general fuzzy rough set model, the greater the local fuzzy boundary region of a local fuzzy rough set over two universes, the lower the accuracy of the model. Therefore, in order to reflect this characteristic more clearly in the local fuzzy rough set over two universes, we first give the concept of accuracy and roughness. Then, based on these two measures, we compare the model presented in this paper with other models.
be a fuzzy approximation space on two universes, ∀Ỹ ∈ F(V ), the accuracy and roughness ofỸ on (U , V ,R) are defined as follows: where | · | represents the cardinality of the fuzzy set. Especially, ifR βT (Ỹ ) = ∅, then ρ(Ỹ ) = 1. There are some relevant properties about the accuracy and the roughness are given as follows.

Proposition 3.2 Let U and V be two non-empty finite universes andR is a fuzzy relation from U to V , that isR
is called a fuzzy approximation space on two universes, 0 ≤ β < α ≤ 1,Ỹ 1 ⊆Ỹ 2 ∈ F(V ), the accuracy and roughness ofỸ have follow properties: , then according to Definition 3.3, x 10.7 0.6 0.7 0.4 0 0.9 x 20.7 0.7 0 0.9 0.8 0.7 x 3 0.5 0.2 0.6 0.4 0.7 0.9 x 30.9 0.9 0.2 0.1 0.5 0.4 Then we use a specific case to analyze the effectiveness of the proposed model, and the comparison results between this model and other fuzzy rough set models over two universes are obtained. The comparison models are the fuzzy probabilistic rough set model over two universes in Yan et al. (2022), the fuzzy rough set model over two universes in Yang et al. (2013), and the variable precision fuzzy rough set model over two universes in Zhou et al. (2021). These models are abbreviated as TFPRS, TFRS and TVPFRS in this paper. The abbreviations here are given for the convenience of the following description.
Example 3.2 (Continued from Example 3.1) According to Table 2, the threshold can be selected as α = 0.9, = 0.1, and the corresponding accuracy can be calculated. Here T (x, y) = max{x + y − 1, 0}. According to TLFRS model, from the calculated result in Example 3.1, there areR αθ (Ỹ ) = 0.7 x 1 + 0.7 x 2 + 0.5 x 3 and R βT (Ỹ ) = 0.7 x 1 + 0.7 x 2 + 0.9 x 3 . Therefore, ρ(Ỹ ) = 1.9 2.3 ≈ 0.83. In a fuzzy information system on two universes, for example, disease diagnosis information system. Doctors need to first analyze their own data to obtain the relationship between each disease and each symptom. But different doctors may get different relationships. The following is a method for calculating relation: Correl formula: whered i andȳ j are the average values of samples. Then the relationship induced by Correl formula: Then the final disease degree is calculated into λ = 1 and became a general fuzzy rough set model over two universes according to the Correl formula, the new information system is obtained, as shown in Table 3.
1. Calculate the upper, lower approximations and accuracy according to TFRS (Yang et al. 2013). It is calculated that For comparison with TVPFRS, if the method in this paper is applied to the crisp target concept Y , we haveD( Table 4 is the calculation results of these models. From Table 4, when the target concept is fuzzy, the B N D of T F RS and T F P RS is greater than the B N D of our model. Moreover, the accuracy ρ of the proposed model is higher than the other two models. And when the target concept is crisp, B N D of the model proposed in this paper is smaller than the B N D of T V P F RS. Moreover, the accuracy ρ of our model is higher than the T V P F RS model. Therefore, the classification results show that the local fuzzy rough set model over two universes can effectively make decisions on fuzzy information system over two universes and have better classification ability in applications.
In conclusion, by comparing the results, it can be found that our method always achieves higher accuracy in computational decision-making, regardless of whether it is a fuzzy target concept or a crisp target concept.

The reduction in local fuzzy rough sets over two universes
Similar to the fuzzy rough set on single universe, the fuzzy rough set on two universes also has redundancy. For example, we know that the existing medical resources are often unable to meet the needs of the public. The lack of doctors often makes the hospital overcrowded. In the hospital, time is life, so the time for each doctor to consult is often very limited and needs to be efficient. According to the above example, for a new patient, we can consider whether it is necessary to ask for all the symptoms to get the results when diagnosing these diseases of the universe U . If only a few symptoms are asked, can the general diagnosis result be obtained? This is the purpose of reduction in local fuzzy rough set model over two universes. First of all, we know that to keep the classification ability of the universe V to the objects in the universe U unchanged, which is equivalent to keeping the lower approximation of the universe V to a certain target concept unchanged. Therefore, the reduction method based on the significance degree is generated. In this paper, since the each object x λ of the universe U is a decision, the following definitions are given first.
Definition 4.1 Let (U , V ,R) be a fuzzy approximation space on two universes, 0 ≤ β < α ≤ 1. For any V ⊆ V , the θlocal fuzzy lower approximation and the T -local fuzzy upper approximation under V about U ofỸ V are defined as follows: When α = 1, β = 0, the lower approximation of the model at this time represents all the objects that can be completely included inỸ V , and the upper approximation represents all the related objects ofỸ V under V .
Because all objects in the universe U are a single decision, the feature selection method here needs to ensure that each x λ ∈ F(U ) about V , if x λ are separable, then V can still be distinguished after the reduction. On single universe, if an object the identification ability of the data in V is not changed, then for any Here, need to definition the classes of each object x λ as [x λ ]R V (y) =R V (x λ , y). So if the identification ability of the data in V is not changed, then for any constant. Secondly, if the classification ability of each object is required to be unchanged, it may be difficult, and errors will inevitably occur in the data processing process. Therefore, introducing the parameters α and β can make the model have a certain fault-tolerant rate. It is necessary to define the corresponding fuzzy decision diversity sets of V , and then give the corresponding reduction definition.

Definition 4.2 Let (U , V ,R) be a fuzzy approximation space on two universes
). However, it is difficult for two fuzzy sets to be completely equal. Therefore, here set an average fault tolerance threshold μ and the number N of objects x λ ∈ U . The above reduction rules are modified as follows.

Definition 4.3 Let (U , V ,R) be a fuzzy approximation space on two universes
In the heuristic algorithm for finding a local reduction, there are two measures to judge the importance of attributes used for heuristic functions, namely inner significance and outer significance. In the following, according to the reduction Definition 4.3, the corresponding definitions of the inner and outer significance can be given.

Definition 4.4
Let (U , V ,R) be a fuzzy approximation space on two universes, which V is a set of all attributes, attribute subsets V ⊆ V , attributes v m ∈ V . The inner significance of attributes v m relative to the set V is defined as: In general, inner significance is used to measure the importance of each attribute relative to a attribute set. In the reduction process, we always preferentially choose attributes corresponding to the maximum inner significance and put them into the reduction set first.
Definition 4.5 Let (U , V ,R) be a fuzzy approximation space on two universes, which V is a set of all attributes, attribute subsets V ⊆ V , attributes v m ∈ V − V . The outer significance of attributes v m relative to the set V is defined as: As for the outer significance, it is often used in forward feature selection process. Similarly, these attributes corresponding to maximum outer significance are preferentially added to the reduction set until the reduction set satisfies the stop criterion.
Algorithm 2 is the corresponding reduction algorithm given according to the above definition. The algorithm is a forward reduction algorithm, that is, in each iteration, the most important attribute is selected first.

Algorithm 2 The reduction of local fuzzy rough sets on two universes(TLFRS).
Input: Fuzzy information system (U , V ,R), parameters α, β, μ and . Output: A local reduct Red. 1: Initialize: Red = ∅. 2: Calculate the relationshipR V −vm corresponding to the attribute set v m and its inner significance S I G inner In Algorithm 2, step 2 needs to compute |V | local lower approximations using Algorithm 1. Then the time complexity in step 2 is O(|V | 2 · |U |).
Step 3 only scans the |V | attributes, hence its time complexity is O(|V |). In steps 4-6, we start from the core attribute set obtained in step 3 and add an attribute with the maximal outer significance to the reduction set Red in each subsequent iteration until a reduction is found. This process is called a forward reduction algorithm whose time complexity is O( |Red| i=1 (|U |·|V i | 2 )), here V i is the reduction result of ith iteration and Red is the final reduction result of the algorithm. The time complexity of each other step of Algorithm 2 is constant.
Then in order to highlight the advantages of the local reduction algorithm, the corresponding global reduction algorithm is given here. Firstly, the corresponding lower approximation and upper approximation are defined as follows: Definition 4.6 Let (U , V ,R) be a fuzzy approximation space on two universes, 0 ≤ β < α ≤ 1. For any V ⊆ V , the θglobal fuzzy lower approximation and the T -global fuzzy upper approximation under V about U ofỸ V are defined as follows: Algorithm 3 The reduction of global fuzzy rough sets on two universes(TGFRS).
Similar to Algorithm 2, the time complexity of Algorithm (|U | · |V i | 2 )), here V i is the reduction result of ith iteration and Red is the final reduction result of the algorithm.
It is easy to see that for many real large-scale data sets in real life, the universe U that needs to be considered in the local idea is often much smaller than the whole universe U in the process of forward searching. Therefore, for large data reduction, the TLFRS algorithm will have better time reduction performance. That is to say, the larger the scale of a data set, the more time the TLFRS algorithm reduces. In short, through the above analysis, it can be found that compared with the TGFRS algorithm, the TLFRS algorithm can greatly improve the running speed.
From Algorithm 2 and Algorithm 3, we can know that both local and global reduction algorithms can obtain more accurate reduction results, because they are given according to the definition of not changing the classification ability of each object. However, due to the large data set, the time complexity is often high. Through the time complexity analysis of algorithm 2 and algorithm 3, we can find that the time complexity of local reduction algorithm is greatly improved compared with global reduction algorithm.

Numerical experiment analysis
In this section, in order to verify the effectiveness and advantage of the proposed model, we present a series of experiments conducted on ten data sets from the UCI Machine Learning Repository. 1 The basic data sets information are shown in Table 5. The experiments are performed on a PC with Windows 7 and Intel(R) Core(TM)i7-6700 CPU @ 3.40 GHz with 8GB of memory.
For numeric data, because considering the requirements for the use of the T -norm and most data sets are continuous. Therefore, each data set needs to be normalized first. Here, we normalized the numerical attribute v into the interval where v(u) max represents the maximum value of all objects u under attribute v and v(u) min represents the minimum value of all objects u under attribute v.
In order to facilitate model comparison, this paper constructs the corresponding two universes data set. It is known that in the fuzzy information system over two universes, one universe is generally a collection of different fuzzy decisions in the general fuzzy information system, and the other universe is generally composed of the feature sets in the general fuzzy information system. There are corresponding relationships between the objects that belong to the two universes respectively. This relationship matrix constitutes a fuzzy information system over two universes. So in this paper, the way to make the data set into a two universes information system is to integrate several fuzzy decision information systems with the same features into a two universes information system. Among them, the object of this two universes information system is the object of those fuzzy decision information systems and their decisions membership, with common characteristics. For the object with the same decision attribute and decision membership, the mean value of the membership degree of each feature is integrated into one object.
We made an experimnetal comparison in two parts. In Sect. 5.1, we compared the decision results without reduction. The fuzzy model TLFRS1 (fuzzy target concept) and fuzzy model TLFRS2 (clear target concept) in this paper are compared with the TFRS model in document (Yang et al. 2013), the TFPRS model in document (Yan et al. 2022) and the TVPFRS model in document (Zhou et al. 2021). By comparing the accuracy ρ, boundary region B N D and running time, it is concluded that the model in this paper can obtain higher accuracy in decision-making. And the model classification ability is strong. And in Sect. 5.2, we compared the local reduction model with the global reduction model. By comparing the experimental results, we can find that the local reduction algorithm proposed in this paper has faster computing time and similar or higher accuracy.

Comparison of decision results
Firstly, the universes of fuzzy decision data set need to be processed before calculation, and the common data set needs to be converted into a data set on two universes. The processing method is to use Correl formula. Only in this way can we compare them with other models.
Objects in the boundary region indicate that the model is unable to classify these objects based on existing knowledge. It is known that a model should classify as many objects as possible in a classification problem. Therefore, when applying a model to a practical problem, the model should have as few objects in the boundary region as possible, that is, the boundary region should be as small as possible, so that we can classify more objects. This is consistent with the role of accuracy, which is also a measurement formula to judge the classification ability of a model. Precision is a measure of the approximate proximity between upper and lower judgements. B N D size in classification data is often a positive integer without an upper boundary, while accuracy is a decimal in the [0, 1] interval. If the accuracy is 1, the model has strong classification ability, and can distinguish any object, that is, the boundary region is 0; the accuracy of 0 represents that the classification ability of the model is poor, and each object is indistinguishable, that is, the boundary region is a collection of all objects in the entire universe. To compare the effects of decision on TFRS, TFPRS, TVPFRS, TLFRS1 and TLFRS2, we select different parameters for each data set. For Concrete, Two universes concrete flow and Concrete flow 2, the is take 0.2 and the α is take 0.8; for Concrete flow 3 and Housing, the is take 0.25 and the α is take 0.8, for Wine-quality red and Two universes wine-quality, the is take 0.4 and the α is take 0.8; for Winequality white, the is take 0.4 and the α is take 0.55; for WDBC and WPBC, the is take 0.2 and the α is take 0.55. Table 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15 are the comparison of accuracy ρ, boundary region B N D and run time Time of the five models, where the computational time is measured in seconds. Table 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15 show that for most data sets, the algorithms TLFRS1 and TLFRS2 proposed in this paper have higher accuracy ρ and smaller boundary region B N D. For the data set of Housing, although the accuracy of TLFRS2 is less than that of TFRS algorithm, because it's target concept is fuzzy, the TLFRS2 methods is a crisp case of this paper that is extended for the comparison of TVPFRS. Therefore, if we compare TFRS and TLFRS1 under the same fuzzy target concept, we can find that they have the same accuracy and boundary region. The same comparison between TLFRS2 and TVPFRS also shows that the model in this data set still has good classification ability. By comparing the running times, we can find that in most cases, the running time of the model in this paper is at a moderate speed, and even has a slight advantage in the results of some data sets, such as the Concrete flow 2, the Concrete flow 3 and the WPBC. Therefore, to sum up, the models in this paper often have better classification ability in decisionmaking, and the running time will not be too long. Figure 1 is the accuracy ρ comparison of the five models on ten datasets of Table 5. The color is cyan and the shape is triangle, which is the accuracy of TFRS model in ten data sets; the color is magenta and the shape is diamond, which is the accuracy of TFPRS model in ten data sets; the color is green and the shape is square, which is the accuracy of TVPFRS model in ten data sets; the color is red and the shape is pentagram, which is the accuracy of TLFRS1 model in ten data sets; the color is blue and the shape is hexagon, which is the accuracy of TLFRS2 model in ten data sets. Fig. 1 shows that when compared with the TFRS model and the TFPRS model, the accuracy of the TLFRS1 is much higher, and when compared with the TVPFRS model, the accuracy of the TLFRS2 is much higher. And the accuracy of the TFPRS model and the TVPFRS model is much lower. It can be seen that TLFRS1 and TLFRS2 have higher accuracy than other three models, which indicates that the classification ability of TLFRS model is superior to other models. Figure 2 is the boundary region B N D comparison of the five models on ten datasets of Table 5. The color of every model are consistent with Fig. 1. Figure 2 shows that the length of boundary region of TLFRS1 is the shortest. There- Fig. 2 Contrast the size of the BND fore, TLFRS is capable of classifying as many objects as possible while ensuring classification accuracy.
In summary, the final classification results show that the new models presented in this paper can effectively make decisions and have better classification ability than other models in applications.

Comparison of reduction results
To show the different patterns of efficiency changes in local and global algorithms as the sample size gradually increases, we divide each of these ten data sets in Table 5 into ten parts of equal size. And we randomly select a sub dataset as the initial set U 1 to be calculated. Then add a sub dataset to the set at each step, that is, |U 1| = |U |/10, |U 2| = 2|U |/10, …, |U 10| = |U |. At each step, calculate the reduction result of the set and record the corresponding running time.
Tables 16,17,18,19,20,21,22,23,24,25 are the result of the global and local reduction algorithm with the increase of the sample scale. Tables 16,17,18,19,20,21,22,23,24,25 show that in most cases, the number of attributes in the local reduction algorithm is often less than the number of attributes in the global reduction algorithm. In most cases, the running time of local reduction algorithm is relatively short. Figures 3, 4, 5, 6, 7 are the broken line graphs drawn by the running time recorded in turn for each dataset as the dataset gradually increases. From the figures, it can be clearly seen that compared with the global reduction algorithm, the local reduction algorithm can greatly improve the running speed  of the model. With the increase of the dataset, the higher the growth in efficiency, the faster the reduction in running time increases. When the entire dataset is finally taken for reduction calculation, it is often able to maximize the running time. Therefore, the local reduction algorithm improves the computational efficiency of reduction.
Through the analysis of Figs 3, 4, 5, 6, 7, we can find that the calculation efficiency of the local reduction algorithm increases with the increase of the data set. Therefore, we extract the classification accuracy results obtained from the calculation of the full set of each data set for analysis. First of all, because the most data sets used in this experiment are fuzzy decision data sets, that is, the decision of the data is a continuous number, so if we use the KNN classifier, it may not be appropriate or may affect the judgment results. Therefore, this paper proposes a new KNN algorithm based on the original KNN algorithm. FKNN is proposed by predecessors in order to reduce the error rate of KNN algo-rithm and reflect the different functions of different samples in K nearest neighbors. This method uses the fuzzy theory in mathematics to represent the membership degree of each sample belonging to a certain class, thus improving the accuracy of classification.
In order to distinguish from the FKNN, the new classifier is called NFKNN. The NFKNN algorithm uses the mean value of the decision values of K closest samples to the target samples as the predicted decision attribute value of the target samples. This method can also be applied to multiple fuzzy decisions, and if the decision value is crisp, it will degenerate into FKNN method. Therefore, this algorithm is more applicable and accurate than the classical KNN algorithm, because the membership is obtained from the specific attribute values, rather than subjectively given. The computational efficiency of the TLFRS algorithm is verified by comparing the classification accuracy with the TGFRS algorithm. NFKNN (K=3) and FC-means (C=5) classification accuracy of different attribute subsets selected by the two algorithms. Table 26 shows the results obtained using the 10-fold cross-validation method. Figure 8 shows that the NFKNN (K=3) and FC-means (C=5) classification accuracy of TLFRS algorithm is basically the same as the TGFRS algorithm, and is slightly higher than the TGFRS algorithm. Evidently, TLFRS is an effective attribute reduction method for all data sets.

Conclusions
In order to improve the classification ability and practicability of the fuzzy rough set model over two universes, a local fuzzy rough set model over two universes is proposed. To further explain the model we proposed, several properties and examples are also provided. In addition, a novel local reduction algorithm was designed using the TLFRS model. This algorithm only needs to calculate part of the objects, which can evaluate the features more efficiently. Experimental comparisons show that our approaches are stable, feasible and efficient in calculating decision results and reductions. Therefore, the local fuzzy rough set model over two universes can improve the existing models effectively. There are, however, also several remaining issues that deserve further investigation. For example, considering the relationship that the union of fuzzy sets is the max value of membership degree, we will study the accelerated case of the reduction methods in the future. Another part of future work would be to consider combining the proposed model with other reduction algorithms.