key: cord-0922291-4ji25n7c
authors: Al-shami, Tareq M.
title: Maximal rough neighborhoods with a medical application
date: 2022-05-02
journal: J Ambient Intell Humaniz Comput
DOI: 10.1007/s12652-022-03858-1
sha: 4b692119566133bf50b18a09aa9ff676b192973b
doc_id: 922291
cord_uid: 4ji25n7c

In this paper, we focus on the main concepts of rough set theory induced from the idea of neighborhoods. First, we put forward new types of maximal neighborhoods (briefly, [Formula: see text] -neighborhoods) and explore master properties. We also reveal their relationships with foregoing neighborhoods and specify the sufficient conditions to obtain some equivalences. Then, we apply [Formula: see text] -neighborhoods to define [Formula: see text] -lower and [Formula: see text] -upper approximations and elucidate which one of Pawlak’s properties are preserved (evaporated) by these approximations. Moreover, we research [Formula: see text] -accuracy measures and prove that they keep the monotonic property under any arbitrary relation. We provide some comparisons that illustrate the best approximations and accuracy measures are obtained when [Formula: see text] . To show the importance of [Formula: see text] -neighborhoods, we present a medical application of them in classifying individuals of a specific facility in terms of their infection with COVID-19. Finally, we scrutinize the strengths and limitations of the followed technique in this manuscript compared with the previous ones.

familiarized the idea of rough sets theory as a mathematical tool to address incomplete information systems and uncertainty. The essential notions in this theory such as upper and lower approximations and accuracy measures are firstly characterized using an equivalence relation. In past few years, many novel models have been espoused, by changing or relaxing some underlying conditions.

After the superb start of studying rough sets using right and left neighborhoods (Yao 1996 (Yao , 1998 , some rough set models have been established such as minimal left neighborhoods ) and minimal right neighborhoods . Then, intersection and union neighborhoods, intersection minimal and union minimal neighborhoods were introduced in (Abd El-Monsef et al. 2014) . Generating right neighborhoods from finite class of arbitrary relations was investigated in (Abu-Donia 2008). Mareay (2016) studied four types of neighborhoods using of neighborhoods system called where ∈ {l, ⟨r⟩, ⟨l⟩, i, u, ⟨i⟩, ⟨u⟩} . In Sect. 4 , we mainly aim to introduce different types of approximations and accuracy measures. We compare between them and elucidate that the best case given by ⟨i⟩ . In Sect. 5, we propose an algorithm to classify the individuals who work in a specific facility with respect to corona-virus COVID-19 infection. Finally, discussions and conclusions are provided in Sects. 6 and 7, respectively.

In this section, we provide some definitions and results of different types of neighborhoods to make this manuscript self-contained and also to rationalize the need to introduce the concept of M -neighborhoods.

A binary relation on B is a subset of B × B . We write if ( , ) ∈ . is called reflexive (resp., symmetric, transitive) if for each ∈ B (resp., ⟺ , whenever and ). If is reflexive, symmetric and transitive, then it called an equivalence relation. Also, we call a comparable relation if or for each , ∈ B.

Definition 2 (Pawlak 1982 (Pawlak , 1991 Let be an equivalence relation on B . We associate every W ⊆ B with two subsets: The following result lists the core properties of Pawlak's rough set model. In the previous generalized rough set models, the validity of these properties was examined. Of course, some of them are evaporated in these models; however, keeping as many as possible of these properties is considered as an advantage of the model. Proposition 1 (Pawlak 1982 (Pawlak , 1991 Let V, W be two subsets of B . The following properties are satisfied if is an equivalence relation on B .

We can also numerically describe rough sets by using the following two measures.

Definition 3 (Pawlak 1982 (Pawlak , 1991 Let be an equivalence relation on B . The A-accuracy and R-roughness measures of a nonempty subset W of B with respect to an equivalence relation are respectively defined by

The equivalence relations are a strict condition and cannot be satisfied in many circumstances. So that, the classical model has been extended using weaker relations than the equivalence. El-Monsef et al. 2014; Abo-Tabl 2011; Allam et al. 2005 Allam et al. , 2006 Yao 1996 Yao , 1998 The N -neighborhoods of an ∈ B (denoted by N ( ) ) are defined under an arbitrary relation on B , where ∈ {r, l, ⟨r⟩, ⟨l⟩ , i, u, ⟨i⟩, ⟨u⟩} , as follows.

Henceforth, we consider ∈ {r, l, ⟨r⟩, ⟨l⟩, i, u , ⟨i⟩, ⟨u⟩} , unless stated otherwise.

The foregoing types of neighborhoods were applied to define new types of lower and upper approximations and accuracy (roughness) measures. Comparisons between them in terms of improving the approximations and increasing the accuracy measures were done in various papers. El-Monsef et al. 2014; Abo-Tabl 2011; Allam et al. 2005 Allam et al. , 2006 Al-shami 2021a; Al-shami et al. 2021b; Yao 1996 Yao , 1998 Let be an arbitrary relation on B and K ∈ {N, E, C} . We define lower and upper approximations of each W ⊆ B with respect to the types of neighborhoods as follows.

Definition 9 (Abd El- Monsef et al. 2014; Allam et al. 2005 Allam et al. , 2006 Al-shami 2021a; Al-shami et al. 2021b; Yao 1996 Yao , 1998 Let be an arbitrary relation on B . The A K and A C -accuracy and R K -roughness measures of a nonempty subset W of B with respect to are respectively defined by

where It should be noted that the above approximations and accuracy measures were also studied via topological spaces; see, (Al-shami 2021b; Al-shami et al. 2021a; El-Bably and Al-shami 2021; Lashin et al. 2005 ; Salama 2020; Singh and Tiwari 2020).

Definition 10 (see, (Dai et al. 2018) ) Let K ∈ {N, E, C} and consider 1 and 2 are two relations on B such that 1 ⊆ 2 . We say that the approximations induced from K-neighborhoods have the property of monotonicity-accuracy (resp.,

Definition 11 (Dai et al. 2018) The maximal right neighborhood of an ∈ B (briefly, M r ( ) ) is defined under an arbitrary relation on B as follows.

In this section, we introduce the concept of maximal neighborhoods of an element with respect to any binary relation. We explore their main properties and determine the conditions under which some of them are identical. We compare between them as well as compare them with the previous ones.

The maximal neighborhoods of an element ∈ B , denoted by M ( ) , induced from any binary relation on B are defined for each as follows.

We display the following example which will helps us to show the obtained relationships as well as makes some comparisons in the next section. Table 1 .

In Table 1 , note that N r ( ) ≠ � and N l ( ) ≠ � , but ∉ N r ( ) and ∉ N l ( ) . This means that the above property does not hold for N -neighborhoods when ∈ {r, l, i, u}.

Similarly, one can prove the proposition when = l . ◻

Proof Consider = r and let , ∈ N r (x) . Then x and x . Since is symmetric and transitive, we obtain and . Now, suppose that z ∈ M r ( ) . Then there exists y ∈ B such that z ∈ N r (y) , where ∈ N r (y) as well. This implies that z . It follows from that x and is transitive that

Hence, the coveted result is obtained.

The other cases are proved in a similar way. ◻ Now, we show the relationships between the different types of M -neighborhoods as well as we determine their relationships with some neighborhoods already introduced in the literature.

To prove (iii), if M ⟨r⟩ ( ) = � , then the proof is trivial. So, suppose that ∈ M ⟨r⟩ ( ) . Then there exists x ∈ B such that , ∈ M r (x) . It follows from Proposition 3 that x ∈ M r ( ) . According to Proposition 2, ∈ M r ( ) . This implies that ∈ M r ( ) . This proves that M ⟨r⟩ ( ) ⊆ M r ( ) . Following similar manner, one can prove the other cases of .

To prove (iv), if M r ( ) = � , then the proof is trivial. So, suppose that ∈ M r ( ) . Then there exists x ∈ B such that , ∈ N r (x) . Since is symmetric and transitive, it follows from Proposition 5 that 

Then there exists y ∈ B such that ∈ M r (y) and ∉ M r (y) . According to Proposition 5, y ∈ M r ( ) . It follows from equality (1) that y ∈ M r ( ) as well. This means that ∈ M r (y) . But this contradicts assumption. Therefore, ∈ M ⟨r⟩ ( ) . Thus,

Hence, the proof is complete. ◻

The following corollary gives one of the unique characterizations of M -neighborhoods which is the determination of the smallest and largest one of neighborhoods from all . The previous types of neighborhoods do not have this characterization; their determination is limited on disjoint two sets {r, l, i, u} and {⟨r⟩, ⟨l⟩, ⟨i⟩, ⟨u⟩}.

The converses of (i), (ii) and (iv) in the Proposition 6 fail as illustrated in Example 1. To show the converse of (iii) need not be true, we provide the next example.

Then we calculate the maximal neighborhoods of each element of B in the Table 2 .

Since is symmetric, it is sufficient to compare between them when = r.

If is an equivalence, then all kinds of maximal neighborhoods are equal.

If is an equivalence, then all kinds of N -neighborhoods of any element are equal. Hence, all kinds of M -neighborhoods of any element are also equal.

◻ In the next results, we scrutinize the interrelations between M -neighborhoods and some of the previous ones.

Proposition 9 Let (B, , ) be a -NS and ∈ B . Then. 

The proof comes from the fact that every N -neighbourhood forms a partition of B under an equivalence relation. ◻

If we want to compute M ( ) from two different -NSs (B, 1 , ) and (B, 2 , ) we write M 1 ( ) and M 2 ( ).

Proposition 12 Let (B, 1 , ) and (B, 2 , ) be two -NSs such that 1 ⊆ 2 . Then M 1 ( ) ⊆ M 2 ( ) for each ∈ B and ∈ {r, l, i, u}.

Then there exists y ∈ B such that , ∈ N 1r (y) . Since 1 ⊆ 2 , N 1r (y) ⊆ N 2r (y) . This implies that ∈ M 2u ( ) . Hence, M 1u ( ) ⊆ M 2u ( ) , as required. Following similar technique, one can prove the other cases. (ii): Let ∈ B . Since is comparable, x or x for each ≠ x . This means that x ∈ M r ( ) or x ∈ M l ( ) . Therefore,

In a similar way, one can prove the following result. 

In the following four results, we examine which one of Pawlak's properties are preserved by M -lower and Mupper approximations.

Theorem 15 Let (B, , ) be a -NS and V, W ⊆ B . The next statements hold true. 

Proof The proof is similar to that of Theorem 15. 

, we have 0 ≤ A M (W) ≤ 1 for any subset W ≠ ∅ of B. Proof Let W be a nonempty subset of B . Then H M (W) ⋃ W ≠ � for each . It is clear that � ⊆ H M (W) ⋂ W ⊆ H M (W) ⋃ W . This automatically means that 0 ≤| H M (W) ⋂ W |≤| H M (W) ⋃ W | . There- fore, 0 ≤ |H M (W) ⋂ W| |H M (W) ⋃ W| ≤ 1 . Hence, A M (W) ∈ [0, 1] . ◻

We devoted the rest of this section to comparing between the given approximations and accuracy measures.

Proof To prove (i), let ∈ H Mu (W) . Then M u ( ) ⊆ W . As we showed in Proposition 6 that M r ( ) ⊆ M u ( ) , so ∈ H Mr (W 

w h e r e ∈ {r, l, i, ⟨r⟩, ⟨l⟩, ⟨u⟩}.

To confirm the obtained results and to illustrate that the converses of the above Proposition 21, Corollary 22, Proposition 23 and Corollary 24 fail we provide the following two examples.

Example 5 Let a -NS (B, , ) as given in Example 2. Note that in this example all M -neighborhoods are identical and all M ⟨ ⟩ -neighborhoods are identical, where ∈ {u, r, l, i} . Therefore, the calculations in Table 3 show that the approximations and accuracy values are obtained from ⟨ ⟩ are better than their counterparts obtained from .

Example 6 Let a -NS (B, , ) as given in Example 1. Note that in this example M -neighborhood and M ⟨ ⟩ -neighborhood are identical for all ∈ {u, r, l, i} . Therefore, in Table 4 , we suffice by comparing the cases of ∈ {u, r, l, i}.

Also, the calculations in Table 4 show that the best approximations and accuracy values are obtained in cases of ∈ {i, ⟨i⟩} . This means that our method is better than the method given by Dai et al. (2018) . Table 3 The approximations and accuracy measures in cases of and ⟨ ⟩ 

At present, COVID-19 pandemic occupies a great interest in all areas of the world because of its rapid negative influences which affect the health system, society, economy, and even politics. According to the experts' speech, the physical contact or nearness between individuals is the main way of transmission of COVID-19. Regrettably, there is no successful remedy yet. Only you can stay safe following some simple precautions according to WHO's recommendations such as wearing a mask, physical distancing (preserve at least a one-meter distance between yourself and others), avoiding crowds, and cleaning your hands. World Health Organization showed that the suspicious individual is quarantined for fourteen days. Then, if no symptoms develop through the quarantine period, the procedures of infection prevention and control measures are continued.On the other hand, isolation continues if the symptoms develop through the quarantine period. Herein, we applied M -neighborhoods to classify the community sample under study (such as patients, medical staff, school staff, etc.) in terms of suspicion of infected them by COVID-19. Note that we can apply this technique to any contagious diseases such as influenza.

To explain our model to quarantine the COVID-19 patients, let us consider B = { 1 , 2 , … , m } as a group of individuals who works at a specific facility such as school, hospital, bank, etc. The facility administration does a periodic check for its employees from time to time (this period is depending on available capabilities) to check their medical status with respect to COVID-19.

Let X ⊆ B be a group of individuals having a positive COVID-19 test. Directly, this group is quarantined. Now, It is natural to ask what about W = B⧵X ? The answer to this question according to our model is as follows, we determine the individuals of W who had contact with individuals of X and vise versa. As we know the contact between individuals leads to spread COVID-19 among them, so that, we consider as a relation on B as following: i j iff i had a contact with j , where i, j ∈ {1, 2, … , m} . is a symmetric relation (but is not transitive) because i is in contact with j iff j is in contact with i . According to Proposition 8, there are only two kinds of M-neighborhoods. Now, we compute N -neighborhood and M -neighborhood for all members in the infected set X. Then, we divide the suspected set W into three groups W 1 , W 2 and W 3 . Group 1 (under high suspicion): The individuals W 1 of W who belong to M -neighborhood of a member ∈ X , where ∈ {⟨r⟩, ⟨l⟩, ⟨i⟩, ⟨u⟩} . This implies that for each w ∈ W 1 belongs to M ( ) all individuals of X who contact infected person are in contact with w as well. This matter increases the probability of infection of w with COVID-19. Group 2 (under suspicion): The individuals W 2 of W who belong to M -neighborhood⧵ M ⟨ ⟩ -neighborhood of a member ∈ X (M ( ) ⧵ M ⟨ ⟩ ( )) , where ∈ {r, l, i, u} . This implies that for each w ∈ W 2 belongs to M ( ) ⧵ M ⟨ ⟩ ( ) there exists an individual of X who is in contact with the infected person contacts w as well. Group 3 (no suspicion): The individuals W 3 of W who do not belong to M -neighborhood of any member ∈ X . This implies that the individuals of W 3 do not contact any suspected person of X or his/her contacts. Therefore, we can say that the individuals of W 3 do not have the COVD-19.

After this classification, the quarantine is applied according to the capabilities available; for instance, it suffices to quarantine the individuals who belong to the high suspicion group if the available capabilities is not enough. Otherwise, we quarantine the two suspicion groups (high suspicion and suspicion). In this technique, our role stops at the classification of suspicion individuals into three groups.

Finally, we present the algorithm (Algorithm 1) arising from the discussion above, in order to classify the members with respect to the degree/probability of their infection with COVID-19. Set the group W ⊆ B who is in contact with the group of positive COVID-19 individuals X.

Define the binary relation on the set B, that takes into account the contacts among the given facility members. 7: for α ∈ X ⊆ B do 8:

compute N σ -neighborhoods.

compute M σ (α)-neighborhood.

compute M σ (α)-neighborhood. 11: end if 12: Classify the elements of W in terms of M σ and M σ neighborhoods into three groups:

High suspicion: The individuals who belong to M σ -neighborhood of at least one α ∈ X

Suspicion: The individuals who belong to M σ -neighborhood of at least one α ∈ X

No suspicion: The individuals who do not belong to M σ -neighborhood of any α ∈ X 16: 17: return the obtained classification. 

One of the considerable approaches to address the issues of vagueness and uncertain knowledge is rough set theory. In this article, we have established novel kinds of neighborhood systems namely M -neighborhoods. We have deliberated some properties concomitant with M -neighborhoods and elucidated their relationships with some neighborhoods types introduced in the literature. Then, we have exploited them to introduce some types of lower and upper approximations. We have compared between them and showed that an M ⟨i⟩ -neighborhood produces the best approximations and highest accuracy measure. Also, we have demonstrated that M -accuracy and M -roughness values are monotonic under any arbitrary relation, where ∈ {r, l, i, u}. Moreover, we have compared the followed technique with its counterparts induced from the method given in (Dai et al. 2018 ) in terms of approximations and accuracy measures of subsets. Also, we have proved the identity between our technique and those given in (Amer et al. 2017 ) under a symmetric relation. Finally, we applied our approach to protect people from infection with the corona-virus COVID-19.

Possible forthcoming works are • Benefit from M -neighborhoods to generate different topologies which are applied to establish new approximations similar to those given in (Abo-Tabl 2013; Dai et al. 2018; Sun et al. 2019 ); • Explore M -neighborhoods system in the areas of soft rough set and fuzzy rough sets.

Determination of the smallest and largest M -neighborhoods from all under any arbitrary relation (see, item (iii) of Proposition 6 and Corollary 7). In fact, this is a unique characterization of M -neighborhoods. The previous types of neighborhoods

This matter leads to compare M -approximations and M ⟨ ⟩ -approximations as investigated in Proposition 23 and to compare A -accuracy and M ⟨ ⟩ -accuracy measures as investigated in Corollary 24. Whereas, we cannot compare between these two types of neighborhoods (approximations, accuracy measures) generated from

Our method keeps the monotonic property for the accuracy and roughness values under any arbitrary relation as illustrated in Proposition 25 and Corollary 26. Whereas, this property is losing or References Abd El-Monsef ME, Embaby OA, El-Bably MK (2014) Comparison between rough set approximations based on different topologies

A comparison of two kinds of definitions of rough approximations based on a similarity relation

Rough sets and topological spaces based on similarity

Comparison between different kinds of approximations by using a family of binary relations

An improvement of rough sets' accuracy measure using containment neighborhoods with a medical application

Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets

Subset neighborhood rough sets

A comparison of two types of rough approximations based on n j -neighborhoods

New rough approximations based on e-neighborhoods

International workshop on rough sets, fuzzy sets, data mining, and granular computing

New approach for closure spaces by relations

On j-near concepts in rough sets with some applications

Generalized rough set models determined by multiple neighborhoods generated from a similarity relation

Different kinds of generalized rough sets based on neighborhoods with a medical application

Rough set theory for topological spaces

Generalized rough sets based on neighborhood systems and topological spaces

Rough sets

Sequences of topological near open and near closed sets with rough applications

Topological structures in rough set theory: a survey

A new approach to rough set based on remote neighborhood systems

Two views of the theory of rough sets in finite universes

Relational interpretations of neighborhood operators and rough set approximation operators

The author is extremely grateful to the editor and anonymous three reviewers for their valuable comments and helpful suggestions which helped to improve the presentation of this paper.Funding This research received no external funding.

No data were used to support this study.

The author declares that there is no conflict of interests regarding the publication of this article.

We only prove (i) and one may prove the other cases following similar technique.SinceB y assumption, W is nonempty; therefore, | H M (W) ⋃ W |> 0 for all . Thus, By (2) and (3) we find Hence, we obtain the desired result. ◻The proofs of the next two results follow from item (iii) of Proposition 6 and Corollary 7.Proposition 23 Let (B, , ) be a -NS and W ⊆ B . Thenw h e r e ∈ {r, l, i, ⟨r⟩, ⟨l⟩, ⟨u⟩}.Corollary 24 Let (B, , ) be a -NS and W be a nonempty subset of B . Then Table 4 The approximations and accuracy measures for all2 keeping under strict conditions in some foregoing methods. This is due to that our method is depending only on the union of N -neighborhoods which is proportional to the size of the given relations. In contrast, the relations used in the definitions of some methods (i.e.; equality, subset, superset) randomly works with respect to the size of the given relations. To illustrate this matter in case of C l -neighborhoods, let 1 3. The M -lower and M -upper approximations keep most Pawlak's properties as we investigated in Proposition 15, Corollary 16, Proposition 17 and Corollary 18. Also, the properties of Pawlak which are losing by M -approximations are preserved under a reflexive condition. 4. Our method produces approximations and accuracy values better than the methods given in (Dai et al. 2018 ) in cases of ∈ {i, ⟨i⟩, ⟨r⟩} . Also, the approximations and accuracy values generated by our methods are equal to those generated by E -neighborhoods under a symmetric relation for ∈ {i, r, l, u} (see, item (ii) of Corollary 10).• Limitations