Multi-users S-rough sets model based on three-way decisions

Abstract

The paper focuses on the question that ensures the completeness of the information and the authenticity of the conclusions, which is based on S-rough sets from the perspective of three-way decisions. In order to make a rational decision by the dynamic information, the multi-users S-rough sets model based on three-way decisions is proposed. Subsequently, in the case of incomplete information, the paper gives an application of the model, which may indicate usefulness of the model and prove that it is the new approach to deal with the relevant problem of uncertainty information provided by multiple users.

Keywords

Completeness of the information multi-users decision making probabilistic rough sets S-rough sets three-way decisions rules

1 Introduction

Rough Sets theory is mainly used in expressing and processing the uncertain, imprecise and vague information [1, 2]. It uses the equivalence relations to divide the universe into a number of equivalence classes. The concepts are approximated by the upper and lower approximations. Namely, the concepts are divided by three regions, which are the positive, boundary and negative regions. The positive region is constituted by the equivalence classes that belong to the approximated entirely, while the negative region is obtained by the equivalence classes that not belong to approximated. The equivalence classes constitute the boundary region if it belongs to the set partly. Based on three regions above, Yao proposed the three-way decision rules and explored a new semantic model [3 –5]. In a sense, the classical rough set is the static rough sets, but the information has the dynamic nature in reality. In order to ensure the completeness of the information, literatures [6 –13] proposed S-rough sets.

In order to ensure the completeness of information and authenticity of conclusion, this paper applies the ideology of three-way decisions to the S-rough sets and extends it to multi-users. In addition, different kinds of multi-users models are also discussed, and application conditions and scope of this model are presented. Subsequently, the paper applies the model to an instance, which verifies the practicality of the model. Eventually, this paper gives the definition and method of attribute reduction. Therefore, we can make a final reasonable decision in the case of existence of incomplete information.

Based on the above analyses, this paper is divided into six parts. In the first part the three-way decision rules and S-rough sets are briefly reviewed. In the second part the S-rough Sets Model based on Three-way Decisions is presented. In the third part extend the model to multi-users. In the fourth part a model instance is given. In the fifth part proposes the attribute reduction of S-rough sets model based on three-way decisions. And In the sixth part theconclusion of this paper is drawn and the further research is pointed out.

2 Three-way decisions and S-rough sets

Pawlak rough sets theory does not take the decision rules of fault-tolerant into account, which requires to introduce the concept of probabilistic rough set [14, 15].

Definition 1. Let S = {U, A, V, f} be an information table, ∀X ⊆ U, 0 ≤ α < β ≤ 1, then the (α, β) probabilistic lower and upper approximations are defined as: $\underline{{apr}_{α, β}} (X) = {x \in U : Pr (X | [x]_{R}) \geq α}$ (1) $\bar{{apr}_{α, β}} (X) = {x \in U : Pr (X | [x]_{R}) > β}$ (2)

In which Pr(X| [x] _R) = | [x] _R ∩ X| ∖ | [x] _R| denote the conditional probability of the classification, | · | represents the cardinality of a set. The (α, β)-probabilistic lower and upper approximations divide the universe into three regions: the (α, β)-probabilistic positive, boundary and negative regions, which are defined as: ${POS}_{(α, β)} (X) = {x \in U : Pr (X | [x]_{R}) \geq α}$ (3) ${BND}_{(α, β)} (X) = {x \in U : β < Pr (X | [x]_{R}) \leq α}$ (4) ${NEG}_{(α, β)} (X) = {x \in U : Pr (X | [x]_{R}) \leq β}$ (5) Accordingly, the basic ideology of three-way decisions can be obtained, which is described as: the positive rule is used for accepting, the negative rule for rejecting, and the boundary rule for abstaining [3 –5].

Definition 2. Given a finite nonempty universe U, R is an equivalence relation defined in U, f_i is element of U. F is the element transfer clan in U which is the set from f₁ to f_n, namely, F = {f₁, f₂, …, f_n}, which is called that X° ⊂ U is one direction singular set in U. And if f ∈ F, X° is one direction S-Sets, namely, one direction S-rough Sets, which is denoted as: $X^{\circ} = X \cup {x : u \in U, u \notin X, f (u) \in X}$ (6) Obviously, the static X gets unidirectional dynamic singular changes because of the presence of F and generates X°. For X°, the lower approximations is(R, F) _∘ (X°), and its upper approximations is (R, F) ° (X°). $\begin{matrix} (R, F)_{\circ} (X^{\circ}) = \cup [x]_{R} \\ = {x : (x \in [f (x)]_{R}) \cup ([x]_{R} \subseteq X^{\circ})} \end{matrix}$ (7) $\begin{matrix} (R, F)^{\circ} (X^{\circ}) = \cup [x]_{R} \\ = {x : (x \in [f (x)]_{R} \cap X^{\circ} \neq φ) \land ([x]_{R} \cap X^{\circ} \neq φ)} \end{matrix}$ (8)

Definition 3. Given a finite nonempty universe U, R is an equivalence relation defined in U, f_i is element of U. F and F′ are the element transfer clan in U, in which F is the set from f₁ to f_n and F′ is the set from $f_{1}^{'}$ to $f_{m}^{'}$ . γ is the union of F and F′. Then X^*⊂ U is two direction Singular Set in U. If f, f′ ∈ γ, X^* is two direction S-Sets, namely, two directions S-rough Sets, and X′ is called the loss set of X. X′ and X^* are denoted as: $X^{'} = X \ {x : x \in X, f (x) \notin X}$ (9) $X^{*} = X^{'} \cup {x : u \in U, u \notin X, f (u) \in X}$ (10)

For X^*, its lower approximations is (R, F) _∘ (X^*) and its upper approximations is (R, F) ° (X^*). $\begin{matrix} (R, γ)_{\circ} (X^{*}) = \cup [x]_{R} \\ = {x : x \in [f (x)]_{R} \cup [x]_{R} \subseteq X^{*}} \end{matrix}$ (11) $\begin{matrix} (R, γ)^{\circ} (X^{*}) = \cup [x]_{R} \\ = {x : (x \in [f (x)]_{R} \cap X^{*} \neq φ) \land ([x]_{R} \cap X^{*} \neq φ)} \end{matrix}$ (12)

The study of the paper is based on the two direction S-rough Sets.

3 S-rough sets model based on three-way decisions

Let U be a finite non-empty set and R be the equivalence relation defined in U, then (U, R) is the approximate space, which is denoted as: apr = (U, R). The division of U in the equivalence relation R is recorded as U/R = {[x] _R : x ∈ U}, in which [x] _R is the equivalence class including x. However, for S-rough sets, due to a family of the element migration γ, it makes [x] _R change.

If u ∈ U, u ∉ X, f (u) = x ∈ X, the universe U is reclassified, which is denoted as: $[x]_{R}^{'} = [x]_{R} \cup {f (u)}_{R}$ (13)

If x ∈ Xf′(x) = u ∉ X, the universe U is reclassified, which is denoted as: $[x]_{R}^{*} = [x]_{R}^{'} - {f^{'} (u)}_{R}$ (14)

Then the universe U is reclassified as $[x]_{R}^{*}$ . Due to the above, it makes the lower and upper approximation of X^* change. So for X^*, the lower approximations of X^*, namely (R, γ) _* (X^*), and the upper approximations of X^*, namely (R, γ)^*(X^*) can be redefined.

Definition 4. For two direction Singular Set X^*, the lower approximations of X^*, namely (R, γ) _* (X^*),and the upper approximations of X^*, namely (R, γ)^*(X^*) are defined as: $(R, γ)_{*} (X^{*}) = \cup [x]_{R}^{*} = {x \in U : [x]_{R}^{*} \subseteq X^{*}}$ (15) $(R, γ)^{*} (X^{*}) = \cup [x]_{R}^{*} = {x \in U : [x]_{R}^{*} \cap X^{*} \neq φ}$ (16)

For two direction Singular Set X^*, the positive region of X^*, namely POS (X^*), the boundary region of X^*, namely BND (X^*), and the negative region of X^*, namely NEG (X^*) are defined as: $POS (X^{*}) = (R, γ)_{*} (X^{*})$ (17) $BND (X^{*}) = (R, γ)^{*} (X^{*}) - (R, γ)_{*} (X^{*})$ (18) $NEG (X^{*}) = U - (R, γ)^{*} (X^{*})$ (19)

The S-rough sets model based on three-way decisions describe the decision-making process by using a finite set of two states and a finite set of three possible actions. According to Definition 3, X^*⊂ U is two direction Singular Set in U. Given the set of states Ω, Ω = {X^*, - X^*} X^* denote that an event belongs to X^* and -X^* denote that an event doesn’t belong to X^*; Given the set of actions A, A = {a_p, a_B, a_N}, in which a_P, a_B and a_N represent the three actions in classifying an object, namely, accepting an event, delaying the decision of an event and refusing an event. Different actions can produce different losses. When x ∈ X, λ_PP, λ_BP and λ_NP respectively denote the loss of taking actions a_P, a_B and a_N; Likewise, when x ∉ X^*, λ_PN, λ_BN and λ_PN respectively denote the loss of taking actions α_P, α_B and α_N. And $[x]_{R}^{*}$ is the new equivalence classes of x migrating by the migration family of γ. When taking the three different actions of α_P, α_B, α_N, the expected loss can be expressed as: $\begin{matrix} R (a_{P} | [x]_{R}^{*}) = λ_{PP} Pr (X^{*} | [x]_{R}^{*}) \\ + λ_{PN} Pr (- X^{*} | [x]_{R}^{*}) \end{matrix}$ (20) $\begin{matrix} R (a_{B} | [x]_{R}^{*}) = λ_{BP} Pr (X^{*} | [x]_{R}^{*}) \\ + λ_{BN} Pr (- X^{*} | [x]_{R}^{*}) \end{matrix}$ (21) $\begin{matrix} R (a_{N} | [x]_{R}^{*}) = λ_{PN} Pr (X^{*} | [x]_{R}^{*}) \\ + λ_{NN} Pr (- X^{*} | [x]_{R}^{*}) \end{matrix}$ (22)

According to the Bayesian decision rules, choosing the minimum loss as the optimal plan of the set of actions is needed, then the following minimum-risk decision rules are obtained:

If $R (a_{P} | [x]_{R}^{*})$ ≤ $R (a_{B} | [x]_{R}^{*})$ and $R (a_{P} | [x]_{R}^{*})$ ≤ $R (a_{N} | [x]_{R}^{*})$ , decide x ∈ POS (X^*);

If $R (a_{B} | [x]_{R}^{*})$ ≤ $R (a_{P} | [x]_{R}^{*})$ and $R (a_{B} | [x]_{R}^{*})$ ≤ $R (a_{N} | [x]_{R}^{*})$ , decide x ∈ BND (X^*);

If $R (a_{N} | [x]_{R}^{*})$ ≤ $R (a_{B} | [x]_{R}^{*})$ and $R (a_{N} | [x]_{R}^{*})$ ≤ $R (a_{P} | [x]_{R}^{*})$ , decide x ∈ NEG (X^*).

Because $Pr (X^{*} | [x]_{R}^{*}) + Pr (- X^{*} | [x]_{R}^{*}) = 1$ , the above decision rules is related to the condition probabilities $Pr (X^{*} | [x]_{R}^{*})$ and the involved loss function λ. Moreover, considered that the loss of accepting the right things is less than the loss of delaying to accept the right things, both of them are less than the loss of refusing to accept the right things; similarly, the loss of rejecting the wrong things is not greater than the loss of delaying to reject the wrong things, both of them are less than the loss of accepting the wrong things. Therefore, a reasonable assumption is $0 \leq λ_{PP} \leq λ_{BP} \leq λ_{NP}, 0 \leq λ_{NN} \leq λ_{BN} \leq λ_{PN}$

Here, we make $\begin{matrix} α = \frac{(λ_{PN} - λ_{BN})}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})} \\ = {(1 + \frac{λ_{BP} - λ_{PP}}{λ_{PN} - λ_{BN}})}^{- 1} \end{matrix}$ (23) $\begin{matrix} β = \frac{(λ_{BN} - λ_{NN})}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})} \\ = {(1 + \frac{λ_{NP} - λ_{BP}}{λ_{BN} - λ_{NN}})}^{- 1} \end{matrix}$ (24) $\begin{matrix} γ = \frac{(λ_{PN} - λ_{NN})}{(λ_{PN} - λ_{NN}) + (λ_{NP} - λ_{PP})} \\ = {(1 + \frac{λ_{NP} - λ_{PP}}{λ_{PN} - λ_{NN}})}^{- 1} \end{matrix}$ (25)

From the rule of (B), β < α, $\frac{(λ_{BP} - λ_{PP})}{(λ_{PN} - λ_{BN})}$ < $\frac{(λ_{NP} - λ_{BP})}{(λ_{BN} - λ_{NN})}$ and β < α, $\frac{(λ_{BP} - λ_{PP})}{(λ_{PN} - λ_{BN})}$ < $\frac{(λ_{NP} - λ_{PP})}{(λ_{PN} - λ_{NN})}$ < $\frac{(λ_{NP} - λ_{BP})}{(λ_{BN} - λ_{NN})}$ can be easily obtained; By calculating, 0 ≤ β < γ < α ≤ 1 can be obtained. Thus, the rule (P), (B), (N) can be rewritten as:

If $Pr (X^{*} | [x]_{R}^{*}) \geq α$ , decide x ∈ POS (X^*);

If β < $Pr (X^{*} | [x]_{R}^{*})$ < α, decide x ∈ BND (X^*);

If $Pr (X^{*} | [x]_{R}^{*}) \leq β$ , decide x ∈ NEG (X^*).

So the semantics of the S-rough sets model based on three-way decisions can be described as:

If the occurrence probability of X^* is not less than the value of α in the description of $[x]_{R}^{*}$ , then $[x]_{R}^{*}$ can be divided into the positive region of X^*, making the positive decisions immediately.

If the occurrence probability of X^* is greater than the value of β and less than the value of α in the description of $[x]_{R}^{*}$ , then $[x]_{R}^{*}$ can be divided into the boundary domain of X^*; Because the evidence of making decisions is insufficient, gathering more information is needed in order to make the right decisions.

If the occurrence probability of X^* is less than or equal to the value of β in the description of $[x]_{R}^{*}$ , then $[x]_{R}^{*}$ can be can be divided into the negative domain of X^*, making the negative decisions immediately.

Thus, in the S-rough sets model based on three-way decisions, its two parameters α and β can be directly calculated by the loss of function. Given the loss of function is needed in advance, so the appropriate prior knowledge is needed, which is limited in the application of many fields. However, there are some different methods [19, 20], which can solve this problem.

According to the Bayes theory, for the rules (P₁)-(N₁), the conditional probability $Pr (X^{*} | [x]_{R}^{*})$ can be calculated by the prior probability Pr(X^*) and the similar ratio $Pr (X^{*} | [x]_{R}^{*}) / Pr (X^{*})$ . $Pr (X^{*} | [x]_{R}^{*}) = Pr (X^{*}) \frac{Pr ([x]_{R}^{*} | X^{*})}{Pr ([x]_{R}^{*})}$ (26) $\begin{matrix} Pr ([x]_{R}^{*}) = Pr (X^{*}) Pr ([x]_{R}^{*} | X^{*}) \\ + Pr (- X^{*}) Pr ([x]_{R}^{*} | - X^{*}) \end{matrix}$ (27) $\begin{matrix} Pr (X^{*} | [x]_{R}^{*}) = \frac{Pr (X^{*}) Pr ([x]_{R}^{*} | X^{*})}{Pr ([x]_{R}^{*})} \\ = \frac{Pr (X^{*}) Pr ([x]_{R}^{*} | X^{*})}{Pr (X^{*}) Pr ([x]_{R}^{*} | X^{*}) + Pr (- X^{*}) Pr ([x]_{R}^{*} | - X^{*})} \end{matrix}$ (28)

Pr(X^*) is the probability of the state property set of X^* in the division of the decision attributes based on S-rough sets; $Pr ([x]_{R}^{*})$ is the probability of the equivalence classes $[x]_{R}^{*}$ in the division of the condition attributes based on S-rough sets. It denote that $Pr ([x]_{R}^{*}) = Pr (v_{1}, v_{2}, \dots, v_{m})$ , in which $Pr ([x]_{R}^{*})$ is the number of the condition attributes and v_i is the value of $[x]_{R}^{*}$ in the i-th (i = 1, 2, …, m) condition attribute. Similarly, the formula $Pr ([x]_{R}^{*} | X^{*}) = Pr (v_{1}, v_{2}, \dots, v_{m} | X^{*})$ can be obtained.

If the attributes are independent in the information system, then the following formula can be obtained: $\begin{matrix} Pr ([x]_{R}^{*}) = Pr (v_{1}, v_{2}, \dots, v_{m}) \\ = Pr (v_{1}) Pr (v_{2}) \dots Pr (v_{m}) = Π_{i = 1}^{m} Pr (v_{i}) \end{matrix}$ (29) $\begin{matrix} Pr ([x]_{R}^{*} | X^{*}) = Pr (v_{1}, v_{2}, \dots, v_{m} | X^{*}) \\ = Pr (v_{1} | X^{*}) Pr (v_{2} | X^{*}) \dots Pr (v_{m} | X^{*}) \\ = Π_{i = 1}^{m} Pr (v_{i} | X^{*}) \end{matrix}$ (30)

Then the related parametersPr(X^*), Pr(v_i) and Pr(v_i|X^*) can be got from the information system directly and be used to calculate the conditional probability $Pr ([x]_{R}^{*} | X^{*})$ . Thus, according to the above analysis, the method can classify and make decisions by calculating the related parameters and using the rules (P₁)-(N₁) for the objects, which provides a way to deal with the practical decision-makingproblems.

4 Multi-users S-rough sets model based on three-way decisions

With the S-rough sets obtained, the three-way decisions will be more reasonable to deal with the existing risky and uncertain information. However, multiple users according to their respective standards of different decisions set, it is necessary to find a reasonable and have the consensus decision criteria and then deduce an agreeable decision which could be accepted by the multiple users. In order to solve the problems of multiuser decision in the real world, the multi-user [16–19 , 38] s-rough sets model based on three-way decisions is proposed.

n is the number of the users, (αⁱ, βⁱ) is the threshold parameters and (Pⁱ, Bⁱ, Nⁱ) is the decision rules obtained by the i-th (i = 1,2, …, n) user. If f (α¹, α², …, αⁿ) and g (β¹, β², …, βⁿ) are procured, which make α^* = f (α¹, α², …, αⁿ), β^* = g (β¹, β², …, βⁿ) and α^* < β^* tenable, new decision rules are obtained.

If $Pr (X^{*} | [x]_{R}^{*}) \geq α^{*}$ , decide x ∈ MPOS (X^*);

If β^* < $Pr (X^{*} | [x]_{R}^{*})$ < α^*, decide x ∈ MBND (X^*);

If $Pr (X^{*} | [x]_{R}^{*})$ ≤ β^*, decide x ∈ MNEG (X^*).

In which, MPOS (X^*), MBND (X^*), and MNEG (X^*) denote the positive region, the boundary region and the negative region, respectively; and they are all decided by each of the multi-users.

Based on different circumstances, the definition of conservative decision-making region and risky decision making region are put forwarded.

Definition 5. Conservative positive region $MPOS (X^{*}) = \cap_{i = 1}^{n} {POS}^{i} (X^{*})$ (31)

conservative boundary region $MBND (X^{*}) = U - \cap_{i = 1}^{n} {POS}^{i} (X^{*}) - \cap_{i = 1}^{n} {NEG}^{i} (X^{*})$ (32)

conservative negative region $MNEG (X^{*}) = \cap_{i = 1}^{n} {NEG}^{i} (X^{*})$ (33)

Definition 6. risky positive region $MPOS (X^{*}) = \cup_{i = 1}^{n} {POS}^{i} (X^{*})$ (34)

risky boundary region $MBND (X^{*}) = U - \cap_{i = 1}^{n} {POS}^{i} (X^{*}) - \cap_{i = 1}^{n} {NEG}^{i} (X^{*})$ (35)

risky negative region $MNEG (X^{*}) = \cap_{i = 1}^{n} {NEG}^{i} (X^{*})$ (36)

In which, POSⁱ(X^*) and NEGⁱ(X^*) denote the positive and the negative region obtained by the i-th (i = 1, 2, …, n) user, respectively; and the MBND (X^*) can be gained as supplementary set of MPOS (X^*) and MNEG (X^*). Values of α^* and β^* can be achieved in many different ways according to different situations of the rules (P^*), (B^*) and (N^*). Such as in the mean acceptance region model and the mean rejection region model, considering each user has different decision proportion, α^* and β^* can be described as $α^{*} = \frac{(k_{1} α^{1} + k_{2} α^{2} + \dots + k_{n} α^{n})}{((k_{1} + k_{2} + \dots + k_{n})}$ (37) $β^{*} = \frac{(k_{1} β^{1} + k_{2} β^{2} + \dots + k_{n} β^{n})}{(k_{1} + k_{2} + \dots + k_{n})}$ (38)

where k_i (i = 1, 2, …, n) is the decision proportion.

5 Model instance

In the trial process of one case, U is the domain constituted by criminal suspects; X ⊂ U and X ≠ φ, in order to prevent the omission or the erroneous judgements, it is essential to add or delete the elements inX. If one suspect proves to commit the crime with sufficient evidence, the suspect should be added to the set X. contrarily, if one suspect proves to be innocent, the suspect should be moved out from X. Then, the result should be submitted to the forum after the final decision is made according to the Three-way Decisions theory [22 –27].

U is supposed to be the domain constituted by criminal suspects.

U = {x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀}, x_i (x_i ∈ U) denote a subset, [x] _i is the equivalence class of x_i.

According to the initial properties of these objects, the initial equivalence classes in U are as follow.

$\begin{matrix} [x]_{1} = {x_{0}, x_{1}}, [x]_{2} = {x_{2}, x_{6}, x_{9}}, [x]_{3} = {x_{3}, x_{5}}, \\ [x]_{4} = {x_{4}, x_{8}}, [x]_{5} = {x_{7}, x_{10}} and \end{matrix}$ $X = {x_{0}, x_{3}, x_{4}, x_{8}, x_{10}} \subset U;$

According to the Definition 3, some elements should be moved into X or out from X. During the investigation of the case, x₃ was forced to get involved in the case but he/she did not violate the law, x₂ and x₉ were escaped criminals.

Therefore $X^{'} = X \ {x : x \in X, f^{'} (x) \notin X} = {x_{0}, x_{4}, x_{8}, x_{10}}$ $\begin{matrix} X^{*} = X^{'} \cup {x : u \in U, u \notin X, f (u) \in X} \\ = {x_{0}, x_{4}, x_{8}, x_{10}, f (x_{2}), f (x_{9})} \end{matrix}$

Depending on the newly-known properties of the objects, the equivalence classes can be reclassified as $\begin{matrix} [x]_{1} = {x_{0}, x_{1}}, [x]_{2} = {f (x_{2}), f (x_{9})}, \\ [x]_{3} = {x_{3}, x_{5}, x_{6}}, [x]_{4} = {x_{4}, x_{8}}, \\ [x]_{5} = {x_{7}, x_{10}} . \end{matrix}$

According to the Definition 4 $\begin{matrix} (R, γ)_{*} (X^{*}) = \cup [x]_{R}^{*} = {x_{4}, x_{8}, f (x_{2}), f (x_{9})} \\ (R, γ)^{*} (X^{*}) = \cup [x]_{R}^{*} \\ = {x_{0}, x_{1}, f (x_{2}), x_{4}, x_{7}, x_{8}, f (x_{9}), f (x_{10})} \end{matrix}$

For one certain type of criminal suspects, the state set Ω = {X^*, - X^*} indicates the state of guilty and innocent; the action set A = {α_P, α_B, α_N} denote three actions, which are detention, delaying decisions and release.

There are six parameters in the model, if [x] _i ∈ X^* let θ_PP, θ_BP, and θ_NP represent the social harmfulness for taking actions of α_P, α_B, and α_N, respectively; if [x] _i ∈ - X^*, let θ_PN, θ_BN, and θ_NN represent the social harmfulness for taking actions of α_P, α_B, and α_N, respectively (see Table 1).

And at the same time these six parameters are assumed to meet the formulas $θ_{PP} \leq θ_{BP} < θ_{NP}; θ_{PN} \geq θ_{BN} \geq θ_{NN} .$

In practical problems, the final decision should be made after the comparison between the value of conditional probability $Pr (X^{*} | [x]_{R}^{*})$ and the value ofα and β. The calculation of $Pr (X^{*} | [x]_{R}^{*})$ requires the priori information which could be carried out by the Equations (26–28). For simplicity, we assume that $Pr (X^{*} | [x]_{R}^{*}) = 0.2650$ , then POS (X^*), BND (X^*) and NEG (X^*) can be conducted from the decision rules (P₁)-(N₁). POS (X^*) = {₁, [x] ₅}, and suspects denoted by [x] ₁, [x] ₅should be detained. BND (X^*) = {[x] ₂}, so how to dispose suspects denoted by [x] ₂ should be decided later. NEG (X^*) = {[x] ₃, [x] ₄}, which means suspects represented by [x] ₃ and [x] ₄ should be released. By Table 2, It is shown that the establishment of three decision regions depends on the value of the conditional probability and the threshold parameters.

6 Attribute reduction of S-rough sets model based on three-way decisions

6.1 Non-monotonicity of the positive region

In the theory of S-rough sets model based on three-way decision, positive region is defined as a universe objects set obtaining correct classifications at a certain probability. Probability threshold value can be acquired by minimizing the expected risk.

Definition 7. Let S = {U, C ∪ D, V, f} be a decision table, where U is a finite nonempty universe set. C and D are condition attributes set and decision attributes set respectively. α-positive region of decision attributes set D relative to condition attributes setC is defined as: ${POS}_{C}^{α} (D) = \cup_{X^{*} \in U / D} \underline{{apr}_{C}^{α}} (X^{*})$ (39)

where $\underline{{apr}_{C}^{α}} (X^{*})$ is the approximate set of X^* under α-, i.e., $\underline{{apr}_{C}^{α}} (X^{*}) = {x \in U : Pr (X^{*} | [x]_{C}^{*}) \geq α}$ (40) $Pr (X^{*} | [x]_{C}^{*}) = \frac{| X^{*} \cap [x]_{C}^{*}}{| [x]_{C}^{*} |}$ (41)

Compared with positive region of Pawlak rough sets theory, the upper α-positive region has a certain error tolerance, which allows a certain degree of misclassification. In this case, classification accuracy used for expressing the classifying ability of attribute set now is not the ratio of the number of objects classified completely correctly to the number of universe objects but the most number of classified correctly objects to the total number of universe objects. Then the performance of classification of attributes set is not always improved as the number of attribute increases.

6.2 An approach for attribute reductionof positive region

According that positive region in S-rough sets model based on three-way decision has the feature of non-monotonicity, a new definition of attribute reduction is given. In this model, due to the characteristic of non-monotonicity, reducing the number of attribute may result in the extension of positive region and expanded positive region has stronger ability of classification. With comprehensive consideration, redefine attribute reduction as follows.

Definition 8. LetS = {U, C ∪ D, V, f} be a decision table and a ∈ [0, 1]. If B ⊆ C is a attribute subset and satisfies the following two conditions:

(1) Non-decreasing feature of the positive region: $| {POS}_{B}^{α} (D) | \geq | {POS}_{C}^{α} (D) |$ (42)

(2) Independence of the attribute: $| {POS}_{B - {α}}^{α} (D) | < | {POS}_{B}^{α} (D) |$ (43) Therefore the attribute subset B is a α-positive region reduction based on a three-way decision S-rough sets model of the attribute set C. The non-decreasing feature of the positive region is non-decreasingly expressed by the cardinal number of the positive region, in the other words, the attributes subset should have a stronger ability of probability classification than the attributes set. The attribute independency is replaced by the feature that positive region will be reduced when any attribute in attribute set is reduced. The new definition of attribute reduction based on the three-way decisions S-rough sets model reflects the non-monotonic feature of the positive region with the change of the attribute. It plays a role in data reduction, which provides a new and effective way for mining essence of the data [29 –37].

7 Conclusion

According to the S-rough Sets model based on three-way decisions, these mantic interpretation of the minimum risk based on the Bayes’ rules is given, and the threshold parameters can be calculated by the action loss functions. Moreover, with the circumstances of multi-users, the threshold parameters can be gained depends on the situation. The conditional probability can be calculated from the information system. The threshold parameters are used to verify the correctness of the conditional probability, meanwhile the conditional probability is used to guide the rationality of the threshold parameters. They oppose each other also complement each other. Thus, the multi-users s-rough sets model based on three-way decisions provides a new perspective to deal with the problem of uncertain information.

Footnotes

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (lzujbky-2012-43). The authors would like to thank the reviewers for their constructive comments and suggestions.

References

Pawlak

, Rough sets, International Journal of Computer and Information Sciences11 (1982), 341–356.

Pawlak

, Rough sets, Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers Press, Boston, 1991.

Yao

, Three-way decision: An interpretation of rules in rough set theory, The 4th International Conference on Rough Sets and Knowledge Technology, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2009, 5589:642–649.

Yao

, Three-way decisions with probabilistic rough sets, Information Sciences180 (2010), 341–353.

Yao

, The superiority of three-way decision in probabilistic rough set models, Information Sciences181 (2011), 1080–1096.

Shi

and Cui

, S-rough set and its general structures, Journal of Shan Dong University (Natural Science)12(6) (2002), 471–474.

Shi

, Singular rough sets and dynamic characteristics of S-rough sets, Computer Science10A (2004), 24–27.

Wang

, Shi

and Lei

, S rough sets, function S rough sets and their relation theorem, Computer Science6 (2007), 156–157.

Shi

, S-rough sets and its applications in diagnosis-recognition for disease, IEEE Proceedings of the First International Conference on Machine Learning and Cybernetics1 (2002), 50–54.

10.

Shi

and Cui

, F-decomposition and F-reduction of S-rough sets, An International Journal Advances in System Science and Applications4 (2004), 487–499.

11.

Shi

and Chang

, One direction S-rough sets, International Journal of Fuzzy Mathematics2 (2005), 239–334.

12.

Shi

, Two direction S-rough sets, International Journal of Fuzzy Mathematics2 (2005), 335–349.

13.

Shi

and Cui

, S-Rough Sets and Rough Decision, Beijing: Science Press, 2006, pp. 101–126.

14.

Yao

, Probabilistic rough set approximations, International Journal of Approximate Reasoning49 (2008), 255–271.

15.

Ziarko

, Probabilistic approach to rough set, International Journal of Approximate Reasoning49(2) (2008), 272–284.

16.

Liu

, Li

, Miao

, Wang

and Liang

, The Three-way Decisions and Granular Computing, Beijing: Science Press, 2013, pp. 40–41, 139–154.

17.

Liu

, Yao

and Li

, Three-way investment decisions with decision theoretic rough sets, International Journal of Computational Intelligence Systems4(1) (2011), 66–74.

18.

Liu

, Li

and Liang

, Three-way decisions in dynamic decision-theoretic rough sets, Lecture Notes in Computer Science8171 (2013), 288–299.

19.

Jia

, Li

, Shang

and Chen

, An adaptive learning parameters algorithm in three- way decisions, Acta Electronica Sinica39(11) (2011), 2520–2525.

20.

Chen

, Liu

and Wu

, New algorithm to get optimal threshold for three decision-making, Journal of Computer Applications32(8) (2012), 2212–2215.

21.

Skowron

, Rauszer

, The discernibility matrices and functions in information systems, Slowinski R. Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory, Dordrecht: Kluwer, 11, 1992, pp. 331–362.

22.

Slezak

, Approximate entropy reducts, Fundamenta Informaticae53(53) (2002), 365–390.

23.

Liang

and Xu

, The algorithm on knowledge reduction in incomplete information systems, International of Uncertainty, Fuzziness and Knowledge-Based Systems10(1) (2002), 95–103.

24.

, Wu

and Zhang

, Approaches to knowledge reduction based on variable precision rough set model, Information Science159 (2004), 255–272.

25.

Yao

, Decision-theoretic rough set models. pp, Proceedings of the 2nd International Conference on Rough Sets and Knowledge Technology (2007), 1–12.

26.

Sang

, Liang

and Qian

, Decision-theoretic rough sets under dynamic granulation, Knowledge-Based Systems91 (2016), 84–92.

27.

Fujita

, Li

and Yao

, Advances in three-way decisions and granular computing, Knowledge-Based Systems91 (2016), 1–3.

28.

Zhang

, Li

and Chen

, Composite rough sets for dynamic data mining, Information Sciences257 (2014), 81–100.

29.

Zhang

, Li

, Ruan

and Liu

, Neighborhood rough sets for dynamic data mining, International Journal of Intelligent Systems27 (2012), 317–342.

30.

Chen

, Li

, Ruan

, Lin

and Hu

, A rough-set based incremental approach for updating approximations under dynamic maintenance environments, IEEE Transactions on Knowledge and Data Engineering25(2) (2013), 274–284.

31.

Zhang

, Wong

J.-S.

, Pan

and Li

, A parallel matrix-based method for computing approximations in incomplete information systems, IEEE Transactions on Knowledge and Data Engineering27(2) (2015), 326–339.

32.

Chen

, Li

, Luo

, Horng

S.-J.

and Wang

, A decision-theoretic rough set approach for dynamic data mining, IEEE Transactions on Fuzzy Systems23(6) (2015), 1958–1970.

33.

, Li

, He

, Zhang

and Li

, A hybrid chemical reaction optimization scheme for task scheduling on heterogeneous computing systems, IEEE Transactions on Parallel and Distributed Systems26(12) (2015), 3208–3222.

34.

, Li

, He

and Truong

T.K.

, A DAG scheduling scheme on heterogeneous computing systems using double molecular structure-based chemical reaction optimization, Journal of Parallel and Distributed Computing73(9) (2013), 1306–1322.

35.

, Li

, Hu

and Li

, A genetic algorithm for task scheduling on heterogeneous computing systems using multiple priority queues, Information Sciences270 (2014), 255–287.

36.

, Li

, Truong

T.K.

and Qiu

, A multiple priority queueing genetic algorithm for task scheduling on heterogeneous computing systems, 2012 IEEE 14th International Conference on High Performance Computing and Communications IEEE, Liverpool, England, UK, 2012, pp. 639–646.

37.

Lin

and Zhou

, Rough Relations, Neighborhood Relations, and Granular Computing, Proceeding the 6th International Conference on Rough Sets and Knowledge Technology, Yao

Jing Tao

, Ramanna

Sheela

, Wang

Guoyin

and Suraj

Zbigniew

, (Eds), Lecture Notes in Artificial Intelligence, 2011, 6954: 689–695.

38.

, Jiao

, Wang

and Yao

, Categorizing Overlapping Regions in Clustering Analysis Using Three-way Decisions, 2014 IEEE/WIC/ACM International Joint Conferences on Web Intelligence and Intelligent Agent Technologies, 2, 2014, pp. 350–357.