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Introduction
Rough set theory, introduced by Pawlak [1], is a kind of important theory about uncertainty information processing. So far, it has been successfully applied in data analysis, pattern recognition, machine learning and knowledge discovery, artificial intelligence, and so on [2, 3, 4, 5, 6, 7]. Using the tool of entropy to deal with the uncertainty problem in rough set theory has been already studied [8, 9, 10, 11], and the combination entropy was proposed in the literature [12].
As it is known that the rough set theory is mainly utilized to address the problem of information granules approximation problem based on a D-Table, and the granularity reflects the different levels of a given problem, then the granular computing (GrC) concerns the processing of complex information entities, information granules, which arise in the process of data abstraction and derivation of knowledge from D-Table. Hence, GrC is a kind of important structure technology, and can be able to resolve the hierarchies problems in rough set theory [13, 14, 15, 16, 17]. Note that three-way decisions serve as a fundamental methodology with extensive applications. Hu [18] discussed three-way decisions based on semi-three-way decision spaces, and Li et al. [17] adopted the multi-granularity to study three-way cognitive concept learning. In particular, Yao [19] pointed out that the three-level analysis falls into the category of three-way decisions, so the three-layer attribute reduction and relevant three-level measure construction become a typical case and a good example of three-way decisions.But the definition of combination entropy proposed in literature [12] did not consider the hierarchical structure of the a decision table (D-Table). Hence, on the basis of Ref. [12, 16], this paper concretely constructs three-way weighted combination-entropies based on the new perspective of a D-Table’s three-layer granular structures and Bayes’ theorem, and reveals the granulation monotonicity and systematic relationships of the weighted combination-entropies.
The relevant conclusions of the study has been deepened information theory of rough set theory, provides a more complete and updated interpretation of granular computing for the uncertainty measurement, and establish more effective basis of the quantitative application for attribute reduction.
Preliminaries
The three-layer granular structures and the three-way probabilities
This section reviews the three-layer granular structures of a given D-Table and the three-way probabilities in Ref. [16].
An information system is a pair S = (U, AT), where,
U is a non-empty finite set of objects;
AT is a non-empty finite set of attributes;
for every a ∈ AT, there is a mapping fa, fa : U → Vfa, where Vfa is called the value set of U.
The D-Table is a special type of information table with AT = C ∪ D and C ∩ D = ∅, where C and D denote the sets of condition attribute and decision attribute, respectively.
Each subset of attributes A ⊆ C determines a binary indistinguishable relation IND(A) as follows IND(A) = {(u, v) ∈ U × U∣∀ a ∈ A, fa(u) = fa(v)}.
IND(A) serves as an equivalence relation to cause C-Class [x]A, which implies a type of basic granule. The classified structure U/IND(A) = {[x]A: x ∈ U} means knowledge or C-Classification. Suppose that U/IND(A) = $U/IND(A) = \{[x]_A^i: i=1,2,\dots,n \}$ , thus |U/IND (A)| = n. Similarly, D can induce the equivalence relation IND (D) and further D-Classification U/IND(D) = {Xj: j = 1, 2, ⋅, m}, thus |U/IND(D)| = m.
Aiming at the D-Table (U, C ∪ D) according to the four basic notions of the D-Table and four granular notions presented in Table 1, the relevant classification and class lead to three-layer granular structures, as shown in Table 2.
Conditional and decisional classifications and classes
Item
C-Classification
C-Class
D-Classification
D-Class
Mathematical symbol
U/IND (A)
$[x]_A^i, i=1,2,\dots,n$
U/IND (D)
Xj, j = 1,2, …, m
Granular essence
Conditional granule set
Conditional granule
Decisional granule set
Decisional granule
Basic descriptions of a D-Table’s three-layer granular structures
Structure
Composition
Granular scale
Granular level
Simple name
(1)
U/IND (A), U/IND (D)
Macro
Top
Macro-Top
(2)
U/IND (A), Xj
Meso
Middle
Meso-Middle
(3)
$[x]_A^{i},X_j$
Micro
Bottom
Micro-Bottom
The three-layer granular structures (Macro-Top, Meso-Middle, and Micro-Bottom) are mainly considered from a systematic viewpoint, the numeric result and hierarchical/granular relationships are described in Fig. 1.
At the Micro-Bottom, C-Class $[x]_A^i$ and D-Class Xj are of concern. They exist in approximate space (U, AT) and can produce some fundamental measures, including probabilities. By connecting the Meso-Middle and its reasoning mechanism, three-way probabilities become bottomed measures that underlie informational construction at higher levels.
Definition 1
At the Micro-Bottom, C-Class $[x]_A^i$ and D-Class Xj are of concern. The three-way probabilities are defined by:
where $C_{|[x]_R^i|}^2 = \frac {\left |[x]_R^i \right | \times \left( \left |[x]_R^i \right | -1 \right)}{2}$ , $\frac{\left| [x]_R^i \right|}{|U|}$ represents the probability of an equivalence Xi within the universe U, and $\frac{C_{|U|}^2 - C_{|[x]_R^i|}^2}{C_{|U|}^2}$ denotes the probability of pairs of the elements which are distinguishable each other within the whole number of pairs of the elements on the universe U.
Proposition 2
LetK1 = (U, R) and K2 = (U, Q) be two approximation spaces, thenCE(P)>CE(Q) ifP ≺ Q.
Three-way weighted combination-entropies at the Meso-Middle
Based on three-way probabilities at the Micro-Bottom, this subsection constructs three-way weighted combination-entropies at the Meso-Middle using the Bayes’ theorem and discusses their granulation monotonicity and systematicness. Relevant results take a link function to underlie the latter informational construction at the Macro-Top.
A promotional measure at the Meso-Middle requires probability fusion when integrating C-Classes into C-Classification. And because Bayes’ theorem provides systematicness of three-way probabilities. So it becomes the starting point. Herein, we first make a key transformation for Bayes’ theorem. According to Theorem 1. with stable Xj,
The above step-by-step deduction implies the hierarchical evolution of Bayes’ theorem. Bayes’ theorem and its three-way probabilities at the Micro-Bottom evolve in the combination-entropy direction, and thus, weight-based combination-entropies and their relationships emerge at the Meco-Middle. Concretely, Eq. (9) provides a constant that is based on Xj, and thus, systematic Eq. (8) concerns three weighted and informational items. In Eq. (8), except the final item, the others three terms derived from the combination entropy proposed in [12] are multiplied by the corresponding weight coefficients of specific probabilities. Next, we introduce the weighted combination-entropy. Suppose that (ξ, pi) denotes a probability distribution and ωi≥0 means the weight, then, the weighted combination-entropy is defined as:
The weighted combination-entropy introduces weights into the combination entropy, where the weights refect the importance degrees for information receivers or attention degrees of information receivers. Concretely, $CE_\omega^{X_j} (A)$ improves absolute $\sum_{i=1}^n P([x]_A^i ) \left( 1-\frac{C_{|[x]_A^i |}^2}{C_{|U|}^2} \right)$ introducing relative $P(X_j/[x]_A^i ) \frac{C_{|U|}^2}{C_{|X_j |}^2}$ to the importance weights, while Cω(Xj/A) and CEω(A/Xj) respectively improve the relative $\sum_{i=1}^{n} \left[ P(X_j/[x]_A^i )\left(1- \frac{C_{\left |[x]_A^i \cap X_j\right |}^2}{C_{\left|[x]_A^i \right|}^2} \right) \right]$ and $\sum_{i=1}^{n} P([x]_A^i / X_j )\left( 1- \frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{|X_j |}^2} \right)$ by introducing absolute $P([x]_A^i ) \frac{C_{\left| [x]_A^i \right |}^2}{C_{|X_j |}^2}$ and P(Xj) . In other words, three-way weighted combination-entropies inherit the essential uncertainty semantics of three-way properties by using different probability weights, and thus, can better describe the system regarding cause A and result Xj, hence, they become robust for uncertainty measurement. Next, we discuss their properties.
Theorem 3
At the Meso-Middle, three-way weighted combination-entropies have granulation monotonicity. Concretely, ifP ⪯ Q, then, (1) $CE_\omega^{(X_j )}(P) \geq CE_\omega^{(X_j )}(Q)$ , (2)CEω(P/Xj) ≥ CEω(Q/Xj), (3)CEω(Xj/P) ≤ CEω(Xj/Q).
proof. Since P ⪯ Q, let $\cup_{t=1}^k[x]_P^t =[x]_Q$ , then we have
Theorem 4 provides an important relationship for the three-way weighted combination-entropies. In other words, CEω(A/Xj) is a linear translation of the sum of $CE_\omega^{X_j}(A)$ and CEω(Xj/A), where $P(X_j)[1-(C_{|U|}^2)/(C_{|X_j |}^2 )]$ is a constant at the Meso-Middle. And it develops Bayes’ theorem at the Micro-Bottom to establish a systematic equation of three-way weighted combination-entropies. Furthermore, eliminating the conversion distance can produce a new measure to simplify the systematic equation.
Definition 4
At the Meso-Middle, the linear weighted combination-entropy with regard to the weighted combination-entropy CEω(Xj/A) is defined as:
At the Meso-Middle, the linear weighted combination-entropy has granulation monotonicity. Concretely, ifP ⪯ Q, then, $CE_\omega^{lin}(X_j/P) \leq CE_\omega^{lin} (X_j/Q)$ .
Corollary 6
Three-way weighted combination-entropies have the equivalent systematicness:
The linear weighted combination-entropy $CE_\omega^{lin} (X_j/A)$ corresponds to CEω(Xj/A) by virtue of a specific linear transformation. The former uses the superscript lin (which means linear) to different from the latter, but both are viewed as only one item for three-way weighted combination-entropies. In contrast to CEω(Xj/A), $CE_\omega^{lin} (X_j/A)$ exhibits same granulation monotonicity, and it simplfies the systematicness of three-way weighted combination-entropies.
In summary, this section at the Meso-Middle becomes important to link the Micro-Bottom and Macro-Top. Bayes’ theorem provides three-way probabilities systematicness, and it further plays a fundamental role in the informational evolution of weighted combination-entropies. It induces essential measures and systematic equations of three-way weighted combination-entropies. Next, three-way weighted combination-entropies are promoted from the Meso-Middle to the Macro-Top. combination
Three-way weighted combination-entropies at the macro-top
For three-way weighted combination-entropies at the Meso-Middle, their monotonicity and systematicness are established. They can hierarchically evolve to Macro-Top by using the natural sum integration with regard to multiple D-Classes. This subsection constructs three-way weighted combination-entropies at the Macro-Top and offers their monotonicity and systematicness.
Definition 5
At Macro-Top, three-way weighted combination-entropies are defined by:
where combination entropy $CE(D)=\sum\limits_{j=1}^{m}P(X_j)[1-(C_{|U|}^2)/(C_{|X_j |}^2 )]$ is a constant.
$CE_\omega^{lin} (D/A)$ and CEω(D/A) exhibit a linear transformation to be viewed as only one item. Three-way weighted combination-entropies at Macro-Top depend on the sum integration to naturally inherit monotonicity and systematicness at the Meso-Middle, and the relevant features are presented as follows.
Theorem 8
At Macro-Top, three-way weighted combination-entropies have granulation monotonicity. Concretely, ifP ⪯ Q, then, $CE_\omega^D (P) \geq CE_\omega^D (Q)$ ; CEω(P/D) ≥ CEω(Q/D), CEω(D/P) ≤ CEω(D/Q), $CE_\omega^{lin}(D/P) \leq$ $CE_\omega^{lin} (D/Q)$ .
Theorem 9
Three-way weighted combination-entropies have systematicness:
At Macro-Top, Theorem 9 describes an important relationship of the three-way weighted combination-entropies by introducing CE (D). Thus, CEω(A/D) is a linear translation of the summation of $CE_\omega^D (A)$ and $CE_\omega^{lin} (D/A)$ or the difference between CEω(D/A) and $CE_\omega^{lin} (D/A)$ .
With regard to the Meso-Middle, the Macro-Top exhibits the hierarchical promotion and systematic integration from D-Classes to D-Classification. Accordingly, three-way weighted combination-entropies at Macro-Top are interestedly fused by three-way weighted combination-entropies at the Meso-Middle, and they exhibit a type of informational summation. The relevant results are well clarified in a relationship as shown Fig. 2.
Conclusion
In summary, based on the new perspective of three-layer granular structures and Bayes’ theorem, this paper concretely constructed three-way weighted combination-entropies, and revealed the granulation monotonicity and systematic relationships of the weighted combination-entropies. The relevant conclusion provided a more complete and updated the interpretation of granular computing for the uncertainty measurement, and established more effective basis of the quantitative application with attribute reduction.