Three-way weighted combination-entropies based on three-layer granular structures

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.

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 X_j,

Because, P([x]Ai/Xj)=P([x]Ai)×P(Xj/([x]Ai))P(Xj),∀i=1,2,…,n $P([x]_A^i/X_j ) = \frac{P([x]_A^i) \times P(X_j/([x]_A^i ))}{P(X_j )}, \forall i=1,2,\dots,n$ . So,

P(Xj)P(([x]Ai)/Xj)=P([x]Ai)×P(Xj/([x]Ai)).$$ \begin{equation} P(X_j )P(([x]_A^i)/X_j )=P([x]_A^i )\times P(X_j/([x]_A^i )). \end{equation} $$(4)

Then, the Eq.(4) on both sides is multiplied by (1−C|[x]Ai∩Xj|2C|Xj|2) $\left (1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2} \right)$ , we have:

P(Xj)×P([x]Ai)/Xj)(1−C|[x]Ai∩Xj|2C|Xj|2)=P([x]Ai)P(Xj/[x]Ai)(1−C|[x]Ai∩Xj|2C|Xj|2)$$ \begin{equation} P(X_j )\times P([x]_A^i)/X_j ) \left (1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2} \right) = P([x]_A^i )P(X_j/[x]_A^i )\left(1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2}\right) \end{equation} $$(5)

then, the right side of Eq.(5) can be calculated as follows:

P([x]Ai)P(Xj/[x]Ai)(1−C|[x]Ai∩Xj|2C|Xj|2)=P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)P(Xj/([x]Ai))C|[x]Ai∩Xj|2C|Xj|2=P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)P(Xj/([x]Ai))C|[x]Ai∩Xj|2C|Xj|2C|[x]Ai|2C|[x]Ai|2=P([x]Ai)P(Xj/([x]Ai))−P(Xj/([x]Ai))(C|[x]Ai∩Xj|2C|[x]Ai|2)P([x]Ai)(C|[x]Ai|2C|Xj|2)=P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)(C|[x]Ai|2C|Xj|2)P(Xj/([x]Ai))+P([x]Ai)(C|[x]Ai|2C|Xj|2)(P(Xj/([x]Ai))−P(Xj/([x]Ai))(C|[x]Ai∩Xj|2C|[x]Ai|2))=P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)(C|[x]Ai|2C|Xj|2)P(Xj/([x]Ai))(C|U|2C|U|2)+P([x]Ai)(C|[x]Ai|2C|Xj|2)×[P(Xj/([x]Ai))−P(Xj/([x]Ai))(C|[x]Ai∩Xj|2C|[x]Ai|2)]=P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)(C|[x]Ai|2C|U|2)P(Xj/([x]Ai)(C|U|2C|Xj|2)+P([x]Ai)P(Xj/([x]Ai))(C|U|2C|Xj|2)−P([x]Ai)P(Xj/([x]Ai))C|U|2C|Xj|2+P([x]Ai)(C|[x]Ai|2C|Xj|2)[P(Xj/[x]Ai)−P(Xj/[x]Ai)(C|[x]Ai∩Xj|2C|[x]Ai|2)]=P(Xj/[x]Ai)(C|U|2C|Xj|2)[P([x]Ai)−P([x]Ai)(C|[x]Ai|2C|U|2)]+P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)P(Xj/([x]Ai))(C|U|2C|Xj|2)+P([x]Ai)(C|[x]Ai|2C|Xj|2)[P(Xj/[x]Ai)−P(Xj/([x]Ai))(C|[x]Ai∩Xj|2C|[x]Ai|2)],$$ \begin{equation}\begin{array}{rl} & P([x]_A^i )P(X_j/[x]_A^i )\left(1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2}\right) \\ = & P([x]_A^i )P \left( X_j/([x]_A^i ))-P([x]_A^i )P(X_j/([x]_A^i ) \right) \frac{C_{|[x]_A^i \cap X_j |}^2} {C_{|X_j |}^2}\\ = & P([x]_A^i )P(X_j/([x]_A^i ))-P([x]_A^i )P(X_j/([x]_A^i )) \frac{C_{\left |[x]_A^i \cap X_j \right |}^2} {C_{|X_j |}^2} \frac{C_{\left |[x]_A^i \right |}^2}{C_{\left |[x]_A^i \right |}^2}\\ = & P([x]_A^i )P \left (X_j/([x]_A^i )\right )-P \left ( X_j/([x]_A^i ) \right) \left(\frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{\left | [x]_A^i \right |}^2}\right) P([x]_A^i) \left( \frac{C_{\left |[x]_A^i \right |}^2}{C_{|X_j |}^2} \right )\\ %= & P([x]_A^i ) P(X_j/([x]_A^i ) - P(X_j/([x]_A^i ) \left ( \frac{C_{\left |[x]_A ^i \cap X_j \right |}^2}{C_{\left | [x]_A^i \right |} ^2} \right ) P([x]_A^i ) \left ( \frac{C_{|[x]_A^i |}^2}{C_{|X_j |^2}\right )+P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2 \right ) P(X_j/([x]_A^i )) - P([x]_A^i ) \left( \frac{C_|[x]_A^i |^2}{C_|X_j |^2} \right )P(X_j/([x]_A^i ) \\ = & P([x]_A^i )P(X_j/([x]_A^i )) - P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right|}^2}{C_{|X_j |}^2} \right ) P(X_j/([x]_A^i )) + P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right ) \left ( P(X_j/([x]_A^i ))-P(X_j/([x]_A^i )) \left ( \frac{C_|[x]_A^i \cap X_j |^2}{C_{\left|[x]_A^i \right |}^2} \right ) \right)\\ = & P([x]_A^i ) P(X_j/([x]_A^i )) - P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right ) P(X_j/([x]_A^i )) \left ( \frac{C_{|U|}^2}{C_{|U|}^2} \right) + P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right) \times \\ & \left [ P(X_j/([x]_A^i )) - P(X_j/([x]_A^i )) \left( \frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{|[x]_A^i |}^2} \right) \right]\\ = & P([x]_A^i ) P(X_j/([x]_A^i )) - P([x]_A^i ) \left ( \frac {C_{\left | [x]_A^i \right |}^2}{C_{|U|}^2} \right ) P(X_j/([x]_A^i ) \left ( \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) + P([x]_A^i )P(X_j/([x]_A^i )) \left ( \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) - \\ & P([x]_A^i ) P(X_j/([x]_A^i )) \frac{C_{|U|}^2}{C_{|X_j |}^2} + P([x]_A^i ) \left( \frac {C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right) \left [ P(X_j/[x]_A^i ) - P(X_j/[x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \cap X_j \right|}^2}{C_{\left | [x]_A^i \right |}^2} \right) \right] \\ = & P(X_j/[x]_A^i ) \left( \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) \left [ P([x]_A^i ) - P([x]_A^i ) \left( \frac {C_{\left| [x]_A^i \right|}^2}{C_{|U|}^2} \right) \right] + P([x]_A^i )P(X_j/([x]_A^i )) - P([x]_A^i )P(X_j/([x]_A^i )) \left( \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) + \\ & P([x]_A^i ) \left( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right) \left [ P(X_j/[x]_A^i )-P(X_j/([x]_A^i )) \left ( \frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{|[x]_A^i |}^2} \right) \right] , \end{array}% \end{equation} $$(6)

thus,

P(Xj)×P([x]Ai)/Xj)(1−C|[x]Ai∩Xj|2C|Xj|2)=P(Xj/[x]Ai)(C|U|2C|Xj|2)[P([x]Ai)−P([x]Ai)(C|[x]Ai|2C|U|2)]+P([x]Ai)(C|[x]Ai|2C|Xj|2)[P(Xj/([x]Ai))−P(Xj/[x]Ai)C|[x]Ai∩Xj|2C|[x]Ai|2)]+P([x]Ai)P(Xj/([x]Ai))−P([x]Ai)P(Xj/[x]Ai)C|U|2C|Xj|2.$$ \begin{equation} \begin{array}{rl} & P(X_j )\times P([x]_A^i)/X_j ) \left (1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2} \right) \\ = & P(X_j/[x]_A^i ) \left( \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) \left [ P([x]_A^i ) - P([x]_A^i ) \left( \frac {C_{\left| [x]_A^i \right|}^2}{C_{|U|}^2} \right) \right] + P([x]_A^i ) \left ( \frac{C_{\left | [x]_A^i \right |}^2}{C_{|X_j |}^2} \right) [ P(X_j/([x]_A^i )) - \\ & P(X_j/[x]_A^i ) \frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{\left | [x]_A^i \right |}^2 )} ]+P([x]_A^i )P(X_j/([x]_A^i ))-P([x]_A^i )P(X_j/[x]_A^i ) \frac{C_{|U|}^2}{C_{|X_j |}^2}. \end{array} \end{equation} $$(7)

According to the i-based summation,

P(Xj)×∑i=1mP([x]Ai)/Xj)(1−C|[x]Ai∩Xj|2C|Xj|2)=∑i=1mP(Xj/[x]Ai)C|U|2C|Xj|2[P([x]Ai)(1−C|[x]Ai|2C|U|2)]+∑i=1mP([x]Ai)C|[x]Ai|2C|Xj|2[P(Xj/([x]Ai)(1−C|[x]Ai∩Xj|2C|[x]Ai|2)]+∑i=1mP([x]Ai)P(Xj/[x]Ai))(1−C|U|2C|Xj|2).$$ \begin{equation} \begin{array}{rl} & P(X_j )\times \sum\limits_{i=1}^m P([x]_A^i)/X_j ) \left (1-\frac{C_{|[x]_A^i \cap X_j |}^2}{C_{|X_j |}^2} \right) \\ = & \sum\limits_{i=1}^{m}P(X_j/[x]_A^i ) \frac{C_{|U|}^2}{C_{|X_j |}^2} \left[ P([x]_A^i ) \left( 1- \frac{C_{|[x]_A^i |}^2}{C_{|U|}^2} \right) \right] + \sum\limits_{i=1}^{m}P([x]_A^i ) \frac{C_{|[x]_A^i |}^2}{C_{|X_j |}^2} \left[ P(X_j/([x]_A^i ) \left ( 1 - \frac{C_{\left | [x]_A^i \cap X_j \right|}^2}{C_{|[x]_A^i |}^2} \right) \right] +\\ & \sum\limits_{i=1}^{m} P([x]_A^i )P(X_j/ [x]_A^i ))\left( 1- \frac{C_{|U|}^2}{C_{|X_j |}^2} \right). \end{array} \end{equation} $$(8)

The final item in Eq. (8) becomes

∑i=1mP([x]Ai)P(Xj/([x]Ai))(1−C|U|2C|Xj|2)=P(Xj)(1−C|U|2C|Xj|2).$$ \begin{equation} \sum\limits_{i=1}^{m}P([x]_A^i )P(X_j/([x]_A^i ))\left( 1 - \frac{C_{|U|}^2}{C_{|X_j |}^2} \right) = P(X_j) \left (1- \frac{C_{|U|}^2}{C_{|X_j |}^2}\right). \end{equation} $$(9)

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 X_j, 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, p_i) denotes a probability distribution and ω_i≥0 means the weight, then, the weighted combination-entropy is defined as:

CEω(ξ)=∑i=1nωiPi(1−CPi).$$ \begin{equation} CE_{\omega}(\xi) = \sum\limits_{i=1}^{n}\omega_{i}P_{i}(1-CP_{i}). \end{equation} $$(10)

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, CEωXj(A) $CE_\omega^{X_j} (A)$ improves absolute ∑i=1nP([x]Ai)(1−C|[x]Ai|2C|U|2) $\sum_{i=1}^n P([x]_A^i ) \left( 1-\frac{C_{|[x]_A^i |}^2}{C_{|U|}^2} \right)$ introducing relative P(Xj/[x]Ai)C|U|2C|Xj|2 $P(X_j/[x]_A^i ) \frac{C_{|U|}^2}{C_{|X_j |}^2}$ to the importance weights, while C_ω(X_j/A) and CE_ω(A/X_j) respectively improve the relative ∑i=1n[P(Xj/[x]Ai)(1−C|[x]Ai∩Xj|2C|[x]Ai|2)] $\sum_{i=1}^{n} \left[ P(X_j/[x]_A^i )\left(1- \frac{C_{\left |[x]_A^i \cap X_j\right |}^2}{C_{\left|[x]_A^i \right|}^2} \right) \right]$ and ∑i=1nP([x]Ai/Xj)(1−C|[x]Ai∩Xj|2C|Xj|2) $\sum_{i=1}^{n} P([x]_A^i / X_j )\left( 1- \frac{C_{\left | [x]_A^i \cap X_j \right |}^2}{C_{|X_j |}^2} \right)$ by introducing absolute P([x]Ai)C|[x]Ai|2C|Xj|2 $P([x]_A^i ) \frac{C_{\left| [x]_A^i \right |}^2}{C_{|X_j |}^2}$ and P(X_j) . 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 X_j, hence, they become robust for uncertainty measurement. Next, we discuss their properties.

proof. Since P ⪯ Q, let ∪t=1k[x]Pt=[x]Q $\cup_{t=1}^k[x]_P^t =[x]_Q$ , then we have

(1)

CEωXj(P)=∑t=1kP(Xj/([x]Pt)(C|U|2C|Xj|2)[P([x]Pt)(1−C|[x]Pt|2C|U|2)]=∑t=1kP(Xj/([x]Pt)P([x]Pt)C|U|2C|Xj|2−∑t=1kP(Xj/[x]Pt)P([x]Pt)C|U|2C|Xj|2C|[x]Pt|2C|U|2=∑t=1kP(Xj∩[x]Pt)C|U|2C|Xj|2−∑t=1kP(Xj∩[x]Pt)C|[x]Pt|2C|Xj|2=∑t=1kP(Xj∩[x]Pt)C|U|2−C|[x]Pt|2C|Xj|2=P(Xj∩[x]P1)(C|U|2−C|[x]P1|2C|Xj|2)+P(Xj∩[x]P2)(C|U|2−C|[x]P2|2C|Xj|2)+⋯+P(Xj∩[x]Pk)(C|U|2−C|[x]Pk|2C|Xj|2)≥P(Xj∩[x]P1)(C|U|2−C|[x]Q|2C|Xj|2)+P(Xj∩[x]P2)(C|U|2−C|[x]Q|2C|Xj|2)+⋯+P(Xj∩[x]Pk)(C|U|2−C|[x]Q|2C|Xj|2)=[P(Xj∩[x]P1)+P(Xj∩[x]P2)+⋯+P(Xj∩[x]Pk)](C|U|2−C|[x]Q|2C|Xj|2)=∑t=1kP(Xj∩[x]Pt)(C|U|2−C|[x]Q|2C|Xj|2)=P(Xj∩[x]Q)(C|U|2−C|[x]Q|2C|Xj|2)=CEωXj(Q).$$ \begin{equation} \begin{array}{rl} CE_\omega^{X_j} (P) & = \sum\limits_{t=1}^k P(X_j/([x]_P^t ) \left( \frac{C_{|U|}^2}{C_{|X_j |}^2}\right) \left[ P([x]_P^t )\left(1- \frac{C_{|[x]_P^t |}^2}{C_{|U|}^2}\right) \right ] \\ & = \sum\limits_{t=1}^kP(X_j/([x]_P^t ) P([x]_P^t ) \frac{C_{|U|}^2}{C_{|X_j |}^2} - \sum\limits_{t=1}^kP(X_j/[x]_P^t )P([x]_P^t ) \frac{C_{|U|}^2}{C_{|X_j |}^2} \frac{C_{\left | [x]_P^t \right |}^2}{C_{|U|}^2}\\ & =\sum\limits_{t=1}^kP(X_j \cap [x]_P^t ) \frac{C_{|U|}^2}{C_{|X_j |}^2} - \sum\limits_{t=1}^k P(X_j \cap [x]_P^t ) \frac{C_{|[x]_P^t |}^2}{C_{|X_j |}^2}\\ &= \sum\limits_{t=1}^k P(X_j \cap [x]_P^t ) \frac{C_{|U|}^2-C_{|[x]_P^t |}^2}{C_{|X_j |}^2}\\ & = P(X_j \cap [x]_P^1 ) \left( \frac{C_{|U|}^2-C_{|[x]_P^1 |}^2}{C_{|X_j |}^2} \right ) + P(X_j \cap [x]_P^2 ) \left(\frac{C_{|U|}^2-C_{|[x]_P^2 |}^2}{C_{|X_j |}^2}\right ) + \cdots + P(X_j \cap [x]_P^k ) \left( \frac{C_{|U|}^2-C_{|[x]_P^k |}^2}{C_{|X_j |}^2}\right )\\ \geq & P(X_j \cap [x]_P^1 ) \left( \frac{C_{|U|}^2-C_{|[x]_Q |}^2}{C_{|X_j |}^2} \right ) + P(X_j \cap [x]_P^2 ) \left( \frac{C_{|U|}^2-C_{|[x]_Q|}^2}{C_{|X_j |}^2} \right )+ \cdots + P(X_j \cap [x]_P^k ) \left( \frac{C_{|U|}^2-C_{|[x]_Q |}^2}{C_{|X_j |}^2}\right )\\ &= \left [ P(X_j \cap [x]_P^1 )+ P(X_j \cap [x]_P^2 )+\cdots+ P(X_j \cap[x]_P^k ) \right] \left( \frac{C_{|U|}^2 - C_{|[x]_Q |}^2}{C_{|X_j |}^2} \right) \\ &= \sum\limits_{t=1}^k P(X_j \cap [x]_P^t ) \left( \frac{C_{|U|}^2-C_{|[x]_Q |}^2}{C_{|X_j |}^2}\right )\\ & = P(X_j \cap [x]_Q ) \left( \frac{C_{|U|}^2-C_{|[x]_Q |}^2}{C_{|X_j |}^2} \right ) \\ & =CE_\omega^{X_j} (Q). \end{array} \end{equation} $$(12)

(2)

CEω(P/Xj)=P(Xj)×∑t=1kP([x]Pt/Xj)(1−C|[x]Pt∩Xj|2C|Xj|2)=P(Xj)×[P([x]Q/Xj)−∑t=1kP([x]Pt/Xj)C|[x]Pt∩Xj|2C|Xj|2]=P(Xj)×P([x]Q/Xj)−P(Xj)×[P([x]P1/Xj)C|[x]P1∩Xj|2C|Xj|2+P([x]P2/Xj)C|[x]P2∩Xj|2C|Xj|2+⋯+P([x]Pk/Xj)C|[x]Pk∩Xj|2C|Xj|2]≥P(Xj)×P([x]Q/Xj)−P(Xj)×[P([x]P1/Xj)C|[x]Q∩Xj|2C|Xj|2+P([x]P2/Xj)C|[x]Q∩Xj|2C|Xj|2+⋯+P([x]Pk/Xj)C|[x]Q∩Xj|2C|Xj|2]=P(Xj)×P([x]Q/Xj)−P(Xj)×[P([x]P1/Xj)+P([x]P2/Xj)+⋯+P([x]Pk/Xj)](C|[x]Q∩Xj|2C|Xj|2)=P(Xj)×P([x]Q/Xj)−P(Xj)×∑i=1kP([x]Pi/Xj)(C|[x]Q∩Xj|2C|Xj|2)=P(Xj)×P([x]Q/Xj)−P(Xj)×P([x]Q/Xj)(C|[x]Q∩Xj|2C|Xj|2)=P(Xj)×P([x]Q/Xj)(1−C|[x]Q∩Xj|2C|Xj|2)=CEω(Q/Xj).$$ \begin{equation} \begin{array}{rl} & CE_\omega(P/X_j ) = P(X_j ) \times \sum\limits_{t=1}^k P([x]_P^t/X_j ) \left( 1- \frac{C_{\left |[x]_P^t \cap X_j \right |}^2}{C_{|X_j |}^2}\right) \\ =& P(X_j ) \times \left [ P([x]_Q/X_j ) - \sum\limits_{t=1}^k P([x]_P^t/X_j ) \frac{C_{\left |[x]_P^t \cap X_j \right |}^2}{C_{|X_j |}^2} \right ]\\ = & P(X_j ) \times P([x]_Q/X_j ) - P(X_j ) \times \left [ P([x]_P^1/X_j ) \frac{C_{\left |[x]_P^1 \cap X_j \right |}^2}{C_{|X_j |}^2} + P([x]_P^2/X_j ) \frac{C_{\left |[x]_P^2 \cap X_j \right |}^2}{C_{|X_j |}^2} + \cdots + P([x]_P^k/X_j ) \frac{C_{\left |[x]_P^k \cap X_j \right |}^2}{C_{|X_j |}^2} \right ]\\ \geq & P(X_j ) \times P([x]_Q/X_j ) - P(X_j ) \times \left [ P([x]_P^1/X_j ) \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2} + P([x]_P^2/X_j ) \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2} + \cdots + P([x]_P^k/X_j ) \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2} \right ]\\ = & P(X_j )\times P([x]_Q/X_j )-P(X_j ) \times \left [ P([x]_P^1/X_j )+P([x]_P^2/X_j )+\cdots+P([x]_P^k/X_j ) \right] \left ( \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2}\right) \\ = & P(X_j )\times P([x]_Q/X_j )-P(X_j ) \times \sum\limits_{i=1}^{k} P([x]_P^i/X_j ) \left ( \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2}\right)\\ = & P(X_j )\times P([x]_Q/X_j )-P(X_j ) \times P([x]_Q/X_j ) \left ( \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2}\right)\\ = & P(X_j ) \times P([x]_Q/X_j ) \left ( 1- \frac{C_{\left |[x]_Q \cap X_j \right |}^2}{C_{|X_j |}^2} \right) = CE_\omega (Q/X_j ). \end{array} \end{equation} $$(13)

(3)

Cω(Xj/P)=∑t=1kP([x]Pt)C|[x]Pt|2C|Xj|2[P(Xj/[x]Pt)(1−C|[x]Pt∩Xj|2C|[x]Pt|2)]=∑t=1kp([x]pt)p(Xj/[x]pt)[C|[x]pt|2C|Xj|2(1−C|Xj∩[x]pt|2C|[x]pt|2)]=∑t=1kp(Xj∩[x]pt)(C|[x]pt|2−C|Xj∩[x]pt|2C|Xj|2)=p(Xj∩[x]p1)(C|[x]p1|2−C|Xj∩[x]p1|2C|Xj|2)+p(Xj∩[x]p2)(C|[x]p2|2−C|Xj∩[x]p2|2C|Xj|2)+⋯+p(Xj∩[x]pk)(C|[x]pk|2−C|Xj∩[x]pk|2C|Xj|2)≤p(Xj∩[x]p1)(C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)+p(Xj∩[x]p2)(C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)+⋯+p(Xj∩[x]pk)(C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)=∑t=1kp(Xj∩[x]pt)(C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)=[∑t=1kp(Xj∩[x]pt)](C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)=p(Xj∩[x]Q)(C|[x]Q|2−C|Xj∩[x]Q|2C|Xj|2)=CWω(Xj/Q).□$$ \begin{equation} \begin{array}{rl} & C_\omega (X_j/P) = \sum\limits_{t=1}^{k} P([x]_P^t ) \frac{C_{\left| [x]_P^t \right |}^2}{C_{|X_j |}^2} \left[ P(X_j/[x]_P^t ) \left(1- \frac{C_{\left |[x]_P^t \cap X_j\right |}^2}{C_{\left|[x]_P^t \right|}^2} \right) \right]\\ = & \sum\limits_{t=1}^{k}p([x]_{p}^{t}) p(X_j/[x]_{p}^{t}) \left[ \frac{C_{\left | [x]_p^t \right |}^2}{C_{|X_j|}^2} \left ( 1 - \frac{C_{\left | X_j \cap [x]_p^t \right|}^{2}}{C_{\left | [x]_p^t \right|}^{2}}\right ) \right] \\ = & \sum\limits_{t=1}^{k} p(X_j \cap [x]_{p}^{t}) \left( \frac{C_{\left | [x]_p^t \right |}^2 - C_{\left | X_j \cap [x]_p^t \right|}^{2}}{C_{|X_j|}^2} \right) \\ = & p(X_j \cap [x]_{p}^{1}) \left( \frac{C_{\left |[x]_p^1\right |}^2 - C_{\left | X_j \cap [x]_p^1 \right|}^{2}}{C_{|X_j|}^2} \right) + p(X_j \cap [x]_{p}^{2}) \left( \frac{C_{\left |[x]_p^2\right |}^2 - C_{\left | X_j \cap [x]_p^2 \right|}^{2}}{C_{|X_j|}^2} \right) +\cdots + p(X_j \cap [x]_{p}^{k}) \left( \frac{C_{\left |[x]_p^k\right |}^2 - C_{\left | X_j \cap [x]_p^k \right|}^{2}}{C_{|X_j|}^2} \right)\\ \leq & p(X_j \cap [x]_{p}^{1}) \left( \frac{C_{\left |[x]_Q\right |}^2 - C_{\left | X_j \cap [x]_Q \right|}^{2}}{C_{|X_j|}^2} \right) + p(X_j \cap [x]_{p}^{2}) \left( \frac{C_{\left |[x]_Q\right |}^2 - C_{\left | X_j \cap [x]_Q \right|}^{2}}{C_{|X_j|}^2} \right) +\cdots + p(X_j \cap [x]_{p}^{k}) \left( \frac{C_{\left |[x]_Q\right |}^2 - C_{\left | X_j \cap [x]_Q \right|}^{2}}{C_{|X_j|}^2} \right)\\ = & \sum\limits_{t=1}^{k} p(X_j \cap [x]_{p}^{t}) \left( \frac{C_{\left | [x]_Q \right |}^2 - C_{\left | X_j \cap [x]_Q\right|}^{2}}{C_{|X_j|}^2} \right)\\ =& \left [ \sum\limits_{t=1}^{k} p(X_j \cap [x]_{p}^{t}) \right ] \left( \frac{C_{\left | [x]_Q \right |}^2 - C_{\left | X_j \cap [x]_Q\right|}^{2}}{C_{|X_j|}^2} \right)\\ =& p(X_j \cap [x]_Q) \left( \frac{C_{\left | [x]_Q \right |}^2 - C_{\left | X_j \cap [x]_Q\right|}^{2}}{C_{|X_j|}^2} \right) = CW_\omega(X_j/Q).\Box \end{array} \end{equation} $$(14)

Theorem 4 provides an important relationship for the three-way weighted combination-entropies. In other words, CE_ω(A/X_j) is a linear translation of the sum of CEωXj(A) $CE_\omega^{X_j}(A)$ and CE_ω(X_j/A), where P(Xj)[1−(C|U|2)/(C|Xj|2)] $P(X_j)[1-(C_{|U|}^2)/(C_{|X_j |}^2 )]$ 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.

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.

At Macro-Top, Theorem 9 describes an important relationship of the three-way weighted combination-entropies by introducing CE (D). Thus, CE_ω(A/D) is a linear translation of the summation of CEωD(A) $CE_\omega^D (A)$ and CEωlin(D/A) $CE_\omega^{lin} (D/A)$ or the difference between CE_ω(D/A) and CEωlin(D/A) $CE_\omega^{lin} (D/A)$ .

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.

Fig 2

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.