A novel attribute reduction method based on intuitionistic fuzzy three-way cognitive clustering

Abstract

Attribute reduction plays a critical role in the Pawlak rough set, which aims to improve the computational efficiency and accuracy of a system by removing redundant attributes. Existing attribute reduction algorithms focus on attribute importance, information entropy and discernibility matrices while ignoring the classification differences of attributes and loss cost in the reduction process. More importantly, people are not “all-around experts”, and their cognition level of domain knowledge (CLDK) may vary, which leads to multiple reduction results based on the same attribute set. In view of this, we integrate learners’ CLDK and present an attribute reduction model based on intuitionistic fuzzy three-way cognitive clustering (IF3WCC). In this scenario, we first present the concept, calculation methods and semantic interpretation of intuitionistic fuzzy cognitive entropy (IFCE) and prove its related properties. Intuitionistic fuzzy cognition similarity (IFCS) is then proposed and utilized to implement IF3WCC. We then introduce the three-way decision (3WD) model to calculate the reduction cost of attributes in various clusters and divide them into irreducible, reducible, and determined reduction sets. Next, we develop a secondary reduction strategy for uncertain reduction attributes and provide a corresponding algorithm. Finally, the rationality and effectiveness of the proposed model are verified by comparing it with existing attribute reduction methods.

Appendices

Appendix A

Proof of Theorem 1

(1) ∀a_j ∈ A, when $\mid {\left[x\right]}_j^{\mathcal{R}}\mid =1$, ${CH}^{\mathcal{R}}=0$, and when $\mid {\left[x\right]}_j^{\mathcal{R}}\mid =U$, ${CH}^{\mathcal{R}}=\frac{1}{\mid U\mid }{\mathit{\log}}_2\mid U\mid$. Because ${CH}^{\mathcal{R}}=\frac{1}{\mid U\mid }{\mathit{\log}}_2\mid U\mid =\frac{\mathit{\log}\mid U\mid }{\mid U\mid \mathit{\log}(2)}$(log(x) is the nature logarithm) and ∣U ∣  ≥ 1, the critical value of ${CH}^{\mathcal{R}}$ is $\frac{d}{d\mid U\mid}\left(\frac{{\mathit{\log}}_2\mid U\mid }{\mid U\mid}\right)=\frac{1-\mathit{\log}\mid U\mid }{{\left|U\right|}^2\mathit{\log}(2)}$, and we can obtain ∣U ∣  = e, i.e., the corresponding value of ${CH}^{\mathcal{R}}$ is $\frac{1}{eln(2)}$. Property (2) can be directly proven by Property (1).

Proof of Theorem 2

The third rule in Definition 2 is obviously satisfied. The following proves rules (1), (2) and (4).

• (1) Let ${\mu}^{\ast }=\left|\Big({\mu}_{\overset{\sim }{A}}(x)-{\mu}_{\overset{\sim }{B}}(x)\right|\in \left[0,1\right]$, ${\nu}^{\ast }=\left|\Big({\nu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{B}}(x)\right|\in \left[0,1\right]$. When μ^∗ = ν^∗ = 0, let ${\mu}^{\ast {CH}^{\mathcal{R}}}=0$. Since Eq. (3) can be transformed into $S\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)=\frac{1}{2}\left(1-{\mathit{\log}}_2\left({\left|\Big({\mu}_{\overset{\sim }{A}}(x)-{\mu}_{\overset{\sim }{B}}(x)\right|}^{CH^{\succ }}+1\right)+1-{\mathit{\log}}_2\left({\left|\Big({\nu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{B}}(x)\right|}^{CH^{\succ }}+1\right)\right)$, we will prove it in two parts: membership and nonmembership. In terms of membership degree, let ${S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)=1-\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)$. When μ^∗ = 0, i.e., ${\mu}_{\overset{\sim }{A}}(x)=1\left(\mathrm{or}{\mu}_{\overset{\sim }{A}}(x)=0\right),{\nu}_{\overset{\sim }{B}}(x)=0\left(\mathrm{or}{\nu}_{\overset{\sim }{B}}(x)=1\right)$, we have $\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)=0$, ${S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)=0$, and when μ^∗ = 1, i.e., ${\mu}_{\overset{\sim }{A}}(x)={\mu}_{\overset{\sim }{B}}(x),{\nu}_{\overset{\sim }{A}}(x)={\nu}_{\overset{\sim }{B}}(x)$, we can obtain $\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)=1$, ${S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)=1-1=0$. For the nonmember part, we obtain ${S}_{\nu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)\in \left[0,1\right]$. Therefore, $S\left(\overset{\sim }{A},\overset{\sim }{B}\right)={S}_{\mu}\left(\overset{\sim }{A},\overset{\sim }{B}\right)+{S}_{\nu}\left(\overset{\sim }{A},\overset{\sim }{B}\right)=\frac{1}{2}\left(1-\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)+1-\mathrm{lo}{\mathrm{g}}_2\left({\nu}^{\ast {CH}^{\succ }}+1\right)\right)$, which simplifies to $S\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)=1-\frac{1}{2}\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)\left({\nu}^{\ast {CH}^{\succ }}+1\right)\in \left[0,1\right]$.

• (2) By rule (1), $S\left(\overset{\sim }{A},\overset{\sim }{B}\right)=1$ holds if and only if $\overset{\sim }{A}=\overset{\sim }{B}$.

• (4) From $\overset{\sim }{A}(x)\subseteq \overset{\sim }{B}(x)\subseteq \overset{\sim }{C}(x)$, we have ${\mu}_{\overset{\sim }{A}}(x)\le {\mu}_{\overset{\sim }{B}}(x)\le {\mu}_{\overset{\sim }{C}}(x),{\nu}_{\overset{\sim }{A}}(x)\ge {\nu}_{\overset{\sim }{B}}(x)\ge {\nu}_{\overset{\sim }{C}}(x)$. Because$\left|{\mu}_{\overset{\sim }{A}}(x)-{\mu}_{\overset{\sim }{C}}(x)\right|\ge$ $\left|{\mu}_{\overset{\sim }{A}}(x)-{\mu}_{\overset{\sim }{B}}(x)\right|$ and $\mathrm{lo}{\mathrm{g}}_2\left({\mu}^{\ast {CH}^{\succ }}+1\right)$ is an increasing function in the definition field [0, 1],$\mathrm{lo}{\mathrm{g}}_2\left(\left|{\mu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{C}}(x)\right|+1\right)\ge$ $\mathrm{lo}{\mathrm{g}}_2\left(\left|{\mu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{B}}(x)\right|+1\right)$ and ${S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)\le {S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)$ hold. Meanwhile, $\left|{\nu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{C}}(x)\right|\ge \left|{\nu}_{\overset{\sim }{A}}(x)-{\nu}_{\overset{\sim }{B}}(x)\right|$; we have ${S}_{\nu}\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)\le {S}_{\nu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)$. Therefore, ${S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)+{S}_{\nu}\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)\le {S}_{\mu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)+{S}_{\nu}\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)$, i.e., $S\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)\le S\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)$. In an analogous manner, we can obtain $S\left(\overset{\sim }{A}(x),\overset{\sim }{C}(x)\right)\le S\left(\overset{\sim }{B}(x),\overset{\sim }{C}(x)\right)$.

In summary, $S\left(\overset{\sim }{A}(x),\overset{\sim }{B}(x)\right)$ is an IFCS between intuitionistic fuzzy sets (IFSs) $\overset{\sim }{A}(x)$ and $\overset{\sim }{B}(x)$.

Appendix B

1.1 Abbreviations

3WD	three-way decision	POS	positive
IFCS	intuitionistic fuzzy cognition similarity	NEG	negative
IFCE	intuitionistic fuzzy cognitive entropy	BND	boundary
CLDK	cognition level of domain knowledge	IS	irreducible set
CDF	cumulative distribution function	RS	reducible set
IF3WCC	intuitionistic fuzzy three-way cognitive clustering	PRS	possibly reducible set