Three-level and three-way uncertainty measurements for interval-valued decision systems

Abstract

Uncertainty measurements underlie the system interaction and data learning. Their relevant studies are extensive for the single-valued decision systems, but become relatively less for the interval-valued decision systems. Thus, three-level and three-way uncertainty measurements of the interval-valued decision systems are proposed, mainly by systematically constructing vertical-horizontal weighted entropies. Firstly, the interval-valued decision systems are endowed with three-level structures, including Micro-Bottom, Meso-Middle, and Macro-Top. Secondly, a three-level decomposition is hierarchically made for the existing conditional entropy. Thirdly, three-way weighted entropies are systematically and hierarchically constructed at the three levels, and they achieve their hierarchy, systematicness, algorithm, boundedness, and granulation monotonicity/non-monotonicity. The three-level and three-way weighted entropies deepen and extend the conditional entropy, and they realize the ingenious criss-cross informatization for the interval-valued decision systems. Their effectiveness of uncertainty measurements is ultimately verified by table examples and data experiments.

Appendices

Appendix A. Proof of Proposition 2

Proof

Regarding $H_{lklhd}(D_{j}/{S_B^\theta (u_i)})$ and by Lemma 2, we have $0\le -\frac{1}{n}P({S_B^\theta (u_i)})P({D_{j}/S_B^\theta (u_i)})\log _{2}P({D_{j}/S_B^\theta (u_i)})\le \frac{1}{n}\times 1\times \frac{1}{e ln2}$. Next prove the relevant realizability. Possible value $P({D_{j}/S_B^\theta (u_i)})=0$ guides $H_{lklhd}(D_{j}/{S_B^\theta (u_i)})=0$. On the other hand, possible value $P({S_B^\theta (u_i)})=1$ and approximate value $P({D_{j}/S_B^\theta (u_i)})\rightarrow \frac{1}{e}$ can realize $H_{lklhd}(D_{j}/{S_B^\theta (u_i)})\rightarrow \frac{1}{n\times eln2}$, and the relevant case corresponds to ${S_B^\theta (u_i)}=U$, $|D_{j}|\rightarrow \frac{n}{e}$. The other measures $H_{prior}({S_B^\theta (u_i)})_{D_{j}}$ and $H_{pstrr}({S_B^\theta (u_i)}/D_{j})$ have the similar proof. $\square$

Appendix B. Proof of Lemma 3

Proof

(1) This proof mainly comes from Ref. [31]. Let $z=\frac{x}{{x+y}}\in (0,1]$ in f(x, y), and then we have

$$\begin{aligned} \frac{\partial {f(x,y)}}{\partial {y}}=\frac{x}{(x+y)\ln 2}>0,\,&\frac{\partial {f(x,y)}}{\partial {x}}=\frac{1}{\ln 2}{(\frac{x}{{x+y}}-1)}-\log _{2} \frac{x}{{x+y}}=\frac{1}{\ln 2}{(z-1)}-\log _{2} z=F(z)\ge 0,\nonumber \\&\mathrm{where}\,F'(z)=\frac{1}{\ln 2}(1-\frac{1}{z})\le 0\Rightarrow F(z)\ge F(1)=0. \end{aligned}$$

(B.1)

(2) g(x, y) has partial derivatives:

$$\begin{aligned} \frac{\partial {g(x,y)}}{\partial {y}}=-\frac{x}{(x+y)\ln 2}\le 0,\, \frac{\partial {g(x,y)}}{\partial {x}}=-\log _2\frac{x+y}{n}-\frac{1}{ln2}\frac{x}{x+y}=\frac{1}{ln2}(\frac{y}{x}+1)-\log _2\frac{x+y}{n}=G(x,y). \end{aligned}$$

(B.2)

Thus, we concern only $G(x,y)>0$, $G(x,y)=0$, $G(x,y)<0$ to present $\frac{\partial {g(x,y)}}{\partial {x}}$, and fixed $y>0$ is a premise. For this purpose, suppose $g_1(x,y)=\frac{1}{ln2}(\frac{y}{x}+1)$ and $g_2(x,y)=\log _2\frac{x+y}{n}$ to offer $G(x,y)=g_1(x,y)-g_2(x,y)$, and $G(x,y)=0$ (i.e., Eq. (36)) becomes a basic equation to solve x. Intuitively, the decreasing hyperbolic-curve $g_1(x,y)$ and the increasing logarithmic-function $g_2(x,y)$ have a sole intersection point with horizontal ordinate $x^\dag$, so $G(x,y)\ge 0$ and $G(x,y)\le 0$ are accordingly realized by $x\le x^\dag$ and $x\ge x^\dag$. More deeply,

$$\begin{aligned} \lim \limits _{x\rightarrow 0^+} G(x,y)&=+\infty -\log _2\frac{y}{n}=+\infty >0,\,\lim \limits _{x\rightarrow +\infty } G(x,y)=0-\infty =-\infty<0,\nonumber \\ \frac{\partial {G(x,y)}}{\partial {x}}&=\frac{\partial {g_1(x,y)}}{\partial {x}}-\frac{\partial {g_2(x,y)}}{\partial {x}}=-\frac{y}{x^2 ln2}-\frac{1}{(x+y)ln2}<0; \end{aligned}$$

(B.3)

because of G(x, y)’s x-continuity and $\frac{\partial {G(x,y)}}{\partial {x}}=\frac{\partial ^2 {g(x,y)}}{\partial {x}^2}<0$, key equation $G(x,y)=0$ (i.e., Eq. (36)) has a unique zero-point $x^\dag$ (i.e., the sole x-univariate stationary point of g(x, y)), while $x\le x^\dag$ and $x\ge x^\dag$ respectively correspond to $G(x,y)\ge 0$ and $G(x,y)\le 0$. In a word, $\frac{\partial {g(x,y)}}{\partial {x}}=G(x,y)$ acquires the required positive-negative identification by the sole root $x^\dag \in (0,+\infty )$. $\square$

Appendix C. Three experimental Interval-VDSs from Ref. [31]: Tables 7, 8, 9

Table 7 Interval-VDS I for Experiment I

Table 8 Interval-VDS II for Experiment II

Table 9 Interval-VDS III for Experiment III