Fuzzy entropy functions based on perceived uncertainty

Abstract

The paper presents fuzzy entropy functions based on perceived vagueness. The proposed entropy functions are based on the principle that different agents may perceive a membership grade differently. The perceived uncertainty for a membership grade is determined through a gain function. In this light, new variants of the popular fuzzy entropy functions are developed. Inspired by these variants, a novel fuzzy entropy function is also developed. The proposed functions are extended to the probabilistic fuzzy domain. A case study is included to illustrate applicability of the work.

Appendices

Appendix A. Proofs for the properties of P-Sh entropy

The basic properties are investigated with $\gamma$ as 1.

Proof of Property 1

From (7), we know that $H_{\text {P-Sh}, d} = \frac{1}{n}\sum _{i = 1}^{n} (1 - \Delta \mu _i){\mathcal {G}}_i$ . Since $\mu _i \in [0, 1]$, $\Delta \mu _i \in [0, 1]$ [see (3)]. Hence $(1 - \Delta \mu _i) \ge 0$, and ${\mathcal {G}}_i = -\log (\Delta \mu _i) \ge 0$. Since both $(1 - \Delta \mu _i) \ge 0$, and ${\mathcal {G}}_i \ge 0$, $H_{\text {P-Sh}, d} \ge 0$.

Proof of Property 2

The proof follows trivially. When $\forall x_i \in X = 0$ or 1, $\Delta \mu _i = 1$. Therefore, $\log \left( \Delta \mu _i\right) = 0$, $\forall i$, and hence, $H_{\text {P-Sh}, d} = 0$.

Proof of Property 3

At $\mu _i = 0.5$, $\Delta \mu _i = 0$, i.e. the minimum. As $\mu _i$ moves towards 0.5, $\Delta \mu _i$ decreases. As $\Delta \mu _i$ decreases, ${\mathcal {G}}_i$ increases. Therefore, the maximum value for ${\mathcal {G}}_i$ is obtained at $\mu _i = 0.5$. As $\log$ is undefined at 0, we consider $\mu _i \rightarrow 0.5$, or $\Delta \mu _i \rightarrow 0$. Similarly, the weight $(1 - \Delta \mu _i)$ attains its maximum value at $\mu _i = 0.5$. Therefore, $H_{\text {P-Sh}, d}$ is maximum at $\mu _i \rightarrow 0.5, \forall x_i \in U$.

Proof of Property 4

$H[X,Y] = H[X] + H[Y]$. Let $\Delta {\mathcal {M}}^X$ and $\Delta {\mathcal {M}}^Y$ denote the sets for relative membership grades for the fuzzy sets X and Y defined on U. Then from (6), we can write:

$$\begin{aligned} H[X]&= -\frac{1}{n}\sum _{i = 1}^{n} (1 - \Delta \mu _i^X)\log ({\Delta \mu _i^X}) \nonumber \\ H[Y]&= -\frac{1}{n}\sum _{i = 1}^{n} (1 - \Delta \mu _i^Y)\log ({\Delta \mu _i}^Y) \end{aligned}$$

(A.1)

Since $\Delta \mu _i^X$ and $\Delta \mu _i^Y$ are independent of each other, we can write:

$$\begin{aligned} H^{X, Y}&= -\frac{1}{n}\sum _{i = 1}^{n} \left( (1 - \Delta \mu _i^X)\log ({\Delta \mu _i^X}) + (1 - \Delta \mu _i^Y)\log ({\Delta \mu _i^Y})\right) \nonumber \\&= -\frac{1}{n}\sum _{i = 1}^{n} (1 - \Delta \mu _i^X)\log ({\Delta \mu _i^X}) - \sum _{i = 1}^{n} (1 - \Delta \mu _i^Y)\log ({\Delta \mu _i}^Y) \nonumber \\&= H[X] + H[Y] \end{aligned}$$

(A.2)

Proof of Property 5

The proof follows straightforward from proofs of Properties 3 and 4.

Proof of Property 6

From (3) : $\Delta \mu _i = 2*|0.5 - \mu _i|$. For a complement set $\mu _i = 1 - \mu _i, \, \forall i$. Hence, $\Delta \mu _i$ for the complement set: $\Delta \mu _i^{'} = 2*|0.5 - (1 - \mu _i)| = \Delta \mu _i$. Therefore, $H = H^{'}$.

Appendix B. Proofs for the properties of P-PP fuzzy entropy

The basic properties are investigated taking $\gamma$ as 1.

Proof of Property 1

We know that the entropy is minimum when $\mu _i = 0$ or 1, or $\Delta \mu _i = 1$, $\forall x_i \in X$. Recalling (9): $H = \frac{1}{n}\sum _{i = 1}^n (1 - \Delta \mu _i)e^{1 - \Delta \mu _i}$, we know that H is minimum when both $(1 - \Delta \mu _i)$ and $e^{1 - \Delta \mu _i}$ are at their respective minimum values, for all $x_i \in X$. Since $\mu _i \in [0, 1]$, we have $\Delta \mu _i \in [0, 1]$. At the upper bound $\Delta \mu _i = 1$, or $\mu _i = 0$ or 1, both $(1 - \Delta \mu _i)$ and $e^{1 - \Delta \mu _i}$ are at their minimums. The minimum H is thus $\frac{1}{n}\left( n(1 - 1)e^{1 - 1}\right) = 0$.

Proof of Property 2

From (9), we know that H is maximum when ${\mathcal {G}}_i$ is maximum, and its weight $(1 - \Delta \mu _i)$ is maximum, for all $x_i \in X$. Since ${\mathcal {G}}_i = e^{1 - \Delta \mu _i}$. Hence, ${\mathcal {G}}_i$ attains its maxima at $\Delta \mu _i = 0$. The weight of ${\mathcal {G}}_i$, i.e. $(1 - \Delta \mu _i)$, is also maximum at $\Delta \mu _i = 0$, or $\mu _i = 0.5$. Hence, H attains the maximum value, when $\mu _i = 0.5$ (or $\Delta \mu _i = 0$), $\forall x_i \in X$. Replacing $\Delta \mu _i = 0$, $\forall x_i \in X$ in (9), we obtain $H_{max} = e$.

Proof of Property 3

We know that $\Delta \mu _i$ decreases as $\mu _i$ increases in the interval [0, 0.5], and $\Delta \mu _i$ increases as $\mu _i$ increases in the interval [0.5, 1]. Therefore, both $(1 - \Delta \mu _i)$ and ${\mathcal {G}}_i = e^{(1 - \Delta \mu _i)}$ are monotonically increasing for $\mu _i \in [0, 0.5]$, are monotonically decreasing for $\mu _i \in [0.5, 1]$, and attain the maximum value at $\mu _i = 0.5$. Resultantly, $H = \frac{1}{n}\sum _{i = 1}^n(1 - \Delta \mu _i)(e^{(1 - \Delta \mu _i)})$ has the same trend.

Proof of Property 4

The proof is same as for that of Property 6 of P-Sh entropy.

Appendix C. Proofs for the properties of the new fuzzy entropy

Proof of Property 1

If $\mu _i = 0$ or 1, then $\Delta \mu _i = 1$. Hence ${\mathcal {G}}_i = e - e^{\Delta \mu _i} = 0$. Therefore, if $\mu _i = 0$ or 1, for all $x_i \in U$, then $H = \frac{1}{n}\sum (1 - 1)0 = 0$, the minimum value for H (Since $\Delta \mu _i \in [0, 1]$, $e - e^{\Delta \mu _i}$ has a lower bound at 0.).

Proof of Property 2

Since $\mu _i \in [0, 1]$, $\Delta \mu _i \in [0, 1]$. At $\mu _i = 0$ or 1, $\Delta \mu _i = 1$. As $\mu _i$ moves towards 0.5, $\Delta \mu _i$ decreases towards 0, and ${\mathcal {G}}_i$ increases towards $e - 1$, the upper bound for ${\mathcal {G}}_i$. If $\mu _i = 0.5$, for all $x_i \in U$, then $H = \frac{1}{n}\sum _{i = 1}^n(1 - 0)(e - 1) = (e - 1)$, the maximum possible value for H.

Proof of Property 3

We know that $\Delta \mu _i$ decreases as $\mu _i$ increases in the interval [0, 0.5], and $\Delta \mu _i$ increases as $\mu _i$ increases in the interval [0.5, 1]. For a given $\gamma$, therefore, both $(1 - \Delta \mu _i)$ and ${\mathcal {G}}_i = (e - e^{(\Delta \mu _i)^{\gamma }})$ are monotonically increasing for $\mu _i \in [0, 0.5]$, are monotonically decreasing for $\mu _i \in [0.5, 1]$, and attain the maximum value at $\mu _i = 0.5$. Resultantly, $H = \frac{1}{n}\sum _{i = 1}^n(1 - \Delta \mu _i)(e - e^{(\Delta \mu _i)^{\gamma }})$ varies in the same way.

Proof of Property 4

We know that $\Delta \mu _i = 2*|0.5 - \mu _i|$. Therefore, $\Delta {\overline{\mu }}_i = 2*|0.5 - (1 - \mu _i)| = |-(0.5 - \mu _i)| = |0.5 - \mu _i| = \Delta \mu _i$. Hence, $H = {H}^{'}$.