3. A new parametric measure of fuzzy information measure involving two parameters α and β

A new generalised fuzzy information measure of order α and type β has been suggested and their necessary and required properties are examined. Thereafter, its validity is also verified. Also, the monotonic behaviour of fuzzy information measure of order α and type β has been conferred.

The generalised measure of fuzzy information of order α and type β is given by,

$$H\_{\alpha}^{\beta}(A) = \frac{1}{(1-\alpha)\beta} \sum\_{i=1}^{n} \left[ \left(\mu\_{A}^{a\mu\_{A}(\mathbf{x}\_{i})} + \left(1 - \mu\_{A}(\mathbf{x}\_{i})\right)^{a\left(1 - \mu\_{A}(\mathbf{x}\_{i})\right)}\right)^{\beta} - 2^{\beta} \right],\tag{26}$$
 
$$\alpha > 0, \alpha \neq 1, \beta \neq 0.$$

#### 3.1. Properties of H<sup>β</sup> <sup>α</sup>ð Þ A

Hβ

<sup>H</sup>αð Þ¼� <sup>P</sup> <sup>X</sup><sup>n</sup>

<sup>H</sup>αð Þ¼� <sup>A</sup> <sup>X</sup><sup>n</sup>

<sup>α</sup>ð Þ¼ <sup>A</sup> <sup>1</sup>

i¼1

we get the measure

i¼1

þ 1 a Xn i¼1

� 1 a

we get the measure

ð Þ¼ <sup>P</sup> <sup>1</sup>

<sup>β</sup> � <sup>α</sup> log

H<sup>α</sup>,<sup>β</sup>

α þ β � 2

Xn i¼1

3. Corresponding to Kapur's [12] measure of entropy

1 a Xn i¼1

μAð Þ xi log μAð Þþ xi 1 � μAð Þ xi

� � log 1 <sup>þ</sup> <sup>a</sup>μAð Þ xi � � � � <sup>þ</sup>

H<sup>α</sup>, <sup>β</sup>

P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> μα

P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup><sup>β</sup>

pi log pi þ

1 þ aμAð Þ xi

ð Þ 1 � a log 1ð Þ � a

μα

14 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

<sup>A</sup>ð Þþ xi 1 � μAð Þ xi

1 þ api

� � � �

4. Corresponding to Kapur's [4] measure of entropy of degree α and type β

ð Þ¼ <sup>P</sup> <sup>1</sup>

<sup>β</sup> � <sup>α</sup> log

<sup>A</sup>ð Þþ xi <sup>1</sup> � μα

<sup>A</sup>ð Þþ xi 1 � μAð Þ xi

Kosko [13] introduced fuzzy entropy and conditioning. Pal and Pal [14] gave object background segmentation using new definition of entropy. Parkash [15] proposed new measures of weighted fuzzy entropy and their applications for the study of maximum weighted fuzzy entropy principle. Parkash and Gandhi [16] suggested new generalised measures of fuzzy entropy and properties. Parkash and Singh [17] gave characterization of useful information theoretic measures. Taneja [18] introduced generalised information measures and their applications. Taneja and Tuteja [19] gave characterization of quantitative-qualitative measure of relative information. Tuteja [20] introduced characterization of nonadditive measures of relative information and accuracy. Tuteja and Hooda [21] proposed generalised useful information measure of type α and degree β. Tuteja and Jain [22, 23] gave characterization of relative useful information having utilities as monotone functions and an axiomatic characterization of relative useful information. Tahayori [24] presented a universal methodology for generating an

� � log 1 <sup>þ</sup> api

� � log 1 � <sup>μ</sup>Að Þ xi

1 a Xn i¼1

P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>p</sup><sup>α</sup> i

P<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>p</sup> β i

<sup>A</sup>ð Þ xi � �<sup>α</sup>

� � " #

� �<sup>α</sup> � <sup>μ</sup><sup>β</sup>

h i

� � � <sup>1</sup>

� �

a

½ 1 þ a � aμAð Þ xi

<sup>A</sup>ðÞþ xi 1 � μAð Þ xi

ð Þ 1 þ a log 1ð Þ þ a

� � log 1 <sup>þ</sup> <sup>a</sup> � <sup>a</sup>μAð Þ xi

, α 6¼ β (24)

� �<sup>β</sup> , <sup>α</sup> <sup>≥</sup> <sup>1</sup>, <sup>β</sup> <sup>≤</sup> <sup>1</sup> or <sup>α</sup> <sup>≤</sup> <sup>1</sup>, <sup>β</sup> <sup>≥</sup> <sup>1</sup>: (25)

� �<sup>β</sup> � <sup>2</sup>

(21)

(23)

, a ≥ 0 (22)

We have supposed that, 00:<sup>α</sup> <sup>¼</sup> 1, we study the following properties:

Property 1: H<sup>β</sup> <sup>α</sup>ð Þ <sup>A</sup> <sup>≥</sup> 0 i.e. <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A is nonnegative.

Property 2: H<sup>β</sup> <sup>α</sup>ð Þ A is minimum if A is a non-fuzzy set.

For <sup>μ</sup>Að Þ¼ xi 0, it implies <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ <sup>A</sup> = 0 and <sup>μ</sup>Að Þ¼ xi 0 we have <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A = 0.

Property 3: H<sup>β</sup> <sup>α</sup>ð Þ A is maximum if A is most fuzzy set.

$$\begin{split} & \text{We have, } \frac{\mathrm{d}\vartheta^{\sharp}\_{\boldsymbol{A}}(\boldsymbol{A})}{\vartheta\mu\_{\boldsymbol{A}}(\boldsymbol{x})} = \frac{a}{1-a} \left[ \left\{ \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right\}^{a\mu\_{\boldsymbol{A}}(\boldsymbol{x})} + \left\{ 1 - \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right\}^{a\left(1-\mu\_{\boldsymbol{A}}(\mathbf{x}\_{i})\right)} \right]^{\beta-1} \left[ \left\{ \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right\}^{a\mu\_{\boldsymbol{A}}(\mathbf{x}\_{i})} \left\{ 1 + \log \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right\} \right] \\ & \left( \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right) - \left\{ 1 - \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right\}^{a\left(1-\mu\_{\boldsymbol{A}}(\mathbf{x}\_{i})\right)} \left\{ 1 + \log \left( 1 - \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \right) \right\}. \\ & \text{Taking } \frac{\mathrm{d}\vartheta^{\sharp}\_{\boldsymbol{A}}(\boldsymbol{A})}{\vartheta\mu\_{\boldsymbol{A}}(\mathbf{x})} = 0 \text{ which is possible } \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) = 1 - \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) \text{ that is if } \mu\_{\boldsymbol{A}}(\mathbf{x}\_{i}) = \frac{1}{2}. \end{split}$$
  $\text{Now, we have } \frac{\mathrm{d}^{2}\mathcal{H}\_{\boldsymbol{A}}^{\ell}(\boldsymbol{A})}{\delta^{\dagger}\mu\_{\boldsymbol{A}}(\mathbf{x}\_{i})}$ 

$$\frac{\partial^2 H\_{\alpha}^{\p}(A)}{\partial^2 \mu\_A(\alpha\_i)} = \frac{-1}{1 - \frac{1}{\alpha}} \left( 2^{1 - \frac{\mu}{2}} \right)^{\beta - 1} \left[ 2^{2 - \frac{\mu}{2}} + \alpha . 2^{1 - \frac{\mu}{2}} (1 - \log 2)^2 \prec 0 \right],$$

Hence, the maximum value exists at μAð Þ¼ xi 1 2.

Property 4: H<sup>β</sup> <sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ <sup>A</sup> , where <sup>A</sup><sup>∗</sup> be sharpened version of A.

When μAð Þ¼ xi 1 2,

$$H\_a^{\mathfrak{f}}(A) = \frac{n.2^{\mathfrak{f}}}{(1-\alpha)\beta} \left(\frac{1-2^{\frac{\alpha\mathfrak{f}}{2}}}{2^{\frac{\alpha\mathfrak{f}}{2}}}\right).$$

When <sup>μ</sup>Að Þ xi lies between 0 and 1/2 then <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A is an increasing function whereas when μAð Þ xi lies between 1/2 and 1 then H<sup>β</sup> <sup>α</sup>ð Þ A is a decreasing function of μAð Þ xi

Let A<sup>∗</sup> be sharpened version of A which means that

i. If μAð Þ xi < 0:5 then μA<sup>∗</sup> ð Þ xi ≤ μAð Þ xi for all i = 1, 2, …, n

ii. If μAð Þ xi > 0:5 then μA<sup>∗</sup> ð Þ xi ≥ μAð Þ xi for all i = 1, 2, …, n

Since H<sup>β</sup> <sup>α</sup>ð Þ <sup>A</sup> is an increasing function of <sup>μ</sup>Að Þ xi for 0 <sup>≤</sup> <sup>μ</sup>Að Þ xi <sup>≤</sup> <sup>1</sup> <sup>2</sup> and decreasing function of <sup>μ</sup>Að Þ xi for <sup>1</sup> <sup>2</sup> ≤ μAð Þ xi ≤ 1, therefore

Case II: For α > 1, 0 < β < 1 we have compiled the values of H<sup>β</sup>

(a) (b) (c)

<sup>α</sup>ð Þ <sup>A</sup> <sup>μ</sup>Að Þ xi <sup>H</sup><sup>β</sup>

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.5419 0.1 0.5074 0.1 2.0337 0.2 0.7749 0.2 0.7763 0.2 2.7289 0.3 0.9075 0.3 0.9534 0.3 3.0732 0.4 0.9778 0.4 1.0517 0.4 3.2411 0.5 1.0 0.5 1.0844 0.5 3.2916 0.6 0.9778 0.6 1.0517 0.6 3.2411 0.7 0.9075 0.7 0.9534 0.7 3.0732 0.8 0.7749 0.8 0.7763 0.8 2.7289 0.9 0.5419 0.9 0.5074 0.9 2.0337 1.0 0.0 1.0 0.0 1.0 0.0

for other values of α and β, we get different concave curves.

Case III: For α > 1, β > 1 we have compiled the values of H<sup>β</sup>

a concave function.

μAð Þ xi H<sup>β</sup>

concave function.

For <sup>α</sup> = 1.5 and <sup>β</sup> <sup>¼</sup> 0.1, values of <sup>H</sup><sup>β</sup>

α = 1.5 and β = 0.1; and (C) For, α = 1.5 and β = 2.5.

the fuzzy entropy in the Figure 1(b) which unambiguously illustrates that the fuzzy entropy is

Table 1. The values of fuzzy information measure for α = 2 and β = 1; α = 1.5 and β = 0.1; and α = 1.5 and β = 2.5.

α = 1.5 and β = 0.1 which implies that the proposed measure is a concave function. Similarly,

fuzzy entropy in Figure 1(c) which unambiguously illustrates that the fuzzy entropy is a

Figure 1. Representation of the monotonic behaviour of fuzzy information measure for (a) For, α = 2 and β = 1; (b) For,

<sup>α</sup>ð Þ A in Table 1, (b) and presented

<sup>α</sup>ð Þ <sup>A</sup> <sup>μ</sup>Að Þ xi <sup>H</sup><sup>β</sup>

Fuzzy Information Measures with Multiple Parameters http://dx.doi.org/10.5772/intechopen.78803

<sup>α</sup>ð Þ A

17

<sup>α</sup>ð Þ A in Table 1, (c) and presented the

<sup>α</sup>ð Þ A have been represented with the help of graph for


Hence, H<sup>β</sup> <sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A .

Property 5: H<sup>β</sup> <sup>α</sup>ð Þ¼ <sup>A</sup> <sup>H</sup><sup>β</sup> <sup>α</sup> A � �, where <sup>A</sup> is the compliment of A i.e. <sup>μ</sup>Að Þ xi =1- <sup>μ</sup>Að Þ xi .

Thus when <sup>μ</sup>Að Þ xi is varied to (1 � <sup>μ</sup>Að Þ xi ) then <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A does not change.

Under the above conditions, the generalised measure proposed in (26) is a valid measure of fuzzy information measure.

#### 3.2. Monotonic behaviour of fuzzy information measure

In this section we study the monotonic behaviour of the fuzzy information measure. For this, diverse values of H<sup>β</sup> <sup>α</sup>ð Þ A by assigning various values to α and β has been calculated and further the generalised measure has been presented graphically.

Case I: For α > 1, β =1, we have compiled the values of H<sup>β</sup> <sup>α</sup>ð Þ A in Table 1, (a) and presented the fuzzy entropy in Figure 1(a) which unambiguously illustrates that the fuzzy information measure is a concave function.

For α = 2, β = 1, values of H<sup>β</sup> <sup>α</sup>ð Þ A have been represented with the help of graph for α = 2 and β = 1 which implies that the proposed measure is a concave function. Similarly, for other values of α and β, we get different concave curves.


∂2 Hβ <sup>α</sup>ð Þ A

<sup>¼</sup> �<sup>1</sup> <sup>1</sup> � <sup>1</sup> α

16 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

Hβ

21�<sup>α</sup> 2 � �<sup>β</sup>�<sup>1</sup> 22�<sup>α</sup>

<sup>2</sup> <sup>þ</sup> <sup>α</sup>:21�<sup>α</sup>

1 � 2 αβ 2

!

2 αβ 2

1 2.

ð Þ 1 � α β

<sup>α</sup>ð Þ A is a decreasing function of μAð Þ xi

<sup>α</sup>ð Þ A in [0, 0.5]

<sup>α</sup>ð Þ A in [0.5, 1]

Under the above conditions, the generalised measure proposed in (26) is a valid measure of

In this section we study the monotonic behaviour of the fuzzy information measure. For this,

fuzzy entropy in Figure 1(a) which unambiguously illustrates that the fuzzy information

β = 1 which implies that the proposed measure is a concave function. Similarly, for other values

� �, where <sup>A</sup> is the compliment of A i.e. <sup>μ</sup>Að Þ xi =1- <sup>μ</sup>Að Þ xi .

<sup>α</sup>ð Þ A does not change.

<sup>α</sup>ð Þ A by assigning various values to α and β has been calculated and further

<sup>α</sup>ð Þ A have been represented with the help of graph for α = 2 and

<sup>α</sup>ð Þ <sup>A</sup> , where <sup>A</sup><sup>∗</sup> be sharpened version of A.

<sup>α</sup>ð Þ¼ <sup>A</sup> <sup>n</sup>:2<sup>β</sup>

<sup>2</sup> ð Þ <sup>1</sup> � log 2 <sup>2</sup> <sup>≺</sup><sup>0</sup>

<sup>α</sup>ð Þ A is an increasing function whereas when μAð Þ xi

<sup>2</sup> and decreasing function of

<sup>α</sup>ð Þ A in Table 1, (a) and presented the

h i

∂2 μAð Þ xi

<sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup>

1 2,

lies between 1/2 and 1 then H<sup>β</sup>

Property 4: H<sup>β</sup>

When μAð Þ¼ xi

Since H<sup>β</sup>

<sup>μ</sup>Að Þ xi for <sup>1</sup>

Hence, H<sup>β</sup>

Property 5: H<sup>β</sup>

Hence, the maximum value exists at μAð Þ¼ xi

When <sup>μ</sup>Að Þ xi lies between 0 and 1/2 then <sup>H</sup><sup>β</sup>

<sup>2</sup> ≤ μAð Þ xi ≤ 1, therefore

<sup>α</sup>ð Þ A .

<sup>α</sup> A

Thus when <sup>μ</sup>Að Þ xi is varied to (1 � <sup>μ</sup>Að Þ xi ) then <sup>H</sup><sup>β</sup>

3.2. Monotonic behaviour of fuzzy information measure

the generalised measure has been presented graphically.

Case I: For α > 1, β =1, we have compiled the values of H<sup>β</sup>

i. <sup>μ</sup>A<sup>∗</sup> ð Þ xi <sup>≤</sup> <sup>μ</sup>Að Þ xi this implies <sup>H</sup><sup>β</sup>

ii. <sup>μ</sup>A<sup>∗</sup> ð Þ xi <sup>≤</sup> <sup>μ</sup>Að Þ xi this implies <sup>H</sup><sup>β</sup>

<sup>α</sup>ð Þ¼ <sup>A</sup> <sup>H</sup><sup>β</sup>

<sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup>

fuzzy information measure.

measure is a concave function.

For α = 2, β = 1, values of H<sup>β</sup>

of α and β, we get different concave curves.

diverse values of H<sup>β</sup>

Let A<sup>∗</sup> be sharpened version of A which means that

i. If μAð Þ xi < 0:5 then μA<sup>∗</sup> ð Þ xi ≤ μAð Þ xi for all i = 1, 2, …, n

ii. If μAð Þ xi > 0:5 then μA<sup>∗</sup> ð Þ xi ≥ μAð Þ xi for all i = 1, 2, …, n

<sup>α</sup>ð Þ <sup>A</sup> is an increasing function of <sup>μ</sup>Að Þ xi for 0 <sup>≤</sup> <sup>μ</sup>Að Þ xi <sup>≤</sup> <sup>1</sup>

<sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup>

<sup>α</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup><sup>β</sup>

Table 1. The values of fuzzy information measure for α = 2 and β = 1; α = 1.5 and β = 0.1; and α = 1.5 and β = 2.5.

Case II: For α > 1, 0 < β < 1 we have compiled the values of H<sup>β</sup> <sup>α</sup>ð Þ A in Table 1, (b) and presented the fuzzy entropy in the Figure 1(b) which unambiguously illustrates that the fuzzy entropy is a concave function.

For <sup>α</sup> = 1.5 and <sup>β</sup> <sup>¼</sup> 0.1, values of <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A have been represented with the help of graph for α = 1.5 and β = 0.1 which implies that the proposed measure is a concave function. Similarly, for other values of α and β, we get different concave curves.

Case III: For α > 1, β > 1 we have compiled the values of H<sup>β</sup> <sup>α</sup>ð Þ A in Table 1, (c) and presented the fuzzy entropy in Figure 1(c) which unambiguously illustrates that the fuzzy entropy is a concave function.

Figure 1. Representation of the monotonic behaviour of fuzzy information measure for (a) For, α = 2 and β = 1; (b) For, α = 1.5 and β = 0.1; and (C) For, α = 1.5 and β = 2.5.

For <sup>α</sup> = 1.5 and <sup>β</sup> <sup>¼</sup> 2.5, values of <sup>H</sup><sup>β</sup> <sup>α</sup>ð Þ A have been represented with the help of graph for α = 1.5 and β = 2.5 which implies that the proposed measure is a concave function. Similarly, for other values of α and β, we get different concave curves.

Hence, <sup>H</sup>α, <sup>β</sup>,<sup>γ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup>α, <sup>β</sup>,γð Þ <sup>A</sup>

fuzzy information measure.

μAð Þ¼ xi

1

Property 4: Hα, <sup>β</sup>,γð Þ¼ A Hα,β,<sup>γ</sup> A

, where A

Thus, when μAð Þ xi is varied to 1 � μAð Þ xi then Hα,β,γð Þ A does not change.

4.2. Monotonic behaviour of fuzzy information measure

further the generalised measure has been presented graphically.

Under the above conditions, the generalised measure proposed in (27) is a valid measure of

In this section we study the monotonic behaviour of the fuzzy information measure. For this, diverse values of Hα, <sup>β</sup>,γð Þ A by assigning various values to α, β and γ have been calculated and

Case I: For α > 1, β = 2, γ = 3, we have compiled the values of Hα,β,γð Þ A in Table 2, (a)–(e) and presented the fuzzy entropy in Figure 2(a)–(e) which unambiguously illustrates that the fuzzy information measure is a concave function. For α = 1.5, β = 2, γ = 3, values of Hα,β,γð Þ A have been represented with the help of graph γ ¼ 3 which implies that the proposed measure is a concave function. Similarly, for other values of α, β and γ we get different concave curves. Further it has been shown that Hα,β,γð Þ A is a concave function obtaining its maximum value at

function of μAð Þ xi in interval (0.5, 1]. Similarly, for α = 2, β = 2 and γ = 3, α = 2.5, β = 2 and γ = 3, α = 3, β = 2 and γ = 3, α = 3.5, β = 2 and γ = 3, γ ¼ 3 values of Hα,β,γð Þ A have been represented with the help of graph which implies that the proposed measure is a concave function.

μAð Þ xi Hα,β,γð Þ A μAð Þ xi Hα,β,γð Þ A μAð Þ xi Hα,β,γð Þ A μAð Þ xi Hα,β,γð Þ A μAð Þ xi Hα,β,γð Þ A 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 12.8444 0.1 6.7318 0.1 4.6517 0.1 3.5865 0.1 2.9316 0.2 14.7302 0.2 7.5514 0.2 5.1212 0.2 3.8867 0.2 3.1353 0.3 15.3147 0.3 7.7794 0.3 5.2385 0.3 3.9540 0.3 3.1762 0.4 15.5250 0.4 7.8559 0.4 5.2750 0.4 3.9734 0.4 3.1870 0.5 15.5795 0.5 7.875 0.5 5.2837 0.5 3.9779 0.5 3.1894 0.6 15.5250 0.6 7.8559 0.6 5.2750 0.6 3.9734 0.6 3.1870 0.7 15.3147 0.7 7.7794 0.7 5.2385 0.7 3.9540 0.7 3.1762 0.8 14.7302 0.8 7.5514 0.8 5.1212 0.8 3.8867 0.8 3.1353 0.9 12.8444 0.9 6.7318 0.9 4.6517 0.9 3.5865 0.9 2.9316 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0

(a) (b) (c) (d) (e)

α = 3, β = 2 and γ = 3; and α = 3.5, β = 2 and γ = 3.

Table 2. The values of fuzzy information measure for α = 1.5, β = 2 and γ = 3; α = 2, β = 2 and γ = 3; α = 2.5, β = 2 and γ = 3;

<sup>2</sup>. Hence Hα, <sup>β</sup>,γð Þ A is increasing function of μAð Þ xi in interval [0, 0.5) and decreasing

is the compliment of A i.e. <sup>μ</sup>Að Þ¼ xi <sup>1</sup> � <sup>μ</sup>Að Þ xi

Fuzzy Information Measures with Multiple Parameters http://dx.doi.org/10.5772/intechopen.78803 19
