4.1. Properties of Hα,β,γð Þ A

We have supposed that, 0<sup>0</sup>:<sup>α</sup> <sup>¼</sup> 1, we study the following properties:

Property 1: Hα, <sup>β</sup>,γð Þ A ≥ 0 i.e. Hα,β,γð Þ A is nonnegative.

Property 2: Hα, <sup>β</sup>,γð Þ A is minimum if A is a non-fuzzy set.

for μAð Þ¼ xi 0, it implies Hα,β,γð Þ¼ A 0 and μAð Þ¼ xi 1 we have Hα, <sup>β</sup>,γð Þ¼ A 0.

Property 3: <sup>H</sup>α, <sup>β</sup>,<sup>γ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup>α,β,γð Þ <sup>A</sup> , where <sup>A</sup><sup>∗</sup> be sharpened version of A.

When μAð Þ xi lies between 0 and 1/2 then Hα,β,γð Þ A is an increasing function whereas when μAð Þ xi lies between 1/2 and 1 then Hα,β,γð 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 <sup>H</sup>α,β,γð 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


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>

For <sup>α</sup> = 1.5 and <sup>β</sup> <sup>¼</sup> 2.5, values of <sup>H</sup><sup>β</sup>

three parameters α, β and γ

α, β and γ has been introduced.

<sup>H</sup>α,β,γð Þ¼ <sup>A</sup> <sup>1</sup>

4.1. Properties of Hα,β,γð Þ A

<sup>μ</sup>Að Þ xi for <sup>1</sup>

1 � α

Xn i¼1

Property 1: Hα, <sup>β</sup>,γð Þ A ≥ 0 i.e. Hα,β,γð Þ A is nonnegative.

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

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

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 <sup>H</sup>α,β,γð 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>

i. <sup>μ</sup>A<sup>∗</sup> ð Þ xi <sup>≤</sup> <sup>μ</sup>Að Þ xi this implies <sup>H</sup>α,β,<sup>γ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup>α,β,γð Þ <sup>A</sup> in [0, 0.5]

ii. <sup>μ</sup>A<sup>∗</sup> ð Þ xi <sup>≤</sup> <sup>μ</sup>Að Þ xi this implies <sup>H</sup>α,β,<sup>γ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup>α,β,γð Þ <sup>A</sup> in [0.5, 1]

Property 2: Hα, <sup>β</sup>,γð Þ A is minimum if A is a non-fuzzy set.

α≻0, α 6¼ 1, β 6¼ 0, γ 6¼ 0

We have supposed that, 0<sup>0</sup>:<sup>α</sup> <sup>¼</sup> 1, we study the following properties:

<sup>μ</sup>ð Þ <sup>α</sup>þ<sup>β</sup> <sup>μ</sup>Að Þ xi

for μAð Þ¼ xi 0, it implies Hα,β,γð Þ¼ A 0 and μAð Þ¼ xi 1 we have Hα, <sup>β</sup>,γð Þ¼ A 0.

μAð Þ xi lies between 1/2 and 1 then Hα,β,γð Þ A is a decreasing function of μAð Þ xi

Property 3: <sup>H</sup>α, <sup>β</sup>,<sup>γ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>≤</sup> <sup>H</sup>α,β,γð Þ <sup>A</sup> , where <sup>A</sup><sup>∗</sup> be sharpened version of A.

by,

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

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

<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,

Further, a new generalised fuzzy information measure involving three parameters α, β and γ 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

The generalised measure of fuzzy information involving three parameters α, β and γ is given

<sup>A</sup> þ 1 � μAð Þ xi

When μAð Þ xi lies between 0 and 1/2 then Hα,β,γð Þ A is an increasing function whereas when

� �ð Þ <sup>α</sup>þ<sup>β</sup> <sup>1</sup>�μ<sup>A</sup> ð Þ ð Þ xi

� <sup>2</sup><sup>γ</sup>

,

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

(27)

� �<sup>γ</sup>

� �

4. A new parametric measure of fuzzy information measure involving

Property 4: Hα, <sup>β</sup>,γð Þ¼ A Hα,β,<sup>γ</sup> A , where A is the compliment of A i.e. <sup>μ</sup>Að Þ¼ xi <sup>1</sup> � <sup>μ</sup>Að Þ xi Thus, when μAð Þ xi is varied to 1 � μAð Þ xi then Hα,β,γð Þ A does not change.

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

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

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

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 μAð Þ¼ xi 1 <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 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.


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

validity is also verified. Also, the monotonic behaviour of fuzzy information measure of order

Fuzzy Information Measures with Multiple Parameters http://dx.doi.org/10.5772/intechopen.78803 21

Fuzzy sets are indispensable in fuzzy system model and fuzzy system design, while the measurement of fuzziness in fuzzy sets is the fuzzy entropy or fuzzy information measure. Therefore, fuzzy information measures occupy important place in the processing of system design. Thus there are enormous applications of fuzzy information in the design of neural

[1] Shannon C. A mathematical theory of communication. Bell System Technical Journal.

[3] Renyi A. On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability; University of California Press;

[4] Havrada J, Charvat F. Quantification methods of classification processes: Concept of

[6] Kapur J. A comparative assessment of various measures of directed divergence. Advances

[7] Kapur J. Measures of Fuzzy Information. New Delhi: Mathematical Sciences Trust Society;

[8] Kaufmann A. Fuzzy Subsets Fundamental Theoretical Elements. New York: Academic

[5] Behara M, Chawla J. Generalized γ-entropy. Selecta Statistica Canadiana. 1974;2:15-38

α, β and γ has been conferred.

network classifiers.

Author details

Anjali Munde

References

1997

Press; 1980

Conflict of interest

I declare that I have no conflict of interest.

Amity University, Uttar Pradesh, India

1948;379–423:623-659

1961. pp. 547-561

Address all correspondence to: anjalidhiman2006@gmail.com

[2] Zadeh L. Fuzzy sets. Information and Control. 1966;8:94-102

structural α-entropy. Kybernetika. 1967;3:30-31

in Management Studies. 1984;3:1-16

Figure 2. Representation of the monotonic behaviour of fuzzy information measure for (a) For, α = 1.5, β = 2 and γ = 3; (b) For, α = 2, β = 2 and γ = 3; (c) For, α = 2.5, β = 2 and γ = 3; (d) For, α = 3, β = 2 and γ = 3; (e) For, α = 3.5, β = 2 and γ = 3.

Further it has been shown that Hα,β,γð Þ A is a concave function obtaining its maximum value at μAð Þ¼ xi 1 <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 function of μAð Þ xi in interval (0.5, 1].

### 5. Conclusions

In this chapter, after reviewing some literatures on measures of information for fuzzy sets, a new generalised fuzzy information measure involving two parameters α and β has been introduced.

The necessary properties of the proposed measure have been verified and further it has been studied that the proposed measure is a concave function as it has shown monotonicity.

Further, a new generalised fuzzy information measure involving three parameters α, β and γ 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, β and γ has been conferred.

Fuzzy sets are indispensable in fuzzy system model and fuzzy system design, while the measurement of fuzziness in fuzzy sets is the fuzzy entropy or fuzzy information measure. Therefore, fuzzy information measures occupy important place in the processing of system design. Thus there are enormous applications of fuzzy information in the design of neural network classifiers.
