A Profile of Trigonometric Measures of Fuzzy Information and Discrimination

In the literature of fuzzy information measures, there exist many well-known parametric and non-parametric measures with their own merits and demerits. One important category of these measures is transcendental measures consisting of logarithmic, exponential and trigonometric fuzzy information and discrimination. These measures have wide applications in pattern recognition, medical diagnosis and MCDM problems. However, trigonometric fuzzy information measures have their own importance for application point of view particularly to geometry.

In present communication the concept of fuzzy information measure is introduced with a few generalizations. A brief survey of trigonometric measures of fuzzy information and discrimination characterized by various authors is described. The Application of one generalized measure of fuzzy discrimination in strategic decision making is explained and illustrated with an example. Conclusion with an exhaustive list of references is also given in the end.

Keywords: Trigonometric Measures; Fuzzy Information

The concept of entropy was introduced and developed to measure the uncertainty of a probability distribution by Shannon [1]. In 1965, Zadeh introduced the concept of fuzzy set and that was extended to measure uncertainty in ambiguous or imprecision linguistic statement [2]. A fuzzy set on a universe of discourse X={x₁,x₂,....,x_n} was defined by Zadeh as given below [2]:

where is the membership function of . The membership value describes the degree of the belongingness of in . The closer the value of μ_A(X) is to 1, the more x belongs to A.

The specificity of fuzzy sets is to capture the idea of partial membership. The characteristic function of a fuzzy set is often called membership function and the role of that has well been explained by Singpurwalla and Booker [3] in probability measures of fuzzy sets.

Fuzzy set theory, in one or other way, is wide applications in many areas of science and technology e.g. clustering, image processing, decision making, pattern recognition, medical diagnosis and multi-criteria decision making, because of its capability to model non-statistical imprecision or vague concepts. When proposing fuzzy set, Zadeh’s [2] concerns were explicitly centred on their potential contribution in the domain of pattern classification, processing and communication of information, abstraction, summarization, etc.

A generalized theory of uncertainty was well explained by Zadeh [4] where he remarked that uncertainty was an attribute of information. Before that the path breaking work of Shannon had led to a universal acceptance of the theory that information was statistical in nature [1]. However, a perception-based theory of probabilistic reasoning of imprecise or vague concepts was explained by Zadeh [5].

Some work related to fuzzy uncertainty management for intelligence analysis was reported by Yager, [6] whereas the generalized fuzzy information theory, its aims, results and some open problems were discussed by Klir [7]. Nanda and Paul redefined Khichin’s [8] version of entropy and discussed the properties of the aging classes based on their generalized entropy.

Taking into consideration the concept of fuzzy set, De Luca and Termini [9] suggested that corresponding to Shannon’s [1] probabilistic entropy, the measure of fuzzy information could be defined as follows:

Fuzzy entropy is a measure of fuzziness of a set which arises from the intrinsic ambiguity or vagueness possessed in the fuzzy set. Bhandari and Pal [10] defined logarithmic fuzzy information, which is a generalization of (1.1) and is given below [10]:

On the same lines many researchers have studied various generalized fuzzy information measures. Hooda, [11] Hooda and Bajaj [12] and many other authors introduced and characterized various generalized additive and non-additive fuzzy information measures and their applications. Mishra et al. [13] proposed exponential fuzzy measures

Hooda and Jain [14] characterized a sub additive trigonometric measure of fuzzy information of type α and degree β corresponding to trigonometric probabilistic entropy studied by Sharma and Taneja [15]. It may be noted that trigonometric measures have their own importance in application point of view, particularly, in geometry.

Monte et al. [16] introduced an axiomatic definition of divergence measure for fuzzy sets and discussed its several properties. They pointed out that divergence measures were described to compute the discrimination among fuzzy sets that was why these were named as fuzzy measures of discrimination. Interestingly, divergence measure is also a measure of dissimilarity which possesses a set of important properties and was applied in evaluation of discrimination measure for fuzzy sets.

In the present communication a survey of trigonometric fuzzy information measures has been made. In section 2, two sine and cosine trigonometric fuzzy information measures are described, while the characterization of a new cosine fuzzy information measure is explained in section 3. In section 4, a new tangent inverse trigonometric fuzzy information measures is described along with a generalized measure of fuzzy discrimination in section 5. In section 6, application of fuzzy discrimination measure in strategic decision making is explained and illustrated with numerical example.

Most of fuzzy information measures have been defined analogous to probabilistic entropies. Hooda and Mishra [17] introduced two trigonometric fuzzy information measures which have no analogous probabilistic entropies as given below:

First of all we check the validity of the proposed measures (2.1) and (2.2).

Theorem 2.1[9]: The fuzzy information measure given by (2.1) and (2.2) are valid measures.

Proof: To prove that the given measure is a valid measure, we shall show that (2.1) satisfies the four properties (P1) to (P4).

(P1).H₁(A)=0 if and only if A is a crisp set.

It implies H₁ (A)=0 if and only if A is a crisp set.

(P2). H₁ (A) is maximum if and only if A is the fuzziest set i.e.,μ_A(x_i)=0.5 for all i=1,2,....,n.

Differentiating H₁ (A) with respect to μ_A(x_i), we have

which vanishes at μ_A(x_i)=0.5.

Again differentiating (2.3) with respect to μ_A(x_i) get

which is less than zero (< 0) at μ_A(x_i) =0.5.

Hence H₁(A) is maximum at μ_A(x_i)=0.5 for i= 1,2,...,n

Further from (2.3) we see that H1(A) is an increasing function of μ_A(x_i)in the region 0≤μ_A(x_i)≤0.5 and H₁(A) is a decreasing function of μ_A(x_i)in the region 0.5≤μ_A(x_i)≤1.

(P3). Let A* be sharpened version of A, which means that

if 0≤ μ_A (x_i) ≤ 0.5, μ_A (x_i) ≤ μ_A (x_i) for all i = 1,2,....n.

and if 0.5≤μ_A (x_i) ≤1, μ_A (x_i) ≥ μ_A (x_i) for all i = 1,2,...n.

Since H1(A) is an increasing function of μ_A(x_i) in the region 0 ≤ μ_A (x_i) ≤ 0.5 and H₁(A) is a decreasing function of μ_A (x_i) in the region 0≤μ_A(x_i)≤0.5, therefore

(i) μ_{A^* (xi)≤ μA(xi)⇒ ≤ H1 (A^*) ≤ H1(A) in [0,0.5] (2.5)}

(ii) μ_{A^* (xi) ≤ μA (xi)⇒ ≤ H1 (A^*) ≤ H1(A) in [0.5,1] (2.6)}

Thus, (2.5) and (2.6) together give

H₁(A^*)≤ H₁(A)

(P4). From the definition it is evident that

H₁(A)= H₁(A^c)

where A^c is complement of A obtained by replacing μ_A(x_i) by 1−μ_A(x_i).

Hence H₁(A) satisfies all the essential four properties of fuzzy information measures. Thus it is a valid measure of fuzzy information. On the same line it can be proved that H₂(A) is a valid fuzzy information measure.

By considering a concave function sinπx,∀x∈[0,1] Hooda and Mishra [17] defined the following fuzzy information measure and proved its validity:

Theorem 2.2[8]: The fuzzy information measure given by (2.7) is valid measure.

Proof: To prove that the given measure is a valid measure, we shall show that (2.7) satisfies the four properties (P1) to (P4).

(P1). H₃(A)=0 if and only if A is a crisp set.

It implies H₃(A)=0 if and only if A is a crisp set.

(P2).H₃(A) is maximum if and only if A is the fuzziest set i.e.,μ_A(x_i)=0.5 for all i=1,2,.....n.

Differentiating H₃(A) with respect to μ_A(x_i), we have

which vanishes at μ_A(x_i)=0.5.

Again differentiating (2.8) with respect to μ_A(x_i),we get

which is less than zero(<0)at μ_A(x_i)=0.5.

Hence H₃(A) is maximum at μ_A(x_i)=0.5 for i=1,2,....n.

H₃(A) is an increasing function of μ_A(x_i) in the region 0≤μ_A(x_i)≤0.5 and H₃(A) is a decreasing function of μ_A(x_i) in the region 0.5≤μ_A(x_i)≤1.

(P3). Let A^* be sharpened version of A, which means that

if 0≤μ_A(x_i)<0.5,μ_A^*(x_i)≤ for all i=1,2,....,n

And if 0.5<μ_A(x_i)≤1,μ_A(x_i)≥μ_A(x_i) for all i=1,2,...,n.

Since H₃(A) is an increasing function of μ_A(x_i) in the region 0≤μ_A(x_i)≤0.5 and H₃(A) is a decreasing function of μ_A(x_i) in the region 0.51≤μ_A(x_i)≤1.

Therefore,

(i) μ_A^*(x_i≤μ_A(x_i)⇒H₃(A^*)≤H₃(A) in [0,0.5] (2.9)

(ii) μ_A^*(x_i≥μ_A(x_i)⇒H₃(A^*)≤H₃(A) in [0.5,1].

(2.10)

Hence (2.9) and (2.10) together give

H₃(A^*)≤H₃(A)

(P4). It is evident from the definition that

H₃(A)≤H₃(A^c)

where A^c is complement of A obtained by replacing μ_A(x_i) by 1−μ_A(x_i).

Hence H₃(A) satisfies all the essential four properties of fuzzy information measures. Thus it is a valid measure of fuzzy information.

Further they defined generalized sine trigonometric fuzzy information measure as given below:

It may be noted that (2.11) reduces to (2.7) when β=π and reduces to (2.1) , when β=π/2

Another generalized sine trigonometric measure of fuzzy information is

In particular when α=0, (2.12) reduces to (2.11) and reduces to (2.7) when α=0 and β=π.

A generalized cosine trigonometric measure of fuzzy information is defined as

which is a new generalized cosine trigonometric measure of fuzzy information and it reduces to the following fuzzy information measure when β=π/2

It can be easily verified that these fuzzy information measure satisfy the essential properties of validity

Let X={x₁,x₂,.....,x_n}be a set of universe and A be a fuzzy set in X having membership function μ_A(x_i) defined on X. Then cosine fuzzy information measure for fuzzy set A was defined as

Theorem3.1[11]: H₈(A) is a valid fuzzy information measure.

Proof: For validity of the measure we shall prove that the following four essential properties (P1) to (P4) are satisfied:

(P1). H₈(A)=0 if and only if A is a crisp set.

Evidently H₈(A) = 0 if and only if either μ_A(x_i)=0 or 1−μ_A(x_i)=0 for i=1,2,......,n.

(P2). H₈(A) is maximum iff μ_A(x_i)=0.5 for all i=1,2,....,n.

Differentiating H₈(A) with respect to μ_A(x_i), we have

Hence H₈(A) is maximum when μ_A(x_i=0.5 for all i=1,2,....,n or A is the fuzziest set.

Further it may be noted that H₈(A) is an increasing function of μ_A(x_i) in the region 0≤μ_A(x_i)≤0.5 and H₈(A) is a decreasing function of μ_A(x_i) in the region ()0.5≤μ_A(x_i)≤1.

(P3). Let A^* be sharpened version of A, then

(i) μ_A^*(x_i)≤μ_A(x_i)⇒H₈(A^*)≤H₈(A) in [0,0.5] (3.3)

μ_A^*(x_i)≥μ_A(x_i)⇒H₈(A^*)≤H₈(A) in [0.5,1] (3.4)

Hence from (3.3) and (3.4) we can conclude that

H₈(A^*)≤H₈(A)

(P4). It is evident that if μ_A(x_i) replacing by 1−μ_A(x_i) for all i=1,2,...,n. then H₈(A^c)=H₈(A), where A^c is complement of A.

Theorem3.2[11]: Let X={x₁,x₂,....,x_n} be the universe of discourse. Let A and B be two fuzzy set in a fixed universe of discourse X. Let A(x_i)=μ_A(x_i) and B(x_i)=μ_B(x_i) satisfying either A⊂B or B⊂A then the following holds:

Corollary 1: Let A be a fuzzy set on X and be its complement, then

H₈(A)=H₈(A^c (3.8) and

Prakash and Gandhi [9] proposed the following tangent fuzzy information measure:

They proved that (4.1) was a valid fuzzy measure and applied to the geometry to estimate the perimeter and area of polygon.

Note: Hooda and Mishra also defined tangent inverse trigonometric fuzzy information measure and prove its validity [17].

Definition4.1 Let X={x₁,x₂,....,x_n} be a set of universe and A be a fuzzy set in having membership function μ_A(x_i) defined on X. Then tangent inverse trigonometric fuzzy information measure for fuzzy set A is defined as

Theorem 4.1 [13]: The measure in (4.2) is a valid fuzzy information measure.

Proof: To prove that the measure (4.2) is a valid measure, we shall show that it is satisfying the four properties (P1) to (P4).

(P1).H_α(A)=0 if and only if A is a crisp set.

If μ_A(x_i=0 or μ_A(x_i)=1,H_α(A)=0.

Hence H_α(A)=0 if A is crisp set,i.e.,μ_A(x_i)=1 or μ_A(x_i=1 for i=1,2,...,n.

Conversely,H_α(A)=0, then

The equality holds if either μ_A(x_i=0 or μ_A(x_i)=1 i.e. A is a crisp set.

Thus H_α(A)=0 if and only if A is a crisp set,i.e.,μ_A(x_i)=0 or μ_A(x_i)=1 for i=1,2,....,n

(P2).H_α(A) is maximum if and only if A is the fuzziest set,i.e.,μ_A(x_i)=0.5 for i=1,2,...,n.

Differentiating (4.2) with respect to μ_A(x_i), we have

Again differentiating (4.3) with respect to μ_A(x_i), we have

Here two cases arise:

(i) μ_A^*(x_i)≤ μ_A(x_i)⇒ H_α(A^*)≤H_α(A) in [0,0.5] (4.6)

(ii) μ_A^*(x_i)≥ μ_A(x_i)⇒ H_α(A^*)≤H_α(A) in [0.5,1] (4.7)

Thus from (4.6) and (4.7) together gives

H_α(A^*)≤H_α(A)

(P4). Let Ac be a complement of A.

Therefore, from the definition, we have

H_α(A)=H_α(A^c)

Since H_α(A) satisfies all the properties of fuzzy information measure, therefore, it is a valid measure of fuzzy information.

Particular Case: When α→1,(4.2) reduces to De Luca and Termini fuzzy information measure [9].

Analogous to Bhatia and Singh, [19] Bhandari et al. [20] suggested the simplest measure of fuzzy discrimination as follows:

Analogous to (5.1), Hooda and Mishra [17] defined a new inverse tangent fuzzy discrimination measure as follows:

Definition5.1. The fuzzy discrimination measure is said to be valid if it satisfies

(i) Non negativity i.e. I(A,B)≥0,

(ii) I(A,B)=0, if A = B,

(iii) I(A,B)≥ 0 is convex function in [0,1]

(iv) I(A,B)≥0 should not change, when μ_A(x_i) is changed to 1-μ_A(x_i)and μ_A(x_i) is changed to 1-μ_B(x_i)

Theorem 5.1 [17]: Let X={x₁,x₂,......,x_n},A,B

&euro

FS(X) then

where α>0,α≠1 is a valid fuzzy discrimination measure

Proof: It can be easily verified that I_α(A,B) satisfy all four properties from (i) to (IV). Thus it is a valid generalized measure of fuzzy discrimination.

In current scenario, the applications of the fuzzy discrimination measure in various fields have been studied, namely, in bio-informatics by Poletti et al. [21] and Fan et al. [22].

Bhatia and Singh [19] also discussed in image thresholding, while Ghosh et al. [23] applied it in automated leukocyte recognition. In this section, the application of the above fuzzy discrimination measure in strategic decision making is proposed and studied.

Decision making problem is the process of finding the optimal solution from all the existing feasible alternatives. It is assumed that a firm Y desires to apply strategies to meet its goal. Let each strategy {S₁,S₂,.....,S_m}has different degree of effectiveness. If it has different input associated with it, then let it be {l₁,l₂,....,l_n}. The fuzzy set Y denotes the effectiveness of a particular strategy with uniform input. Then,

It is assumed that I_t(1≤t≤n) determines the minimum value of {I_α(Y,I)}_0<α<1. With this we find max_{1{μIt(Si)}, let it correspond to Sp(1≤p≤m). Hence, if the strategy is implemented with input budget of the firm will meet its goal in the most input effective manner.}

Let m = n = 5 in the above model. Table 1 shows the efficiency of different strategies at uniform inputs. Table 2 illustrates the efficiency of different strategies at particular inputs and Table 3 the numerical values of discrimination measure {I_α(Y,I)}1≤j≤n<0<α<1.

The calculated numerical data of the proposed fuzzy discrimination measure indicates that input l₁ is more suitable. The assessment of the results existing in Table 2 and Table 3 points out that strategy S₄ is most effectual. Thus a firm will achieve its goal most efficiently if the strategy S₄ is applied with an input 1_l.

The present article is one kind of survey of research work undertaken by me and my research scholars in transcendental fuzzy measures of information and discrimination. These measures have very interesting properties and have found wide applications in Science and Engineering problems.

The above mentioned work was extended further by Mishra et al. [24] by defining and characterized ‘Weighted Trigonometric and Hyperbolic Fuzzy Information Measures’. They also studied their application in optimization theory.

Prakash et al. [25] also studied new measures of weighted fuzzy entropy and their applications for the study of maximum weighted fuzzy entropy principle. Recently, transcendental fuzzy measures of information and discrimination have invited attention of researchers and hence, a lot of work in this area of research is in progress.

The author is extremely thankful to the reviewers for their comments and suggestions for improvement of the article.