Vague Entropy Measure for Complex Vague Soft Sets

The complex vague soft set (CVSS) model is a hybrid of complex fuzzy sets and soft sets that have the ability to accurately represent and model two-dimensional information for real-life phenomena that are periodic in nature. In the existing studies of fuzzy and its extensions, the uncertainties which are present in the data are handled with the help of membership degree which is the subset of real numbers. However, in the present work, this condition has been relaxed with the degrees whose ranges are a subset of the complex subset with unit disc and hence handle the information in a better way. Under this environment, we developed some entropy measures of the CVSS model induced by the axiomatic definition of distance measure. Some desirable relations between them are also investigated. A numerical example related to detection of an image by the robot is given to illustrate the proposed entropy measure.


Introduction
Classical information measures deal with information which is precise in nature, while information theory is one of the trusted ways to measure the degree of uncertainty in data. In our day-to-day life, uncertainty plays a dominant role in any decision-making process. In other words, due to an increase of the system day-by-day, decision makers may have to give their judgments in an imprecise, vague and uncertain environment. To deal with such information, Zadeh [1] introduced the theory of fuzzy sets (FSs) for handling the uncertainties in the data by defining a membership function with values between 0 and 1. In this environment, Deluca and Termini [2] proposed a set of axioms for fuzzy entropy. Liu [3] and Fan and Xie [4] both studied information measures related to entropy, distance, and similarity for fuzzy sets. With the growing complexities, researchers are engaged in extensions such as intuitionistic fuzzy set (IFS) [5], vague set (VS) [6], interval-valued IFS [7] to deal with the uncertainties. Under these extensions, Szmidt and Kacprzyk [8] extended the axioms of Deluca and Termini [2] to the IFS environment. Later on, corresponding to Deluca and Termini's [2] fuzzy entropy measure, Vlachos and Sergiadis [9] extended their measure in the IFS environment. Burillo and Bustince [10] introduced the entropy of intuitionistic fuzzy sets (IFSs), as a tool to measure the degree of intuitionism associated with an IFS. Garg et al. [11] presented a generalized intuitionistic fuzzy entropy measure of order α and degree β to solve decision-making problems. In addition to the mentioned examples, other authors have also addressed the problem of decision-making by using the different information measures [12][13][14][15][16][17][18][19][20][21][22][23][24][25].

Preliminaries
In this section, we briefly reviewed some basic concepts related to the VSs, SSs, CVSSs defined over the universal set U. Definition 1 [6]. A vague set (VS) V in U is characterized by the truth and falsity membership functions t V , f V : U→[0.1] with t V (x) + f V (x) ≤ 1 for any x ∈ U. The values assigned corresponding to t V (x) and f V (x) are the real numbers of [0, 1]. The grade of membership for x can be located in [t V (x), 1 − f V (x)] and the uncertainty of x is defined as ( It is clearly seen from the definition that VSs are the generalization of the fuzzy sets. If we assign 1 − f V (x) to be 1 − t V (x) then VS reduces to FS. However, if we set 1 − t V (x) to be ν A (x) (called the non-membership degree) then VS reduces to IFS. On the other hand, if we set t V (x) = µ L V (x) and 1 − f V (x) = µ U V (x) then VS reduces to interval-valued FS. Thus, we conclude that VSs are the generalization of the FSs, IFSs and interval-valued FSs. Definition 3 [26]. Let P(U) denote the power set of U. A pair (F, A) is called a soft set (SS) over V where F is a mapping given by F : A → P(U) . Definition 4 [42]. Let V(U) be the power set of VSs over U. A pair (F, A) is called a vague soft set (VSS) over U, whereF is a mapping given byF : A → V(U). Mathematically, VSS can be defined as follows: It is clearly seen that this set is the hybridization of the SSs and VSs.
Definition 5 [57]. A complex vague set (CVS) is defined as an ordered pair defined as are the truth and falsity membership functions with unit disc and are defined as The major advantages of the CVSS are that it represents two-dimensional information in a single set and each object is characterized in terms of its magnitude as well as its phase term. Further, the soft set component in CVSS provides an adequate parameterization tool to represent the information.
Definition 7 [54]. Let two CVSSs (F, A) and (G, B) over U, the basic operations between them are defined as (i) (F, A) ⊂ (G, B) if and only if the following conditions are satisfied for all x ∈ U :

Axiomatic Definition of Distance Measure and Vague Entropy
Let E be a set of parameters and U be the universe of discourse. In this section, we present some information measures namely distance and entropy for the collections of CVSSs, which are denoted by CVSS(U).
Next, we define the axiomatic definition for the vague entropy for a CVSS.
If the following two cases holds for all a ∈ A and x ∈ U, Based on this definition, it is clear that a value close to 0 indicates that the CVSS has a very low degree of vagueness whereas a value close to the 1 implies that the CVSS is highly vague. For all x ∈ U, the nearer r t Fa (x) is to k f Fa (x), the larger the vague entropy measure and it reaches a maximum when r t Fa (x) = k f Fa (x). Condition M5 on the other hand, is slightly different as it is constructed using the sharpened version of a vague soft set as explained in Hu et al. [58], instead of the usual condition of (F, A) ⊂ (G, B) implies that the entropy of (F, A) is higher than the entropy of (G, B). In [58], Hu et al. proved that this condition is inaccurate and provided several counter-examples to disprove this condition. Subsequently, they replaced this flawed condition with two new cases. We generalized these two cases to derive condition (M5) in this paper, in a bid to increase the accuracy of our proposed vague entropy. We refer the readers to [58] for further information on these revised conditions.

Relations between the Proposed Distance Measure and Vague Entropy
In the following, let U be universal and φ be empty over CVSSs. Then based on the above definition, we define some of the relationship between them as follows: Theorem 1. Let (F, A) be CVSS and d is the distance measure between CVSSs, then the equations M 1 , M 2 and M 3 defined as below are the valid vague entropies of CVSSs.
Proof. Here, we shall prove only the part (i), while others can be proved similarly.
It is clearly seen from the definition of vague entropies that M 1 satisfies conditions (M1) to (M4). So we need to prove only (M5). For it, consider the two cases stated in Definition 9. We only prove that the condition (M5) is satisfied for Case 1; the proof for Case 2 is similar and is thus omitted.
From the conditions given in Case 1 of (M5), we obtain the following relationship: Therefore, we have: Hence, it follows that: Now, by definition of M 1 , we have: This completes the proof.

Theorem 2.
If d is the distance measure between CVSSs, then: is a vague entropy of CVSSs.

Proof. For two CVSSs
, clearly seen that M 4 satisfies conditions (M1)-(M4). So, it is enough to prove that M 4 satisfy the condition (M5). Consider the case: which implies that: Thus, we obtain: Therefore, we have: Hence, by definition of M 4 , we have: Similarly, we can obtain for other case i.e., when Therefore, M 4 is a valid entropy measure.

Theorem 3.
For CVSS (F, A) and if d is the distance measure between CVSSs, then: is a vague entropy of CVSSs.
Proof. It can be obtained as similar to Theorem 2, so we omit here. Proof. By definition of M 4 and M 5 , we have: Then: is a vague entropy of CVSSs.
Theorem 6. If d is the distance measure between CVSSs and satisfies d((F, A), U) = d((F, A), φ), then: is a vague entropy of CVSSs.

Theorem 7.
If d is a distance measure between CVSSs that satisfies: Proof. The proof of the Theorems 5-7 can be obtained as similar to above, so we omit here.

Theorem 8.
If d is a distance measure between CVSSs, then: is a vague entropy of CVSSs.

Theorem 9.
If d is a distance measure between CVSSs, then: is a vague entropy of CVSSs.
Theorem 10. If d is a distance measure between CVSSs (F, A) and (G, B) such that: Theorem 11. If d is a distance measure between CVSSs, then: is a vague entropy of CVSSs.
Theorem 12. If d is a distance measure between CVSSs, then: is a vague entropy of CVSSs.
Proof. The proof of these Theorems can be obtained as similar to above, so we omit here.

Illustrative Example
In this section, we present a scenario which necessitates the use of CVSSs. Subsequently, we present an application of the entropy measures proposed in Section 4 to an image detection problem to illustrate the validity and effectiveness of our proposed entropy formula.
Firstly, we shall define the distance between any two CVSSs as follows: Definition 10. Let (F, A) and (G, B) be two CVSSs over U. The distance between (F, A) and (G, B) is as given below: In order to demonstrate the utility of the above proposed entropy measures M i (i = 1, 2, . . . , 11), we demonstrate it with a numerical example. For it, consider a CVSS (F, A) whose data sets are defined over the parameters e 1 , e 2 ∈ E and x 1 , x 2 , x 3 ∈ U as follows: and hence the complement of CVSS is: Then, the distance measure based on the Definition 10, we get ( 2167. Therefore, the values of the entropy measures defined on the Theorem 1 to Theorem 11 are computed as: Next, we give an illustrative example from the field of pattern recognition which are stated and demonstrated as below.

The Scenario
A type of robot has a single eye capable of capturing (and hence memorizing) things it sees as an 850 × 640, 24 bit bitmap image. The robot was shown an object (a pillow with a smiley), and the image that was captured by the robot's eye at that instant is shown in Figure 1. This image was saved as pic001.bmp in the memory of the robot. Next, we give an illustrative example from the field of pattern recognition which are stated and demonstrated as below.

The Scenario
A type of robot has a single eye capable of capturing (and hence memorizing) things it sees as an 850 × 640, 24 bit bitmap image. The robot was shown an object (a pillow with a smiley), and the image that was captured by the robot's eye at that instant is shown in Figure 1. This image was saved as pic001.bmp in the memory of the robot. The robot was then given a way (in this example, it is done by human input) to recognize the object, whenever the robot encounters the object again, by retrieving the colors at certain coordinates of its field of vision, and then comparing this with the same coordinates from image pic001.bmp stored in its memory. In order to distinguish noises, the coordinates are chosen in clusters of four, as shown in Figure 2. The coordinates of the clusters of the images are summarized in Table 1. The robot was then given a way (in this example, it is done by human input) to recognize the object, whenever the robot encounters the object again, by retrieving the colors at certain coordinates of its field of vision, and then comparing this with the same coordinates from image pic001.bmp stored in its memory. In order to distinguish noises, the coordinates are chosen in clusters of four, as shown in Figure 2. The coordinates of the clusters of the images are summarized in Table 1.            From a human perspective, it is clear that the object shown in image A will be recognized as the same object shown in image pic001.bmp, and it will be concluded that the object is shown in image B (a red airplane) is not the image pic001.bmp stored in the memory of the robot. No conclusion can be deduced from image C as it is made up of only noise, and therefore we are unable to deduce the exact object behind the noise. By retrieving the coordinates from Table 1, we now obtain the following sets of colors which are given in Table 2. From a human perspective, it is clear that the object shown in image A will be recognized as the same object shown in image pic001.bmp, and it will be concluded that the object is shown in image B (a red airplane) is not the image pic001.bmp stored in the memory of the robot. No conclusion can be deduced from image C as it is made up of only noise, and therefore we are unable to deduce the exact object behind the noise. By retrieving the coordinates from Table 1, we now obtain the following sets of colors which are given in Table 2.    The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively    The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively    The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively    The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively  The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively. Luminosity,   The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively    The luminosity and hue of the pixels are obtained using a picture editing program, and these are given in Tables 3 and 4, respectively 3  3  162  163  122  120  58  63  24  31  62  24  96  100  106  106  91  88  78  78  55  56  95  95  3  3  163  163  125  122  65  67  14  61  40  96   CF  85  88  80  83  78  79  97  102  103  104  102  101  6  5  79  77  24  24  49  28  0  22  109  56  88  89  81  82  78  78  97  98  103  104  99  100  5  5  81  81  33  32  60  81  3  139  27  96   RF  83  82  64  64  60  59  119  119  136  136  139  141  6  6  16  15  62  63  98  40  93  94  56  16  84  84  65  63  59  59  120  119  135  134  138  139  6  6  16  14  61  64  64  0  20  30  45  44   T  60  60  59  59  58  59  142  143  144  144  152  151  116  117  27  27  127  127  74  28  0  75  81  198  59  61  61  62  60  59  144  144  145  144  150  148  120  119  25  24  125  125  120  25  104  3  0  75   M  26  23  15  14  9  9  19  21  34  40  33  35  81  87  175  191  36  42  77  22  112  15  14  24  25  26  13  13  8  8  19  20  43  40  36  36  86  93  181  145  79  66  113  84  121  115  31 33  CF  29  29  29  28  29  29  30  29  30  30  31  30  160  160  8  8  224  230  155  86  160  214  177  0  29  29  29  29  29  29  30  29  30  30  30  30  180  180  8  8  213  220  42  51  200  107  97  165   RF  29  30  29  29  31  31  30  30  32  32  33  33  160  160  160  160  139  139  154  5  238  68  94  36  30  30  29  29  31  31  30  30  32  32  32  33  160  160  160  153  137  141  68  160  192  76  131  158   T  11  11  11  12  10  10  12  12  12  12  13  12  13  9  168  165  7  7  119  212  160  45  160  130  11  11  9  9  9  10  12  12  11  12  13  12  9  13  164  160  8  10  29  205  31  67  160  200   M  11  10  13  13  18  14  16  15  18  20  17  16  10  14  19  18  208  183  145  118  195  26  124  160  11  12  13  13  14  18  15  16  21  19  16  17  13  14  17  15  184  5  42  29  49  181 115 220 We choose = 6400 and = 6400 for this s scenario are as given in Tables 5-7. We choose = 6400 and = 6400 for scenario are as given in Tables 5-7. ∈ {1, 2, 3}, which denote image A, B and C, respectively using the formula given below: F Let = { , , }, and = {LE, RE, LF, CF, RF, T, M}. We now form three CVSSs (ℱ , ), ∈ {1,2,3}, which denote image , and , respectively using the formula given below: where: The luminosity and hue of the pixels are obtained using a are given in Tables 3 and 4, respectively. Luminosity, , , , , where ∈ {LE, RE, LF, CF, RF, T, pixels). Hue, ℌ , , , , where ∈ {LE, RE, LF, CF, RF, T, pixels). We choose = 6400 and = 6400 for this scenario. The scenario are as given in Tables 5-7. 0,k,n,q : p, q ∈ {0, 1, 2, 3} .

Formation of CVSS and Calculation of Entropies
We choose ρ = 6400 and ρ = 6400 for this scenario. The CVSSs that were formed for this scenario are as given in Tables 5-7.   By using Definition 10, the entropy values for images A, B and C are as summarized in Table 8. From these values, it can be clearly seen that M i (F 3 , A) > M i (F 2 , A) > M i (F 1 , A) for all i = 1, 2, . . . , 11. Hence it can be concluded that Image A is the image that is closest to the original image pic001.bmp that is stored in the memory of the robot, whereas Image C is the image that is the least similar to the original pic001.bmp image. The high entropy value for (F 3 , A) is also an indication of the abnormality of Image C compared to Images A and B. These entropy values and the results obtained for this scenario prove the effectiveness of our proposed entropy formula. The entropy values obtained in Table 8 further verifies the validity of the relationships between the 11 formulas that was proposed in Section 4.

Conclusions
The objective of this work is to introduce some entropy measures for the complex vague soft set environment to measure the degree of the vagueness between sets. For this, we define firstly the axiomatic definition of the distance and entropy measures for two CVSSs and them some desirable relations between the distance and entropy are proposed. The advantages of the proposed measures are that they are defined over the set where the membership and non-membership degrees are defined as a complex number rather than real numbers. All of the information measures proposed here complement the CVSS model in representing and modeling time-periodic phenomena. The proposed measures are illustrated with a numerical example related to the problem of image detection by a robot. Furthermore, the use of CVSSs enables efficient modeling of the periodicity and/or the non-physical attributes in signal processing, image detection, and multi-dimensional pattern recognition, all of which contain multi-dimensional data. The work presented in this paper can be used as a foundation to further extend the study of the information measures for complex fuzzy sets or its generalizations. On our part, we are currently working on studying the inclusion measures and developing clustering algorithms for CVSSs. In the future, the result of this paper can be extended to some other uncertain and fuzzy environment [59][60][61][62][63][64][65][66][67][68].