An Intelligent Similarity Model between Generalized Trapezoidal Fuzzy Numbers in Large Scale

Mohamedou Cheikh Tourad; Abdelmounaim Abdali

doi:10.5391/IJFIS.2018.18.4.303

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International Journal of Fuzzy Logic and Intelligent Systems 2018; 18(4): 303-315

Published online December 31, 2018

https://doi.org/10.5391/IJFIS.2018.18.4.303

An Intelligent Similarity Model between Generalized Trapezoidal Fuzzy Numbers in Large Scale

Mohamedou Cheikh Tourad, and Abdelmounaim Abdali

Applied Mathematics and Computer Science Laboratory, Cadi Ayyad University, Marrakech, Morocco

Correspondence to :
Mohamedou Cheikh Tourad, (cheikhtouradmohamedou@gmail.com)

Received: November 13, 2018; Revised: December 19, 2018; Accepted: December 24, 2018

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The rapid expansion of data published on the web has given rise to the similarity problem on a large scale, a very important subject for scientific research in the field of computer science. Several methods have been developed for this. In this paper, we propose the first mathematical model to find the similarity value between generalized trapezoidal fuzzy numbers (GTFNs). This model employs fuzzy inference systems to find the value of an effective weighting, the weights to be associated to different kinds of methods that can handle an important scale of the data. This model will allow us to develop intelligent systems. A comparative study based on 21 sets of GTFNs has been carried out to demonstrate the difference between our approach and existing methods. This study shows that our model is more reasonable than existing methods.

Keywords: Cosine coefficient, Jaccard index, GTFNs, FIS, Similarity, Large scale

1. Introduction

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The concept of fuzzy logic, proposed in 1965 by Zadeh [1], has been used to manage a kind of probability. To employ this concept, the generalized trapezoidal fuzzy numbers (GTFNs) are most the most popular in practice. In the literature, several similarity methods between GTFNs have been introduced (e.g., [2–10]). But, these existing methods of similarity measures have many weaknesses. In many situations, such methods cannot appropriately find the similarity between two GTFNs. In the present study, a novel mathematical model for a fuzzy-number similarity method between GTFNs has been created, based on the weights associated to each similarity measure. This model uses the cosine coefficient and the Jaccard Index. Additionally, we describe and provide three characteristics of the proposed model. A comparative study has been carried out, based on 21 sets of GTFNs, to show that the model can surmount the limitations of the existing measures.

The remainder of this article is structured as follows. Section 2 presents a summary of the fundamental notions of existing methods. Section 3 introduces the novel approach, presenting the similarity methods employed and finding the weights associated to each similarity method. Many properties are suggested and proven. Section 4 compares it with the existing similarity methods. We give our conclusions in Section 5.

2. Related Work

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The notion of a GFN T̃ is presented by Chen [11, 12] as follows: T̃ = {t₁, t₂, t₃, t₄, w_T̃ }, where the t_kk = 1, 2, 3, 4 denote real numbers, and we take into account a weight w_T̃ with 0 < w_T̃ < 1. When w_T̃ = 1, T̃ is a normal trapezoidal fuzzy number (NTFN). When t₂ = t₃, T̃ is called a GTFN. When t₁ = t₂ and t₃ = t₄, T̃ is called a crisp interval. When w_T̃ = 1 and t₁ = t_k (where k =1,2,3,4), T̃ is a real number.

Several Similarity methods between GTFNs T̃ and H̃ = {h₁, h₂, h₃, h₄, w _H̃ } have been introduced:

Chen [2] proposed a new method to solve the similarity problems of the GFNs. This method is based on the geometric distance. The author used this method to clear the problems of fuzzy risk analysis and subjective mental workload assessment.

(1)SC(T˜,H˜)=1-(∑k=14tk-hk4).

Hsieh and Chen [3] introduced a new approach to find a solution to the similarity problems of the GFNs. This approach is based on the average integration representation distance.

(2)SHC(T˜,H˜)=11+d(T,H),

where d(T̃, H̃) = |P(T) − P(H)|, P(T˜)=t1+2t2+2t3+t46, P(H˜)=h1+2h2+2h3+h46.

Lee [4] presented a new measure to answer the similarity problems of the GFNs. This measure is based on the the L_P norm. The author used this measure to solve the problems of aggregating individual opinions into group consensus under group decision.

(3)SL(T˜,H˜)=1-‖T˜-H˜‖LP‖U‖×4-1P,

where ‖T˜-H˜‖LP=(∑k=14(∣tk-hk∣)P)1p, taking into account a positive integer P. ||U|| = max U − min U where U indicates the universe of discourse. Chen and Chen [5] defined a new proposed method to clear the similarity problems of GFNs. It is based on the simple center of gravity method (SCGM). The authors used this proposed to solve the problems of fuzzy risk analysis.

(4)SCC(T˜,H˜)=1-(∑k=14tk-hk4)×(1-∣XT˜*-XH˜*∣)B(ST˜,SH˜)×min(yT˜*,yH˜*)max(yT˜*,yH˜*),

where

yT˜*={wT˜6×(2+t3-t2t4-t1)if t4≠t1,wT˜2if t4-t1,yH˜*={wH˜6×(2+h3-h2h4-h1)if h4≠h1,wH˜2if h4-h1,xT˜*={(yT˜*×(t3+t2))+(t4+t1)×(wT˜-yT˜*)2wT˜if wT˜≠0t4+t12if wT˜=0,xH˜*={(yH˜*×(h3+h2))+(h4+h1)×(wH˜-yH˜*)2wH˜if wH˜≠0h4+h12if wH˜=0,B(ST˜,SH˜)={1if ST˜+SH˜>0,0if ST˜+SH˜=0,

where S_T̃ = t₄ − t₁, S_H̃ = h₄ − h₁. Wei and Chen [7] proposed a new approach in order to solve the similarity problems of GFNs. It is based on the concepts of geometric distance, the perimeter and the height of GFNs. The authors used this approach to to answer the problems of fuzzy risk analysis.

(5)SWC(T˜,H˜)=(1-(∑k=14tk-hk4))×(min(P(T˜),P(H˜))+min(wT˜,wH˜)max(P(T˜),P(H˜))+max(wT˜,wH˜)),

where

P(T˜)=((t1-t2)2+(wT˜)2)+(t3-t2)+((t4-t3)2+(wT˜)2))+(t4-t1),P(H˜)=((h1-h2)2+(wH˜)2)+(h3-h2)+((h4-h3)2+(wH˜)2))+(h4-h1).

Recently, Xu et al. [8] changed the similarity method SCC proposed by Chen and Chen [5], introducing the new similarity S_X. The authors used this method to answer the problems of fuzzy risk analysis.

(6)SX(T˜,H˜)=1-(∑k=14tk-hk8)-(d(T˜,H˜)2),

where d(T˜,H˜)=((xT˜*-xH˜*)2+(yT˜*-yH˜*)2)1.25.

Chen [9] introduced a new proposed method to find a solution to the similarity problems of GFNs. This method is based on the geometric mean operator. The author used this approach to clear the problems of fuzzy-number used in the information retrieval.

(7)SSJ(T˜,H˜)=[(∏k=14(2-∣tk-hk∣))14-1]×min(yT˜*,yH˜*)max(yT˜*,yH˜*),

where yT˜* and yH˜* are calculated as in Chen and Chen [5].

3. Approach

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

We will now present a novel mathematical model, MCESTA (Mohamedou Cheikh Elghotob Cheikh Saad bouh Cheikh Tourad Abass), Abdelmounaim Abdali, a large scale similarity method between GTFNs, call them T̃ and H̃. It is a kind of hybrid of the similarity measures S_k and Skq, where each measure is associated with an importance weight (α_k and β_q). MCESTA is an efficient development of the formula FBHSM=∑k=1nαk×Sk proposed by Gupta et al. [13] to calculate similarity in IR (information retrieval). MCESTA employs a fuzzy inference system (FIS) to find the value of the effective weighting. This hybrid similarity measure is calculated as follows:

MCESTA(T˜,H˜)={∑k=1nαk×Sk(T˜,H˜),Sk(T˜,H˜)=∑q=1mkβq×Skq(T˜,H˜),

where ∑k=1nαk≤1 and ∑q=1mkβq≤1.

Our model will be as follows:

(8)MCESTA(T˜,H˜)=∑K=1nαK×[∑q=1mKβqSkq(T˜,H˜)],

where S_k is a similarity method between T̃ and H̃, Skq is a similarity sub-measure between T̃ and H̃ that should take into account the two criteria of their relative location and size. This measure is calculated to determine a reasonable value of similarity S_k. α_k is an importance weight associated with S_k and β_q is an importance weight associated with Skq. The number n defines the number of similarity methods S_k used to calculate the model. The number m_k defines the number of similarity sub-measures Skq employed to calculate S_k.

3.1 Similarity Measures S_k and Skq

The descriptions of the similarity methods employed, the cosine coefficient and the Jaccard Index [14], are given below. This choice of similarity measures will be validated in Section 4. We have n = 2, S₁ = cos and S₂ = Jaccard and we choose m₁ = 1 and m₂ = 2, which implies, in (9),

MCESTA(T˜,H˜)=∑k=12αk×[∑q=1mkβqSkq(T˜,H˜)].

for each k we provide a corresponding m_k.

MCESTA(T˜,H˜)=α1×[∑q=1m1βqS1q(T˜,H˜)]+α2×[∑q=1m2βqS2q(T˜,H˜)],MCESTA(T˜,H˜)=α1×[β1S11(T˜,H˜)]+α2×[β1S21(T˜,H˜)+β2S22(T˜,H˜)],MCESTA(T˜,H˜)=α1×β1 cos(T˜,H˜)+α2×β1Jaccard1(T˜,H˜)+α2×β2Jaccard2(T˜,H˜),

where S11=cos, S21=Jaccard1 and S22=Jaccard2. To facilitate the calculation of this approach, one needs to make the following changes of variable: C = α₁ × β₁, J₁ = α₂ × β₁ and J₂ = α₂ × β₂. So this equation will become

(9)MCESTA(T˜,H˜)=C×cos(T˜,H˜)+J1×Jaccard1(T˜,H˜)+J2×Jaccard2(T˜,H˜).

3.1.1 The cosine coefficient cos(T̃, H̃)

The cosine coefficient calculates the similarity between vectors in an easy and more intelligent way: it is by the determination of the direction. cos(T̃, H̃) is calculated as follows:

xk=tk,

where [k = {1, 2, 3, 4}], x₅ = w_T, x₆ = t₂ −t₁, x₇ = t₃ −t₂, x₈ = t₄ − t₃. Also, y_k = h_k where [k = {1, 2, 3, 4}], y₅ = w_H, y₆ = h₂ − h₁, y₇ = h₃ − h₂, y₈ = h₄ − h₃.

(10)cos(T˜,H˜)=∑k=18xk×yk(∑k=18xk2)12×(∑k=18yk2)12.

3.1.2 The Jaccard Index

The Jaccard Index, Jaccard(T̃, H̃) [10], measures the similarity as follows: the size of the intersection is divided by the size of the union.

Jaccard₁(T̃, H̃) is calculated as follows: x_k = t_k − m, where [k = {1, 2, 3, 4}], x₅ = w_T̃, x₆ = t₂ −t₁, x₇ = t₃ −t₂, x₈ = t₄ − t₃, and y_k = h_k − m, where [k = {1, 2, 3, 4}], y₅ = w_H̃, y₆ = h₂ − h₁, y₇ = h₃ − h₂, y₈ = h₄ − h₃, where m = min(t₁, h₁).

(11)Jaccard1(T˜,H˜)=∑k=18xk×yk(∑k=18xk2)+(∑k=18yk2)-∑k=18xk×yk.

Jaccard₂(T̃, H̃) is calculated as following: x5-k*=Mk-xk, where [k = {1, 2, 3, 4}], x5*=wT,x6*=t2-t1,x7*=t3-t2,x8*=t4-t3, and y5-k*=Mk-yk, where [k = {1, 2, 3, 4}], y5*=wH,y6*=h2-h1,y7*=h3-h2,y8*=h4-h3, where M = max(t₄, h₄).

(12)Jaccard2(T˜,H˜)=∑k=18xk*×yk*∑k=18(xk*)2+∑k=18(yk*)2-∑i=k8xk*×yk*.

3.2 The Methods to Define the Weights C, J₁ and J₂

Our choice is based on an FIS, such systems are recognized and have been used in many different fields [15, 16]. To calculate the weights mentioned in Eq. 9, we use the Mamdani-type FIS mentioned in [13, 17–19] adapted to our model. Figure 1 shows the Mamdani for MCESTA.

In Table 1 there is presented the FIS fuzzy rule for the MCESTA, which is a set of semantic declarations that define how the FIS-Mamdani must carry out its decision making for input state (Cosine, Jaccard₁, and Jaccard₂) or controlling an output(C, J₁ and J₂). This table uses the values of the functions presented in Figure 2.

In Figure 3, the FIS-diagram for the MCESTA defined as a procedure for developing the association relation (from a given value to an output value) based on the concepts of fuzzy logic. This figure shows the values which are the most important for the weights following C, J₁ and J₂.

Figure 4 to Figure 9 show the evolution of the FIS-Rule surface diagram for the weights (C, J₁ and J₂) in MCESTA.

3.3 Properties

3.3.1 Identical

T˜=H˜⇔MCESTA(T˜,H˜)=1.

3.3.2 Symmetric relation

MCESTA(T˜,H˜)=MCESTA(H˜,T˜).

3.3.3 Scale-free

This property has been used by Hwang and Yang [10] for validating their proposed similarity method. We have that

MCESTA(T˜1,H˜1)=MCESTA(T˜,H˜)

for T̃₁ = α × T̃ = (α×t₁, α×t₂, α×t₃, α×t₄, α×w_T̃) and H̃₁ = α × H̃ = (α × h₁, α × h₂, α × h₃, α × h₄, α × w_H̃), where α ≥ 0. Normally, the best similarity should be scale free. This is a very significant property for the large scale. According to this result, we deduce that a very large data Set₁ = (T̃₁, H̃₁) can be analyzed using another batch of data of small size Set = (T̃, H̃). But according to the scale relation T̃₁ = α × T̃ and H̃₁ = α × H̃, the interpretation and the analysis of Set gives a global idea of the interpretation of Set₁, which is important when it can be difficult to treat everything in the context of big data or the machines don’t have sufficient capacity. This property is very important for large scale studies.

3.4 Proof of Properties

To see that our model verifies the three properties, it is enough that every similarity (Cosine, Jaccard₁ and Jaccard₂) used in this model satisfies these properties. For the sub-measures Jaccard₁ and Jaccard₂, this is proved by Hwang and Yang [10]. It remains to treat Cosine.

3.4.1 MCESTA is identical

T˜=H˜⇔cos(T˜,H˜)=1.

The ⇒ is proved by observing that since T̃ = H̃, then x_k = y_k, k = {1, 2, 3, 4, 6, 7, 8}. Therefore

∑k=18xk×yk(∑k=18xk2)12×(∑k=18yk2)12=∑k=18xk×xk(∑k=18xk2)12×(∑k=18xk2)12=∑k=18xk2∑k=18xk2=1.

Thus,

tildeT=H˜⇒cos(T˜,H˜)=1.
The ⇐ is proved as follows: Since

cos(T˜,H˜)=1⇒∑k=18(xk×yk)(∑k=18xk2)12×(∑k=18yk2)12=1,∑k=18(xk×yk)=(∑k=18xk2)12×(∑k=18yk2)12⇒(∑k=18xk×yk)2=(∑k=18xk2)×(∑k=18yk2),

we have that:

(∑k=18xk2)×(∑k=18yk2)=∑k=18xk2×yk2+(∑k=18xk2)×(∑q=1,q≠k8yq2)=∑k=18xk2×yk2+∑k=18∑q=1,q≠k8xk2×yq2,

(a) (∑k=18xk2)×(∑k=18yk2)=∑k=18xk2×yk2+∑k=18∑q=1,q≠k8xk2×yq2, and we have that:

(b) (∑k=18xk×yk)2=∑k=18xk2×yk2+∑k=18∑q=1,q≠k8(xk×yk)×(xq×yq), the intersection between (a) and (b) is 0:

∑k=18∑q=1,q≠k8xk2×yk2-∑k=18∑q=1,q≠k8(xk×yk)×(xq×yq)=0,(∑k=18∑q=1,q≠k8(xk×yq))×[(xk×yq)-(xq×yk)]=0,

we have:

0≤∑k=18xk×yk(∑k=18xk2)12×(∑k=18yk2)12≤1⇒0≤(∑k=18xk×yk)2(∑k=18xk2)×(∑k=18yk2)≤1⇒(a)-(b)≥0,

so (a)–(b)= 0 ⇒ (x_k × y_q)) × [(x_k × y_q) − (x_q × y_k)] = 0, we have x_k ≠ 0 and y_k ≠ 0. Thus (a)–(b)= 0 ⇒ [(x_k ×y_q)− (x_q × y_k)] = 0. ⇒ ∃λ/∀λ ≥ 0, xkyk=xqyq=λ. particularly λ = 1 ⇒ x_k = y_k.

Therefore, we have that x_k = y_k, [k = 1, 2, 3, 4, 5, 6, 7, 8], T̃= H̃.

3.4.2 MCESTA is a symmetric relation

cos(T˜,H˜)=cos(H˜,T˜).

This follows since ∑k=18xk×yk=∑k=18yk×xk and (∑k=18xk2)12×(∑k=18yk2)12=(∑k=18yk2)12×(∑k=18xk2)12. It simple to show that cos(T̃, H̃) = cos(H̃, T̃).

3.4.3 MCESTA is scale-free

cos(T˜1,H˜1)=cos(T˜,H˜).

Suppose T̃₁ = α × T̃ = (α × t₁, α × t₂, α × t₃, α × t₄, α × w_T̃) and H̃₁ = α × H̃ = (α×h₁, α×h₂, α×h₃, α×h₄, α×w_H̃), where α ≥ 0.

We have cos(T˜1,H˜1)=∑k=18(α×xk)×(α×yk)(∑k=18((α×xk)2)12×(∑k=18((α×yk)2)12=∑k=18xk×yk(∑k=18xk2)12×(∑k=18yk2)12=cos(T˜,H˜).

4. A Comparative Study

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2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
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4.1 Implementation

Our approach to similarity and the existing methods S_C (Eq. 1), S_HC (Eq. 2), S_L (Eq. 3), S_CC (Eq. 4), S_WC (Eq. 5), S_X (Eq. 6), S_SJ (Eq. 7) are implemented employing the Python 3.6 with libraries data science (NumPy, SciPy). All calculations were done with a quad-core processor 3.60 GHz and a 16 GB memory for the JVM. The sets of GTFNs are implemented as associated lists of arrays (Array List).

4.2 Examples and Comparisons

To validate and compare our contributions with the existing methods, we made the calculations on the sets already used by Hwang and Yang [10] in 2014. There are 21 sets (Set₁ to Set₂₁) of fuzzy numbers, shown in Figure 10 to Figure 30 respectively. The estimated times taken by our approach and by the existing methods are given in Table 2. We can demonstrate and analyze the weaknesses and limitations of each of the existing similarity measures from the table.

We can analyze Table 2 in terms of three types: incorrect results, scale-dependent results, and direction.

4.2.1 Incorrect results

We have in Set₁, S_HC(Ã,Ã₁) =1: we can judge the result as incorrect.
For Set₃ and Set₄, we have that the similarity of Set₄ should be more similar than that of Set₃, but the similarity methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁) and S_L(Ã,Ã₁) produce the same result: we can judge the result as incorrect.
We have in Set₅, Ã ≠ Ã₁. The similarity produced by S_C(Ã,Ã₁), S_C(Ã,Ã₁) and S_L(Ã,Ã₁) is 1: we can say the result is incorrect.
We have in Set₆ for S_L(Ã,Ã₁), it can’t evaluate the similarity degree. we can judge the result as incorrect.
For Set₇, the similarity produced by S_L(Ã,Ã₁) is 0: the result is incorrect.
For Set₈ and Set₉, we have that the similarity of Set₄ should be different from the similarity of those with Set₃, but the similarity methods S_C(Ã,Ã₁) and S_C(Ã,Ã₁) produce the same result: we can say the result is incorrect.
For Set₉ and Set₁₀, we have that the similarity of Set₉ should be more similar than that Set₁₀, however, Table 2 proves that the similarity produced by the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁) and S_X(Ã,Ã₁) are incorrect results.
We have in Set₁₁, Ã ≠ Ã₁. The similarity produced by S_HC(Ã,Ã₁) is 1: we can judge the result as incorrect.
We have in Set₁₄, Ã ≠ Ã₁. The similarity produced by S_C(Ã,Ã₁), S_HC(Ã,Ã₁) and S_L(Ã,Ã₁) is 1: the result is incorrect.
For Set₁₄ and Set₁₅, we have that the similarity of Set₁₄ should be more similar than that of Set₁₅. However, Table 2 proves that the similarity produced by the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁), S_X(Ã,Ã₁) and S_SJ(Ã,Ã₁) are incorrect results.
It can be seen for the two different Set₁₆ and Set₁₇ that the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_CC(Ã,Ã₁) and S_X(Ã,Ã₁) produce incorrect results.
For Set₁₈ and Set₁₉, we have that the similarity of Set₁₈ should be more similar than that of Set₁₉, but the similarity methods S_C(Ã, Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã, Ã₁), S_WC(Ã,Ã₁), S_X(Ã, Ã₁) and S_SJ(Ã, Ã₁) produce the same result: we can say the result is incorrect.

4.2.2 Scale-dependent results

For Set₂₀ and Set₂₁, we have that Set₂₀ and Set₂₁ are in double-scale relation. Usually, the best similarity methods must verify property 3 (scale-free), but the similarity methods S_C(Ã, Ã₁), S_HC(Ã, Ã₁), S_L(Ã, Ã₁), S_CC(Ã, Ã₁), S_WC(Ã Ã₁), S_X(Ã, Ã₁) and S_SJ(Ã, Ã₁) are scaledependent (are not scale-free), while our approach is scale-free.

4.2.3 Direction(Cosine) passing to large-scale

The direction similarity is very important. This measurement is used in the big data framework when the data tends towards the infinite. But the similarity methods S_C(Ã,Ã₁), S_HC(AÃ,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁), S_WC(Ã,Ã₁), S_X(Ã,Ã₁)and S_SJ(Ã,Ã₁) do not use the direction, while our approach uses this measurement with a weight C = 0.508, see Figure 3.

After the analysis of Table 2, the three properties and the direction based on the cosine coefficient help this similarity approach to be the best choice for the large scale.

5. Conclusion

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
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Tables
References
Biographies

A novel mathematical model MCESTA for GTFNs has been is presented (as well as the existing methods). This model is based on using weights associated with each one of several differeur nt similarity measures. We have been able to infer the importance weights by a Mamdani-type FIS [17]. This model uses the cosine coefficient and Jaccard Index. Three properties of the model are proved, one property is advantageous for being used with large scale datasets. A comparative study has also been presented to explain how onovel approach can overcome the limitations and weaknesses of the existing methods. This approach will help us develop an intelligent filtering of the pub-sub system [20, 21].

Conflict of Interest

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5. Conclusion
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No potential conflict of interest relevant to this article was reported.

Figures

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Fig. 1.

The Mamdani for MCESTA.

Fig. 2.

FIS value membership functions.

Fig. 3.

FIS-diagram for the MCESTA.

Fig. 4.

FIS-rule surface diagram for C by Cosine and Jaccard1 in MCESTA.

Fig. 5.

FIS-rule surface diagram for C by Cosine and Jaccard2 in MCESTA.

Fig. 6.

FIS-rule surface diagram for J1 by Jaccard1 and Cosine in MCESTA.

Fig. 7.

FIS-rule surface diagram for J1 by Jaccard1 and Jaccard2 in MCESTA.

Fig. 8.

FIS-rule surface diagram for J2 by Jaccard2 and Cosine in MCESTA.

Fig. 9.

FIS-rule surface diagram for J2 by Jaccard2 and Jaccard2 in MCESTA.

Fig. 10.

Set₁.

Fig. 11.

Set₂.

Fig. 12.

Set₃.

Fig. 13.

Set₄.

Fig. 14.

Set₅.

Fig. 15.

Set₆.

Fig. 16.

Set₇.

Fig. 17.

Set₈.

Fig. 18.

Set₉.

Fig. 19.

Set₁₀.

Fig. 20.

Set₁₁.

Fig. 21.

Set₄.

Fig. 22.

Set₁₃.

Fig. 23.

Set₁₄.

Fig. 24.

Set₁₅.

Fig. 25.

Set₁₆.

Fig. 26.

Set₁₇.

Fig. 27.

Set₁₈.

Fig. 28.

Set₁₉.

Fig. 29.

Set₂₀.

Fig. 30.

Set₂₁.

Tables

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. FIS fuzzy rule for MCESTA.

IF (Jaccard₁ is L) and (Jaccard₂ is L) and (Cos is L) THEN (J₁ is L) (J₂ is L) (C is L) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is M) and (Cos is L) THEN (J₁ is L) (J₂ is M) (C is L) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is H) and (Cos is L) THEN (J₁ is L) (J₂ is H) (C is L) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is L) and (Cos is L) THEN (J₁ is M) (J₂ is L) (C is L) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is M) and (Cos is L) THEN (J₁ is M) (J₂ is M) (C is L) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is H) and (Cos is L) THEN (J₁ is M) (J₂ is H) (C is L) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is L) and (Cos is L) THEN J₁ is H) (J₂ is L) (C is L) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is M) and (Cos is L) THEN (J₁ is H) (J₂ is M) (C is L) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is H) and (Cos is L) THEN (J₁ is H) (J₂ is H) (C is L) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is L) and (Cos is M) THEN (J₁ is L) (J₂ is L) (C is M) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is M) and (Cos is M) THEN (J₁ is L) (J₂ is M) (C is M) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is H) and (Cos is M) THEN (J₁ is L) (J₂ is H) (C is M) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is L) and (Cos is M) THEN (J₁ is M) (J₂ is L) (C is M) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is M) and (Cos is M) THEN (J₁ is M) (J₂ is M) (C is M) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is H) and (Cos is M) THEN (J₁ is M) (J₂ is H) (C is M) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is L) and (Cos is M) THEN (J₁ is H) (J₂ is L) (C is M) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is M) and (Cos is M) THEN (J₁ is H) (J₂ is M) (C is M) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is H) and (Cos is M) THEN (J₁ is H) (J₂ is H) (C is M) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is L) and (Cos is H) THEN (J₁ is L) (J₂ is L) (C is H) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is M) and (Cos is H) THEN (J₁ is L) (J₂ is M) (C is H) (1).

IF (Jaccard₁ is L) and (Jaccard₂ is H) and (Cos is H) THEN (J₁ is L) (J₂ is H) (C is H) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is L) and (Cos is H) THEN (J₁ is M) (J₂ is L) (C is H) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is M) and (Cos is H) THEN (J₁ is M) (J₂ is M) (C is H) (1).

IF (Jaccard₁ is M) and (Jaccard₂ is H) and (Cos is H) THEN (J₁ is M) (J₂ is H) (C is H) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is L) and (Cos is H) THEN (J₁ is H) (J₂ is L) (C is H) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is M) and (Cos is H) THEN (J₁ is H) (J₂ is M) (C is H) (1).

IF (Jaccard₁ is H) and (Jaccard₂ is H) and (Cos is H) THEN (J₁ is H) (J₂ is H) (C is H) (1).

L, Low; M, Medium; H, High..

Table. 2.

Table 2. Comparison.

Method/Set_i	S_L	S_HC	S_C	S_CC	S_WC	S_X	S_SJ	Our approach
1	0.9167	1.0	0.975	0.8357	0.95	0.9627	0.8356	0.9877
2	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0
3	0.5	0.7692	0.7	0.42	0.682	0.7136	0.5997	0.8475
4	0.5	0.7692	0.7	0.49	0.7	0.7158	0.7	0.8551
5	1.0	1.0	1.0	0.8	0.8248	0.9652	0.8	0.9797
6	#	1.0	1.0	1.0	1.0	1.0	1.0	1.0
7	0.0	0.9091	0.9	0.9	0.9	0.9053	0.9	0.9725
8	0.5	0.9091	0.9	0.54	0.8411	0.8631	0.5991	0.9482
9	0.6667	0.9091	0.9	0.81	0.9	0.9053	0.9	0.9756
10	0.8333	1.0	0.9	0.9	0.7833	0.95	0.8974	0.9311
11	0.75	1.0	0.9	0.72	0.8003	0.9127	0.72	0.9572
12	0.8	0.9375	0.9	0.78	0.8309	0.8904	0.8959	0.9068
13	0.75	0.9091	0.9	0.81	0.9	0.9053	0.9	0.979
14	1.0	1.0	1.0	0.7	0.7209	0.9553	0.7	0.9484
15	0.75	1.0	0.95	0.9048	0.6215	0.9675	0.9042	0.9187
16	0.4	0.7692	0.7	0.49	0.6222	0.7158	0.6971	0.8125
17	0.25	0.7692	0.7	0.49	0.7	0.7158	0.7	0.828
18	0.5	0.7692	0.7	0.49	0.7	0.7158	0.7	0.8551
19	0.5	0.7692	0.7	0.49	0.7	0.7158	0.7	0.7636
20	0.5	0.7692	0.7	0.49	0.7	0.7158	0.7	0.8551
21	0.81	0.8696	0.85	0.7225	0.85	0.8579	0.85	0.8551

Bold text represents incorrect results and italicized text, scale-dependent results..

References

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Chen, SM (1996). New methods for subjective mental workload assessment and fuzzy risk analysis. Cybernetics and Systems. 27, 449-472. https://doi.org/10.1080/019697296126417
Hsieh, CH, and Chen, SH . Similarity of generalized fuzzy numbers with graded mean integration representation., Proceedings of the 8th International Fuzzy Systems Association World Congress, 1999, Taipei, Taiwan, pp.551-555.
Lee, HS 1999. An optimal aggregation method for fuzzy opinions of group decision., Proceedings of 1999 IEEE International Conference on Systems, Man, and Cybernetics, Tokyo, Japan, Array, pp.314-319. https://doi.org/10.1109/ICSMC.1999.823219
Chen, SJ, and Chen, SM (2003). Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. IEEE Transactions on Fuzzy Systems. 11, 45-56. https://doi.org/10.1109/TFUZZ.2002.806316
Yang, MS, Hung, WL, and ChangChien, SJ (2005). On a similarity measure between LR-type fuzzy numbers and its application to database acquisition. International Journal of Intelligent Systems. 20, 1001-1016. https://doi.org/10.1002/int.20102
Wei, SH, and Chen, SM (2009). A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. Expert Systems with Applications. 36, 589-598. https://doi.org/10.1016/j.eswa.2007.09.033
Xu, Z, Shang, S, Qian, W, and Shu, W (2010). A method for fuzzy risk analysis based on the new similarity of trapezoidal fuzzy numbers. Expert Systems with Applications. 37, 1920-1927. https://doi.org/10.1016/j.eswa.2009.07.015
Chen, SJ (2011). Fuzzy information retrieval based on a new similarity measure of generalized fuzzy numbers. Intelligent Automation & Soft Computing. 17, 465-476. https://doi.org/10.1080/10798587.2011.10643161
Hwang, CM, and Yang, MS (2014). New similarity measures between generalized trapezoidal fuzzy numbers using the Jaccard Index. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 22, 831-844. https://doi.org/10.1142/S0218488514500445
Chen, SH (1998). Operations of fuzzy numbers with step form membership function using function principle. Information Sciences. 108, 149-155. https://doi.org/10.1016/S0020-0255(97)10070-6
Chiao, KP 2016. Ranking interval type 2 fuzzy sets using parametric graded mean integration representation., Proceedings of 2016 International Conference on Machine Learning and Cybernetics, Jeju, Korea, Array, pp.606-611. https://doi.org/10.1109/ICMLC.2016.7872956
Gupta, Y, Saini, A, and Saxena, AK (2014). Fuzzy logic-based approach to develop hybrid similarity measure for efficient information retrieval. Journal of Information Science. 40, 846-857. https://doi.org/10.1177/0165551514548989
Real, R, and Vargas, JM (1996). The probabilistic basis of Jaccards index of similarity. Systematic Biology. 45, 380-385. https://doi.org/10.1093/sysbio/45.3.380
Samiri, MY, Najib, M, Elfazziki, A, Abourraja, MN, Boudebous, D, and Bouain, A (2017). APRICOIN: an adaptive approach for prioritizing high-risk containers inspections. IEEE Access. 5, 18238-18249. https://doi.org/10.1109/ACCESS.2017.2746838
Tsai, JT, Chou, JH, and Lin, CF (2015). Designing microstructure parameters for backlight modules by using improved adaptive neuro-fuzzy inference system. IEEE Access. 3, 2626-2636. https://doi.org/10.1109/ACCESS.2015.2508144
Mamdani, EH, and Assilian, S (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies. 7, 1-13. https://doi.org/10.1016/S0020-7373(75)80002-2
Gupta, Y, Saini, A, and Saxena, AK (2015). A new fuzzy logic based ranking function for efficient information retrieval system. Expert Systems with Applications. 42, 1223-1234. https://doi.org/10.1016/j.eswa.2014.09.009
Lee, E, Choi, C, and Kim, P (2017). Intelligent handover scheme for drone using fuzzy inference systems. IEEE Access. 5, 13712-13719. https://doi.org/10.1109/ACCESS.2017.2724067
Tourad, MC, and Abdali, A 2017. Toward efficient ranked-key algorithm for the web notification of big data systems., Proceedings of the 2nd International Conference on Big Data, Cloud and Applications, Tetouan, Morocco, Array. https://doi.org/10.1145/3090354.3090386
Tourad, MC, Abdali, A, and Outfarouin, A 2016. On a new index of publish/subscribe system in the context of big data., Proceedings of 2016 IEEE/ACS 13th International Conference of Computer Systems and Applications, Agadir, Morocco, Array, pp.1-2. https://doi.org/10.1109/AICCSA.2016.7945756

Biographies

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Abstract
1. Introduction
2. Related Work
3. Approach
4. A Comparative Study
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Mohamedou Cheikh Tourad received the degree of specialized Engineer from the Faculty of Sciences and Technology, Marrakesh, Morocco, in 2014, where he is currently pursuing a Ph.D. degree with the Computer Science Department of the Faculty of Sciences and Technology. His research interests are in Intelligent Publish/Subscribe System, Big Data and Fuzzy logic, Data science, Machine learning, Deep learning.

E-mail: cheikhtouradmohamedou@gmail.com

Abdelmoumaim Abdali received a Ph.D. in Solid Mechanics and Structures from the University of Amiens, France, in 1996. He is a professor of Computer Science at the University Cadi Ayyad, Faculty of Sciences and Technology, Marrakech, Morocco, and a member of the Laboratory of Applied Mathematics and Computer Science (LAMAI) Marrakesh, Morocco. His research interests are: Publish/Subscribe System, software engineering, Big Data, Fuzzy logic, computer science, DTN network, business intelligence, design and implementation in data warehouse, ubiquitous systems, modeling & design, metamodel design, model transformation, model verification & validation methods, numerical simulation, smart cities, biomechanics, and bone remodeling.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2018; 18(4): 303-315

Published online December 31, 2018 https://doi.org/10.5391/IJFIS.2018.18.4.303

An Intelligent Similarity Model between Generalized Trapezoidal Fuzzy Numbers in Large Scale

Mohamedou Cheikh Tourad, and Abdelmounaim Abdali

Applied Mathematics and Computer Science Laboratory, Cadi Ayyad University, Marrakech, Morocco

Correspondence to: Mohamedou Cheikh Tourad, (cheikhtouradmohamedou@gmail.com)

Received: November 13, 2018; Revised: December 19, 2018; Accepted: December 24, 2018

Abstract

Keywords: Cosine coefficient, Jaccard index, GTFNs, FIS, Similarity, Large scale

1. Introduction

2. Related Work

Several Similarity methods between GTFNs T̃ and H̃ = {h₁, h₂, h₃, h₄, w _H̃ } have been introduced:

(1)

S_{C} (\tilde{T}, \tilde{H}) = 1 - (\sum_{k = 1}^{4} \frac{t_{k} - h_{k}}{4}) .

Hsieh and Chen [3] introduced a new approach to find a solution to the similarity problems of the GFNs. This approach is based on the average integration representation distance.

(2)

S_{H C} (\tilde{T}, \tilde{H}) = \frac{1}{1 + d (T, H)},

where d(T̃, H̃) = |P(T) − P(H)|, $P (\tilde{T}) = \frac{t_{1} + 2 t_{2} + 2 t_{3} + t_{4}}{6}, P (\tilde{H}) = \frac{h_{1} + 2 h_{2} + 2 h_{3} + h_{4}}{6}$ .

(3)

S_{L} (\tilde{T}, \tilde{H}) = 1 - \frac{{‖ \tilde{T} - \tilde{H} ‖}_{L_{P}}}{‖ U ‖} \times 4^{- \frac{1}{P}},

where ${‖ \tilde{T} - \tilde{H} ‖}_{L_{P}} = {(\sum_{k = 1}^{4} {(∣ t_{k} - h_{k} ∣)}^{P})}^{\frac{1}{p}}$ , taking into account a positive integer P. ||U|| = max U − min U where U indicates the universe of discourse. Chen and Chen [5] defined a new proposed method to clear the similarity problems of GFNs. It is based on the simple center of gravity method (SCGM). The authors used this proposed to solve the problems of fuzzy risk analysis.

(4)

S_{C C} (\tilde{T}, \tilde{H}) = 1 - (\sum_{k = 1}^{4} \frac{t_{k} - h_{k}}{4}) \times {(1 - ∣ X_{\tilde{T}}^{*} - X_{\tilde{H}}^{*} ∣)}^{B (S_{\tilde{T}}, S_{\tilde{H}})} \times \frac{min (y_{\tilde{T}}^{*}, y_{\tilde{H}}^{*})}{max (y_{\tilde{T}}^{*}, y_{\tilde{H}}^{*})},

where

\begin{array}{l} y_{\tilde{T}}^{*} = {\begin{array}{l} \frac{w_{\tilde{T}}}{6} \times (2 + \frac{t_{3} - t_{2}}{t_{4} - t_{1}}) & if t_{4} \neq t_{1}, \\ \frac{w_{\tilde{T}}}{2} & if t_{4} - t_{1}, \end{array} \\ y_{\tilde{H}}^{*} = {\begin{array}{l} \frac{w_{\tilde{H}}}{6} \times (2 + \frac{h_{3} - h_{2}}{h_{4} - h_{1}}) & if h_{4} \neq h_{1}, \\ \frac{w_{\tilde{H}}}{2} & if h_{4} - h_{1}, \end{array} \\ x_{\tilde{T}}^{*} = {\begin{array}{l} \frac{(y_{\tilde{T}}^{*} \times (t_{3} + t_{2})) + (t_{4} + t_{1}) \times (w_{\tilde{T}} - y_{\tilde{T}}^{*})}{2 w_{\tilde{T}}} & if w_{\tilde{T}} \neq 0 \\ \frac{t_{4} + t_{1}}{2} & if w_{\tilde{T}} = 0, \end{array} \\ x_{\tilde{H}}^{*} = {\begin{array}{l} \frac{(y_{\tilde{H}}^{*} \times (h_{3} + h_{2})) + (h_{4} + h_{1}) \times (w_{\tilde{H}} - y_{\tilde{H}}^{*})}{2 w_{\tilde{H}}} & if w_{\tilde{H}} \neq 0 \\ \frac{h_{4} + h_{1}}{2} & if w_{\tilde{H}} = 0, \end{array} \\ B (S_{\tilde{T}}, S_{\tilde{H}}) = {\begin{array}{l} 1 & if S_{\tilde{T}} + S_{\tilde{H}} > 0, \\ 0 & if S_{\tilde{T}} + S_{\tilde{H}} = 0, \end{array} \end{array}

(5)

S_{W C} (\tilde{T}, \tilde{H}) = (1 - (\sum_{k = 1}^{4} \frac{t_{k} - h_{k}}{4})) \times (\frac{min (P (\tilde{T}), P (\tilde{H})) + min (w_{\tilde{T}}, w_{\tilde{H}})}{max (P (\tilde{T}), P (\tilde{H})) + max (w_{\tilde{T}}, w_{\tilde{H}})}),

where

\begin{array}{l} P (\tilde{T}) = (\sqrt{{(t_{1} - t_{2})}^{2} + {(w_{\tilde{T}})}^{2}}) + (t_{3} - t_{2}) + (\sqrt{{(t_{4} - t_{3})}^{2} + {(w_{\tilde{T}})}^{2}})) + (t_{4} - t_{1}), \\ P (\tilde{H}) = (\sqrt{{(h_{1} - h_{2})}^{2} + {(w_{\tilde{H}})}^{2}}) + (h_{3} - h_{2}) + (\sqrt{{(h_{4} - h_{3})}^{2} + {(w_{\tilde{H}})}^{2}})) + (h_{4} - h_{1}) . \end{array}

(6)

S_{X} (\tilde{T}, \tilde{H}) = 1 - (\sum_{k = 1}^{4} \frac{t_{k} - h_{k}}{8}) - (\frac{d (\tilde{T}, \tilde{H})}{2}),

where $d (\tilde{T}, \tilde{H}) = \frac{(\sqrt{{(x_{\tilde{T}}^{*} - x_{\tilde{H}}^{*})}^{2} + {(y_{\tilde{T}}^{*} - y_{\tilde{H}}^{*})}^{2}})}{\sqrt{1.25}}$ .

(7)

S_{S J} (\tilde{T}, \tilde{H}) = [{(\prod_{k = 1}^{4} (2 - ∣ t_{k} - h_{k} ∣))}^{\frac{1}{4}} - 1] \times \frac{min (y_{\tilde{T}}^{*}, y_{\tilde{H}}^{*})}{max (y_{\tilde{T}}^{*}, y_{\tilde{H}}^{*})},

where $y_{\tilde{T}}^{*}$ and $y_{\tilde{H}}^{*}$ are calculated as in Chen and Chen [5].

3. Approach

We will now present a novel mathematical model, MCESTA (Mohamedou Cheikh Elghotob Cheikh Saad bouh Cheikh Tourad Abass), Abdelmounaim Abdali, a large scale similarity method between GTFNs, call them T̃ and H̃. It is a kind of hybrid of the similarity measures S_k and $S_{k}^{q}$ , where each measure is associated with an importance weight (α_k and β_q). MCESTA is an efficient development of the formula $FBHSM = \sum_{k = 1}^{n} α_{k} \times S_{k}$ proposed by Gupta et al. [13] to calculate similarity in IR (information retrieval). MCESTA employs a fuzzy inference system (FIS) to find the value of the effective weighting. This hybrid similarity measure is calculated as follows:

M C E S T A (\tilde{T}, \tilde{H}) = {\begin{array}{l} \sum_{k = 1}^{n} α_{k} \times S_{k} (\tilde{T}, \tilde{H}), \\ S_{k} (\tilde{T}, \tilde{H}) = \sum_{q = 1}^{m_{k}} β_{q} \times S_{k}^{q} (\tilde{T}, \tilde{H}), \end{array}

where $\sum_{k = 1}^{n} α_{k} \leq 1$ and $\sum_{q = 1}^{m_{k}} β_{q} \leq 1$ .

Our model will be as follows:

(8)

M C E S T A (\tilde{T}, \tilde{H}) = \sum_{K = 1}^{n} α_{K} \times [\sum_{q = 1}^{m_{K}} β_{q} S_{k}^{q} (\tilde{T}, \tilde{H})],

where S_k is a similarity method between T̃ and H̃, $S_{k}^{q}$ is a similarity sub-measure between T̃ and H̃ that should take into account the two criteria of their relative location and size. This measure is calculated to determine a reasonable value of similarity S_k. α_k is an importance weight associated with S_k and β_q is an importance weight associated with $S_{k}^{q}$ . The number n defines the number of similarity methods S_k used to calculate the model. The number m_k defines the number of similarity sub-measures $S_{k}^{q}$ employed to calculate S_k.

3.1 Similarity Measures S_k and $S_{k}^{q}$

M C E S T A (\tilde{T}, \tilde{H}) = \sum_{k = 1}^{2} α_{k} \times [\sum_{q = 1}^{m_{k}} β_{q} S_{k}^{q} (\tilde{T}, \tilde{H})] .

for each k we provide a corresponding m_k.

\begin{array}{l} M C E S T A (\tilde{T}, \tilde{H}) = α_{1} \times [\sum_{q = 1}^{m_{1}} β_{q} S_{1}^{q} (\tilde{T}, \tilde{H})] + α_{2} \times [\sum_{q = 1}^{m_{2}} β_{q} S_{2}^{q} (\tilde{T}, \tilde{H})], \\ M C E S T A (\tilde{T}, \tilde{H}) = α_{1} \times [β_{1} S_{1}^{1} (\tilde{T}, \tilde{H})] + α_{2} \times [β_{1} S_{2}^{1} (\tilde{T}, \tilde{H}) + β_{2} S_{2}^{2} (\tilde{T}, \tilde{H})], \\ M C E S T A (\tilde{T}, \tilde{H}) = α_{1} \times β_{1} cos (\tilde{T}, \tilde{H}) + α_{2} \times β_{1} J a c c a r d_{1} (\tilde{T}, \tilde{H}) + α_{2} \times β_{2} J a c c a r d_{2} (\tilde{T}, \tilde{H}), \end{array}

where $S_{1}^{1} = cos, S_{2}^{1} = J a c c a r d_{1}$ and $S_{2}^{2} = J a c c a r d_{2}$ . To facilitate the calculation of this approach, one needs to make the following changes of variable: C = α₁ × β₁, J₁ = α₂ × β₁ and J₂ = α₂ × β₂. So this equation will become

(9)

M C E S T A (\tilde{T}, \tilde{H}) = C \times cos (\tilde{T}, \tilde{H}) + J_{1} \times J a c c a r d_{1} (\tilde{T}, \tilde{H}) + J_{2} \times J a c c a r d_{2} (\tilde{T}, \tilde{H}) .

3.1.1 The cosine coefficient cos(T̃, H̃)

The cosine coefficient calculates the similarity between vectors in an easy and more intelligent way: it is by the determination of the direction. cos(T̃, H̃) is calculated as follows:

x_{k} = t_{k},

(10)

cos (\tilde{T}, \tilde{H}) = \frac{\sum_{k = 1}^{8} x_{k} \times y_{k}}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}}} .

3.1.2 The Jaccard Index

The Jaccard Index, Jaccard(T̃, H̃) [10], measures the similarity as follows: the size of the intersection is divided by the size of the union.

(11)

J a c c a r d_{1} (\tilde{T}, \tilde{H}) = \frac{\sum_{k = 1}^{8} x_{k} \times y_{k}}{(\sum_{k = 1}^{8} x_{k}^{2}) + (\sum_{k = 1}^{8} y_{k}^{2}) - \sum_{k = 1}^{8} x_{k} \times y_{k}} .

Jaccard₂(T̃, H̃) is calculated as following: $x_{5 - k}^{*} = M_{k} - x_{k}$ , where [k = {1, 2, 3, 4}], $x_{5}^{*} = w_{T}, x_{6}^{*} = t_{2} - t_{1}, x_{7}^{*} = t_{3} - t_{2}, x_{8}^{*} = t_{4} - t_{3}$ , and $y_{5 - k}^{*} = M_{k} - y_{k}$ , where [k = {1, 2, 3, 4}], $y_{5}^{*} = w_{H}, y_{6}^{*} = h_{2} - h_{1}, y_{7}^{*} = h_{3} - h_{2}, y_{8}^{*} = h_{4} - h_{3}$ , where M = max(t₄, h₄).

(12)

J a c c a r d_{2} (\tilde{T}, \tilde{H}) = \frac{\sum_{k = 1}^{8} x_{k}^{*} \times y_{k}^{*}}{\sum_{k = 1}^{8} {(x_{k}^{*})}^{2} + \sum_{k = 1}^{8} {(y_{k}^{*})}^{2} - \sum_{i = k}^{8} x_{k}^{*} \times y_{k}^{*}} .

3.2 The Methods to Define the Weights C, J₁ and J₂

Figure 4 to Figure 9 show the evolution of the FIS-Rule surface diagram for the weights (C, J₁ and J₂) in MCESTA.

3.3 Properties

3.3.1 Identical

\tilde{T} = \tilde{H} \Leftrightarrow M C E S T A (\tilde{T}, \tilde{H}) = 1.

3.3.2 Symmetric relation

M C E S T A (\tilde{T}, \tilde{H}) = M C E S T A (\tilde{H}, \tilde{T}) .

3.3.3 Scale-free

This property has been used by Hwang and Yang [10] for validating their proposed similarity method. We have that

M C E S T A ({\tilde{T}}_{1}, {\tilde{H}}_{1}) = M C E S T A (\tilde{T}, \tilde{H})

3.4 Proof of Properties

3.4.1 MCESTA is identical

\tilde{T} = \tilde{H} \Leftrightarrow cos (\tilde{T}, \tilde{H}) = 1.

The ⇒ is proved by observing that since T̃ = H̃, then x_k = y_k, k = {1, 2, 3, 4, 6, 7, 8}. Therefore

$\begin{array}{l} \frac{\sum_{k = 1}^{8} x_{k} \times y_{k}}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}}} = \frac{\sum_{k = 1}^{8} x_{k} \times x_{k}}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}}} \\ = \frac{\sum_{k = 1}^{8} x_{k}^{2}}{\sum_{k = 1}^{8} x_{k}^{2}} = 1. \end{array}$

Thus,

$t i l d e T = \tilde{H} \Rightarrow cos (\tilde{T}, \tilde{H}) = 1.$
The ⇐ is proved as follows: Since

$\begin{array}{l} cos (\tilde{T}, \tilde{H}) = 1 \Rightarrow \frac{\sum_{k = 1}^{8} (x_{k} \times y_{k})}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}}} = 1, \\ \sum_{k = 1}^{8} (x_{k} \times y_{k}) = {(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}} \\ \Rightarrow {(\sum_{k = 1}^{8} x_{k} \times y_{k})}^{2} = (\sum_{k = 1}^{8} x_{k}^{2}) \times (\sum_{k = 1}^{8} y_{k}^{2}), \end{array}$

we have that:

\begin{array}{l} (\sum_{k = 1}^{8} x_{k}^{2}) \times (\sum_{k = 1}^{8} y_{k}^{2}) = \sum_{k = 1}^{8} x_{k}^{2} \times y_{k}^{2} + (\sum_{k = 1}^{8} x_{k}^{2}) \times (\sum_{q = 1, q \neq k}^{8} y_{q}^{2}) \\ = \sum_{k = 1}^{8} x_{k}^{2} \times y_{k}^{2} + \sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} x_{k}^{2} \times y_{q}^{2}, \end{array}

(a) $(\sum_{k = 1}^{8} x_{k}^{2}) \times (\sum_{k = 1}^{8} y_{k}^{2}) = \sum_{k = 1}^{8} x_{k}^{2} \times y_{k}^{2} + \sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} x_{k}^{2} \times y_{q}^{2}$ , and we have that:

(b) ${(\sum_{k = 1}^{8} x_{k} \times y_{k})}^{2} = \sum_{k = 1}^{8} x_{k}^{2} \times y_{k}^{2} + \sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} (x_{k} \times y_{k}) \times (x_{q} \times y_{q})$ , the intersection between (a) and (b) is 0:

\begin{array}{l} \sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} x_{k}^{2} \times y_{k}^{2} - \sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} (x_{k} \times y_{k}) \times (x_{q} \times y_{q}) = 0, \\ (\sum_{k = 1}^{8} \sum_{q = 1, q \neq k}^{8} (x_{k} \times y_{q})) \times [(x_{k} \times y_{q}) - (x_{q} \times y_{k})] = 0, \end{array}

we have:

\begin{array}{l} 0 \leq \frac{\sum_{k = 1}^{8} x_{k} \times y_{k}}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}}} \leq 1 \\ \Rightarrow 0 \leq \frac{{(\sum_{k = 1}^{8} x_{k} \times y_{k})}^{2}}{(\sum_{k = 1}^{8} x_{k}^{2}) \times (\sum_{k = 1}^{8} y_{k}^{2})} \leq 1 \\ \Rightarrow (a) - (b) \geq 0, \end{array}

so (a)–(b)= 0 ⇒ (x_k × y_q)) × [(x_k × y_q) − (x_q × y_k)] = 0, we have x_k ≠ 0 and y_k ≠ 0. Thus (a)–(b)= 0 ⇒ [(x_k ×y_q)− (x_q × y_k)] = 0. ⇒ ∃λ/∀λ ≥ 0, $\frac{x_{k}}{y_{k}} = \frac{x_{q}}{y_{q}} = λ$ . particularly λ = 1 ⇒ x_k = y_k.

Therefore, we have that x_k = y_k, [k = 1, 2, 3, 4, 5, 6, 7, 8], T̃= H̃.

3.4.2 MCESTA is a symmetric relation

cos (\tilde{T}, \tilde{H}) = cos (\tilde{H}, \tilde{T}) .

This follows since $\sum_{k = 1}^{8} x_{k} \times y_{k} = \sum_{k = 1}^{8} y_{k} \times x_{k}$ and ${(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}} = {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}}$ . It simple to show that cos(T̃, H̃) = cos(H̃, T̃).

3.4.3 MCESTA is scale-free

cos ({\tilde{T}}_{1}, {\tilde{H}}_{1}) = cos (\tilde{T}, \tilde{H}) .

Suppose T̃₁ = α × T̃ = (α × t₁, α × t₂, α × t₃, α × t₄, α × w_T̃) and H̃₁ = α × H̃ = (α×h₁, α×h₂, α×h₃, α×h₄, α×w_H̃), where α ≥ 0.

We have $cos ({\tilde{T}}_{1}, {\tilde{H}}_{1}) = \frac{\sum_{k = 1}^{8} (α \times x_{k}) \times (α \times y_{k})}{(\sum_{k = 1}^{8} {({(α \times x_{k})}^{2})}^{\frac{1}{2}} \times (\sum_{k = 1}^{8} {({(α \times y_{k})}^{2})}^{\frac{1}{2}}} = \frac{\sum_{k = 1}^{8} x_{k} \times y_{k}}{{(\sum_{k = 1}^{8} x_{k}^{2})}^{\frac{1}{2}} \times {(\sum_{k = 1}^{8} y_{k}^{2})}^{\frac{1}{2}}} = cos (\tilde{T}, \tilde{H})$ .

4. A Comparative Study

4.1 Implementation

4.2 Examples and Comparisons

We can analyze Table 2 in terms of three types: incorrect results, scale-dependent results, and direction.

4.2.1 Incorrect results

We have in Set₁, S_HC(Ã,Ã₁) =1: we can judge the result as incorrect.
For Set₃ and Set₄, we have that the similarity of Set₄ should be more similar than that of Set₃, but the similarity methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁) and S_L(Ã,Ã₁) produce the same result: we can judge the result as incorrect.
We have in Set₅, Ã ≠ Ã₁. The similarity produced by S_C(Ã,Ã₁), S_C(Ã,Ã₁) and S_L(Ã,Ã₁) is 1: we can say the result is incorrect.
We have in Set₆ for S_L(Ã,Ã₁), it can’t evaluate the similarity degree. we can judge the result as incorrect.
For Set₇, the similarity produced by S_L(Ã,Ã₁) is 0: the result is incorrect.
For Set₈ and Set₉, we have that the similarity of Set₄ should be different from the similarity of those with Set₃, but the similarity methods S_C(Ã,Ã₁) and S_C(Ã,Ã₁) produce the same result: we can say the result is incorrect.
For Set₉ and Set₁₀, we have that the similarity of Set₉ should be more similar than that Set₁₀, however, Table 2 proves that the similarity produced by the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁) and S_X(Ã,Ã₁) are incorrect results.
We have in Set₁₁, Ã ≠ Ã₁. The similarity produced by S_HC(Ã,Ã₁) is 1: we can judge the result as incorrect.
We have in Set₁₄, Ã ≠ Ã₁. The similarity produced by S_C(Ã,Ã₁), S_HC(Ã,Ã₁) and S_L(Ã,Ã₁) is 1: the result is incorrect.
For Set₁₄ and Set₁₅, we have that the similarity of Set₁₄ should be more similar than that of Set₁₅. However, Table 2 proves that the similarity produced by the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁), S_X(Ã,Ã₁) and S_SJ(Ã,Ã₁) are incorrect results.
It can be seen for the two different Set₁₆ and Set₁₇ that the methods S_C(Ã,Ã₁), S_HC(Ã,Ã₁), S_CC(Ã,Ã₁) and S_X(Ã,Ã₁) produce incorrect results.
For Set₁₈ and Set₁₉, we have that the similarity of Set₁₈ should be more similar than that of Set₁₉, but the similarity methods S_C(Ã, Ã₁), S_HC(Ã,Ã₁), S_L(Ã,Ã₁), S_CC(Ã, Ã₁), S_WC(Ã,Ã₁), S_X(Ã, Ã₁) and S_SJ(Ã, Ã₁) produce the same result: we can say the result is incorrect.

4.2.2 Scale-dependent results

For Set₂₀ and Set₂₁, we have that Set₂₀ and Set₂₁ are in double-scale relation. Usually, the best similarity methods must verify property 3 (scale-free), but the similarity methods S_C(Ã, Ã₁), S_HC(Ã, Ã₁), S_L(Ã, Ã₁), S_CC(Ã, Ã₁), S_WC(Ã Ã₁), S_X(Ã, Ã₁) and S_SJ(Ã, Ã₁) are scaledependent (are not scale-free), while our approach is scale-free.

4.2.3 Direction(Cosine) passing to large-scale

The direction similarity is very important. This measurement is used in the big data framework when the data tends towards the infinite. But the similarity methods S_C(Ã,Ã₁), S_HC(AÃ,Ã₁), S_L(Ã,Ã₁), S_CC(Ã,Ã₁), S_WC(Ã,Ã₁), S_X(Ã,Ã₁)and S_SJ(Ã,Ã₁) do not use the direction, while our approach uses this measurement with a weight C = 0.508, see Figure 3.

After the analysis of Table 2, the three properties and the direction based on the cosine coefficient help this similarity approach to be the best choice for the large scale.

Fig 1.

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