Intuitionistic Fuzzy Similarity Measures and Their Role in Classification

Leila Baccour; Adel M. Alimi; Robert I. John

doi:10.1515/jisys-2015-0086

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Intuitionistic Fuzzy Similarity Measures and Their Role in Classification

Leila Baccour , Adel M. Alimi and Robert I. John

Published/Copyright: October 24, 2015

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From the journal Journal of Intelligent Systems Volume 25 Issue 2

Abstract

We present some similarity and distance measures between intuitionistic fuzzy sets (IFSs). Thus, we propose two semi-metric distance measures between IFSs. The measures are applied to classification of shapes and handwritten Arabic sentences described with intuitionistic fuzzy information. The experimental results permitted to do a comparative analysis between intuitionistic fuzzy similarity and distance measures, which can facilitate the selection of such measure in similar applications.

1 Introduction

Similarity and distance measures between IFSs [1, 2] have been proposed and discussed using different approaches. Some of the proposed measures are generalizations of crisp or fuzzy distance measures. Atanassov [2] and Szmidt and Kacprzyk [39] generalized crisp Hamming and Euclidean distance measures to IFSs, and Hung and Yang [24, 25] generalized Hausdorff distance and L_p metric to propose similarity measures between IFSs. Hung and Yang [26] generalized similarity measures between type-1 fuzzy sets to propose intuitionistic fuzzy similarity measures (IFSMs).

The existing intuitionistic fuzzy similarity measures (IFSMs) in literature are examined and compared [8, 29], and many measures are proposed to resolve deficiencies of existing measures. For instance, Hong and Kim [22] proposed IFSMs to resolve deficiencies found in the measures presented in Refs. [14, 15], but these IFSMs have some counter-intuitive cases resolved by Fan and Zhangian [20] who proposed another IFSM. Szmidt and Kacprzyk [40] proposed IFSMs using the complementary of intuitionistic fuzzy sets to measure the dissimilarity between IFSs in addition to their similarity. More details about IFSMs are presented in Ref. [11].

In general, the existing IFSMs are demonstrated to validate properties. However, some of them do not validate one or more properties, and others are shown to have some counter-intuitive cases. A comparative analysis by Li et al. [29] shows the problems found in some IFSMs from literature, and the relationship between some IFSMs are done in Ref. [10].

In crisp logic, a distance measure, which does not validate triangular inequality, is considered as semi-metric distance measure (see Appendices). Some measures between intuitionistic fuzzy sets, in literature, do not validate the fourth property of similarity or distance measure. Could these measures be used in operations of classification? Is the satisfaction of the fourth property mandatory to obtain good results in classification? Through analogy to crisp distance measures, we propose semi-metric distance measures, and we apply them to classification to answer the precedent questions. Intuitionistic fuzzy similarity measures are crucial operators for comparison between two patterns and are used in many systems such as recognition systems, classification systems, etc.; as examples, applications include handwritten Arabic sentences recognition [5, 9], logic programming [3, 4], decision-making problems [38], and medical diagnosis [27, 37]. Despite the numerous works on IFSMs and the different propositions of similarity and distance measures, most of them did numerical applications on some artificial samples and did not use a large data set to show the validation of IFSMs for classifying patterns or another domain in which similarity can influence results. Hung and Yang [26] applied some IFSMs to students’ evaluation, but the used data set was not large. A comparative analysis between IFSMS in practical and in real applications is needed. This is the objective of the paper in which we apply IFSMs to two applications of classifications using large data sets.

In Section 2, we present the definition of IFSs followed by distance and similarity measures from the literature. In Section 4, we propose two semi-metric distance measures, and we apply them to some numerical examples. Therefore, we apply them and the measures from literature to the classification of shapes and recognition of handwritten Arabic sentences. The set of sentences is extracted from the IFN/ENIT data set (set A) [36], and the shapes represent the SQUID data set. The extraction process of intuitionistic fuzzy features of the two data sets is described in Sections 5 and 6, respectively. The results of the classification obtained with all measures are compared in Sections 5 and 6.

2 Intuitionistic Fuzzy Sets

An intuitionistic fuzzy set A (IFS) in a universe of discourse X is defined by Ref. [1] as:

A={〈x, μ_A(x), ν_A(x)〉|x∈X}, where μ_A(x):X→[0,1] and ν_A(x):X→[0,1] with the condition 0≤μ_A(x)+ν_A(x) ≤1 ∀x∈X. IFSs(X) is the set of all IFSs in X.

The numbers μ_A(x), ν_A(x) denote, respectively, the degrees of membership and of non-membership of the element x∈X to A. Also, μ_A and ν_A are called, respectively, truth membership function and false membership function.

In addition to membership and non-membership functions, a function of hesitancy or uncertainty of x to A denoted by π_A(x) must be taken into consideration. π_A(x) is computed as:

πA(x) = 1 − μA(x) − νA(x) with 0 ≤ πA(x) ≤ 1

3 Similarity, Distance, and Semi-metric Measures Between IFSs

An intuitionistic fuzzy similarity measure S(A, B) serves to match between two IFSs A and B of IFSs(X) and should satisfy the following properties

(P1) 0 ≤ S(A, B) ≤ 1(P2) S(A, B) = 1 if and only if A = B(P3) S(A, B) = S(B, A)(P4) S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C) if A ⊆ B ⊆ C. C ∈ IFSs(X)

Note that similarity and distance (dissimilarity) measures are complementary; when the first increases, the second decreases. Normalized distance measure and similarity measure are dual concepts. So, S(A, B)=1−d(A, B) and vice versa. The properties of distance measures below are complementary to those of similarity measures:

(DP1) 0 ≤ d(A, B) ≤ 1(DP2) d(A, B) = 0 if and only if A = B(DP3) d(A, B) = d(B, A)(DP4) If A ⊆ B ⊆ C, A, B, C ∈ IFSs(X), then d(A, C) ≥ d(A, B) and d(A, C) ≥ d(B, C)

In crisp logic, distance measures are classified to distance, index of distance, index of dissimilarity, semi-metric distance, or distance ultra-metric (see Appendices) according to their validations of the whole or some of properties. For IFSs, with reference and analogy to crisp distance, measures that do not validate the property DP4 of distance measures can also be considered if they show correct classification, and thus, they will be called semi-metric distance measures.

Notation: The following notations are used to simplify the formulas of distance and similarity measures:

μAi = μA(xi) and μBi = μB(xi)νAi = νA(xi) and νBi = νB(xi)Δμi = μAi − μBi, Δνi = νAi − νBi and Δπi = πAi − πBi,

In the sequel, we present existing similarity and distance measures in Tables 1 and 2 between tow IFSs A and B of n elements.

Table 1

Similarity Measures Between IFSs.

Authors	IFSMs
Chen [14, 15]	SC(A, B) = 1 − ∑i = 1n\|SA(xi) − SB(xi)\|2n	(1)
	where S_A(x_i)=μ_A(x_i)−ν_A(x_i) and S_B(x_i)=μ_B(x_i)−ν_B(x_i).
Hong and Kim [22]	SH(A, B) = 1 − ∑i = 1n\|Δμi\| + \|Δνi\|2n.	(2)
Fan and Zhangian [20]	SL(A, B) = 1 − ∑i = 1n\|SA(xi) − SB(xi)\|4n − ∑i = 1n\|Δμi\| + \|Δνi\|4n	(3)
Dengfeng and Chuntian [19]	Sdp(A, B) = 1 − 1np∑i = 1n\|φA(xi) − φB(xi)\|pp	(4)
	φA(xi) = μA(xi) + 1 − νA(xi)2 and φB(xi) = μB(xi) + 1 − νB(xi)2 with 1≤p<+∞
Mitchell [32]	Smod(A, B) = 12(φμ(A, B) + φΦ(A, B))	(5)
	where φ_μ(A, B) and φ_Φ(A, B) denote, respectively, the similarity measures between the low membership functions μ_A and μ_B and the high membership functions Φ_A=1−ν_A and Φ_B=1−ν_B as follows:
	φμ(A, B) = Sdp(μA(xi), μB(xi)) = 1 − 1np∑i = 1n\|Δμi\|pp
	φΦ(A, B) = Sdp(ΦA(xi), ΦB(xi)) = 1 − 1np∑i = 1n\|Δνi\|pp
Liang and Shi [28]	Sep(A, B) = 1 − 1np∑i = 1n(φμAB(xi) + φνAB(xi))pp	(6)
	where φμAB(xi) = \|Δμi\|2 and φνAB(xi) = \|1 − νA(xi)2 − 1 − νB(xi)2\|
	Ssp(A, B) = 1 − 1np∑i = 1n(φs1(xi) + φs2(xi))pp	(7)
	with φs1(xi) = \|mA1(xi) − mB1(xi)\|2 and φs2(xi) = \|mA2(xi) − mB2(xi)\|2
	where A₁ and A₂ are the sub-intervals from A divided by the median value denoted mA(xi) = μA(xi) + 1 − νA(xi)2 of the interval [μ_A(x_i), 1−ν_A(x_i)]. The median values of A₁ and A₂ are mA1(xi) = mA(xi) + μA(xi)2 and mA2(xi) = mA(xi) + 1 − νA(xi)2. The median values mB1(xi) and mB2(xi) of B₁ and B₂ are computed similarly.
	Shp(A, B) = 1 − 1np∑i = 1n(∑m = 13ωmφm(xi))pp	(8)
	where: 0 ≤ ωm ≤ 1, ∑m = 13ωm = 1, φ1(xi) = φs1(xi) + φs2(xi), φ2(xi) = \|φA(xi) − φB(xi)\|, φA(xi) and φB(xi) are defined in formula (4).
	φ3(xi) = max(πA(xi)2, πB(xi)2) − min(πA(xi)2, πB(xi)2).
Zhang and Fu [44]	SZF(A, B) = 1 − 12n∑i = 1n(\|δA(xi) − δB(xi)\| + \|αA(xi) − αB(xi)\|)	(9)
	where: δ_A(x_i)=μ_A(x_i)+π_A(x_i)μ_A(x) and α_A(x_i)=ν_A(x_i)+π_A(x_i)ν_A(x_i)
Hung and Yang [26]	Sw1(A, B) = 1n∑i = 1nmin(μA(xi), μB(xi)) + min(νA(xi), νB(xi))max(μA(xi), μB(xi)) + max(νA(xi), νB(xi))	(10)
	Spk1(A, B) = ∑i = 1nmin(μA(xi), μB(xi)) + min(νA(xi), νB(xi))∑i = 1nmax(μA(xi), μB(xi)) + max(νA(xi), νB(xi))	(11)
	Spk2(A, B) = 1 − 12(maxi\|μA(xi) − μB(xi)\| + maxi\|νA(xi) − νB(xi)\|)	(12)
	Spk3(A, B) = 1−∑i = 1n\|μA(xi) − μB(xi)\| + \|νA(xi) − νB(xi)\|∑i = 1n(\|μA(xi) + μB(xi)\| + \|νA(xi) + νB(xi)\|)	(13)
	Snew1(A, B) = 1 − 1 − exp(−12∑i = 1n\|μA(xi) − μB(xi)\| + \|νA(xi) − νB(xi)\|)1 − exp(−n)	(14)
	Snew2(A, B) = 1 − 1 − exp(−12∑i = 1n\|μA(xi) − μB(xi)\| + \|νA(xi) − νB(xi)\|)1 − exp(−n)	(15)
Szmidt and Kacprzyk [40]	Similarity measures between elements x and y:
	Sim1(x, y) = 1 − f(dHSK(x, y), dHSK(x, yc))	(16)
	Sim2(x, y) = 1 − f(dHSK(x, y), dHSK(x, yc))1 + f(dHSK(x, y), dHSK(x, yc))	(17)
	Sim3(x, y) = 1 − f(dHSK(x, y), dHSK(x, yc))21 + f(dHSK(x, y),dHSK(x, yc))2	(18)
	Sim4(x, y) = exp(−f(dHSK(x, y), dHSK(x, yc))) − exp(−1)1 − exp(−1)	(19)
	where: f(dHSK(x, y), dHSK(x, yc)) = dHSK(x, y)dHSK(x, y) + dHSK(x, yc) and y^c the complement of y and d_{H_SK} (28) is defined in Section 3.2.
	Thus, a measure between two IFSs A and B is defined as:
	Simk(A, B) = 1n∑i = 1nsimk(dHSK(xi, yi), dHSK(xi, yic)), for k = 1 … 4	(20)
Ye [43]	Cosine similarity measure between IFSs:
	SY(A, B) = 1n∑i = 1nμA(xi)μB(xi) + νA(xi)νB(xi)μA2(xi) + νA2(xi)μB2(xi) + νB2(xi)	(21)
Park et al. [35]	SP = 1 − 1n∑i = 1n\|Δπi\|	(22)
Chu et al. [16]	SC1p = 1 − ∑i = 1n wi(δ1\|Δμi\|p + δ2\|Δνi\|p + δ3\|Δπi\|p)1p	(23)
	where ∑j = 13 δj = 1, 0 < δj < 1 and ∑i = 1nwi = 1
	SC2p = 1 − [∑i = 1nwj(δ1{\|Δμi\|max(μAi, μBi)}p + δ2{\|Δνi\|max(νAi, νBi)}p + δ3{\|Δπi\|max(πAi, πBi)}p)]1p	(24)
	It is considered that 00 = 0
Mukherjee and Basu [33]	SMB = ∑i = 1n(μAiμBi + νAiνBi + πAiπBi)max(∑i = 1n(μAi2 + νAi2 + πAi2), ∑i = 1n(μBi2 + νBi2 + πBi2))	(25)

Table 2

Distance Measures Between IFSs.

Authors	Distance measures
Atanassov [2]	Hamming and Euclidean distances
	dHA(A, B) = 12n∑i = 1n\|Δμi\| + \|Δνi\|	(26)
	dEA(A, B) = 12n∑i = 1n(Δμi)2 + (Δνi)2.	(27)
Szmidt and Kacprzyk [39]	Hamming and Euclidean distances
	dHSK(A, B)=12n∑i = 1n\|Δμi\| + \|Δνi\| + \|Δπi\|.	(28)
	dESK(A, B)=12n∑i = 1n(Δμi)2 + (Δνi)2 + (Δπi)2.	(29)
Hung and Yang [24]	Hausdorff distance
	dHd(A, B) = 1n∑i = 1n max(\|Δμi\|, \|Δνi\|)	(30)
	Similarity measures using d_Hd(A, B):
	SHY1(A, B) = 1 − dHd(A, B)	(31)
	SHY2(A, B) = exp−dHd(A,B) − exp−11 − exp−1	(32)
	SHY3(A, B) = 1 − dHd(A, B)1 + dHd(A, B)	(33)
Wang and Xin [42]	dWX1(A, B) = 1n∑i = 1n[\|Δμi\| + \|Δνi\|4] + 1n∑i = 1n[max(\|Δμi\|, \|Δνi\|)2]	(34)
	dWX2p(A, B) = 1np∑i = 1n(φμ(xi) + φν(xi))pp	(35)
	where φ_μ(x_i)=\|μ_A(x_i)−μ_B(x_i)\|/2, φ_ν(x_i)=\|ν_A(x_i)−ν_B(x_i)\|/2 and p>0
Huang et al. [23]	dH1p(A, B) = (∑i = 1n\|Δμi\|p + \|Δνi\|p2n)1P	(36)
	dH2(A, B) = 1n∑i=1nλ1 max(\|Δμi\|,\|Δνi\|)+λ2 min(\|Δμi\|,\|Δνi\|)	(37)
	dH3p(A, B) = 1n∑i=1nλ1 max(\|Δμi\|p, \|Δνi\|p) + λ2 min(\|Δμi\|p, \|Δνi\|p)p	(38)
	where 0≤λ₁≤1, 0≤λ₂≤1, λ₁≥λ₂, λ₁+λ₂=1
	If λ₁=1, λ₂=0 or λ1 = λ2 = 12,p=1 the d_H₂ and d_H₃ distances lead to the Hamming distances d_{H_A} and d_{H_SK}, respectively. Having the same hypothesis and for p=2, the distance d_H₃ leads to Euclidean distance d_{E_A}
	dH4(A, B) = 1n∑i = 1n2(\|Δμi\| + \|Δνi\|)2+\|Δμi\| + \|Δνi\|	(39)
	dH5(A, B) = 2∑i = 1n(\|Δμi\| + \|Δνi\|)2n + ∑i = 1n\|Δμi\| + \|Δνi\|	(40)
	dH6(A, B) = 2dH1(A, B)1 + dH4(A, B)	(41)
Hung and Yang [25]	The following IFSMs are proposed based on L_p metric
	SHY4p(A, B) = 21p − Lp(A, B)21p	(42)
	SHY5p(A, B) = exp(−Lp(A, B)) − exp(−21p)1 − exp(−21p)	(43)
	SHY6p(A, B) = 21p − Lp(A, B)21p(1 + Lp(A, B))	(44)
	where: Lp(A, B) = 1n∑i = 1ndp(IA, IB)
	dp(IA(xi), IB(xi)) = (\|Δμi\|p + \|Δνi\|p)1/p, p ≥ 1
	and I_A(x_i)=[μ_A(x_i), 1−ν_A(x_i)] and I_B(x_i)=[μ_B(x_i), 1−ν_B(x_i)], i=1, 2, …, n
Boran and Akay [13]	dBAp(A, B) = 12n(t + 1)p∑i = 1n\|t ∗ Δμi − Δνi)\|p + \|t ∗ Δνi − Δμi\|pp	(45)
	where p=1, 2, 3, … and t=2, 3, 4, … identifies the level of uncertainty

3.1 Similarity Measures Between Intuitionistic Fuzzy Sets

3.1.1 Comments on Measures From Literature

The measures SC1p (23) and SC2p (24) can be correct only if each of the sum is divided by n (second part of the formulas after 1-).
In the measures SC1p and SC2p, it is very difficult to determine the weight for n elements especially if the sum of the weights is one. So, the weight could be eliminated in these measures, that it will be possible to compute them.
SC1p = dHSK when δ₁=δ₂=δ₃ and when ω_i is omitted.
S_P cannot classify patterns when they are described with both fuzzy and intuitionistic fuzzy information.
The measure S_pk3 (13) does not satisfy the property (P4) of similarity measures between IFSs.

Proof 1:Let A⊆B⊆C, then μ_A(x)≤μ_B(x)≤μ_C(x) and ν_A(x)≥ν_B(x)≥ν_C(x). Therefore,

|μA(x) − μB(x)| ≤ |μA(x) − μC(x)|, |νA(x) − νB(x)| ≤ |νA(x) − νC(x)|,|μA(x) + μB(x)| ≤ |μA(x) + μC(x)| and |νA(x) + νB(x)| ≥ |νA(x) + νC(x)|

According to the above hypothesis, we have:

Consequently, this inequality is not true, too:

1 − |μA(x) − μB(x)||μA(x) + μB(x)| + |νA(x) − νB(x)||νA(x) + νB(x)| ≥ 1 − |μA(x) − μC(x)||μA(x) + μC(x)| + |νA(x) − νC(x)||νA(x) + νC(x)|,

and then S(A, B)≥S(A, C) is not true. Thus, the property P4 is not verified for the similarity measure S_pk3.

Other comments about similarity measures and their relationships are given in Ref. [10].

3.2 Distance Measures Between Intuitionistic Fuzzy Sets

Some distance measures are proposed based on crisp distances, and some similarity measure propositions are based on distance measures. We cite the following measures in Table 2 from literature.

4 Proposition of Semi-Metric Distance Measures

In the following, we propose two semi-metric distance measures between IFSs. The first semi-metric distance measure is defined as:

(46)dSMp(A, B) = 12n∑i = 1n(|μA(xi) − μB(xi)|pμA(xi) + μB(xi) + |νA(xi) − νB(xi)|p2 − νA(xi)−νB(xi)). (46)

where p≥1. If μ_A(x_i)+μ_B(x_i)=0 or 2−ν_A(x_i)−ν_B(x_i)=0, then, for the i^th value x_i, dSMp(A, B) = 0.

It is easy to remark that dSMp satisfies the properties DP1−DP3. Next, we verify if it satisfies the property DP4.

Proof 2:Let A⊆B⊆C, then, μ_A(x_i)≤μ_B(x_i)≤μ_C(x_i) and ν_A(x_i)≥ν_B(x_i)≥ν_C(x_i). Therefore,

|μA(xi) − μB(xi)| ≤ |μA(xi) − μC(xi)|, |νA(xi) − νB(xi)| ≤ |νA(xi) − νC(xi)|,|μA(xi) + μB(xi)| ≤ |μA(xi) + μC(xi)|, and |2 − νA(xi) − νB(xi)| ≥ |2 − νA(xi) − νC(xi)|

as p≥0, we have:

|μA(xi) − μB(xi)|p ≤ |μA(xi) − μC(xi)|p, |νA(xi) − νB(xi)|p ≤ |νA(xi) − νC(xi)|p.

According to the above hypothesis, we have:

|νA(xi) − νB(xi)|p|2 − νA(xi) + νB(xi)| ≤ |νA(xi) − νC(xi)|p|2 − νA(xi) + νC(xi)|,

|μA(xi) − μB(xi)|p|μA(xi) + μB(xi)| + |νA(xi) − νB(xi)|p|2 − νA(xi) − νB(xi)| ≤ |μA(xi) − μC(xi)|p|μA(xi) + μC(xi)| + |νA(xi) − νC(xi)|p|2 − νA(xi) − νC(xi)|.

SodSMp(A, B) ≤ dSMp(A, C)is not true. Thus, the property (DP4) is not verified for the distance measuredSMp.

The second semi-metric distance measure is extending a similarity measure defined between two type-1 fuzzy sets A′ and B′ in Ref. [34] as:

(47)L(A′, B′) = 1 − maxi(|μA′(xi) − μB′(xi)|). (47)

Based on the measure L(A′, B′), we propose the following measure:

The distance d_L is a distance that satisfies the properties DP1−DP3, but it does not satisfy the property DP4 of distance measures between IFSs as shown by the following proof:

4.1 Numerical Examples

In this section, we apply the proposed similarity and semi-metric distance measures to some numerical examples in which patterns are borrowed from Ref. [28]. The results of comparisons between the patterns A_i, i=1, 2, 3 and B of the two data sets are given in Table 3.

Table 3

Numerical Examples and Comparisons Between Proposed Distance Measures.

IFSM	Example 1			Example 2
IFSM	A₁, B	A₂, B	A₃, B	A₁, B	A₂, B	A₃, B
dSMp = 1	0.0	0.110	0.250	0.381	0.018	0.539
dSMp = 2	0.0	0.011	0.06	0.065	0.006	0.238
d_L	0.0	0.15	0.35	0.45	0.15	0.199

Example 1

Let A1 = {(x1, 0.3, 0.3), (x2, 0.2, 0.2), (x3, 0.1, 0.1)}

A2 = {(x1, 0.2, 0.2), (x2, 0.2, 0.2), (x3, 0.2, 0.2)}

A3 = {(x1, 0.4, 0.4), (x2, 0.4, 0.4), (x3, 0.4, 0.4)}

B = {(x1, 0.3, 0.3), (x2, 0.2, 0.2), (x3, 0.1, 0.1)}

From this data, it is clear that B=A₁, so the sample B belongs to the pattern A₁.

Example 2

Let A1 = {(x1, 0.1, 0.1), (x2, 0.5, 0.1), (x3, 0.1, 0.9)}

A2 = {(x1, 0.5, 0.5), (x2, 0.7, 0.3), (x3, 0.0, 0.8)}

A3 = {(x1, 0.7, 0.2), (x2, 0.1, 0.8), (x3, 0.4, 0.4)}

B = {(x1, 0.4, 0.4), (x2, 0.6, 0.2), (x3, 0.0, 0.8)}

According to results of Ref. [24], sample B belongs to the pattern A₂. On examining the table below, we remark that all the proposed measures classify patterns correctly.

5 Application of Similarity Measures to Shape Classification

5.1 Data Set Presentation

We used the SQUID data set (http://www.ee.surrey.ac.uk/CVSSP/demos/css/demo.html), which contains about 1100 images of edges of fishes. Every image is represented with an extended curvature scale space (CSS) descriptors [8, 30, 31] of original shape to identify concavities of the shapes’ edges and CSS descriptors of dual shapes to identify convexities of shapes; for more details, see Ref. [12]. In addition to the extended CSS descriptors, two features are used: the eccentricity and the frequency of codes in Freeman Chain Code [21] of shapes’ edges. In conclusion, the features used to describe the shapes are:

one value of eccentricity
a vector of concavity points (ordinates points)
a vector of convexity points (ordinates points)
Freeman chain code description

There are 30 classes of shapes, and every shape is labeled with the cluster to which it belongs. The data set of the shapes is divided into two data sets: The first is constituted of 738 shapes that serves as reference data set, and the second is constituted of 362 shapes that serves as test data set.

5.2 Constitution of Intuitionistic Fuzzy Features of Shapes

In this section, we are interested in creating intuitionistic fuzzy features of shapes. This stage allows us to pass from a crisp data set constituted with crisp values to an intuitionistic fuzzy data set. The construction of this data set is described in the next paragraphs.

5.2.1 Constitution of Intuitionistic Fuzzy Values of Freeman Chain Code

We extract Freeman chain codes for every shape edge. Freeman codes are a succession of numbers from zero to seven. The numbers indicate directions of a pixel constructing the connected compound edge. This directions can be vertical, horizontal, or representing a curvature. The codes 1, 3, 5, and 7 indicate the pixel belongingness to a curvature.

The number (N) of each direction in the chain code is computed. Then, the frequency F of each direction is computed as N divided by the total number (NT₁) of codes indicating curvatures. Otherwise, N is divided by the total number (NT₂) of codes not indicating a curvature. So, we obtain eight information. It is obvious that all obtained information are in the interval [0,1] (N is lower than NT₁ or NT₂).

We consider two intuitionistic fuzzy sets A and B in the discourse universe X={0, 1, 2, 3, 4, 5, 6, 7} with A={1, 3, 5, 7} and B={0, 2, 4, 6}. Set A concerns codes indicating curvature in Freeman chain, and set B concerns codes not indicating curvatures. Thus, membership and non-membership degrees are computed for two sets A and B according to the following functions:

A = {〈x, μA(x), νA(x)〉|x ∈ X}B = {〈x, μA(x), νA(x)〉|x ∈ X}where (μA(x) = N/NT1νA(x) = 0where(μB(x) = 0νB(x) = N/NT2

5.2.2 Constitution of Intuitionistic Fuzzy Values of Eccentricity and CSS Descriptors

The CSS descriptors and the eccentricity are first fuzzified according to type-1 fuzzy sets (see [12]). The obtained results serve to constitute membership and non-membership degrees using these formulas proposed in Ref. [41].

Let A={〈x, μ_A(x), ν_A(x)〉|x∈X} an IFS where μ_A and ν_A are defined by these formulas:

(49)(μA(x) = (1 − μA′(x))λνA(x) = λ ∗ μA′(x). (49)

where λ∈[0,1] and A′ a type-1 fuzzy set. In our application λ=0.8.

The process of representation with intuitionistic fuzzy information of the features’ eccentricity and CSS descriptors is described in Figure 1.

Figure 1:

Process of Representation with IF Information of the Feature Eccentricity and CSS Descriptors.

5.2.3 Experimentation and Results

We applied the proposed semi-metric distance measures and similarity measures from literature to classify the data test of shapes using KNN classifier. The process of classification is represented in Figure 2, and the results are given in Table 4 below. The 1-best presents the shape number found in the first position, and the 10-best is the sum of found shapes in the i^th position where i∈1…10.

Figure 2:

Classification of Shapes Using an Intuitionistic Fuzzy Similarity Measure.

Table 4

Shape Classification Results Obtained with Intuitionistic Fuzzy Similarity Measures.

IFSM	1-best (%)	2-best (%)	10-best (%)	Error rate
S_Y (21)	22.93	41.71	83.43	16.57
dSMp = 2 (46)	21.82	40.1	82.32	17.68
d_{E_A} (27)	21.55	40.88	82.32	17.68
d_{E_SK} (29)	21.54	40.33	82.32	17.68
S_MB (25)	21	36.19	82.04	17.96
Sep = 1 (6), dWX2p = 1 (35)	22.65	39.78	82.22	18.78
d_L (48)	20.99	33.98	81.77	18.23
S_P (22)	23.2	37.85	81.49	18.51
S_mod (5), Ssp = 1 (7), Sdp=1 (4)	22.65	39.78	81.22	18.78
Shp = 1 (8)	22.65	39.5	81.22	18.78
dBAp = 1 (45)	22.65	39.78	81.22	18.78
S_pk2 (12)	18.23	34.25	80.94	19.06
S_new1 (14)	22.93	38.67	80.66	19.34
S_pk3 (13)	23.48	40.61	80.39	19.61
S_pk1 (11)	23.2	40.88	80.39	19.61
S_ZF (9)	23.2	40.88	80.39	19.61
d_WX₁ (34), d_{H_SH} (28)	22.93	40.61	80.39	19.61
S_HY₁ (31)	22.93	40.61	80.39	19.61
S_HY₃ (33)	22.38	40.33	80.11	19.89
SHY4p = 1 (42), d_{H_A} (26), S_H (2)	23.48	40.61	80.11	19.89
SHY5p = 1 (43)	23.2	40.88	79.83	20.17
S_HY₂ (32)	23.2	40.61	79.83	20.17
S_new₂ (15)	22.1	37.57	79.83	20.17
d_H₅ (40)	18.78	38.4	79.56	20.44
S_L (3)	22.93	40.88	79.28	20.72
dSMp = 1 (46)	24.86	39.23	78.73	21.27
S_w1 (10)	22.65	40.06	78.73	21.27
SHY6p = 1 (44)	22.38	41.16	78.73	21.27
d_H4 (39)	22.65	39.78	78.45	21.55
SC2p = 1 (24)	20.17	35.08	77.9	22.1
Sim_k (20)	19.34	38.4	77.62	22.38

In Table 4, results are classified in descending order of 10-best. The results are very close to one another, varying from 83.43% to 77.62% with a difference of 5.81%. The best results are given by S_Y followed by those of dSMp = 2, dEA, dESK, SMB,Sep = 1, dWX2p = 1 with slight differences. The lowest results are obtained by sim_k given after those of dSMp = 1, Sw1, SHY6p = 1. The remaining measures give results between 81.77% and 79.28% with a difference of 1.74%. The results on 1-best are also very close situated between 24.86% produced by dSMp = 1 and 19.34% given by sim_k with a difference of 5.52%. Similarly, results in 2-best are close with sim_k in the last rank and S_Y in the first rank. The classification results give an idea of measure applications to a large data set. Unfortunately, the results of the classification are not very high for all measures, so the used features may not fit the data set of shapes. In addition, the results given by the similarity measures are close, which puts all IFSMs in the same level with a bit of difference. We note that SC2p = 1 (24) is computed in the shape classification and Arabic sentence recognition with omission of ω. In the second section, we apply IFSMs to Arabic sentence classification, and we compare the results to have more ideas about the results of IFSMs.

6 Application of Similarity Measures to Arabic Sentence Recognition

Handwritten Arabic words present many difficulties for recognition because of the variations in information between different writers and the overlapping between the word characters. So, features representing the words cannot be exact; they only contain imprecision. This last grows when the words are not normalized. In our work, we use an extract of the IFN/ENIT data set [36] constituted of different names of Tunisian towns. The names of the towns can be formed with one word or more; so, they are considered as sentences. In this section, we describe a data set of Arabic sentences with intuitionistic fuzzy information. Thus, we apply fuzzy similarity measures from literature and the proposed semi-metric distance measures to recognize Arabic sentences [5, 9]. In the following subsections, we describe these processes with more details.

6.1 Data Set Sentence Presentation

We use 6537 images of 824 handwritten Tunisian town/village names extracted from the IFN/ENIT data set [36]. The data set images are written by different writers and undergo some preprocessing like noise reduction but are not normalized. The data set is divided into two data sets: reference data set constituted of 4357 sentence images and test data set constituted of 2180 images of sentences [5, 6, 12]. Recognition process is done into two steps:

description of images by features using intuitionistic fuzzy sets
comparison between test data set and reference data set using an intuitionistic fuzzy similarity measure

The features are first constituted according to crisp sets from word connected compounds and are:

sum of pixel distances from the base or the top lines
higher black pixel coordinates
direction frequency of Freeman chain code

Thus, the features are fuzzified according to type-1 fuzzy sets, then, according to intuitionistic fuzzy sets. The descriptions of crisp and fuzzy features are presented in detail in Ref. [12]. In the following, we describe the process of constitution of intuitionistic fuzzy features.

6.1.1 Sum of Pixel Distances From the Base or the Top Lines

For the black pixel in the edge of connected compounds, three sums of mathematical distances are computed, d₁, d₂, and d₃, as described in Ref. [12]. The sum of the distances of the black pixels is divided by the sum of all black pixels of connected compound edge denoted as S. So, we obtain the distance sums as d1 = d1s,d2 = d2s and d3=d3s. These results are fuzzified by computing their membership degrees to trapezoidal functions. To obtain intuitionistic information, the formulas of Ref. [41] given in Section 5.2.1 is used, to compute membership and non-membership degrees as:

(50)(μA(x) = (1 − μA′(x))λνA(x) = λ ∗ μA′(x). (50)

where λ∈[0,1] (in this application, λ=0.8; this value is chosen to minimize the value of π) and A′ a type-1 fuzzy set.

The membership and non-membership degrees are computed for each sum of distances.

6.1.2 Higher Black Pixel Coordinates

The abscissa of the higher black pixel can indicate its position (at the beginning, in the middle, or at the end) in the connected compound, and the ordinate comparing with its position on the top line can indicate a degree of membership to a triangular function, representing type-1 fuzzy set, to be a stroke. Three type-1 fuzzy sets are then obtained. Consequently, three intuitionistic fuzzy sets are defined and have the following description:

A={〈x,μA(x),νA(x)〉|x∈X}with μA(x) = μA′(x) and νA(x) = 1 − μA(x)

where μ_A′ is the membership degree of x to a type-1 fuzzy set represented by a triangular membership.

For each connected compound, six degrees of belongingness and non-belongingness are retained. The two first degrees represent the membership degrees of a pixel in the beginning of a connected compound, the two second degrees represent the membership degrees of a pixel in the middle of a connected compound, and the two last degrees represent the membership degrees of a pixel at the end of a connected compound.

6.1.3 Direction Frequency of Freeman Chain Code

Connected compounds are described by chain codes of Freeman. The process of computation of membership and non-membership degrees is the same presented in the Section 5.2.1.

6.2 Experimentation and Results

The similarity measures from the literature and the proposed semi-metric measure are applied to match between test data set and reference data set of Arabic sentences using the KNN classifier. The recognition results for each applied IFSM are presented in Table 5. The 1-best represents the rate of sentences found in the first position, and the 10-best is the rate of the sum of found sentences in the i^th position where i∈1…10.

Table 5

Arabic Sentence Recognition Results Obtained with Intuitionistic Fuzzy Similarity Measures.

IFSM	1-best (%)	2-best (%)	10-best (%)	Error rate (%)
dSMp = 1 (46)	67.98	77.11	91.88	8.11
dSMp = 2 (46)	69.5	78.03	91.47	8.53
SC2p = 1 (24)	68.17	77.06	91.38	8.62
S_mod (5)	67.38	76.88	91.19	8.81
Sep = 1 (6), dWX2p = 1 (35)	66.61	76.56	91	9
dBAp = 1 (45)	67.16	76.65	90.92	9.08
Sdp = 1 (4)	66.61	76.19	91	9
S_pk3 (13)	67.16	76.69	90.83	9.17
Ssp = 1 (7)	67.2	76.79	90.78	9.22
S_w1 (10)	68.66	78.11	90.64	9.35
Shp = 1 (8)	66.51	75.77	90.46	9.54
S_ZF (9)	67.8	76.51	90.46	9.54
SHY5p = 1 (43)	66.56	75.73	90.32	9.68
SHY6p = 1 (44)	67.11	76.19	90.32	9.68
SHY4p = 1 (42), d_{H_A} (26), S_H (2)	66.28	75.41	90.32	9.68
S_L (3)	65.04	75.41	89.68	10.32
d_WX1 (34)	66.06	74.59	89.45	10.55
d_{E_A} (27)	63.94	73.53	89.22	10.78
S_HY3 (33)	66.19	74.45	89.17	10.83
S_HY2 (32)	66.06	74.4	88.94	11.06
S_pk1 (11)	67.52	75.91	88.67	11.33
S_HY1 (31)	65.64	74.04	88.53	11.47
S_pk2 (12)	63.21	72.8	88.17	11.83
d_{H_SK} (28)	64.86	73.58	88.07	11.93
d_{E_SK} (29)	64.22	72.39	87.89	12.11
d_L (48)	58.58	68.76	84.91	15.09
S_P (22)	57.94	67.11	84.59	15.41
Sim_k (20)	48.39	60.27	84.58	15.51
S_Y (21)	45.87	55.83	79.95	20.05
S_new1 (14)	52.48	57.48	62.29	37.7
S_MB (25)	22.06	31.83	61.19	38.81
d_H4 (39)	22.48	29.4	59.22	40.78
d_H5 (40)	20.87	27.8	58.17	41.83
S_new2 (15)	40.05	43.76	46.23	53.76

The obtained results are classified in descending order of the 10-best (see Table 5). dSMp = 1 is in the first rank with recognition results equal to 91.88%, and S_new2 is in the last rank with results of 46.23%. These results show an important difference of 45.65% between the best and worst results, which can explain the differences between the results of the IFSMs. dSMp = 1, Smod, Se1, dWX2p = 1, and Sd1 are in the second rank with bit differences in their results followed by those of Spk3, Ssp, Sw1, Shp, S, SHY5p = 1, SHY5p = 1, SHY4p = 1,d_{H_A}, and S_H with results varying from 90.83% to 90.32%. The results decrease with measures d_L, sim_k and continue decreasing with measures S_Y and S_new1. The lowest results are given by d_H4, d_H5, S_new2. We note that Sep is proposed to resolve weakness of Sdp, but their results are the same when p=1. The IFSMs d_{H_A} and d_{E_A} produce better results than d_{H_SK} and d_{E_SK}, respectively, despite that the latter are proposed to rectify d_{H_A} and d_{E_A}.

In addition, some measures represent some counter-intuitive cases such as S_mod and Sdp and do not verify some properties; however, they produce the best results. Similarly, the semi-metric dSMp do not satisfy the fourth property of distance measure; however, it produces the best results when p=1 and p=2. In the same way, the measure S_pk3 does not verify the fourth property, but it produces acceptable results. The proposed measures d_L produces modest results. In 1-best and 2-best, the results differ slightly from the 10-best; dSMp and S_w1 are in the first rank, and d_H4, d_H5, S_new2 are in the last rank.

7 Conclusion

We presented intuitionistic fuzzy similarity and distance measures from the literature. We proposed two semi-metric distance measures. Thus, we applied them and measures from the literature to shape the classification and Arabic sentence recognition. In both applications, most measures produced acceptable results, the proposed measures dSMp gave the best results when p=1 and p=2 in handwritten Arabic sentence classification. d_H4, d_H5, sim_k produced low results in both applications, S_Y produced the best results in the shape classification and low results in the handwritten Arabic sentence classification. S_new1 and S_new2 produced low results in the handwritten Arabic sentence classification and acceptable results in the shape classification. dSMp = 1 produced modest results in the first application with best results in the 1-best and best results in the second application. d_L produced modest results in both applications.

From the obtained results, we can conclude that the validation of the fourth property of similarity or distance measures has no influence on the results. Some of the measures produced best results in both applications such as dSMp = 2; consequently, it must produce good results in similar applications. Similarly, d_H4, d_H5, and sim_k produced the worst results in both applications, so, they can produce low results in similar applications. The remaining similarities produced acceptable results in both applications.

The experimental results showed that a semi-metric distance measure can produce good results. In addition, some IFSMs are proposed to rectify or correct existing IFSMs, produced inferior results to those of these measures. To produce good results, an IFSM may not depend on some properties and may contain some counter-intuitive cases. IFSMs are numerous; some classification algorithms need IFSM, and an application of any IFSM may affect results. Thus, an IFSM, which satisfies all properties, does not imply that it produces the best results. The choice should only be based on experimentation. In future work, the proposed measures will be implemented on other data sets and other research topics [7, 17, 18] to assess their results.

Acknowledgments

The authors would like to acknowledge the financial support of this work by grants from the General Direction of Scientific Research (DGRST), Tunisia, under the ARUB program.

Appendix

Properties of Crisp Distance Measures

A distance measure between two crisp vectors x and y, in the discourse universe E, is defined as a function d:E²→ℝ⁺ and have the following properties:

∀ x ∈ E, d(x, x) = 0 (minimality)
∀ x, y ∈ E, d(x, y) = 0 ⇒ x = y (identity)
∀ x, y ∈ E, d(x, y) = d(y, x) (symmetry)
∀ x, y, z ∈ E, d(x, y) ≤ d(x, z) + d(z, y) (triangular inequality)

According to the properties, a function d is called distance, index of distance, index of dissimilarity, semi-metric distance, or distance ultra-metric as:

sign of dissimilarity: if the function d satisfies the properties 1 and 2
index of distance or semi-metric distance: if the function d satisfies the properties 1, 2, and 3
distance: if the function d satisfies the properties 1, 2, 3, and 4
distance ultra-metric: if the function d verifies the property:
∀ x, y, z ∈ E, d(x, y) ≤ max(d(x, z), d(z, y))

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Received: 2015-8-13

Published Online: 2015-10-24

Published in Print: 2016-4-1

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles in the same Issue

https://doi.org/10.1515/jisys-2015-0086

Keywords for this article

Similarity measures; distance measures; intuitionistic fuzzy sets; semi-metric distance measures; handwritten Arabic sentence classification; shape classification; type-1 fuzzy sets

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