Chordal distance and non-Archimedean chordal distance between Atanassov’s intuitionistic fuzzy set

Abstract

The non-Archimedean normed space theory is an important research object in mathematical physics whose triangle inequality holds in a stronger form. In this note, we propose a generalized chordal distance and a non-Archimedean chordal distance for intuitionistic fuzzy sets. An illustrative example is given to calculate the constructed distances with the different parameters. And some experiments show that the new distances based on chordal distance and non-Archimedean distance are more efficient than the Euclidean-like distances in pattern recognition.

Keywords

Non-Archimedean norm chordal distance intuitionistic fuzzy set

1 Introduction

Distance (metric) plays an important role in various fields such as signal processing, pattern recognition, machine learning and data modeling applications. How to construct a universal distance for the practical problem is still an open problem. For example, it is hard to give a universal distance to measure the distance between two fuzzy sets, particularly in two intuitionistic fuzzy sets.

Zadeh first proposed fuzzy set model as a extension of the classical notion of set in [1]. Atanassov introduced the concept of intuitionistic fuzzy set [2 –5]. Generally speaking, there are two strategies to construct a distance between two intuitionistic fuzzy sets. One strategy is based on topology theory (deduced by a topology), and the another strategy is based on normed space theory such as Banach space theory (deduced by a norm). The common used distance functions are the Minkowski distance, the Euclidean distance, the Pearson correlation distance, the Mahalanobis distance and some special metrics [6 –8]. Notice that some of these distances are based on l_p norm, and they are Archimedean and Euclidean-like whose triangle inequality holds. We know that the distances deduced by l_α norm are linear. Then there may have bad results by using linear distances on the nonlinear datasets.

The Euclidean-like distance is a good choice for symmetry sphery datasets, but may have a bad effect for unbalanced nonlinear datasets. Because of this, we try to construct some nonlinear distances for the special dataset such as intuitionistic fuzzy set. In pattern recognition, by using the Euclidean-like distance, we may not recognize the right pattern, and a simple example is given in Section 4. Inspired by this, we introduce the notions of chordal distanceand non-Archimedea valuation to modify the classical Euclidean-like distance in this paper. Where the first nonlinear distance is a chordal distance which comes from complex analysis, and the another is a non-Archimedean distance which comes from non-Archimedean normed space theory. Non-Archimedean normed space theory has been widely used in physical and mathematics [9]. The non-Archimedean valuation is different from the absolute value function and has some nice properties such as |n|_nA ≤ 1, $n \in ℕ$ . And the triangle inequality holds in a stronger form under the non-Archimedean frame, which is nonlinear and can be used to speed up some clustering algorithms such as K-Means and Fuzzy C-Means. Some experiments show that the proposed distances based on chordal distance and non-Archimedean distance are more efficient than the Euclidean-like distances in pattern recognition.

2 Mathematical foundation

In this section, we give some notions, which will be used in this paper. Let $ℕ$ denote the set of positive integers, and let $ℕ_{0} : = ℕ \cup {0}, ℝ^{+} : = [0, \infty)$ .

2.1 Generalized chordal distance

A mapping $N : K \to R^{+}$ is called a norm if it satisfies the following conditions:

$N (a) = 0$ if and only if a = 0;

$N (ab) = N (a) N (b), a, b \in K;$

$N (a + b) \leq N (a) + N (b), a, b \in K;$

is called the triangle inequality.

A mapping on a set $𝕂$ is called a distance or metric, $D : K \times K \to [0, \infty)$ , and for $a, b, c \in 𝕂$ , if the following conditions hold:

$D (a, b) \geq 0;$

$D (a, b) = 0 \Leftrightarrow a = b;$

$D (a, b) = D (b, a);$

$D (a, b) \leq D (a + c) + D (b + c) .$

The standard l_p norm is defined as $∥ x ∥_{p} = {(\sum_{i = 1}^{n} | x_{i} |^{p})}^{1 / p} .$

For p = 1, the l₁ norm is also called the taxicab norm. For p = 2, l₂ norm is called the Euclidean norm and the l_∞ norm is called the maximum norm or infinity norm.

We can obtain a distance (or metric), if we let $D (a, b) = N (a - b)$ . In fact, it provides a way to construct a distance. For example, the taxicab distance is deduced by l₁-norm and the Euclidean distance is deduced by l₂-norm. Similarly, if (2) is replaced by $D (a, a) = 0$ , then we can get a pseudodistance. In functional analysis, pseudodistance are referred to as semidistance because of their relation to seminorms. We drop the condition (3), we will get a quasidistance. The different distances may have different effects on datasets.

Now, the generalized definition of the chordal distance is given by the following:

Definition 1. The generalized chordal distance is defined as $D_{C} : 𝕂 \times 𝕂 \to [0, \infty),$ where $D_{C} (a, b) = \frac{∥ a - b ∥}{\sqrt{1 + ∥ a ∥^{2}} \sqrt{1 + ∥ b ∥^{2}}},$ (2.1) and ∥· ∥ is a norm or an absolute value function.

For $b = \infty, a \in 𝕂$ , $D_{C} (a, b) = \frac{1}{\sqrt{1 + ∥ a ∥^{2}}} .$ And $D_{C} (a, b) = 0,$ for b =∞ , a = ∞.

2.2 Non-Archimedean chordal distance

The norm $N$ is called non-Archimedean if the condition (III) is replaced by the ultrametric inequality: $N (a + b) \leq max {N (a), N (b)}, a, b \in 𝕂 .$

A mapping $| \cdot | : 𝕂 \to ℝ^{+}$ is a non-Archimedean valuation in a field $𝕂$ if it satisfies the following conditions:

|a|=0 ⇔ a = 0,

$| ab | = | a | | b |, a, b \in 𝕂$ ,

$| a + b | \leq max {| a |, | b |}, a, b \in 𝕂 .$

In addition, we assume that | · | is non-trivial, i.e.,

there exists $a \in 𝕂$ such that |a|≠0, 1.

The condition (ii) implies that the non-Archimedean valuation is a homomorphism of groups.

Based on the above, we have the following results: (1) |n · 1|≤1 for any $n \in ℕ$ ; (2) For $a, b \in 𝕂$ , if |a| ≠ |b| then the strict triangle (ultrametric) inequality actually can be be strengthened to |a+ b| = max {|a|, |b|} ; (3) if $a, b, c \in 𝕂$ and |a - b| < |a - c|, then |a - c| = |b - c|.

Let $D (a, b) = | a - b |$ . It is easy to show that $D$ is a non-Archimedean metric whose triangle (ultrametric) inequality is replaced by $D (a, b) \leq max {D (a, c), D (b, c)} .$

Next, we give the another definition in the paper.

Definition 2. For $a, b \in 𝕂$ , the non-Archimedean chordal distance is defined as $D_{nC} (a, b) = \frac{| a - b |}{max {1, | a |} max {1, | b |}}$ (2.2) where | · | is a non-Archimedean valuation.

For $a \in 𝕂$ and b =∞, $D_{nC} (a, b) = \frac{1}{max {1, | a |}},$ and for a = b =∞, $D_{nC} (a, b) = 0 .$

By the definition of the non-Archimedean chordal distance, it is easy to prove the following results:

if |b|≤1, |a|≤1, then $D_{nC} (a, b) = | a - b |$ ;

if |a|>1, |b|≤1, then by the strong triangle inequality, we have |a - b| = |a|, and $D_{nC} (a, b) = \frac{| a - b |}{| b |} = 1$ ;

if |a|>1, |b|>1, then $D_{nC} (a, b) = \frac{| a - b |}{| a | | b |} = | \frac{1}{a} - \frac{1}{b} |$ .

Example 1. Assume that $𝕂 = ℚ$ and that p > 2 is a prime number. The p-adic valuation for $ℤ$ is defined as $v_{p} : ℤ \to ℕ$ $v_{p} (n) = {\begin{matrix} max {r \in ℕ : p^{r} | n}, & n \neq 0, \\ \infty, & n = 0 . \end{matrix}$ (2.3)

It is easy to prove that the p-adic valuation is a non-Archimedean valuation. Moreover, the p-adic valuation can be extended into the rational numbers $ℚ$ . That is $v_{p} : ℚ \to ℕ$ $v_{p} (\frac{a}{b}) = v_{p} (a) - v_{p} (b) .$ (2.4)

A function $| \cdot |_{p} : ℚ \to ℝ$ is called a p-adic norm if it satifies $| x |_{p} = {\begin{matrix} p^{- v_{p} (x)}, & x \neq 0, \\ 0, & x = 0 . \end{matrix}$ (2.5)

Notice that the p-adic norm is nonlinear.

To simplify the calculation, we can replace | · | in (2.2) by the absolute value function. We need to modify Definition 2.2 because 1 is always larger than |μ_A|, |ν_A| and |π_A| for an intuitionsti fuzzy set A = (μ_A, ν_A, π_A). Then we get the following modified definition of the generalized non-archimedean distance.

Definition 3. For $a, b \in 𝕂$ , the non-Archimedean chordal distance is defined as $D_{nC}^{λ} (a, b) = \frac{| a - b |}{max {λ, | a |} max {λ, | b |}},$ (2.6) where | · | is a non-Archimedean valuation or valuation (absolute value function) and λ ∈ [0, 1].

For $a \in 𝕂$ and b =∞, $D_{nC}^{λ} (a, b) = \frac{1}{max {λ, | a |}},$ and for a = b =∞, $D_{nC}^{λ} (a, b) = 0 .$

3 The distances for intuitionistic fuzzy set

Definition 4. [2] An intuitionistic fuzzy set A in $X$ is defined as $A = {< a, μ_{A} (a), ν_{A} (a) > | a \in X},$ where $μ_{A} : X \to [0, 1]$ and $ν_{A} : X \to [0, 1]$ represent the degree of membership and the degree of non-membership of the element $a \in X$ , respectively, and for every $a \in X$ , $0 \leq μ_{A} (a) + ν_{A} (a) \leq 1 .$

Definition 5. [2] Let A be an intuitionistic fuzzy set. The intuitionistic fuzzy index (hesitation margin) of the element a ∈ A is defined as $π_{A} (a) = 1 - μ_{A} (a) - ν_{A} (a) .$

We will review some classical distances between two intuitionistic fuzzy sets A, B in $X = {x_{1}, x_{2}, . . ., x_{n}}$ . In [10], Szmidt and Kacprzyk proposed the following distances: the Hamming distance $D_{H}$ : $\begin{matrix} D_{H} (A, B) \\ = \frac{1}{2} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | π_{A} (x_{i}) - π_{B} (x_{i}) |); \end{matrix}$ the Euclidean distance $D_{E}$ : $\begin{matrix} D_{E} (A, B) \\ = (\frac{1}{2} \sum_{i = 1}^{n} (μ_{A} (x_{i}) - μ_{B} (x_{i}))^{2} \\ {+ (ν_{A} (x_{i}) - ν_{B} (x_{i}))^{2} + (π_{A} (x_{i}) - π_{B} (x_{i}))^{2})}^{\frac{1}{2}}; \end{matrix}$ the normalized Hamming distance $D_{nH}$ : $\begin{matrix} D_{nH} (A, B) & = & \frac{1}{2 n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) |); \end{matrix}$ the normalized Euclidean distance $D_{nE}$ : $\begin{matrix} D_{nE} (A, B) & = & (\frac{1}{2 n} \sum_{i = 1}^{n} (μ_{A} (x_{i}) - μ_{B} (x_{i}))^{2} \\ + (ν_{A} (x_{i}) - ν_{B} (x_{i}))^{2} \\ + (π_{A} (x_{i}) - π_{B} (x_{i}))^{2})^{\frac{1}{2}} . \end{matrix}$

It is easy to see that the four distances are all linear Euclidean-like distances. We rewrite them by

$\begin{matrix} D_{E, p} (A, B) & = & (\frac{1}{2 n} \sum_{i = 1}^{n} | μ_{A} (x_{i}) - μ_{B} (x_{i}) |^{p} \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |^{p} \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) |^{p})^{\frac{1}{p}} . \end{matrix}$ (3.1)

The absolute value function | · | in (3.1) plays an important role in the proposed distances. And it is easy to see that the distance $D_{E, p}$ is Archimedean. we then replace the function | · | by the generalized chordal distance which is constructed in the second section. Next, we obtain the following modified distances:

Definition 6. The chordal distance in intuitionistic fuzzy set $D_{Ch, p}$ is defined as

$\begin{matrix} D_{Ch, p} (A, B) & = & (\frac{1}{2 n} \sum_{i = 1}^{n} D_{C} (μ_{A} (x_{i}), μ_{B} (x_{i}))^{p} \\ + D_{C} (ν_{A} (x_{i}), ν_{B} (x_{i}))^{p} \\ + D_{C} (π_{A} (x_{i}), π_{B} (x_{i}))^{p})^{\frac{1}{p}} . \end{matrix}$ (3.2)

Definition 7. The non-Archimedean chordal distance in intuitionistic fuzzy set $D_{nCh, p}^{λ}$ is defined as

$\begin{matrix} D_{nCh, p}^{λ} (A, B) & = & (\frac{1}{2 n} \sum_{i = 1}^{n} D_{nC}^{λ} (μ_{A} (x_{i}), μ_{B} (x_{i}))^{p} \\ + D_{nC}^{λ} (ν_{A} (x_{i}), ν_{B} (x_{i}))^{p} \\ + D_{nC}^{λ} (π_{A} (x_{i}), π_{B} (x_{i}))^{p})^{\frac{1}{p}} . \end{matrix}$ (3.3)

Assume that A, B are two fuzzy sets. Let d (A, B), l (A, B), e (A, B), q (A, B) denote the Hamming distance, the normalized Hamming distance, the Euclidean distance and the normalized Euclidean distance proposed by Szmidt and Kacprzyk in [10]. Taking into account an intuitionistic-type representation of a fuzzy set, Szmidt and Kacprzyk gave the following distances, the generalized Hamming distance d^' (A, B), the generalized normalized Hamming distance l^' (A, B), the generalized Euclidean distance e^' (A, B) and the generalized normalized Euclidean distance q^' (A, B). And they obtained the following results: $\begin{array}{l} d^{'} (A, B) = 2 \cdot d (A, B); \\ l^{'} (A, B) = 2 \cdot l (A, B); \\ e^{'} (A, B) = \sqrt{2} \cdot e (A, B); \\ q^{'} (A, B) = \sqrt{2} \cdot q (A, B) . \end{array}$

The essence of the four distances is the same. It is easy to verify that for the constructed distances $D_{nCh, p}^{λ} (A, B)$ and $D_{Ch, p} (A, B)$ (the two term appraoch) the above linear equations do not hold in this case. In fact, the distance $D_{nCh, p}^{λ}$ extends the normalized Hamming distance l (A, B) and the normalized Euclidean distance q (A, B). Let λ = 1 and p = 1, we get $D_{nCh, p}^{λ} = l (A, B)$ . Let λ = 1 and p = 2, we get $D_{nCh, p}^{λ} = q (A, B)$ .

Example 2. Assume that A, B, C in $X = {1, 2, 3, 4, 5, 6, 7}$ are three intuitionistic fuzzy sets. Let $\begin{matrix} A & = & {< 1, 0.5, 0.3 >, < 2, 0.2, 0.8 >, \\ < 4, 0.3, 0.4 >, < 5, 0.2, 0.4 >, < 6, 1, 0 >}, \\ B & = & {< 1, 0.1, 0.3 >, < 4, 0.5, 0.4 >, \\ < 5, 0.2, 0.2 >, < 7, 0.4, 0.4 >} \end{matrix}$ and $\begin{matrix} C & = & {< 1, 0.2, 0.3 >, < 2, 0.3, 0.3 >, \\ < 3, 0.1, 0.7 >, < 4, 0.4, 0.4 >, \\ < 5, 0.3, 0.6 >, < 6, 0.1, 0.1 >, \\ < 7, 0.4, 0.5 >} . \end{matrix}$

Table 1 shows the different distances between institutionstic fuzzys with different parameters. As can be seen form Table 1, with different parameters we can get different scaling distances.

Table 1

The distances with different parameters

$D$	$D_{Ch, 1}$	$D_{Ch, 2}$	$D_{nCh, 1}^{0.5}$	$D_{nCh, 1}^{0.4}$	$D_{nCh, 1}^{0.2}$	$D_{nCh, 2}^{0.5}$	$D_{nCh, 2}^{0.4}$	$D_{nCh, 2}^{0.2}$
(A, B)	0.1571	0.3162	0.4857	0.6607	1.4524	0.8619	1.1669	2.5743
(B, C)	0.0571	0.1000	0.2286	0.3482	1.0476	0.4000	0.6169	1.8279
(A, C)	0.1429	0.2803	0.4429	0.6250	1.3690	0.7502	1.0394	2.1835

4 Some applications

Experiment 1. We give some experiments to compare the capabilities of $D_{Ch, p}$ and $D_{nCh, p}^{λ}$ . Assume that there exist three intuitionistic fuzzy sets on a set $X = {x_{1}, x_{2}, x_{3}}$ , $\begin{matrix} A : & {< x_{1}, 0.5, 0.5 >, < x_{2}, 0.3, 0.2 >, \\ < x_{3}, 0.4, 0.3 >}, \\ B : & {< x_{1}, 0.3, 0.6 >, < x_{2}, 0.7, 0.3 >, \\ < x_{3}, 0.2, 0.7 >} \end{matrix}$ and $\begin{matrix} C : & {< x_{1}, 0.5, 0.3 >, < x_{2}, 0.1, 0.8 >, \\ < x_{3}, 0.3, 0.1 >} . \end{matrix}$

Suppose that A, B, C represent three patterns and let D denote a new sample $\begin{matrix} D : & {< x_{1}, 0.1, 0.3 >, < x_{2}, 0.2, 0.8 >, \\ < x_{3}, 0.3, 0.3 >} . \end{matrix}$

As can be seen form Table 2, we obtain that (i) the new constructed distances, $D_{Ch, 2}$ , $D_{nCh, 2}^{0.4}$ and $D_{nCh, 2}^{0.2}$ , have obtain the results consistent with results from the Euclidean-like distance; and (ii) the sample D belongs to the pattern C.

Table 2
Comparison of different distances

$D$ Euclidean-like $D_{Ch, 2}$ $D_{nCh, 2}^{0.4}$ $D_{nCh, 2}^{0.2}$

(A, D) 0.4472 0.3778 1.9230 3.7861

(B, D) 0.4359 0.3569 1.7815 3.3866

(C, D) 0.2646 0.2313 1.2791 2.9274

$D$	Euclidean-like	$D_{Ch, 2}$	$D_{nCh, 2}^{0.4}$	$D_{nCh, 2}^{0.2}$
(A, D)	0.4472	0.3778	1.9230	3.7861
(B, D)	0.4359	0.3569	1.7815	3.3866
(C, D)	0.2646	0.2313	1.2791	2.9274

Next, the following example shows that the Euclidean-like distance has a bad effect for recognizing patterns. We choice the other patterns to compare the capabilities of different distances.

Experiment 2. We give some experiments to compare the capabilities of $D_{Ch, p}$ and $D_{nCh, p}^{λ}$ . Assume that there exist three intuitionistic fuzzy sets on a set $X = {x_{1}, x_{2}, x_{3}}$ , $\begin{matrix} A : & {< x_{1}, 0.5, 0.5 >, < x_{2}, 0.3, 0.2 >, \\ < x_{3}, 0.4, 0.3 >}, \\ B : & {< x_{1}, 0.3, 0.6 >, < x_{2}, 0.7, 0.3 >, \\ < x_{3}, 0.2, 0.7 >} \end{matrix}$ and $\begin{matrix} C : & {< x_{1}, 0.5, 0.3 >, < x_{2}, 0.1, 0.8 >, \\ < x_{3}, 0.3, 0.1 >} . \end{matrix}$

Suppose that A, B, C represent three patterns and let D denote a new sample $\begin{matrix} D : & {< x_{1}, 0.4, 0.55 >, < x_{2}, 0.5, 0.25 >, \\ < x_{3}, 0.3, 0.5 >} . \end{matrix}$

As can be seen form Table 3, we have the following results: (i) For the above patterns, the distance from the pattern A to the sample D is equal to the distance from the pattern B to the sample D, thus the Euclidean-like distance cannot recognize the pattern of the sample D; (ii) $D_{Ch, 2}$ , $D_{nCh, 2}^{0.4}$ and $D_{nCh, 2}^{0.6}$ all can recognize the pattern; (iii) Form the data represented by $D_{Ch, 2}$ , $D_{nCh, 2}^{0.4}$ and $D_{nCh, 2}^{0.6}$ , we obtain that the sample D belongs to the pattern B, and that is correspond with our intuitive analysis.

Table 3

Comparison of different distances

$D$	Euclidean-like	$D_{Ch, 2}$	$D_{nCh, 2}^{0.4}$	$D_{nCh, 2}^{0.6}$
(A, D)	0.1826	0.1576	0.9358	0.5072
(B, D)	0.1826	0.1551	0.8769	0.4714
(C, D)	0.3873	0.3250	1.6848	0.9936

5 Conclusion

In this paper, we propose two nonlinear distances, the chordal distance and non-Archimedean chordal distance. By complex analysis and non-Archimedean space theory, it is easy to see that the constructed distance are both nonlinear and non-Euclidean. By using the proposed distances, we then obtain tow generalized distance between intuitionistic fuzzy sets, and some experiments show that the new distances based on chordal distance and non-Archimedean distance are more efficient than the Euclidean-like distances in pattern recognition.

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgement

This work is supported by Business College of Shanxi University (Grant No. 2015036). All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Chordal distance and non-Archimedean chordal distance between Atanassov’s intuitionistic fuzzy set

Abstract

Keywords

1 Introduction

2 Mathematical foundation

2.1 Generalized chordal distance

Table 2 Comparison of different distances D Euclidean-like D Ch , 2 D nCh , 2 0.4 D nCh , 2 0.2 (A, D) 0.4472 0.3778 1.9230 3.7861 (B, D) 0.4359 0.3569 1.7815 3.3866 (C, D) 0.2646 0.2313 1.2791 2.9274

Competing interests

Footnotes

Acknowledgement

References

Table 2
Comparison of different distances

$D$ Euclidean-like $D_{Ch, 2}$ $D_{nCh, 2}^{0.4}$ $D_{nCh, 2}^{0.2}$

(A, D) 0.4472 0.3778 1.9230 3.7861

(B, D) 0.4359 0.3569 1.7815 3.3866

(C, D) 0.2646 0.2313 1.2791 2.9274