Distance Measures Based on Metric Information Matrix for Atanassov’s Intuitionistic Fuzzy Sets

Abstract

The metric matrix theory is an important research object of metric measure geometry and it can be used to characterize the geometric structure of a set. For intuitionistic fuzzy sets (IFS), we defined metric information matrices (MIM) of IFS by using the metric matrix theory. We introduced the Gromov–Hausdorff metric to measure the distance between any two MIMs. We then constructed a kind of metric information matrix distance knowledge measure for IFS. The proposed distance measures have the ability to measure the distance between two incomplete intuitionistic fuzzy sets. In order to reduce the information confusion caused by the disorder of MIM, we defined a homogenous metric information matrix distance by rearranging MIM. Some theorems are given to show the properties of the constructed distance measures. At the end of the paper, some numerical experiments are given to show that the proposed distances can recognize different patterns represented by IFS.

1. Introduction

Zadeh proposed a new field of fuzzy set theory [1], which was further advanced by other researchers [2,3,4,5]. Atanassov defined the intuitionistic fuzzy set model as a generalization of the classical concept of the fuzzy set [6,7,8]. Further mathematical definitions and theorems on intuitionistic fuzzy sets (IFS) were given (see [9,10,11,12]).

In the field of intuitionistic fuzzy sets, the problem of constructing a metric distance and knowledge measure is an important topic. Guo proposed some knowledge measures for Atanassov’s intuitionistic fuzzy sets and applied them in pattern recognition [13]. With an axiomatic framework, Liu systematically reviewed the basic relations between the different distances and measures [14]. For interval-valued fuzzy sets, Zhang et al. gave another axiomatic definition of distance measures, and in the paper [15] they provided two kinds of measures, one based on numerical integration and the other constructed using the Hausdorff distance. From a probabilistic perspective, Hung et al. introduced two kinds of entropy measures for intuitionistic fuzzy sets [16]. Applications can be found in various areas, such as group decision making problems, intuitionistic fuzzy evidence theory, and medical diagnosis [17,18,19,20]. The degree of fuzziness technology of sequence membership grades was used in the study of the Kolmogorov–Sinai entropy to measure the entropy rate of imprecise systems [21]. Some Shapley-weighted similarity measures of Atanassov’s interval-valued intuitionistic fuzzy sets were given based on the Shapley function [22,23]. Many research papers have been published that consider the knowledge measures of the intuitionistic fuzzy set [24,25,26,27].

Garg et al. proposed some distance measures for connection number sets based on set pair analysis theory and investigated their applications to decision-making processes [28]. Using single-valued neutrosophic sets theory, they gave some new biparametric distance measures and compared them to existing measures in terms of counter-intuitive cases to show the validity of the proposed measures [29]. Under the environment of a type-2 intuitionistic fuzzy set, a family of distance measures based on Hamming, Euclidean, and Hausdorff metrics were presented [30]. The authors proposed four different notions of centers, namely centroid, orthocenter, circumcenter, and incenter of the triangle, and  constructed a few distance measures for IFSs for decision-making problems [31]. Yu et al. conducted a comprehensive literature review of the papers related to intuitionistic fuzzy sets from 1984 to 2019 and made some very meaningful conclusions [32]. Alghazzawi et al. proposed the concept of $\rho$ anti-intuitionistic fuzzy sets and $\rho$ anti-intuitionistic fuzzy subgroups, and proved some of their algebraic properties [33]. In the paper [34], the authors proposed the concept of a t-intuitionistic fuzzy order of an element of a t-intuitionistic fuzzy subgroup (t-IFSG) of a finite group and examined different important algebraic properties of this phenomenon.

Wang et al. introduced axiomatic definitions of entropy, similarity measures, and distance for vague soft sets, and some formulas were deduced to calculate the entropy and the measures [35]. Lorentzian metric theory is an important tool for studying the Lorentzian mainfold. By using the Lorentzian inner product, Qi et al. proposed some Lorentzian-like knowledge measures for Atanassov’s intuitionistic fuzzy sets and gave some generalized partially ordered relations to reduce the drawback of the classical order [36]. The axiomatization of the entropy of intuitionistic fuzzy sets has great application value in theory and practical applications. Al-Shami defined a new orthopair fuzzy set named $(2,1)$-Fuzzy set and systematically studied its properties and weighted aggregated operators [37]. Through numerical experiments, the effectiveness of the proposed method in applying it to fuzzy multi-criteria decision-making was verified. Furthermore, the authors further extended orthopair fuzzy sets to $(m,n)$-fuzzy sets and presented some application cases [38]. For more articles, please refer to [39,40,41,42,43,44].

For incomplete intuitionistic fuzzy sets, the results of distance-based pattern discrimination are often confused due to the lack of some information. Chen et al. presented a new approach to group decision-making problems with incomplete intuitionistic fuzzy relations [45]. To handle multicriteria fuzzy decision-making problems, a new multicriteria decision-making method was proposed, in which the information about criteria’s weights was not completely certain, and the criteria values of alternatives were Atanassov’s intuitionistic fuzzy sets [46].

Although there have been many achievements in the research of similarity and measurement of IFS, there are still many areas for improvement. Our analysis is as follows:

The first thing to be improved is that, in the application of pattern recognition, the existing distances or measures are usually induced by Euclidean distance, cosine distance, entropy, etc., and the geometric features between IFS are not considered. The geometric characteristics between IFS often have some relatively stable characteristics, such as rotation and translation invariance, which can be used to better measure the similarity relationship of data during rotation and translation.

The second area that can be improved is that, due to the complexity of the calculation of IFS, the definition of a strict metric becomes difficult. Some counterexamples will appear when the distance or similarity measures are applied to the pattern recognition of IFS.

The third is the measurement of the distance between incomplete intuitionistic fuzzy sets. Due to the lack of information in incomplete IFS, it will be difficult to calculate the distance with homogeneous properties. The usual processing method is to complete the default value. However, this method of introducing the default value will change the physical meaning of the IFS, resulting in unreasonable results.

Based on the above analysis, we propose a class of IFS distance metrics based on distance matrices [47,48,49]. Our main contributions are summarized as follows in three respects:

(1) The metric information theory (MIM) introduced by us induces the metric information matrix to measure the distance between IFS. Because the metric information matrix has geometric rotation invariability and translation invariability, the new distance has a certain geometric relationship description ability, which can better measure the internal relationship between IFS;

(2) The newly constructed metric information matrix distance can identify different patterns in the framework of IFS. Numerical experiments were conducted to show the effectiveness of the proposed distance measure;

(3) The MIM distance measure provides a method of measuring between incomplete IFS, which can be used to solve the pattern recognition problem under the framework of incomplete IFS.

This paper is organized as follows. In Section 2, we reviewed related basic definitions and theoretical backgrounds needed in this paper. In Section 3, we defined metric information distance measures and proved the principal theorems. Some numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion.

2. Preliminaries

In this section, we first review some concepts and definitions that will need to be used later. Let $\mathcal{X}$ be the universal set.

2.1. Intuitionistic Fuzzy Set

Definition 1 

([6]). An intuitionistic fuzzy set E in $\mathcal{X}$ is defined as

$$E=\{<x_0,{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right)>|x_0\in \mathcal{X}\},$$

where ${\mu }_E:\mathcal{X}\to [0,1]$ and ${\nu }_E:\mathcal{X}\to [0,1]$ represent the degree of membership and the degree of non-membership of the element $x_0\in \mathcal{X}$, respectively. Moreover, for every $x_0\in \mathcal{X}$ we have

$$0\le {\mu }_E\left(x_0\right)+{\nu }_E\left(x_0\right)\le 1.$$

Definition 2 

([6]). Let E be an intuitionistic fuzzy set. The intuitionistic fuzzy index (hesitation margin) of the element $x_0\in E$ is defined as

$${\pi }_E\left(x_0\right)=1-{\mu }_E\left(x_0\right)-{\nu }_E\left(x_0\right).$$

Suppose that $E=\{<x_0,{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right),{\pi }_E\left(x_0\right)$$>|x_0\in \mathcal{X}\}$ is an intuitionistic fuzzy set. The complement $E^c$ is defined as

$$E^c=\{<x_0,{\nu }_E\left(x_0\right),{\mu }_E\left(x_0\right),{\pi }_E\left(x_0\right)>|x_0\in \mathcal{X}\}.$$

We will use $\mathbb{F}\left(\mathcal{X}\right)$ to denote the space of intuitionistic fuzzy sets. Let $E,F$ be intuitionistic fuzzy sets. We also give some operations defined by Atanassov [8] as follows:

(H1)

$E\subset F$ if and only if ${\mu }_E\left(x_0\right)\le {\mu }_F\left(x_0\right),{\nu }_E\left(x_0\right)\ge {\nu }_F\left(x_0\right)$ for $x_0\in \mathcal{X};$

(H2)

$E=F$ if and only if $E\subset F$ and $F\subset E$;

(H3)

$E\cup F:=\{<x_0,max\{{\mu }_E\left(x_0\right),{\mu }_F\left(x_0\right)\},$

• $min\{{\nu }_E\left(x_0\right),{\nu }_F\left(x_0\right)\}>|x_0\in \mathcal{X}\}$;

(H4)

E is less fuzzy than F ($E⪯F$), i.e., for $x_0\in \mathcal{X}$,

• if ${\mu }_F\left(x_0\right)\le {\nu }_F\left(x_0\right)$, then ${\mu }_E\left(x_0\right)\le {\mu }_F\left(x_0\right)$ and ${\nu }_E\left(x_0\right)\ge {\nu }_F\left(x_0\right)$;
• if ${\mu }_F\left(x_0\right)\ge {\nu }_F\left(x_0\right)$, then ${\mu }_E\left(x_0\right)\ge {\mu }_F\left(x_0\right)$ and ${\nu }_E\left(x_0\right)\le {\nu }_F\left(x_0\right)$.

2.2. Gromov–Hausdorff Distance

The Gromov–Hausdorff distance was introduced by Gromov in the original French version of his green book, which has many important applications to various topics in geometry [50].

How far two compact metric spaces are from being isometric is defined by using the classical Hausdorff distance and measures.

Definition 3. 

Let $\mathcal{Z}$ be a metric space and $\mathcal{X},\mathcal{Y}\subset \mathcal{Z}$ be two subsets. The Hausdorff distance $D_H(\mathcal{X},\mathcal{Y})$ between $\mathcal{X}$ and $\mathcal{Y}$ is defined to be the infimum of $\epsilon \ge 0$ such that $\mathcal{X}\subset B_{\epsilon }\left(\mathcal{Y}\right)$ and $\mathcal{Y}\subset B_{\epsilon }\left(\mathcal{X}\right)$.

Definition 4. 

Let $\mathcal{X}$ and $\mathcal{Y}$ be two metric spaces. We embed $\mathcal{X}$ and $\mathcal{Y}$ into some metric space $\mathcal{Z}$ isometrically and define the Gromov–Hausdorff distance $D_{GH}(\mathcal{X},\mathcal{Y})$ between $\mathcal{X}$ and $\mathcal{Y}$ to be the infimum of $D_H(\mathcal{X},\mathcal{Y})$ over all such $\mathcal{Z}$ and all such isometric embeddings $\mathcal{X},\mathcal{Y}\hookrightarrow \mathcal{Z}$.

Definition 5. 

Let $\mathcal{X}$ be a metric space. For a natural number N, the distance matrix $K_N\left(\mathcal{X}\right)$ of $\mathcal{X}$ of order N is defined to be the set of symmetric matrices ${\left\{d_{\mathcal{X}}(x_i,x_j)\right\}}_{ij}$ of order N, where $x_i,i=1,2,...,N$, run over all points in $\mathcal{X}$.

If $\mathcal{X}$ is compact, then $K_N\left(\mathcal{X}\right)$ is compact for any N.

The $l_{\infty }$ norm of a square matrix $A=\left(a_{ij}\right)$ of order N is defined as

$${\parallel A\parallel }_{\infty }:=\underset{i,j=1}{\overset{N}{max}}\left|a_{ij}\right|.$$

3. Gromov–Hausdorff Metric Information Matrix

In this section, we propose the notions of Gromov–Hausdorff metric information matrix distance and a homogeneous metric information matrix. We then provide some theorems to demonstrate the main results of this paper. We first define the intuitionistic fuzzy metric information matrix based on the metric matrix for an intuitionistic fuzzy set on a singleton set.

3.1. Gromov–Hausdorff Metric Information Matrix

Definition 6. 

Let $\mathcal{X}=\left\{x_0\right\}$ be a singleton set and $E=\{<{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right),{\pi }_E\left(x_0\right)>\}$ be an intuitionistic fuzzy set. The intuitionistic fuzzy metric information matrix $\mathcal{K}\left(E\right)$ is defined as

$$\mathcal{K}\left(E\right):={\left[d_{ij}\right]}_{3\times 3}=\left[\begin{array}{ccc}0& d({\nu }_E,{\mu }_E)& d({\pi }_E,{\mu }_E)\\ d({\mu }_E,{\nu }_E)& 0& d({\pi }_E,{\nu }_E)\\ d({\mu }_E,{\pi }_E)& d({\nu }_E,{\pi }_E)& 0\end{array}\right],$$

(1)

where d is a distance or a norm on $\mathbb{R}$.

Remark 1. 

Matrix $\mathcal{K}\left(E\right):={\left[d_{ij}\right]}_{N\times N}$ has $N\times N$ entries but only $N(N-1)/2$ pieces of information. Suppose that d is the Euclidean metric on $\mathbb{R}$. We have

I 

$d_{ij}\ge 0$ (non-negativity);

II 

$d_{ij}=d_{ji}$ (symmetry);

III 

$d_{ij}=0\iff {\mu }_E={\nu }_{E}or{\mu }_E={\pi }_{E}or{\nu }_E={\pi }_E$ (selfdistance);

IV 

$d_{ij}\le d_{ik}+d_{kj},i\ne j\ne k$ (triangle inequality).

Example 1. 

Let $\mathcal{X}=\left\{x_0\right\}$ be a singleton set and $E=\{<{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right),{\pi }_E\left(x_0\right)>\}=\{<0.4,0.3,0.1>\}$ be an intuitionistic fuzzy set. Then, the metric information matrix $\mathcal{K}\left(E\right)$ can be obtained as the following:

$$\begin{array}{cc}\hfill \mathcal{K}\left(E\right)& =\left[\begin{array}{ccc}0& d({\nu }_E,{\mu }_E)& d({\pi }_E,{\mu }_E)\\ d({\mu }_E,{\nu }_E)& 0& d({\pi }_E,{\nu }_E)\\ d({\mu }_E,{\pi }_E)& d({\nu }_E,{\pi }_E)& 0\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{ccc}0& 0.1& 0.3\\ 0.1& 0& 0.2\\ 0.3& 0.2& 0,\end{array}\right]\hfill \end{array}$$

(2)

where d is the absolute value distance $\parallel ·\parallel$.

For generalized intuitionistic fuzzy sets, we give the following generalized definition.

Definition 7. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E=\left\{E_i\right|E_i=<{\mu }_E\left(x_i\right),{\nu }_E\left(x_i\right),{\pi }_E\left(x_i\right)>,x_i\in \mathcal{X}\}$ be an intuitionistic fuzzy set. For a natural number $m\le n$, the intuitionistic fuzzy metric information matrix ${\mathcal{DM}}_m\left(E\right)$ of E of order m is defined to be a set of symmetric matrices ${\left(dm(E_i,E_j)\right)}_{ij}$ of order m, where $E_i,i=1,...,m$, run over all points of E

$${\left[d{m}_{ij}\right]}_{i,j=1}^m={\left[\mathcal{D}(E_i,E_j)\right]}_{i,j=1}^m,$$

(3)

where $\mathcal{D}$ is a distance or a metric between two intuitionistic fuzzy sets.

We give another definition of the metric information matrix in the view of vector space theory, and we can use it to construct the Euclidean-like distance.

Definition 8. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E=\left\{E_i\right|E_i=<{\mu }_E\left(x_i\right),{\nu }_E\left(x_i\right),{\pi }_E\left(x_i\right)>,x_i\in \mathcal{X}\}$ be an intuitionistic fuzzy set. For a natural number m, the intuitionistic fuzzy metric information matrix ${\mathcal{DM}}_m^E\left(E\right)$ of E of order m is defined to be a set of symmetric matrices $d_m$ of order m, where $x_i,i=1,...,n$, run over all points of $\mathcal{X}$

$$d_m=\left[\begin{array}{ccc}0& \mathcal{D}({\nu }_E,{\mu }_E)& \mathcal{D}({\pi }_E,{\mu }_E)\\ \mathcal{D}({\mu }_E,{\nu }_E)& 0& \mathcal{D}({\pi }_E,{\nu }_E)\\ \mathcal{D}({\mu }_E,{\pi }_E)& \mathcal{D}({\nu }_E,{\pi }_E)& 0,\end{array}\right],$$

(4)

where $\mathcal{D}$ is a distance or a metric on ${\mathbb{R}}^m$ and ${\mu }_E=[{\mu }_E\left(x_1\right),...,{\mu }_E\left(x_m\right)],$ ${\nu }_E=$$[{\nu }_E\left(x_1\right),...,{\nu }_E\left(x_m\right)],$ ${\pi }_E=[{\pi }_E\left(x_1\right),...,{\pi }_E\left(x_m\right)]$.

We can visualize the metric information matrix in the form of a heat map. In Figure 1, we get an intuitionistic fuzzy set by blurring the color image of three channels. We calculate its metric information matrix with the $l_{\infty }$ norm. Figure 2 shows the heat map of the metric information matrices for the intuitionistic fuzzy set.

We will give two kinds of distances with different strategies: one is the direct application of Gromov–Hausdorff distance, and the other is based on a homogenous strategy.

Definition 9. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets. For a natural number $m\le n$, the Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GH}^m(E,F)$ of order m is defined as

$${\mathcal{D}}_{GH}^m(E,F):=D_{gh}({\mathcal{DM}}_m\left(E\right),{\mathcal{DM}}_m\left(F\right)),$$

(5)

where $D_{gh}$ is a Gromov–Hausdorff distance.

Remark 2. 

If we replace ${\mathcal{DM}}_m$ with ${\mathcal{DM}}_m^E$, then we have a Euclidean-like Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GHE}^m(E,F)$ of order m. Moreover, if the distance used in ${\mathcal{D}}_{GH}$ is replaced with the matrix norm used in ${\mathcal{D}}_{GHE}$ such as the Euclidean norm, then the distances ${\mathcal{D}}_{GH}$ and ${\mathcal{D}}_{GHE}$ are equal.

Furthermore, we can define the following normalized distance.

Definition 10. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets. For a natural number $m\le n$, the Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GH}^m(E,F)$ of order m is defined as

$${\mathcal{D}}_{GH}^m(E,F):=C\times D_{gh}({\mathcal{DM}}_m\left(E\right),{\mathcal{DM}}_m\left(F\right)),$$

(6)

where $D_{gh}$ is a Gromov–Hausdorff distance and C is the normalized coefficient associated with the information metric matrix ${\mathcal{DM}}_m$ and the distance $D_{gh}$.

Remark 3. 

The coefficient C is decided by m and the distance $D_{gh}$. When the norm in the distance $D_{gh}$ is ${\parallel ·\parallel }_2$, we can choose $C=\frac{1}{\sqrt{m}}$. Moreover, if we replace ${\mathcal{DM}}_m$ with ${\mathcal{DM}}_m^E$, then we have a Euclidean-like normalized Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GHE}^m(E,F)$ of order m and the normalized coefficient $C=\frac{1}{\sqrt{3}}$.

Since the sets ${\mathcal{DM}}_m\left(E\right)$ and ${\mathcal{DM}}_m\left(F\right)$ are unordered, it can be seen from the definition of the distance ${\mathcal{D}}_{GH}^m(E,F)$ that the distance ${\mathcal{D}}_{GH}^m(E,F)$ is independent of the order of ${\mathcal{DM}}_m\left(E\right)$ and ${\mathcal{DM}}_m\left(F\right)$. In practical applications, it often involves the comparison of intuitionistic fuzzy sets, such as intuitionistic fuzzy decision-making problems. In practical applications, such as intuitionistic fuzzy decision-making problems, order relations are often involved. Therefore, we introduced order-preserving into the definition of ${\mathcal{D}}_{GH}^m$ to obtain a new metric information matrix distance with order-preserving ability.

Definition 11. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets. For a natural number $m\le n$, the homogeneous information matrix distance ${\mathcal{D}}_H^m(E,F)$ of order m is defined as

$${\mathcal{D}}_H^m(E,F):=\underset{i=1}{\overset{c_n^m}{min}}\left\{D_h\left(sort{\left({\mathcal{DM}}_m\left(E\right)\right)}_i,sort{\left({\mathcal{DM}}_m\left(F\right)\right)}_i\right)\right\},$$

(7)

where $D_h$ is a matrix distance and $sort(·)$ is a high-dimensional matrix formed by splicing a metric information matrix constructed by orderly extracting and combining elements of an intuitionistic fuzzy set.

Remark 4. 

If we replace ${\mathcal{DM}}_m$ with ${\mathcal{DM}}_m^E$, then we have a Euclidean-like homogeneous information matrix distance ${\mathcal{D}}_{HE}^m$ of order m.

Definition 12. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets. For a natural number $m\le n$, the normalized homogeneous information matrix distance ${\mathcal{D}}_H^m(E,F)$ of order m is defined as

$${\mathcal{D}}_H^m(E,F):=C\times \underset{i=1}{\overset{c_n^m}{min}}\left\{D_h\left(sort{\left({\mathcal{DM}}_m\left(E\right)\right)}_i,sort{\left({\mathcal{DM}}_m\left(F\right)\right)}_i\right)\right\},$$

(8)

where $D_h$ is a matrix distance and $sort(·)$ is a high-dimensional matrix formed by splicing a metric information matrix constructed by orderly extracting and combining elements of an intuitionistic fuzzy set, and C is the normalized coefficient associated with the metric information matrix ${\mathcal{DM}}_m$.

Remark 5. 

The coefficient C is decided by m and the distance $D_h$. When the norm in the distance $D_h$ is ${\parallel ·\parallel }_2$, we can choose $C=\frac{1}{\sqrt{m}}$. Moreover, if we replace ${\mathcal{DM}}_m$ with ${\mathcal{DM}}_m^E$, then we have a Euclidean-like normalized homogeneous information matrix distance ${\mathcal{D}}_{HE}^m$ of order m with coefficient $C=\frac{1}{\sqrt{3}}$.

3.2. Information Fusion

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets on $\mathcal{X}$. With the above notions, for any $m\in \mathbb{R}$, we can obtain an information matrix distance $d_m(E,F)$, which can be seen as partial distance information between E and F. For $m\in \{1,2,3,...,n\}$, we can obtain an information matrix distance vector,

$$\overline{DV}(E,F)=<d_1(E,F),d_2(E,F),...,d_n(E,F)>,$$

(9)

where $d_m(E,F)$ can be chosen from ${\mathcal{D}}_{HE}^m(E,F),$${\mathcal{D}}_H^m(E,F),$${\mathcal{D}}_{GH}^m(E,F)$ and ${\mathcal{D}}_{GHE}^m(E,F)$ for $m\in \{1,2,...,n\}$.

When $m=1$, we can obtain the metric information matrix $d_1(E,F)$ by using Definition 6. It is noticeable that $d_1(E,F)$ is a different measure compared with $d_i(E,F)$ for $i\ge 2$ by Definition 7 or Definition 8. Therefore, we did not adopt this definition here.

We need a frame of information fusion to measure the total information from the partial distance information. In this paper, we adopted a weight method to fuse the partial distance information. Intuitively, if m is smaller, then the information matrix distance $d_m$ carries less information. So, we set the weight coefficient $c_m=\frac{m}{n}$. The weight W can be given as

$$W_m=\frac{c_m^2}{{\sum }_{i=1}^n{c}_i^2}.$$

(10)

Definition 13. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$ and $E,F$ be two intuitionistic fuzzy sets on $\mathcal{X}$. The Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GH}$ is defined as

$${\mathcal{D}}_{GH}(E,F)=W^T\overline{DV}(E,F),$$

(11)

where W is the weight and $\overline{DV}$ is the vector of the information matrix distance.

Remark 6. 

By using the metric information matrix ${\mathcal{D}}_{HE}^m(E,F),$${\mathcal{D}}_H^m(E,F)$ or ${\mathcal{D}}_{GHE}^m(E,F)$ in Equation (9), we can obtain the other three different information matrix distances ${\mathcal{D}}_{HE}$, ${\mathcal{D}}_{GHE}$, and${\mathcal{D}}_H$.

Algorithm 1 shows the basic steps for constructing metric information matrix distances.  

Algorithm 1: Metric information matrix distances.

Input: Set E and F to be two IFS on $\mathcal{X}$. Let the number of elements in $\mathcal{X}$ be n. Output: The distance ${\mathcal{D}}_{GH}(E,F)$ between E and F.     for $k=1,2,...n$         Calculate the k-order metric information matrix ${\mathcal{DM}}_k\left(E\right)$           and ${\mathcal{DM}}_k\left(F\right)$, respectively.         Calculate the Gromov–Hausdorff information matrix distance            ${\mathcal{D}}_{GH}^k(E,F)$ of order k.         Calculate the information fusion weight $W_i$.       end     Calculate the information matrix distance vector $\overline{DV}(E,F)$.     Calculate the Gromov–Hausdorff information matrix distance.        ${\mathcal{D}}_{GH}(E,F)$. end

Remark 7. 

It is worth noting that, for a given set $\mathcal{X}$, calculating the Gromov–Hausdorff distance can have a high computational cost if the number of elements n is large. When n is large, the order of the metric information matrix can be sampled to reduce computational complexity, although this will lose some information, so there needs to be a trade-off between the amount of information and the computational cost.

3.3. The Main Results

In this subsection, we give some propositions and theorems on the proposed distances.

Proposition 1 

([50]). Let $\mathcal{X}$ and $\mathcal{Y}$ be two compact metric spaces. If ${\mathcal{DM}}_m\left(\mathcal{X}\right)={\mathcal{DM}}_m\left(\mathcal{Y}\right)$ (or ${\mathcal{DM}}_m^E$) for every natural number m, then $\mathcal{X}$ and $\mathcal{Y}$ are isometric to each other.

The following theorem characterizes the relation between the Hausdorff distance and the Gromov–Hausdorff distance with a control inequality.

Proposition 2 

([50]). For any two compact metric spaces $\mathcal{X}$ and $\mathcal{Y}$ and for any natural number m, we have

$$d_H({\mathcal{DM}}_m\left(\mathcal{X}\right),{\mathcal{DM}}_m\left(\mathcal{Y}\right))\le 2d_{GH}(\mathcal{X},\mathcal{Y}),$$

(12)

where $d_H$ in the left hand is the Hausdorff distance defined by the $l_{\infty }$ metric on the set of square matrices of order m.

The Proposition 2 also holds for ${\mathcal{DM}}_m^E$. In the next theorem, we will show that some of the above distances are pseudometrics.

Theorem 1. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$. The Gromov–Hausdorff information matrix distance measure ${\mathcal{D}}_{GH}:\mathbb{F}\left(\mathcal{X}\right)\times \mathbb{F}\left(\mathcal{X}\right)\to {\mathbb{R}}_{+}$ is a pseudometric.

Proof. 

We only need to show the axioms for the pseudometric.

(i)

For any $E,F\in \mathbb{F}\left(\mathcal{X}\right)$, it is easy to see that ${\mathcal{D}}_{GH}(E,F)\ge 0;$

(ii)

For any $E\in \mathbb{F}\left(\mathcal{X}\right)$ we have ${\mathcal{D}}_{GH}(E,E)=0$;

(iii)

For any $E,F\in \mathbb{F}\left(\mathcal{X}\right)$, by the definition of the metric information matrix, we have that metric information matrices are all symmetrical. Then, it is easy to get ${\mathcal{D}}_{GH}(E,F)={\mathcal{D}}_{GH}(F,E)$;

(iv)

For any $E,F,G\in \mathbb{F}\left(\mathcal{X}\right)$, we have

$$\begin{array}{cc}\hfill & d_m(E,F)=D_{gh}({\mathcal{DM}}_m\left(E\right),{\mathcal{DM}}_m\left(F\right))\hfill \\ \hfill & \le D_{gh}({\mathcal{DM}}_m\left(E\right),{\mathcal{DM}}_m\left(G\right))\hfill \\ \hfill & +D_{gh}({\mathcal{DM}}_m\left(G\right),{\mathcal{DM}}_m\left(F\right))\hfill \\ \hfill & =d_m(E,G)+d_m(G,F).\hfill \end{array}$$

(13)

Suppose that

$$\overline{DV}(E,F)=<d_1(E,F),d_2(E,F),...,d_n(E,F)>.$$

Assume that

$${\mathcal{D}}_{GH}(E,F)=W^T\overline{DV}(E,F),$$

where W is the fixed weight vector. It is easy to see that

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{GH}(E,F)=W^T\overline{DV}(E,F)\hfill \\ \hfill & \le W^T\overline{DV}(E,G)+W^T\overline{DV}(G,F)\hfill \\ \hfill & ={\mathcal{D}}_{GH}(E,G)+{\mathcal{D}}_{GH}(G,F)\hfill \end{array}$$

(14)

for any $E,G,F\in \mathbb{F}\left(\mathcal{X}\right)$.

Then, ${\mathcal{D}}_{GH}$ is a pseudometric. □

Theorem 2. 

Let $\mathcal{X}=\{x_1,x_2,...,x_n\}$. The Euclidean-like Gromov–Hausdorff information matrix distance measure ${\mathcal{D}}_{GHE}:\mathbb{F}\left(\mathcal{X}\right)\times \mathbb{F}\left(\mathcal{X}\right)\to {\mathbb{R}}_{+}$ is a pseudometric.

Proof. 

The proof is similar to Theorem 1, so we omit it here. □

Remark 8. 

The homogeneous information matrix distance ${\mathcal{D}}_H$ or ${\mathcal{D}}_{HE}$ is not a pseudometric, because the triangle inequality does not hold under the operator $\mathrm{sort}(·)$.

4. Numerical Examples

In this section, we will give some comparison experiments of the constructed distance with those of the Euclidean distance $d_{eu}$ [11], Hamming distance $d_{hm}$ [11], chordal distance $d_{ch}$ [26], and the distance measures $d_{ye},d_{fa}$ [18,51].

Example 2. 

Let $\mathcal{X}=\{x_i,i\in \{1,2,...,5\}\}$ and $E=\{<x_0,{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right)>|x_0\in \mathcal{X}\}$ be an intuitionistic fuzzy set. Assume that

$E_1=\{<x_1,0.1,0.8>,<x_2,0.3,0.5>,<x_3,0.6,0.2>,<x_4,0.9,0.1>,<x_5,1.0,0.0>\}$,

$E_2=\{<x_1,0.5,0.3>,<x_2,0.6,0.3>,<x_3,0.8,0.2>,<x_4,0.4,0.5>,<x_5,0.2,0.2>\}$,

$E_3=\{<x_1,0.5,0.2>,<x_2,0.3,0.4>,<x_3,0.7,0.3>,<x_4,0.1,0.9>,<x_5,0.2,0.3>\}$.

Assume that $E_1,E_2,E_3$ represents three patterns. Let $E_0$ be a new sample,

$E_0=\{<x_1,0.4,0.4>,<x_2,0.5,0.5>,<x_3,0.6,0.4>,<x_4,0.3,0.6>,<x_5,0.3,0.2>\}$.

As can be seen from

Table 1. The comparison results for different distances or measures on Example 2.

$\mathbb{F}\left(\mathcal{X}\right)$	${\mathcal{D}}_{\mathit{H}}$	${\mathcal{D}}_{{\mathit{H}}_{\mathit{Ch}}}$	${\mathcal{D}}_{\mathit{GH}}$	${\mathcal{D}}_{{\mathit{GH}}_{\mathit{Ch}}}$	${\mathit{d}}_5$	${\mathit{d}}_{\mathit{eu}}$ [11]	${\mathit{d}}_{\mathit{hm}}$ [11]	${\mathit{d}}_{\mathit{ch}}$[26]	${\mathit{d}}_{\mathit{fa}}$ [18]	${\mathit{d}}_{\mathit{ye}}$ [51]
$(E_0,E_1)$	1.06	0.64	1.00	0.58	1.60	0.95	2.10	1.04	0.31	0.82
$(E_0,E_2)$	0.24	0.14	0.22	0.13	0.36	0.32	0.70	1.39	0.12	0.96
$(E_0,E_3)$	0.66	0.41	0.62	0.37	1.03	0.44	1.00	2.46	0.14	0.95

, we can obtain the following results:

(1)

All of the distances in this example can recognize the right pattern of the unknown sample on this simple dataset;

(2)

$E_0$ belongs to the pattern $E_2$, which corresponds with our intuitive analysis;

(3)

As can be seen from Figure 3, the graph form of the metric matrix ${\mathcal{DM}}_5\left(E_0\right)$ is closest to the graph of metric matrix ${\mathcal{DM}}_5\left(E_2\right)$;

(4)

The Gromov–Hausdorff metric information matrix distance ${\mathcal{D}}_{GH}$ and the homogeneous metric information matrix distance ${\mathcal{D}}_H$ can be used in pattern recognition;

(5)

The homogenous metric information matrix distance ${\mathcal{D}}_H$ is not less than the Gromov–Hausdorff metric information matrix distance ${\mathcal{D}}_{GH}$ between any two intuitionistic fuzzy sets;

(6)

By replacing the Euclidean norm with the chordal distance, the generalized Gromov–Hausdorff metric information matrix distance ${\mathcal{D}}_{G{H}_{Ch}}$ and the modified homogeneous metric information matrix distance ${\mathcal{D}}_{H_{Ch}}$ still have the ability to recognize the different patterns.

Example 3. 

Let $\mathcal{X}=\{x_i,i\in \{1,2,...,7\}\}$ and $E=\{<x_0,{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right)>|x_0\in \mathcal{X}\}$ be an intuitionistic fuzzy set. Assume that

$E_1=\{<x_1,0.25,0.75>,<x_2,0.42,0.63>,<x_3,0.15,0.37>,<x_4,0.34,0.47>,<x_5,1.0,0.0>,<x_6,0.45,0.45>,<x_7,0.73,0.23>\}$,

$E_2=\{<x_1,0.44,0.35>,<x_2,0.32,0.65>,<x_3,0.42,0.54>,<x_4,0.91,0.11>,<x_5,0.35,0.35>,<x_6,0.25,0.48>,<x_7,0.58,0.33>\}$,

$E_3=\{<x_1,0.53,0.34>,<x_2,0.12,0.32>,<x_3,0.22,0.34>,<x_4,0.34,0.55>,<x_5,0.22,0.82>,<x_6,0.53,0.44>,<x_7,0.73,0.23>\}$.

And assume that $E_1,E_2,E_3$ represent three patterns. Let $E_0$ be a new sample

$E_0=\{<x_1,0.23,0.45>,<x_2,0.35,0.44>,<x_3,0.53,0.23>,<x_4,0.62,0.33>,<x_5,0.24,0.55>,<x_6,0.73,0.22>,<x_7,0.81,0.10>\}$.

As can be seen from

Table 2. The comparison results for different distances or measures on Example 3.

$\mathbb{F}\left(\mathcal{X}\right)$	${\mathcal{D}}_{\mathit{H}}$	${\mathcal{D}}_{{\mathit{H}}_{\mathit{Ch}}}$	${\mathcal{D}}_{\mathit{GH}}$	${\mathcal{D}}_{{\mathit{GH}}_{\mathit{Ch}}}$	${\mathit{d}}_7$	${\mathit{d}}_{\mathit{eu}}$ [11]	${\mathit{d}}_{\mathit{hm}}$ [11]	${\mathit{d}}_{\mathit{ch}}$[26]	${\mathit{d}}_{\mathit{fa}}$ [18]	${\mathit{d}}_{\mathit{ye}}$ [51]
$(E_0,E_1)$	0.85	0.52	0.78	0.47	1.56	0.93	2.41	3.56	0.25	0.84
$(E_0,E_2)$	0.84	0.53	0.69	0.43	1.22	0.69	1.93	2.82	0.21	0.89
$(E_0,E_3)$	0.79	0.54	0.67	0.46	1.11	0.65	1.86	2.74	0.18	0.92

, we can obtain the following results:

(1)

By using the Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GH}$ with the Euclidean norm, we can obtain a uniform result with Euclidean-like distances $d_{eu},d_{hm},d_{fa},d_{ye}$ and the chordal distance $d_{ch}$;

(2)

By using the homogeneous information matrix distance ${\mathcal{D}}_H$ with the Euclidean norm we can also obtain a uniform result with Euclidean-like distances $d_{eu},d_{hm},d_{fa},d_{ye}$ and the chordal distance $d_{ch}$;

(3)

The results on the Gromov–Hausdorff information matrix distance ${\mathcal{D}}_{GH}$ with the chordal distance and the homogeneous information matrix distance ${\mathcal{D}}_H$ with chordal distance are different;

(4)

As can be seen from Figure 4, the graph form of the metric information matrix ${\mathcal{DM}}_7\left(E_0\right)$ of order 7 is closer to the graphs of metric information matrices ${\mathcal{DM}}_7\left(E_2\right)$ and ${\mathcal{DM}}_7\left(E_3\right)$;

(5)

The homogenous metric information matrix distance ${\mathcal{D}}_H$ is not less than the Gromov–Hausdorff metric information matrix distance ${\mathcal{D}}_{GH}$ between any two intuitionistic fuzzy sets;

(6)

In pattern recognition, we will achieve different results when we use different distances between two intuitionistic fuzzy sets in the constructed distances;

(7)

By using information matrix distance $d_7$, we can obtain the same result as the distance ${\mathcal{D}}_H$ or ${\mathcal{D}}_{GH}$, which shows that sometimes the partial information distance can also be used in pattern recognition;

(8)

The distance approach is not the perfect solution for pattern recognition.

Example 4. 

Let $\mathcal{X}=\{x_i,i\in \{1,2,...,5\}\}$ and $E=\{<x_0,{\mu }_E\left(x_0\right),{\nu }_E\left(x_0\right)>|x_0\in A⫋\mathcal{X}\}$ be an incomplete intuitionistic fuzzy set. Assume that

$E_1=\{<x_1,0.1,0.8>,<x_2,0.3,0.5><x_4,0.9,0.1>,<x_5,1.0,0.0>\}$,

$E_2=\{<x_2,0.6,0.3>,<x_3,0.8,0.2>,<x_4,0.4,0.5>\}$,

$E_3=\{<x_1,0.5,0.2>,<x_2,0.3,0.4>,<x_4,0.1,0.9>,<x_5,0.2,0.3>\}$.

And assume that $E_1,E_2,E_3$ represent three patterns. Let $E_0$ be a new sample

$E_0=\{<x_1,0.4,0.4>,<x_2,0.5,0.5>,<x_3,0.6,0.4>,<x_5,0.3,0.2>\}$.

As can be seen from

Table 3. The comparison results for different distances or measures in Example 4.

$\mathbb{F}\left(\mathcal{X}\right)$	${\mathcal{D}}_{\mathit{H}}$	${\mathcal{D}}_{{\mathit{H}}_{\mathit{Ch}}}$	${\mathcal{D}}_{\mathit{GH}}\mathit{ori}$	${\mathcal{D}}_{\mathit{GH}}$	${\mathcal{D}}_{{\mathit{GH}}_{\mathit{Ch}}}\mathit{ori}$	${\mathcal{D}}_{{\mathit{GH}}_{\mathit{Ch}}}$	${\mathit{d}}_5$	${\mathit{d}}_{\mathit{eu}}$ [11]	${\mathit{d}}_{\mathit{hm}}$ [11]	${\mathit{d}}_{\mathit{ch}}$ [26]	${\mathit{d}}_{\mathit{fa}}$ [18]	${\mathit{d}}_{\mathit{ye}}$ [51]
$(E_0,E_1)$	1.33	0.66	0.54	1.24	0.98	0.60	2.02	1.49	3.30	4.48	0.33	/
$(E_0,E_2)$	1.63	0.85	0.09	1.17	0.10	0.62	2.26	1.16	2.60	4.49	0.19	/
$(E_0,E_3)$	0.75	0.45	0.31	0.69	0.50	0.42	1.15	1.33	2.60	4.16	0.22	/

and Figure 5, we can obtain the following results:

(1)

The Gromov–Hausdorff metric information matrix distance ${\mathcal{D}}_{GH}$ and its extension ${\mathcal{D}}_{G{H}_{Ch}}$ can work on incomplete intuitionistic fuzzy sets;

(2)

The results on the distances ${\mathcal{D}}_{GH}ori,{\mathcal{D}}_{G{H}_{Ch}}ori,d_{eu}$ and $d_{fa}$ show that the new sample belongs to the pattern $E_2$;

(3)

The results on the distances ${\mathcal{D}}_H,{\mathcal{D}}_{H_{Ch}},{\mathcal{D}}_{GH}$, ${\mathcal{D}}_{G{H}_{Ch}}$,$d_5$ and $d_{ch}$ show that the new sample belongs to the pattern $E_3$;

(4)

The measure $d_{ye}$ cannot be calculated in this case;

(5)

The new constructed distance ${\mathcal{M}}_1$ can be used to measure the distance between two incomplete intuitionistic fuzzy sets;

(6)

By padding in the incomplete part of the incomplete intuitionistic fuzzy sets with <0, 0, 0>, we will obtain the different results;

(7)

The Hamming distance cannot recognize the pattern of the new sample $E_0$, because the distances of $E_0$ between the pattern $E_2$ and the pattern $E_3$ are equal;

(8)

In fact, the intuitionistic fuzzy sets in this example are the incomplete ones corresponding to those in Example 2. By padding in the incomplete part of the incomplete intuitionistic fuzzy sets with <0, 0, 0>, we will obtain the different results compared with the results on Example 2, which also shows that padding in incomplete intuitionistic fuzzy sets with 0 will change the practical significance of the original information of intuitionistic fuzzy sets.

5. Comprehensive Analysis

Numerical experiments show that the newly proposed distances can effectively identify the pattern differences between different intuitionistic fuzzy sets. Due to the translation invariance and rotation invariance of the metric information matrix, the metric information distance also has this geometric invariance, which can better characterize the geometric structure information within intuitionistic fuzzy sets. Moreover, the metric information matrix distance can measure the distance metric of incomplete intuitionistic fuzzy sets, and can be used to identify different intuitionistic fuzzy set patterns. However, the distance constructed based on the metric information matrix also has an obvious shortcoming, namely, its high computational cost. Due to the need to construct the metric information matrix first, and then calculate the Gromov–Hausdorff distance of the metric information matrix, it has higher computational complexity compared to traditional distances, making it unsuitable for situations where there are too many elements in an intuitionistic fuzzy set. However, this defect can be solved through order sampling, which is the next step that needs to be optimized.

6. Conclusions

In this paper, we proposed some intuitionistic fuzzy set distances based on metric information matrices. By making full use of the geometric relationships between elements of intuitionistic fuzzy sets, one can better measure the differences between them. The newly proposed distances can be used to solve the problem of fuzzy pattern recognition. Metric information matrix distance is more intuitive in pattern recognition, but like other distances or similarity measures, there are also some shortcomings, such as counter examples in special situations. For various order relationships within a fuzzy set, the metric information matrix cannot fully adapt, and it is necessary to optimize the construction of different methods for defining metric information matrices for various specially defined orders. In addition, the distance metric for incomplete intuitionistic fuzzy sets is also a problem worthy of further research in the future.