Fuzzy c-means clustering method with the fuzzy distance definition applied on symmetric triangular fuzzy numbers

Abstract

The conventional fuzzy c-means (FCM) clustering method can be applied on data, where data features are crisp; however, when the features are fuzzy, the conventional FCM cannot be utilized. Recently, some researchers applied FCM on fuzzy numbers when the used metric has a crisp value. Since difference between two fuzzy numbers can be represented by a fuzzy value better than crisp one, in this paper, it is going to extend the FCM method for clustering symmetric triangular fuzzy numbers, where the used metric has a fuzzy value. It will be shown that the proposed fuzzy distance expresses the distance between two fuzzy numbers much better than crisp metrics. Then the proposed method has been applied on simulated and various real data, where it is compared with several new methods. The experimental results show better performance of the proposed method in compare to other ones.

Keywords

Clustering fuzzy c-means clustering method triangular fuzzy number fuzzy smaller

1 Introduction

Fuzzy data analysis has been widely used in numerous fields including pattern recognition, data mining and image processing. Progresses in fuzzy mathematics have encouraged engineers and scientists in applied fields to present real models such as fuzzy image filtering [1], fuzzy clustering [2 –10], fuzzy regression analysis [11, 12] and fuzzy classification [13, 14].

A fuzzy number (in compare to a crisp one) works better in declaring the difference between two fuzzy numbers [15 –17]. For example the difference between “about 2” and “about 4” is better to be said “about 2” rather than “exact 2” (which is a crisp number). This idea is considered in some pattern recognition fields such as classification [18]. In spite of this advantage, calculation complexities dealing with fuzzy logic way have become a pitfall.

Nevertheless, all of the following reasons are the major points of the authors to focus on the triangular fuzzy numbers in this paper:

The simplifications done on some kinds of fuzzy numbers like triangular, LR-type and trapezoidal numbers [19 –21];

Frequently use of triangular numbers in processing dates and due times [22, 23];

Triangular fuzzy numbers belong to a typical class of fuzzy numbers, which can be regarded as a general form of the symbolic interval and the crisp numbers [7].

It must be mentioned, the method introduced here is applicable on the other kinds of fuzzy numbers by the same structure as well.

The rest of paper is organized as follows: preliminary discussions, i.e the conventional FCM clustering method (applied on crisp numbers) and some fuzzy conceps are considered in Section 2. In section3, a brief review of valid literature has been described. The proposed FCM clustering method to apply on symmetric triangular fuzzy numbers is presented in Section 4. Section 5 presents simulation results of the proposed method applied on simulated and real (Taiwanese tea, students and satellite images) fuzzy numbers. Finally the conclusion is given in section 6.

1 Preliminaries

As the proposed method is based on the conventional FCM, a description of FCM will be given concerning the special manner in which it is applied on crisp numbers. Then some fuzzy concepts are reviewed in this section for better understanding of the proposed method, which will be explained in the next section.

1.1 The conventional FCM applied on crisp numbers

One of the most widely used fuzzy clustering method is FCM [24, 25] which is the fuzzy equivalent of the nearest mean “hard” clustering method. The FCM clustering method assigns fuzzy memberships to each input number. The goal of the FCM algorithm is to minimize the following objective function with respect to fuzzy membership $u_{i, k}$ and clusters centers v_i: $J (U, V) = \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} d^{2} (x_{k}, v_{i})$ (2.1.1) $d^{2} (x_{k}, v_{i}) = {(x_{k} - v_{i})}^{T} (x_{k} - v_{i}) = {∥ x_{k} - v_{i} ∥}^{2}$ (2.1.2)

where, X ={ x₁, ⋯ , x_k, ⋯ , x_n } denotes the set of input numbers, $x_{k} = [x_{k, 1}, \dots, x_{k, p}] \in ℝ^{p}$ ( $ℝ$ denotes real numbers set) for k = 1, ⋯ , n are p-dimensional vector numbers on the real numbers, c is the number of clusters, and θ > 1 is the fuzziness index. The matrix $U = {[u_{i, k}]}_{c \times n}$ is called a constrained fuzzy c partition of X, if the entries of U satisfy the following equation:

${\begin{matrix} u_{i, k} \in [0, 1], for i = 1, \dots, c a n d k = 1, \dots, n \\ \sum_{i = 1}^{c} u_{i, k} = 1, k = 1, \dots, n \end{matrix}$ (2.1.3) where $u_{i, k}$ is the membership value of the k-th number to i-th cluster, V ={ v₁, ⋯ , v_i, ⋯ , v_c } is cluster centers set and $v_{i} = [v_{i, 1}, \dots, v_{i, p}] \in ℝ^{p}$ for i = 1, ⋯ , c is the center of i-th cluster (in the FCM clustering method we suppose c ⩽ n or the number of input numbers are greater or equal than the number of clusters).

Using Lagrange function, we can minimize the objective function J (U, V) in (2.1.1) which is subjected to the constraints (2.1.3) and conclude the following updated function:

$\begin{matrix} L (U, V, λ) = \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} {(x_{k} - v_{i})}^{T} \\ (x_{k} - v_{i}) - \sum_{k = 1}^{n} λ_{k} (\sum_{i = 1}^{c} u_{i, k} - 1) \end{matrix}$ (2.1.4)

$\begin{matrix} u_{i, k} = {(\sum_{r = 1}^{c} {(\frac{d^{2} (x_{k}, v_{i})}{d^{2} (x_{k}, v_{r})})}^{\frac{1}{θ - 1}})}^{- 1}, \\ for i = 1, \dots, candk = 1, \dots, n \end{matrix}$ (2.1.5) $v_{i} = \frac{\sum_{k = 1}^{n} u_{i, k}^{θ} x_{k}}{\sum_{k = 1}^{n} u_{i, k}^{θ}}, for i = 1, \dots, c$ (2.1.6)

1.2 Some fuzzy concepts

Definition 2.2.1. Let $ℝ$ denote the set of all real numbers. A fuzzy number is a mapping $\tilde{x} : ℝ \to [0, 1]$ with the following properties:

The membership degree $μ_{\tilde{x}} (x) \in [0, 1]$ quantifies the membership grade of the element x to the fuzzy set $\tilde{x}$ . The value 0 means that x is not a member of the fuzzy set; the value 1 means that x is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, belonging partially to the fuzzy set.

$\tilde{x}$ is a normal fuzzy set if the core of $\tilde{x}$ , $C (\tilde{x}) = {x \in ℝ : μ_{\tilde{x}} (x) = 1}$ , is not empty,

$μ_{\tilde{x}} (.)$ is upper semi-continuous,

$\tilde{x}$ is a convex fuzzy set if $μ_{\tilde{x}} (x + (1 -) y) ⩾ \min {μ_{\tilde{x}} (x), μ_{\tilde{x}} (y)}$ , for all $x, y \in ℝ, \in [0, 1]$ ,

The support of $\tilde{x}$ is $S (\tilde{x}) = {x \in ℝ : μ_{\tilde{x}} (x) > 0}$ and its closure $cl (S (\tilde{x}))$ is compact.

Definition 2.2.2. The LR-type fuzzy number is a special type of representation for a fuzzy number. It is defined by two functions L (and R) which map $ℝ^{+} \to [0, 1]$ and are decreasing shape functions with constraints: L (0) = 1, L (1) = 0, ∀ x 〈 1 : L (x) 〉 0 and ∀x > 0 : L (x) < 1 . A fuzzy number $\tilde{x}$ is LR-type if there exist reference functions L (for left), R (for right) and scalars $α_{\tilde{x}}, β_{\tilde{x}} > 0$ , with $μ_{\tilde{x}} (x) = {\begin{matrix} \begin{matrix} L (\frac{m_{\tilde{x}} - x}{α_{\tilde{x}}}) & x ⩽ m_{\tilde{x}}, \end{matrix} \\ \begin{matrix} R (\frac{x - m_{\tilde{x}}}{β_{\tilde{x}}}) & x ⩾ m_{\tilde{x}} . \end{matrix} \end{matrix}$

Here, $m_{\tilde{x}}$ called the mean value of $\tilde{x}$ , is a real number and $α_{\tilde{x}}$ and $β_{\tilde{x}}$ are called the left and right spreads, respectively. Also, $μ_{\tilde{x}} (x)$ is the membership function of LR-type fuzzy number $\tilde{x}$ , denoted by ${(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{LR}$ illustrated in Fig. 1.

Fig. 1

An LR-type fuzzy number.

Definition 2.2.3. Let $\tilde{x} = {(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{LR}$ be a LR-type fuzzy number. If L (.) = R (.) = T (.), where $T (x) = {\begin{matrix} 1 - x & 0 ⩽ x ⩽ 1, \\ 0 & Otherwise \end{matrix}$ then $\tilde{x}$ is called triangular fuzzy number, denoted by $\tilde{x} = {(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{T}$ . Furtheremore $\tilde{x}$ is also called symmetric triangular fuzzy number if $α_{\tilde{x}} = β_{\tilde{x}} = s_{\tilde{x}}$ , denoted by $\tilde{x} = (m_{\tilde{x}}, s_{\tilde{x}})_{T}$ .

Theorem 2.2.1 [26]. Let $\tilde{x} = {(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{LR}$ and $\tilde{y} = {(m_{\tilde{y}}, α_{\tilde{y}}, β_{\tilde{y}})}_{LR}$ be two LR-type fuzzy numbers, then $\begin{matrix} \tilde{x} \times \tilde{y} \approx (m_{\tilde{x}} m_{\tilde{y}}, m_{\tilde{x}} α_{\tilde{y}} + m_{\tilde{y}} α_{\tilde{x}}, m_{\tilde{x}} β_{\tilde{y}} + \\ {m_{\tilde{y}} β_{\tilde{x}})}_{LR} if m_{\tilde{x}} ⩾ 0 and m_{\tilde{y}} ⩾ 0, \end{matrix}$ $\begin{matrix} \tilde{x} \times \tilde{y} \approx (m_{\tilde{x}} m_{\tilde{y}}, - m_{\tilde{x}} α_{\tilde{y}} + m_{\tilde{y}} α_{\tilde{x}}, - m_{\tilde{x}} β_{\tilde{y}} + \\ {m_{\tilde{y}} β_{\tilde{x}})}_{LR} if m_{\tilde{x}} < 0 and m_{\tilde{y}} ⩾ 0, \end{matrix}$ $\begin{matrix} \tilde{x} \times \tilde{y} \approx (m_{\tilde{x}} m_{\tilde{y}}, - m_{\tilde{x}} α_{\tilde{y}} - m_{\tilde{y}} α_{\tilde{x}}, - m_{\tilde{x}} β_{\tilde{y}} - \\ {m_{\tilde{y}} β_{\tilde{x}})}_{LR} if m_{\tilde{x}} < 0 and m_{\tilde{y}} < 0 . \end{matrix}$

Definition 2.2.4 [27]. For any of the two fuzzy numbers; $\tilde{x}$ , $\tilde{y}$ and α ∈ (0, 1], where ${(\tilde{x})}_{α} = [(x)_{α}^{L}, (x)_{α}^{U}]$ and ${(\tilde{y})}_{α} = [(y)_{α}^{L}, (y)_{α}^{U}]$ denote the α-cut of $\tilde{x}$ and $\tilde{y}$ , respectively, if we define the partial ordering of closed intervals in the usual way, that is $[(x)_{α}^{L}, (x)_{α}^{U}] ⩽ [(y)_{α}^{L}, (y)_{α}^{U}]$ iff $(x)_{α}^{L} ⩽ (y)_{α}^{L}$ and $(x)_{α}^{U} ⩽ (y)_{α}^{U}$ , then, we have $\tilde{x} \underset{f}{⩽} \tilde{y}$ iff ${(\tilde{x})}_{α} ⩽ {(\tilde{y})}_{α}$ for all α ∈ (0, 1], where $\underset{f}{⩽}$ denotes the fuzzy smaller than.

Theorem 2.2.2. From definition (2.2.4), for any triangular fuzzy numbers $\tilde{x} = {(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{T}$ and $\tilde{y} = {(m_{\tilde{y}}, α_{\tilde{y}}, β_{\tilde{y}})}_{T}$ , we have $\tilde{x} \underset{f}{⩽} \tilde{y}$ iff $m_{\tilde{x}} ⩽ m_{\tilde{y}}$ , $m_{\tilde{x}} - α_{\tilde{x}} ⩽ m_{\tilde{y}} - α_{\tilde{y}}$ and $m_{\tilde{x}} + β_{\tilde{x}} ⩽ m_{\tilde{y}} + β_{\tilde{y}}$ ; as a result, these three conditions guaranty the correctness of the relation ${(\tilde{x})}_{α} ⩽ {(\tilde{y})}_{α}$ for all α ∈ (0, 1].

This fact is illustrated in Fig. 2 through an example.

Fig. 2

The fuzzy smaller concept for $\tilde{x} = {(m_{\tilde{x}}, α_{\tilde{x}}, β_{\tilde{x}})}_{T}$ and $\tilde{y} = {(m_{\tilde{y}}, α_{\tilde{y}}, β_{\tilde{y}})}_{T}$ , where $\tilde{x} \underset{f}{⩽} \tilde{y}$ .

Definition 2.2.5. Let each of p features (dimensions) of number $\tilde{x}$ , “ ${\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{p}$ ”, be a fuzzy number with probably a different width of uncertainty (Fig. 3 shows the uncertainty widths for a 1-dimensional fuzzy number). This number can be shown by a hyper-rectangle (HR) in p dimensions at each α-cut, α ∈ (0 , 1]. Let’s name this hyper-rectangle as α-cut hyper-rectangle or α-cut HR. In this diagram, each dimension width of α-cut HR for this number shows the uncertainty width of the corresponding feature at α-cut i.e. ${({\tilde{x}}_{j})}_{α} = [(x_{j})_{α}^{L}, (x_{j})_{α}^{U}]$ for j = 1, ⋯ , p (See Fig. 4). In other words, we ignore the membership value of this fuzzy number and show it later at α-cut by the set ${(x_{1}, \dots, x_{p}) | (x_{j})_{α}^{L} ⩽ x_{j} ⩽ (x_{j})_{α}^{U}, j = 1, \dots, p}$ , which is a hyper-rectangle, called α-cut HR.

Fig. 3

Width of uncertainty.

Fig. 4

0-cut HR of a 2-dimensional fuzzy number $\tilde{x} = {{\tilde{x}}_{1}, {\tilde{x}}_{2}}$ .

1.3 The FCM applying on non-crisp numbers

There are various papers to apply FCM on the non-crisp (fuzzy) numbers [2–8 , 28–30]. They can be summarized in one structure as Fig. 5, where the mapping function changes respect to influenced fuzzy number type and utilized metric definition [8].

Fig. 5

FCM applying on any crisp and non-crisp numbers in one structure [8].

2 Literature review

Fuzzy clustering is an important tool for discovering data structure. The fuzzy c-means (FCM) algorithm [25], the best-known clustering algorithm, has been used in a wide range of engineering and scientific disciplines such as medicine imaging, bioinformatics, pattern recognition, and data mining. Hathaway et al. [2] proposed a new distance definition for fuzzy LR-type data and applied FCM on them. Yang et al. [3] described a class of fuzzy clustering procedure for fuzzy data. Considering most of fuzzy clustering techniques, that imposed on crisp data based on the concept of membership function which use the idea of fuzzy set theory, Yang et al. [3] derived new types of fuzzy clustering procedures dealing with fuzzy data, called fuzzy c-numbers (FCN) clustering. They construct these FCNs especially for LR-type, triangular, trapezoidal and normal fuzzy numbers. Yang et al. [4] also presented fuzzy clustering algorithms for a combination of symbolic and fuzzy data features. Besides, they gave a modified dissimilarity measurement for symbolic and fuzzy data and then applied FCM clustering algorithm on these combined data types. Finally, they applied suggested clustering algorithm to real data with combined feature variables of both symbolic and fuzzy data. Hung and Yang [5], also suggested a kind of fuzzy clustering algorithm, called the alternative fuzzy c-numbers (AFCN), for LR-type fuzzy numbers based on an exponential distance function. On the basis of the gross error sensitivity and influence function, this distance has been claimed to be robust with respect to noise and outliers. Hence, the AFCN clustering algorithm is more robust than FCN presented by Yang et al. [3]. Yang et al. [6] applied a mixed-variable fuzzy clustering algorithm, called mixed-variable fuzzy c-means (MVFCM), for cell formation (CF) in Group technology (GT). A fuzzy clustering algorithm for triangular fuzzy numbers is studied by Rong et al. [7]. They introduced a new distance between triangular fuzzy numbers using three interval number parameters, and proved that the presented distance is a complete metric on the set of triangular fuzzy numbers. Consequently, based on this new distance, they presented two fuzzy c-means types of clustering algorithms to deal with triangular fuzzy numbers. Finally Hadi et al. [8] suggested Vector form of fuzzy c-means (VFCM) that simplified the FCM clustering method applying to non-crisp numbers. They showed that the VFCM method is a simple and general form of FCM that applies the FCM clustering method to various types of numbers (crisp and non-crisp, with different correspondent metrics) in a single structure, and without any complex calculations and exhaustive derivations. They also suggested the meta-fusion, that uses fuzzy (symbolic-interval) numbers to demonstrate the output of (multi panchromatic satellite images) fusion process in order to multi images segmentation [30]. They showed this fusion method and conventional FCM method applying on the fuzzy results of fusion process leads to better performance. Wang et al. [10] presented a fuzzy-based customer clustering algorithm with a hierarchical analysis structure for heterogeneity and high-dimension of customers’ characteristics. They developed a fuzzy clustering algorithm based on Axiomatic Fuzzy Set to group the customers into multiple clusters. Wang et. al. also, presented an approach to optimize the vehicle routing problem based on customer fuzzy clustering with similar characteristics under a hierarchical analysis structure [9].

In the overviewed clustering-based researches, the distance between two fuzzy numbers is assumed to gain a crisp value [2–8 , 30]. Considering the distance of two fuzzy numbers as a fuzzy number (not a crisp one), the conventional FCM clustering method objective function (in fuzzy state) has been utilized in the proposed method. Therefore unknown parameters in the proposed method are found through minimizing a fuzzy objective function. It will be shown that the crisp metric (e.g. the Yang distance [3, 5]) is unable to express the distance between two triangular fuzzy numbers as well as the proposed fuzzy metric.

3 The proposed method

As mentioned earlier, the reason of adapting triangular fuzzy numbers as the affected elements for clustering is to use them frequently in processing dates and due times [22, 23]. Note that, triangular fuzzy numbers belong to a typical class of fuzzy numbers, which can be regarded as a general form of the symbolic interval and the crisp numbers [7]. Furthermore, in the case of having symmetric triangular fuzzy numbers, which is mentioned in section 2.2, computations become very simple. Therefore we choose symmetric triangular fuzzy numbers to analyze the proposed method as well [11, 12].

Let ${\tilde{x}}_{k} = {{\tilde{x}}_{k, 1}, \dots, {\tilde{x}}_{k, p}}$ , for k = 1, ⋯ , n, be input symmetric triangular fuzzy numbers (vectors) in p dimensions ${\tilde{x}}_{k, j} = {(m_{{\tilde{x}}_{k, j}}, s_{{\tilde{x}}_{k, j}})}_{T}$ , j = 1, ⋯ , p. Furthermore, ${\tilde{v}}_{i} = {{\tilde{v}}_{i, 1}, \dots, {\tilde{v}}_{i, p}}$ , for i = 1, ⋯ , c are supposed to be the desired symmetric triangular clusters centers having p dimensions ${\tilde{v}}_{i, j} = {(m_{{\tilde{x}}_{k, j}}, s_{{\tilde{x}}_{k, j}})}_{T}$ , j = 1, ⋯ , p. We define following expression to apply the proposed FCM to input numbers of which the features are symmetric triangular fuzzy numbers: $\begin{matrix} min_{\tilde{V}, U} \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} {\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k}) \\ subjectto \sum_{i = 1}^{c} u_{i, k} = 1, fork = 1, \dots, n \end{matrix}$ (4.1)

where ${\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k}) = \sum_{j = 1}^{p} {\tilde{d}}^{2} ({\tilde{v}}_{i, j}, {\tilde{x}}_{k, j})$ . According to subtraction definition in symmetric triangular fuzzy numbers case, [31, 32], we have:

$\begin{matrix} {\tilde{v}}_{i, j} - {\tilde{x}}_{k, j} = {(m_{{\tilde{v}}_{i, j}}, s_{{\tilde{v}}_{i, j}})}_{T} - {(m_{{\tilde{x}}_{k, j}}, s_{{\tilde{x}}_{k, j}})}_{T} \\ = {(m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}}, s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}})}_{T} \end{matrix}$ (4.2)

If we define $\tilde{d} ({\tilde{v}}_{i, j}, {\tilde{x}}_{k, j}) ≜ {\tilde{v}}_{i, j} - {\tilde{x}}_{k, j} = {(m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}}, s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}})}_{T}$ as a fuzzy distance, the used squared distance in the FCM cost function (3.1) will express by using theorem (2.2.1) as follows:

$\begin{matrix} {\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k}) \approx \sum_{j = 1}^{p} ({(m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}})}^{2}, 2 | m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}} | \\ {(s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}}))}_{T} \end{matrix}$ (4.3)

where | · | applies element absolute value on a vector. The mentioned distance definition $\tilde{d} ({\tilde{v}}_{i, j}, {\tilde{x}}_{k, j})$ does not satisfy all metric conditions. Therefore, in this paper we propose a fuzzy distance (fuzzy metric) definition for symmetric triangular fuzzy numbers as follows: $\tilde{d} ({\tilde{v}}_{i, j}, {\tilde{x}}_{k, j}) = {(| m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}} |, s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}})}_{T}$

It can be that equation (3.4) satisfies all metric conditions. However, this metric do not change the used squared distance in the proposed FCM cost function (equation (3.1)) and ${\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k})$ will be as equation (4.3).

If $m_{{\tilde{x}}_{k}} ≜ {[m_{{\tilde{x}}_{k, 1}}, \dots, m_{{\tilde{x}}_{k, p}}]}^{T}$ , $m_{{\tilde{v}}_{i}} ≜ [m_{{\tilde{v}}_{i, 1}}, {\dots, m_{{\tilde{v}}_{i, p}}]}^{T}$ , $s_{{\tilde{x}}_{k}} ≜ [s_{{\tilde{x}}_{k, 1}}, \dots, {s_{{\tilde{x}}_{k, p}}]}^{T}$ and $s_{{\tilde{v}}_{i}} ≜ {[s_{{\tilde{v}}_{i, 1}}, \dots, s_{{\tilde{v}}_{i, p}}]}^{T}$ be defined as vectors then the vectorized demonstration of ${\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k})$ in (3.3) is as follows:

${\tilde{d}}^{2} ({\tilde{v}}_{i}, {\tilde{x}}_{k}) \approx {(∥ m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} ∥^{2}, 2 {| m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} |}^{T} (s_{{\tilde{v}}_{i}} + s_{{\tilde{x}}_{k}}))}_{T},$ (4.5) where ∥x - y ∥ ² = (x - y) ^T (x - y). According to the lemma used in [11], the following expression can be used instead of expression (4.1): $\begin{matrix} \min_{m_{\tilde{v}} s_{\tilde{v}}, U} \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} \\ [\begin{matrix} {(m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}})}^{T} (m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}}) \\ + 2 K {| m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} |}^{T} (s_{{\tilde{v}}_{i}} + s_{{\tilde{x}}_{k}}) \end{matrix}] \\ subject to {\begin{matrix} \sum_{i = 1}^{c} u_{i, k} = 1, for k = 1, \dots, n \\ s_{{\tilde{v}}_{i, j}} ⩾ 0, for i = 1, \dots, c and j = 1, \dots, p \end{matrix} \end{matrix}$

where $\sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} {(m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}})}^{T} (m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}})$ characterizes the method complexity and $2 \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{i, k}^{θ} {| m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} |}^{T} (s_{{\tilde{v}}_{i}} + s_{{\tilde{x}}_{k}})$ characterizes the method vagueness (see [11]). More vagueness in the fuzzy data clustering methods means more inexactness in the data clustering results. The trade off parameter K ⩾ 0 could be chosen by the decision-maker [11]. More vagueness in the fuzzy data clustering methods means more inexactness in the data clustering results and it is possible to achieve the output with high inexactness (high Membership grade) or low inexactness (low membership grade) by adjust the k parameter. Therefore, as it can be seen in the simulated and real data clustering results, by adjust the k parameter the inexactness of output can be controlled.

The corresponding Lagrange function of (0.6) is as follows:

$\begin{matrix} L (m_{\tilde{v}} s_{\tilde{v}}, U) = \sum_{i = 1}^{c} \sum_{k = 1}^{n} \sum_{j = 1}^{p} \\ \begin{matrix} u_{i, k}^{θ} [\begin{matrix} {(m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}})}^{2} \\ + 2 K | m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}} | (s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}}) \end{matrix}] \\ - \sum_{k = 1}^{n} λ_{k} ((\sum_{i = 1}^{c} u_{i, k}) - 1) \\ - \sum_{i = 1}^{c} \sum_{j = 1}^{p} β_{i, j} s_{{\tilde{v}}_{i, j}} \end{matrix} \end{matrix}$ (4.7)

To obtain the optimal solution, the following conditions should be satisfied:

$\begin{matrix} \frac{\partial L}{\partial u_{i, k}} = 0 \to u_{i, k} = \\ {[\sum_{r = 1}^{c} {(\frac{| | m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} | |^{2} + 2 K | m_{{\tilde{v}}_{i}} - m_{{\tilde{x}}_{k}} |^{T} (s_{{\tilde{v}}_{i}} + s_{{\tilde{x}}_{k}})}{| | m_{{\tilde{v}}_{r}} - m_{{\tilde{x}}_{k}} | |^{2} + 2 K | m_{{\tilde{v}}_{r}} - m_{{\tilde{x}}_{k}} |^{T} (s_{{\tilde{v}}_{r}} + s_{{\tilde{x}}_{k}})})}^{\frac{1}{θ - 1}}]}^{- 1} \end{matrix}$ (4.8)

$\begin{matrix} \frac{\partial L}{\partial m_{{\tilde{v}}_{i, j}}} = 0 \to 2 \sum_{k = 1}^{n} u_{i, k}^{θ} [(m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}}) \\ + K (sgn (m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}})) (s_{{\tilde{v}}_{i, j}} + s_{{\tilde{x}}_{k, j}})] = 0 \end{matrix}$ (4.9) $\frac{\partial L}{\partial s_{{\tilde{v}}_{i, j}}} = 0 \to 2 K \sum_{k = 1}^{n} u_{i, k}^{θ} | m_{{\tilde{v}}_{i, j}} - m_{{\tilde{x}}_{k, j}} | = β_{i, j}$ (4.10) $β_{i, j} s_{{\tilde{v}}_{i, j}} = 0$ (4.11) $β_{i, j} ⩾ 0$ (4.12)

where k = 1, ⋯ , n, i = 1, ⋯ , c, j = 1, ⋯ , p and $sgn (x) = {\begin{matrix} 1, x > 0 \\ 0, x = 0 \\ - 1, x < 0 \end{matrix}$ . According to equation (4.10), equation (4.12) will always be satisfied. According to (4.11) we can conclude that ∀i, j, one of the β_i,j or $s_{{\tilde{v}}_{i, j}}$ or both are zero as a result. If β_i,j = 0, from (4.10) it can be concluded that c = n (the number of clusters is equal to the number of input numbers or each cluster have only one member) and acording to (4.8) fuzzy c-means would be converted to the nearest mean “hard” clustering method (k-means). In this case from (4.10), the centers means ( $m_{{\tilde{v}}_{i}}$ ) and input numbers means ( $m_{{\tilde{x}}_{k}}$ ) would become the same. According to (4.9) $s_{{\tilde{v}}_{i}}$ can take any arbitrary non-negative value. Therefore we can summarize this case as: $\begin{matrix} m_{{\tilde{v}}_{i, j}} = m_{{\tilde{x}}_{k_{i}, j}}, {k^{'}}_{i} \in {1, \dots, n} and {k^{'}}_{i_{1}} \neq {k^{'}}_{i_{2}} \\ for i_{1} \neq i_{2} \\ u_{i, k} = {\begin{matrix} 1, {k = k^{'}}_{i} \\ 0, k \neq {k^{'}}_{i} \end{matrix} \\ s_{{\tilde{v}}_{i, j}} has an arbitrary non - negative value \end{matrix}$

However, the mentioned case is not our major concern, so we will not examine it any more.

Considering the relation c < n, which is common in the clustering method, with equation (4.10), we can conclude that β_i,j > 0, therefore (4.11) results in $s_{{\tilde{v}}_{i, j}} = 0$ for i = 1, ⋯ , c and j = 1, ⋯ , p. It means that the centers here cannot be triangular fuzzy numbers and are crisp. Other unknown parameters u_i,k and $m_{{\tilde{v}}_{i, j}}$ can be obtained from (4.8) and (4.9) respectively. As the proposed method is an iterative method, let’s assume obtaining the unknown parameters u_i,k and $m_{{\tilde{v}}_{i, j}}$ for k = 1, ⋯ , n, i = 1, ⋯ , c and j = 1, ⋯ , p in the l-th iteration i.e. $u_{i, k}^{(l)}$ and $m_{{\tilde{v}}_{i, j}}^{(l)}$ considering the unknown parameters in the previous iteration i.e. $u_{i, k}^{(l - 1)}$ and $m_{{\tilde{v}}_{i, j}}^{(l - 1)}$ . The practical equations to achieve $u_{i, k}^{(l)}$ and $m_{{\tilde{v}}_{i, j}}^{(l)}$ can be obtained from (4.8) and (4.9) respectively as follows:

$u_{i, k}^{(l)} = {[\sum_{r = 1}^{c} {(\frac{\sum_{j = 1}^{p} [{(m_{{\tilde{v}}_{i, j}}^{(l - 1)} - m_{{\tilde{x}}_{k, j}})}^{2} + 2 K | m_{{\tilde{v}}_{i, j}}^{(l - 1)} - m_{{\tilde{x}}_{k, j}} | s_{{\tilde{x}}_{k, j}}]}{\sum_{j = 1}^{p} [{(m_{{\tilde{v}}_{r, j}}^{(l - 1)} - m_{{\tilde{x}}_{k, j}})}^{2} + 2 K | m_{{\tilde{v}}_{r, j}}^{(l - 1)} - m_{{\tilde{x}}_{k, j}} | s_{{\tilde{x}}_{k, j}}]})}^{\frac{1}{θ - 1}}]}^{- 1}$ (4.14)

$m_{{\tilde{v}}_{i, j}}^{(l)} = \frac{\sum_{k = 1}^{n} {(u_{i, k}^{(l)})}^{θ} [m_{{\tilde{x}}_{k, j}} - K (sgn (m_{{\tilde{v}}_{i, j}}^{(l - 1)} - m_{{\tilde{x}}_{k, j}})) s_{{\tilde{x}}_{k, j}}]}{\sum_{k = 1}^{n} {(u_{i, k}^{(l)})}^{θ}}$ (4.15)

Finally, we can summarize the proposed method and write its semi-code through the following steps:

Step 1. Set l = 1. If c = n use (4.13) to calculate $u_{i, k}^{(l)}$ , $m_{{\tilde{v}}_{i, j}}^{(l)}$ and $s_{{\tilde{v}}_{i, j}}^{(l)}$ ; go to Step 4. Otherwise, if c < n choose a number for l_Max∈ { 1, 2, ⋯ }, θ > 1, K ⩾ 0 and $m_{{\tilde{v}}_{i, j}}^{(0)}$ randomly for i = 1, …, c and j = 1, ⋯ , p. l is iteration number and l_Max is the Maximum Iteration.

Step 2. Calculate $u_{i, k}^{(l)}$ and $m_{{\tilde{v}}_{i, j}}^{(l)}$ for k = 1, ⋯ , n, i = 1, ⋯ , c and j = 1, ⋯ , p using (4.14) and (4.15) respectively.

Step 3. If l = l_Max or $u_{i, k}^{(l)}$ and $m_{{\tilde{v}}_{i, j}}^{(l)}$ for k = 1, ⋯ , n, i = 1, ⋯ , candj = 1, ⋯ , p were not changed with respect to $u_{i, k}^{(l - 1)}$ and $m_{{\tilde{v}}_{i, j}}^{(l - 1)}$ , set $s_{{\tilde{v}}_{i, j}}^{(l)} = 0$ for i = 1, …, c and j = 1, ⋯ , p. Otherwise l → l + 1 and go to Step 2.

Step 4. Let $u_{i, k}^{(l)}$ , ${\tilde{v}}_{i, j} = {(m_{{\tilde{v}}_{i, j}}^{(l)}, s_{{\tilde{v}}_{i, j}}^{(l)})}_{T}$ for k = 1, ⋯ , n, i = 1, ⋯ , candj = 1, ⋯ , p as the output of proposed clustering method.

Note: Why fuzzy distance has better performance relative to the Yang distance [3, 5] in clustering?

Let $\tilde{x} = {(m_{\tilde{x}}, s_{\tilde{x}})}_{T}$ , $\tilde{y} = {(m_{\tilde{y}}, s_{\tilde{y}})}_{T}$ and $\tilde{y} + \tilde{Δ y} = {(m_{\tilde{y}} + m_{\tilde{Δ y}}, s_{\tilde{y}} + s_{\tilde{Δ y}})}_{T}$ be three triangular fuzzy numbers. The Yang distance definition [3, 5] for $\tilde{x}$ and $\tilde{y}$ is: $\begin{matrix} d^{2} (\tilde{x}, \tilde{y}) = {(m_{\tilde{x}} - m_{\tilde{y}})}^{2} \\ + {((m_{\tilde{x}} - \frac{1}{2} s_{\tilde{x}}) - (m_{\tilde{y}} - \frac{1}{2} s_{\tilde{y}}))}^{2} \\ + \dots {((m_{\tilde{x}} + \frac{1}{2} s_{\tilde{x}}) - (m_{\tilde{y}} + \frac{1}{2} s_{\tilde{y}}))}^{2} \end{matrix}$

In the proposed fuzzy distance, ${\tilde{d}}^{2} (\tilde{x}, \tilde{y}) \neq {\tilde{d}}^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y})$ , $\forall \tilde{Δ y} = (m_{\tilde{Δ y}} \neq 0,$ ${s_{\tilde{Δ y}} \neq 0)}_{T}$ While there exist a nonzero fuzzy

number like $\tilde{Δ y} = {(m_{\tilde{Δ y}} \neq 0, s_{\tilde{Δ y}} \neq 0)}_{T}$ for the case of Yang’s crisp distance definition (4.16) where $d^{2} (\tilde{x}, \tilde{y}) = d^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y})$ .

For simplicity, suppose $m_{\tilde{x}} = \frac{1}{2} s_{\tilde{x}}$ and $m_{\tilde{y}} = \frac{1}{2} s_{\tilde{y}}$ . If we rewrite the equation $d^{2} (\tilde{x}, \tilde{y}) = d^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y})$ using the above assumption, we will get:

$\begin{matrix} - 2 (m_{\tilde{x}} - m_{\tilde{y}}) m_{\tilde{Δ y}} + m_{\tilde{Δ y}}^{2} \\ + {(m_{\tilde{Δ y}} - \frac{1}{2} s_{\tilde{Δ y}})}^{2} + {(m_{\tilde{Δ y}} + \frac{1}{2} s_{\tilde{Δ y}})}^{2} \\ - 4 (m_{\tilde{x}} - m_{\tilde{y}}) (m_{\tilde{Δ y}} + \frac{1}{2} s_{\tilde{Δ y}}) = 0 \end{matrix}$ (4.17)

To make it more simple, let $m_{\tilde{Δ y}} = - \frac{1}{2} s_{\tilde{Δ y}}$ . Therefore, considering the previously equation we have:

$\begin{array}{l} 5 m_{\tilde{Δ} y}^{2} - 2 (m_{\tilde{x}} - m_{\tilde{y}}) m_{\tilde{Δ} y} = 0 \to m_{\tilde{Δ} y} \\ = 0 o r m_{\tilde{Δ} y} = 0.4 * (m_{\tilde{x}} - m_{\tilde{y}}) \end{array}$ (4.18)

Hence for two fuzzy numbers $\tilde{x} = {(m_{\tilde{x}}, 2 m_{\tilde{x}})}_{T}$ and $\tilde{y} = {(m_{\tilde{y}}, 2 m_{\tilde{y}})}_{T}$ where $m_{\tilde{x}} < m_{\tilde{y}}$ , if we define $\tilde{Δ y} = {(0.4 (m_{\tilde{x}} - m_{\tilde{y}}), 0.8 (m_{\tilde{y}} - m_{\tilde{x}}))}_{T}$ , according to the Yang distance definition (4.16) we will have: $d^{2} (\tilde{x}, \tilde{y}) = d^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y}) = 5 {(m_{\tilde{x}} - m_{\tilde{y}})}^{2}$

While in the proposed method and with fuzzy distance definition (4.4) we have: ${\begin{matrix} {\tilde{d}}^{2} (\tilde{x}, \tilde{y}) = {({(m_{\tilde{x}} - m_{\tilde{y}})}^{2}, 2 | m_{\tilde{x}} - m_{\tilde{y}} | (s_{\tilde{x}} + s_{\tilde{y}}))}_{T} \\ {\tilde{d}}^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y}) = {(\begin{matrix} {(m_{\tilde{x}} - m_{\tilde{y}} - m_{\tilde{Δ y}})}^{2}, \\ 2 | m_{\tilde{x}} - m_{\tilde{y}} - m_{\tilde{Δ y}} | (s_{\tilde{x}} + s_{\tilde{y}} + s_{\tilde{Δ y}}) \end{matrix})}_{T} \end{matrix}$

It can be proved that there not exist $\tilde{Δ y} = {(m_{\tilde{Δ y}} \neq 0, s_{\tilde{Δ y}} \neq 0)}_{T}$ that ${\tilde{d}}^{2} (\tilde{x}, \tilde{y}) = {\tilde{d}}^{2} (\tilde{x}, \tilde{y} + \tilde{Δ y})$ . Therefore, we can conclude the proposed fuzzy distance is more exact comparing the Yang distance.

Even using the defuzzifying process (see equation (4.6)), in the condition of repeating the procedure, just like Yang distance, it will result in the following equation: $\begin{matrix} m_{\tilde{Δ y}}^{2} + 2 K | m_{\tilde{x}} - m_{\tilde{y}} - m_{\tilde{Δ y}} | (s_{\tilde{x}} + s_{\tilde{y}} + s_{\tilde{Δ y}}) \\ = 2 m_{\tilde{Δ y}} (m_{\tilde{x}} - m_{\tilde{y}}) + 2 K | m_{\tilde{x}} - m_{\tilde{y}} | (s_{\tilde{x}} + s_{\tilde{y}}) \end{matrix}$

As it can easily be seen, the equality condition depends on K parameter. On the other hand, since the proposed method is a clustering algorithm and clustering is an unsupervised partitioning method (when all input numbers are known and there will not be any unknown test data in clustering methods) therefore, the range of input numbers would be accessible. Since K parameter can be selected by the user, we select it in a way that the satisfied $\tilde{y} + \tilde{Δ y} = {(m_{\tilde{y}} + m_{\tilde{Δ y}}, s_{\tilde{y}} + s_{\tilde{Δ y}})}_{T}$ in above equation, wouldn’t be placed in the input data range.

4 Experimental results

In this section, the performance of the proposed method will be examined using some fuzzy numbers through two subsections. In the first subsection we use the proposed method for clustering simulated fuzzy numbers. In the next subsection, the proposed method is applied on Taiwanese tea dataset to detect their grades. Consequently, the Yang method is applied on Taiwanese tea dataset and the results of the two clustering methods, the proposed one and the Yang method, will be compared.

1.4 Simulated fuzzy data clustering using the proposed method

In the first subsection, to make demonstration easier, a 2-dimensional data is used. The cores of the fuzzy numbers are shown by small directional triangles in the fig.s. Moreover, only the 0-cut HR of each triangular fuzzy number is shown in the fig.s to make it clearer. The resulted centers in each fig. are indicated by directional triangles that have magenta cores and cyan screens.

Generating 6 fuzzy numbers, we obtained the clustering results for the proposed method for c = 2 and θ = 2. Figure 6 shows the results for K = 0, 0.3, 0.6, 1, 2 and 10 respectively. Table 1 shows the fuzzy membership grade of input numbers for two resulted centers corresponding each value of K parameter. As it can be seen, for the best performance; 0 ⩽ K ⩽ 1. For K > 1, as the centers recede from input numbers, the input numbers get the same membership values for each center. This is due to the increase of vagueness in the proposed method as the K parameter takes a larger number. Obviously, this fact can be easily seen through equations (4.14) and (4.15), where by increasing K, the second term of numerator and denominator in (4.14) and the second term of numerator in (4.15) increase with respect to the corresponding first term resulting in an increase in the vagueness of the proposed method.

Fig. 6

Resulted clusters for K = 0, 0.3, 0.6, 1, 2 and 10.

Table 1

Membership values of input numbers to the clusters

		Membership values ([u_i,k] _c×n, c = 2, n = 6)
K = 0	0.8119	0.8112	0.9225	0.3008	0.2467	0.0822
	0.1881	0.1888	0.0775	0.6992	0.7533	0.9178
K = 0.3	0.8287	0.7873	0.8697	0.3680	0.3290	0.1127
	0.1713	0.2127	0.1303	0.6320	0.6710	0.8873
K = 0.6	0.8372	0.7679	0.8142	0.4016	0.3687	0.1576
	0.1628	0.2321	0.1858	0.5984	0.6313	0.8424
K = 1	0.7413	0.7509	0.7497	0.3846	0.3627	0.2465
	0.2587	0.2491	0.2503	0.6154	0.6373	0.7535
K = 2	0.4594	0.4911	0.4868	0.5477	0.5104	0.5183
	0.5406	0.5089	0.5132	0.4523	0.4896	0.4817
K = 10	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000

1.5 The methods application in Taiwanese tea dataset clustering

In this sub-section, we are going to use the proposed method for clustering Taiwanese tea, the defined dataset in [5]. Furthermore, the performance of proposed method will be compared with Yang FCM which is called FCN (see [3, 5]).

Tea has become an important agricultural product in Taiwan and there exist currently 20,000 ha of tea farms with an annual production of 21,000 tons. Different types of tea produced in Taiwan include green tea, Paochong, Oolong and black tea. Recently, Paochong and Oolong varieties have cornered the other two markets. Because the tea varieties and prices are numerous and complicated, many consumers have been confused. To give consumers a better understanding of Taiwanese tea, the Taiwan Tea Experiment Station (TTES) has been making serious attempts to formulate an evaluation system for tea quality. Generally speaking there are four criteria used to evaluate tea quality: appearance, tincture, liquid color and aroma.

Sincetea evaluation is always judged by experts, its quality levels are described by terms: perfect, good, medium, poor and bad. These five quality levels cause an inherent ambiguity in human perception. Since fuzzy sets are proper for describing ambiguity and imprecision in natural language, these terms can be defined using triangular fuzzy numbers as follows: X_perfect = (1, 0.25, 0) _T, X_good =(0.75, 0.25, 0.25) _T, X_medium = (0.5, 0.25, 0.25) _T, X_poor = (0.25, 0.25, 0.25) _T and X_bad = (0, 0, 0.25) _T. These representations are shown in Fig. 7. Because tea evaluation varies by the criteria of each individual expert, 10 experts were assigned to evaluate each kind of tea and assign the quality levels of perfect, good, medium, poor and bad to the four criteria of appearance, tincture, liquid color and aroma. For each criterion, a fuzzy arithmetic average was used to obtain a fuzzy number and then perfect, good, medium, poor or bad labels were assigned to that fuzzy number. The final evaluation data is shown in Table 2. Let ${\tilde{x}}_{k, j} = {(m_{k, j}, l_{k, j}, r_{k, j})}_{T}$ be assessed by k-th criterion of the j-th type of tea, j = 1, 2, 3, 4, k = 1, …, 69. The overall performance for k-th type of tea is determined as ${\bar{x}}_{k} = {({\bar{m}}_{k}, {\bar{l}}_{k}, {\bar{r}}_{k})}_{T}$ where ${\bar{m}}_{k} = \frac{1}{4} \sum_{j = 1}^{4} m_{k, j}, {\bar{l}}_{k} = \frac{1}{4} \sum_{j = 1}^{4} l_{k, j}, {\bar{r}}_{k} = \frac{1}{4} \sum_{j = 1}^{4} r_{k, j} .$ (5.2.1)

Fig. 7

Five triangular fuzzy numbers for a particular criterion.

Here the input numbers are fuzzy triangular but, our proposed method is applicable to the samples of which features are symmetric triangular fuzzy numbers. Therefore, we approximate each input member by the nearest corresponding symmetric triangular fuzzy number as follows [5]: ${\bar{x}}_{k} = {({\bar{m}}_{k}, ({\bar{l}}_{k} + {\bar{r}}_{k}) / 2)}_{T} .$

Now, we apply the proposed method and FCN to cluster resulted Taiwanese teas symmetric triangular fuzzy numbers assuming θ = 2 and c = 5. Furthermore, in the proposed method we assign two values for K ; 0.01 and 0.3. Resulted clusters centers and membership values of each tea type in clusters are obtained in Table 2 for each clustering method.

Table 2

Proposed clustering method and FCN applying over 69 types of Taiwan tea tree

It must be mentioned, in order to achieve a valid K, in the proposed method, where $0 ⩽ m_{\tilde{x}}, m_{\tilde{y}}, m_{\tilde{y} + \tilde{Δ y}} ⩽ 1$ , $0 < s_{\tilde{x}} ⩽ 0.25, s_{\tilde{y}} = s_{\tilde{y} + \tilde{Δ y}} = 0$ ( $\tilde{x}$ is supposed to be derived from input data while $\tilde{y}$ and $\tilde{y} + \tilde{Δ y}$ are from prototypes), the equation (3.21) results K as follows: $K = \frac{m_{\tilde{Δ y}} (2 m_{\tilde{x}} - 2 m_{\tilde{y}} - m_{\tilde{Δ y}})}{2 s_{\tilde{x}} (| m_{\tilde{x}} - m_{\tilde{y}} - m_{\tilde{Δ y}} | - | m_{\tilde{x}} - m_{\tilde{y}} |)}$

Making a full search in the mentioned intervals, it can be concluded that there is no K > 0 to satisfy the above equation.

To calculate the amount of fuzziness in clustering results, the following validity index is proposed: $XB = (\sum_{k = 1}^{n} \sum_{i = 1}^{c} \min (u_{i, k}, 1 - u_{i, k})) / n$

In the k-means clustering method of which the clustering method is crisp and fuzziness value is equal to zero, we have XB = 0. When the fuzziness in the clustering results increases, the index XB grows up. Therefore, in the FCM clustering method and when $u_{i, k} = \frac{1}{c}$ for i = 1, ⋯ , c and k = 1, ⋯ , n and consequently the maximum fuzziness is achieved, XB = 1.

The corresponding grades of each tea type using three methods are bolded in Table 2. It can be easily observed that choosing K = 0.01, the proposed method becomes the “nearest mean “hard” clustering method (k-means)” applied to tea fuzzy numbers (because the corresponding fuzziness validity index XB is 0.0861, which is near to zero) and taking K = 0.3 it becomes a fuzzy c-means clustering method (because the corresponding fuzziness validity index XB is 0.9577, which is near to one), in which the vagueness of the proposed method increases as K parameter takes a larger number. However, the FCN works as a “nearest mean “hard” clustering method (k-means)” on tea fuzzy numbers (because the corresponding fuzziness validity index XB is 0.1458, which is near to zero). As a conclusion it can be inferred from above that the proposed clustering method is a general form of FCN concerning the change in the K parameter.

After testing the proposed method on several datasets, it is possible to formulate some conclusions on the value of K parameter as:

The proposed method $(K) = {\begin{matrix} nearest mean “ {hard}^{″} clustering method (k - means), K < 0.1 \\ fuzzy c - means clustering method, K ⩾ 0.1 \end{matrix}$

In the Table 2 last column, the proposed method has been compared with FCN method [2] and the results show that the proposed method (by adjust k) is a global mode for fuzzy clustering.

The student dataset [28] consists of six students and four attributes. Table 3 shows the corresponding fuzzy data. Wen et al. [29] implement the SCA for the data set and then single linkage hierarchical algorithm on the final data. They achieve two well-separated clusters and hence the optimal cluster number c^* is 2. The clustering results show that the data points 1, 2 and 4 belong to cluster 1 and the data points 3, 5 and 6 belong to cluster 2. That is, Tom, David and Jane belong to cluster 1 and Bob, Joe and Jack belong to cluster 2. They find that (i) the members of cluster 1 have high marks in Mathematics 2 and Physics and the moderate mark in Mathematics 1, (ii) the members of cluster 2 have very lower mark in Mathematics1, moderate mark in Mathematics2 and higher mark in Physics. In the sense, the students in cluster2 have bad performance in Mathematics.

Table 3

Student dataset

Student	Physics 1	Mathematics 1	Physics 2	Mathematics 2
Tom	(10,10)	(15,0)	(15,1)	(12,2)
David	(12,2)	(9,0)	(10,0)	(16,2)
Bob	(16.5,3.5)	(6,0)	(16,2)	(10.5,0.5)
Jane	(19,0)	(12,2)	(11,1)	(19,1)
Joe	(12,2)	(1,1)	(14,0)	(8,2)
Jack	(9,0)	(1,0)	(7.5,1.5)	(5,1)

The proposed method has been tested on the student dataset too. Table 4 shows the result for two different values of k. It can be seen for k = 0.05 two clusters has been achieved: the members of cluster 1 have higher average marks in mathematics (Tom, David and Jane) and the members of cluster 2 have higher average marks in physics (Bob, Joe and Jack) and for k = 0.4 the result is fuzzy clustering which obtained in [29].

Table 4

Resulted overall accuracy (OA) and Kappa (Ka) indices in the accuracy assessment of proposed method and other literature results

Index Method	% OA	% Ka
The Proposed method	62.34	46.12
Segmentation using Meta fusion [30]	57.35	41.06
Kernel Level Set Segmentation [34]	57.15	40.67
Multilevel thresholding based on Electro-magnetism Optimization [35]	55.78	38.36
Multilevel thresholding based on harmony search optimization [36]	57.11	40.84

To demonstrate the efficiency of the proposed method, its performance is evaluated in segmentation of land-cover using satellite images. For this purpose two panchromatic satellite images Geoey-1 (with 0.5 m spatial resolution) and IRS-P5 (with 2.5 m spatial resolution) are utilized that are located in south of Iran, with geographical location of 26.67–26.8 N and 56.04–56.06 E. These images are resampled and registered together and have same size (1000*1000 pixels) [30].

To obtain the ground truth (GT) data for accuracy assessment, as same as [30], the correspondent map is used where four classes (sea with black DN, main road with new and dark asphalt, sidetrack with old and bright asphalt and building’s roof with white DN) are selected and labeled to 1, 2, 3 and 4 respectively; according to DN increasing manner. Table 4 shows the simulation results of proposed (with K = 0.11, obtained by trial and error method) and comparing methods where overall accuracy (OA) and Kappa (Ka) validity indices are utilized.

In the aspect of complexity analysis, the proposed method for each K-parameter value is as same as the conventional FCM method [37, 38]. Since, to get the best K-parameter the rial and error method (by varying from 0 to 1) is utilized in this paper and wide range values of K-parameter must be evaluated, the overall complexity of proposed method is very large and multiple value of conventional FCM method, depending on number of utilized different K-parameter values.

5 Conclusion

In this paper, we applied FCM clustering method on symmetric triangular fuzzy numbers based on the definition of a fuzzy distance between two fuzzy numbers. We used fuzzy smaller concept for symmetric triangular fuzzy numbers and defined a fuzzy metric for them. Similar to the conventional FCM, we found the unknown parameters in clustering process by minimizing a fuzzy objective function using a wellknown defuzzification procedure. It is proved that the proposed fuzzy distance expresses the distance of two fuzzy numbers better than crisp metric (the Yang metric [3, 5]). It is shown that the proposed method is a general form of FCN [3, 5], base on the changes in the proposed method parameters.

K parameter in the proposed method has been obtained with trial and error method and this is main problem in this work. In the future research, this parameter can be obtained with smart algorithm systems. It is to be noted that, the proposed algorithm is potential to apply in other wide datasets too.

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