Intuitionistic fuzzy transform and its application 1

Abstract

Intuitionistic fuzzy transform is an approximate method based on the intuitionistic fuzzy partition. To begin with, a novel definition of intuitionistic fuzzy partition is given, and the properties of triangular intuitionistic fuzzy partition are also given. Secondly, the method of intuitionistic fuzzy transform is introduced, which transforms a continuous function into two gravity vectors according to the membership and non-membership functions that based on intuitionistic fuzzy partition. Some fundamental properties of intuitionistic fuzzy partition are surveyed. Thirdly, the method of inverse intuitionistic fuzzy transform is established by using the previous gravity vectors corresponding to the intuitionistic fuzzy partition. The results show that the approximate function of the original one can be rebuilt by the membership and non-membership functions respectively. even a hybrid approximate function can be rebuilt by both the membership and non-membership functions. Finally, some elementary properties of the inverse intuitionistic fuzzy transform are studied and the method is illustrated by a specific example.

Keywords

Intuitionistic fuzzy set intuitionistic fuzzy partition intuitionistic fuzzy transform inverse intuitionistic fuzzy transform

1. Introduction

Since the fuzzy set was first proposed by Zadeh [29] in 1965, fuzzy modeling, as the resulting methodology of fuzzy set, became a very effective and flexible technique for the solution of complex practical application. Among which, fuzzy transform (F-transform) is the most popular one and has attracted considerable attention in the fuzzy theory community. F-transform has solved many problems such as data analysis [7 , 21], images processing [18 , 24] and differential equations solving [13, 14], et al.

F-transform was proposed by Perfilieva in [16, 17]. It was an operator which transforms a continuous function on an interval [a, b] into a n-dimensional vector by taking advantage of fuzzy partition. It offers a new setting to obtain fine approximation of functions of single and multiple variables. On the contrary, an inverse F-transform(i-F-transform) converts a n-dimensional vector into a new continuous function which approximates the original one. The advantage of i-F-transform is that the approximate function is more simple and easy to compute than the original one. The membership degree plays a key role in both F-transform and i-F-transform, especially for fuzzy number, which is a special case of fuzzy set [6, 9].

Because of the complexity of data and the ambiguity of human mind, it is difficult to make use of a membership degree function to characterize the accurate membership degrees in the classical fuzzy set theory. As an extension of Zadeh’s fuzzy set, Atanassov introduced the concept of intuitionistic fuzzy set (IFS) in [1 –3]. IFS fully reflects the ambiguity of things. IFS may express more abundant and flexible information as compared with the fuzzy set [28]. For example, it can express support, objection, and hesitation in reality. Up to now, the studies of IFS have attracted great attention and have been applied to various fields, such as clustering analysis, decision making, medical diagnosis, pattern recognition, and so on.

Nevertheless, there is no literature to discuss the method of intuitionistic fuzzy transform as far as we know. The question is how to establish the intuitionistic fuzzy transform. For this reason, the main purpose of this paper is to establish the direct and inverse intuitionistic fuzzy transform by IFS theory. Thus, the advantage of IFS in dealing with uncertain and vague information, allows us to obtain the more powerful approximate function which can substitute the original one.

The paper is organized as follows. In Section 2, we remind necessary basic concepts of intuitionistic fuzzy set, triangular intuitionistic fuzzy number, intuitionistic fuzzy partition and their fundamental properties. In Section 3, motivated by earlier research, we propose the intuitionistic fuzzy transform and study some basic properties of it. In Section 4, we present the inverse intuitionistic fuzzy transform. Finally, we draw the conclusions and give a specific example to illustrate our method.

2. Preliminaries

In this section, we give a brief review of some preliminaries.

2.1. Intuitionistic fuzzy set, intuitionistic fuzzy number and triangular intuitionistic fuzzy number

To begin with, we introduce some basic notions related to IFSs.

Definition 2.1. [3] Let X be a nonempty set. An IFS in X is defined by A = (μ_A, ν_A), where ν_A : X → [0, 1] and ν_A : X → [0, 1] such that 0 ≤ μ_A (x) + ν_A (x) ≤1, ∀ x ∈ X. The numbers μ_A (x) , ν_A (x) ∈ [0, 1] denote the degree of membership and non-membership of x ∈ A respectively. For each intuitionistic fuzzy subset A in X, π_A (x) =1 − μ_A (x) − ν_A (x) stands for the degree of hesitation or uncertainty for the object x to belong to A.

Intuitionistic fuzzy set was extended naturally from fuzzy set by adding an additional non-membership degree, which may express more abundant and flexible information as compared with the fuzzy set.

As a extension of fuzzy numbers, an intuitionistic fuzzy number (IFN) is easier to deal with the fuzzy and the uncertain information. Recently, the researches on IFNs have received a little attention and several definitions of IFNs and ranking methods have been proposed [5 , 27]. For the purpose of describing fuzzy information, the results can be expressed by some special forms of IFNs, such as triangular intuitionistic fuzzy numbers(TIFNs), trapezoidal intuitionistic fuzzy numbers(TrIFNs).

Definition 2.2. [27] An IFN A = (μ_A, ν_A) in set of real numbers R, is defined as

$\begin{matrix} μ_{A} (x) = {\begin{matrix} f_{A} (x), & a \leq x < b_{1}, \\ 1, & b_{1} \leq x < b_{2}, \\ g_{A} (x), & b_{2} \leq x \leq c, \\ 0, & x < a or x > c . \end{matrix} \\ ν_{A} (x) = {\begin{matrix} h_{A} (x), & e \leq x < f_{1}, \\ 0, & f_{1} \leq x < f_{2}, \\ k_{A} (x), & f_{2} \leq x \leq g, \\ 1, & x < e or x > g . \end{matrix} \end{matrix}$ (1) where 0 ≤ μ_A (x) + ν_A (x) ≤1 and a, b₁, b₂, c, e, f₁, f₂, g ∈ R such that e ≤ a, f₁ ≤ b₁ ≤ b₂ ≤ f₂, c ≤ g, and four functions f_A, g_A, h_A, k_A : R → [0, 1] are called the legs of membership function μ_A and non-membership function ν_A. f_A and k_A are non-decreasing continuous functions. h_A and g_A are non-increasing continuous functions.

The triangular fuzzy number(TFN) was first introduced by Dubois and Prade [9]. The concept of a TIFN is defined in a similar way as follows:

Fig. 1

An IFN and a TIFN 〈 (a, b, c) ; w, u〉.

Definition 2.3. [10] TIFN A TIFN A is a special IFS in set of real numbers R, whose membership function μ (x) and the non-membership function ν (x) are defined respectively as follows: $\begin{matrix} μ (x) = {\begin{matrix} 0, & x < a or x > c; \\ \frac{w}{b - a} (x - a), & a \leq x < b; \\ \frac{w}{c - b} (c - x), & b \leq x \leq c . \end{matrix} \end{matrix}$ $ν (x) = {\begin{matrix} 1, & x < a or x > c; \\ \frac{1 - u}{b - a} (b - x) + u, & a \leq x < b; \\ \frac{1 - u}{c - b} (x - b) + u, & b \leq x \leq c . \end{matrix}$ (2)

In this paper, we denote a TIFN by A = 〈 (a, b, c) ; w, u〉, where w represents the maximum degree of membership and u represents the minimum degree of non-membership such that satisfied 0 ≤ w, u ≤ 1 and 0 ≤ w + u ≤ 1.

2.2. Intuitionistic Fuzzy partition

Intuitionistic fuzzy partition (IF-Partition) plays a vital role in both direct and inverse intuitionistic fuzzy transform(i-IF-transform). Being similar to the definition of IF-Partition in [25, 26], we give another novel one as follows:

Definition 2.4. def.IF-Partition Let $ℙ = {x_{1} = a < x_{2} < \dots <$ x_n = b} be fixed points in [a, b]. IFNs $A = {A_{1} (x), A_{2} (x), \dots, A_{n} (x)}$ , identified with their membership functions μ₁ (x) , …, μ_n (x) and non-membership functions ν₁ (x) , …, ν_n (x), i.e., A_k (x) = 〈μ_k (x) , ν_k (x) 〉 defined on [a, b]. The IFNs form an IF-Partition of [a, b], denoted by $(ℙ, A)$ , if they fulfill the following conditions for k = 1, …, n:

μ_k, ν_k : [a, b] → [0, 1];

μ_k (x) and ν_k (x) are continuous functions on [0, 1];

μ_k (x) , k = 2, …, n, strictly increases on [x_k-1, x_k] and strictly decreases on [x_k, x_k+1]. ν_k (x) , k = 2, …, n, strictly decreases on [x_k-1, x_k] and strictly increases on [x_k, x_k+1];

$\sum_{k = 1}^{n} μ_{k} (x) \leq 1$ , for all x ∈ [a, b];

$\sum_{k = 1}^{n} (μ_{k} (x) + ν_{k} (x)) \geq 1$ , for all x ∈ [a, b];

| {k|μ_k (x) + ν_k (x) ≤1, x ∈ [a, b]} |≤1(i.e., there is at most one IFS such μ_k (x) + ν_k (x) ≤1).

For the rest of this paper, $(ℙ, A)$ denotes an IF-Partition. $(ℙ, A_{T})$ denotes a triangular intuitionistic fuzzy partition (TIF-Partition) when IFNs are TIFNs $A = {A_{1} (x), A_{2} (x), \dots, A_{n} (x)}$ where A₁ (x) = 〈 (x₁, x₁, x₂) , w₁, u₁〉, A_k (x) = 〈 (x_k-1, x_k, x_k+1) , w_k, u_k〉, k = 2, …, n − 1, A_n (x) = 〈 (x_n-1, x_n, x_n) , w_n, u_n〉.

Let us consider an example of TIF-Partition on [a, b].

Example 2.1. exmp.1 Let $(ℙ, A_{T})$ be a TIF-Partition. Then the formal representation of such triangular membership functions μ_k (x) and non-membership functions ν_k (x) are given by following:

If k = 1, then $μ_{1} (x) = {\begin{matrix} \frac{w_{1}}{h_{1}} (x_{2} - x), & x \in [x_{1}, x_{2}]; \\ 0, & x \notin [x_{1}, x_{2}] . \end{matrix}$ (3) $ν_{1} (x) = {\begin{matrix} \frac{1 - u_{1}}{h_{1}} (x - x_{1}) + u_{1}, & x \in [x_{1}, x_{2}]; \\ 0, & x \notin [x_{1}, x_{2}] . \end{matrix}$ (4)

If k = 2, 3, …, n − 1 $μ_{k} (x) = {\begin{matrix} \frac{w_{k}}{h_{k - 1}} (x - x_{k - 1}), & x \in [x_{k - 1}, x_{k}); \\ \frac{w_{k}}{h_{k}} (x_{k + 1} - x), & x \in [x_{k}, x_{k + 1}); \\ 0, & x \notin [x_{k - 1}, x_{k + 1}) . \end{matrix}$ (5) $ν_{k} (x) = {\begin{matrix} \frac{1 - u_{k}}{h_{k - 1}} (x_{k} - x) + u_{k}, & x \in [x_{k - 1}, x_{k}); \\ \frac{1 - u_{k}}{h_{k}} (x - x_{k}) + u_{k}, & x \in [x_{k}, x_{k + 1}); \\ 0, & x \notin [x_{k - 1}, x_{k + 1}) . \end{matrix}$ (6)

If k = n, then $μ_{n} (x) = {\begin{matrix} \frac{w_{n}}{h_{n - 1}} (x_{n} - x), & x \in [x_{n - 1}, x_{n}]; \\ 0, & x \notin [x_{n - 1}, x_{n}] . \end{matrix}$ (7) $ν_{n} (x) = {\begin{matrix} \frac{1 - u_{n}}{h_{n - 1}} (x - x_{n - 1}) + u_{n}, & x \in [x_{n - 1}, x_{n}]; \\ 0, & x \notin [x_{n - 1}, x_{n}] . \end{matrix}$ (8)

Fig. 2 shows a non-uniform TIF-Partition on [a, b].

Fig. 2

TIF-Partition on [a, b].

Theorem 2.1. thm1 Let $(ℙ, A_{T})$ be a TIF-Partition. Then the following properties hold

$U_{1} = \int_{a}^{b} μ_{1} (x) d x = \int_{x_{1}}^{x_{2}} μ_{1} (x) d x = \frac{h_{1} w_{1}}{2}$ .

$V_{1} = \int_{a}^{b} ν_{1} (x) d x = \int_{x_{1}}^{x_{2}} ν_{1} (x) d x = \frac{h_{1} (1 + u_{1})}{2}$ .

$U_{k} = \int_{a}^{b} μ_{k} (x) d x = \int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) d x = \frac{w_{k} (h_{k - 1} + h_{k})}{2}$ .

$V_{k} = \int_{a}^{b} ν_{k} (x) d x = \int_{x_{k - 1}}^{x_{k + 1}} ν_{k} (x) d x = \frac{(1 + u_{k}) (h_{k - 1} + h_{k})}{2}$ .

$U_{n} = \int_{a}^{b} μ_{n} (x) d x = \int_{x_{n - 1}}^{x_{n}} μ_{n} (x) d x = \frac{h_{n} w_{n}}{2}$ .

$V_{n} = \int_{a}^{b} ν_{n} (x) d x = \int_{x_{n - 1}}^{x_{n}} ν_{n} (x) d x = \frac{h_{n} (1 + u_{n})}{2}$ .

Proof. Since IFNs are TIFNs, it is easy to show that functions in Equations(3)-(8) are integrable. Therefore, the results can be obtain by integration.□

Theorem 2.2. Let $(ℙ, A_{T})$ be a TIF-Partition and f be a continuous function on [a, b]. Then the following property holds: $\frac{\int_{a}^{b} f (x) μ_{k} (x) d x}{\int_{a}^{b} μ_{k} (x) d x} = \frac{\int_{a}^{b} f (x) (1 - ν_{k} (x)) d x}{\int_{a}^{b} (1 - ν_{k} (x)) d x} .$ (9)

Proof. Without loss of generality, we may prove the theorem in case of k = 2, 3, ⋯ , n − 1. Given a TIF-Partition $(ℙ, A_{T})$ , it is obvious that Equations (5) and (6). By integration, we get that $\begin{matrix} \frac{\int_{a}^{b} f (x) μ_{k} (x) d x}{\int_{a}^{b} f (x) (1 - ν_{k} (x)) d x} = \\ \frac{\int_{x_{k - 1}}^{x_{k}} f (x) \frac{w_{k}}{h_{k - 1}} (x - x_{k - 1}) d x + \int_{x_{k}}^{x_{k + 1}} f (x) \frac{w_{k}}{h_{k}} (x_{k + 1} - x) d x}{\int_{x_{k - 1}}^{x_{k}} f (x) \frac{1 - u_{k}}{h_{k - 1}} (x - x_{k - 1}) d x + \int_{x_{k}}^{x_{k + 1}} f (x) \frac{1 - u_{k}}{h_{k}} (x_{k + 1} - x) d x} \\ = \frac{w_{k}}{1 - u_{k}} . \end{matrix}$

On the other hand, according to Equations (5) and (6) and Theorem (2.1), we obtain $\begin{matrix} \frac{\int_{a}^{b} μ_{k} (x) d x}{\int_{a}^{b} (1 - ν_{k} (x)) d x} = \\ \frac{\frac{w_{k} (h_{k - 1} + h_{k})}{2}}{\int_{x_{k - 1}}^{x_{k}} \frac{(1 - u_{k})}{h_{k - 1}} (x - x_{k - 1}) d x + \int_{x_{k}}^{x_{k + 1}} \frac{(1 - u_{k})}{h_{k}} (x_{k + 1} - x) d x} \\ = \frac{w_{k}}{1 - u_{k}} . \end{matrix}$

Therefore, the proof is completed by comparing the above formulas.□

Definition 2.5. Let $(ℙ, A_{T})$ be a TIF-Partition of [a, b] be given by fixed points set $ℙ = {x_{1} = a < x_{2} < \dots < x_{n} = b}$ and IFSs $A = {A_{1} (x), \dots, A_{n} (x)}$ , where A_k (x) are TIFNs on [a, b]. The intuitionistic fuzzy partition is said uniform triangular intuitionistic fuzzy partition (UTIF-Partition) if the points x₁, x₂, …, x_n, n ≥ 3, are equidistant, which means that x_k = a + h (k − 1) , k = 1, 2, …, n, where h = (b − a)/(n − 1), and the following additional properties are fulfilled:

μ_k (x_k − x) = μ_k (x_k + x) , ν_k (x_k − x) = ν_k (x_k + x), ∀x ∈ [0, h] , k = 1, …, n;

μ_k (x) = μ_k+1 (x + h) , ν_k (x) = ν_k+1 (x + h), ∀x ∈ [x_k, x_k+1] , k = 1, 2, …, n − 1.

For the rest of this paper, $(ℙ_{U}, A_{T})$ denotes a UTIF-Partition on [a, b].

Corollary 2.3. Let $(ℙ_{U}, A_{T})$ be a UTIF-Partition, i.e., x_k = a + h (k − 1), k = 1, 2, ⋯ , n, where h = (b − a)/(n − 1). Then the following properties hold

$\int_{a}^{b} μ_{1} (x) d x = \int_{a}^{b} μ_{n} (x) d x = \frac{hw}{2}$ .

$\int_{a}^{b} μ_{k} (x) d x = \int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) d x = hw$ , for k = 2, ⋯ , n − 1.

$\int_{a}^{b} ν_{1} (x) d x = \int_{a}^{b} ν_{n} (x) d x = \frac{h (1 + u)}{2}$ .

$\int_{a}^{b} ν_{k} (x) d x = \int_{x_{k - 1}}^{x_{k + 1}} ν_{k} (x) d x = h (1 + u)$ , for k = 2, ⋯ , n − 1.

$\int_{a}^{b} (1 - ν_{1} (x)) d x = \int_{x_{1}}^{x_{2}} (1 - ν_{k} (x)) d x = \frac{h (1 - u)}{2}$ .

$\int_{a}^{b} (1 - ν_{k} (x)) d x = \int_{x_{k - 1}}^{x_{k + 1}} (1 - ν_{k} (x)) d x = h (1 - u)$ , for k = 2, ⋯ , n − 1.

Proof. This theorem can be proved in the same way as shown in Theorem 2.2. □

3. Intuitionistic fuzzy transform and its properties

In this section, we introduce the method of intuitionistic fuzzy transform (IF-transform), which transforms a continuous function f on [a, b] into two sets of n-dimensional gravity vectors, corresponding to the membership and non-membership.

3.1. IF-transform

Definition 3.1. Let f be any function from C ([a, b]) and $(ℙ, A)$ be an IF-Partition, in which A_k (x) = 〈μ_k (x) , ν_k (x) 〉, k = 1, …, n be IFNs that constitute $A$ of [a, b]. The n-tuple of real numbers F_μ,n [f] = [F_μ,1, …, F_μ,n] and F_ν,n [f] = [F_ν,1, …, F_ν,n] given by $\begin{matrix} F_{μ, k} = \frac{\int_{a}^{b} f (x) μ_{k} (x) d x}{\int_{a}^{b} μ_{k} (x) d x}, \\ F_{ν, k} = \frac{\int_{a}^{b} f (x) ν_{k} (x) d x}{\int_{a}^{b} ν_{k} (x) d x} . \end{matrix}$ (10) are called gravity vector of the membership and the non-membership of the intuitionistic fuzzy transform (IF-transform) with respect to IF-Partition $(ℙ, A)$ .

It is worth noticing that IF-transform establishes a correspondence between a set of continuous functions on an interval of real numbers and the set of n-dimensional (real) vectors.

Remark 3.1. Let $(ℙ_{U}, A_{T})$ be a TIF-Partition, i.e., A₁ (x) = 〈 (x₁, x₁, x₂) , w₁, u₁〉, A_k (x) = 〈 (x_k-1, x_k, x_k+1) , w_k, u_k〉, k = 2, …, n − 1, A_n (x) = 〈 (x_n-1, x_n, x_n) , w_n, u_n〉. Then we havethat

$F_{μ, 1} = \frac{2}{h_{1} w_{1}} \int_{a}^{b} f (x) μ_{1} (x) d x$ , $F_{ν, 1} = \frac{2}{(1 + u) (h_{k - 1} + h_{k})} \int_{a}^{b} f (x) ν_{1} (x) d x$ .

$F_{μ, k} = \frac{2}{w_{k} (h_{k - 1} + h_{k})} \int_{a}^{b} f (x) μ_{k} (x) d x$ , $F_{ν, k} = \frac{2}{(1 + u) (h_{k - 1} + h_{k})} \int_{a}^{b} f (x) ν_{k} (x) d x$ .

$F_{μ, n} = \frac{2}{h_{n} w_{n}} \int_{a}^{b} f (x) μ_{n} (x) d x$ , $F_{ν, n} = \frac{2}{(1 + u) (h_{k - 1} + h_{k})} \int_{a}^{b} f (x) ν_{n} (x) d x$ .

3.2. Properties of IF-transform

Theorem 3.1 Given any functions f and g from C ([a, b]). Let F_μ,n [f] = [F_μ,1, …, F_μ,n] and F_ν,n [f] = [F_ν,1, …, F_ν,n] be two n-dimensional (real) vectors which defined by Definition 3.1. Then, for any real number C, we have that $\begin{matrix} (1) F_{μ, n} [f + g] = F_{μ, n} [f] + F_{μ, n} [g], \\ F_{ν, n} [f + g] = F_{ν, n} [f] + F_{ν, n} [g] . \end{matrix}$ $(2) F_{μ, n} [Cf] = C F_{μ, n} [f], F_{ν, n} [Cf] = C F_{ν, n} [f] .$

Proof. Since the definition of n-dimensional vectors F_μ,n [f] and F_ν,n [f], the proof is straight forward.□

Theorem 3.2 Let f ∈ C ([a, b]) and $(ℙ_{U}, A_{T})$ be a UTIF-Partition on [a, b].

The k-th element of F_μ,k [f], i.e., F_μ,k, gives the minimum solution to function $Φ_{1} (y) = \int_{a}^{b} (f (x) - y)^{2} μ_{k} (x) d x .$ (11)

The k-th element of F_ν,k [f], i.e., F_ν,k, gives the minimum solution to function $Φ_{2} (y) = \int_{a}^{b} (f (x) - y)^{2} ν_{k} (x) d x .$ (12)

Proof. Here we only give the first case and another is in a similar approach. It is clear that the function (f (x) − y) ²μ_k (x) is continuously differentiable because of the continuously differentiable of functions f, μ_k (x) and ν_k (x). Therefore, we have that $Φ_{1}^{'} (y) = - 2 \int_{a}^{b} (f (x) - y) μ_{k} (x) d x .$ Let $Φ_{1}^{'} (y) = 0$ . Then we have that $y = \frac{\int_{a}^{b} f (x) μ_{k} (x) d x}{\int_{a}^{b} μ_{k} (x) d x}$ reaches the minimum solution of function Φ ₁ (y). □

Theorem 3.3 Given any continuous function f on [a, b] and a UTIF-Partition $(ℙ_{U}, A_{T})$ on [a, b]. Let F_μ,k and F_ν,k be the k-th component of F_μ,n [f] and F_ν,n [f], respectively. Then the following estimations hold $\begin{matrix} | f (t) - F_{μ, k} | \leq ω (2 h, f), | f (t) - F_{ν, k} | \leq ω (2 h, f) \\ for all t \in [x_{k}, x_{k + 1}], \end{matrix}$ where h = (b − a)/(n − 1) and $ω (2 h, f) = max_{| δ | \leq 2 h} max_{x \in [a, b - δ]} | f (x + δ) - f (x) |$ is the modules of continuity of function f on [a, b].

Proof. Without loss of generality, let us consider the first case and 1 ≤ k ≤ n − 1. Given any continuous function f on [a, b]. For all t ∈ [x_k, x_k+1]., then we can obtain that $\begin{matrix} | f (t) - F_{μ, k} | = | \frac{f (t) \int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) d x}{\int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) d x} - \frac{\int_{x_{k - 1}}^{x_{k + 1}} f (x) μ_{k} (x) d x}{\int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) d x} | \\ \leq \frac{\int_{x_{k - 1}}^{x_{k + 1}} | f (t) - f (x) | μ_{k} (x) D x}{\int_{x_{k - 1}}^{x_{k + 1}} μ_{k} (x) D x} \leq ω (2 h, f) . \end{matrix}$ The second can be proved by the similar way.□

4. Inverse IF-transform and its properties

In contrast to IF-transform, the method of inverse intuitionistic fuzzy transform (i-IF-transform) turns the two sets of gravity vectors into an approximate functions of the original function f.

The IF-transform converts a continuous function f on [a, b] with the IF-Partition $(ℙ, A)$ into two sets of n-dimensional real vector F_μ,n [f] and F_ν,n [f]. We now turn to the inverse problem, in other words, how to reconstructed approximate functions of the original function f. i-IF-transform

Definition 4.1. Let f be any continuous function on [a, b], which followed by Definition 2.4. Suppose that F_μ,n [f] and F_ν,n [f] be gravity vector of the membership and non-membership with respect to IF-Partition $(ℙ, A)$ on [a, b]. Then the functions $\begin{matrix} {\hat{f}}_{μ, n} (x) = \sum_{k = 1}^{n} F_{μ, k} \cdot μ_{k} (x), \\ {\hat{f}}_{ν, n} (x) = \sum_{k = 1}^{n} F_{ν, k} \cdot (1 - ν_{k} (x)), k = 1, 2, \dots, n . \end{matrix}$ (13) are called the membership-based i-IF-transform and the non-membership-based i-IF-transform.

Definition 4.2. Let f, F_μ,n [f] and F_ν,n [f] satisfy the Definition 4.1. Then the function $\begin{matrix} {\hat{f}}_{α, n} (x) = \sum_{k = 1}^{n} [α \cdot F_{μ, k} μ_{k} (x) + (1 - α) \cdot F_{ν, k} \cdot (1 - ν_{k} (x))], \\ k = 1, \dots, n \end{matrix}$ (14) are called the α-hybrid i-IF-transform, denoted with α-i-IF-transform, where α is the motive force of the membership and 1 − α of the non-membership.

Remark 4.1 Let us consider the special case as follows:

If α = 0, then ${\hat{f}}_{0, n} (x) = {\hat{f}}_{ν, n} (x) = \sum_{k = 1}^{n} F_{ν, k} \cdot (1 - ν_{k} (x)), k = 1, 2, \dots, n$ .

If α = 1, then ${\hat{f}}_{1, n} (x) = {\hat{f}}_{μ, n} (x) = \sum_{k = 1}^{n} F_{μ, k} \cdot μ_{k} (x), k = 1, 2, \dots, n$ .

Generally speaking, due to the loss of information, it results in that μ_k (x) + ν_k (x) notequiv1 in IF-Partition $(ℙ, A)$ .Thus, the IF-transform and i-IF-transform are no longer precisely. Despite all this, the original function f can be reconstructed by the α-i-IF-transform Eq.(14).

Example 4.1. By considering the function f (x) = x − sin(3x) +2 on [1, 6], we give the UTIF-Partition $(ℙ_{U}, A_{T})$ based on Definition 2.5 as show in Figs. 3, 4 and 5:

Fig. 3

UTIF-Partition of [1, 6] when n = 6, w = 0.85, u = 0.10

Fig. 4

Two i-IF-transforms with respect to n = 6, 20, 40, respectively.

Fig. 5

The i-IF-transform with respect to n = 6, 20, 40, respectively and α-hybrid (α = 0.6) i-IF-transform.

Fig. 4 (a) , (b) and (c) show the i-IF-transform ${\hat{f}}_{μ, n} (x)$ and ${\hat{f}}_{ν, n} (x)$ with respect to n = 6, 20 and 40, respectively.

Fig. 5 (a) , (b) and (c) show the special case that w = 1 and u = 0 with respect to n = 6, 20 and 40, respectively.

Fig. 5 (d) , (e) and (f) show that 0.6-hybrid i-IF-transform (red line) when the motive force α = 0.6, w = 0.85 and u = 0.10 with respect to n = 6, 20 and 40, respectively.

When the maximum membership w of TIFN increases, it is notice that i-IF-transform effect is better than the small one. The inverse IF-transform practically coincides with the original function f (x) when n = 40.

4.1. Properties of inverse IF-transform

The following properties are the fundamentals of the i-IF-transform setting.

Theorem 4.1. Let f be any continuous function on [a, b] and $(ℙ, A)$ be an IF-Partition. Assume that F_μ,k and F_ν,k are the k-th component of gravity vector F_μ,n [f] and F_ν,n [f]. Then, there exists fixed points ξ_k, η_k ∈ [x_k-1, x_k+1] such that F_μ,k = f (ξ_k) and F_ν,k = f (η_k) and for k = 1, 2, ⋯ , n ${\hat{f}}_{μ, n} (x) = \sum_{k = 1}^{n} f (ξ_{k}) μ_{k} (x), {\hat{f}}_{ν, n} (x) = \sum_{k = 1}^{n} f (η_{k}) (1 - ν_{k} (x)) .$ (15)

Proof. According to the mean value theorem and Theorem 2.1, it is easy to verify that there exists fixed points ξ₁, η₁ ∈ [x₁, x₂], ξ_k, η_k ∈ [x_k-1, x_k+1] and ξ_n, η_n ∈ [x_n-1, x_n] with respect to the IF-Partition $(ℙ, A)$ such that $\begin{matrix} \int_{a}^{b} f (x) μ_{1} (x) d x = \frac{2}{hw} f (ξ_{1}), & \int_{a}^{b} f (x) ν_{1} (x) d x = \frac{2}{h (1 + u)} f (η_{1}), \\ \int_{a}^{b} f (x) μ_{k} (x) d x = \frac{1}{hw} f (ξ_{k}), & \int_{a}^{b} f (x) ν_{k} (x) d x = \frac{1}{h (1 + u)} f (η_{k}), \\ \int_{a}^{b} f (x) μ_{n} (x) d x = \frac{2}{hw} f (ξ_{n}), & \int_{a}^{b} f (x) ν_{n} (x) d x = \frac{2}{h (1 + u)} f (η_{n}) . \end{matrix}$ It can easily be shown that F_μ,k = f (ξ_k) , F_ν,k = f (η_k). Therefore, it is now obvious that the theorem holds.□

Theorem 4.2. Let f be any continuous function and $(ℙ, A)$ be an IF-Partition. Let ${\hat{f}}_{α, n} : [a, b] \to R$ be the α-hybrid i-IF-transform of function f. Then, there exists fixed points ξ_k, η_k ∈ [x_k-1, x_k+1] such that F_μ,k = f (ξ_k) and F_ν,k = f (η_k) and the following formula holds for k = 1, 2, ⋯ , n ${\hat{f}}_{α, n} (x) = \sum_{k = 1}^{n} [α f (ξ_{k}) μ_{k} (x) + (1 - α) f (η_{k}) (1 - ν_{k} (x)) .$ (16)

Proof. The proof of this theorem can be completed by the method analogous to that used in Theorem 4.1.□

Theorem 4.3. Let f and g be any two continuous functions on [a, b] and $(ℙ_{U}, A_{T})$ be a UTIF-Partition on [a, b] which be in osculated with Definition 2.5. Suppose that ${\hat{f}}_{μ, n} (x)$ , ${\hat{g}}_{μ, n} (x)$ , ${\hat{f}}_{ν, n} (x)$ and ${\hat{g}}_{ν, n} (x)$ are component of gravity vector of the membership and non-membership with respect to UTIF-Partition $(ℙ, A)$ . Then, the following result holds $\begin{matrix} \int_{a}^{b} {\hat{f}}_{μ, n} (x) g (x) d x = \int_{a}^{b} f (x) {\hat{g}}_{μ, n} (x) d x, \\ \int_{a}^{b} {\hat{f}}_{ν, n} (x) g (x) d x = \int_{a}^{b} f (x) {\hat{g}}_{ν, n} (x) d x . \end{matrix}$ (17)

Proof. Here we only give the proof of first case and another can be prove in a similar way. On the one hand, according to Definition 3.1, we can obtain the gravity vector F_u,n and G_u,n of membership and non-membership with respect to UTIF-Partition $(ℙ_{U}, A_{T})$ . Then it follows that, $\begin{matrix} \int_{a}^{b} {\hat{f}}_{μ, n} (x) g (x) d x \\ = \sum_{k = 1}^{n} \int_{a}^{b} F_{μ, k} μ_{k} (x) g (x) d x \\ = \sum_{k = 1}^{n} F_{μ, k} \int_{a}^{b} g (x) μ_{k} (x) d x \\ = \sum_{k = 1}^{n} F_{μ, k} G_{μ, k} \int_{a}^{b} μ_{k} (x) d x . \end{matrix}$

On the other hand, $\begin{matrix} \int_{a}^{b} f (x) {\hat{g}}_{μ, n} (x) d x \\ = \sum_{k = 1}^{n} \int_{a}^{b} f (x) G_{μ, k} μ_{k} (x) d x \\ = \sum_{k = 1}^{n} G_{μ, k} \int_{a}^{b} f (x) μ_{k} (x) d x \\ = \sum_{k = 1}^{n} G_{μ, k} F_{μ, k} \int_{a}^{b} μ_{k} (x) d x . \end{matrix}$ Owing to the above two formulas are equal, the proof is completed.□

5. Conclusions

In this paper, to begin with, we introduce a novel definition of IF-Partition. And we discuss some properties of TIF-Partition even including the special case, i.e., of UTIF-Partition. Secondly, we propose the method of IF-transform, which based on the membership and non-membership function related to the UTIF-Partition $(ℙ_{U}, A_{T})$ . We give its fundamental properties. Thirdly, we establish the method of i-IF-transform, which based on gravity vectors of membership and non-membership functions respectively, even the case of α-hybrid i-IF-transform. The results show that the original function can be substituted by the approximate one which is reconstructed using the method of i-IF-transform. Next, we will discuss the method of multidimensional i-IF-transform and its applications.

References

K.T.

Atanassov , Intuitionistic fuzzy sets, Fuzzy Sets Syst 20 (1) (1986), 87–96.

K.T.

Atanassov , Intuitionistic fuzzy sets, in: Intuitionistic Fuzzy Sets, Springer, 1999.

K.T.

Atanassov , On Intuitionistic Fuzzy Sets Theory, Springer-Verlag Berlin, Berlin, 2012.

Bodjanova , Fuzzy Sets and Fuzzy Partitions, In:

Opitz ,

Lausen and

Klar , (eds) Information and Classification. Studies in Classification, Data Analysis and Knowledge Organization. Springer, Berlin, Heidelberg, 1993.

Burillo ,

Bustince and

Mohedano , Some definition of intuitionistic fuzzy number, Fuzzy based Expert Systems, Fuzzy Bulgarian Enthusiasts, Sofia, Bulgaria, 1994.

Coroianu and

Stefanini , General approximation of fuzzy numbers by F-transform, Fuzzy Sets Syst 288 (2016), 46–74.

Di Martino ,

Loia and

Sessa , Fuzzy transform method and attribute dependency in data analysis, Information Sciences 180 (2010), 405–493.

Di Martino ,

Loia and

Sessa , Fuzzy transforms method in prediction data analysis, Fuzzy Sets Syst 180 (2011), 146–163.

D.J.

Dubois , Fuzzy Sets and Systems: Theory and Applications, 144, Academic Press, 1980.

10.

D.F.

Li , A ratio ranking method of triangular intuitionis-tic fuzzy numbers and its application to MADM problems, Comput Math Appl 60 (6) (2010), 1557–1570.

11.

Mitchell , Ranking-intuitionistic fuzzy numbers, Int J Uncertain Fuzz 12 (2004), 377–386.

12.

Novak and

Perfilieva , Fuzzy transform in the analysis of data, Int J Approx Reason 48 (2008), 36–46.

13.

Perfilieva and

Chaldeeva , Fuzzy transformation, in: Proc of IFSA'2001 World Congress, Vancouver, Canada, 2001.

14.

Perfilieva , Fuzzy transform: Application to reef growth problem, in:

R.B.

Demicco and

G.J.

Klir (Eds.), Fuzzy Logic in Geology, Academic Press, Amsterdam, 2003.

15.

Perfilieva , Fuzzy transforms and their applications to data compression, The 14th IEEE International Conference on Fuzzy Systems, 2005 FUZZ'05, 2005.

16.

Perfilieva , Fuzzy transforms: Theory and applications, Fuzzy Sets Syst 157 (8) (2006), 993–1023.

17.

Perfilieva , Fuzzy transforms: A challenge to conventional transforms, Adv Imag Elect Phys 47 (2007), 137–196.

18.

Perfilieva and

B.De.

Baets , Fuzzy transforms of monotone functions with application to image compression, Inform Sciences 180 (2010), 3304–3315.

19.

Perfilieva ,

Holčapek and

Kreinovich , A new reconstruction from the f-transform components, Fuzzy Sets Syst Special Issue on F-Transform: Theoretical Aspects and Advanced Applications 288 (2016), 3–25.

20.

Perfilieva ,

Hodáková and

Hurtík , Differentiation by the f-transform and application to edge detection, Fuzzy Sets Syst Special Issue on F-Transform: Theoretical Aspects and Advanced Applications 288 (2016), 96–114.

21.

Štepnička and

Valóšek , Numerical solution of partial differential equations with help of fuzzy transform, in: Proc Int Confon FUZZ195 IEEE 2005, Reno, USA, 2005, pp. 1104–1109.

22.

Shabani and

Baloui Jamkhaneh , A new generalized intuitionistic fuzzy number, Journal of Fuzzy Set Valued Analysis 2014 (2014), 1–10.

23.

Stefanini , F-transform with parametric generalized fuzzy partitions, Fuzzy Sets Syst 180 (1) (2011), 98–120.

24.

Stepnicka and

Polakovic , A neural network approach to the fuzzy transform, Fuzzy Sets Syst 160 (2009), 1037–1047.

25.

Torra and

J.H.

Mi , I-Fuzzy Partitions for Representing Clustering Uncertainties, Springer Berlin Heidelberg, 2010.

26.

Torra and

Miyamoto , A definition for I-fuzzy partitions, Soft Comput 15 (2011), 363–369.

27.

Z.S.

Xu and

R.R.

Yager , Some geometric aggregation operators based on intuitionistic fuzzy sets, lut J Gen Syst 35 (2006), 417–433.

28.

R.R.

Yager , Some aspects of intuitionistic fuzzy sets, Fuzzy OptimDecisMa 8 (1) (2009), 67–90.

29.

L.A.

Zadeh , Fuzzy sets, Inform Control 8 (3) (1965), 338–353.