A new type of fuzzy systems in terms of vague partitions

Abstract

This paper focuses on simplifying the structure of fuzzy systems and improving the precision. By regarding the fuzzy rule base as a mapping from the vague partition on the input universe to the vague partition on the output universe, we first design a new type of fuzzy system using the complete and continuous fuzzy rule base in terms of vague partitions. We then exploit Weierstrass’s approximation theorem to show that this new type of fuzzy system can approximate any real continuous function on a closed interval to arbitrary accuracy and provide the corresponding approximation accuracy with respect to infinite norms. We also provide two numerical examples to illustrate the effectiveness of this new type of fuzzy system. Both theoretical and numerical results show that this new type of fuzzy system achieves the quite approximation effect with a few fuzzy rules.

Keywords

Vague partition Fuzzy system Fuzzy rule base Approximation

1 Introduction

Fuzzy systems have been a significant field in the application of fuzzy set theory since Zadeh [1] first proposed fuzzy sets in 1965. In the past decades, the study of fuzzy systems mainly included how to design fuzzy systems and analyze the approximation ability of the designed fuzzy systems.

When designing fuzzy systems, the construction of membership functions is fundamental work. The appropriate selection or construction of membership functions plays a vital role in ensuring the system’s capability. Therefore, many scholars have studied the influence of different membership functions on the approximation ability of fuzzy systems. Zeng and Singh [12, 23] studied the approximation properties of fuzzy systems derived from fuzzy sets with triangular and trapezoidal membership functions, including uniform approximation and universal approximation. Chen and Gang [18] proposed the fuzzy system based on fuzzy sets with cosine and quadratic polynomial membership functions, and discussed the universal approximation and approximation accuracy. Yuan et al. [19] developed fuzzy systems based on parameter singleton fuzzifier methods. The fuzzy systems have been proven to equip with first-order and second-order accuracy. Sadjadi et al. [20, 21] proposed smooth fuzzy models, and discussed the accuracy of fuzzy models based on smooth functions. Jiang and Yuan [11] proposed a new kind of fuzzy systems derived from pyramid membership functions, and discussed its approximation properties. The above research mainly focuses on the approximation properties of those fuzzy systems derived from fuzzy sets with specific types of membership functions. Therefore, the systematic study of fuzzy systems based on fuzzy sets with different types of fuzzy membership functions is one of the motivations of this paper.

The approximation properties of fuzzy systems, which contain universal approximation property and approximation precision, are the theoretical basis for applying fuzzy systems. Wang [16] and Kosko [22] proved that fuzzy systems can be universal approximators through Stone-Weierstrass Theorem [17]. Theoretically, a continuous function on a closed interval could be approximated by a fuzzy system with any precision, which explains the fuzzy controller’s ability to achieve satisfactory performance in application. However, a closer examination of these results reveals that not being the case, and in fact, there is a gap between the theoretical results and the reported applications. The gap has to do with the fact that often in applications a small number of fuzzy rules are used. On the other hand, the reported results on function approximation may require a large number of fuzzy rules [2, 3]. To simplify the fuzzy rule base, Song et al. [4] reduced the number of fuzzy rules by using iterative parameter adaptation techniques. Yen and Wang [5] applied orthogonal transformation methods to simplify fuzzy rule-based systems. The method of fuzzy similarity is used in [6] and [7] to leave out the redundant fuzzy rules. Chen and Saif [8] gave a fuzzy system equipped with a dynamic rule base that performs well in a few fuzzy rules. Jiang and Yuan [10] proposed the cone fuzzy sets and applied them to construct fuzzy systems so that the structure of fuzzy systems is simplified. In fact, if a large number of fuzzy rules are already known, we can use interpolation or other numerical methods to obtain the approximate of the function without resorting to the fuzzy system. Therefore, fuzzy systems using a small number of fuzzy rules to achieve the desired result on function approximation is another motivation for this paper.

The important difference between our fuzzy system and the existing fuzzy systems is that our proposed fuzzy systems can process information by using fuzzy rule base directly. Therefore, the new fuzzy system weakens the subjectivity caused by fuzzification, defuzzification and fuzzy reasoning methods selection to a certain extent. For this reason, we need to make vague partitions for the input and output universes. Up to now, there have existed many vague partition methods [13 , 25–29]. We choose the method mentioned in [13, 14]. One of the reasons is that, based on this method, Pan and Xu proposed an axiomatic system to govern membership degrees, and then introduced the axiomatic definition of fuzzy sets derived from vague partitions. Axiomatic fuzzy sets provide an effective mathematical tool for understanding and describing fuzzy phenomena. Another reason is that the fuzzy rule base is complete and continuous based on vague partitions. Then the fuzzy rule base can be viewed as a mapping between two vague partitions. Consequently, the mapping from the input universe to the output universe can be obtained by mapping between two vague partitions. Our key contributions are stated in this article as mentioned below:

As the theoretical basis of the new fuzzy system, some properties of regular vague partitions are investigated, and five typical vague partitions are presented.

A new type of fuzzy system is proposed, and the structure of the fuzzy system is simplified.

The obtained results from illustrated numerical examples show that the proposed fuzzy system based on five typical vague partitions could perform better with a few fuzzy rules than other fuzzy systems.

The rest of this article is organized as follows: Several concepts related to this article are recalled and some new insights are given in section 2, In section 3, some mathematical properties of vague partitions are investigated, and we give five typical vague partitions. In section 4, the construction process of the new fuzzy system is displayed, and its approximation effect is discussed theoretically. In section 5, two numerical examples are presented illustrating the new fuzzy systems can perform well with a few fuzzy rules. In section 6, we summarize this article and indicate future research direction.

2 Preliminaries

In this section, unless otherwise stated, $ℕ^{+}$ refers to the collection of nonzero natural numbers. For any $n \in ℕ^{+}$ , the set {1, 2, ⋯ , n} is denoted by $\bar{n}$ .

2.1 The vague partition and some terminologies

Definition 2.1. [15] Let $U = [α, β] \subset ℝ$ . A vague partition of U is an object of the form

$\tilde{U} = {A_{1}, A_{2} \dots, A_{n}}, n \in ℕ^{+},$

where the degrees of memberships of the point x ∈ U to the class A_i is defined by the function μ_{A
_i} : U → [0, 1] , i = 1, ⋯ , n, respectively, that meet the following conditions:

for any $i \in \bar{n}$ , μ_{A
_i} (x) is continuous or has finite first-class broken points on U;

for any $i \in \bar{n}$ , there is at least one x₀ ∈ U such that μ_{A
_i} (x₀) =1;

for any $i \in \bar{n}$ , if μ_{A
_i} (x₀) =1 for x₀ ∈ U, then μ_{A
_i} (x) is not decreasing on [α, x₀], and is not increasing on [x₀, β];

0 < μ_{A
₁} (x) + ⋯ + μ_{A
_n} (x) ≤1 holds for any x ∈ U.

Based on the vague partition, the redefinition of fuzzy sets is as follows.

Definition 2.2. [14] Let $U = [α, β] \subset ℝ$ and $\tilde{U} = {A_{1}, A_{2}, \dots, A_{n}}$ , $n \in ℕ^{+}$ , be a vague partition of U, N be a strong negation on [0, 1], ⊗ be a triangular norm, ⊕ be a triangular conorm, where ⊗ and ⊕ are dual with regard to N. The set $F (\tilde{U})$ of fuzzy sets on U with regard to $\tilde{U}$ are constituted by the following elements:

if there exists $i \in \bar{n}$ making that μ_A (x) = μ_{A
_i} (x) for all x ∈ U, then $A = {(x, μ_{A} (x)) ∣ x \in U} \in F (\tilde{U})$ ;

if $μ_{A} (x) = \bar{μ} (x) = 1$ for all x ∈ U, then $A = {(x, μ_{A} (x)) ∣ x \in U} \in F (\tilde{U})$ ;

if $μ_{A} (x) = \underline{μ} (x) = 0$ for all x ∈ U, then $A = {(x, μ_{A} (x)) ∣ x \in U} \in F (\tilde{U})$ ;

if $A = {(x, μ_{A} (x)) ∣ x \in U} \in F (\tilde{U})$ and $r \in ℝ$ , then $A^{r} = {(x, (μ_{A} (x))^{r}) ∣ x \in U} \in F (\tilde{U})$ ;

if $A = {(x, μ_{A} (x)) ∣ x \in U} \in F (\tilde{U})$ , then $A^{N} = {(x, (μ_{A} (x))^{N}) ∣ x \in U} \in F (\tilde{U})$ ;

if A ={ (x, μ_A (x)) ∣ x ∈ U }, $B = {(x, μ_{B} (x)) ∣ x \in U} \in F (\tilde{U})$ , then $A \cap_{\otimes} B = {(x, μ_{A} (x) \otimes μ_{B} (x)) ∣ x \in U} \in F (\tilde{U})$ ;

if A = {(x, μ_A (x)) ∣ x ∈ U}, $B = {(x, μ_{B} (x)) ∣ x \in U} \in F (\tilde{U})$ , then $A \cup_{\oplus} B = {(x, μ_{A} (x) \oplus μ_{B} (x)) ∣ x \in U} \in F (\tilde{U})$ ;

$F (\tilde{U})$ do not contain other elements.

$F (\tilde{U})$ is able to be seen as a function space derived from $\tilde{U}$ . All the fuzzy sets in $F (\tilde{U})$ are called axiomatic fuzzy sets.

For $A \in F (\tilde{U})$ , its support set and kernel set are $supp A = {x \in U ∣ μ_{A} (x) > 0},$ $\ker A = {x \in U ∣ μ_{A} (x) = 1},$ respectively. A is called a single-kernel fuzzy set if kerA is a singleton set. Otherwise, A is called a muti-kernel fuzzy set. [A] ^λ is called a λ-cut set of A if, for every 0 < λ ≤ 1,

${[A]}^{λ} = {x \in U ∣ μ_{A} (x) \geq λ}$ and ${[A]}^{0} = \bar{supp A} .$

2.2 Fuzzy system

The configurations of the fuzzy system are shown in Fig. 1. The readers can refer to [9, 16] for more on fuzzy systems.

Fig. 1

Basic structure of fuzzy systems.

The fuzzy rule base consists of the following rules: $R^{i} : IF x is A^{i} THEN y is B^{i}$ (1) where x, y refer the input and the output respectively, Aⁱ and Bⁱ, i = 1, 2, ⋯ , l are linguistic terms, and l refers the number of fuzzy rules. The following concepts are introduced with regard to the fuzzy rule base.

Definition 2.3. [9] For any x ∈ U, the fuzzy rule base is called complete if there exists a fuzzy rule Rⁱ form as (1) such that μ_{A
ⁱ} (x) ≠0.

Definition 2.4. [9] The fuzzy rule base is called consistent if there exists no fuzzy rule such that the same antecedent with different consequents.

Definition 2.5. [9] The fuzzy rule base is called continuous if, for fuzzy rules with non-empty antecedents intersection, the intersection of consequents is not empty.

3 Some typical vague partitions

In this section, we add some restrictions to Definition 2.1 and discuss some properties of the vague partition.

Definition 3.1. Let $U = [α, β] \subset ℝ$ . A vague partition of U is an object of the form $\tilde{U} = {A_{1}, A_{2} \dots, A_{n}}, n \in ℕ^{+},$ where the degrees of memberships of the point x ∈ U to the class A_i is defined by the function μ_{A
_i} : U → [0, 1] , i = 1, ⋯ , n, respectively, that satisfies the following conditions:

for any $i \in \bar{n}$ , μ_{A
_i} (x) is continuous on U;

for any $i \in \bar{n}$ , there is only one point x_i ∈ U such that μ_{A
_i} (x_i) =1. Additionally, for $j \in \bar{n}$ , and i ≠ j, x_i < x_j if i < j;

for any $i \in \bar{n}$ , if μ_{A
_i} (x_i) =1 for x_i ∈ U, μ_A
₁ (x) is strictly decreasing on [α, x₂], μ_{A
_i} (x), i = 2, ⋯ , n - 1, is strictly increasing on [x_i-1, x_i] and strictly decreasing on [x_i, x_i+1], μ_{A
_n} (x) is strictly on [x_n-1, β];

0 < μ_A
₁ (x) + ⋯ + μ_{A
_n} (x) =1 holds for any x ∈ U.

In the sequel, unless otherwise stated, the vague partition $\tilde{U}$ of U, $U \subset ℝ$ , satisfies Definition 3.1. Obviously, all the fuzzy sets in $\tilde{U}$ are single-kernel fuzzy sets.

Corollary 3.2. Let $U = [α, β] \subset ℝ$ , $\tilde{U} = {A_{1}, \dots, A_{n}}$ , $n \in ℕ^{+}$ , be a vague partition of U, and let kerA_i = {x_i}, $i \in \bar{n}$ , x_i ∈ U. Then

kerA₁ = {α} and kerA_n = {β}, i.e., x₁ = α and x_n=β;

suppA_i ∩ suppA_i+1 = (x_i, x_i+1);

A_i∩ _⊗A_i+1 ≠ ∅, where ∩_⊗ is the maximum triangular norm.

Lemma 3.3. Let $U = [α, β] \subset ℝ$ , $\tilde{U}$ be a vague partition of U. Then, for any fuzzy set $A \in \tilde{U}$ , A is a convex fuzzy set, that is, for any t₁, t₂ ∈ U, we have $μ_{A} (p t_{1} + q t_{2}) \geq min {μ_{A} (t_{1}), μ_{A} (t_{2})},$ where p, q ∈ [0, 1] , p + q = 1 .

Proof. By (2) of Definition 3.1, there exists a point t₀ ∈ U such that μ_A (t₀) =1. If t₁ = t₂, then it is obvious. If t₁ ≠ t₂, without losing generality, we suppose that t₁ < t₂. Thus, t₁ ≤ pt₁ + qt₂ ≤ t₂, where ^″ = ″ cannot be taken at the same time. The proof can be divided into the following three cases.

(i) If t₁ < t₂ ≤ t₀, then μ_A (pt₁ + qt₂) ≥ μ_A (t₁) = min {μ_A (t₁) , μ_A (t₂)} since μ_A (x) is not decreasing on [t₁, t₀].

(ii) If t₀ ≤ t₁ < t₂, then μ_A (pt₁ + qt₂) ≥ μ_A (t₂) = min {μ_A (t₁) , μ_A (t₂)} since μ_A (x) is not increasing on [t₀, t₂].

(iii) If t₁ < t₀ < t₂, then μ_A (x) is not decreasing on [t₁, t₀] and not increasing on [t₀, t₂]. If t₁ ≤ pt₁ + qt₂ ≤ t₀, then μ_A (pt₁ + qt₂) ≥ μ_A (t₁). If t₀ < pt₁ + qt₂ ≤ t₂, then μ_A (pt₁ + qt₂) ≥ μ_A (t₂). Therefore, μ_A (pt₁ + qt₂) ≥ min {μ_A (t₁) , μ_A (t₂)}.

Consequently, for p, q ∈ [0, 1] and p + q = 1, we have μ_A (pt₁ + qt₂) ≥ min {μ_A (t₁) , μ_A (t₂)}. □

Theorem 3.4. Let $U = [α, β] \subset ℝ$ , $\tilde{U}$ be a vague partition of U. Then, for $A \in \tilde{U}$ , [A] ^λ is a closed interval on $ℝ$ for each 0 ≤ λ ≤ 1.

Proof. By (2) of Definition 3.1, there exists a point t₀ ∈ U such that μ_A (t₀) =1. Since μ_A (x) is continuous on U, it follows that there exist the maximum point t₁ ∈ [α, t₀] and the minimum point t₂ ∈ [t₀, β] such that μ_A (t₁) = λ and μ_A (t₂) = λ.

In what follows, the key is to show [A] ^λ is a closed interval rather than the union of multiple closed intervals. For each 0 ≤ λ ≤ 1, if t₁, t₂ ∈ [A] ^λ, then μ_A (t₁) ≥ λ and μ_A (t₂) ≥ λ. For any t ∈ [t₁, t₂], we have $μ_{A} (t) \geq min {μ_{A} (t_{1}), μ_{A} (t_{2})}$ by Lemma 3.3. Therefore, x ∈ [A] ^λ. These imply that if two points in [A] ^λ, then a closed interval with two points as the endpoints is also contained in [A] ^λ.

Consequently, [A] ^λ is a closed interval on $ℝ$ .

□

From the above theorem, we denote [A] ^λ by $[a_{λ}^{-}, a_{λ}^{+}]$ . What is more, $a_{λ}^{-}$ is increasing and $a_{λ}^{+}$ is decreasing with regard to λ.

Theorem 3.5. Let $U = [α, β] \subset ℝ$ , $\tilde{U} = {A_{1}, A_{2}, \dots,$ A_n}, $n \in ℕ^{+}$ , be a vague partition of U, and let kerA_i = {x_i}, $i \in \bar{n}$ , x_i ∈ U. Then

μ_{A
_i} (x) + μ_{A
_i+1} (x) =1 for x ∈ (x_i, x_i+1);

[A_i] ^λ ∩ [A_i+1] ^η is a point, where λ ∈ [0, 1] and λ + η = 1.

Proof. (1) Suppose μ_{A
_i} (x) + μ_{A
_i+1} (x) <1 for x ∈ (x_i, x_i+1). Then at least a fuzzy set $A_{k} \in \tilde{U}$ exists making μ_{A
_i} (x) + μ_{A
_k} (x) + μ_{A
_i+1} (x) =1 for x ∈ (x_i, x_i+1). Clearly, μ_{A
_k} (x) >0 for x ∈ (x_i, x_i+1). From the part (4) of Definition 3.1, we obtain that μ_{A
_k} (x_i) =1 and μ_{A
_k} (x_i+1) =0. Therefore, x_k ∈ (x_i, x_i+1) since μ_{A
_k} (x) needs to satisfy the part (3) of Definition 3.1. Furthermore, we have i < k < i + 1 contracting to $k \in \bar{n}$ . Consequently, μ_{A
_i} (x) + μ_{A
_i+1} (x) =1 for x ∈ (x_i, x_i+1).

(2) If λ = 1 or λ = 0, then it is obvious. We consider λ ∈ (0, 1). For $[A_{i}]^{λ} = [a_{i_{λ}}^{-}, a_{i_{λ}}^{+}]$ and $[A_{i + 1}]^{η} = [a_{i + 1_{η}}^{-}, a_{i + 1_{η}}^{+}]$ , we have $a_{i_{λ}}^{+}, a_{i + 1_{η}}^{-} \in (x_{i}, x_{i + 1})$ and $μ_{A_{i + 1}} (a_{i_{λ}}^{+}) = 1 - μ_{A_{i}} (a_{i_{λ}}^{+}) = 1 - λ = η .$ Thus, [A_i] ^λ∩ [A_i+1] ^1-λ ≠ ∅.

Next, we show [A_i] ^λ ∩ [A_i+1] ^1-λ is a point rather than a closed interval. From the part (3) of Definition 3.1, there are no other points belonging to $(a_{i_{λ}}^{-}, a_{i_{λ}}^{+})$ to make μ_{A
_i+1} (x) = η, and no other points belonging to $(a_{i + 1_{η}}^{-}, a_{i + 1_{η}}^{+})$ to make μ_{A
_i+1} (x) = η. Thus $a_{i_{λ}}^{+} = a_{i + 1_{η}}^{-}$ .

Consequently, [A_i] ^λ ∩ [A_i+1] ^1-λ is a point. □

Then support sets of fuzzy sets in $\tilde{U}$ can be represented as follows. $\begin{matrix} supp A_{1} = (α, x_{2}), \\ supp A_{i} = (x_{i - 1}, x_{i + 1}), i = 2, \dots, n - 1, \\ supp A_{n} = (x_{n - 1}, β) . \end{matrix}$

For the vague partition $\tilde{U}$ , we introduce the following notations,

$supp (\tilde{U}) = {supp A_{i} ∣ i \in \bar{n}},$ $\ker (\tilde{U}) = {\ker A_{i} ∣ i \in \bar{n}} .$

From the above statements, the following proposition is obtained immediately.

Proposition 3.6. Let $U = [α, β] \subset ℝ$ , and ${\tilde{U}}^{1}$ , ${\tilde{U}}^{2}$ be two vague partitions of U. Then $\ker ({\tilde{U}}^{1}) = \ker ({\tilde{U}}^{2})$ if and only if $supp ({\tilde{U}}^{1}) = supp ({\tilde{U}}^{2})$ .

The above analysis reveals that a vague partition $\tilde{U}$ can be characterized roughly by $\ker (\tilde{U})$ .

We introduce five kinds of typical regular single-kernel vague partitions. The membership functions are shown below respectively.

1. Triangular vague partition

$μ_{A_{1}} (x) = {\begin{matrix} \frac{x}{α - x_{2}} - \frac{x_{2}}{α - x_{2}}, & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} \frac{x}{x_{i} - x_{i - 1}} - \frac{x_{i - 1}}{x_{i} - x_{i - 1}}, & x \in [x_{i - 1}, x_{i}), \\ \frac{x}{x_{i} - x_{i + 1}} - \frac{x_{i + 1}}{x_{i} - x_{i + 1}}, & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1, $μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ \frac{x}{β - x_{n - 1}} - \frac{x_{n - 1}}{β - x_{n - 1}}, & x \in [x_{n - 1}, β] . \end{matrix}$

2. Sinusoial vague partition

$μ_{A_{1}} (x) = {\begin{matrix} 1 - sin \frac{π (x - α)}{2 (x_{2} - α)}, & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} sin \frac{π (x - x_{i - 1})}{2 (x_{i} - x_{i - 1})}, & x \in [x_{i - 1}, x_{i}), \\ 1 - sin \frac{π (x - x_{i})}{2 (x_{i + 1} - x_{i})}, & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1, $μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ sin \frac{π (x - x_{n - 1})}{2 (β - x_{n - 1})}, & x \in [x_{n - 1}, β] . \end{matrix}$ 3. Cosine vague partition

$μ_{A_{1}} (x) = {\begin{matrix} cos \frac{π (x - α)}{2 (x_{2} - α)}, & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} 1 - cos \frac{π (x - x_{i - 1})}{2 (x_{i} - x_{i - 1})}, & x \in [x_{i - 1}, x_{i}), \\ cos \frac{π (x - x_{i})}{2 (x_{i + 1} - x_{i})}, & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1,

$μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ 1 - cos \frac{π (x - x_{n - 1})}{2 (β - x_{n - 1})}, & x \in [x_{n - 1}, β] . \end{matrix}$ 4. Exponential vague partition

$μ_{A_{1}} (x) = {\begin{matrix} 2 - e^{\frac{ln 2 \cdot (x - α)}{x_{2} - α}}, & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} e^{\frac{ln 2 \cdot (x - x_{i - 1})}{x_{i} - x_{i - 1}}} - 1, & x \in [x_{i - 1}, x_{i}), \\ 2 - e^{\frac{ln 2 \cdot (x - x_{i})}{x_{i + 1} - x_{i}}}, & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1,

$μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ e^{\frac{ln 2 \cdot (x - x_{n - 1})}{β - x_{n - 1}}} - 1, & x \in [x_{n - 1}, β] . \end{matrix}$ 5. Logarithmic vague partition

$μ_{A_{1}} (x) = {\begin{matrix} 1 - ln (\frac{(e - 1) (x - α)}{x_{2} - α} + 1), & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} ln (\frac{(e - 1) (x - x_{i - 1})}{x_{i} - x_{i - 1}} + 1), & x \in [x_{i - 1}, x_{i}), \\ 1 - ln (\frac{(e - 1) (x - x_{i})}{x_{i + 1} - x_{i}} + 1), & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1,

$μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ ln (\frac{(e - 1) (x - x_{n - 1})}{β - x_{n - 1}} + 1), & x \in [x_{n - 1}, β] . \end{matrix}$

According to practical problems, we can build vague partitions by combining the membership functions in the above five typical fuzzy partitions, called mixed vague partitions. The graphs of the vague partitions mentioned above are shown in Figs. 2 –6.

Fig. 2

Graph of a triangular vague partition.

Fig. 3

Graph of a sinusoial vague partition.

Fig. 4

Graph of a cosine vague partition.

Fig. 5

Graph of an exponentia vague partition.

Fig. 6

Graph of a logarithmic vague partition.

4 A new type of fuzzy system

In this section, let $U = [α, β] \subset ℝ$ , $\tilde{U} = {A_{1}, \dots, A_{n}}$ , $n \in ℕ^{+}$ , a vague partition of U, $V = [ω, υ] \subset ℝ$ , $\tilde{V} = {B_{1}, \dots, B_{k}}$ , $k \in ℕ^{+}$ , a vague partition of V, where both $\tilde{U}$ and $\tilde{V}$ are one of the above five typical vague partitions. Then we propose the fuzzy system based on vague partitions.

4.1 The fuzzy rule base

R_i: IF A_i THEN B_{s
_i},

where $R_{i}, i \in \bar{n}$ refers the i-th rule, n refers the number of fuzzy rules, $A_{i} \in \tilde{U}$ , $B_{s_{i}} \in \tilde{V}$ , $s_{i} \in \bar{k}$ .

A complete and consistent fuzzy rule base can be seen as a mapping $\tilde{f}$ from $\tilde{U}$ to $\tilde{V}$ . Reversely, $\tilde{f}$ can represent a fuzzy rule base. Moreover, $\tilde{f}$ is said to be continuous if the fuzzy rule base is continuous. In what follows, we introduce the concept of monotonicity with regard to $\tilde{f}$ to further characterize its continuity. Let $\tilde{f} : \tilde{U} \to \tilde{V}$ . For any $A_{i}, A_{i + 1} \in \tilde{U}$ , $\tilde{f}$ is said to be increasing, or decreasing if $\tilde{f} (A_{i}) < \tilde{f} (A_{i + 1})$ , or $\tilde{f} (A_{i}) > \tilde{f} (A_{i + 1})$ .

Theorem 4.1. Let $\tilde{f} : \tilde{U} \to \tilde{V}$ be a surjection. If $\tilde{f}$ is monotonous, then $\tilde{f}$ is continuous.

Proof. Suppose that $\tilde{f}$ is discontinuous. From the part (2) of Corollary 3.2, there exist $A_{i}, A_{i + 1} \in \tilde{U}$ such that $\tilde{f} (A_{i}) \cap_{\otimes} \tilde{f} (A_{i + 1}) = \emptyset$ , in which ∩_⊗ is the maximum triangular norm. Thus, at least one fuzzy set $B_{j} \in \tilde{V}$ exists making $\tilde{f} (A_{i}) < B_{j} < \tilde{f} (A_{i + 1})$ . Since $\tilde{f}$ is surjection, it follows that ${\tilde{f}}^{- 1} (B_{j}) < A_{i} < A_{i + 1}$ or $A_{i} < A_{i + 1} < {\tilde{f}}^{- 1} (B_{j})$ , contracting to $\tilde{f}$ is monotonous.

□

For the reverse, $\tilde{f}$ may not be monotonous if $\tilde{f}$ is continuous for it is a surjection, as presented in the example below.

Example 4.2 Let $U = [α, β] \in ℝ$ and $\tilde{U} = {A_{1}, A_{2}, A_{3}}$ be a vague partition of U, $V = [ω, υ] \in ℝ$ and $\tilde{V} = {B_{1}, B_{2}}$ be a vague partition of V, $\tilde{f} : \tilde{U} \to \tilde{V}$ and $\tilde{f} (A_{1}) = B_{1}$ , $\tilde{f} (A_{2}) = B_{2}$ , $\tilde{f} (A_{3}) = B_{1}$ . Clearly, $\tilde{f}$ is continuous but not monotonous.

Theorem 4.3. Let $\tilde{f} : \tilde{U} \to \tilde{V}$ be a bijection. Then $\tilde{f}$ is continuous is equivalent to $\tilde{f}$ is monotonous.

Proof. The sufficiency is clearly from Theorem 4.1.

Necessity. Since $\tilde{f}$ is bijective, it follows that the number of elements in $\tilde{U}$ is equal to that in $\tilde{V}$ . Let $\tilde{U} = {A_{1}, A_{2}, \dots, A_{n}}$ and $\tilde{V} = {B_{1}, B_{2}, \dots, B_{n}}$ .If n = 2, then it is evident. If n ≥ 3, suppose that $\tilde{f}$ is not monotonous. Then fuzzy sets A_i, A_i+1, A_i+2 make $\tilde{f} (A_{i})$ , $\tilde{f} (A_{i + 1})$ , $\tilde{f} (A_{i + 2})$ dissatisfy that $\tilde{f} (A_{i}) < \tilde{f} (A_{i + 1}) < \tilde{f} (A_{i + 2})$ or $\tilde{f} (A_{i}) > \tilde{f} (A_{i + 1}) > \tilde{f} (A_{i + 2})$ . Therefore, there are four cases with respect to the order relationship of $\tilde{f} (A_{i})$ , $\tilde{f} (A_{i + 1})$ , $\tilde{f} (A_{i + 2})$ .

$\tilde{f} (A_{i + 1}) > \tilde{f} (A_{i + 2}) > \tilde{f} (A_{i})$ ,

$\tilde{f} (A_{i + 2}) > \tilde{f} (A_{i}) > \tilde{f} (A_{i + 1})$ ,

$\tilde{f} (A_{i}) > \tilde{f} (A_{i + 2}) > \tilde{f} (A_{i + 1})$ ,

$\tilde{f} (A_{i + 1}) > \tilde{f} (A_{i}) > \tilde{f} (A_{i + 2})$ .

From case (i), we obtain $supp \tilde{f} (A_{i}) \cap supp \tilde{f} (A_{i + 1}) = \emptyset$ , furthermore, $\tilde{f} (A_{i}) \cap_{\otimes} \tilde{f} (A_{i + 1}) = \emptyset$ , in which ∩_⊗ is the maximum triangular norm, contracting to $\tilde{f}$ is continuous. Similarly, cases (ii), (iii), (iv) are contracting to $\tilde{f}$ is continuous.

Consequently, $\tilde{f}$ is monotonous if $\tilde{f}$ is continuous.

□

The preceding theorem indicates that the monotonicity and continuity are equivalent if $\tilde{f}$ is bijective.

4.2 Basic configuration of the new fuzzy system

Let $\tilde{f} : \tilde{U} ⟶ \tilde{V}$ be a continuous. Then we construct f : U → V. Furthermore, a new type of fuzzy system derived from vague partitions is designed, and its configuration is displayed schematically in Fig. 8

Fig. 8

Basic configuration of the new type of fuzzy system.

According to the configuration of the new fuzzy system as displayed in Fig. 8, let kerA_i = {x_i}, x_i ∈ U, $i \in \bar{n}$ , kerB_{s
_i} = {y_{s
_i}}, y_{s
_i} ∈ V, $s_{i} \in \bar{k}$ . The method of the new fuzzy system dealing with data is described in Algorithm 1.

Algorithm 1
Input: A set of rules R_i $(i \in \bar{n})$ with $\tilde{U}$ and $\tilde{V}$ , and a point x ∈ U
Output:f (x)
1: ifx = x_ithen
2: f (x) = y_{s _i}
3: else
4: There exists $A_{i}, A_{i + 1} \in \tilde{U}$ such that x ∈ suppA_i ∩ suppA_i+1 and μ_{A _i} (x) = λ_i, μ_{A _i+1} (x) =1 - λ_i
5: if $\tilde{f} (A_{i}) \neq \tilde{f} (A_{i + 1})$ then
6: $f (x) = [\tilde{f} (A_{i})]^{λ_{i}} \cap [\tilde{f} (A_{i + 1})]^{1 - λ_{i}}$
7: else
8: $f (x) = max {\tilde{f} (A)]^{λ_{i}} \cap [\tilde{f} (A_{i + 1})]^{1 - λ_{i}}}$
9: end if
10: end if
11: returnf (x)

Taking triangular vague partitions as an example, the above algorithm is displayed schematically in Fig. 7a and Fig. 7b.

Fig. 7

Graph of Algorithm 1.

Remark 4.4. For the preceding algorithm, we have the following notes.

The part (2) of Theorem 3.5 guarantees that $f (x) = [\tilde{f} (A_{i})]^{λ} \cap [\tilde{f} (A_{i + 1})]^{1 - λ}$ is a point on V.

$\tilde{f} : \tilde{U} \to \tilde{V}$ is a Zadeh extension for f : U → V since $μ_{\tilde{f} (A_{i})} (y) = max_{f (x) = y} μ_{A_{i}} (x)$ , $i \in \bar{n}$ .

Consequently, the proposed fuzzy system could be represented by f (x) as follows.

equation 4.1

$f (x) = {\begin{matrix} I_{1} (x), & x \in [α, x_{2}), \\ I_{i} (x), & x \in [x_{i}, x_{i + 1}), i = 2, \dots, n - 2, \\ I_{n - 1} (x), & x \in [x_{n - 1}, β], \end{matrix}$ (2) where

equation 4.2 $I_{l} (x) = {\begin{matrix} max μ_{\tilde{f} (A_{l})}^{- 1} (μ_{A_{l}} (x)), & if \tilde{f} (A_{l}) < \tilde{f} (A_{l + 1}), \\ min μ_{\tilde{f} (A_{l})}^{- 1} (μ_{A_{l}} (x)), & if \tilde{f} (A_{l}) > \tilde{f} (A_{l + 1}), \\ max μ_{\tilde{f} (A_{l})}^{- 1} (max {μ_{A_{l}} (x), μ_{A_{l + 1}} (x)}), & if \tilde{f} (A_{l}) = \tilde{f} (A_{l + 1}), \end{matrix}$ (3)

l = 1, 2, ⋯ , n.

4.3 Approximation properties of the new fuzzy system

Based on previous sections, we see that f (x) is a piecewise function. The basic mechanism of the new fuzzy system f (x) is its interpolation property. Derived from computational mathematics theory, the new fuzzy system can approximate a given continuous function on a closed interval with arbitrary precision. For investigating approximation properties of the new fuzzy system based on five typical vague partitions mentioned in Section 3, the ∞-norm for a bounded function g (x) on U is defined as $∣ ∣ g (x) ∣ ∣_{\infty} = sup_{x \in U} ∣ g (x) ∣$ . We first show the following lemmas.

Lemma 4.5. Let f (x) be a fuzzy system form as Eq. (4.1). Then f (x) is second-order differentiable on (x_i, x_i+1), $i \in \bar{n - 1}$ .

Proof. If $\tilde{f} (A_{i}) < \tilde{f} (A_{i + 1})$ , then $f (x) = max μ_{\tilde{f} (A_{i})}^{- 1} (μ_{A_{i}} (x))$ for x ∈ (x_i, x_i+1). Considering five typical vague partitions, we have μ_{A
_i} (x) is second-order differentiable. Additionally, $μ_{\tilde{f} (A_{i})} (y)$ is second-order differentiable and strictly monotonous on (f (x_i) , f (x_i+1)).

Similarly, we have f (x) is second-order differentiable on (x_i, x_i+1) for $\tilde{f} (A_{i}) > \tilde{f} (A_{i + 1})$ and $\tilde{f} (A_{i}) = \tilde{f} (A_{i + 1})$ .

Consequently, f (x) is second-order differentiable on (x_i, x_i+1). □

Since the number of fuzzy rules is finite, it follows that the number of inflection points of f (x) is finite. Then following lemma is obtained immediately.

Lemma 4.6 Let f (x) be a fuzzy system form as Eq. (4.1). Then f (x) is almost everywhere second-order differentiable on U, where nondifferentiable points set is $\ker (\tilde{U})$ .

Lemma 4.7. Let P (x) be a polynomial on U. Then for any ɛ > 0 there exists a new fuzzy system f (x) such that $∣ ∣ f (x) - P (x) ∣ ∣_{\infty} < ɛ .$

Proof. For any x ∈ U, there exists $i \in \bar{n - 1}$ such that x ∈ [x_i, x_i+1]. Using the interpolation property of f (x), $\begin{matrix} ∣ f (x) - P (x) ∣ & = ∣ f (x) - f (x_{i}) + P (x_{i}) - P (x) ∣ \\ \leq ∣ f (x) - f (x_{i}) ∣ + ∣ P (x_{i}) - P (x) ∣ . \end{matrix}$

By Lagrange’s Mean Theorem, there exist φ ∈ (x_i, x_i+1) and γ ∈ (x_i, x_i+1) such that $\begin{matrix} ∣ f (x) - p (x) ∣ & \leq ∣ f^{'} (φ) (x_{i} - x) ∣ + ∣ P^{'} (γ) (x_{i} - x) ∣ \\ \leq (∣ f^{'} (φ) ∣ + ∣ P^{'} (γ) ∣) M, \end{matrix}$ where $M = max_{i \in \bar{n - 1}} ∣ x_{i} - x_{i + 1} ∣$ . If h is sufficiently small, then for any ɛ > 0 it holds that $∣ f (x) - P (x) ∣ < ɛ$ for x ∈ U.

Consequently, $∣ ∣ f (x) - P (x) ∣ ∣_{\infty} = sup_{x \in U} ∣ f (x) - P (x) ∣ < ɛ .$ □

Lemma 4.8. (Weierstrass’s Approximation Theorem) Let g (x) be a continuous function on U. Then for any ɛ > 0 there exists a polynomial P (x) such that $∣ P (x) - g (x) ∣ < ɛ$ for all x ∈ U.

Theorem 4.9. Let h (x) be an arbitrary continuous function on U. Then for any ɛ > 0 there exists a fuzzy system f (x) form as Eq.(4.1) such that

$∣ ∣ f (x) - h (x) ∣ ∣_{\infty} < ɛ .$

Proof. For any ɛ > 0, there is a polynomial P (x) on U such that $∣ P (x) - h (x) ∣ < \frac{ɛ}{2}$ for all x ∈ U by Lemma 4.8.

For P (x), there is a fuzzy system f (x) form as Eq.(4.1) such that

$∣ f (x) - P (x) ∣ < \frac{ɛ}{2}$ for all x ∈ U by Lemma 4.7.

Therefore,

$\begin{matrix} ∣ h (x) - f (x) ∣ & = ∣ h (x) - P (x) + P (x) - f (x) ∣ \\ \leq ∣ h (x) - P (x) ∣ + ∣ P (x) - f (x) ∣ \\ < ɛ \end{matrix}$ for all x ∈ U.

Consequently, $∣ ∣ f (x) - h (x) ∣ ∣_{\infty} = sup_{x \in U} ∣ f (x) - h (x) ∣ < ɛ .$ □

Theorem 4.9 shows that the new fuzzy system is able to be a universal approximator.

Theorem 4.10. Let g (x) be continuous and differentiable on U satisfying h (x_i) = y_{s
_i}. For the fuzzy system f (x) form as Eq.(4.1), it holds that

$∣ ∣ f (x) - h (x) ∣ ∣_{\infty} \leq M ∣ ∣ f^{'} (x) - h^{'} (x) ∣ ∣_{\infty},$ where $M = max_{i \in \bar{n - 1}} ∣ x_{i} - x_{i + 1} ∣$ , $x \in U ∖ \ker (\tilde{U})$ .

Proof. For any x ∈ U, there exists $i \in \bar{n - 1}$ such that x ∈ [x_i, x_i+1]. Let R₁ (x) = h (x) - f (x), x ∈ [x_i, x_i+1]. Since h (x_i) = f (x_i), it follows that R₁ (x) = T₁ (x) (x - x_i), where T₁ (x) is continuous on [x_i, x_i+1] and differentiable on (x_i, x_i+1). For a fixed point x, we define a function as $φ_{1} (s) = g (s) - f (s) - T_{1} (x) (s - x_{i}), s \in [x_{i}, x_{i + 1}] .$ Obviously, x_i and x are two roots of φ₁ (s) =0. From Rolle’s theorem, there exits α₁ ∈ (x_i, x_i+1) related to x such that $φ_{1}^{'} (α_{1}) = 0$ . Therefore, $T_{1} (x) = (h^{'} (α_{1}) - f^{'} (α_{1})), α_{1} \in (x_{i}, x_{i + 1}) .$ Furthermore, $\begin{matrix} ∣ R_{1} (x) ∣ & = ∣ h^{'} (α_{1}) - f^{'} (α_{1}) ∣ (x - x_{i}) \\ \leq (∣ ∣ f^{'} (x) - h^{'} (x) ∣ ∣_{\infty}) \cdot (max ∣ x_{i} - x_{i + 1} ∣), \end{matrix}$ for x ∈ (x_i, x_i+1). Consequently, from arbitrariness of x and Lemma 4.6, we obtain $∣ ∣ f (x) - h (x) ∣ ∣_{\infty} \leq M ∣ ∣ f^{'} (x) - h^{'} (x) ∣ ∣_{\infty},$ where $M = max_{i \in \bar{n - 1}} ∣ x_{i} - x_{i + 1} ∣$ , $x \in U ∖ \ker (\tilde{U})$ .

□

Theorem 4.11. Let h (x) be continuous and second-order differentiable on U satisfying h (x_i) = y_{s
_i}. For the new fuzzy system f (x) form as Eq. (4.1), it holds that

$∣ ∣ f (x) - h (x) ∣ ∣_{\infty} \leq \frac{M^{2}}{2} ∣ ∣ f^{″} (x) - h^{″} (x) ∣ ∣_{\infty},$ where $M = max_{i \in \bar{n - 1}} ∣ x_{i} - x_{i + 1} ∣$ , $x \in U ∖ \ker (\tilde{U})$ .

Proof. For any x ∈ U, there exists $i \in \bar{n - 1}$ such that x ∈ [x_i, x_i+1]. Let R₂ (x) = h (x) - f (x), x ∈ [x_i, x_i+1]. Since h (x_i) = f (x_i) and f (x_i+1) = h (x_i+1), it follows that R₂ (x) = T₂ (x) (x - x_i) (x - x_i+1), where T₂ (x) is continuous on [x_i, x_i+1] and second-order differentiable on (x_i, x_i+1). For a fixed point x, we define a function as $φ_{2} (s) = g (s) - f (s) - T_{2} (x) (s - x_{i}) (s - x_{i + 1}),$ s ∈ [x_i, x_i+1] . Obviously, x_i, x_i+1 and x are three roots of φ₂ (t) =0. From Rolle’s theorem, there exists α₂ ∈ (x_i, x_i+1) related to x such that φ″ (α₂) =0. Therefore $T_{2} (x) = \frac{1}{2} (h^{″} (α_{2}) - f^{″} (α_{2})), α_{2} \in (x_{i}, x_{i + 1}) .$ Furthermore, for x ∈ (x_i, x_i+1). Consequently, from arbitrariness of x and Lemma 4.6, we have $∣ ∣ f (x) - h (x) ∣ ∣_{\infty} \leq \frac{M^{2}}{2} ∣ ∣ f^{″} (x) - h^{″} (x) ∣ ∣_{\infty},$ where $M = max_{i \in \bar{n - 1}} ∣ x_{i} - x_{i + 1} ∣$ , $x \in U ∖ \ker (\tilde{U})$ .

□

Theorem 4.10 and Theorem 4.11 claim a fact that two factors determine approximation accuracy, that is, one is the number of fuzzy rules, the other is the supremum of ∣f′ (x) - h′ (x)∣, ∣f″ (x) - h″ (x)∣ for $x \in U ∖ \ker (\tilde{U})$ . In general, increasing the number of rules can reduce the approximation error but will increase the complexity of the fuzzy system. The new type of fuzzy system reduces the approximation error by selecting the appropriate vague partition according to the approximated function.

5 Numerical examples

In this section, we employ six types of elementary functions and two types of functions compounded from the elementary functions to test the proposed fuzzy system (System I), and compare the approximation ability of the fuzzy system with the following fuzzy systems:

the fuzzy system with triangular fuzzification and central average defuzzification (System II) [9].

the fuzzy system with Gauss fuzzification and central average defuzzification (System III) [9]

the fuzzy system with quadratic polynomial fuzzification and central average defuzzification (System IV) [18]

equation 5.1 The center of a fuzzy set can be defined as if the average of all points that maximize the membership of a fuzzy set is a finite value, and then the average is called the center of the fuzzy set. The final output of the above three types of fuzzy systems is $f (x) = \frac{\overset{n}{\underset{i = 1}{Σ}} \bar{y_{i}} μ_{A_{i}} (x)}{\overset{n}{\underset{i = 1}{Σ}} μ_{A_{i}} (x)}, x \in U,$ (4) where $\bar{y_{i}}$ is the center of the fuzzy set Bⁱ in fuzzy rule (1).

For System II, let U = [α, β]. The expression for μ_{A
_i} (x) in Eq.(5.1) is $μ_{A_{i}} (x) = {\begin{matrix} I (x), & x \in [α, γ), \\ 1, & x = γ, \\ D (x), & x \in [γ, β], \end{matrix}$ i = 1, 2, ⋯ , n, where α ≤ γ ≤ β, I (x) ∈ [0, 1] is a non-decreasing function on [α, γ), and D (x) ∈ [0, 1] is a non-increasing function on [γ, β].

For System III, the expression for μ_{A
_i} (x) in Eq.(5.1) is $μ_{A_{i}} (x) = a_{i} e^{[- (\frac{x - \bar{x_{i}}}{δ_{i}})^{2}]}, x \in U,$ i = 1, 2, ⋯ , n, where a_i ∈ (0, 1], δ_i ∈ (0, + ∞), $\bar{x_{i}} \in ℝ$ is the center of the fuzzy set A_i.

For System IV, let U = [α, β] and kerA_i = {x_i}, i = 1, 2, ⋯ , n. The expression for μ_{A
_i} (x) in Eq.(5.1) is $μ_{A_{1}} (x) = {\begin{matrix} 1 - (x - α)^{2}, & x \in [α, x_{2}), \\ 0, & x \in [x_{2}, β], \end{matrix}$

$μ_{A_{i}} (x) = {\begin{matrix} a_{i} (x - x_{i - 1})^{2}, & x \in [x_{i - 1}, x_{i}), \\ 1 - (x - x_{i})^{2}, & x \in [x_{i}, x_{i + 1}], \\ 0, & otherwise, \end{matrix}$ i = 2, 3, ⋯ , n - 1, where a_i > 0 is a constant, $μ_{A_{n}} (x) = {\begin{matrix} 0, & x \in [α, x_{n - 1}), \\ a_{n} (x - x_{n - 1})^{2}, & x \in [x_{n - 1}, β], \end{matrix}$ where a_n > 0 is a constant.

Example 5.1. Consider the following six types of elementary function:

h₁ (x) = x³, x ∈ [0, 2];

h₂ (x) = e^x, x ∈ [0, 2];

h₃ (x) =3, x ∈ [0, 2];

h₄ (x) = ln x, x ∈ [1, e];

h₅ (x) = sin πx, x ∈ [1.5, 2.5];

h₆ (x) = arcsin x, x ∈ [-1, 1].

We, respectively, design System I, System II, System III, and System IV to approximate these functions with three rules and compare the approximation capability of these fuzzy systems. For System I, the vague partition of the input universe is denoted by

{\tilde{U_{j}}}^{1}

, j = 1, ⋯ , 6, and its fuzzy rule base with respect to the above six functions are described respectively in Appendix A. We denote partitions of the input universes of System II, System III, System IV by

{\tilde{U_{j}}}^{2}

{\tilde{U_{j}}}^{3}

{\tilde{U_{j}}}^{4}

, j = 1, ⋯ , 6, respectively. For System II, let

{\tilde{U_{j}}}^{2}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{2})

, j = 1, ⋯ , 6. For System III, let

{\tilde{U_{j}}}^{3}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{3})

, j = 1, ⋯ , 6, and a_i = 1, δ_i = 0.3, i = 1, 2, 3. For System IV, let

{\tilde{U_{j}}}^{4}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{4})

, j = 1, ⋯ , 6, and a_i = 1, i = 1, 2, 3. By carrying out equidistant partitions of input universes, we select 500 sample points to calculate approximation errors of their fuzzy systems. These results are shown in Table 1.

Table 1

Maximum errors of different systems

Functions\ Errors\ Systems	System I	System II	System III	System IV
h₁ (x)	0.4345	1.1284	2.4245	0.6311
h₂ (x)	0.1747	0.5759	1.4677	0.5993
h₃ (x)	0	0	0	0
h₄ (x)	0.0271	0.0478	0.1286	0.2025
h₅ (x)	0	0.2105	0.2966	0.4587
h₆ (x)	0.2062	0.3307	0.6424	0.6760

As shown in Table 1, System I has better approximation capability than the other three types of fuzzy systems. This also can be seen in the approximation of systems in Figs. 9 –13

Fig. 9

Approximation effect of Systems on h₁(x).

Fig. 10

Approximation effect of Systems on h₂(x).

Fig. 11

Approximation effect of Systems on h₄(x).

Fig. 12

Approximation effect of Systems on h₅(x).

Fig. 13

Approximation effect of Systems on h₆(x).

Remark 5.2.

For h₃ (x) =3, the maximum errors of the above four fuzzy systems are all 0. Therefore, we do not show the approximation effect of the four systems on this function.

For h₅ (x) = sinπx, the maximum error of System I is 0. Therefore, the images representing System I and this function respectively coincide in Fig. 12(a).

Next, we test System I using two functions compounded from the elementary functions and compare the approximation ability of the other three fuzzy systems.

Example 5.3

Consider the following two function:

g₁ (x) = cos ³x, x ∈ [0, 1.5708];

$g_{2} (x) = (lg x)^{\frac{3}{2}}$ , x ∈ [1, 10].

{\tilde{U}}_{j}^{1}

, j = 1, 2, and its fuzzy rule base with respect to the above two functions are described respectively in Appendix B. We denote partitions of the input universes of System II, System III, System IV by

{\tilde{U_{j}}}^{2}

{\tilde{U_{j}}}^{3}

{\tilde{U_{j}}}^{4}

, j = 1, 2, respectively. For System II, let

{\tilde{U_{j}}}^{2}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{2})

, j = 1, 2. For System III, let

{\tilde{U_{j}}}^{3}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{3})

, j = 1, 2, and a_i = 1, δ_i = 0.3, i = 1, 2, 3. For System IV, let

{\tilde{U_{j}}}^{4}

satisfy

\ker ({\tilde{U_{j}}}^{1}) = \ker ({\tilde{U_{j}}}^{4})

, j = 1, 2, and a_i = 1, i = 1, 2, 3. By carrying out equidistant partitions of input universes, we select 500 sample points to calculate approximation errors of their fuzzy systems. These results are shown in Table 2.

Table 2

Maximum errors of different systems

Functions\ Errors\ Systems	System I	System II	System III	System IV
g₁ (x)	0.0632	0.1632	0.2289	0.2884
g₂ (x)	0.0177	0.1401	0.6290	0.4693

Table 2 shows that, with three rules, the approximation performance of System I is better than the other three types of systems. This also can be seen in the approximation of systems shown in Fig. 14 and Fig. 15.

Fig. 14

Approximation effect of Systems on g₁(x).

Fig. 15

Approximation effect of Systems on g₂(x).

If we preset the precision as 0.0632 and 0.0177 for g₁ (x) and g₂ (x) respectively, we can calculate the minimum number of rules required by System II and System IV, as displayed in Table 3.

Table 3

The minimum number of rules for systems

Functions\ Number of rules\ Systems	System II	System IV
g₁ (x)	5	9
g₂ (x)	5	24

Table 3 shows that, for g₁ (x) System II and System IV need 5 and 9 rules, respectively, to achieve the same approximation effect for System I with three fuzzy rules, for g₂ (x) System II and System IV need 5 and 24 rules, respectively, to achieve the same approximation effect for System I with three fuzzy rules. It implies that uppercaseexpandafterromannumeral1 can perform well with a few fuzzy rules.

Remark 5.4. Since System III does not satisfy the interpolation mechanism, its error will not decrease with increasing the number of rules. Therefore, the minimum number of rules required by System III under the preset accuracy is not considered here.

6 Conclusion

In this article, we investigated the mathematical properties of regular vague partitions. On the basis of these, we proposed a new type of fuzzy system derived from five typical single-kernel vague partitions. The new fuzzy system can deal with the information from the fuzzy rule base directly, then structures of fuzzification, fuzzy reasoning engine and defuzzification are omitted. Experiments on approximating six types of elementary functions and two functions compounded from elementary functions claimed that the new fuzzy system could achieve the reported results with a few fuzzy rules. In the future, we will establish a multidimensional fuzzy rule base by building vague partitions on $ℝ^{n}$ , then the new fuzzy systems can be employed to solve MISO problems.

Footnotes

Acknowledgments

This work was partially funded by the National Natural Science Foundation of China (Grant Nos. 61673320).

Appendix A

(1) h₁ (x) = x³, x ∈ [0, 2]. U₁ = [0, 2], then ${\tilde{U_{1}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} cos \frac{π}{2} x, & x \in [0, 1), \\ 0, & x \in [1, 2], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} 1 - cos \frac{π}{2} x, & x \in [0, 1), \\ cos \frac{π}{2} (x - 1), & x \in [1, 2], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [0, 1), \\ 1 - cos \frac{π}{2} (x - 1), & x \in [1, 2] . \end{matrix}$

V₁ = [1, e²], then ${\tilde{V_{1}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} - y + 1, & y \in [0, 1), \\ 0, & y \in [1, 8], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} y, & y \in [0, 1), \\ - \frac{1}{7} y + \frac{8}{7}, & y \in [1, 8], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [0, 1), \\ \frac{1}{7} y - \frac{1}{7}, & y \in [1, 8] . \end{matrix}$ $\tilde{f_{1}} : {\tilde{U_{1}}}^{1} \to {\tilde{V_{1}}}^{1}$ and $\tilde{f_{1}} (A_{1}) = B_{3}$ , $\tilde{f} (A_{2}) = B_{2}$ and $\tilde{f_{1}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{1} (x) = {\begin{matrix} 1 - cos \frac{π}{2} x, & x \in [0, 1), \\ 8 - 7 cos \frac{π}{2} (x - 1), & x \in [1, 2] . \end{matrix}$

(2) h₂ (x) = e^x, x ∈ [0, 2]. U₂ = [0, 2], then ${\tilde{U_{2}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} 2 - e^{ln 2 \cdot x}, & x \in [0, 1), \\ 0, & x \in [1, 2], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} e^{ln 2 \cdot x} - 1, & x \in [0, 1), \\ 2 - e^{ln 2 \cdot (x - 1)}, & x \in [1, 2], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [0, 1), \\ e^{ln 2 \cdot (x - 1)} - 1, & x \in [1, 2] . \end{matrix}$

V₂ = [1, e²], then ${\tilde{V_{2}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} - \frac{y}{e - 1} + \frac{e}{e - 1}, & y \in [1, e), \\ 0, & y \in [e, e^{2}], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} \frac{y}{e - 1} - \frac{1}{e - 1}, & y \in [1, e), \\ - \frac{y}{e^{2} - e} + \frac{e^{2}}{e^{2} - e}, & y \in [e, e^{2}], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [1, e), \\ \frac{y}{e^{2} - e} - \frac{e}{e^{2} - e}, & y \in [e, e^{2}] . \end{matrix}$ $\tilde{f_{2}} : {\tilde{U_{1}}}^{1} \to {\tilde{V_{2}}}^{1}$ and $\tilde{f_{2}} (A_{1}) = B_{3}$ , $\tilde{f_{2}} (A_{2}) = B_{2}$ and $\tilde{f_{2}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{2} (x) = {\begin{matrix} (e - 1) (e^{ln 2 \cdot x} - 1) + 1, & x \in [0, 1), \\ e^{2} - (e^{2} - e) (2 - e^{ln 2 \cdot (x - 1)}), & x \in [1, 2] . \end{matrix}$

(3) h₃ (x), x ∈ [0, 2]. U₃ = [0, 2], then ${\tilde{U_{3}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} - x + 1, & x \in [0, 1), \\ 0, & x \in [1, 2], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} x, & x \in [0, 1), \\ - x + 2, & x \in [1, 2], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [0, 1), \\ x - 1, & x \in [1, 2] . \end{matrix}$

V₃ = {3}, then ${\tilde{V_{3}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = 1, y = 3$ $μ_{B_{2}} (y) = 1, y = 3$ $μ_{B_{3}} (y) = 1, y = 3 .$ $\tilde{f_{3}} : {\tilde{U_{3}}}^{1} \to {\tilde{V_{3}}}^{1}$ and $\tilde{f_{3}} (A_{1}) = B_{3}$ , $\tilde{f_{3}} (A_{2}) = B_{2}$ and $\tilde{f_{3}} (A_{3}) = B_{1}$ . The final output of System I is $f_{3} (x) = 3, x \in [0, 2] .$

(4) h₄ (x) = ln x, x ∈ [1, e]. U₄ = [1, e], then ${\tilde{U_{4}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} 1 - ln (2 x - 1), & x \in [1, \frac{e + 1}{2}), \\ 0, & x \in [\frac{e + 1}{2}, e], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} ln (2 x - 1), & x \in [1, \frac{e + 1}{2}), \\ 1 - ln (2 x - e), & x \in [\frac{e + 1}{2}, e], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [0, 1), \\ ln (2 x - e), & x \in [\frac{e + 1}{2}, e] . \end{matrix}$

V₄ = [0, 1], then ${\tilde{V_{4}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} - \frac{1}{ln \frac{e + 1}{2}} y + 1, & y \in [0, ln \frac{e + 1}{2}), \\ 0, & y \in [ln \frac{e + 1}{2}, 1], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} \frac{1}{ln \frac{e + 1}{2}} y, & y \in [0, ln \frac{e + 1}{2}), \\ - \frac{1}{1 - ln \frac{e + 1}{2}} (x - 1), & y \in [ln \frac{e + 1}{2}, 1], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [0, ln \frac{e + 1}{2}), \\ 1 + \frac{1}{1 - ln \frac{e + 1}{2}} (x - 1), & y \in [ln \frac{e + 1}{2}, 1] . \end{matrix}$ $\tilde{f_{4}} : {\tilde{U_{4}}}^{1} \to {\tilde{V_{4}}}^{1}$ and $\tilde{f_{4}} (A_{1}) = B_{3}$ , $\tilde{f_{4}} (A_{2}) = B_{2}$ and $\tilde{f_{4}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{4} (x) = {\begin{matrix} ln \frac{e + 1}{2} ln (2 x - 1), & x \in [0, \frac{e + 1}{2}), \\ ln \frac{e + 1}{2} (1 - ln (2 x - e)) + ln (2 x - e), & x \in [\frac{e + 1}{2}, e] . \end{matrix}$ (5) h₅ (x) = sin πx, x ∈ [1.5, 2.5]. U₅ = [1.5, 2.5], then ${\tilde{U_{5}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} cos π (x - 1.5), & x \in [1.5, 2), \\ 0, & x \in [2, 2.5], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} 1 - cos π (x - 1.5), & x \in [1.5, 2), \\ 1 - sin π (x - 2), & x \in [2, 2.5], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [1.5, 2), \\ sin π (x - 2), & x \in [2, 2.5] . \end{matrix}$

V₅ = [-1, 1], then ${\tilde{V_{5}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} - y, & y \in [- 1, 0), \\ 0, & y \in [1, 8], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} y + 1, & y \in [- 1, 0), \\ - y + 1, & y \in [0, 1], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [- 1, 0), \\ y, & y \in [0, 1] . \end{matrix}$ $\tilde{f_{5}} : {\tilde{U_{5}}}^{1} \to {\tilde{V_{5}}}^{1}$ and $\tilde{f_{5}} (A_{1}) = B_{3}$ , $\tilde{f_{5}} (A_{2}) = B_{2}$ and $\tilde{f_{5}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{5} (x) = {\begin{matrix} - cos π (x - 1.5), & x \in [1.5, 2), \\ sin π (x - 2), & x \in [2, 2.5] . \end{matrix}$

(6) h₆ (x) = arcsin x, x ∈ [-1, 1]. U₆ = [-1, 1], then ${\tilde{U_{6}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where

$μ_{A_{1}} (x) = {\begin{matrix} 1 - sin \frac{π}{2} (x + 1), & x \in [- 1, 0), \\ 0, & x \in [0, 1], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} sin \frac{π}{2} (x + 1), & x \in [- 1, 0), \\ cos \frac{π}{2} x, & x \in [0, 1], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [- 1, 0), \\ 1 - cos \frac{π}{2} x, & x \in [0, 1] . \end{matrix}$

$V_{6} = [- \frac{π}{2}, \frac{π}{2}]$ , then ${\tilde{V_{6}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} - \frac{2}{π} y, & y \in [- \frac{π}{2}, 0), \\ 0, & y \in [0, \frac{π}{2}], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} \frac{2}{π} y + 1, & y \in [- \frac{π}{2}, 0), \\ 1 - \frac{2}{π} y, & y \in [0, \frac{π}{2}], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [- \frac{π}{2}, 0), \\ \frac{2}{π} y, & y \in [0, \frac{π}{2}] . \end{matrix}$ $\tilde{f_{6}} : {\tilde{U_{6}}}^{1} \to {\tilde{V_{6}}}^{1}$ and $\tilde{f_{6}} (A_{1}) = B_{3}$ , $\tilde{f_{6}} (A_{2}) = B_{2}$ and $\tilde{f_{6}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{6} (x) = {\begin{matrix} \frac{π}{2} sin \frac{π}{2} (x + 1) - \frac{π}{2} & x \in [- 1, 0), \\ \frac{π}{2} - \frac{π}{2} cos \frac{π}{2} x, & x \in [0, 1] . \end{matrix}$

Appendix B

(1) g₁ (x) = cos ³x, x ∈ [0, 1.5708]. U₁ = [0, 1.5708], then ${\tilde{U_{1}}}^{1} = {A_{1}, A_{2}, A_{3}}$ , where $μ_{A_{1}} (x) = {\begin{matrix} cos (2.402 x), & x \in [0, 0.6539), \\ 0, & x \in [0.6539, 1.5708], \end{matrix}$

$μ_{A_{2}} (x) = {\begin{matrix} 1 - cos (2.402 x), & x \in [0, 0.6539), \\ 1 - sin (1.7132 x - 1.1203), & x \in [0.6539, 1.5708], \end{matrix}$

$μ_{A_{3}} (x) = {\begin{matrix} 0, & x \in [0, 0.6539), \\ sin (1.7132 x - 1.1203), & x \in [0.6539, 1.5708] . \end{matrix}$ V₁ = [0, 1], then ${\tilde{V_{1}}}^{1} = {B_{1}, B_{2}, B_{3}}$ , where

$μ_{B_{1}} (y) = {\begin{matrix} 1 - 2 y, & y \in [0, 0.5), \\ 0, & y \in [0.5, 1], \end{matrix}$ $μ_{B_{2}} (y) = {\begin{matrix} 2 y, & y \in [0, 0.5), \\ - 2 y + 2, & y \in [0.5, 1], \end{matrix}$ $μ_{B_{3}} (y) = {\begin{matrix} 0, & y \in [0, 0.5), \\ 2 y - 1, & y \in [0.5, 1] . \end{matrix}$ $\tilde{f_{11}} : {\tilde{U_{1}}}^{1} \to {\tilde{V_{1}}}^{1}$ and $\tilde{f_{11}} (A_{1}) = B_{3}$ , $\tilde{f_{11}} (A_{2}) = B_{2}$ and $\tilde{f_{11}} (A_{3}) = B_{1}$ . The final output of System I is

$f_{11} (x) = {\begin{matrix} 0.5 cos (2.0402 x) + 0.5, & x \in [0, 0.6539), \\ 0.5 - 0.5 sin (1.7132 x - 1.1203), & x \in [0.6539, 1.5708] . \end{matrix}$ (2) $g_{2} (x) = (lg (x))^{\frac{3}{2}}$ , x ∈ [1, 10]. U₁ = [1, 10], then ${\tilde{U_{1}}}^{1} = {A_{1}, A_{2}}$ , where $μ_{A_{1}} (x) = 1 - ln (\frac{(e - 1) (x - 1)}{9} + 1), x \in [1, 10],$

$μ_{A_{2}} (x) = ln (\frac{(e - 1) (x - 1)}{9} + 1), x \in [1, 10] .$

V₂ = [0, 1],then ${\tilde{V_{2}}}^{1} = {B_{1}, B_{2}}$ , where $μ_{B_{1}} (y) = - y + 1, y \in [0, 1],$

$μ_{B_{2}} (y) = y, y \in [0, 1] .$ $\tilde{f_{21}} : {\tilde{U_{2}}}^{1} \to {\tilde{V_{2}}}^{1}$ and $\tilde{f_{21}} (A_{1}) = B_{1}$ , $\tilde{f_{21}} (A_{2}) = B_{2}$ . The final output of System I is $f_{21} (x) = ln (\frac{(e - 1) (x - 1)}{9} + 1), x \in [0, 1] .$

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