Study on weighted Nagar-Bardini algorithms for centroid type-reduction of general type-2 fuzzy logic systems

Abstract

General type-2 fuzzy logic systems (GT2 FLSs) have drawn great attentions since the alpha-planes representation of general type-2 fuzzy sets (GT2 FSs) was proposed. The iterative of type-reduction (TR) algorithms are difficult to apply in practical applications. In the enhanced types of algorithms, the Nagar-Bardini (NB) algorithms decrease the computation complexity greatly. In terms of the Newton-Cotes quadrature formulas of numerical integration techniques, the paper extends the NB algorithms to three different forms of weighted NB (WNB) algorithms according to the comparisons between the sum operation in NB algorithms and the integral operation in continuous version of NB (CNB) algorithms. The NB algorithms just become a special case of the WNB algorithms. Four simulation examples are used to illustrate and analyze the performances of the WNB algorithms while performing the centroid TR of GT2 FLSs. It also shows that, in general, the WNB algorithms have smaller absolute error and faster convergence speed compared with the NB algorithms, which provides the potential value for T2 FLSs designers and users.

Keywords

alpha-planes representation general type-2 fuzzy logic systems type-reduction weighted Nagar-Bardini algorithms simulation

1 Introduction

As we all know, the membership grades type-2 fuzzy sets (T2 FSs) are themselves type-1 fuzzy sets (T1 FSs). Interval type-2 fuzzy logic systems (IT2 FLSs) [1 –4] based on IT2 FSs [5, 6] are most commonly used T2 FLSs for applications as the membership grades of IT2 FSs all equal to 1 uniformly. However, researching on GT2 FLSs has been increased rapidly since the alpha-planes or z-slices [7 –9] representations of GT2 FSs were raised. The computation complexity of GT2 FLSs [10 –14] has been greatly decreased due to the proposed representation of G2 FSs, therefore, they are gradually applied to many applications in recent years. As the secondary membership grades of GT2 FSs lie on the interval [0, 1], GT2 FSs [15] can be considered as higher order uncertain parameter models compared with the IT2 FSs. With the increment of design degrees of freedom, GT2 FLSs based GT2 FSs may outperform IT2 FLSs in many fields.

Generally speaking, a T2 FLS is made up of five blocks, they are fuzzifier, inference, rules, type-reducer, and defuzzifier. Among which, the block of type-reduction acts as the central role of transforming the T2 FS to the T1 FS. Then the block of defuzzification convert the T1 FS to the crisp output. The type-reduction and defuzzification compose of the output processing. T1 FLSs do not involve the process of type-reduction. While the type-reduction involves comprehensive computations. If all the uncertainties of T2 membership functions were disappeared, the T2 FLSs [16] naturally collapse to the T1 FLSs. This paper focuses only on the FLSs that use type-reduction followed by defuzzification.

Usually, the most popular Karnik-Mendel (KM) algorithms [17, 18] were used to compute the centroid of IT2 FSs or perform the centroid TR of IT2 FLSs. However, it usually needs two to six iterations for each KM iteration. In order to reduce the computational cost, Wu and Mendel developed the enhanced KM (EKM) algorithms [19], which can save about two iterations on average. Then the continuous version of KM and EKM (CKM and CEKM [20]) were proposed for theoretical analysis. In addition, the monotoncity and super-exponential convergence of CKM algorithms were proved by Mendel and Liu [21]. According to the numerical integration technique, Liu extended the EKM algorithms to the weighted EKM (WEKM) algorithms [22] to compute the left centroid endpoints of IT2 FSs. Moreover, the simulation results verified that the WEKM algorithms have smaller absolute error and faster convergence speed compared with the EKM algorithms. Recent studies by Nagar and Bardini (NB) verified that IT2 FLSs based on the NB algorithms [23] have superior performances to respond the affect of uncertainties in the systems’ parameters than the IT2 FLSs based on other type-reduction (TR) algorithms like EKM, Nie-Tan (NT) [24], Begian-Melek-Mendel (BMM [25, 26]), and Greenfield-Chiclana Collapsing Defuzzifier (GCCD [27]). All above works laid a rich foundation for performing centroid type-reduction of T2 FLSs based on type-reduction algorithms.

Inspired by references [5 , 28–30], the paper performs the centroid type-reduction of alpha-planes based GT2 FLSs with weighted Nagar-Bardini (WNB) algorithms. Firstly, the continuous version of NB (CNB) algorithms is provided. By observing and comparing the sum operation in discrete version of NB algorithms and the integral operation in CNB algorithms, the NB algorithms are extended to the WNB algorithms resort to the Newton-Cotes quadrature formulas in numerical integration. While computing the centroid type-reduced set and defuzzified value of GT2 FLSs, the WNB algorithms can achieve more accurate results compared with the NB algorithms. Whereas the NB algorithms turn into a special case of the WNB algorithms as the weights of the latter are constants.

The rest of this paper is organized as follows. Section 2 gives the background of GT2 FSs and GT2 FLSs. Section 3 provides the Newton-Cotes quadrature formulas, the NB and CNB based GT2 FLSs, and how to perform the centroid TR of GT2 FLSs by means of the WNB algorithms. Through four simulation examples, the performances of the WNB and NB algorithms are given in Section 4. Finally, Section 4 gives the conclusions.

2 Background

2.1 GT FSs

Definition 1. A GT2 FS $\tilde{B}$ can be described by its T2 membership function (MF) $μ_{\tilde{B}} (y, v)$ , i.e., $\tilde{B} = \int_{y \in Y} \int_{v \in J_{y}} μ_{\tilde{B}} (y, v) / (y, v)$ (1) where ∬ denotes all the admissible y and v, and v ∈ [0, 1]. The point-value representation of $\tilde{B}$ is as: $\overset{\leftrightarrow}{B} = {((y, v), μ_{\tilde{B}} (y, v)) | \forall y \in Y, \forall v \in [0, 1]}$ (2)

Definition 2. The secondary MF [31] of $\tilde{B}$ is a vertical slice of $μ_{\tilde{B}} (y, v)$ , i.e., $μ_{\tilde{B}} (y = y^{'}, v) \equiv μ_{\tilde{B}} (y^{'}) = \int_{v \in J_{y^{'}}} f_{y^{'}} (v) / v$ (3) where ∫ denotes all the possible union operations. Here we can express $μ_{\tilde{B}} (y^{'})$ by $\tilde{B} (y)$ for simplicity. Therefore, Equation (2) can also be expressed as: $\tilde{B} = \int_{y \in Y} \tilde{B} (y) / y$ (4)

Definition 3. Let ${\tilde{B}}_{α} (y)$ be the α-cut of $\tilde{B} (y)$ , α ∈ [0, 1], i.e., [32] ${\tilde{B}}_{α} (y) = {v | f_{y} (v) \geq α} = [a_{α} (y), b_{α} (y)]$ (5) where f_y (v) is the secondary membership grade, for every y ∈ Y, ${\tilde{B}}_{α}$ can be expressed by the following α-cut decomposition as: ${\tilde{B}}_{α} = ⋃_{\forall α \in [0, 1]} [α / {\tilde{B}}_{α} (y)] = sup {α / [a_{α} (y), b_{α} (y)]}$ (6) where ⋃denotes the union operation. Then we can substitute Equation (6) into Equation (4) to obtain the vertical slice representation of $\tilde{B}$ , i.e., $\tilde{B} = \int_{\forall y \in Y} {\underset{\forall α \in [0, 1]}{\cup} [α / {\tilde{B}}_{α} (y)]} / y .$ (7)

Definition 4. The alpha-planes (z-slices or horizontal slices) [7 , 33] representation of $\tilde{B}$ is as: $\tilde{B} = ⋃_{\forall α \in [0, 1]} {α / [\int_{\forall y \in Y} {\tilde{B}}_{α} (y) / y]} .$ (8) where ${\tilde{B}}_{α}$ is the union of all primary MFs whose secondary membership grades are greater than or equal to α, i.e., ${\tilde{B}}_{α} = \int_{\forall y \in Y} \int_{\forall v \in [0, 1]} {(y, v) | f_{y} (v) \geq α} .$ (9)

Definition 5. The two dimensional support of $μ_{\tilde{B}} (y, v)$ is called the footprint of uncertainty (FOU) of $\tilde{B}$ , i.e., [7 , 27] $FOU (\tilde{B}) = {(y, v) \in Y \times [0, 1] | μ_{\tilde{B}} (y, v) > 0}$ (10) where the FOU $(\tilde{B})$ is bounded by the lower MF ${\underline{μ}}_{\tilde{B}} (y)$ and the upper MF ${\bar{μ}}_{\tilde{B}} (y)$ , i.e., ${\underline{μ}}_{\tilde{B}} (y) = inf {v | v \in [0, 1], μ_{\tilde{B}} (y, v) > 0}$ (11) ${\bar{μ}}_{\tilde{B}} (y) = sup {v | v \in [0, 1], μ_{\tilde{B}} (y, v) > 0} .$ (12)

2.2 GT2 FLSs

In general, GT2 FLSs can be divided into Takagi-Sugeno-Kang (TSK) type and Mamdani type from the aspect of structure. Here we consider a Mamdani type GT2 FLS [9] with p inputs x₁ ∈ X₁, ⋯ , x_p ∈ X_p and one output y ∈ Y, which can be described by M fuzzy rules, where the sth rule is of the form: $R^{s} : If x_{1} is {\tilde{F}}_{1}^{s} and \dots and x_{p} is {\tilde{F}}_{p}^{s}, then y is {\tilde{G}}^{s}$ (13) where ${\tilde{F}}_{i}^{s} (i = 1, \dots, p; s = 1, \dots, M)$ is the antecedent GT2 FS, ${\tilde{G}}^{s} (s = 1, \dots, M)$ is the consequent GT2 FS.

Here we adopt the singleton fuzzifier for simplicity, i.e., as $x_{i} = x_{i}^{'}$ , only the vertical slice ${\tilde{F}}_{i}^{s} (x_{i}^{'})$ is activated, and the α-cut decomposition is as: ${\tilde{F}}_{i}^{s} (x_{i}^{'}) = sup_{\forall α \in [0, 1]} α / [a_{i, α}^{s} (x_{i}^{'}), b_{i, α}^{s} (x_{i}^{'})] .$ (14)

Then the firing interval of each fuzzy rule at the corresponding α-level can be calculated as: $F_{α} : {\begin{matrix} F_{α}^{s} (x^{'}) \equiv [{\underline{f}}_{α}^{s} (x^{'}), {\bar{f}}_{α}^{s} (x^{'})] \\ {\underline{f}}_{α}^{s} (x^{'}) \equiv T_{i = 1}^{p} a_{i, α}^{s} (x_{i}^{'}) \\ {\bar{f}}_{α}^{s} (x^{'}) \equiv T_{i = 1}^{p} b_{i, α}^{s} (x_{i}^{'}) \end{matrix}$ (15) where T represents the minimum or product t-norm operation [9].

For each fuzzy rule, calculate the corresponding α-level horizontal slice ${\tilde{G}}_{α}^{s}$ of consequent ${\tilde{G}}^{s}$ as: ${\tilde{G}}_{α}^{s} = \int_{\forall y \in Y} {\tilde{G}}_{α}^{s} (y) / y = \int_{\forall y \in Y} [g_{l, α}^{s} (y), g_{r, α}^{s} (y)] / y .$ (16)

Then combine the firing interval of every fuzzy rule with its corresponding consequent horizontal slice ${\tilde{G}}_{α}^{l}$ to obtain the firing rule horizontal slice (α-plane) ${\tilde{B}}_{α}^{l}$ as: ${\tilde{B}}_{α}^{l} : {\begin{matrix} FOU ({\tilde{B}}_{α}^{l}) = [{\underline{μ}}_{{\tilde{B}}_{α}^{l}} (y | x^{'}), {\bar{μ}}_{{\tilde{B}}_{α}^{l}} (y | x^{'})] \\ {\underline{μ}}_{{\tilde{B}}_{α}^{l}} (y | x^{'}) = {\underline{f}}_{α}^{l} (x^{'}) * g_{L, α}^{l} (y) \\ {\bar{μ}}_{{\tilde{B}}_{α}^{l}} (y | x^{'}) = {\bar{f}}_{α}^{l} (x^{'}) * g_{R, α}^{l} (y) \end{matrix}$ (17) where * represents the minimum or product t-norm operation [9].

Next, the aggregated output horizontal slice ${\tilde{B}}_{α}$ can be obtained by merging all the horizontal slices as: ${\tilde{B}}_{α} : {\begin{matrix} FOU ({\tilde{B}}_{α}) = [{\underline{μ}}_{{\tilde{B}}_{α}} (y | x^{'}), {\bar{μ}}_{{\tilde{B}}_{α}} (y | x^{'})] \\ {\underline{μ}}_{{\tilde{B}}_{α}} (y | x^{'}) = {\underline{μ}}_{{\tilde{B}}_{α}^{1}} (y | x^{'}) \lor \dots \lor {\underline{μ}}_{{\tilde{B}}_{α}^{M}} (y | x^{'}) \\ {\bar{μ}}_{{\tilde{B}}_{α}} (y | x^{'}) = {\bar{μ}}_{{\tilde{B}}_{α}^{1}} (y | x^{'}) \lor \dots \lor {\bar{μ}}_{{\tilde{B}}_{α}^{M}} (y | x^{'}) \end{matrix}$ (18) where ∨ denotes the maximum operation [9].

Finally we compute the centroid of ${\tilde{B}}_{α}$ to obtain the type-reduced set Y_C,α (x′) at a certain α-level, i.e., [9] $Y_{C, α} (x^{'}) = C_{{\tilde{B}}_{α}} (x^{'}) = α / [l_{{\tilde{B}}_{α}} (x^{'}), r_{{\tilde{B}}_{α}} (x^{'})]$ (19) where the two end points $l_{{\tilde{B}}_{α}} (x^{'})$ and $r_{{\tilde{B}}_{α}} (x^{'})$ can be computed by many TR algorithms [17–22 , 30] as: $l_{{\tilde{B}}_{α}} (x^{'}) = min_{μ_{{\tilde{B}}_{α}} (y_{i}) \in [{\underline{μ}}_{{\tilde{B}}_{α}} (y_{i}), {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})]} \frac{\sum_{i = 1}^{N} y_{i} μ_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} μ_{{\tilde{B}}_{α}} (y_{i})}$ (20) $r_{{\tilde{B}}_{α}} (x^{'}) = max_{μ_{{\tilde{B}}_{α}} (y_{i}) \in [{\underline{μ}}_{{\tilde{B}}_{α}} (y_{i}), {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})]} \frac{\sum_{i = 1}^{N} y_{i} μ_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} μ_{{\tilde{B}}_{α}} (y_{i})}$ (21) where N denotes the number of discrete points of y.

Note that the final output of Mamdani type GT2 FLSs can be derived from the defuzzification method in the next section.

3 WNB algorithms

Before introducing the proposed WNB algorithms, I first give two parts of preliminary contents: Newton-Cotes quadrature formulas [22, 34], NB and CNB algorithms [23, 28].

3.1 Newton-Cotes quadrature formulas

The thought of estimation theory is to approximate the definite integral $\int_{a}^{b} f (x) dx$ as the linear combination of functional values f (x_i). Consequently, the computations of definite integral can be changed to compute functional values.

Definition 6 (Numerical integration [22, 28]) Suppose that a = x₀ < x₁ < x₂ < ⋯ < x_N = b, and the formula: $Q (f) = \sum_{i = 0}^{N} w_{i} f (x_{i}) = w_{0} f (x_{0}) + \dots + w_{N} f (x_{N})$ (22) which satisfies: $\int_{a}^{b} f (x) dx = Q (f) + E (f)$ (23) Equation (23) is referred to as the numerical integration or quadrature formula, in which E (f) is called as the remainder or the error term of quadrature formula, ${w_{i}}_{i = 0}^{N}$ is the weight coefficient, and ${x_{i}}_{i = 0}^{N}$ is the quadrature node.

Definition 7. (Accuracy order [22]) An integer n can be defined as the accuracy of quadrature formula if it satisfies: for any order polynomials which meet i ≤ n, the remainder E (P_i) =0; and for some polynomial P_i+1 (x) whose order is n + 1, E (P_i+1) ≠0, i.e., when the order i ≤ n, $\int_{a}^{b} P_{i} (x) dx = Q (P_{i})$ , and when the order i = n + 1, $\int_{a}^{b} P_{i + 1} (x) dx = Q (P_{i + 1})$ .

Here the composite trapezoidal rule, composite Simpson rule, and composite Simpson 3/8 rule are used to approximate the f (x)as the straight line, quadrature polynomial function, and cubic polynomial function, respectively.

Lemma 1. (Composite trapezoidal rule [22, 28]) Let the function y = f (x) be defined on [a, b]. Divide [a, b] into N subintervals ${x_{i - 1}, x_{i}}_{i = 1}^{N}$ with the equi-width $h = \frac{b - a}{N}$ , where the equidistant node is as x_i = x₀ + ih (i = 0, 1, ⋯ , N), then the numerical approximation of definite integral with the composite trapezoidal rule is as $\int_{a}^{b} f (x) dx = \frac{h}{2} [f (a) + f (b) + 2 \sum_{i = 1}^{N - 1} f (x_{i})] + E_{T} (f, h)$ (24)

If the function f is second order continuous differentiable on [a, b], then the error term $E_{T} (f, h) = - \frac{(b - a) f^{″} (ζ)}{12} h^{2}$ , where ζ ∈ (a, b).

Lemma 2. (Composite Simpson rule [22, 28]) Let the function y = f (x) be defined on [a, b]. Divide [a, b] into 2N subintervals ${x_{i - 1}, x_{i}}_{i = 1}^{2 N}$ with the equi-width $h = \frac{b - a}{2 N}$ , where the equidistant node is as x_i = x₀ + ih (i = 0, 1, ⋯ , 2N), then the numerical approximation of definite integral with the composite Simpson rule is as

$\begin{matrix} \int_{a}^{b} f (x) dx = \frac{h}{3} [f (a) + f (b) + 2 \sum_{i = 1}^{N - 1} f (x_{2 i}) \\ + 4 \sum_{i = 0}^{N - 1} f (x_{2 i + 1})] + E_{S} (f, h) \end{matrix}$ (25)

If the function f is fourth order continuous differentiable on [a, b], then the error term $E_{S} (f, h) = - \frac{(b - a) f^{(4)} (ζ)}{180} h^{4}$ , where ζ ∈ (a, b).

Lemma 3. (Composite Simpson 3/8 rule [22, 28]) Let the function y = f (x) be defined on [a, b]. Divide [a, b] into 3N subintervals ${x_{i - 1}, x_{i}}_{i = 1}^{3 N}$ with the equi-width $h = \frac{b - a}{3 N}$ , where the equidistant node is as x_i = x₀ + ih (i = 0, 1, ⋯ , 3N), then the numerical approximation of definite integral with the composite Simpson 3/8 rule is as

$\begin{matrix} \int_{a}^{b} f (x) dx = \frac{3 h}{8} [f (a) + f (b) + \sum_{i = 1}^{N} 2 f (x_{3 i}) \\ + \sum_{i = 1}^{N} 3 f (x_{3 i - 2}) + \sum_{i = 1}^{N} 3 f (x_{3 i - 1})] + E_{SC} (f, h) \end{matrix}$ (26)

If the function f is fourth order continuous differentiable on [a, b], then the error term $E_{SC} (f, h) = - \frac{(b - a) f^{(4)} (ζ)}{80} h^{4}$ , where ζ ∈ (a, b).

Suppose that the formulas (24)–(26) be measurable, i.e., all the integrals are in the Lebesque sense.

3.2 NB and CNB algorithms

NB algorithms [23, 28] can perform the TR and defuzzification simultaneously. For the centroid of ${\tilde{B}}_{α}$ , $Y_{NB, α} = α / [l_{{\tilde{B}}_{α}}, r_{{\tilde{B}}_{α}}]$ , in which the two end points can be calculated by the discrete version of NB algorithms as (see Equations (20) and (21)): $l_{{\tilde{B}}_{α}} = \frac{\sum_{i = 1}^{N} y_{i} {\underline{μ}}_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} {\underline{μ}}_{{\tilde{B}}_{α}} (y_{i})}, and r_{{\tilde{B}}_{α}} = \frac{\sum_{i = 1}^{N} y_{i} {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})}$ (27)

Finally the T1 FS Y_NB can be obtained by aggregating all the α-planes Y_NB,α, i.e., $Y_{NB} = sup_{\forall α \in [0, 1]} α / Y_{NB, α} (x^{'}) .$ (28)

In the practical calculations, if the number of α-planes is m, that is to say, α = α₁, ⋯ , α_m, and α is uniformly divided. Then the crisp output of GT2 FLSs is as: $y_{NB} (x^{'}) = \sum_{i = 1}^{m} α_{i} [(l_{{\tilde{B}}_{α_{i}}} (x^{'}) + r_{{\tilde{B}}_{α_{i}}} (x^{'})) / 2] / \sum_{i = 1}^{m} α_{i}$ (29) this Equation [5, 17] was referred to as the average of endpoints defuzzification.

Similar to the continuous version of KM or EKM algorithms [6 , 22–24], the continuous version of NB (CNB) algorithms can be adopted for studying the TR and defuzzification of GT2 FLSs. Here we let the GT2 FS $\tilde{B}$ be the centroid output FS by means of the inference engine [29]. For each α_i (i = 1, ⋯ , m), the corresponding IT2 FS is $R_{{\tilde{B}}_{α_{i}}}$ . Suppose that the primary variable of $\tilde{B}$ satisfy that a = y₁ < y₂ < ⋯ < y_N = b, the CNB algorithms for the centroid two end points should be as: $l_{{\tilde{B}}_{α}} = \frac{\int_{a}^{b} y {\underline{μ}}_{{\tilde{B}}_{α}} (y) dy}{\int_{a}^{b} {\underline{μ}}_{{\tilde{B}}_{α}} (y) dy}, and r_{{\tilde{B}}_{α}} = \frac{\int_{a}^{b} y {\bar{μ}}_{{\tilde{B}}_{α}} (y) dy}{\int_{a}^{b} {\bar{μ}}_{{\tilde{B}}_{α}} (y) dy} .$ (30)

The non-iterative NB algorithm provides a close form of TR. Moreover, the output is a linear combination of two outputs of T1 FLSs: one is from the LMF, and the other is from the UMF.

Finally, the type-reduced set and output of GT2 FLSs can be calculated as in Equations (28) and (29), respectively.

3.3 WNB algorithms

From the aspect of theory, the CNB algorithms make us better understand the NB algorithms. According to the former two subsections, a type of weighted NB-WNB algorithms is proposed for performing the centroid TR of GT2 FLSs in comparison with the NB algorithms. In this section, the contents are adapted from [22, 23].

Here the centroid two end points of ${\tilde{B}}_{α}$ is calculated by the WNB algorithms as: $l_{{\tilde{B}}_{α}} = \frac{\sum_{i = 1}^{N} w_{i} y_{i} {\underline{μ}}_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} w_{i} {\underline{μ}}_{{\tilde{B}}_{α}} (y_{i})}, and r_{{\tilde{B}}_{α}} = \frac{\sum_{i = 1}^{N} w_{i} y_{i} {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})}{\sum_{i = 1}^{N} w_{i} {\bar{μ}}_{{\tilde{B}}_{α}} (y_{i})}$ (31) where w_i denotes the weight (see Table 1).

Table 1

Weights for the WNB algorithms

Algotihms	Integration rule	Weights
NB	—	w_i = 1 (i = 1, …, N)
TWNB	Composite Trapzoidal rule	$w_{i} = {\begin{matrix} 1 / 2, i = 1, N, \\ 1, i \neq 1, N . \end{matrix}$
SWNB	Composite Simpson rule	$w_{i} = {\begin{matrix} 1 / 2, i = 1, N, \\ 1, i = 1 mod (2), i \neq 1, N, \\ 2, i = 0 mod (2), i \neq N . \end{matrix}$
S3/8WNB	Composite Simpson 3/8 rule	$w_{i} = {\begin{matrix} 1 / 3, i = 1, N, \\ 2 / 3, i = 1 mod (3), i \neq 1, N, \\ 1, i = 2 mod (3), i \neq N, \\ 1, i = 0 mod (2), i \neq N . \end{matrix}$

Then we still calculate the type-reduced set and the output as in Equations (28) and (29), respectively.

From the viewpoint of numerical integration, the WNB algorithms are the numerical implementation of CNB algorithms. Comparing Equations (27) and (30), it can be found that the continuous version of NB algorithms is similar to the discrete version of NB algorithms. However, the sum operations in NB algorithms are transformed to the definite integral operations in CNB algorithms, that is to say, the sum operations on the discrete nodes y_i act as the integration operations of corresponding function. By means of Equation (22) in numerical integration [35], the related weight w_i for each MF of sampling points y_i can be assigned to obtain the more accurate type-reduced set and output. Here the NB algorithms are a special case of WNB algorithms as the weights of the latter are selected as w_i = 1 (i = 1, 2, ⋯ , N). Furthermore, although many types of weights assignment can be adopted, this paper only use the Newton-Cotes quadrature formulas based numerical integration approaches which are introduced in section 3.1, and they are referred to as the composite trapezoidal rule, composite Simpson rule, and Simpson 3/8 rule, respectively. The sampling points for three types of WNB algorithms are equally distributed on the interval [a, b], i.e., $y_{i} = a + \frac{i - 1}{N - 1} (b - a) (i = 1, \dots, N)$ .

As shown in Table 1, except for the NB algorithms, formulas (24)–(26) are adopted to assign the weights for three types of WNB algorithms in terms of the following steps:

We substitute x₀ = a, x_N = b, x_i (i = 0, 1, 2, ⋯, N) in Equation (24), x₀ = a, x_2N = b, x_i (i = 0, 1, 2, ⋯ , 2N) in Equation (25), and x₀ = a, x_3N = b, x_i (i = 0, 1, ⋯ , 3N) in Equation (26) all by x₁ = a, x_N = b, x_i (i = 1, ⋯ , N).

The coefficients h/2, h/3, and 3h/8 in Equations (24)–(26) cal all be cancelled by the quotient of two integrals as in Equation (30).

See Table 1, the weights for TWNB and SWNB are assigned as one-half of the coefficients in Equations (23) and (24), whereas the weights of S3/8WNB are assigned as one-third of the coefficients in Equation (25).

The number of sampling points of SWNB and S3/8WNB algorithms can not be only restricted to N = 2n + 1 and N = 3n + 1, but as required by the Equations (24) and (25), where n is an integer, N = 1 mod (2) and N = 1 mod (3).

According to Equations (23)–(25) and Table 1, the TWNB, SWNB and S3/8WNB algorithms can approximate the MF of numerical integration as first order, second order, and third order polynomial functions, respectively. However, they are only special cases of Newton-Cotes quadrature formulas. Although the more generalized WNB algorithms can be obtained according to any higher order of Newton-Cotes quadrature formulas, they may greatly affect by the Runge’s phenomenon.

Next we summarize the relations between CNB and WNB algorithms as they complete the TR and defuzzification of GT2 FLSs:

WNB algorithms can calculate the centroid values according to the sum operation of sampling points y_i (i = 1, ⋯ , N). While the CNB algorithms adopt the integral operation to compute the comparatively accurate centroid values. In theory, the solutions of WNB algorithms will approach CNB algorithms as the number of sampling point N→ ∞.

In general, the computation results of WNB algorithms will be more accurate as the number of sampling point increases.

The numerical computations of WNB algorithms are based on the sum operations. Whereas the computations of CNB algorithms are in terms of the integral operations. In conclusion, the WNB algorithms can be considered as the numerical implementation of CNB algorithms according to the numerical integration approaches.

Finally, we can make a conclusion about computing the centroid TR and defuzzification of GT2 FLSs by means of WNB and CNB algorithms, and the steps are as follows:

Step 1: According to the fuzzy reasoning [29], combine the GT2 FSs in fuzzy rules to obtain the centroid type-reduced set $\tilde{B}$ for the GT2 FLSs.

Step 2: Break the value of α into 0, 1/Δ, ⋯ , (Δ - 1)/Δ, 1. And decompose the GT2 FS $\tilde{B}$ into multiple α-planes with the corresponding α values.

Step 3: By means of the CNB and WNB algorithms, calculate the centroid $α / [l_{{\tilde{B}}_{α}}, r_{{\tilde{B}}_{α}}]$ for each of the corresponding IT2 FS $R_{{\tilde{B}}_{α}}$ , where $R_{{\tilde{B}}_{α}} = α / {\tilde{B}}_{α}$ .

Step 4: Union all the centroids to obtain the output T1 FS (see Equation (28)).

Step 5: Consider the CNB algorithms as the benchmark, and compare the performances of WNB algorithms with the NB algorithms.

4 Simulation studies

In this section, four numerical simulation examples are provided for studying. Before performing the TR and defuzzification, here we make the premise that the centroid output GT2 FS of GT2 FLSs has been known by means of weighting and merging all the fuzzy rules. For the first example, the FOU is composed of piecewise linear functions [6 , 22–24], while the corresponding secondary MF (vertical slice) is the trapezoidal type. For the second example, the FOU is Gaussian functions and piecewise linear functions [19 , 25], while the corresponding vertical slice is also the trapezoidal type. For the third example, the FOU is composed of Gaussian functions [6 , 22–24], while the corresponding vertical slice is the triangular type. For the last example, the FOU is the Gaussian MF with uncertain derivation [19 , 25], while the corresponding vertical slice is also the triangular type.

In the simulations, the α is decomposed into Δ effective values as 0, 1/Δ, ⋯ , (Δ - 1)/Δ, 1. And let the number of Δ be varied from 1 to 100 with the step size of one. Here we denote the primary variable of centroid output GT2 FS by the letter x. Furthermore, the number of sampling points of x is selected as 1000. In order to improve the calculation accuracy, we compare the NB and WNB algorithms. Figure 1 and Table 2 provide the defined FOUs for four examples. In addition, Fig. 2 and Table 3 give the defined secondary MFs (vertical slices) for four examples.

Fig. 1

Graphs of FOUs; (a) example 1, (b) example 2, (c) example 3, and (d) example 4.

Table 2

The FOU MF expressions for four examples [14, 28]

Num	FOU MF expressions
1	${\underline{μ}}_{{\tilde{A}}_{1}} (x) = max {[\begin{matrix} \frac{x - 1}{6}, 1 \leq x \leq 4 \\ \frac{7 - x}{6}, 4 < x \leq 7 \\ 0, otherwise \end{matrix}], [\begin{matrix} \frac{x - 3}{6}, 3 \leq x \leq 5 \\ \frac{8 - x}{9}, 5 < x \leq 8 \\ 0, otherwise \end{matrix}]}$
	${\bar{μ}}_{{\tilde{A}}_{1}} (x) = max {[\begin{matrix} \frac{x - 1}{2}, 1 \leq x \leq 3 \\ \frac{7 - x}{4}, 3 < x \leq 7 \\ 0, otherwise \end{matrix}], [\begin{matrix} \frac{x - 2}{5}, 2 \leq x \leq 6 \\ \frac{16 - 2 x}{5}, 6 < x \leq 8 \\ 0, otherwise \end{matrix}]}$
2	${\underline{μ}}_{{\tilde{A}}_{2}} (x) = {\begin{matrix} \frac{0.6 (x + 5)}{19}, - 5 \leq x \leq 2.6 \\ \frac{0.4 (14 - x)}{19}, 2.6 < x \leq 14 \end{matrix}$
	${\bar{μ}}_{{\tilde{A}}_{2}} (x) = {\begin{matrix} exp [- \frac{1}{2} {(\frac{x - 2}{5})}^{2}], - 5 \leq x \leq 7.185 \\ exp [- \frac{1}{2} {(\frac{x - 9}{1.75})}^{2}], 7.185 < x \leq 14 \end{matrix}$
3	${\underline{μ}}_{{\tilde{A}}_{3}} (x) = max {0.5 exp [- \frac{(x - 3)^{2}}{2}], 0.4 exp [- \frac{(x - 6)^{2}}{2}]}$
	${\bar{μ}}_{{\tilde{A}}_{3}} (x) = max {exp [- 0.5 \frac{(x - 3)^{2}}{4}], 0.8 exp [- 0.5 \frac{(x - 6)^{2}}{4}]}$ x ∈ [0, 10]
4	${\underline{μ}}_{{\tilde{A}}_{4}} (x) = exp [- \frac{1}{2} {(\frac{x - 3}{0.25})}^{2}]$ ${\bar{μ}}_{{\tilde{A}}_{4}} (x) = exp [- \frac{1}{2} {(\frac{x - 3}{1.75})}^{2}]$ x ∈ [0, 10].

Fig. 2

Graphs of vertical slice; (a) example 1, (b) example 2, (c) example 3, and (d) example 4.

Table 3

The second MF expressions for four examples [14, 28]

Num	Second MF expressions
1	Top left end point: $L (x) = \underline{u} (x) + 0.6 w (\bar{u} (x) - \underline{u} (x))$ top right end point: $R (x) = \bar{u} (x) - 0.6 (1 - w) (\bar{u} (x) - \underline{u} (x)) w = 0$
2	The same as in example 1.
3	Apex: $Apex = \underline{u} (x) + w (\bar{u} (x) - \underline{u} (x)) w = 0.25$
4	The Apex is the same as in example three. w = 0.75

In the first example, we first use the CNB algorithms to calculate the centroid type-reduced T1 FS for Δ = 100 and the centroid defuzzified values for Δ ranging from 1 to 100, and the graphs are shown in Figs. 3 and 4. Furthermore, the computation results of CNB algorithms are considered as the benchmark.

Fig. 3

The centroid type-reduced T1 FS for example 1(computed by the CNB algorithms).

Fig. 4

The centroid defuzzified values for example 1(computed by the CNB algorithms).

Next, the type-reduced T1 FS for four types of WNB algorithms is provided in Fig. 5(a), in addition, the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms is shown in Fig. 5(b), in which the grades of centroid type-reduced MF and the absolute error |C_WNB - C_CNB| are selected as the independent and dependent variables, respectively.

Fig. 5

(a) The type-reduced T1 FS for four types of WNB algorithms (b) the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms for example 1.

Moreover, we give the centroid defuzzified values of four types of WNB algorithms in Fig. 6(a), and the absolute error of centroid defuzzified value between the CNB and WNB algorithms in Fig. 6(b), in which the Δ and the absolute error |y_WNB - y_CNB| are chosen as the in dependent and dependent variables, respectively.

Fig. 6

(a) The centroid defuzzified values of four types of WNB algorithms (b) the absolute error of centroid defuzzified value between the CNB and WNB algorithms for example 1.

In the second example, we still use the CNB algorithms to calculate the centroid type-reduced T1 FS for Δ = 100 and the centroid defuzzified values for Δ ranging from 1 to 100, which are shown in Figs. 7 and 8.

Fig. 7

The centroid type-reduced T1 FS for example 2(computed by the CNB algorithms).

Fig. 8

The centroid defuzzified values for example 2(computed by the CNB algorithms).

Then the type-reduced T1 FS for four types of WNB algorithms is provided in Fig. 9(a), and the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms is shown in Fig. 9(b). In addition, the centroid defuzzified values of four types of WNB algorithms is shown in Fig. 10(a), and the absolute error of centroid defuzzified value between the CNB and WNB algorithms is shown in Fig. 10(b).

Fig. 9

(a) The type-reduced T1 FS for four types of WNB algorithms (b) the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms for example 2.

Fig. 10

(a) The centroid defuzzified values of four types of WNB algorithms (b) the absolute error of centroid defuzzified value between the CNB and WNB algorithms for example 2.

In the third example, we adopt the CNB algorithms to compute the centroid type-reduced T1 FS for Δ = 100 and the centroid defuzzified values for Δ ranging from 1 to 100, and the graphs are shown in Figs. 11 and 12.

Fig. 11

The centroid type-reduced T1 FS for example 3(computed by the CNB algorithms).

Fig. 12

The centroid defuzzified values for example 3(computed by the CNB algorithms).

Then the type-reduced T1 FS for four types of WNB algorithms is provided in Fig. 13(a), in addition, the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms is shown in Fig. 13(b). In addition, we provide the centroid defuzzified values of four types of WNB algorithms in Fig. 14(a), and the absolute error of centroid defuzzified value between the CNB and WNB algorithms in Fig. 14(b).

Fig. 13

(a) The type-reduced T1 FS for four types of WNB algorithms (b) the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms for example 3.

Fig. 14

(a) The centroid defuzzified values of four types of WNB algorithms (b) the absolute error of centroid defuzzified value between the CNB and WNB algorithms for example 3.

In the last example, we use the CNB algorithms to calculate the centroid type-reduced T1 FS for Δ = 100 and the centroid defuzzified values for Δ ranging from 1 to 100, which are shown in Figs. 15 and 16.

Fig. 15

The centroid type-reduced T1 FS for example 4(computed by the CNB algorithms).

Fig. 16

The centroid defuzzified values for example 4(computed by the CNB algorithms).

Then the type-reduced T1 FS for four types of WNB algorithms is given in Fig. 17(a), and the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms is shown in Fig. 17(b). The centroid defuzzified values of four types of WNB algorithms is shown in Fig. 18(a), and the absolute error of centroid defuzzified value between the CNB and WNB algorithms is given in Fig. 18(b).

Fig. 17

(a) The type-reduced T1 FS for four types of WNB algorithms (b) the absolute error of centroid type-reduced T1 between the CNB and WNB algorithms for example 4.

Fig. 18

(a) The centroid defuzzified values of four types of WNB algorithms (b) the absolute error of centroid defuzzified value between the CNB and WNB algorithms for example 4.

In order to further analyze the WNB algorithms, we use the Table 4 and Table 5 to provide the relative error |y_{WNB
_i} - y_{CNB
_i}|/|y_{CNB
_i}| (i = 1, 2, ⋯ , 4) between WNB and CNB algorithms for the centroid type-reduced FS at Δ = 100, and the defuzzified values for Δ varying from 1 to 100, respectively.

Table 4

Mean of the centroid type-reduced sets relative errors as Δ = 100

Algorithm	NB	TWNB	SWNB	S3/8WNB
Example 1	0.00000004	0.00000004	0.00000002	0.00000006
Example 2	0.00028132	0.00001599	0.00001039	0.00001917
Example 3	0.00008479	0.00000071	0.00000076	0.00003008
Example 4	0.00000033	0.00000033	0.00000033	0.00000794
Total mean	0.00009162	0.00000427	0.00000288	0.00001431

Table 5

Mean of the centroid defuzzified values relative errors for Δ varying from 1 to 100

Algorithm	NB	TWNB	SWNB	S3/8WNB
Example 1	0.00000004	0.00000004	0.00000002	0.00000005
Example 2	0.00026998	0.00001532	0.00000996	0.00001830
Example 3	0.00008462	0.00000071	0.00000076	0.00003018
Example 4	0.00000020	0.00000020	0.00000020	0.00000866
Total mean	0.00008871	0.00000407	0.00000274	0.00001430

Seeing from the Tables 4 and 5, and the Figs. 5, 6, 9, 10, 13, 14, and 17, 18, we can get the following conclusions:

For four types of WNB algorithms, the computation results of both absolute errors of centroid type-reduced FS and defuzzified values converge as Δ = 100 and Δ varies from 1 to 100, respectively. For the first example, the results calculated by the SWNB algorithms obtain the minimum absolute errors; whereas the results obtained by the other three types of WNB algorithms almost get the same. For both example 2 and example 3, the results computed by the proposed three types of WNB algorithms are obviously smaller than the NB algorithms, in which the SWNB algorithms obtain the smallest errors in example 2 and the TWNB algorithms obtain the smallest errors in example 3. For the last example, the results computed by the S3/8WNB algorithms are the maximum, whereas the other three types of WNB algorithms can get the same computation results.

When Δ = 100, the NB algorithms can obtain the largest relative error as 0.028132%, whereas the proposed WNB algorithms obtain the largest relative error as 0.003008%. The total mean relative error calculated by the NB algorithms is 0.009162%, whereas the largest total mean relative error computed by the proposed WNB algorithms is 0.001431%.

As Δ varies from 1 to 100, the NB algorithms can obtain the largest relative error as 0.026998%, whereas the proposed WNB algorithms obtain the largest relative error as 0.003018%. The total mean relative error calculated by the NB algorithms is 0.008817%, whereas the largest total mean relative error computed by the proposed WNB algorithms is 0.001430%.

In a word, the proposed WNB algorithms have better performances than the NB algorithms in terms of the calculation accuracy, in which the SWNB algorithms calculate the optimal results.

Then the computation time of the WNB algorithms is studied for real applications. Generally speaking, the calculation results are unrepeatable as they refer to the specific hardware and software environments. Here we perform the simulations by a dual-core CPU dell desktop with E5300@2.60 GHz and 2.00GB memory, running Windows XP, and programming on Matlab 2013a. The number of Δ is chosen as 200, while the number of primary variable is discretized as N = 50 : 50 : 2000. Then we give the figures of computation time in Figs. 19 –22.

Fig. 19

Computation time for example 1 (as Δ = 200).

Fig. 20

Computation time for example 2 (as Δ = 200).

Fig. 21

Computation time for example 3 (as Δ = 200).

Fig. 22

Computation time for example 4 (as Δ = 200).

Moreover, the computation time of four examples as Δ varies from 1 to 100 is studied. And the comparisons of computation time are shown in Figs. 23 –26.

Fig. 23

Computation time for example 1 (as Δ varies from 1 to 100).

Fig. 24

Computation time for example 2 (as Δ varies from 1 to 100).

Fig. 25

Computation time for example 3 (as Δ varies from 1 to 100).

Fig. 26

Computation time for example 4 (as Δ varies from 1 to 100).

Observing from Figs. 19 –22, we find that the computation time of four types of WNB algorithms emerge linear variation except for small fluctuations. Here we can adopt the least square regression model for four types of WNB algorithms as t = a + bN, in which t denotes the computation time, the unit of t is second, N represents the number of discretized primary variable, and a and b denote the regression coefficients. The computation time different rate can be defined as: $(max_{i = 1, \dots, 4} {t_{i}} - min_{i = 1, \dots, 4} {t_{i}}) / max_{i = 1, \dots, 4} {t_{i}}$ (32) where t_i (i = 1, ⋯ , 4) denotes the computation time of WNB algorithms. Table 6 provides the regression coefficients for the linear model.

Table 6

Regression model coefficients of WNB algorithms (as Δ = 200)

Regression	NB	TWNB	SWNB	S3/8WNB
Coefficient	a/10^-3b/10^-3	a/10^-3b/10^-3	a/10^-3b/10^-3	a/10^-3b/10^-3
Example 1	0.0016 1.9498	0.0019 2.2665	0.0027 2.2240	0.0030 2.1676
Example 2	0.0014 2.0988	0.0019 2.1965	0.0027 2.0697	0.0032 1.9967
Example 3	0.0016 1.9215	0.0021 2.2272	0.0027 2.1382	0.0034 1.9114
Example 4	0.0014 1.9783	0.0020 2.0920	0.0026 2.0611	0.0030 2.0460
Total mean	0.0015 1.9871	0.0020 2.1956	0.0027 2.1233	0.0032 2.0304

As shown in Figs. 19 –22 and Table 6, the convergence speeds of the proposed WNB algorithms are slightly faster than the NB algorithms. When the number of sampling points of primary variable is fixed, the size relation of computation time of four types of WNB algorithms is as: NB < TWNB<SWNB<S3/8WNB. This may due the fact that the weights of WNB algorithms are more complicated than the NB algorithms. Moreover, the calculation time difference rate for them is between 6.59% and 51.39%. Then we observe the Figs. 23 –26, the same conclusions can be obtained that the convergence speeds of the proposed WNB algorithms are faster than the NB algorithms.

According to the above analysis, it can find that the proposed WNB algorithms generated from the numerical integration technique could be adopted for performing the centroid TR of GT2 FLSs. As shown in Tables 4 and 5, the SWNB algorithms are the best choice if only the calculation accuracy is taken into account. If both the calculation accuracy and the computation efficiency were considered, the SWNB algorithms are suggested to perform the centroid TR of GT2 FLSs as in example 1, the TWNB or SWNB algorithms are advised to perform the centroid TR of GT2 FLSs as in examples 2 and 3, and the NB or TWNB algorithms are suggested to perform the centroid TR of GT2 FLSs as in example 4.

Finally it should be indicated that the paper only pays attention to the performances of WNB algorithms in theoretical studies. Then four simulation examples illustrate that the proposed WNB algorithms are superior to the NB algorithms. However, if the calculation precision requirement is not very high, the proposed WNB algorithms can not show their advantages. Then the simple and computational faster NB algorithms will work.

5 Conclusions

In this paper, we apply the WNB algorithms that are derived for an IT2 FS at each alpha-plane to perform the centroid TR of GT2 FLSs. According to the Newton-Cotes quadrature formulas in numerical integration, the NB algorithms are extended to three different forms of WNB algorithms. In view of the centroid type-reduced sets and defuzzified values in four simulation examples, the proposed WNB algorithms can obtain better absolute error and faster convergence speed compared with the NB algorithms.

Many interesting works still lie ahead, including studying the center-of-sets TR of IT2 and GT2 FLSs [6 , 36], and the research on designing and applying intelligent optimization algorithms-based IT2 and GT2 FLSs [9 , 37–40] for control and forecasting problems. Future studies will be focused on uncertain fuzzy rules-based T2 FLSs design and applications.

Footnotes

Acknowledgments

This paper is sponsored by the National Natural Science Foundation of China (No. 61973146, No. 61773188, No. 61803189) and Liaoning Province Natural Science Foundation Guidance Project (No. 20180550056). The author is very thankful to Professor J. M. Mendel, who has provided the author some valuable suggestions generously.

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