A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area

Wenqing WU; Xin MA; Bo ZENG; Yuanyuan ZHANG

doi:10.21078/JSSI-2023-0119

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Journal of Systems Science and Information ›› 2024, Vol. 12 ›› Issue (2) : 245-273. DOI: 10.21078/JSSI-2023-0119

A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area

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Abstract

This study considers a nonlinear grey Bernoulli forecasting model with conformable fractional-order accumulation, abbreviated as CFNGBM $(1, 1, λ)$ , to study the gross regional product in the Cheng-Yu area. The new model contains three nonlinear parameters, the power exponent $γ$ , the conformable fractional-order $α$ and the background value $λ$ , which increase the adjustability and flexibility of the CFNGBM $(1, 1, λ)$ model. Nonlinear parameters are determined by the moth flame optimization algorithm, which minimizes the mean absolute prediction percentage error. The CFNGBM $(1, 1, λ)$ model is applied to the gross regional product of 16 cities in the Cheng-Yu area, which are Chongqing, Chengdu, Mianyang, Leshan, Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Yibin, Ya'an, Dazhou and Ziyang. With data from 2013 to 2021, several grey models are established and results show that the new model has higher accuracy in most cases.

Key words

nonlinear grey Bernoulli model / conformable fractional-order operator / moth flame optimization algorithm / gross regional product / the Cheng-Yu area

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Wenqing WU , Xin MA , Bo ZENG , Yuanyuan ZHANG. A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area. Journal of Systems Science and Information, 2024, 12(2): 245-273 https://doi.org/10.21078/JSSI-2023-0119

1 Introductions

Professor Deng^[1] in his monograph stated that nonlinear models are an important part of grey forecasting models owing to the large amount of nonlinear data in our daily life. According to the structure of the GM

(1, 1)

model, Deng proposed the nonlinear grey Bernoulli model, abbreviated as NGBM

(1, 1)

, which is an essential nonlinear grey forecasting model and widely used in various aspects of science and technology. Chen, et al.^[2] in 2008 further studied a novel NGBM

(1, 1)

model where the power exponent

γ

and the adjustable variable

λ

in the background value are two undetermined nonlinear parameters. A fluctuating raw data sequence and two practical applications are considered to show the accuracy of the new model. Zhou, et al.^[3] presented a parameter optimization scheme of the NGBM

(1, 1)

model by the particle swarm optimization algorithm. Wang^[4] developed an optimized Nash NGBM

(1, 1)

model to study the main economic indices of China's high-technology enterprises. The calculation results illustrate that the new model can significantly improve the accuracy of simulation and fitting. Lu, et al.^[5] investigated an ONGBM

(1, 1)

model where the initial condition is the sum of the

n

th component of the first-order accumulated generating sequence

x^{(1)} (n)

and a correction term

c

. The new model is used to predict the traffic flow. Xiao, et al.^[6] established a new BC-NGBM

(1, 1)

model with Box-Cox transformation and quantum adiabatic evolution to study the annual biomass energy consumption of China, US, Brazil and Germany. Yang and Xie^[7] proposed an integral matching-based NGBM

(1, 1)

model for predicting China's coal consumption. For more analysis of the NGBM

(1, 1)

model, the reader is referred to ^[8–13].

To further improve the accuracy of grey models, Wu, et al.^[14] introduced the definitions of the fractional-order operator. In his work, the accuracy of the FAGM

(1, 1)

model are verified by five numerical examples. Due to this advantage, a variety of fractional-order grey forecasting models have been proposed, and applied in different branches of science and industry, see ^[15–22]. In 2020, Ma, et al.^[23] proposed a conformable fractional-order operator based on the work of Khalil, et al.^[24], and introduced it into the GM

(1, 1)

model to develop the CFGM

(1, 1)

model. The feasibility and applicability of the new model are illustrated with a series of examples. Compared with the fractional-order operator, the conformable fractional-order operator is more simple in calculation. Since 2020, a great number of grey models with conformable fractional-order operator are developed, see ^[25–32]. This two types of fractional-order operators are also introduced to the NGBM

(1, 1)

model. Wu, et al.^[33] predicted short-term China's renewable energy consumption by a fractional nonlinear grey Bernoulli model named FANGBM

(1, 1)

. Şahin used fractional FANGBM

(1, 1)

model to study the cumulative number of confirmed cases of COVID-19 in Italy, UK and USA^[34], Turkey's electricity generation and installed capacity^[35], and renewable energy of consumption in France, Germany, Italy, Spain, Turkey and UK^[36]. Zheng, et al.^[37] studied natural gas production and consumption by a CFNHGBM

(1, 1, k)

model. Wang, et al.^[38] proposed the FBTDGM

(1, 1)

model to forecast energy production and consumption, the ANGBM

(1, 1, T_{n})

model to forecast China's renewable energy production^[39]. Xie and Yu^[40] studied the CFNGBM

(1, 1)

model and applied it to the CO

_{2}

emissions of China and India. Results show that the conformable nonlinear grey Bernoulli model is able to provide very promising and competitive results.

The year after the proposal of the classical GM

(1, 1)

model, were accompanied by the highest attention to such new forecasting theory, which resulted in the proposal of various types of grey forecasting models. Despite the merits of these grey models, there is a fundamental question here as if there is any grey forecasting model for solving all data sequence with satisfactory results. Actually, this is impossible. We should propose new grey forecasting models that are highly suitable for solving new problems. Therefore, this paper introduces the conformable fractional-order operator into the classical nonlinear grey Bernoulli model to establish a new model termed CFNGBM

(1, 1, λ)

, which contains three nonlinear parameters to increase model's adjustability and flexibility. The study mainly includes two parts: The proposed CFNGBM

(1, 1, λ)

model, the application in the gross regional product in the Cheng-Yu area. Details of every part are demonstrated as follows. 1) We transform the whitening differential equation of the CFNGBM

(1, 1, λ)

model to a linear one, and deduce its solutions of the time response function, the restored values, system linear parameters

a

and

b

. We further prove the optimization problem

Q (a, b)

has a local minimum at point

(a, b)

by using the properties of the Hessian matrix. In addition, the moth flame optimization (MFO) algorithm is selected to search optimal values of power exponent

γ

, the conformable fractional-order

α

and the background value

λ

under an optimization problem. 2) We employ the new CFNGBM

(1, 1, λ)

model to the gross regional product of all cities in Cheng-Yu area. The results are compared with the GM

(1, 1)

, CFGM

(1, 1)

, NGBM

(1, 1)

and CFNGBM

(1, 1)

models. These models are selected for comparison is that the GM

(1, 1)

, CFGM

(1, 1)

, NGBM

(1, 1)

, and CFNGBM

(1, 1)

models are special cases of the CFNGBM

(1, 1, λ)

model. From the computational results, we observe that the new model with three nonlinear parameters is suitable for the gross regional product in the Cheng-Yu area. The contents described above two parts are shown in the following Figure 1.

Figure 1 The main contents of the study

Full size|PPT slide

The rest of this paper is organized as follows. Section 2 discusses the conformable fractional-order nonlinear grey Bernoulli model. Its solution, linear parameters and nonlinear parameters are all derived. The new model is applied in the gross regional product of all cities in the Cheng-Yu area in Section 3. Conclusions are drawn in Section 4.

2 The Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation

This section first gives some abbreviations for various grey forecasting models, notations for different variables, definitions of conformable fractional-order accumulation and difference, and the classical nonlinear grey Bernoulli model NGBM

(1, 1)

. And then we construct the new CFNGBM

(1, 1, λ)

model along with some properties.

2.1 Abbreviations, Notations and Definitions

Different abbreviations for various grey forecasting models will be used are listed in the following Table 1.

Table 1 Abbreviations for various grey forecasting models

Abbreviation	Definition
GM $(1, 1)$	Grey model with integer-order accumulation
CFGM $(1, 1)$	Grey model with conformable fractional-order accumulation
NGBM $(1, 1)$	Nonlinear grey Bernoulli model with integer-order accumulation
CFNGBM $(1, 1)$	Nonlinear grey Bernoulli model with conformable fractional-order accumulation
CFNGBM $(1, 1, λ$ )	Nonlinear grey Bernoulli model with conformable fractional-order accumulation with background value $λ$

For ease of reference, some notations used in this paper are given in Table 2.

Table 2 Notations for different variables

Symbol	Meaning
CFA	The conformable fractional-order accumulation
CFD	The conformable fractional-order difference
$b o l d s y m b o l X^{(0)}$	The non-negative original data sequence
$X^{(α)}$	The $α$ th-order accumulated generating sequence
$ν$	The number of data used to build models
$n$	The length of the total sample data
$a, b$	The linear parameters of grey models
$γ$	The exponent power in the nonlinear grey Bernoulli model
$⌈ \cdot ⌉$	The ceil function
$T$	The transpose operator
MFO	The moth flame optimization algorithm

Further, the definitions of the conformable fractional-order accumulation and difference are provided below.

Definition 1 (see Ma, et al. ^[23]). Let series $X^{(0)} = {x^{(0)} (l) | l = 1, 2, \dots, n}$ , the $α^{t h}$ conformable fractional-order accumulation $($ CFA) series of $X^{(0)}$ be $X^{(α)} (α > 0)$ where $x^{(α)} (l), l = 1, 2, \dots, n$ are expressed as

\begin{array}{rcl} x^{(α)} (l) = \sum_{i = 1}^{l} [\begin{array}{c} ⌈ α ⌉ \\ l - i \end{array}] \frac{1}{i^{⌈ α ⌉ - α}} x^{(0)} (i), α > 0, \end{array}

(1)

\begin{array}{rcl} w h e r e [\begin{array}{c} ⌈ α ⌉ \\ l - i \end{array}] = (\binom{⌈ α ⌉ - 1 + l - i}{l - i}) = \frac{(⌈ α ⌉ - 1 + l - i) (⌈ α ⌉ - 2 + l - i) \dots ⌈ α ⌉}{(l - i)!} . \end{array}

Definition 2 (see Ma, et al. ^[23]). Set the $α^{t h}$ conformable fractional-order difference $($ CFD) sequence of $X^{(0)}$ be $X^{(- α)} (α > 0)$ where $x^{(- α)} (l), l = 1, 2, \dots, n$ satisfy

\begin{array}{rcl} x^{(- α)} (l) = l^{⌈ α ⌉ - α} \sum_{i = 1}^{l} [\begin{array}{c} - ⌈ α ⌉ \\ l - i \end{array}] x^{(0)} (i), α > 0, \end{array}

(2)

where $[\begin{matrix} - ⌈ α ⌉ \\ l - i \end{matrix}] = (- 1)^{l - i} [\begin{matrix} ⌈ α ⌉ \\ l - i \end{matrix}]$ .

Further, denote by

A_{α}

the

α^{t h}

CFA matrix, and by

D_{α}

the

α^{t h}

CFD matrix, and Eqs.(1) and (2) are described as

\begin{array}{rcl} X^{(α)} = A_{α} X^{(0)}, \end{array}

(3)

\begin{array}{rcl} X^{(- α)} = D_{α} X^{(0)}, \end{array}

(4)

where

\begin{array}{rcl} A_{α} = {(\begin{array}{ccccc} \frac{[\begin{matrix} ⌈ α ⌉ \\ 0 \end{matrix}]}{1^{⌈ α ⌉ - α}} & 0 & 0 & \dots & 0 \\ \frac{[\begin{matrix} ⌈ α ⌉ \\ 1 \end{matrix}]}{1^{⌈ α ⌉ - α}} & \frac{[\begin{matrix} ⌈ α ⌉ \\ 0 \end{matrix}]}{2^{⌈ α ⌉ - α}} & 0 & \dots & 0 \\ \frac{[\begin{matrix} ⌈ α ⌉ \\ 2 \end{matrix}]}{1^{⌈ α ⌉ - α}} & \frac{[\begin{matrix} ⌈ α ⌉ \\ 1 \end{matrix}]}{2^{⌈ α ⌉ - α}} & \frac{[\begin{matrix} ⌈ α ⌉ \\ 0 \end{matrix}]}{3^{⌈ α ⌉ - α}} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{[_{n - 1}^{⌈ α ⌉}]}{1^{⌈ α ⌉ - α}} & \frac{[_{n - 2}^{⌈ α ⌉}]}{2^{⌈ α ⌉ - α}} & \frac{[_{n - 3}^{⌈ α ⌉}]}{3^{⌈ α ⌉ - α}} & \dots & \frac{[\begin{matrix} ⌈ α ⌉ \\ 0 \end{matrix}]}{n^{⌈ α ⌉ - α}} \end{array})}_{n \times n}, \\ D_{α} = {(\begin{array}{ccccc} [\begin{matrix} ⌈ - α ⌉ \\ 0 \end{matrix}] 1^{⌈ α ⌉ - α} & 0 & 0 & \dots & 0 \\ [\begin{matrix} ⌈ - α ⌉ \\ 1 \end{matrix}] 2^{⌈ α ⌉ - α} & [\begin{matrix} ⌈ - α ⌉ \\ 0 \end{matrix}] 2^{⌈ α ⌉ - α} & 0 & \dots & 0 \\ [\begin{matrix} ⌈ - α ⌉ \\ 2 \end{matrix}] 3^{⌈ α ⌉ - α} & [\begin{matrix} ⌈ - α ⌉ \\ 1 \end{matrix}] 3^{⌈ α ⌉ - α} & [\begin{matrix} ⌈ - α ⌉ \\ 0 \end{matrix}] 3^{⌈ α ⌉ - α} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ [_{n - 1}^{- ⌈ α ⌉}] n^{⌈ α ⌉ - α} & [_{n - 2}^{- ⌈ α ⌉}] n^{⌈ α ⌉ - α} & [_{n - 3}^{- ⌈ α ⌉}] n^{⌈ α ⌉ - α} & \dots & [\begin{matrix} ⌈ - α ⌉ \\ 0 \end{matrix}] n^{⌈ α ⌉ - α} \end{array})}_{n \times n} . \end{array}

Next, we will use these definitions to construct the nonlinear grey Bernoulli model with conformable fractional-order accumulation in what follows.

2.2 The NGBM $(1, 1)$ Model with Integer-Order Accumulation

In this subsection, we will briefly review the definition of the NGBM

(1, 1)

model, along with the time response function and its parameters estimation.

Definition 3 (see Deng^[1], Chen, et al. ^[2]). The whitening differential equation of the NGBM $(1, 1)$ model is outlined as

\begin{array}{rcl} \frac{d x^{(1)} (t)}{d t} + a x^{(1)} (t) = b {(x^{(1)} (t))}^{γ}, \end{array}

(5)

where $a$ and $b$ are linear parameters, $γ$ is nonlinear parameter. Further, the grey basic form of the NGBM $(1, 1)$ model is

\begin{array}{rcl} x^{(1)} (k) - x^{(1)} (k - 1) + a z^{(1)} (k) = b {(z^{(1)} (k))}^{γ}, \end{array}

(6)

where $z^{(1)} (k) = 0.5 x^{(1)} (k - 1) + 0.5 x^{(1)} (k), k = 2, 3, \dots, n .$

It follows from Eq.(5) and initial condition

x^{(1)} (1) = x^{(0)} (1)

that the time response function of the NGBM

(1, 1)

model is

\begin{array}{rcl} {\hat{x}}^{(1)} (k) = {[{(x^{(0)} (1))}^{1 - γ} - \frac{b}{a}] e^{- a (1 - γ) (k - 1)} + \frac{b}{a}}^{\frac{1}{1 - γ}}, k = 1, 2, \dots, \end{array}

(7)

and the restored values

{\hat{x}}^{(0)} (k) = {\hat{x}}^{(1)} (k) - {\hat{x}}^{(1)} (k - 1), k = 1, 2, \dots

It follows from Eq.(6) and the least squares estimation method that linear parameters

a

and

b

are calculated as

\begin{array}{rcl} (a, b)^{T} = {(Λ^{T} Λ)}^{- 1} Λ^{T} η, \end{array}

(8)

where the matrixes are

\begin{array}{rcl} Λ = (\begin{array}{c} - z^{(1)} (2) & {(z^{(1)} (2))}^{γ} \\ - z^{(1)} (3) & {(z^{(1)} (3))}^{γ} \\ ⋮ & ⋮ \\ - z^{(1)} (ν) & {(z^{(1)} (ν))}^{γ} \end{array}), η = (\begin{array}{c} x^{(1)} (2) - x^{(1)} (1) \\ x^{(1)} (3) - x^{(1)} (2) \\ ⋮ \\ x^{(1)} (ν) - x^{(1)} (ν - 1) \end{array}), \end{array}

with

ν

is the number of data for modelling.

2.3 The CFNGBM $(1, 1, λ)$ Model

This subsection studies the conformable fractional nonlinear grey Bernoulli forecasting model with changeable background value, along with parameters estimation and time response function.

Definition 4 From the $α^{t h}$ -CFA sequence $X^{(α)}$ and the NGBM $(1, 1)$ model, the whitening equation of the CFNGBM $(1, 1, λ)$ model is defined as

\begin{array}{rcl} \frac{d x^{(α)} (t)}{d t} + a x^{(α)} (t) = b {(x^{(α)} (t))}^{γ}, α > 0, \end{array}

(9)

where $a$ and $b$ are linear system parameters, $γ$ and $α$ are nonlinear system parameters.

We next derive the expressions of the time response function and system parameters of the CFNGBM

(1, 1, λ)

model. To achieve this, we first translate the nonlinear CFNGBM

(1, 1, λ)

model to a linear form by variable substitution.

Multiplying both side of the whitening differential equation Eq.(9) with factor

{(x^{(α)} (t))}^{- γ}

, it obtains that

\begin{array}{rcl} \frac{d x^{(α)} (t)}{d t} \frac{1}{{(x^{(α)} (t))}^{γ}} + a \frac{1}{{(x^{(α)} (t))}^{γ - 1}} = b . \end{array}

(10)

Let

y^{(α)} (t) = \frac{1}{{(x^{(α)} (t))}^{γ - 1}}

, then we have

\begin{array}{rcl} \frac{d y^{(α)} (t)}{d t} = (1 - γ) \frac{1}{{(x^{(α)} (t))}^{γ}} \frac{d x^{(α)} (t)}{d t}, \end{array}

(11)

which yields

\frac{d x^{(α)} (t)}{d t} = \frac{1}{1 - γ} {(x^{(α)} (t))}^{γ} \frac{d y^{(α)} (t)}{d t}

. And substituting it into Eq.(10) which arrives at

\begin{array}{rcl} \frac{d y^{(α)} (t)}{d t} + a (1 - γ) y^{(α)} (t) = b (1 - γ), \end{array}

(12)

which is a linear equaiton, and is called the linearized form of the CFNGBM

(1, 1, λ)

model.

Theorem 1 The expression of the time response function of CFNGBM $(1, 1, λ)$ model is

\begin{array}{rcl} {\hat{x}}^{(α)} (k) = {[{(x^{(0)} (1))}^{1 - γ} - \frac{b}{a}] e^{- a (1 - γ) (k - 1)} + \frac{b}{a}}^{\frac{1}{1 - γ}}, k = 1, 2, \dots, \end{array}

(13)

and the restored values ${\hat{x}}^{(0)} (k), k = 2, 3, \dots$ are calculated by

\begin{array}{rcl} {\hat{x}}^{(0)} (k) = k^{⌈ α ⌉ - α} \sum_{i = 1}^{k} [\begin{array}{c} - ⌈ α ⌉ \\ k - i \end{array}] x^{(α)} (i), α > 0. \end{array}

(14)

Proof Multiplying both side of the linearized form of the CFNGBM

(1, 1, λ)

model in Eq.(12) with

e^{a (1 - γ) t}

, we have that

\begin{array}{rcl} e^{a (1 - γ) t} \frac{d y^{(α)} (t)}{d t} + a (1 - γ) e^{a (1 - γ) t} y^{(α)} (t) = b (1 - γ) e^{a (1 - γ) t}, \end{array}

(15)

which is also

\begin{array}{rcl} \frac{d [y^{(α)} (t) e^{a (1 - γ) t}]}{d t} = b (1 - γ) e^{a (1 - γ) t} . \end{array}

(16)

Taking integration on Eq.(16) within

[1, t]

, we obtain that

\begin{array}{rcl} \int_{1}^{t} \frac{d [y^{(α)} (s) e^{a (1 - γ) s}]}{d s} d s = \int_{1}^{t} b (1 - γ) e^{a (1 - γ) s} d s . \end{array}

(17)

It follows from Eq.(17) that

\begin{array}{rcl} y^{(α)} (t) e^{a (1 - γ) t} - y^{(α)} (1) e^{a (1 - γ)} = \frac{b}{a} [e^{a (1 - γ) t} - e^{a (1 - γ)}] . \end{array}

(18)

Noting that

y^{(α)} (t) = {(x^{(α)} (t))}^{1 - γ}

and

y^{(α)} (1) = {(x^{(α)} (1))}^{1 - γ} = {(x^{(0)} (1))}^{1 - γ}

, we get that

\begin{array}{rcl} x^{(α)} (t) = {[{(x^{(0)} (1))}^{1 - γ} - \frac{b}{a}] e^{- a (1 - γ) (t - 1)} + \frac{b}{a}}^{\frac{1}{1 - γ}} . \end{array}

(19)

Set

t = k

in Eq.(19), we obtain the expression of the time response function of the CFNGBM

(1, 1, λ)

model, and the expression of the restored values by the CFD. This completes the proof.

Theorem 2 Once original sequence $X^{(0)}$ , the values of $γ$ , $α$ and $γ$ are given, the linear system parameters $a$ and $b$ of the CFNGBM $(1, 1, λ)$ model are

\begin{array}{rcl} (a, b)^{T} = {(B^{T} B)}^{- 1} B^{T} Y, \end{array}

(20)

where matrixes $B$ and $Y$ are

\begin{array}{rcl} B = (1 - γ) (\begin{array}{c} λ y^{(α)} (1) + (1 - λ) y^{(α)} (2) & - 1 \\ λ y^{(α)} (2) + (1 - λ) y^{(α)} (3) & - 1 \\ ⋮ & ⋮ \\ λ y^{(α)} (ν - 1) + (1 - λ) y^{(α)} (ν) & - 1 \end{array}), Y = (\begin{array}{c} y^{(α)} (1) - y^{(α)} (2)) \\ y^{(α)} (2) - y^{(α)} (3)) \\ ⋮ \\ y^{(α)} (ν - 1) - y^{(α)} (ν)) \end{array}) . \end{array}

Further, the sum of squares of all the deviations from the given modelling data has a strict local minimum at point $(a, b)$ .

Proof Considering integration on Eq.(12) in

[k - 1, k]

, we obtain that

\begin{array}{rcl} \int_{k - 1}^{k} \frac{d y^{(α)} (t)}{d t} d t + \int_{k - 1}^{k} a (1 - γ) y^{(α)} (t) d t = \int_{k - 1}^{k} b (1 - γ) d t . \end{array}

(21)

Applying the trapezoid formula to the term

\int_{k - 1}^{k} y^{(α)} (t) d t = λ y^{(α)} (k) + (1 - λ) y^{(α)} (k - 1) \overset{Δ}{=} z_{y}^{(α)} (k)

, and

y^{(α - 1)} (k) = y^{(α)} (k) - y^{(α)} (k - 1)

, Eq.(21) is

\begin{array}{rcl} y^{(α)} (k) - y^{(α)} (k - 1) + a (1 - γ) z_{y}^{(α)} (k) = b (1 - γ), \end{array}

(22)

which is also

\begin{array}{rcl} y^{(α - 1)} (k) = - a (1 - γ) z_{y}^{(α)} (k) + b (1 - γ) . \end{array}

(23)

According to the least squares method, the sum of squares of all the deviations

Q (a, b)

from the given modelling data is constructed with the following formula.

\begin{array}{rcl} min_{a, b} Q (a, b) = \frac{1}{2} \sum_{k = 2}^{ν} {(y^{(α - 1)} (k) + a (1 - γ) z_{y}^{(α)} (k) - b (1 - γ))}^{2} . \end{array}

(24)

It follows from Eq.(24) that

\begin{array}{rcl} {\begin{cases} \frac{\partial Q (a, b)}{\partial a} = \sum_{k = 2}^{ν} (y^{(α - 1)} (k) + a (1 - γ) z_{y}^{(α)} (k) - b (1 - γ)) (1 - γ) z_{y}^{(α)} (k), \\ \frac{\partial Q (a, b)}{\partial b} = \sum_{k = 2}^{ν} (y^{(α - 1)} (k) + a (1 - γ) z_{y}^{(α)} (k) - b (1 - γ)) [- (1 - γ)] . \end{cases} \end{array}

(25)

We write the above equations in matrix form with the following expression

\begin{array}{rcl} (\begin{array}{c} \frac{\partial Q (a, b)}{\partial a} \\ \frac{\partial Q (a, b)}{\partial b} \end{array}) \\ = {(\begin{array}{c} (1 - γ) z_{y}^{(α)} (2) & - (1 - γ) \\ (1 - γ) z_{y}^{(α)} (3) & - (1 - γ) \\ ⋮ & ⋮ \\ (1 - γ) z_{y}^{(α)} (ν) & - (1 - γ) \end{array})}^{T} (\begin{array}{c} y^{(α - 1)} (2) + a (1 - γ) z_{y}^{(α)} (2) - b (1 - γ) \\ y^{(α - 1)} (3) + a (1 - γ) z_{y}^{(α)} (3) - b (1 - γ) \\ ⋮ \\ y^{(α - 1)} (ν) + a (1 - γ) z_{y}^{(α)} (ν) - b (1 - γ) \end{array}) \\ = (1 - γ) {(\begin{array}{c} z_{y}^{(α)} (2) & - 1 \\ z_{y}^{(α)} (3) & - 1 \\ ⋮ & ⋮ \\ z_{y}^{(α)} (ν) & - 1 \end{array})}^{T} [(\begin{array}{c} y^{(α - 1)} (2) \\ y^{(α - 1)} (3) \\ ⋮ \\ y^{(α - 1)} (ν) \end{array}) + (1 - γ) (\begin{array}{c} z_{y}^{(α)} (2) & - 1 \\ z_{y}^{(α)} (3) & - 1 \\ ⋮ & ⋮ \\ z_{y}^{(α)} (ν) & - 1 \end{array}) (\begin{array}{c} a \\ b \end{array})] \\ = B^{T} (- Y + B θ), \end{array}

(26)

where the matrixes

B

Y

and

θ

are

\begin{array}{rcl} B = (1 - γ) (\begin{array}{c} z_{y}^{(α)} (2) & - 1 \\ z_{y}^{(α)} (3) & - 1 \\ ⋮ & ⋮ \\ z_{y}^{(α)} (ν) & - 1 \end{array}), Y = - (\begin{array}{c} y^{(α - 1)} (2) \\ y^{(α - 1)} (3) \\ ⋮ \\ y^{(α - 1)} (ν) \end{array}), θ = (\begin{array}{c} a \\ b \end{array}) . \end{array}

The system parameters

θ = (a, b)^{T}

making

Q (a, b)

minimum should satisfy

\begin{array}{rcl} B^{T} (- Y + B θ) = 0, \end{array}

(27)

which yields

\begin{array}{rcl} θ = (a, b)^{T} = {(B^{T} B)}^{- 1} B^{T} Y . \end{array}

(28)

Moreover, From Eq.(25), we have

\begin{array}{rcl} {\begin{cases} \frac{\partial^{2} Q (a, b)}{\partial a \partial a} = \sum_{k = 2}^{ν} (1 - γ) z_{y}^{(α)} (k) (1 - γ) z_{y}^{(α)} (k), \\ \frac{\partial^{2} Q (a, b)}{\partial a \partial b} = \frac{\partial^{2} Q (a, b)}{\partial b \partial a} = \sum_{k = 2}^{ν} - (1 - γ) (1 - γ) z_{y}^{(α)} (k), \\ \frac{\partial^{2} Q (a, b)}{\partial b \partial b} = \sum_{k = 2}^{ν} [- (1 - γ)] [- (1 - γ)], \end{cases} \end{array}

(29)

which means that the Hessian matrix of

Q (a, b)

\begin{array}{rcl} H = (\begin{array}{c} \frac{\partial^{2} Q (a, b)}{\partial a \partial a} & \frac{\partial^{2} Q (a, b)}{\partial a \partial b} \\ \frac{\partial^{2} Q (a, b)}{\partial b \partial a} & \frac{\partial^{2} Q (a, b)}{\partial b \partial b} \end{array}) = (\begin{array}{c} \sum_{k = 2}^{ν} {((1 - γ) z_{y}^{(α)} (k))}^{2} & \sum_{k = 2}^{ν} - {(1 - γ)}^{2} z_{y}^{(α)} (k) \\ \sum_{k = 2}^{ν} - {(1 - γ)}^{2} z_{y}^{(α)} (k) & \sum_{k = 2}^{ν} {(1 - γ)}^{2} \end{array}) . \end{array}

(30)

From the expressions of matrix

B

and matrix

H

, it is easily found that

\begin{array}{rcl} H = B^{T} B . \end{array}

(31)

Once

H

is a positive-definite matrix, the

Q (a, b)

has a strict local minimum at point

(a, b)

, thus the remaining task is to determine whether the Hessian matrix is positive-definite. For any vector

u_{2 \times 1}

, we have

u^{T} H u = u^{T} B^{T} B u = (B u)^{T} (B u)

v - 1 < 2

, there exists nonzero vector

u \neq 0

making

B u = 0

, which results

u^{T} H u ⩾ 0

, then

H

is not a positive-definite matrix, and the

Q (a, b)

does not have a local minimum or a local maximum at point

(a, b)

v - 1 ⩾ 2

, there is only zero vector

u = 0

making

B u = 0

. For any nonzero vector

u \neq 0

B u \neq 0

. This implies that

u^{T} H u = (B u)^{T} (B u) = {∥ B u ∥}_{2}^{2} > 0

, which shows

H

is a positive-definite matrix, and the

Q (a, b)

has a local minimum at point

(a, b)

. Thus the proof is completed.

It follows from Definition 4 that the CFNGBM

(1, 1, λ)

model is a general nonlinear grey model, and some existing grey forecasting models are special cases of it, see Figure 2.

Figure 2 The relationship among the CFNGBM $(1, 1, λ)$ , CFNGBM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , GVM(1, 1) and GM $(1, 1)$ models

Full size|PPT slide

∙

When

λ = 0.5

in Eq.(9), the CFNGBM

(1, 1, λ)

model degenerates to the CFNGBM

(1, 1)

model^[40] with the following expression

\begin{array}{rcl} \frac{d x^{(α)} (t)}{d t} + a x^{(α)} (t) = b {(x^{(1)} (t))}^{γ}, α > 0. \end{array}

(32)

∙

When

γ = 0

λ = 0.5

in Eq.(9), the CFNGBM

(1, 1, λ)

model degenerates to the CFGM

(1, 1)

model^[23] with the following expression

\begin{array}{rcl} \frac{d x^{(α)} (t)}{d t} + a x^{(α)} (t) = b, α > 0. \end{array}

(33)

∙

When

α = 1

λ = 0.5

in Eq.(9), the CFNGBM

(1, 1, λ)

model reduces to the nonlinear grey Bernoulli model NGBM

(1, 1)

^[1] with the following formula

\begin{array}{rcl} \frac{d x^{(1)} (t)}{d t} + a x^{(1)} (t) = b {(x^{(1)} (t))}^{γ} . \end{array}

(34)

∙

When

γ = 2, α = 1

λ = 0.5

in Eq.(9), the CFNGBM

(1, 1, λ)

model reduces to the grey Verhulst model GVM

(1, 1)

^{[41, 42]} with the following formula

\begin{array}{rcl} \frac{d x^{(1)} (t)}{d t} + a x^{(1)} (t) = b {(x^{(1)} (t))}^{2} . \end{array}

(35)

∙

When

γ = 0, α = 1

λ = 0.5

in Eq.(9), the CFNGBM

(1, 1, λ)

model becomes the GM

(1, 1)

model^[43] with the following expression

\begin{array}{rcl} \frac{d x^{(1)} (t)}{d t} + a x^{(1)} (t) = b . \end{array}

(36)

2.4 An Optimization Problem for System Nonlinear Parameters by the Moth Flame Optimization Algorithm

It can be seen from Theorem 2 that parameters

a

and

b

can be straightforwardly computed by Eq.(20) when parameters

γ

α

and

λ

are given. Therefore an optimization problem is established where the mean absolute prediction percentage error (MAPE

_{p r e d}

) is the objective function, and

γ

α

and

λ

are decision variables.

\begin{array}{rcl} min_{γ, α, λ} {M A P E}_{p r e d} (γ, α, λ) = \frac{1}{n - ν} \sum_{k = ν + 1}^{n} | \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} | \times 100 %, \end{array}

(37)

where MAPE is defined as

\begin{array}{rcl} M A P E = \frac{1}{m - ℓ + 1} \sum_{k = ℓ}^{m} | \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} | \times 100 %, ℓ ⩽ m ⩽ n . \end{array}

When

ℓ = 2, m = ν

, the MAPE is the mean absolute simulation percentage error MAPE

_{s i m u}

; when

ℓ = 2, m = n

, the MAPE is the total mean absolute percentage error MAPE

_{t o t a l}

Our purpose is to find optimal values of

γ

α

and

λ

which minimize the performance MAPE

_{p r e d}

. However it is difficult for us to derive the closed-form expressions of

γ

α

λ

in such an optimization problem. In the present study, a novel nature-inspired algorithm moth flame optimization algorithm is chose to determine

γ

α

and

λ

The moth flame optimization algorithm is a novel nature-inspired population-based algorithm, which models the behavior of moths' transverse orientation for travelling^[44]. The MFO algorithm has high ability to avoid local optimal since a set of solutions are involved during optimization. Information can be exchanged between the candidate solutions, which assist them to overcome different difficulties of search spaces. The main idea of the MFO algorithm is presented below.

ⅰ) Initially, the position of each moth is generated by

\begin{array}{rcl} X (:, i) = r a n d (n u m, 1) \times (u b (i) - l b (i)) + l b (i), \end{array}

where num is the number of moths,

i

is the

i

th variable/dimension,

u b (i)

is the upper bound of the

i

th variable, and

l b (i)

is the lower bound of the

i

th variable. We then calculate the fitness values of all moths and store them in an array.

ⅱ) Using the following formula to generate the number of flames

\begin{array}{rcl} f l a m e n o = r o u n d (n u m - l \times \frac{n u m - 1}{m a x_{i t e r}}), \end{array}

where

l

is the current number of iteration. According to the value of the fitness function for each moth, the flames are assigned. The first flame has the best fitness value and the last flame has the worst fitness value.

ⅲ) A logarithmic spiral is chosen as the main update mechanism of moths in this algorithm as follows.

\begin{array}{rcl} M_{i + 1} = S (M_{i}, F_{j}) = | F_{j} - M_{i} | e^{θ} \cos (2 π θ) + F_{j}, \end{array}

where

M_{i}

indicates the position of the

i

th moth,

F_{j}

indicates the position of the

j

th flame,

θ

is a random number in

[r, 1]

where

r = - 1 - \frac{l}{{m a x}_{i t e r}}

is a linearly decreased from

- 1

- 2

over the course of iteration.

ⅳ) Using the new positions of moths, we compute the fitness values of all moths, and update the number and positions of flames. This pattern continuous until the termination conditions are met.

In this study, it is assumed that the number of moths is 90, and the number of variables is 3. In MFO algorithm, the moths are search agents, and the flames are flags. Every moth searches around a flame to update its position. The first moth updates its position with respect to the first flame, the second moth updates its position with respect to the second flame, and so on. The general framework of MFO is provided in Figure 3.

Figure 3 The general framework of the MFO algorithm

Full size|PPT slide

3 Application in the Gross Regional Product in the Cheng-Yu Area

This section applies grey forecasting models including the GM

(1, 1)

model, the CFGM

(1, 1)

model, the NGBM

(1, 1)

model, the CFNGBM

(1, 1)

model and the CFNGBM

(1, 1, λ)

model to study the gross regional product of 16 cities in the Cheng-Yu area.

3.1 Background

On January 3, 2020, General Secretary Xi Jinping chaired the sixth meeting of the Central Finance and Economic Commission and proposed that promoting the construction of Chengdu Chongqing double city economic circle is conducive to forming an important growth pole of high-quality development in the West and building a strategic highland of inland opening up. The construction of the Chengdu Chongqing economic circle is a systematic project. It is necessary to strengthen top-level design and overall coordination, highlight the driving role of central cities, firmly establish the concept of integrated development, achieve unified planning, integrated deployment, mutual cooperation, and joint implementation, accelerate the construction of the modern industrial system, and enhance the capacity for collaborative innovation and development. We will optimize the spatial distribution of land, strengthen the protection of the ecological environment, promote institutional innovation, and strengthen the joint construction and sharing of public services, so as to make the Cheng-Yu area an important economic center with national influence, a scientific and technological innovation center, a new highland of reform and opening up, and a livable place with high quality of life, and boost high-quality development. Located in the upper reaches of the Yangtze River and in the Sichuan Basin, Cheng-Yu area is adjacent to Hunan and Hubei in the east, Qinghai Xizang in the west, Yunnan and Guizhou in the south, and Shaanxi and Gansu in the north. It is the highest level of development and a large potential urbanization area in western China, and an important part of the implementation of the Yangtze River Economic Belt, and the Belt and Road Strategy. Among the four major regions in China, namely the northwest region, north region, south region, Qinghai Xizang region, the Cheng-Yu area is the only one located in the western inland region. Some recent research work on the Cheng-Yu area can be found in ^[45–48].

The Cheng-Yu Economic Circle covers an area of 185000 square kilometers, involving Chongqing, Chengdu, Mianyang, Leshan, Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Yibin, Ya'an, Dazhou and Ziyang. In this area, the periodicity of economic development is highly complex and uncertain. This paper is based on the economic status quo in the Cheng-Yu area and uses the grey forecasting theory to study the stage characteristics and the latest trend of the economy. The calculation results can provide scientific and reasonable basic data and empirical conclusions for the development and improvement of the economic sustainable development in the construction of the Chengdu Chongqing double city economic circle. The raw data of gross regional product of the 16 cities in the Cheng-Yu area are listed in Table 3, while the description of the numerical design is shown in Figure 4.

Table 3 Raw data of gross regional product in the Cheng-Yu area (million yuan)

City	Chongqing	Chengdu	Mianyang	Leshan	Zigong	Deyang	Meishan	Luzhou
2013	13027.60	9450.66	1470.41	1096.93	913.37	1403.84	821.09	1156.59
2014	14623.78	10368.43	1612.08	1195.81	942.09	1465.24	900.39	1279.12
2015	16040.54	10662.31	1743.00	1280.63	970.27	1525.77	958.67	1369.31
2016	18023.04	11874.07	1957.91	1335.40	1020.84	1692.82	1011.94	1500.64
2017	20066.29	13931.39	2313.57	1481.61	1166.17	1907.43	1149.22	1698.91
2018	21588.80	15698.94	2613.3	1709.81	1314.74	2148.39	1269.90	1895.55
2019	23605.77	17012.65	2856.2	1863.31	1428.49	2335.91	1380.20	2081.26
2020	25002.79	17716.70	3010.08	2003.42	1458.44	2404.10	1423.74	2157.20
2021	27894.02	19916.98	3350.3	2205.15	1601.31	2656.56	1547.87	2406.10

City	Suining	Neijiang	Nanchong	Guang'an	Yibin	Ya'an	Dazhou	Ziyang
2013	706.69	915.11	1283.71	790.91	1293.47	428.95	1159.16	507.51
2014	809.00	973.86	1391.70	838.46	1411.37	475.79	1272.12	537.09
2015	871.36	1018.00	1463.40	874.39	1470.10	519.02	1366.56	572.76
2016	926.84	1086.43	1592.97	928.03	1609.56	561.65	1495.49	643.84
2017	1046.43	1182.11	1838.25	1047.67	1862.19	608.54	1697.58	688.54
2018	1230.85	1318.83	2115.73	1157.00	2349.31	653.34	1879.53	728.63
2019	1345.73	1433.30	2322.22	1250.44	2601.89	723.79	2041.49	777.80
2020	1403.18	1465.88	2401.10	1301.60	2802.10	754.59	2117.80	807.50
2021	1519.87	1605.30	2601.98	1417.80	3148.08	840.56	2351.70	890.50

Figure 4 The numerical design of the application in the gross regional product

Full size|PPT slide

3.2 The Gross Regional Product in the Cheng-Yu Area

∙

Chongqing

We use the MFO algorithm to seek the optimal nonlinear parameters

γ

α

, or

λ

of grey models based on the historical data from 2013 to 2018 in Chongqing. The minimum MAPE

_{p r e d}

and the corresponding nonlinear parameters are displayed in Table 4 and Figures 5

\sim

Table 4 The minimum MAPE $_{p r e d}$ and corresponding nonlinear parameters of grey models

Model	MAPE $_{p r e d}$	$α$	$γ$	$λ$
CFGM $(1, 1)$	0.9687	0.8709	–	0.5000
NGBM $(1, 1)$	0.9275	–	0.0944	0.5000
CFNGBM $(1, 1)$	0.9454	0.9972	0.0921	0.5000
CFNGBM $(1, 1, λ$ )	0.9104	0.8298	$- 0.0455$	0.4715

Figure 5 The convergence curve of MAPE $_{p r e d}$ and $α$ of the CFGM $(1, 1)$ model in Chongqing case

Full size|PPT slide

Figure 6 The convergence curve of MAPE $_{p r e d}$ and $γ$ of the NGBM $(1, 1)$ model in Chongqing case

Full size|PPT slide

Figure 7 The convergence curve of MAPE $_{p r e d}$ , $α$ and $γ$ of the CFNGBM $(1, 1)$ model in Chongqing case

Full size|PPT slide

Figure 8 The convergence curve of MAPE $_{p r e d}$ , $α$ , $γ$ and $λ$ of the CFNGBM $(1, 1, λ)$ model in Chongqing case

Full size|PPT slide

Further, the computational values of the gross regional product of Chongqing are obtained which are tabulated in Table 5, which show the new CFNGBM

(1, 1, λ)

model obtains more competitive results in this application.

Table 5 The results of the gross regional product of Chongqing by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	13027.60	13027.6000	13027.6000	13027.6000	13027.6000	13027.6000
2014	14623.78	14660.3057	14474.6793	14419.7501	14460.8603	14473.9196
2015	16040.54	16188.5170	16284.5387	16257.7120	16294.1753	16230.6969
2016	18023.04	17876.0313	18044.2505	18019.4259	18049.5199	17979.0338
2017	20066.29	19739.4544	19828.2366	19804.8763	19826.8504	19771.5042
2018	21588.80	21797.1235	21674.0857	21658.1397	21670.1344	21639.4848
2019	23605.77	24069.2869	23605.7700	23605.7697	23605.7700	23604.7394
2020	25002.79	26578.3039	25641.2743	25667.3082	25653.1080	25684.5408
2021	27894.02	29348.8645	27795.7346	27859.2019	27828.3720	27894.0199
MAPE $_{s i m u}$ (%)		0.9163	0.8480	0.8786	0.8824	0.8316
MAPE $_{p r e d}$ (%)		4.4935	0.9687	0.9275	0.9454	0.9104
MAPE $_{t o t a l}$ (%)		2.2578	0.8932	0.8970	0.9061	0.8612

∙

Chengdu, Mianyang, Leshan, Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Yibin, Ya'an, Dazhou and Ziyang

By a similar argument used in Chongqing, we obtain values of system parameters of these grey forecasting models for the gross regional product of Chengdu, Mianyang, Leshan, Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Yibin, Ya'an, Dazhou and Ziyang. And then the values of the gross regional product of these cities are computed which tabulated in the following Tables 6

\sim

20.

Table 6 The results of the gross regional product of Chengdu by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	9450.66	9450.6600	9450.6600	9450.6600	9450.6600	9450.6600
2014	10368.43	9802.0478	9697.7092	9451.2892	9988.7884	9872.8210
2015	10662.31	10988.5260	11136.1566	11011.4264	11035.3619	11440.4793
2016	11874.07	12318.6202	12510.5143	12459.4319	12416.0122	12869.0126
2017	13931.39	13809.7142	13886.5734	13881.3030	13862.1596	14248.9026
2018	15698.94	15481.2961	15297.2487	15317.4627	15336.5046	15622.0957
2019	17012.65	17355.2127	16763.0519	16791.7802	16834.8710	17012.6500
2020	17716.70	19455.9556	18298.9072	18320.7100	18360.1984	18436.4628
2021	19916.98	21810.9806	19916.9800	19916.9799	19916.9801	19905.2171
MAPE $_{s i m u}$ (%)		2.9051	3.8307	3.9678	2.9060	4.6452
MAPE $_{p r e d}$ (%)		7.1134	1.5844	1.5692	1.5590	1.3739
MAPE $_{t o t a l}$ (%)		4.4832	2.9883	3.0683	2.4009	3.4185

Table 7 The results of the gross regional product of Mianyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1470.41	1470.4100	1470.4100	1470.4100	1470.4100	1470.4100
2014	1612.08	1556.5752	1527.5220	1478.8851	1561.8342	1538.7390
2015	1743.00	1768.8603	1800.4645	1772.6133	1787.6091	1819.5812
2016	1957.91	2010.0967	2054.8157	2041.7779	2042.5873	2077.6356
2017	2313.57	2284.2329	2304.0142	2301.1721	2300.8738	2327.4276
2018	2613.30	2595.7557	2554.7134	2558.0619	2559.9964	2576.0353
2019	2856.20	2949.7638	2810.9210	2816.7482	2820.5628	2827.6328
2020	3010.08	3352.0514	3075.4077	3080.1174	3083.6476	3085.0265
2021	3350.30	3809.2028	3350.3000	3350.2997	3350.3000	3350.3000
MAPE $_{s i m u}$ (%)		1.9063	3.2293	3.3789	2.5179	3.4166
MAPE $_{p r e d}$ (%)		9.4447	1.2519	1.2360	1.2306	1.1633
MAPE $_{t o t a l}$ (%)		4.7332	2.4878	2.5753	2.0352	2.5716

Table 8 The results of the gross regional product of Leshan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ and CFNGBM $(1, 1)$ models

Year	Data	GM (1, 1)	CFGM (1, 1)	NGBM (1, 1)	CFNGBM (1, 1)	CFNGBM (1, 1, λ)
2013	1096.93	1096.9300	1096.9300	1096.9300	1096.9300	1096.9300
2014	1195.81	1159.1940	1159.1940	1164.5985	1198.1427	1208.9946
2015	1280.63	1268.2728	1268.2729	1266.9683	1244.4876	1292.7850
2016	1335.40	1387.6159	1387.6159	1384.8165	1370.1718	1418.1893
2017	1481.61	1518.1889	1518.1889	1516.9931	1516.7485	1557.4231
2018	1709.81	1661.0488	1661.0488	1663.9501	1674.9166	1706.0332
2019	1863.31	1817.3515	1817.3515	1826.7023	1842.5091	1863.3100
2020	2003.42	1988.3622	1988.3622	2006.5796	2019.1730	2029.4984
2021	2205.15	2175.4648	2175.4648	2205.1499	2205.1500	2205.1500
MAPE _simu (%)		2.6516	2.6516	2.4895	2.0067	2.7178
MAPE _pred (%)		1.5214	1.5214	0.7075	0.6342	0.4339
MAPE_total (%)		2.2278	2.2278	1.8213	1.4920	1.8614

Table 9 The results of the gross regional product of Zigong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	913.37	913.3700	913.3700	913.3700	913.3700	913.3700
2014	942.09	897.1021	892.5980	882.2174	920.5321	910.2606
2015	970.27	981.0165	987.7061	983.2918	976.4196	1018.0488
2016	1020.84	1072.7802	1081.0203	1079.6067	1071.6232	1116.0641
2017	1166.17	1173.1275	1176.0576	1176.2722	1174.1035	1210.9265
2018	1314.74	1282.8612	1274.5933	1275.5903	1278.9027	1305.5336
2019	1428.49	1402.8593	1377.7640	1378.9231	1385.0334	1401.4902
2020	1458.44	1534.0820	1486.4275	1487.2367	1492.4379	1499.8324
2021	1601.31	1677.5792	1601.3100	1601.3101	1601.3100	1601.3100
MAPE $_{s i m u}$ (%)		2.7984	3.3694	3.4596	2.2606	4.4338
MAPE $_{p r e d}$ (%)		3.9146	1.8233	1.8148	1.7911	1.5761
MAPE $_{t o t a l}$ (%)		3.2170	2.7896	2.8428	2.0845	3.3622

Table 10 The results of the gross regional product of Deyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1403.84	1403.8400	1403.8400	1403.8400	1403.8400	1403.8400
2014	1465.24	1408.5109	1393.3698	1368.5588	1428.9336	1415.3150
2015	1525.77	1559.7469	1576.9797	1565.1279	1561.2600	1617.1574
2016	1692.82	1727.2217	1750.2702	1745.7042	1737.2249	1797.7584
2017	1907.43	1912.6786	1921.9280	1921.6597	1919.4231	1969.8143
2018	2148.39	2118.0487	2096.1861	2098.1887	2102.6458	2139.0512
2019	2335.91	2345.4699	2275.5909	2278.2705	2286.2953	2308.6602
2020	2404.10	2597.3101	2461.9196	2463.8964	2470.7439	2480.6818
2021	2656.56	2876.1911	2656.5600	2656.5579	2656.5600	2656.5600
MAPE $_{s i m u}$ (%)		1.9636	2.9690	3.0769	2.0370	3.8602
MAPE $_{p r e d}$ (%)		5.5712	1.6624	1.6516	1.6320	1.4507
MAPE $_{t o t a l}$ (%)		3.3165	2.4790	2.5424	1.8851	2.9567

Table 11 The results of the gross regional product of Meishan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM (1, 1)	CFGM (1, 1)	NGBM (1, 1)	CFNGBM (1, 1)	CFNGBM (1, 1, λ)
2013	821.09	821.0900	821.0900	821.0900	821.0900	821.0900
2014	900.39	876.8595	869.5616	859.8532	891.8928	878.8334
2015	958.67	958.8279	966.1924	962.2983	953.0116	983.8864
2016	1011.94	1048.4587	1058.4132	1057.0600	1049.0351	1079.0862
2017	1149.22	1146.4682	1150.4431	1150.3403	1149.5357	1171.0158
2018	1269.90	1253.6396	1244.3314	1244.7960	1250.0800	1262.5357
2019	1380.20	1370.8292	1341.3153	1341.9144	1349.9120	1355.2231
2020	1423.74	1498.9738	1442.2657	1442.6880	1449.0904	1450.0914
2021	1547.87	1639.0972	1547.8700	1547.8694	1547.8700	1547.8700
MAPE _simu (%)		1.5517	2.1842	2.2827	1.3576	2.8273
MAPE _pred (%)		3.9523	1.3728	1.3683	1.3250	1.2202
MAPE_total (%)		2.4519	1.8799	1.9398	1.3454	2.2246

Table 12 The results of the gross regional product of Luzhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1156.59	1156.5900	1156.5900	1156.5900	1156.5900	1156.5900
2014	1279.12	1247.1494	1235.0474	1221.2838	1265.4024	1281.4050
2015	1369.31	1381.5558	1392.3454	1386.3383	1373.8933	1394.3745
2016	1500.64	1530.4473	1545.9866	1543.3726	1533.0141	1554.6806
2017	1698.91	1695.3849	1702.3368	1701.4357	1701.6386	1722.3719
2018	1895.55	1878.0981	1864.6450	1864.5976	1873.7637	1891.7590
2019	2081.26	2080.5024	2035.0130	2035.3003	2048.4054	2061.9954
2020	2157.20	2304.7199	2215.0420	2215.3136	2225.7087	2233.2913
2021	2406.10	2553.1016	2406.1000	2406.0998	2406.1000	2406.1000
MAPE $_{s i m u}$ (%)		1.3016	1.9963	2.0789	0.9749	1.4382
MAPE $_{p r e d}$ (%)		4.3281	1.6345	1.6341	1.5848	1.4843
MAPE $_{t o t a l}$ (%)		2.4366	1.8606	1.9121	1.2036	1.4555

Table 13 The results of the gross regional product of Suining by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	706.69	706.6900	706.6900	706.6900	706.6900	706.6900
2014	809.00	777.7660	770.9007	753.7539	791.9635	804.3043
2015	871.36	865.8257	875.9895	868.3060	868.0328	878.4366
2016	926.84	963.8556	976.7779	974.0244	969.8482	981.1922
2017	1046.43	1072.9846	1077.9478	1077.8531	1076.3927	1088.0237
2018	1230.85	1194.4693	1181.8337	1182.8942	1184.7636	1195.5076
2019	1345.73	1329.7087	1289.8908	1290.9318	1294.6332	1303.1767
2020	1403.18	1480.2601	1403.1800	1403.1792	1406.2225	1411.1963
2021	1519.87	1647.8571	1522.5689	1520.5702	1519.8700	1519.8700
MAPE $_{s i m u}$ (%)		2.7966	3.5246	3.8339	2.7471	2.8206
MAPE $_{p r e d}$ (%)		5.0349	1.4423	1.3727	1.3379	1.2445
MAPE $_{t o t a l}$ (%)		3.6360	2.7437	2.9109	2.2187	2.2296

Table 14 The results of the gross regional product of Neijiang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	915.11	915.1100	915.1100	915.1100	915.1100	915.1100
2014	973.86	948.3987	945.7211	942.7257	971.0887	980.0283
2015	1018.00	1025.1867	1027.7195	1026.6834	1010.5014	1028.1554
2016	1086.43	1108.1919	1111.6211	1111.2706	1100.5145	1117.2273
2017	1182.11	1197.9176	1199.2976	1199.2222	1199.1230	1213.0931
2018	1318.83	1294.9081	1291.7955	1291.8225	1299.8282	1310.4811
2019	1433.30	1399.7515	1389.8733	1389.9212	1401.1927	1408.3254
2020	1465.88	1513.0836	1494.1747	1494.2055	1502.9811	1506.5395
2021	1605.30	1635.5918	1605.3000	1605.2999	1605.3000	1605.3000
MAPE $_{s i m u}$ (%)		1.6949	1.9333	1.9664	1.0395	1.5439
MAPE $_{p r e d}$ (%)		2.4826	1.6534	1.6529	1.5904	1.5054
MAPE $_{t o t a l}$ (%)		1.9903	1.8284	1.8488	1.2461	1.5295

Table 15 The results of the gross regional product of Nanchong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1283.71	1283.7100	1283.7100	1283.7100	1283.7100	1283.7100
2014	1391.70	1325.2404	1310.1724	1270.8811	1310.8242	1328.7180
2015	1463.40	1481.5184	1504.8037	1484.8532	1504.8385	1541.6037
2016	1592.97	1656.2254	1685.6414	1677.9793	1685.5553	1729.9674
2017	1838.25	1851.5345	1862.5012	1862.6038	1862.4290	1907.5826
2018	2115.73	2069.8754	2040.1491	2044.5204	2040.1238	2080.6926
2019	2322.22	2313.9639	2221.4081	2227.0460	2221.4228	2252.7710
2020	2401.10	2586.8364	2408.2002	2412.3461	2408.2274	2426.0208
2021	2601.98	2891.8872	2601.9800	2601.9809	2601.9800	2601.9800
MAPE $_{s i m u}$ (%)		2.5749	3.8793	4.0349	3.8688	4.7795
MAPE $_{p r e d}$ (%)		6.4109	1.5456	1.5223	1.5458	1.3428
MAPE $_{t o t a l}$ (%)		4.0134	3.0042	3.0927	2.9977	3.4907

Table 16 The results of the gross regional product of Guang'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	790.91	790.9100	790.9100	790.9100	790.9100	790.9100
2014	838.46	810.4209	806.1722	800.2023	828.3993	836.3542
2015	874.39	882.5480	887.0569	884.7228	874.6430	882.6788
2016	928.03	961.0944	967.0751	966.2840	958.2327	966.6667
2017	1047.67	1046.6313	1048.9844	1048.9358	1048.1555	1056.3424
2018	1157.00	1139.7811	1134.1914	1134.4722	1139.6606	1146.8736
2019	1250.44	1241.2211	1223.6111	1223.9654	1231.7841	1237.2998
2020	1301.60	1351.6893	1317.9459	1318.1933	1324.4446	1327.5527
2021	1417.80	1471.9890	1417.8000	1417.8004	1417.7999	1417.8000
MAPE $_{s i m u}$ (%)		1.8855	2.3207	2.3869	1.2057	1.4131
MAPE $_{p r e d}$ (%)		2.8025	1.1338	1.1307	1.0824	1.0149
MAPE $_{t o t a l}$ (%)		2.2294	1.8756	1.9158	1.1594	1.2638

Table 17 The results of the gross regional product of Yibin by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models.

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1293.47	1293.4700	1293.4700	1293.4700	1293.4700	1293.4700
2014	1411.37	1292.9026	1288.3990	1249.8215	1343.8687	1384.6234
2015	1470.10	1482.8115	1501.6987	1480.1593	1491.3364	1546.3541
2016	1609.56	1700.6154	1722.8719	1712.4056	1710.9608	1770.4955
2017	1862.19	1950.4115	1959.5802	1956.5309	1957.4288	2015.6089
2018	2349.31	2236.8992	2216.6377	2218.1700	2224.2990	2275.8978
2019	2601.89	2565.4678	2497.9579	2501.5271	2510.9988	2550.8446
2020	2802.10	2942.2984	2807.2055	2810.3028	2818.4509	2841.1975
2021	3148.08	3374.4800	3148.0800	3148.0789	3148.0800	3148.0799
MAPE $_{s i m u}$ (%)		4.8876	5.7559	5.8337	4.5925	5.6889
MAPE $_{p r e d}$ (%)		4.5316	1.3922	1.3834	1.3589	1.1191
MAPE $_{t o t a l}$ (%)		4.7541	4.1195	4.1648	3.3799	3.9752

Table 18 The results of the gross regional product of Ya'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	428.95	428.9500	428.9500	428.9500	428.9500	428.9500
2014	475.79	478.2150	478.2150	479.9233	486.9937	490.1761
2015	519.02	517.4285	517.4285	516.7580	504.8068	508.8211
2016	561.65	559.8575	559.8575	558.7418	552.7979	557.0901
2017	608.54	605.7656	605.7656	605.3078	606.8385	610.9874
2018	653.34	655.4382	655.4382	656.4937	663.2052	666.8358
2019	723.79	709.1839	709.1839	712.5323	721.0297	723.7900
2020	754.59	767.3368	767.3368	773.7550	780.1368	781.6854
2021	840.56	830.2582	830.2582	840.5598	840.5600	840.5594
MAPE $_{s i m u}$ (%)		0.3825	0.3825	0.5672	1.6918	1.6537
MAPE $_{p r e d}$ (%)		1.6443	1.6443	1.3651	1.2556	1.1969
MAPE $_{t o t a l}$ (%)		0.8557	0.8557	0.8664	1.5282	1.4824

Table 19 The results of the gross regional product of Dazhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	1159.16	1159.1600	1159.1600	1159.1600	1159.1600	1159.1600
2014	1272.12	1244.4203	1229.2475	1212.6902	1257.0795	1255.4409
2015	1366.56	1377.3405	1390.5689	1383.1476	1373.5877	1426.2945
2016	1495.49	1524.4584	1543.5620	1540.4179	1531.4351	1584.8349
2017	1697.58	1687.2904	1695.7064	1694.8231	1694.8179	1737.7772
2018	1879.53	1867.5149	1850.6647	1850.9296	1858.7678	1888.9372
2019	2041.49	2066.9897	2010.6590	2011.3416	2022.6662	2040.6604
2020	2117.80	2287.7711	2177.2576	2177.8171	2186.8134	2194.5292
2021	2351.70	2532.1347	2351.7000	2351.6988	2351.7000	2351.7000
MAPE $_{s i m u}$ (%)		1.2298	1.9975	2.1148	1.0735	2.9050
MAPE $_{p r e d}$ (%)		5.6491	1.4392	1.4369	1.3936	1.2212
MAPE $_{t o t a l}$ (%)		2.8870	1.7882	1.8606	1.1935	2.2736

Table 20 The results of the gross regional product of Ziyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Year	Data	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
2013	507.51	507.5100	507.5100	507.5100	507.5100	507.5100
2014	537.09	538.7081	530.9667	528.8336	526.8612	526.6206
2015	572.76	582.5615	586.7672	585.8403	596.1857	594.6668
2016	643.84	629.9847	636.9127	636.1696	635.7901	634.3269
2017	688.54	681.2684	684.5869	683.9341	679.5462	678.5930
2018	728.63	736.7269	731.2724	730.8426	726.9286	726.5053
2019	777.80	796.6999	777.8000	777.8000	777.8000	777.8000
2020	807.50	861.5551	824.6961	825.3635	832.1741	832.4418
2021	890.50	931.6897	872.3266	873.9157	890.1395	890.4930
MAPE $_{s i m u}$ (%)		1.2664	1.1197	1.1970	1.7569	1.7976
MAPE $_{p r e d}$ (%)		4.5832	1.3901	1.3582	1.0320	1.0299
MAPE $_{t o t a l}$ (%)		2.5102	1.2211	1.2574	1.4851	1.5097

3.3 Further Discussions

According to the results listed in Tables 5

\sim

20, we summarized the performance measures of model's accuracy for the gross regional product in the Cheng-Yu area. The ranks of the MAPE in the simulation phase, prediction phase and total phase are presented in Tables 21

\sim

23, and Figures 9

\sim

11.

Table 21 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
Chongqing	5	2	3	4	1
Chengdu	1	3	4	2	5
Mianyang	1	3	4	2	5
Leshan	3	3	2	1	4
Zigong	2	3	4	1	5
Deyang	1	3	4	2	5
Meishan	2	3	4	1	5
Luzhou	2	4	5	1	3
Suining	2	4	5	1	3
Neijiang	3	4	5	1	2
Nanchong	1	3	4	2	5
Guang'an	3	4	5	1	2
Yibin	2	4	5	1	3
Ya'an	1	1	2	4	3
Dazhou	2	3	4	1	5
Ziyang	3	1	2	4	5

Table 22 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
Chongqing	5	4	2	3	1
Chengdu	5	4	3	2	1
Mianyang	5	4	3	2	1
Leshan	4	4	3	2	1
Zigong	5	4	3	2	1
Deyang	5	4	3	2	1
Meishan	5	4	3	2	1
Luzhou	5	4	3	2	1
Suining	5	4	3	2	1
Neijiang	5	4	3	2	1
Nanchong	5	3	2	4	1
Guang'an	5	4	3	2	1
Yibin	5	4	3	2	1
Ya'an	4	4	3	2	1
Dazhou	5	4	3	2	1
Ziyang	5	4	3	2	1

Table 23 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

	GM $(1, 1)$	CFGM $(1, 1)$	NGBM $(1, 1)$	CFNGBM $(1, 1)$	CFNGBM $(1, 1, λ)$
Chongqing	5	2	3	4	1
Chengdu	5	2	3	1	4
Mianyang	5	2	4	1	3
Leshan	4	4	2	1	3
Zigong	4	2	3	1	5
Deyang	5	2	3	1	4
Meishan	5	2	3	1	4
Luzhou	5	3	4	1	2
Suining	5	3	4	1	2
Neijiang	5	3	4	1	2
Nanchong	5	2	3	1	4
Guang'an	5	3	4	1	2
Yibin	5	3	4	1	2
Ya'an	1	1	2	4	3
Dazhou	5	2	3	1	4
Ziyang	5	1	2	3	4

Figure 9 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Full size|PPT slide

Figure 10 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Full size|PPT slide

Figure 11 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Full size|PPT slide

It can be seen from Tables 21

\sim

23, and Figures 9

\sim

11 that the CFNGBM

(1, 1, λ)

model has the highest accuracy in the prediction stage, followed by CFNGBM

(1, 1)

, NGBM

(1, 1)

, CFGM

(1, 1)

and GM

(1, 1)

models. However, the new model does not have the best modeling accuracy, or even the worst among these models. There is a trade-off between the MAPE

_{s i m u}

and MAPE

_{p r e d}

that requires us to select appropriate performance as the objective function in the optimization problem. Results also show grey model with three adjustable parameters can get better results than that has less adjustable parameters.

Another interesting result is that the MAPE

_{p r e d}

of the NGBM

(1, 1)

model is almost better than that of the CFGM

(1, 1)

model. These two models have one adjustable parameter, the NGBM

(1, 1)

model is the power exponent

γ

which mainly changes the structure of a grey model, while the CFGM

(1, 1)

model is the conformable fractional-order

α

which mainly changes the structure of original data. Thus changing the structure of a grey model can improve accuracy more than changing the structure of original data.

Moreover, with the obtained system parameters of the CFNGBM

(1, 1, λ)

model and historical data of gross regional product of each city in the Cheng-Yu area, we calculate the future values of these cities from the year 2022 to 2026, and the results are displayed in Table 24 and Figure 12, while the annual increase rates of them are listed in Table 25.

Table 24 The gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years

	Chongqing	Chengdu	Mianyang	Leshan	Zigong	Deyang	Meishan	Luzhou
2023	32758.4681	23013.5111	3910.9382	2587.5760	1815.9540	3024.1179	1754.3615	2758.1571
2024	35441.3813	24668.1669	4209.0188	2795.8363	1930.0736	3217.4890	1863.9880	2938.2714
2025	38310.5444	26398.8970	4520.5674	3016.5059	2049.2949	3418.2224	1978.4109	3121.6196
2026	41381.0072	28212.1365	4846.7353	3250.4055	2174.0219	3626.9770	2098.0124	3308.5406

	Suining	Neijiang	Nanchong	Guang'an	Yibin	Ya'an	Dazhou	Ziyang
2022	1629.5114	1704.8469	2781.8074	1508.2638	3472.7440	900.5124	2513.0765	952.0670
2023	1740.4108	1805.4224	2966.4320	1599.1651	3816.4961	961.6624	2679.4062	1017.3084
2024	1852.8290	1907.2528	3156.6376	1690.7075	4180.6765	1024.1286	2851.3369	1086.3838
2025	1966.9995	2010.5444	3353.1147	1783.0741	4566.6562	1088.0258	3029.4519	1159.4775
2026	2083.1325	2115.4848	3556.4926	1876.4290	4975.8412	1153.4637	3214.2933	1236.7888

Figure 12 The plots of the gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years

Full size|PPT slide

Table 25 The annual increase rate of the gross regional product in the next several years (%)

	Chongqing	Chengdu	Mianyang	Leshan	Zigong	Deyang	Meishan	Luzhou
2022	8.4367	7.6514	8.2031	8.4250	6.5700	6.8074	6.5420	7.2650
2023	8.3019	7.3980	7.8842	8.2245	6.4129	6.5805	6.3809	6.8679
2024	8.1900	7.1899	7.6217	8.0485	6.2843	6.3943	6.2488	6.5302
2025	8.0955	7.0160	7.4019	7.8928	6.1770	6.2388	6.1386	6.2400
2026	8.0147	6.8686	7.2152	7.7540	6.0863	6.1071	6.0453	5.9879
mean	8.2078	7.2248	7.6652	8.0689	6.3061	6.4256	6.2711	6.5782

	Suining	Neijiang	Nanchong	Guang'an	Yibin	Ya'an	Dazhou	Ziyang
2022	7.2139	6.2011	6.9112	6.3806	10.3131	7.1325	6.8621	6.9146
2023	6.8057	5.8994	6.6369	6.0269	9.8986	6.7906	6.6186	6.8526
2024	6.4593	5.6403	6.4119	5.7244	9.5423	6.4956	6.4167	6.7900
2025	6.1620	5.4157	6.2243	5.4632	9.2325	6.2392	6.2467	6.7282
2026	5.9041	5.2195	6.0653	5.2356	8.9603	6.0144	6.1015	6.6678
mean	6.5090	5.6752	6.4499	5.7661	9.5893	6.5345	6.4491	6.7906

From the computational results, it is known that the gross regional product of Chongqing will soar from 30247.3604 million yuan in 2022 to 41381.0072 million yuan in 2026 with an average annual growth rate of 8.2078%, and the values of Chengdu will increase from 21428.2456 million yuan in 2022 to 28212.1365 million yuan in 2026 with a yearly growth rate 7.2248%, the values of Mianyang will rise from 3625.1269 million yuan in 2022 to 4846.7353 million yuan in 2026 with an average annual growth rate of 7.6652%, the values of Leshan will grow from 2390.9342 million yuan in 2022 to 3250.4055 million yuan in 2026 with a yearly growth rate 8.0689%.

In Zigong city, the gross regional product will rise from 1706.5162 million yuan in 2022 to 2174.0219 million yuan in 2026, with an average annual growth rate of 6.3061%. In Deyang city, the gross regional product will increase from 2837.4029 million yuan in 2022 to 3626.9770 million yuan in 2026 with a yearly growth rate of 6.4256%. The gross regional product of Meishan will rise from 1649.1323 million yuan in 2022 to 2098.0124 million yuan in 2026 with an average annual increase rate of 6.2711%, while Luzhou city increase from 2580.9041 million yuan in 2022 to 3308.5406 million yuan in 2026 with a yearly growth rate 6.5782%.

It can be seen in Tables 24, 25 and Figure 12 that the gross regional product of Suining city will soar from 1629.5114 million yuan in 2022 to 2083.1325 million yuan in 2026, with an average increase rate of 6.5090%, the values of Neijiang city will increase from 1704.8469 million yuan in 2022 to 2115.4848 million yuan in 2026 with a yearly growth rate 5.6752%, the values of Nanchong will rise from 2781.8074 million yuan in 2022 to 3556.4926 million yuan in 2026 with an average annual growth rate of 6.4499%, the values of Guang'an will grow from 1508.2638 million yuan in 2022 to 1876.4290 million yuan in 2026 with a yearly increase rate 5.7661%.

In Yibin city, the gross regional product will soar from 3472.7440 million yuan in 2022 to 4975.8412 million yuan in 2026, with an average growth rate of 9.5893%. Moreover, the minimum growth rate is 8.9603% in 2026, and the other yearly growth rates are all above 9%. In Ya'an city, the gross regional product will increase from 900.5124 million yuan in 2022 to 1153.4637 million yuan in 2026 with a yearly growth rate of 6.5345%. In Dazhou city, the gross regional product will increase from 2513.0765 million yuan in 2022 to 3214.2933 million yuan in 2026 with an average increase rate of 6.4491%, while the gross regional product of Ziyang city will grow from 952.0670 million yuan in 2022 to 1236.7888 million yuan in 2026 with a yearly growth rate 6.7906%.

These results show that Yibin city has the highest growth rate, Chongqing, Chengdu, Mianyang, Leshan have a medium growth rate, and the left cities including Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Ya'an, Dazhou and Ziyang have the lowest yearly growth rate in the next several years. These computational results can provide a reference guide both for governments and entrepreneurs in Cheng-Yu area to adjust their economic and social development plans.

4 Conclusions

This work considered a nonlinear grey Bernoulli model CFNGBM

(1, 1, λ)

model with three adjustable parameters which is a generalized model. The GM

(1, 1)

, CFGM

(1, 1)

, NGBM

(1, 1)

, CFNGBM

(1, 1)

modles are all special cases of it. Applying the grey technique, the theory of differential equation and the definition of the conformable fractional-order operator, the expressions of the time response function and system parameters are all deduced. Meanwhile, three nonlinear system parameters are numerically determined by the MFO algorithm. According to these theoretical expressions, the gross regional product of Cheng-Yu area including Chongqing, Chengdu, Mianyang, Leshan, Zigong, Deyang, Meishan, Luzhou, Suining, Neijiang, Nanchong, Guang'an, Yibin, Ya'an, Dazhou and Ziyang are studied. The values from 2022 to 2026 are also calculated by the CFNGBM

(1, 1, λ)

model. From the computational results, it is meaningful to develop such a general nonlinear model, and it can obtain satisfactory results in the gross regional product of all cities in the Cheng-Yu area.

Due to the fact that the calculation of the conformable fractional-order operators is simpler and more direct than the classical fractional-order operators, the entire calculation process is not very complex. In addition, the mean absolute prediction percentage error (MAPE

_{p r e d}

) is taken as the objective function, which allows the new model to have higher accuracy in prediction and obtain satisfactory results. On the other hand, the new model also has some disadvantages. It should be noted that the time response function is derived by the whitening differential equation, while the linear system parameters are derived by the grey basic form. But the inconsistency between the whitening differential equation and the grey basic form due to the trapezoid approximation formula, which may cause large errors in some applications. Further, the CFNGBM

(1, 1, λ)

model is a univariate grey forecasting model, and other factors may influence the trend of the gross regional product, such as the fixed asset investment, the local government budgetary income, the labor force, the total wages of works and staff, are not considered in such a model. Therefore nonlinear multivariate grey models with conformable fractional-order operators should be investigated in the future.

References

Publishing order | Descend order by publishing year | Descend order by cited within

1	Deng J. Fundations of grey theory. Wuhan: Huazhong University of Science and Technology Press, 2002. Cited in this article [3]

Chen C I, Chen H, Chen S. Forecasting of foreign exchange rates of Taiwan's major trading partners by novel nonlinear grey Bernoulli model NGBM

(1, 1)

. Communications in Nonlinear Science and Numerical Simulation, 2008, 13 (6): 1194- 1204.

https://doi.org/10.1016/j.cnsns.2006.08.008

Cited in this article [2]

3	Zhou J, Fang R, Li Y, et al. Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization. Applied Mathematics and Computation, 2009, 207 (2): 292- 299. https://doi.org/10.1016/j.amc.2008.10.045 Cited in this article [1]

4	Wang Z. An optimized nash nonlinear grey Bernoulli model for forecasting the main economic indices of high technology enterprises in China. Computers & Industrial Engineering, 2013, 64 (3): 780- 787. Cited in this article [1]

5	Lu J, Xie W, Zhou H, et al. An optimized nonlinear grey Bernoulli model and its applications. Neurocomputing, 2016, 177 (C): 206- 214. Cited in this article [1]

6	Xiao Q, Shan M, Gao M, et al. Parameter optimization for nonlinear grey Bernoulli model on biomass energy consumption prediction. Applied Soft Computing, 2020, 95, 106538. https://doi.org/10.1016/j.asoc.2020.106538 Cited in this article [1]

7	Yang L, Xie N. Integral matching-based nonlinear grey Bernoulli model for forecasting the coal consumption in China. Soft Computing, 2021, 25, 52095223. Cited in this article [1]

8	Ma X, Liu Z, Wang Y. Application of a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China. Journal of Computational and Applied Mathematics, 2019, 347, 84- 94. https://doi.org/10.1016/j.cam.2018.07.044 Cited in this article [1]

9	Wu W, Ma X, Zeng B, et al. A novel grey bernoulli mode for short-term natural gas consumption forecasting. Applied Mathematical Modelling, 2020, 84, 393- 404. https://doi.org/10.1016/j.apm.2020.04.006

10	Wu W, Ma X, Zeng B, et al. Forecasting short-term solar energy generation in Asia Pacific using a nonlinear grey Bernoulli model with time power term. Energy & Environment, 2021, 32, 759- 783.

11	Zheng C, Wu W, Xie W, et al. Forecasting the hydroelectricity consumption of China by using a novel unbiased nonlinear grey Bernoulli model. Journal of Cleaner Production, 2021, 278, 123903. https://doi.org/10.1016/j.jclepro.2020.123903

12	Ding S, Li R, Wu S. A novel composite forecasting framework by adaptive data preprocessing and optimized nonlinear grey Bernoulli model for new energy vehicles sales. Communications in Nonlinear Science and Numerical Simulation, 2021, 99, 105847. https://doi.org/10.1016/j.cnsns.2021.105847

13	Wang Y, Wang H. Forecasting CO $_{2}$ emissions using a novel fractional discrete grey Bernoulli model: A case of Shaanxi in China. Urban Climate, 2023, 49, 101452. https://doi.org/10.1016/j.uclim.2023.101452 Cited in this article [1]

14	Wu L, Liu S, Yao L, et al. Grey system model with the fractional order accumulation. Communications in Nonlinear Science and Numerical Simulation, 2013, 18 (7): 1775- 1785. https://doi.org/10.1016/j.cnsns.2012.11.017 Cited in this article [1]

15	Wu L, Liu S, Cui W, et al. Non-homogenous discrete grey model with fractional-order accumulation. Neural Computing & Applications, 2014, 25 (5): 1215- 1221. Cited in this article [1]

16	Mao S, Gao M, Xiao X, et al. A novel fractional grey system model and its application. Applied Mathematical Modelling, 2016, 40 (7-8): 5063- 5076. https://doi.org/10.1016/j.apm.2015.12.014

17	Wu W, Ma X, Zeng B, et al. Application of the novel fractional grey model FAGMO $(1, 1, k)$ to predict China's nuclear energy consumption. Energy, 2018, 165, 223- 234. https://doi.org/10.1016/j.energy.2018.09.155

18	Kang Y, Mao S, Zhang Y. Variable order fractional grey model and its application. Applied Mathematical Modelling, 2021, 97, 619- 635. https://doi.org/10.1016/j.apm.2021.03.059

19	Gao M, Yang H, Xiao Q, et al. A novel method for carbon emission forecasting based on Gompertz's law and fractional grey model: Evidence from American industrial sector. Renewable Energy, 2022, 181, 803- 819. https://doi.org/10.1016/j.renene.2021.09.072

20	Yan S, Su Q, Gong Z, et al. Fractional order time-delay multivariable discrete grey model for short-term onlline publlic opinion prediction. Expert Systems with Applications, 2022, 197, 116691. https://doi.org/10.1016/j.eswa.2022.116691

21	Zeng B, Li H, Mao C, et al. Modeling, prediction and analysis of new energy vehicle sales in China using a variable-structure grey model. Expert Systems with Applications, 2022, 22, 118879.

22	Mao C, Gou X, Zeng B. A novel grey prediction model with structure variability and real domain fractional order and its performance comparisons. Grey Systems: Theory and Application, 2023, 13 (2): 381- 405. https://doi.org/10.1108/GS-07-2022-0072 Cited in this article [1]

23	Ma X, Wu W, Zeng B, et al. The conformable fractional grey system model. ISA Transactions, 2020, 96, 255- 271. https://doi.org/10.1016/j.isatra.2019.07.009 Cited in this article [4]

24	Khalil R, Horani M A, Yousef A, et al. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 2014, 264, 65- 70. https://doi.org/10.1016/j.cam.2014.01.002 Cited in this article [1]

25	Xie W, Liu C, Wu W, et al. Continuous grey model with conformable fractional derivative. Chaos Solitons & Fractals, 2020, 139, 110285. Cited in this article [1]

26	Wu W, Ma X, Zhang Y, et al. A novel conformable fractional non-homogeneous grey model for forecasting carbon dioxide emissions of BRICS countries. Science of the Total Environment, 2020, 707, 135447. https://doi.org/10.1016/j.scitotenv.2019.135447

27	Wu W, Zeng L, Liu C, et al. A time power-based grey model with conformable fractional derivative and its applications. Chaos, Solitons & Fractals, 2022, 155, 111657.

28	Wu W, Ma X, Zeng B, et al. A novel multivariate grey system model with conformable fractional derivative and its applications. Computers & Industrial Engineering, 2022, 164, 107888.

29	Wu W, Ma X, Zhang H, et al. A conformable fractional discrete grey model CFDGM $(1, 1)$ and its application. International Journal of Grey Systems, 2022, 2 (1): 5- 15. https://doi.org/10.52812/ijgs.36

30	Dun M, Xu Z, Wu L, et al. The information priority of conformable fractional grey model. Journal of Computational and Applied Mathematics, 2022, 415, 114460. https://doi.org/10.1016/j.cam.2022.114460

31	Liu Y, Yang Y, Pan F, et al. A conformable fractional unbiased grey model with a flexible structure and it's application in hydroelectricity consumption prediction. Journal of Cleaner Production, 2022, 367, 133029. https://doi.org/10.1016/j.jclepro.2022.133029

32	Zhu H, Liu C, Wu W, et al. A novel conformable fractional nonlinear grey multivariable prediction model with marine predator algorithm for time series prediction. Computers & Industrial Engineering, 2023, 180, 109278. Cited in this article [1]

33	Wu W, Ma X, Zeng B, et al. Forecasting short-term renewable energy consumption of China using a novel fractional nonlinear grey Bernoulli model. Renewable Energy, 2019, 140, 70- 87. https://doi.org/10.1016/j.renene.2019.03.006 Cited in this article [1]

34	Şahin U, Şahin T. Forecasting the cumulative number of confirmed cases of COVID-19 in Italy, UK and USA using fractional nonlinear grey Bernoulli model. Chaos, Solitons & Fractals, 2020, 138, 109948. Cited in this article [1]

Şahin U. Projections of Turkey's electricity generation and installed capacity from total renewable and hydro energy using fractional nonlinear grey Bernoulli model and its reduced forms. Sustainable Production and Consumpiton, 2020, 23, 52- 62.

https://doi.org/10.1016/j.spc.2020.04.004

Cited in this article [1]

36	Şahin U. Future of renewable energy consumption in France, Germany, Italy, Spain, Turkey and UK by 2030 using optimized fractional nonlinear Bernoulli model. Sustainable Production and Consumption, 2021, 25, 1- 14. https://doi.org/10.1016/j.spc.2020.07.009 Cited in this article [1]

37	Zheng C, Wu W, Xie W, et al. A MFO-based conformable fractional nonhomogneous grey Bernoulli model for natural gas production and consumption forecasting. Applied Soft Computing Journal, 2021, 99, 106891. https://doi.org/10.1016/j.asoc.2020.106891 Cited in this article [1]

Wang Y, He X, Zhang L, et al. A novel fractional time-delayed grey bernoulli forecasting model and its application for the energy production and consumpiton prediction. Engineering Applications of Artificial Intelligence, 2022, 110, 104683.

https://doi.org/10.1016/j.engappai.2022.104683

Cited in this article [1]

Wang Y, Nie R, Chi P, et al. A novel fractional structural adaptive grey Chebyshev polynomial Bernoulli model and its application in forecasting renewable energy production of China. Expert Systems with Applications, 2022, 210, 118500.

https://doi.org/10.1016/j.eswa.2022.118500

Cited in this article [1]

40	Xie W, Yu G. A novel conformable fractional nonlinear grey Bernoulli model and its application. Complexity, 2020, 9178098. https://doi.org/10.1155/2020/9178098 Cited in this article [2]

41	Zeng B, Tong M, Ma X. A new-structure grey Verhulst model: Development and performance comparison. Applied Mathematical Modelling, 2020, 81, 522- 537. https://doi.org/10.1016/j.apm.2020.01.014 Cited in this article [1]

42	Zeng B, Ma X, Shi J. A new-structure grey Verhulst model for China's tight gas production forecasting. Applied Soft Computing, 2020, 96, 106600. https://doi.org/10.1016/j.asoc.2020.106600 Cited in this article [1]

43	Deng J. Control problems of grey systems. Systems & Control Letters, 1982, 1 (5): 288- 294. Cited in this article [1]

44	Mirjalili S. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowledge-Based Systems, 2015, 89, 228- 249. https://doi.org/10.1016/j.knosys.2015.07.006 Cited in this article [1]

45	Li X, Abdullah L, Sobri S, et al. Long-term air pollution characteristics and multi-scale meteorological factor variability analysis of Mega-mountain Cities in the Chengdu-Chongqing Economic Circle. Water, Air, & Soil Pollution, 2023, 234, 328. Cited in this article [1]

46	Weng G, Li H, Li Y. The temporal and spatial distribution characteristics and influencing factors of tourist attractions in Chengdu-Chongqing economic circle. Environment, Development and Sustainability, 2023, 25, 8677- 8698. https://doi.org/10.1007/s10668-022-02418-z

47	Wang L, Yuan M, Li H, et al. Exploring the coupling coordination of urban ecological resilience and new-type urbanization: The case of China's Chengdu-Chongqing Economic Circle. Environmental Technology & Innovation, 2023, 9, 103372.

48	Liu M, Luo X, Qi L, et al. Simulation of the spatiotemporal distribution of PM2.5 concentration based on GTWR-XGBoost two-stage model: A case study of Chengdu Chongqing Economic Circle. Atmosphere, 2023, 14 (1): 115. https://doi.org/10.3390/atmos14010115 Cited in this article [1]

Funding

the National Natural Science Foundation of China(72001181)

the National Natural Science Foundation of China(71901184)

the Sichuan Federation of Social Science Associations(SC20B122)

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Abstract
Key words
Cite this article
1 Introductions
Figure 1 The main contents of the study
2 The Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation
2.1 Abbreviations, Notations and Definitions
Table 1 Abbreviations for various grey forecasting models
Table 2 Notations for different variables
2.2 The NGBM $(1, 1)$ Model with Integer-Order Accumulation
2.3 The CFNGBM $(1, 1, λ)$ Model
Figure 2 The relationship among the CFNGBM $(1, 1, λ)$ , CFNGBM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , GVM(1, 1) and GM $(1, 1)$ models
2.4 An Optimization Problem for System Nonlinear Parameters by the Moth Flame Optimization Algorithm
Figure 3 The general framework of the MFO algorithm
3 Application in the Gross Regional Product in the Cheng-Yu Area
3.1 Background
Table 3 Raw data of gross regional product in the Cheng-Yu area (million yuan)
Figure 4 The numerical design of the application in the gross regional product
3.2 The Gross Regional Product in the Cheng-Yu Area
Table 4 The minimum MAPE $_{p r e d}$ and corresponding nonlinear parameters of grey models
Figure 5 The convergence curve of MAPE $_{p r e d}$ and $α$ of the CFGM $(1, 1)$ model in Chongqing case
Figure 6 The convergence curve of MAPE $_{p r e d}$ and $γ$ of the NGBM $(1, 1)$ model in Chongqing case
Figure 7 The convergence curve of MAPE $_{p r e d}$ , $α$ and $γ$ of the CFNGBM $(1, 1)$ model in Chongqing case
Figure 8 The convergence curve of MAPE $_{p r e d}$ , $α$ , $γ$ and $λ$ of the CFNGBM $(1, 1, λ)$ model in Chongqing case
Table 5 The results of the gross regional product of Chongqing by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 6 The results of the gross regional product of Chengdu by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 7 The results of the gross regional product of Mianyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 8 The results of the gross regional product of Leshan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ and CFNGBM $(1, 1)$ models
Table 9 The results of the gross regional product of Zigong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 10 The results of the gross regional product of Deyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 11 The results of the gross regional product of Meishan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 12 The results of the gross regional product of Luzhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 13 The results of the gross regional product of Suining by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 14 The results of the gross regional product of Neijiang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 15 The results of the gross regional product of Nanchong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 16 The results of the gross regional product of Guang'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 17 The results of the gross regional product of Yibin by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models.
Table 18 The results of the gross regional product of Ya'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 19 The results of the gross regional product of Dazhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
Table 20 The results of the gross regional product of Ziyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models
3.3 Further Discussions
Table 21 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Table 22 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Table 23 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Figure 9 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Figure 10 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Figure 11 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities
Table 24 The gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years
Figure 12 The plots of the gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years
Table 25 The annual increase rate of the gross regional product in the next several years (%)
4 Conclusions
References
Funding

Received	Accepted	Published
2023-08-11	2023-10-16	2024-04-25
Issue Date
2024-04-23

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Abstract

Key words

Cite this article

1 Introductions

Figure 1 The main contents of the study

2 The Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation

2.1 Abbreviations, Notations and Definitions

Table 1 Abbreviations for various grey forecasting models

Table 2 Notations for different variables

2.2 The NGBM(1,1) Model with Integer-Order Accumulation

2.3 The CFNGBM(1,1,λ) Model

Figure 2 The relationship among the CFNGBM(1,1,λ), CFNGBM(1,1), CFGM(1,1), NGBM(1,1), GVM(1, 1) and GM(1,1) models

2.4 An Optimization Problem for System Nonlinear Parameters by the Moth Flame Optimization Algorithm

Figure 3 The general framework of the MFO algorithm

3 Application in the Gross Regional Product in the Cheng-Yu Area

3.1 Background

Table 3 Raw data of gross regional product in the Cheng-Yu area (million yuan)

Figure 4 The numerical design of the application in the gross regional product

3.2 The Gross Regional Product in the Cheng-Yu Area

Table 4 The minimum MAPEpred and corresponding nonlinear parameters of grey models

Figure 5 The convergence curve of MAPEpred and α of the CFGM(1,1) model in Chongqing case

Figure 6 The convergence curve of MAPEpred and γ of the NGBM(1,1) model in Chongqing case

Figure 7 The convergence curve of MAPEpred, α and γ of the CFNGBM(1,1) model in Chongqing case

Figure 8 The convergence curve of MAPEpred, α, γ and λ of the CFNGBM(1,1,λ) model in Chongqing case

Table 5 The results of the gross regional product of Chongqing by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 6 The results of the gross regional product of Chengdu by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 7 The results of the gross regional product of Mianyang by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 8 The results of the gross regional product of Leshan by the GM(1,1), CFGM(1,1), NGBM(1,1) and CFNGBM(1,1) models

Table 9 The results of the gross regional product of Zigong by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 10 The results of the gross regional product of Deyang by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 11 The results of the gross regional product of Meishan by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 12 The results of the gross regional product of Luzhou by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 13 The results of the gross regional product of Suining by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 14 The results of the gross regional product of Neijiang by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 15 The results of the gross regional product of Nanchong by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 16 The results of the gross regional product of Guang'an by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 17 The results of the gross regional product of Yibin by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models.

Table 18 The results of the gross regional product of Ya'an by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 19 The results of the gross regional product of Dazhou by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

Table 20 The results of the gross regional product of Ziyang by the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models

3.3 Further Discussions

Table 21 Summary of ranks for MAPEsimu of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Table 22 Summary of ranks for MAPEpred of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Table 23 Summary of ranks for MAPEtotal of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Figure 9 Summary of ranks for MAPEsimu of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Figure 10 Summary of ranks for MAPEpred of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Figure 11 Summary of ranks for MAPEtotal of the GM(1,1), CFGM(1,1), NGBM(1,1), CFNGBM(1,1) and CFNGBM(1,1,λ) models in all cities

Table 24 The gross regional product of all cities in Cheng-Yu area by the CFNGBM(1,1,λ) model in the next several years

Figure 12 The plots of the gross regional product of all cities in Cheng-Yu area by the CFNGBM(1,1,λ) model in the next several years

Table 25 The annual increase rate of the gross regional product in the next several years (%)

4 Conclusions

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References

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Footnotes

Funding

2.2 The NGBM $(1, 1)$ Model with Integer-Order Accumulation

2.3 The CFNGBM $(1, 1, λ)$ Model

Figure 2 The relationship among the CFNGBM $(1, 1, λ)$ , CFNGBM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , GVM(1, 1) and GM $(1, 1)$ models

Table 4 The minimum MAPE $_{p r e d}$ and corresponding nonlinear parameters of grey models

Figure 5 The convergence curve of MAPE $_{p r e d}$ and $α$ of the CFGM $(1, 1)$ model in Chongqing case

Figure 6 The convergence curve of MAPE $_{p r e d}$ and $γ$ of the NGBM $(1, 1)$ model in Chongqing case

Figure 7 The convergence curve of MAPE $_{p r e d}$ , $α$ and $γ$ of the CFNGBM $(1, 1)$ model in Chongqing case

Figure 8 The convergence curve of MAPE $_{p r e d}$ , $α$ , $γ$ and $λ$ of the CFNGBM $(1, 1, λ)$ model in Chongqing case

Table 5 The results of the gross regional product of Chongqing by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 6 The results of the gross regional product of Chengdu by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 7 The results of the gross regional product of Mianyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 8 The results of the gross regional product of Leshan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ and CFNGBM $(1, 1)$ models

Table 9 The results of the gross regional product of Zigong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 10 The results of the gross regional product of Deyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 11 The results of the gross regional product of Meishan by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 12 The results of the gross regional product of Luzhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 13 The results of the gross regional product of Suining by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 14 The results of the gross regional product of Neijiang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 15 The results of the gross regional product of Nanchong by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 16 The results of the gross regional product of Guang'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 17 The results of the gross regional product of Yibin by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models.

Table 18 The results of the gross regional product of Ya'an by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 19 The results of the gross regional product of Dazhou by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 20 The results of the gross regional product of Ziyang by the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models

Table 21 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Table 22 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Table 23 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Figure 9 Summary of ranks for MAPE $_{s i m u}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Figure 10 Summary of ranks for MAPE $_{p r e d}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Figure 11 Summary of ranks for MAPE $_{t o t a l}$ of the GM $(1, 1)$ , CFGM $(1, 1)$ , NGBM $(1, 1)$ , CFNGBM $(1, 1)$ and CFNGBM $(1, 1, λ)$ models in all cities

Table 24 The gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years

Figure 12 The plots of the gross regional product of all cities in Cheng-Yu area by the CFNGBM $(1, 1, λ)$ model in the next several years