An Optimized Fractional Nonlinear Grey System Model and Its Application in the Prediction of the Development Scale of Junior Secondary Schools in China

Abstract

As part of China’s compulsory nine-year education system, junior secondary education (JSSE) plays a vital role in supporting students’ physical and mental development. The accurate prediction of the development scale trend of JSSE is helpful for the government to estimate the scale of educational development within a chosen time frame so as to aid decision making.Nevertheless, China’s education system is complex, highly dimensional, and largely influenced by policy and other factors, which results in difficulty in modeling the education sample. Based on gray system theory, this paper proposes an improved fractional-order grey prediction model, OCFNGBM(1,1), to predict the development scale of JSSE. We describe the basic expressions of the model, the parameter estimation method, and the optimization method for hyperparameters and construct a scheme for optimizing the background value coefficients. Data collected from official websites from 2011 to 2021 are used to build the forecasting model, and data from 2011 to 2017 are used to evaluate the model’s accuracy. Our experimental results indicate that the OCFNGBM(1,1) model has higher accuracy than the classical nonlinear gray prediction model. The OCFNGBM(1,1) model was employed to forecast the development scale of JSSE in China from 2022 to 2024, which provided useful information. This research provides a resource to help the national education department to develop a comprehensive and long-term plan for the development goals, scale, speed, steps, and measures of relevant education.

1. Introduction

China’s educational reforms and development have always prioritized compulsory education, also referred to as universal compulsory education, which is a compulsory and free school education provided to school-age children for a specific period of time in accordance with national laws. A compulsory education period of nine years is stipulated by the Compulsory Education Law of China, of which the last three years are dedicated to junior secondary education (JSSE). This is the phase of education when compulsory education becomes non-compulsory and plays an important role in China’s education system. A country’s JSSE quality is largely influenced by the number of high-level reserving talents, which indirectly determines the quality of its workers and plays an important role in economic development. There is no doubt that the education development trend plays an influential role in promoting the reform and development of education, as well as personnel training. The number of schools, rate of enrollment, the number of students enrolled, the graduation rate, and so on all contribute to the formulation of comprehensive and long-term education plans of the national or regional education authorities.

Accurately predicting trends in the scale of educational development requires prejudging the scale of educational development within a certain future time frame, which makes it easier for decision makers to formulate measures to guide the construction of campuses and the introduction of teachers.

China’s education system is complex, with a high number of dimensions each greatly affected by policy and other factors. However, data on JSSE are relatively sparse. Grey prediction is a method for predicting systems with uncertain factors [1]. It identifies the degree of difference between the development trends of system factors; i.e., by performing correlation analysis and analyzing the original data, it determines the rules of system changes, generates data sequences with high regularity, and establishes a differential equation model to predict the development trends of things [2]. Based on a series of quantitative values of response prediction object characteristics observed at equal time intervals, gray prediction predicts the characteristic quantity at a particular point in time. The gray prediction model is based on the GM(1,1) model [3]. An exponential law governs the accumulation and release of gray indices, and it is widely used in sectors such as industry [4], agriculture, energy [5], transportation, education [6], etc.

In this paper, a nonlinear grey prediction model OCFNGBM(1,1) based on grey theory is presented for predicting the development scale trends of JSSE. The fractional order in this paper is simple in terms of calculation, and the background values are optimized, which can further improve the accuracy of the model. Furthermore, in contrast to other classical models, our model can resist disturbance traps, making prediction more accurate under conditions of poor information and small samples.

The following is a summary of the main contributions:

(1) In this paper, we propose a new nonlinear grey prediction model that takes into account the errors inherent in numerical calculations, using discrete conformable fractional-order integrals for the construction of the cumulative operator and a PSO to optimize the model’s background value coefficients.

(2) Using historical data from Chinese junior high schools, we used the newly proposed model and the comparison model separately, and the experimental results showed that the newly proposed model had the best predictive performance.

(3) A newly developed model was used to predict the future number of junior high schools in China as a way to infer the scale of junior high school development and to provide decision making advice to school administrators.

The remainder of this article is organized as follows. This paper’s relevant work is described in Section 2. Section 3 introduces the classical nonlinear grey prediction model, provides some modeling details, and lays the theoretical foundations for building a new model. Section 4 presents a detailed description of our OCFNGBM(1,1) model based on fractional order. In Section 5, we analyze the development situation of JSSE and apply our prediction model to predict the development scale of JSSE. The key issues of this paper are discussed in Section 6. In Section 7, a conclusion is drawn.

2. Related Work

Education has a great influence on the development of a country, and many scholars have studied cases of educational development. For example, Yan et al. [7] predicted the scale of regional higher education using BP Neural Networks taking regional populations, GDP, the proportion contributed by tertiary industries to the GDP, Engel’s coefficient of rural residents, and urban residents’ disposable income as explanatory variables. Shruthi [8] used the Naive Bayes classification algorithm to predict the performance of students using the behaviors and results of previously graduated students stored in the database and by using the behaviors of the present students. Li et al. [9] proposed a deep neural network by extracting informative data as a feature with corresponding weights for the prediction of student performance. In the paper, multiple updated hidden layers were used to design a neural network. Wanjau et al. [10] proposed a general framework for mining student data using performance-weighted ensemble classifiers. They trained an ensemble of classification models from enrollment data streams to improve the quality of student data by eliminating noisy instances. Stallings [11] presents a study on the prediction of university enrollment using three computational intelligence (CI) techniques. Xu et al. [12] applied the R/S analysis of fractal theory to predict the trend of the regional differences in the level of China’s higher education developmental. Zhang et al. [13] put forward the reverse fuzzy number prediction model for fuzzy time series prediction. The model was applied in the simulated prediction of student enrollment from 1997 to 2012 for Guangxi University with historical data. Padmapriya [14] applied data mining techniques to predict higher education admissibility.

However, the development of education is often disturbed by economic policies, and a large amount of data is often unavailable. In the case of small data, machine learning models or other techniques often fail. The grey system theory regards a random quantity as the grey quantity that changes within a certain range, despite the existence of random interference components. After this, a certain technical treatment can always find its regularity. Accordingly, grey models can have a good effect on such data sets and have been subject to a number of important research advances [15]. For example, Ma et al. [16] proposed a novel time-delayed polynomial grey prediction model, abbreviated as TDPGM(1,1), to predict China’s natural gas consumption. Compared to other models, Zhou’s grey forecasting model was able to obtain a high level of forecasting accuracy for seasonal time series [17]. A series of practical problems were solved by Zhang’s nonlinear gray prediction models based on fractal derivatives [18]. Wang proposed a multivariate gray prediction model for forecasting in the energy industry based on the Hausdorff derivative [19]. The generalized fractional-order grey prediction model proposed by Xie extends the use of the fractional-order gray prediction models [20]. Using conformable fractional order derivatives, Wu proposed a gray prediction model containing time power terms. Based on the experimental results, it is evident that this model has many excellent properties and is suitable for practical applications [21]. Even though the model described above can predict small samples to a certain extent, the calculation process is generally complicated. Using conformable fractional-order derivatives [22], a new grey prediction model is proposed to solve problems related to predict the scale of junior high school development. Wang et al. [23] proposed a fractional structural adaptive grey Chebyshev polynomial Bernoulli model and applied it to forecast the renewable energy production of China. He et al. [24] presented a structure adaptive new information priority discrete grey prediction model and forecasted the renewable energy generation. Wang et al. [25] employed a structural adaptive Caputo fractional grey prediction model to forecast China’s energy production and consumption.

Although researchers have proposed many grey models, such models have some limitations. Firstly, the previous fractional nonlinear grey prediction models are complex in terms of calculation, which is not conducive to its popularization. Secondly, the background value is not optimized, which will introduce some errors. Accordingly, to address the issues, we proposed a nonlinear grey prediction model called OCFNGBM(1,1) and applied it in predicting the scale of JSSE development.

3. Preliminaries

3.1. The Basis of the FANGBM(1,1) Model

The purpose of this subsection is to introduce some basic modeling concepts. Using fractional-order cumulative generating operators, Wu first proposed the FANGBM(1,1) model, which has many excellent properties [26]. In order to construct a gray prediction model that is computationally simpler, we improve upon this model using a fractional-order conformable operator [27]. In order to begin, we provide some basic definitions. Throughout this paper, ${\mathbb{N}}_c=\{c,c+1,c+2,\dots \},{\mathbb{N}}_c^d=\{c,c+1,c+2,\dots ,d\}$, where $c,d\in \mathbb{R}$ and $d-c\in {\mathbb{N}}_1.$

Definition 1

([28]).Let ${\mathit{Y}}^{\left(0\right)}={\left[y^{\left(0\right)}\left(1\right),y^{\left(0\right)}\left(2\right),\cdots ,y^{\left(0\right)}\left(n\right)\right]}^{\mathrm{T}}$ be the original sequence of the raw data. The fractional-order accumulated generating operator (FAGO) is defined as ${}^{{\nabla }^{-\mu }}{Y}^{\left(0\right)}:=A^{\mu }{Y}^{\left(0\right)}={[{\nabla }^{-\mu }{y}^{\left(0\right)}\left(1\right),{\nabla }^{-\mu }{y}^{\left(0\right)}\left(2\right),\cdots ,{\nabla }^{-\mu }{y}^{\left(0\right)}\left(n\right)]}^T$, where

$${\left(\begin{array}{ccccc}\left[\begin{array}{c}\mu \\ 0\end{array}\right]& 0& 0& \cdots & 0\\ \left[\begin{array}{c}\hfill \mu \\ \hfill 1\end{array}\right]& \left[\begin{array}{c}\hfill \mu \\ \hfill 0\end{array}\right]& 0& \cdots & 0\\ \left[\begin{array}{c}\mu \\ 2\end{array}\right]& \left[\begin{array}{c}\hfill \mu \\ \hfill 1\end{array}\right]& \left[\begin{array}{c}\mu \\ 0\end{array}\right]& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \left[\begin{array}{c}\hfill \mu \hfill \\ \hfill n-1\hfill \end{array}\right]& \left[\begin{array}{c}\mu \\ n-2\end{array}\right]& \left[\begin{array}{c}\mu \\ n-3\end{array}\right]& \cdots & \left[\begin{array}{c}\mu \\ 0\end{array}\right]\end{array}\right)}_{n\times n},$$

(1)

with $\left[\begin{array}{c}\hfill \mu \hfill \\ \hfill i\hfill \end{array}\right]=\frac{\mu (\mu +1)\cdots (\mu +i-1)}{i!},\left[\begin{array}{c}\hfill 0\hfill \\ \hfill i\hfill \end{array}\right]=0,\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]=1$, where the symbol T indicates a transposition of the data in this paper.

Definition 2.

The FANGBM(1,1) is expressed mathematically as follows:

$$\frac{d{y}^{\left(\mu \right)}\left(\tau \right)}{d\tau }+a{y}^{\left(\mu \right)}\left(\tau \right)=b{\left(y^{\left(\mu \right)}\left(\tau \right)\right)}^{\gamma },$$

(2)

where the power index γ can be taken as an arbitrary real number—it is typically calculated using planning models and intelligent optimization algorithms. According to the least squares algorithm, if n is the amount of data used for modeling, then the parameters a and b of the FANGBM(1,1) model are calculated as follows:

$${[\widehat{a},\widehat{b}]}^T={\left({\mathbf{A}}^{\mathbf{T}}\mathbf{A}\right)}^{-1}{\mathbf{A}}^{\mathbf{T}}\mathbf{W},$$

(3)

where

$$\mathbf{A}=\left(\begin{array}{cc}-{\vartheta }^{\left(\mu \right)}\left(2\right)& {\left({\vartheta }^{\left(\mu \right)}\left(2\right)\right)}^{\gamma }\\ -{\vartheta }^{\left(\mu \right)}\left(3\right)& {\left({\vartheta }^{\left(\mu \right)}\left(3\right)\right)}^{\gamma }\\ ⋮& ⋮\\ -{\vartheta }^{\left(\mu \right)}\left(n\right)& {\left({\vartheta }^{\left(\mu \right)}\left(n\right)\right)}^{\gamma }\end{array}\right),\mathbf{W}=\left(\begin{array}{c}{y}^{\left(\mu \right)}\left(2\right)-y^{\left(\mu \right)}\left(1\right)\\ y^{\left(\mu \right)}\left(3\right)-y^{\left(\mu \right)}\left(2\right)\\ ⋮\\ y^{\left(\mu \right)}\left(n\right)-y^{\left(\mu \right)}(n-1)\end{array}\right),$$

(4)

where ${\vartheta }^{\left(\mu \right)}\left(k\right)=0.5y^{\left(\mu \right)}\left(k\right)+0.5y^{\left(\mu \right)}\left(k-1\right),k\in {\mathbb{N}}_2^n$ is a background value of $y^{\left(\mu \right)}\left(k\right)$; Ref. [26] provides more details on the modeling of FANGBM(1,1).

3.2. Conformable Fractional Accumulation and Difference

In the following step, we introduce a more straightforward fractional order cumulative generating operator that can be used to construct a simpler nonlinear grey prediction model.

Definition 3

([27]).Set a function $f:[0,\infty )⟶\mathbb{R}$; the conformable fractional derivative with μ can be represented as

$$T_{\mu }\left(f\right)\left(\tau \right)=\underset{\epsilon \to 0}{lim}\frac{f\left(\tau +\epsilon {\tau }^{1-\mu }\right)-f\left(\tau \right)}{\epsilon },$$

(5)

where $t>0,\mu \in (0,1).$

Definition 4

([29]).Set $f:\left[t_0,+\infty \right]\to \mathbb{R}$; the conformable fractional derivative (CFD) with $\mu \in (0,1]$ is defined as

$$T_{{\tau }_0}^{\mu }f\left(\tau \right)=\underset{\epsilon \to 0}{lim}\frac{f\left[\tau +\epsilon {\left(\tau -{\tau }_0\right)}^{1-\mu }\right]-f\left(\tau \right)}{\epsilon },$$

(6)

for all $t>t_0$.

Lemma 1

([29]).Let $0<\mu <1$, CFD and classical first-order derivative are related as follows:

$$T_{{\tau }_0}^{\mu }f\left(\tau \right)={\left(\tau -{\tau }_0\right)}^{1-\mu }\frac{df\left(\tau \right)}{d\tau },$$

(7)

in this case, f is a differentiable function.

Definition 5

([30]).Let $\mathbb{N}:=\left\{{\mathbb{N}}^{+}\cup 0\right\}$; then, the discrete conformable fractional derivative (DCFD) is defined as

$${\Delta }^{\mu }f\left(k\right)=k^{\left[\mu \right]-\mu }{\Delta }^{n}f\left(k\right),$$

(8)

where $\mu \in (n,n+1])$ for all $n\in \mathbb{N}$.

Definition 6

([30]).Let $(\mu \in (n,n+1\left]\right)$ be the conformable fractional accumulation defined as

$${\nabla }^{\mu }f\left(k\right)={\nabla }^n\left(\frac{f\left(k\right)}{k^{\left[\mu \right]-\mu }}\right).$$

(9)

4. The Optimized Conformable Fractional Grey Bernoulli Model

4.1. Formulating the OCFNGBM(1,1) Model

Taking advantage of the previous nonlinear grey forecasting models FANGBM(1,1), CFA, and DCFD, we propose an improved grey forecasting model with a simpler structure that facilitates more efficient computation.

Set a time sequence $Y^{\left(0\right)}={\left(y^{\left(0\right)}\left(1\right),y^{\left(0\right)}\left(2\right),y^{\left(0\right)}\left(3\right)\dots y^{\left(0\right)}\left(n\right)\right)}^T$ as a non-negative sequence; then, the basic form of the OCFNGBM(1,1) model is

$$\frac{d{y}^{\left(\mu \right)}\left(\tau \right)}{d\tau }+a{y}^{\left(\mu \right)}\left(\tau \right)=b{\left(y^{\left(\mu \right)}\left(\tau \right)\right)}^{\gamma },$$

(10)

where $y^{\left(\mu \right)}\left(\tau \right)$ is the continuous form of $y^{\left(\mu \right)}\left(k\right)$, and $y^{\left(\mu \right)}\left(k\right)={\displaystyle \sum _{i=1}^k}\frac{y^{\left(0\right)}\left(i\right)}{i^{\left[\mu \right]-\mu }}$. For the sake of simplicity, the order r is taken to be $\mu \in (0,1]$. The discrete form of OCFNGBM(1,1) can be obtained by the discretization method as

$$y^{\left(\mu \right)}\left(k\right)-y^{\left(\mu \right)}(k-1)+a{\vartheta }^{\left(\mu \right)}\left(k\right)=b{\left({\vartheta }^{\left(\mu \right)}\left(k\right)\right)}^{\gamma },$$

(11)

where ${\vartheta }^{\left(\mu \right)}\left(k\right)=\omega y^{\left(\mu \right)}\left(k\right)+\left(1-\omega \right)y^{\left(\mu \right)}\left(k-1\right),k\in {\mathbb{N}}_2^n$ and $\omega \in \left[0,1\right]$ is a background value of $y^{\left(\mu \right)}\left(k\right)$. The least squares estimation matrix form of the parameter is as follows:

$$\widehat{\gamma }={[\widehat{a},\widehat{b}]}^T={\left({\mathbf{B}}^T\mathbf{B}\right)}^{-1}{\mathbf{B}}^{\mathbf{T}}\mathbf{Y},$$

(12)

where

$$\mathbf{B}=\left(\begin{array}{cc}-{\vartheta }^{\left(\mu \right)}\left(2\right)& {\left({\vartheta }^{\left(\mu \right)}\left(2\right)\right)}^{\gamma }\\ -{\vartheta }^{\left(\mu \right)}\left(3\right)& {\left({\vartheta }^{\left(\mu \right)}\left(3\right)\right)}^{\gamma }\\ ⋮& ⋮\\ -{\vartheta }^{\left(\mu \right)}\left(n\right)& {\left({\vartheta }^{\left(\mu \right)}\left(n\right)\right)}^{\gamma }\end{array}\right),\mathbf{W}=\left(\begin{array}{c}{y}^{\left(\mu \right)}\left(2\right)-y^{\left(\mu \right)}\left(1\right)\\ y^{\left(\mu \right)}\left(3\right)-y^{\left(\mu \right)}\left(2\right)\\ ⋮\\ y^{\left(\mu \right)}\left(n\right)-y^{\left(\mu \right)}(n-1)\end{array}\right).$$

(13)

Set $x^{\left(0\right)}\left(1\right)=x^{\left(1\right)}\left(1\right)$; the time response function is

$${\widehat{y}}^{\left(\mu \right)}\left(k\right)={\left[\left({\left(y^{\left(\mu \right)}\left(1\right)\right)}^{1-\gamma }-\frac{\widehat{b}}{\widehat{a}}\right)e^{-\widehat{a}(1-\gamma )(k-1)}+\frac{\widehat{b}}{\widehat{a}}\right]}^{\frac{1}{1-\gamma }}.$$

(14)

Using the DCFD, the restored values can be obtained as follows:

$${\widehat{y}}^{\left(0\right)}\left(k\right)=k^{1-\mu }\left({\widehat{y}}^{\left(\mu \right)}\left(k\right)-{\widehat{y}}^{\left(\mu \right)}(k-1)\right).$$

(15)

4.2. Error Measures

The mean absolute error (MAE) is a measure of the overall level of error. The root mean square error (RMSE) and mean square error (MSE) demonstrate the degree of differences between observed and forecasted values. The mean absolute percent error (MAPE) is a measure of the prediction accuracy of a forecasting method in statistics. These are the four effective error measures that are used to assess the effectiveness of forecasting models:

$$\mathrm{MAPE}=\frac{1}{n-1}\sum _{h=2}^n\frac{\left|{\widehat{y}}^{\left(0\right)}\left(h\right)-y^{\left(0\right)}\left(h\right)\right|}{y^{\left(0\right)}\left(h\right)}\times 100\%,$$

(16)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n-1}\sum _{h=2}^n{\left({\widehat{y}}^{\left(0\right)}\left(h\right)-y^{\left(0\right)}\left(h\right)\right)}^2},$$

(17)

$$\mathrm{MSE}=\frac{1}{n-1}\sum _{h=2}^n{\left({\widehat{y}}^{\left(0\right)}\left(h\right)-y^{\left(0\right)}\left(h\right)\right)}^2,$$

(18)

$$\mathrm{MAE}=\frac{1}{n-1}\sum _{h=2}^n\left|{\widehat{y}}^{\left(0\right)}\left(h\right)-y^{\left(0\right)}\left(h\right)\right|.$$

(19)

4.3. PSO Method for Selecting the Optimal Parameters

To obtain the optimal values of the three parameters—the power exponent of the Bernoulli differential equation, the background value coefficient, and the order—we used the mean absolute percent error (MAPE) as the error function in this study. The modeling details of OCFNGBM(1,1) can be easily obtained using PSO.

For optimization, the order of the search range, background value coefficient, and power index is 0–1. The maximum number of iterations is 100, and in each iteration, the population is 100. The following steps should be taken according to the planning model:

Step 1: Calculate the conformable fractional accumulating the generation sequence of an original data sequence.

Step 2: Go to Step 6 if the iterations of the algorithm reach the upper limit, and go to step 3 otherwise.

Step 3: Perform mutation on three individuals randomly.

Step 4: Individuals are updated in accordance with PSO’s update rules.

Step 5: Construct a new population by selecting new individuals. If the algorithm satisfies the termination condition, it goes to step 6; if not, it goes to step 3.

Step 6: Output the optimal individual and obtain the optimal solution after optimization.

The specific planning models involved in the above steps are as follows:

$$\begin{array}{c}\underset{\mu ,\gamma ,\omega }{min}\frac{1}{n}{\displaystyle \sum _{i=1}^n}\left|\frac{{\widehat{y}}^{\left(\mu \right)}\left(i\right)-y^{\left(0\right)}\left(i\right)}{y^{\left(0\right)}\left(i\right)}\right|\times 100\%,\hfill \\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{y}^{\left(\mu \right)}\left(k\right)={\displaystyle \sum _{i=1}^k}\frac{y^{\left(0\right)}\left(i\right)}{i^{\left[\mu \right]-\mu }}\hfill \\ \mathbf{B}=\left(\begin{array}{cc}-{\vartheta }^{\left(\mu \right)}\left(2\right)& {\left({\vartheta }^{\left(\mu \right)}\left(2\right)\right)}^{\gamma }\\ -{\vartheta }^{\left(\mu \right)}\left(3\right)& {\left({\vartheta }^{\left(\mu \right)}\left(3\right)\right)}^{\gamma }\\ ⋮& ⋮\\ -{\vartheta }^{\left(\mu \right)}\left(n\right)& {\left({\vartheta }^{\left(\mu \right)}\left(n\right)\right)}^{\gamma }\end{array}\right),\mathbf{W}=\left(\begin{array}{c}{y}^{\left(\mu \right)}\left(2\right)-y^{\left(\mu \right)}\left(1\right)\\ y^{\left(\mu \right)}\left(3\right)-y^{\left(\mu \right)}\left(2\right)\\ ⋮\\ y^{\left(\mu \right)}\left(n\right)-y^{\left(\mu \right)}(n-1)\end{array}\right),\hfill \\ {\widehat{y}}^{\left(\mu \right)}\left(k\right)={\left[\left({\left(y^{\left(\mu \right)}\left(1\right)\right)}^{1-\gamma }-\frac{\widehat{b}}{\widehat{a}}\right)e^{-\widehat{a}(1-\gamma )(k-1)}+\frac{\widehat{b}}{\widehat{a}}\right]}^{\frac{1}{1-\gamma }},\hfill \\ {\widehat{y}}^{\left(0\right)}\left(k\right)=k^{1-\mu }\left({\widehat{y}}^{\left(\mu \right)}\left(k\right)-{\widehat{y}}^{\left(\mu \right)}(k-1)\right).\hfill \end{array}\right.\hfill \end{array}$$

(20)

The procedure of the OCFNGBM(1,1) model can be stated as a flowchart depicted in Figure 1.

5. Development Scale Prediction of JSSE

Using the OCFNGBM(1,1) model, this section forecasts the number of junior high schools (NJHS) in four Chinese cities (Jiangxi, Henan, Sichuan, Guizhou); the data of these four cases are from the official website of the National Bureau of statistics of China, which can be downloaded at http://www.stats.gov.cn/. We evaluate the feasibility and effectiveness of our model with the data of NJHS, where the data from 2011 to 2017 are used to build the prediction models, and the data from 2018 to 2021 are used to validate the modeling accuracy. The results are shown in

Table 1. Results of MSE, MAE, RMSE and MAPE for different forecasting models (the best result for each metric is bolded).

	Indices	CFNGBM(1,1) e	FHGM(1,1) d	NGBM(1,1) c	FGM(1,1) b	OCFNGBM(1,1) a
Jiangxi	MAPE	1.5691	1.5517	1.5682	2.1804	0.85103
	MSE	1492.6	1457.1	1487.4	2837.4	497.89
	MAE	34.495	34.114	34.476	47.926	18.736
	RMSE	38.634	38.171	38.567	53.267	22.313
Henan	MAPE	3.5564	3.4622	3.5163	4.1311	3.2397
	MSE	39546	37636	38676	50483	33879
	MAE	166.72	162.32	164.84	193.46	151.97
	RMSE	198.86	194	196.66	224.69	184.06
Sichuan	MAPE	3.3372	3.0408	3.2054	3.2684	2.4364
	MSE	17072	14426	15860	16494	9887.8
	MAE	122.31	111.56	117.54	120.06	89.529
	RMSE	130.66	120.11	125.94	128.43	99.437
Guizhou	MAPE	3.8755	3.8579	3.8755	3.8813	3.5226
	MSE	9320.8	9230	9320.8	9350.4	7580.2
	MAE	78.078	77.723	78.078	78.193	70.963
	RMSE	96.544	96.073	96.544	96.697	87.064

a—Jiangxi’s power index, order, and background value coefficient for OCFNGBM(1,1) are 0, 1, and 0.23897, respectively. The power index, order, and background value coefficient for OCFNGBM(1,1) in Henan are 0, 0.97402, and 0.58214. The power index, order, and background value coefficient for OCFNGBM(1,1) are 0, 0.9206, and 0.56284, respectively, in Sichuan. The power index, order, and background value coefficient of OCFNGBM(1,1) in Guizhou are 0, 0.95862, and 0.42885, respectively; b—In Jiangxi, Henan, Sichuan and Guizhou, the fractional orders of FGM(1,1) are 0.038261, 0.078593, 0.19111, and 0.95367, respectively; c—In Jiangxi, Henan, Sichuan and Guizhou, the background value coefficients of NGBM(1,1) are 0.01238, 0.024849, 0.07347, and 0.042735, respectively; d—In Jiangxi, Henan, Sichuan and Guizhou, the fractional orders of FHGM(1,1) are 0.98827, 0.97663, 0.93008, and 0.95733, respectively; e—Jiangxi’s power index and order for OCFNGBM(1,1) are 0 and 0.98437, respectively. The power index and order for OCFNGBM(1,1) in Henan are 0 and 0.96727, respectively. The power index and order for OCFNGBM(1,1) are 0 and 0.90182, respectively, in Sichuan. The power index, and order of OCFNGBM(1,1) in Guizhou are 0.042735 and 1, respectively.

. The newly proposed model obtained the most accurate predictions based on the four regions’ data. Consequently, the newly proposed and improved OCFNGBM(1,1) can effectively predict the NJHS due to its predictive accuracy and generalization abilities. As can be seen from the table, OCFNGBM(1,1) has better prediction performance compared to CFNGBM(1,1), which proves that our improvement strategy has worked; for example, for the data of the Jiangxi region, the MAPE value of the improved model is 0.85103%, while the MAPE value of CFNGBM(1,1) is 1.5691%. Specifically, OCFNGBM(1,1) is a nonlinear model, and as a result, its prediction performance is generally better than that of a linear model. The FGM(1,1) and FHGM(1,1) models in this study are linear, and their prediction performance on the data from the four regions is lower than that of the model proposed in this paper. It is also worth mentioning that nonlinear models with more parameters are also prone to overfitting, which results in poor prediction accuracy.

The search process for four intelligent optimization algorithms on four regions (Jiangsu, Henan, Sichuan, and Guizhou) is shown in Figure 2.

Based on the above analysis, it can be seen that OCFNGBM(1,1) obtains a better prediction performance on the dataset of the four regions, so we use this model to predict the values for the next three years. According to

Table 2. The number of junior high schools is predicted using a new model in four regions of China (Jiangxi, Henan, Sichuan, and Guizhou).

Year	Jiangxi	Henan	Sichuan	Guizhou
2022	2190.4	4429.6	3391.1	1835.3
2023	2199	4406.3	3317.1	1793.4
2024	2207.7	4382.4	3243.2	1752.1

, the NJHS in the three regions, Henan, Sichuan, and Guizhou, is declining, while that in Jiangxi is rising. Analyzing different trends allows us to allocate resources rationally. More educational resources should be added to prevent a shortage of educational resources in Jiangxi.

6. Discussion

It has been demonstrated in the literature that fractional-order calculus can be used to describe complex laws in complex systems, and it is a generalization of integer-order calculus. It has also been used in recent years to illustrate complex phenomena in natural sciences, successfully solving complex scientific problems that cannot be solved using integer-order derivatives. Fractional-order differential equations are based on fractional-order derivatives, and they play an essential role in natural science and are often used to represent complex laws. However, fractional-order calculus has been used less frequently in social sciences than in natural science. As a result of the perturbation of complex systems by external factors, many problems in social sciences can also be abstracted into complex systems. The change in the educational system can be seen as the result of the evolution of the educational system in the time dimension, and the time series generated by the educational system can be viewed as a map of the educational system in a one-dimensional space.

Education data are usually small, and many of the existing traditional models, such as machine learning, cannot get accurate prediction results. The grey system model is essential for solving minor sample modeling problems. Numerous reports in the literature have demonstrated that grey system models can adequately handle prediction, correlation analysis, and decision-making in small-sample environments. Recent years have seen increased development of grey system models, solutions to many theoretical problems, and the removal of many technical bottlenecks due to the active contributions of many scholars. Many scholars have expressed concern about grey prediction among grey system models. Over the years, linear models have evolved into nonlinear models, integer-order models have evolved into fractional-order models, and univariate models have evolved into multivariate models. All of these strategies have significantly improved prediction accuracy. As grey prediction technology develops, fractional-order gray system models are becoming increasingly capable of handling complex nonlinear problems. They are more likely to achieve better prediction accuracy by continually adjusting the order. Despite this, there is still room for improvement in the fractional-order gray system model. For example, changing the differential equations of the grey model into different equations will introduce specific errors. Furthermore, the background value was not optimized, which will introduce some errors.

Due to the influence of economic policies on issues such as the scale of junior high school development, it is often difficult to directly capture the intrinsic patterns using accurate pre-capture inherent cases and a small amount of data. Furthermore, internal marks are often nonlinear and complex, making them difficult to predict accurately. This paper aims to analyze the classical fractional-order nonlinear grey forecasting model and further improve its forecasting accuracy. We described the basic expressions of the novel model, the parameter estimation method, and the optimization method for hyperparameters and constructed a scheme for optimizing the background value coefficients. Through the experiments, the new model showed a good prediction effect and can thus be used to predict the development scale of junior high schools in the four chosen regions of China. The proposed model and optimization scheme will likely provide policy suggestions to the relevant departments responsible for making decisions. Our model has more freedom than a classical grey forecasting model because it can dynamically change the nonlinear parameters, the cumulative order, and the background value coefficients. It should be noted that although our model has more freedom, which will further improve the modeling range of the gray model, this can lead to overfitting, where the model works well during the modeling phase but the accuracy of the model is affected during the prediction phase. In the next stage of our research, we will study how to avoid overfitting in a small sample environment, such as adding regularization terms during modeling to improve the generalization ability of the model so as to further improve the prediction accuracy of the model.

7. Conclusions

To improve the prediction accuracy of the grey system, a new grey prediction model is proposed in this paper, which is applied to the prediction of the development scale of JSSE. Our study proposes a new nonlinear model for predicting the scale of JSSE development in China with a high level of accuracy. By using fractional-order cumulative generation technology instead of integer-order cumulative technology, we were able to deduce the fractional-order form of the grey model. Additionally, the inheritance and memory of the fractional-order model can improve the predictability of JSSE’s development scale. Using a parameter optimization scheme, we were able to obtain the optimal cumulative order. It is of great significance to China’s education planning to be able to predict the data of JSSE in the next few years. The analysis above indicates that the newly constructed model has the potential to be used to predict some practical problems in the education system. Nevertheless, our model has more freedom and may be overfitted. In future, we will investigate how to avoid overfitting in a small sample environment.