#### Abstract

To further promote the development of the grey system theory, this paper develops a novel conformable fractional-order grey Bernoulli model with a time-delay effect, namely, the CFTDNGBM (1, 1) model. In addition, the arithmetic optimization algorithm (AOA) is incorporated into the system of the model to solve the hyperparameters existing in the model. Compared with the previous grey prediction models, the CFTDNGBM (1, 1) model with a conformable fractional-order accumulation operation (CFAO), time-delay factor, and Bernoulli parameter has stronger compatibility in structure. The proposed model and its nine competitive models with excellent performance are used to predict and analyze the consumption level and per capita consumption expenditure of rural residents in China to verify the feasibility of the proposed method. The case results show that in both cases, the seven descriptive indicators of the CFTDNGBM (1, 1) model are higher than those of its competing models. Therefore, the CFTDNGBM (1, 1) model has a certain application value.

#### 1. Introduction

In recent years, due to the development of deep learning, people pay more attention to the deep learning models based on large sample data sets and ignore the small sample learning method, which is more serious in time series analysis. Prediction is the most important part of time series analysis. An accurate prediction can enable decision-makers to make correct decisions based on the prediction results. For time series with small data sets, the deep learning model based on large samples is obviously no longer applicable. Based on this, we turn our attention to the grey prediction model which can realize the small sample modeling model.

The grey system model, as a small sample learning method, has been praised for its simple modeling mechanism and high predictive performance [1–5]. With the development of society, many derivative models and optimization measures based on the most basic GM (1, 1) model have been proposed. These methods can be roughly divided into three parts, namely, structural optimization, hybrid optimization, and optimization based on data enhancement methods. Table 1 shows some important work.

From Table 1, we can see that the structural optimization of the grey model mainly focuses on the optimization of the grey action of the model. Scholars hope to enhance the adaptive performance of the model by using functions with strong fitting performance to replace the traditional constant grey action. This class of models can be further divided into two parts, namely, polynomial function-based expansion models and time-delay term-based expansion models. Up to now, the polynomial function-based expansion model has gradually converged to perfection because the PTGM (1, 1, *α*) model proposed by Liu et al. is a unified expression of most existing grey prediction models. In contrast, the time-delay term-based extended model has received less attention because of its more complex structure, which has led to its slower progress. It is important to note that extended models based on time-delay terms are not time-delay models in the traditional sense; they are simply prediction algorithms that use a so-called time-delay polynomial instead of the grey action of the GM (1,1) model. Although the extended models based on time delay have received little attention, some existing studies show that the extended models based on time delay usually outperform the extended models based on polynomial functions. Therefore, the time-delay term-based extended models seem to be more desirable, both in terms of refining the grey prediction domain and building efficient prediction models for predictive analysis. In addition, most of the grey models based on structural optimization have corresponding discrete forms, which appear to address the shortcomings of traditional grey forecasting models that do not satisfy unbiasedness. Therefore, their discrete forms are more reasonable and reliable than the traditional grey prediction models based on ordinary differential equations.

Hybrid optimization focuses on combining grey forecasting algorithms with other forecasting models or correction measures, with the aim of developing an efficient new forecasting algorithm by combining the advantages of two or more methods. However, such optimization methods are heavily influenced by the performance of the model used for the combination. Therefore, the establishment of an efficient grey forecasting model is still the top priority. Optimization based on data enhancement methods mainly uses various fractional-order accumulation operations to enhance the initial data and then build the model, which usually has better prediction performance than traditional models due to the strong adaptive performance of fractional-order accumulation operations. However, the method of determining the hyperparameters in such models is still open to discussion.

According to the previous description, we can know that the attention of the time delay grey prediction model is less than that of other models, which may lead to the unbalanced development of this field. In addition, data enhancement and discretization measures can effectively improve the prediction performance of the model. To promote the development of grey system theory, this paper develops a new grey prediction model with a time-delay polynomial based on discrete operations, the conformable fractional accumulation operator (CFAO), and the Bernoulli operator. It can be seen that the constructed model combines all the advantages of the existing optimization methods, which makes it have stronger fitting performance than the earlier grey prediction models.

In particular, to reduce the complexity of the model, we set the conformable fractional accumulation operator and the time-delay operator to the same hyperparameter. In addition, the arithmetic optimization algorithm (AOA) with a relatively simple mechanism is used to solve the hyperparameters existing in the model. The specific contributions of this paper are as follows:(1)We establish a discrete time delay polynomial grey prediction model with the Bernoulli operator(2)The AOA is used to solve the planning problem based on the proposed model(3)The proposed model is used to study the development of China’s rural regional economies

The other parts of this paper are arranged as follows: Section 2 introduces the proposed model, including its modeling steps and solving methods. Section 3 introduces the application of the proposed model in rural regional economies. Section 4 is the summary of this article.

#### 2. Methods

In this section, we will develop a new time-delay polynomial grey prediction model based on the conformable fractional accumulation operator (CFAO) [28], the NGM (1,1,*k*,*c*) [6], and the FTDGM model [14] and introduce its modeling mechanism and solution method in detail.

##### 2.1. Conformable Fractional Accumulation Operator-CFAO

Let be a time series composed of original data, then its conformable fractional accumulation generation sequence is as follows:where means forward rounding, such as [28].

According to [28], the inverse accumulative generating operator of can be represented by the following way:where . In particular, when , the CFAO degenerates into the 1-order accumulation operator (1-AGO).

##### 2.2. Conformable Fractional Grey Bernoulli Model with Time-Delay Polynomial

If is the conformable fractional accumulation sequence of, the following nonlinear ordinary differential equationis called the whitening differential equation of the conformable fractional grey Bernoulli model with a time-delay polynomial (CFTNGBM (1,1) model), where is the time-delay polynomial and is a nonlinear Bernoulli parameter since the general solution of the equation does not exist when , .

By multiplying both sides of Equation (3) by , we can get

We set , then equation (4) converts to

namely,where

The integral form of Equation (6) on an interval is used to estimate the least square parameters of the model, namely,where

Taking as the initial condition to solve Equation (6), we can get

Furthermore, we can obtain the time response function of the model, namely,where .

The output values of this model can be obtained by using Equation (2), namely,

##### 2.3. Discrete CFTNGBM (1, 1) Model Satisfying Unbiasedness

In this section, we will discretize the CFTNGBM (1, 1) model to make it unbiased. First, we conduct an unbiased analysis of the CFTNGBM (1, 1) model, as shown in the following paragraph.

If the proposed model satisfies unbiasedness, then we have the following relationship:

Obviously, we can see from Equation (9) that the previous relationship is not tenable, so the proposed model does not satisfy the unbiasedness. Based on this, we will discretize the model to make it unbiased.

Based on the first-order backward difference, Equation (6) can be expressed as follows:that is,

We set then Equation (14) converts to the following:

which is called the expression of the discrete CFTNGBM (1,1) model-CFTDNGBM (1,1).

To minimize the estimation error, we present the following unconstrained programming problem:where

and are the actual values and estimated values, respectively. According to the extreme value existence condition, we have , namely,

Therefore, we know that the least square estimate of the model is

When , Equation (15) converts to the following:

When , we have

When , we can get

According to this law, we can obtain the recursive formula of Equation (15), namely,then we can use to restore Equation (22) to obtain the time response function of the CFTDNGBM (1,1) model, namely,

According to Equation (2), we can know that the prediction formula of the CFTDNGBM (1,1) model is as follows:

Theorem 1. *CFTDNGBM (1,1) model satisfies unbiasedness.*

*Proof. *Suppose there is a time series satisfying Equation (22), namely,where are the given parameters, then we havewhich means that satisfies the proposed model. We can easily give the matrix form of equation (26), namely, , where the definitions of are similar to equation (16). Furthermore, we can get . Obviously, we have , which means that our given parameters are consistent with the least squares estimate of the parameters of the model.

Based on , we havewhich means that the model is unbiased.

##### 2.4. The Solution Method of the Model

Obviously, due to the existence of hyperparameters and , it is difficult for us to solve the CFTDNGBM (1, 1) model with traditional methods. Considering that the purpose of the model is to minimize the estimation error, here we take MAPE as the objective function and the modeling mechanism of the model as constraints to establish the following planning model.

In the field of grey systems, swarm intelligence optimization algorithms are usually used to solve such complex planning problems. Here, we choose an arithmetic optimization algorithm (AOA) with faster search speed to solve this planning problem to obtain the hyperparameters of the model and the corresponding prediction results. The AOA is an efficient swarm intelligence optimization algorithm developed by Abailigah et al. based on arithmetic operation symbols [29]. It is mainly composed of three parts: the initialization phase, the exploration phase, and the exploitation phase.

###### 2.4.1. Initialization Phase

At this stage, the AOA algorithm iterates based on a matrix composed of a group of candidates, and the optimal candidate solution in each iteration is taken as the approximate optimal solution of the current stage, where

In addition to constructing matrix , determining the search stage is also an important step in the AOA algorithm. As described by Abualigah et al., the math optimizer accelerated function (MOAF) is used for the following search phase, which can be computed by, i.e.,where represents the function value at the ith iteration, is the value between 1 and the maximum number of iterations , which represents the current iteration, and represent the maximum and minimum values of the acceleration function.

###### 2.4.2. Exploration Phase

The exploration stage is the most important part of AOA. In AOA, exploration operators randomly explore the search area over several areas and search for better solutions based on the two main search strategies (division (D) and multiplication (M) search strategy). This stage is constrained by the MOAF for the condition of ( is a random number). The first operator in this phase, D, is constrained by , and the second operator (M) is ignored until D completes its task. If conditions change, then the second operator participates in the task. This process can be expressed as follows:where represents the ith solution in the next iteration, describes the jth position of the ith solution at the current stage, and represents the jth position of the best solution obtained so far. is a small integer, and represent the upper and lower bounds of the *j*th position, respectively, and is a parameter used to adjust the search process. represents the function value of the *t*th iteration, and its mathematical expression is as follows:where represents the current iteration and represents the maximum number of iterations. is a sensitive parameter that defines the exploitation precision of the iterative process.

###### 2.4.3. Exploitation Phase

This subsection will describe the exploitation phase of the AOA. This stage is constrained by the value of the MOAF. In AOA, the exploitation operator (subtraction (S) and addition (A)) of AOA deeply explores the search area in several dense regions and searches for better solutions based on two main search strategies. The mathematical expressions for this stage are shown as follows:

At this stage, when , the first operator (*S*) starts working, otherwise another operator (*A*) will start working. Different from the previous stage, developers try to avoid falling into the local search area, which helps exploration search strategies to find the optimal solution and maintain the diversity of candidate solutions. In addition, the parameters of are carefully designed to produce a random value in each iteration, which allows operators to keep exploring not only during the first iteration but also during the last.

##### 2.5. The Calculation Method of the Model

In order to facilitate readers to understand the calculation steps of the model, here we use a case (China’s oil consumption (Exajoules)) to explain. The original data are the oil consumption from 2002 to 2020 obtained from the World Energy Statistics Review-2021 (https://www.bp.com/statisticalreview), which can be expressed as follows:where the data from 2002 to 2016 are used as the training set and the data from 2017 to 2020 are used to test the performance of the CFTDNGBM (1,1) model.

Firstly, we input the original sequence into the programming model in Subsection 2.4 and obtain the solution () of this programming model through the AOA algorithm.

According to equation (1), we can obtain a conformable fractional accumulation generation sequence of the original time series, that is,then we can construct the matrixused to estimate the least squares parameters.

Furthermore, we have

By inputting the obtained least squares parameters into equation (23), we can obtain the time response sequence

Finally, according to , we can get the final prediction results of the model, namely,

#### 3. Application

This section describes the application of the proposed model and its competitors in the rural regional economy. To facilitate understanding, the application process of the proposed model in the cases is plotted in Figure 1.

##### 3.1. Data Collection, Preprocessing, Description of the Models, and Evaluation Indicators

The accurate prediction of the rural residents’ consumption level and rural per capita consumption index is helpful for the government to formulate corresponding policies for short-term guidance and regulation of rural residents’ consumption according to the forecast results so as to improve rural residents’ consumption levels and regulate the overall operation of the macroeconomy. Therefore, it is of great importance to study the future development trends of RRCL and RRPCCEX. The two sets of data used in this article are the consumption level of rural residents (RRCL) from 2002 to 2020 and the per capita consumption expenditure of rural residents (RRPCCEX) from 2003 to 2021. These two sets of data are from the National Bureau of Statistics of China (https://www.stats.gov.cn/,2020), and the details are shown in Table 2. In particular, the consumption level data of rural residents from 2002 to 2014 and the per capita consumption index of rural residents from 2003 to 2015 are used as training sets, and the data from 2015 to 2020 and 2016 to 2021 are used to test the performance of the model.

According to [30], one of the reasons why the grey prediction model is ill-conditioned is that the original data dimension is too large and multiplicative transformations can alleviate the ill-condition of the system to a certain extent. Therefore, this paper selects multiplicative transformation as the preprocessing operation of the model, which can be expressed as follows:where is an integer, and its function is mainly to reduce the dimensions of the original data to single digits.

It should be mentioned that if there are missing values in the sequence, then we need to carry out nonequidistant conformable fractional accumulation processing on the data and establish a nonequidistant CFTDNGBM (1, 1) model. The construction process of the nonequidistant conformable fractional order accumulation operation is similar to the nonequidistant fractional order accumulation operation proposed in [31]. For space reasons, it is not constructed here.

In addition to the model proposed in this paper, the models used for the comparative analysis also include other nine different types of discrete grey models with hyperparameters, which are LSSVR [32], CFDGM (1, 1) [33], FDGM (1, 1) [34], FNDGM (1, 1, *k*, *c*) [35], FDGMP (1, 1, *N*) [36], FDGM (1, 1, *t*^{α}) [37], FGDGMP (1, 1, *N*, *α*) [38], WDGM (1, 1) [39], and the FTDGM (1, 1) model [40]. The reasons why these models are selected for comparative analysis are that they all belong to excellent prediction models and all contain hyperparameters, which make them have strong competitive ability. It is worth mentioning that all models are implemented in the MATLAB 2019a environment.

To evaluate the performance of the models, we collected seven metrics used to quantify the performance of predictive models, which arewhere *l* is the number of data used to build the proposed model.

##### 3.2. Comparison with Other Optimization Algorithms

In this subsection, we will establish CFTDNGBM (1, 1) models based on different optimization algorithms and compare their prediction results with those of the CFTDNGBM (1, 1) model based on AOA. The selected optimization algorithms for comparison include WOA (Whale Optimization Algorithm) [41], MPA (Marine Predators Algorithm) [42], GWO (Grey Wolf Optimization) [43], GOA (Grasshopper Optimization Algorithm) [44], EOA (Equilibrium Optimizer Algorithm) [45], and ALO (Ant lion optimizer) [46]. The reason why these algorithms are chosen as comparison models is that they have been used to solve grey prediction models and have reference significance.

Table 3 shows the evaluation metrics of the CFTDNGBM (1, 1) model based on the seven intelligent optimization algorithms in the two cases. From Table 3, we can see that in the training phase of the RRCL case, the performance of the seven algorithms is not very different, among which the performance of EOA is slightly better than the other six algorithms, and the performance of the AOA, ALO, and WOA algorithms in the testing phase is much higher than the other algorithms. Taking MAPE as an example, the MAPE of AOA, ALO, and WOA in the test phase are 2.3113, 2.3120, and 2.3508, respectively, which are much higher than the MAPE of the other four algorithms. It seems a strange phenomenon that the performance gap of the algorithm in the training set is small, but the performance gap in the test set is very large. This phenomenon shows that in some cases, when the loss function of the model based on some intelligent algorithms in the training set reaches a certain critical point, the performance of the model in the test set will become worse as the performance in the training set becomes better. Therefore, the best algorithm should be determined by multiple comparisons when selecting the intelligent algorithm for solving the model. In the case of RRPCCEX, although the fitting performance of MPA is better than that of other algorithms, the performance of AOA in the test set is better than that of the other six algorithms and its evaluation index in the training set is very close to that of MPA. Therefore, AOA outperforms the other algorithms in these two cases. By the way, the hyperparameter outputs by the seven algorithms in the two cases are shown in the appendix.

##### 3.3. Comparison with Other Forecasting Models

The evaluation metrics of the CFTDNGBM (1, 1) model and the other nine forecasting models mentioned in Subsection 3.1 in the two cases are shown in Table 4. The modeling details of the ten models are shown in Tables 5–10. Figure 2 reflects the convergence of the AOA-based CFTDNGBM (1, 1) model.

From Table 4, we can see that in the training stage of the RRCL case, the CFTDNGBM model has the best performance, while the LSSVR model has the worst performance. In the test stage, the seven indicators of the CFTDNGBM model are the smallest among the ten models. Although the performance of the CFDGM (1, 1) model in the test set is close to that of the CFTDNGBM model, its fitting performance is lower than that of the CFTDNGBM model. Therefore, the CFTDNGBM model performs best in the RRCL case. Different from the RRCL case, in the RRPCCEX case, the CFTDNGBM model significantly outperforms the other algorithms in both the training and test sets. For example, the MAPE of the CFTDNGBM model in the test set is 3.5174, while the smallest MAPE among the other nine models is 6.4070. In conclusion, the CFTDNGBM model outperforms the other nine models in two cases.

#### 4. Conclusion

In this study, we propose a CFTDNGBM (1,1) model with all the characteristics of a time-delay polynomial, a conformable fractional accumulation operator, and a Bernoulli operator. This model has a simpler modeling mechanism than traditional time-delay grey prediction models. In addition, a novel AOA algorithm is introduced to solve the hyperparameters of the proposed model. In order to verify the validity of the model and expand the application scope of the grey system model, the CFTDNGBM (1,1) model and nine competitive models are used to study the regional economies in rural China. Numerical results show that in both cases, the seven quantitative indexes of the CFTDNGBM (1,1) model are superior to those of its competing models, which shows the superiority of the proposed model. It should be noted that the model proposed in this paper is not applicable to a large sample setting. According to the description in the literature [30], the pathological nature caused by large magnitude gaps in the coefficient matrix can make the prediction results of the model inaccurate, while the gaps in the elements of the coefficient matrix of the proposed model increase rapidly with the increase in the sample size. Therefore, the model proposed in this paper is suitable for a small-sample nonlinear time series.

The limitation of this study is that the randomness of the algorithm is not taken into account. Since the proposed model has two hyperparameters, the prediction results may be unstable. In the future, we will continue to improve this aspect of work in order to enrich the application of the swarm intelligence optimization algorithm in grey system theory.

#### Abbreviation

AOA: | Arithmetic optimization algorithm |

CFTDNGBM (1, 1): | Conformable fractional-order grey Bernoulli model with time-delay effect |

CFAO: | Conformable fractional-order accumulation operation |

GM (1, 1): | The most basic grey prediction model |

NGM (1, 1, k, c): | Nonhomogeneous grey prediction model |

GM (1, 1, t^{α}): | Grey prediction model with time power term |

GMP (1, 1, N): | Grey prediction model with polynomial |

PTGM (1, 1, α): | Generalized grey prediction model with time power term |

FTDGM: | Fractional grey prediction model with a time delay term |

GM (1, N): | Multivariate grey prediction model |

DGM (1, 1): | Discrete grey model |

VAR: | Vector autoregressive |

DEA: | Data envelopment analysis |

NGBM (1, 1, k, c): | Nonlinear nonhomogeneous grey prediction model |

NGBM (1, 1, N): | Nonlinear grey prediction model with polynomial |

1-AGO: | 1-order accumulation operator |

NGBM (1, 1): | Nonlinear grey Bernoulli model |

LSSVM: | Least squares support vector machine |

CFNGM (1, 1, k, ):c | Conformable fractional nonhomogeneous grey prediction model |

CFGM (1, 1): | Conformable fractional grey model |

MOAF: | Math optimizer accelerated function |

RRCL: | Consumption level of rural residents |

RRPCCEX: | Per capita consumption expenditure of rural residents |

FDGM (1, 1): | Fractional discrete GM (1, 1) |

FNDGM (1, 1, k, c): | Fractional discrete nonhomogeneous grey prediction model |

FDGMP (1, 1, N): | Fractional discrete grey prediction model with polynomial |

FDGM (1, 1, t^{α}): | Fractional discrete grey prediction model with time power term |

FGDGMP (1, 1, N, α): | Fractional discrete generalized grey prediction model with time power term |

WDGM (1, 1): | Discrete grey model with the weighted accumulation |

FTDGM (1, 1): | Fractional-order accumulative linear time-varying parameters discrete grey forecasting model |

CFDGM (1, 1): | Conformable fractional discrete grey model |

WOA: | Whale optimization algorithm |

MPA: | Marine predators algorithm |

GWO: | Grey wolf optimization |

GOA: | Grasshopper optimization algorithm |

EOA: | Equilibrium optimizer algorithm |

ALO: | Ant lion optimizer |

AOA: | Arithmetic optimization algorithm |

#### Data Availability

The data used in this article come from the National Bureau of Statistics of China.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

Ran Wang conducted writing of the original draft, methodology, and experimental data validation. Hongmei Du conducted conceptualization, methodology, supervision, writing, review, and editing. Qinwen Yang conducted investigation, software, and data collection. Hui Liu performed investigation, visualization, and collected resources.

#### Acknowledgments

This work was supported by the Natural Social Science Found of China (No. 20BJY046).