A Novel Multivariable MGM (1, m) Direct Prediction Model and Its Optimization

With regard to the traditional MGM (1, m) model having jumping error in solving process, an MGM (1, m) direct prediction model (denoted as DMGM (1, m) model) is proposed and its solution method is put forward at first. Second, considering the inherent time development trend of system behavior sequence is ignored in the DMGM (1, m) model, the DMGM (1, m) model is optimized by introducing a time polynomial term, and the optimized model can be abbreviated as TPDMGM (1, m, 
 
 φ
 
 ) model. Subsequently, it is theoretically proved that the TPDMGM (1, m, 
 
 φ
 
 ) model can achieve mutual transformation with the traditional MGM (1, m) model and the DMGM (1, m) model by adjusting the parameter values. Finally, two case studies about predicting the deformation of foundation pit and Henan’s vehicle ownership have been carried out to validate the effectiveness of proposed models. Meanwhile, the MGM (1, m) model and Verhulst model are established for comparison. Results show that the modeling performance of four models from superior to inferior is ranked as TPDMGM (1, m, 
 
 φ
 
 ) model, DMGM (1, m) model, MGM (1, m) model, and Verhulst model, which on the one hand testifies the correctness of defect analysis of the MGM (1, m) model and on the other hand verifies that the TPDMGM (1, m, 
 
 φ
 
 ) model has advantages in predicting the system variables with mutual relation, mutual restriction, and time development trend characteristic.


Introduction
Grey system theory is an interdisciplinary scientific area that was first proposed in the 1980s by Professor Deng [1]. Since then, the grey system theory has become quite popular in dealing with system analysis, prediction, decision making, and control with partially known information. e grey prediction model is one of the most important components of grey system theory, which can realize the prediction by excavating the internal rules of accumulated sequence [2]. Nowadays, the grey prediction model has been widely used in various fields such as health care [3], agriculture [4], environment [5], and energy [6]. e GM (1, 1) model is the foundation and core of grey prediction models, and it is the simplest and most widely used single variable grey prediction model. Over the past four decades, scholars made a lot of efforts to improve the modeling performance of the GM (1, 1) model, which mainly includes three aspects: parameters optimization [7], modeling range expansion [8], and modeling structure improvement [9]. ese achievements have improved the prediction accuracy of the GM (1, 1) model to a certain extent and enriched the research system of grey prediction theory. By observing the basic forms of the GM (1, 1) model and its extended models, it is known that they are only applicable to the system simulation and prediction with one variable. However, the actual economic and social system often contains multiple variables as they restrict and interact with each other [10]. In order to solve this kind of system prediction problem more accurately, Zhai et al. [10] proposed the MGM (1, m) model based on the GM (1, 1) model. e multivariable MGM (1, m) model is a natural extension of the GM (1, 1) model under m-variables, which is presented in the form of m-variable and first-order ordinary differential equations [10]. Although the MGM (1, m) model has been successfully applied in many cases, there may still be distortions in modeling results. erefore, scholars have conducted many relevant studies to optimize the MGM (1, m) model. In the aspect of modeling rang expansion, due to the continuous improvement of the level of science and technology and the limited cognitive ability of human beings, system information is only known the range instead of the exact value. Hence, Xiong et al. [5] proposed an MGM (1, m) model for interval grey numbers and applied it to air pollution prediction. In terms of parameter optimization, Dai et al. [11] improved the background value by using a nonhomogeneous exponential function to fit the first-order accumulated sequence. In order to reflect the new information priority principle, Wu et al. [12] replaced the traditional first-order accumulation generation operator with fractional-order accumulation generation operator in the MGM (1, m) model and used the particle swarm algorithm to calculate the optimal order. Based on these achievements, the modeling precision of the MGM (1, m) model in practical application is improved. rough observing the basic forms of these optimized models, it is found that they are mainly suitable for the original data matrix with quasi-exponential law, whereas the modeling performance for the general data sequence is not ideal. For this reason, Wang and Zhao [13] proposed a nonhomogeneous multivariate MGM (1, m) prediction model, which improved the consistency between modeling data and model structure. In view of the nonlinear relationship between the variables in system, Xiong et al. [14] constructed a nonlinear multivariable MGM (1, m) prediction model and studied the time lag characteristics of system behaviors. e works presented above studied the modeling rang expansion, parameter optimization, and modeling structure improvement of the traditional MGM (1, m) model, and significant findings were obtained that improved the performance of the MGM (1, m) model. However, these optimized MGM (1, m) models still have some defects in two aspects. (1) Parameter defect: there is a jumping error between parameter estimation and parameter application in the MGM (1, m) model. (2) Structure defect: the model structure does not match the characteristics of the system variables. e traditional MGM (1, m) model and its optimization models only consider that there is an interaction and mutual restrict relationship between each variable, while ignoring the inherent time development trend of each variable. For example, in the research of foundation pit deformation prediction, settlement observation points are not only interrelated and interact with each other but also gradually tend to the saturation state as time changes. is practical application background is not reflected in the existing multivariable MGM (1, m) prediction model.
In this paper, the defects of the traditional MGM (1, m) model are systematically studied; based on this, the definition equation of the DMGM (1, m) model is proposed and the recursive formula of the DMGM (1, m) model is given. On the basis of the DMGM (1, m) model, taking into account the inherent time development trend of system behaviors, an optimized TPDMGM (1, m, φ) model is constructed by introducing a time polynomial term. Finally, the validity of proposed models is verified by predicting the deformation of foundation pit and vehicle ownership in Henan province. e other parts of this research are organized as follows. In Section 2, basic definitions of the MGM (1, m) model and its defects are presented. Section 3 is the detailed modeling procedure of the DMGM (1, m) model. e optimized TPDMGM (1, m, φ) model is put forward in Section 4. Section 5 gives two example analyses about predicting the deformation of foundation pit and vehicle ownership in Henan province to illustrate the practicality and effectiveness of proposed models. e last section is devoted to conclusions.

The MGM (1, m) Model and Its Defect Analysis
e MGM (1, m) model is a grey prediction model with m variables and first-order differential equations [10]. Compared with the traditional grey prediction models with a single variable, such as GM (1, 1) model and DGM (1, 1) model, the MGM (1, m) model takes into account the interrelationship and mutual restriction between variables on the system. erefore, it is more useful for system prediction than those for grey prediction models with one variable. e definition of the MGM (1, m) model is as follows.
For convenience, equation (1) can be written in the matrix, that is, Further, by discretizing equation (1), the discrete form of the MGM (1, m) model is en, the matrix form of equation (3) can be written as equation (4), which is Theorem 1 (see [10]). Assume that y (0) i , y (1) i , and z (1) i are defined as Definition 1 and p i � (a i1 , a i2 , . . . , a im , b i ) T , i � 1, 2, . . . , m is parameter vector, then the identification values of parameters vector can be obtained by the least square method, namely, Theorem 2. Let the identification values of parameters vector be described in eorem 1; assume that the initial value y (1) (1) � y (1) (1) � y (0) (1), that is to say, the fitting curve must pass through the first data point of the original data sequence, then the time response function of the multivariable MGM (1, m) model is given as follows: Proof. According to equation (2), we have dy (1) (t) Integrating the whitening differential equation (8), we get 1 Ay (1) Ay (1) Setting t � 1, we can obtain the value of C, that is, C � e − A (Ay (1) (1) + b).
en, substituting C into equation (9), we obtain the time response function of the MGM (1, m) model as follows: e MGM (1, m) model effectively realizes a unified description of multiple variables from the perspective of system and can better describe the mutual influence and mutual restriction among the variables in system. However, the parameters of the MGM (1, m) model are often deduced from equation (3), whereas the time response function is solved by equation (1). e inconsistency between parameters estimation and parameters application is an important factor leading to a poor prediction accuracy of the MGM (1, m) model. erefore, it is particularly important to construct a multivariable MGM (1, m) direct prediction model that can unify parameters estimation and parameters application.

Construction of the DMGM (1, m) Model
. . , m be the mean sequence generated by consecutive neighbors of x (1) i , then Mathematical Problems in Engineering is called a multivariable MGM (1, m) direct prediction model, denoted as DMGM (1, m) model.

Solution of the DMGM (1, m) Model
Taking i � 1 as an example, then we obtain Similarly, the estimated values of parameters can be solved by the least square method when i � 2, 3, . . . , m.
. . , m, that is to say, the fitting curve must pass through the first data point of the original data sequence, then the recursive formula of the DMGM (1, m) model is given as follows:  m) model not only significantly improves the modeling accuracy in theory but also has the advantage of simple modeling process. However, it is found that the DMGM (1, m) model can only describe the interrelationship and mutual restriction of system variables from equation (11). Because the DMGM (1, m) model is the discrete form of the MGM (1, m) model, therefore, the DMGM (1, m) model is suitable for the homogeneous exponential data prediction as the MGM (1, m) model. In the actual economic and social system, system variables often have a nonhomogeneous exponential characteristic or time power term characteristics, that is to say, system variables have a certain relationship with time term. Hence, considering that the certain relationship between system variables and time items, we will add a time polynomial term into the DMGM (1, m) model. Subsequently, an optimized DMGM (1, m) model, abbreviated as TPDMGM (1, m, φ), is proposed in the following section.

Construction and Solution of the TPDMGM (1, m, φ) Model
In this section, we will present the modeling procedure of the TPDMGM (1, m, φ) model, including definition equation, property, time response function, and modeling steps.

Construction of the TPDMGM (1, m, φ) Model
. . , m be the mean sequence generated by consecutive neighbors of u (1) i , then is called a DMGM (1, m) model with time polynomial term, abbreviated as TPDMGM (1, m, φ) model. In this model, θ 11 t + θ 12 t 2 + · · · + θ 1φ t φ is the time polynomial term that reflects the relationship between system behaviors with time items. Additionally, it is worth noting that when φ � 0, the TPDMGM (1, m, φ) model is transformed into the DMGM (1, m) model, and when Proof. When φ � 0, according to Definition 3, we can obtain Obviously, equation (16) is equal to equation (11), that is, the DMGM (1, m) model. Proof ends.

Mathematical Problems in Engineering
Taking i � 1 as an example, then we obtain w 1 � (c 11 , c 12 , . . . , c 1m , θ 10 , θ 11 , . . . , θ 1φ Similarly, the estimated values of parameters can be solved by the least square method when i � 2, 3, . . . , m. However, as can be seen in equation (21), if we want to solve the estimated values of parameters, we should calculate the value of φ at first. Taking into account the volatility characteristics of high-order polynomials and the principle of thrift (the more complex the model, the higher the probability of overfitting) [16], the value of parameter φ is limited to 0, 1, 2, 3 { }. In practical applications, the minimum average relative error is used as the criterion and the debugging method is used to solve the optimal value of φ.
. . , m, then the recursive formula of the TPDMGM (1, m, φ) model is given as follows:

Modeling Steps of the TPDMGM (1, m, φ) Model.
According to the above studies of the TPDMGM (1, m, φ) model, the whole modeling and optimization process are briefly described as follows: Step 1. System variables selection: system variables that influence, correlate, and restrict with each other are selected.
Step 2. Model construction: the TPDMGM (1, m, φ) model is established according to equation (15) when the system variables have time development trend characteristic; otherwise, the DMGM (1, m) model is built through equation (11).
Step 3. Parameter estimation: the debugging method is used to solve the value of φ, and the least square method is used to estimate the value of parameters.
Step 4. Simulation and prediction: the parameter values in step 3 are substituted into the recursive formula and then the simulation and prediction values of system variables are computed.
Step 5. Accuracy test: the mean relative simulation and prediction percentage errors are calculated.
Note. ere are two methods to testify whether the system variables have a time development trend characteristic. ① Quantitative analysis: Before modeling, MATLAB can be used to fit the original data sequence, and then the fitting functions of data sequences are obtained. According to the function, if it is a homogeneous exponential function, the DMGM (1, m) model will be used for prediction; otherwise, the TPDMGM (1, m, φ) model will be used for prediction. ② Qualitative analysis: In the actual economic and social system, system variables cannot simply be expressed by the homogeneous exponential growth law. ese system variables often have a nonhomogeneous exponential characteristic or time power term characteristics, that is to say, system variables have a certain relationship with time term. erefore, most system variables have a time development trend and can be predicted by the TPDMGM (1, m, φ) model. To clearly illustrate the procedure of TPDMGM (1, m, φ) model construction, a flowchart is drawn in Figure 1.

Case Study and Analysis
In this section, two case studies about forecasting the deformation of foundation pit and Henan's vehicle ownership are given as follows.

Case 1: Forecasting the Deformation of Foundation Pit.
In order to verify the efficiency and practicality of the proposed DMGM (1, m) model and TPDMGM (1, m, φ) model, three groups of original data sequences that are representative and truly reflect the deformation law of foundation pits in Feng et al. [17] are selected in this paper for modeling. e original data are shown in Table 1.
In our modeling process, we use the first seven data for modeling and the last two data for testing. To illustrate the performance of the DMGM (1, m) model and the TPDMGM (1, m, φ) model, the simulation and prediction errors of these models are computed and compared with the commonly used MGM (1, m) model and Verhulst model. e meanings of symbols given in Table 2. e detailed modeling results are shown in Table 3-5, as follows. From these results in Tables 3-5, we can see that the TPDMGM (1, 3, 1) model, both in prediction and simulation, has the smallest errors among four models, the performance of the DMGM (1, 3) model is slightly inferior to the TPDMGM (1, 3, 1) model but better than that of the MGM (1, 3) model. Comparatively, the results generated by the Verhulst model are the worst among four models. e overall average relative simulation and prediction percentage errors of the TPDMGM (1, 3, 1) model are 0.0594% and 0.9029%, while the DMGM (1, 3), MGM (1,3), and Verhulst models are 0.0644% and 0.9428%, 0.8108% and 1.8119%, and 3.4344% and 3.9838%, respectively. In summary, the  Mathematical Problems in Engineering 7 TPDMGM (1, 3, 1) model has the best modeling performance among these four models.

Case 2: Forecasting the Vehicle Ownership in Henan
Province. In the system for predicting vehicle ownership, vehicle ownership (x 1 ), gross domestic product (shorted as GDP) (x 2 ), and population (x 3 ) are three variables that influence and restrict each other [11]. erefore, we use the vehicle ownership prediction in Henan province to testify the performance of the DMGM (1, m) model and the TPDMGM (1, m, φ) model. e raw data of x 1 ∼ x 3 from 2012∼2019, presented in Table 6, are obtained from Henan statistical yearbook [18]. Next, we use the first seven data for modeling and the last data for testing.

Symbols
Meanings Relative simulation percentage error of Relative prediction percentage error of Average relative simulation percentage error for each variable Δ p Average relative prediction percentage error for each variable Δ As Overall average relative simulation percentage error Δ Ap Overall average relative prediction percentage error   Tables 7-9, as follows.
From Tables 7-9, we know that the average relative errors in simulation and prediction of the TPDMGM (1, 3, 1) model are 0.0801% and 0.7143%; the modeling accuracy of the DMGM (1, 3) model is slightly lower than that of the TPDMGM (1, 3, 1) model, which is 0.3983% and 1.4449%. However, the average relative errors in simulation and prediction of the MGM (1,3) model and Verhulst model are 0.8768%, 3.4811%, and 0.6080%, 9.8347%, which are worse than the modeling accuracy of the TPDMGM (1,3,1) model and DMGM (1,3) model. e results show that both the simulation and prediction performance of the TPDMGM (1, 3, 1) model are the best among these four models.

Analysis of Comparison
Results. e errors of four models in two case studies are shown in Table 10, and the comparison results are analyzed as follows.
As shown in Table 10, the modeling performance of four models in two cases from superior to inferior is ranked as the TPDMGM (1, m, φ) model, DMGM (1, m) model, MGM (1, m) model, and Verhulst model. e Verhulst model is a single variable grey prediction model, while the other three models are multivariable grey prediction models. Although the Verhulst model is a commonly used grey prediction model for forecasting the sequence with saturated state, it has a low precision in this study because it cannot describe the variables with mutual correlation and mutual restriction characteristic. e MGM (1, m) model can make up for the shortcoming of the Verhulst model; however, the modeling precision is relatively low due to the jumping error between parameter solution and parameter application. e novel proposed DMGM (1, m) model avoids the jumping error in the MGM (1, m) model by unifying the parameter estimation and application into a discrete form, which improves the modeling accuracy. However, the DMGM (1, m) model is only as suitable for predicting homogeneous exponential sequence as the traditional MGM (1, m) model. Considering the system variables in the actual economic and social system often have a nonhomogeneous exponential characteristic or time power term characteristics, that is to say, system variables have a certain relationship with time term, this paper introduces a time polynomial term into the DMGM (1, m) model and constructs the TPDMGM   Table 6: Raw data of the vehicle ownership system.
Year  erefore, after comparing the performance among these four models, the TPDMGM (1, m, φ) model may be an optimal option to predict the system variables with mutual relation, mutual restriction, and time development trend characteristic.  1, m, φ) model can be applied to predict the system variables with interrelation and mutual restriction characteristics, while the TPDMGM (1, m, φ) model has a higher modeling accuracy in theoretical due to its perfect   However, the current study of the TPDMGM (1, m, φ) model is not comprehensive. Problem, such as the intelligent algorithm optimization of parameter φ, has not been discussed in this paper. erefore, how to construct a constraint equation and use intelligent optimization algorithms to solve the optimal parameters φ is the next step of our work.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.