A novel multi-variable grey forecasting model and its application in forecasting the grain production in China

Highlights

• A novel multi-variable grey forecasting model, NMGM(1,N), is proposed.

• A smooth generation method with variable weight is used to build NMGM(1,N).

• NMGM(1,N) can weaken the influence of extreme value on model performance.

• The novel model is successfully used to forecast the grain production in China.

Abstract

In the multi-variable grey forecasting model GM(1,N), the extreme value of the independent variable is one of the important factors that affect the simulation and prediction results of the dependent variable. In this study, a smooth generation method was used to weaken the influence of the extreme value on the performance of GM(1,N), and a novel multi-variable grey forecasting model NMGM(1,N) based on the smooth generation of independent variable sequences with variable weights was constructed. The parameters of NMGM(1,N) were estimated and proved, and the time-response expression and the final restored expression of NMGM(1,N) were deduced. Besides, the detailed modeling steps were provided, and a MATLAB program for building the new model was developed. Moreover, the evaluation criteria of the model were introduced, and numerical examples were used to test the performance of the model. Lastly, the new model was applied to forecast China’s grain production. The global mean relative percentage error of the new model was only 0.689%, in comparison with the ones obtained from the GM(1,N), GM(0,N) and OGM(1,N), which were 4.268%, 3.480% and 1.302% respectively. The findings show that the new model has the best performance, which confirms the effectiveness of the structure improvement.

Introduction

Grey forecasting models can be divided into two types: univariate and multi-variate models. In recent years, the grey forecasting model has been researched extensively and a series of important achievements have been made (Shaikh et al., 2017, Zeng et al., 2020, Zeng et al., 2020, Zhou et al., 2020, Zeng et al., 2020). The multi-variable grey forecasting model is represented by GM(1,N). The modeling object of the model is composed of a dependent variable (or System characteristic variable) and ‘N-1′ independent variables (or Related factor variable; Driver items) (Liu, Forrest, & Yang, 2012). The modeling process considers fully the influence of independent variables on the change trend of the dependent variable. GM(1,N) makes up the deficiency of single structure and limited simulation ability of single-variable grey prediction models. However, GM(1,N) has only been used as a system analysis tool for a long time, and its important prediction ability has not been widely promoted and applied. The main reason is that the model has many shortcomings in the modeling mechanism and model structure, which often leads to greater model errors than that of the GM(1,1) model in practical applications.

In order to improve the modeling performance of GM(1,N), scholars made a lot of efforts, which mainly include three aspects: expanding the model structure (Xiong, Huang, Peng, & Wu, 2020), optimizing model parameters (Dang et al., 2017, Wang and Hao, 2016, Wu and Zhang, 2018) and broadening the model’s application scope (Zeng, Shu, Yan, Shi, & He, 2019).

(a) Expanding the model structure: By Expanding the structure of GM(1,N), some structural parameters, such as the time lag term (Mao, Gao, & Xiao, 2015), were added into the GM(1,N) model, and then some new GM(1,N) models were proposed. Xie, Wu, Li, and Li (2020) introduced a multivariate nonlinear grey model based on the kernel method to improve the prediction accuracy of traffic-related emissions. Based on a Gaussian vector basis kernel function and a global polynomial kernel function combined with the characteristics of grey prediction models, a new multi-kernel GMC(1,N) model (Duan, Wang, Pang, Liu, & Zeng, 2020), which was more comprehensive and suitable for nonlinearity, was established.

(b) Optimizing model parameters: By optimizing the parameters of GM(1,N), such as the background-value coefficient (Zeng, Yan, He, & Shi, 2020) and initial-condition value (Luo & Liu, 2017), some new GM(1,N) models were established. Based on the whitening equation and recursive method of the GMC(1,N) model, a new multivariate grey prediction model RDGM(1,N) (Ma & Liu, 2016) was proposed. Based on the existing fractional multivariate grey model with convolution integral, a new fractional multivariate discrete grey model (Ma et al., 2019) was proposed.

(c) Broadening the model’s application scope: By broadening the application scope of GM(1,N), some new GM(1,N) models were proposed. The GM(1,N) model based on the virtual variable control (Ding, Dang & Xu, 2018) was established, which provides a quantitative solution to the small sample modeling problem when the system variable is affected by the virtual variable. The IEGM(1,N) model (Ding, Dang, Xu, & Wang, 2018) considering the interaction between driving factors was proposed, which provides an effective analysis tool for solving multi-variable, small-sample system modeling with interaction.

The above researches on GM(1,N) greatly enriched the theoretical system of the multi-variable grey forecasting model, expanded its application scope, and promoted its integration with practical problems. Nowadays, the upgraded GM(1,N) models are widely used in the fields of transportation, agriculture, energy, management, etc. (Ahmed et al., 2020, Wu et al., 2018, Ye et al., 2020), solving practical problems, such as engineering prediction (Guo, Liu, Wu, Gao, & Yang, 2015), simulation of anaerobic digestion system (Ren, 2018) and estimation of electronic waste (Duman, Kongar, & Gupta, 2019).

At present, all GM(1,N) models are established by directly using the sequences of independent variables as drivers. However, there exists a major drawback in the modeling process of these models. The sequences of independent variables are the basis of constructing the parameter matrix of GM(1,N), which determines the size of model parameters and the rationality of simulation and prediction results. It can be seen that the sequence of independent variables is an important factor that affects the modeling performance of GM(1,N). Currently, GM(1,N) models all use the sequences of independent variables for modeling directly. The extreme values in the sequences are likely to affect the rationality of the estimated values of model parameters. Therefore, to weaken the influence of extreme values in sequences of independent variables on the modeling results of GM(1,N), we proposed a new method of smooth generation of independent variable sequences with variable weights and constructed a new multi-variable grey forecasting model NMGM(1,N).

In order to verify the performance of the novel model and compare it with other models, we used NMGM(1,N) to forecast China’s grain production, and compared it with GM(1,N), GM(0,N) (Liu, Yang & Forrest, 2016) and OGM(1,N) (Zeng, Luo, Liu, Bai, & Li, 2016). The results showed that the global mean relative percentage error (GMRPE) of NMGM(1,N), GM(1,N), GM(0,N) and OGM(1,N) were 0.689%, 4.268%, 3.480% and 1.302% respectively. It can be seen that the performance of NMGM(1,N) was much better than that of GM(1,N) and GM(0,N). Even compared with OGM(1,N), the model with the best performance currently, its performance was also better. Similarly, the effectiveness and practicability of NMGM(1,N) model were also verified by other cases and evaluation criteria.

The rest of this paper is organized as follows. In Section 2, we introduce the OGM(1,N) model and analyze its main defect. In Section 3, we propose an improved GM(1,N) model based on the smooth generation method, then study and prove its parameter estimation, and derive its time-response expression and its final restored expression. In Section 4, we provide the detailed modeling steps and the MATLAB program for building the new model and optimizing the weight of the drivers of the model by the PSO algorithm. In Section 5, we introduce the evaluation criteria of the model, and verify the performance and stability of the model by numerical examples. In Section 6, we use the new model to forecast China's grain production. In this application, we describe the modeling and forecasting process in detail. In Section 7, we make a further overview of the improved new model and a prospect of future research work.