Multi-parameter grey prediction model based on the derivation method

doi:10.1016/j.apm.2021.04.016

Applied Mathematical Modelling

Volume 97, September 2021, Pages 588-601

https://doi.org/10.1016/j.apm.2021.04.016 Get rights and content

Highlights

•
Based on the derivation method, a new derived grey multi-parameter prediction model (DMGM (1, n)) is proposed.
•
The mathematical analysis is used to discuss the main reason why DMGM (1, n) is superior to GMC (1, n).
•
DMGM (1, n) outperforms the other grey multivariable models by comparing the results of these models in three cases.

Abstract

In this study, in order to reduce the morbidity and improve the structural stability of the existing grey multivariable convolution forecasting model, a new derived multivariable grey model based on the derivation method, abbreviated as DMGM (1, n), is presented. Firstly, the time response formula of DMGM (1, n) is deduced by derivation method, which can avoid solving the inverse matrix so as to reduce the morbidity of the model. Secondly, the parameter identification of the model is given based on the least-squares method. Then, it is proved theoretically that DMGM (1, n) is superior to GMC (1, n) because the solution of the former overcomes the shortcoming of the latter that the original model does not take full advantage of all the information from the raw data for modeling. Finally, three real cases with different variables were performed. The fitting and prediction results indicate that DMGM (1, n) is better than GMC (1, n) and the other multivariate grey prediction models in these cases, which also demonstrates that this novel model outperforms the other grey models in this paper.

Introduction

In system research, people often get information with some uncertainty due to knowledge limitations and internal external perturbations [1]. An important part of uncertain information theory, namely the grey system, was established by Deng Julong in the 1980s [2]. Over decades of development, grey prediction model, which is a critically important part of grey system [3], has a range of applications covering energy forecasting [4], [5], [6], traffic flow prediction [7], [8], [9], [10], tourism [11], economy [12], [13], [14], and other fields [15], [16], [17], [18]. The commonly used univariate grey model GM (1, 1) [19] is a cornerstone of the grey forecasting model. However, the complicated and changeable nature of real systems cannot be reflected by this univariate model [20].

Thus, professor Deng [21] presented a multi-parameter grey forecasting model termed GM (1, N), which took into account one main feature factor (output) and N-1 action factors (input) in modeling. Although this multi-parameter model is mainly used for state analysis and decision making, it cannot be used as a prediction tool [2]. Therefore, to address this issue, a novel grey multi-parameter convolution forecasting model was proposed by Tien [22].

However, with the appearance of GMC (1, n), many researchers recently found that this model still performed with inaccuracy in some applications. Then several accurate forms of GMC (1, n) models have been presented. The process of development for GMC (1, n) is summarized in Table 1.

As Table 1 indicates, the optimisation methods for GMC (1, n) can be mainly divided into two categories, parameter optimization and the structure optimization. Parameter optimization includes background values and coefficient optimisation. Structure optimization is mainly for the stability of the system. Although these optimization methods have made numerous contributions to improve GMC (1, n) from every side, the morbidity problem of this model cannot be ignored. In grey modeling theory, solving the inverse matrix in the process of solving the time response equation is one of the causes of morbidity. The goal of this study is to establish a new grey multivariable prediction model to reduce the morbidity and improve the structural stability of GMC (1, n).

Thus, a novel multivariable grey prediction model is developed in this paper. The main contributions of this article are shown below:

(1)
A new derived grey multi-parameter prediction model (DMGM (1, n)) based on the derivation method is proposed. The solution of this model is deduced by this method so as to reduce the morbidity of the model.
(2)
Mathematical analysis is used to discuss the main reason why DMGM (1, n) is superior to GMC (1, n), that is, the time response equation of DMGM (1, n), $\sum_{i = 1}^{t - 1} X_{1}^{(0)} (i)$ , takes full advantage of all the information from the raw data for modeling.
(3)
The DMGM (1, n) model outperforms the other multivariable models by comparing the fitting and forecasting effectiveness of these models in three real cases of different variables.

The remainder of this article is organized as follows. A brief introduction about the principle of GMC (1, n) in Section 2. A new grey derived multivariable model DMGM (1, n) is established In Section 3, and the solution and the parameter identification of this model are also given. In Section 4, the difference and relationship between DMGM (1, n) and GMC (1, n) are discussed. Three cases of different variables are adopted to compare the effectiveness of DMGM (1, n) with the other multivariable grey models in Section 5. The conclusions are presented in the last section.

Section snippets

The traditional GMC (1, n) model

In this section, we will give a brief introduction about the grey multivariable convolution model (GMC (1, n)) [22], which is based on the traditional GM (1, n) model by adding a control parameter u. The definition is as follows.

Definition 1

[22]. Assumed that $X^{(0)} = (X_{1}^{(0)}, X_{2}^{(0)}, \dots, X_{n}^{(0)})$ are raw sequences, where $(X_{2}^{(0)}, X_{3}^{(0)}, \dots, X_{n}^{(0)})$ are n − 1 input variables, and $X_{1}^{(0)}$ is one output variable. $X_{i}^{(1)} (t) = \sum_{i = 1}^{t} X_{i}^{(0)} (t)$ and $X_{1}^{(1)} (r p + t) = \sum_{k = r p + 1}^{r p + t} X_{1}^{(0)} (k) (t = 1, 2, \dots, r + r f)$ are one order accumulative generation

The DMGM (1, n) model and its properties

In this section, based on GMC (1, n), we will establish the DMGM (1, n) model, which is a derived multivariable grey model. The modeling mechanism, relevant definitions and properties of the model are discussed. In consideration of the stability of the system, the time response equation of the DMGM (1, n) model is calculated by using the derivation method in the following. The modeling steps and algorithm are also indicated in the last of this section.

The difference and relationship between DMGM(1, n) and GMC(1, n)

In this section, we will theoretically prove the reason why DMGM (1, n) superior to GMC (1, n).

Case study

Three multivariable cases are presented to evaluate the effectiveness of DMGM (1, n) in this section. This new model is then compared with the other multivariable grey models. The data, analysis and additional details of the comparison results are discussed as follows.

Conclusions

This paper developed a novel multivariable grey forecasting model based on the derivation method (DMGM (1, n)). The following conclusions were obtained:

(1)
The solution of the proposed DMGM (1, n) adopted the derivation method in calculating the time response formula so as to reduce the morbidity of the model. In grey modeling theory, one of the causes of morbidity is solving the inverse matrix. The parameter estimation of DMGM (1, n) is also discussed.
(2)
From the stability structure of the model,

Acknowledgments

The authors give sincerest thanks to the editors and the anonymous reviewers for their helpful comments and suggestions. Thanks Prof. Xiao Xinping a lot for much advice and help during the revision of the manuscript. This research is supported by the National Natural Science Foundation of China (71871174).

References (35)

X. Ma et al.
A novel fractional time delayed grey model with grey wolf optimizer and its applications in forecasting the natural gas and coal consumption in Chongqing China
Energy
(2019)
Q.Z. Xiao et al.
Parameter optimization for the nonlinear grey Bernoulli model and its application in predicting biomass energy consumption
Appl. Soft Comput.
(2020)
B. Zeng et al.
Application of a new grey prediction model and grey average weakening buffer operator to forecast China's shale gas output
Energy Rep.
(2020)
X.P. Xiao et al.
A novel car-following inertia grey model and its application in forecasting short-term traffic flow
Appl. Math. Model.
(2020)
H.M. Duan et al.
Tensor alternating least squares grey model and its application to short-term traffic flows
Appl. Soft Comput. J.
(2020)
X.P. Xiao et al.
A new grey model for traffic flow mechanics
Eng. Appl. Artif. Intell.
(2020)
J.S. Lu et al.
An optimized nonlinear grey Bernoulli model and its applications
Neurocomputing
(2016)
S.H. Mao et al.
Lotka–Volterra model for the competition and cooperation between third-party online payment systems and online banking in China
Appl. Soft Comput.
(2020)
L.L. Ye et al.
A novel time-delay multivariate grey model for impact analysis of CO₂ emissions from China's transportation sectors
Appl. Math. Model.
(2021)
B. Zeng et al.
A new multivariable grey prediction model with structure compatibility
Appl. Math. Model.
(2019)

T.L. Tien

The deterministic grey dynamic model with convolution integral DGDMC (1, n)

Appl Math Model

(2009)

T.L. Tien

The indirect measurement of tensile strength by the new model FGMC (1, n)

Measurement

(2011)

Z.X. Wang

Nonlinear grey prediction model with convolution integral NGMC (1, n) and its application to the forecasting of China’s industrial SO₂ emissions

J. Appl. Math.

(2014)

B. Zeng et al.

A novel multi-variable grey forecasting model and its application in forecasting the amount of motor vehicles in Beijing

Comput. Ind. Eng.

(2016)

X. Ma et al.

Research on the novel recursive discrete multivariate grey prediction model and its applications

Appl. Math. Model.

(2016)

L.F. Wu et al.

Grey multivariable convolution model with new information priority accumulation

Appl. Math. Model.

(2018)

X. Ma et al.

Liu A novel kernel regularized nonhomogeneous grey model and its applications

Commun. Nonlinear Sci. Numer. Simul.

(2017)

Cited by (24)

A recursive polynomial grey prediction model with adaptive structure and its application
2024, Expert Systems with Applications
As a sparse data analysis algorithm, ensuring a reasonable model structure is an important challenge for grey models to identify the control mechanism of the uncertain system from observational data. To improve the intelligence and adaptability of the model, this study presents a synchronized optimization strategy for data prioritization and model structure for discrete polynomial grey prediction model. The proposed polynomial grey model contains two hyper-parameters: memory factor parameter and structural parameter. The memory factor is introduced into the discrete model to reconstruct the objective function of structural parameter optimization, thereby avoiding the problem of information superposition. The structural parameter is used to enhance the adaptability of grey prediction model in uncertain data analysis tasks. By employing a recursive estimation approach, an adaptive strategy for estimating model hyper-parameters is proposed, which focuses on minimizing prediction errors within the in-sample data. Additionally, a comparison is made between the proposed improved polynomial grey model and existing polynomial grey models in terms of data information mining, estimation stability, and robustness against measurement noise. The proposed model is applied to the practical engineering application of wear prediction, further validating the effectiveness of the proposed approach in non-equidistant time series prediction tasks.
Exploring the short-term water damage characteristics of asphalt mixtures: The combined effect of salt erosion and dynamic water scouring
2024, Construction and Building Materials
In icy or snowy conditions, improper application and uneven dispersion of deicing salts on roadways can lead to saltwater accumulation, subsequently causing roadway scouring when vehicles transit. This study aims to assess the combined influence of salt erosion and dynamic water scouring on short-term water damage of asphalt mixtures. A custom-designed dynamic water scouring simulation apparatus is used, employing three diverse deicing salt solutions as agents for dynamic water scouring. Numerous iterations of dynamic water scouring tests are conducted on two types of graded asphalt mixtures. Post-scouring, alterations in air void content and water stability are evaluated through void content tests, Marshall water immersion tests, and freeze-thaw splitting tests. Additionally, the GM (1,N) damage prediction model from gray theory is applied for data modeling and prediction. The results reveal a significant impact of the combined effects of salt erosion and dynamic water scouring on the short-term performance of asphalt mixtures. In the early phase, dynamic water scouring primarily influences the asphalt composite, augmenting micro-cracks and causing aggregate disintegration. The intermediate phase sees the disintegration of the asphalt membrane by deicing salts, disrupting adhesion between asphalt and aggregate, and progressively leading to the emergence of new micro-cracks. During later stages, dynamic water scouring continues to deteriorate the asphalt composite, building on the erosion instigated by deicing salts, resulting in the expansion of newly formed micro-cracks and further aggregate disintegration. Among the three deicing salts, CH₄N₂O exhibits the most pronounced impact on short-term water damage, followed by NaCl, while CH₂CH₃OH has the least effect. Increasing the fine aggregate proportion in the asphalt mixture enhances water damage resistance. The GM (1,N) damage prediction model effectively anticipates air void content and water stability of asphalt mixtures. The research outcomes offer theoretical and technical support for understanding the durability and service lifespan of asphalt pavements in cold regions.
Time-lead nonlinear grey multivariable prediction model with applications
2023, Applied Mathematical Modelling
Time lags or leads and non-linearity of complex systems are critical to address in accurate scientific prediction. However, few studies have focused on time-lead and nonlinear models with limited information. Therefore, this paper aims to develop a novel grey multivariate model with time-lead operator τ and nonlinear factor r from the perspective of this problem. Next, the solution of the proposed model is deduced through a derivation method to reduce the morbidity of the model. Then, the parameter identification of the model is given on the basis of the least-squares method. The grey prediction evolution algorithm is employed to find the optimal value of parameters τ and r to meet the minimum errors. Finally, house prices of Wuhan, Beijing and Shanghai in China are performed to verify the validity and reliability of the new model. The results demonstrate the model we proposed has satisfactory prediction precision, demonstrating that time-lead and nonlinear factor in the model is reasonable. At the end of the paper, the house prices of these three cities from 2022 to 2028 are also predicted. The average annual growth rate in the three regions is 2.49%, 3.57% and 2.97%, respectively. These results are useful to policy analysis regarding house price correlations. They provide insights for market investors and policy makers.
Estimation of the respiratory mortality attributable to comprehensive effects of air pollutants in China using an enhanced seasonal and self-adaptive multivariate grey convolution model
2023, Expert Systems with Applications
Exposure to air pollutants has become a major global health threat and is a dominant contributor to increasing respiratory mortality. Accurate projections of respiratory mortality are crucial to mitigating the adverse influences of air pollution on public health and guiding long-term policy implementation. However, the forecasting models' accuracy is difficult to guarantee because seasonal factors perturb the raw series of air pollutants and respiratory mortality. This paper proposes an enhanced seasonal and self-adaptive multivariate grey convolution model to solve this dilemma. The innovations of the proposed model reside in the following areas: firstly, a seasonal operator vector of multidimensional data is initially built to digitize the seasonality of air pollutants and related mortality. Secondly, a multivariate grey convolution model with an adaptive time-lag cumulative optimization item is put forward. The weight coefficients of each driving variable at different periods are intended to adapt dynamically. Thirdly, the self-adaptive time-delay coefficient is determined by the Particle Swarm Algorithm, enhancing the accuracy of predictions. Theoretically, the generalizability and adaptability of the newly presented model are well-validated. Moreover, its practical applicability and higher forecasting performance are further verified in estimating the respiratory mortality attributable to the comprehensive effects of air pollutants in China.
Forecasting China's hydropower generation using a novel seasonal optimized multivariate grey model
2023, Technological Forecasting and Social Change
Global warming and environmental degradation are essential issues that endanger human survival. The conflict between rising carbon emissions and carbon neutrality goals has prompted an urgent need for China's energy sector to step up efforts to develop clean energy generation. As a significant hydropower country, hydropower generation is China's mainstay of clean energy generation. The work contributing to sustainable hydropower development requires reasonable forecasts of clean energy generation. This paper proposes a seasonal optimized multivariate grey model that optimizes background value and supplements dummy variables to explore hidden factors through an optimization algorithm to the related sequences. The novel model improves the fitting and prediction accuracy through the loop supplementation of dummy variables. The model tests the effect of China's hydropower generation prediction and compares results with other methods. The mean absolute percentage error of the model training and test groups is 3.87 % and 0.83 %. Finally, this paper predicts hydropower generation in China from 2022 to 2025 based on the power generation during China's 13th Five-Year Plan Period.
Predicting environmental sustainability performances of firms using trigonometric grey prediction model
2023, Environmental Development
Environmental sustainability performance predictions can help firms to set benchmarks for improvement and comparison. Although several works in literature have attempted in measuring environmental sustainability performances, predicting the same remains an area less explored. This is certainly due to the requirement of large datasets for constructing efficacious time-series prediction models. We attempt in this direction to forecast the environmental sustainability performances of firms based on the three indicators of performance provided by Thomson Reuters, the Resource use score, the Emissions score, and the Environmental innovation score. We propose a prediction model, which is best suited for prediction, when the available datasets are small. A trigonometric grey prediction model is employed, where the GM (1, 1) model for prediction is used initially, and later on, an error prediction model is built, based on a trigonometric residual prediction model. The model has been applied for real data of ten Indian firms to predict their future environmental sustainability performances. From the results, we observe an increasing consideration for environmental sustainability by the majority of Indian firms for the predicted year. The results of the study agree with the remarks reported in existing literatures. Hence, managers of firms are recommended to use the proposed prediction model for forecasting and improving their environmental sustainability performances for the future.

View all citing articles on Scopus

View full text

Multi-parameter grey prediction model based on the derivation method

Highlights

Abstract

Introduction

Section snippets

The traditional GMC (1, n) model

The DMGM (1, n) model and its properties

The difference and relationship between DMGM(1, n) and GMC(1, n)

Case study

Conclusions

Acknowledgments

Energy

Appl. Soft Comput.

Energy Rep.

Appl. Math. Model.

Appl. Soft Comput. J.

Eng. Appl. Artif. Intell.

Neurocomputing

Appl. Soft Comput.

Appl. Math. Model.

Appl. Math. Model.

Appl Math Model

Measurement

J. Appl. Math.

Comput. Ind. Eng.

Appl. Math. Model.

Appl. Math. Model.

Commun. Nonlinear Sci. Numer. Simul.