A novel structure-adaptive intelligent grey forecasting model with full-order time power terms and its application

https://doi.org/10.1016/j.cie.2018.04.016Get rights and content

Highlights

  • A novel grey prediction model with full order time power terms is proposed.

  • Alterable structure of FOTP-GM(1,1) model make it suit more kinds of sequences.

  • Quasi-exponential sequence with velocity, acceleration terms is predicted precisely.

  • FOTP-GM(1,1) has higher accuracy than existing GM(1,1) derived univariate models.

  • Total production volume of hydropower, nuclear power and wind power is predicted.

Abstract

To solve the problem that traditional grey models cannot simulate accurately any given non-homogeneous exponential sequence with velocity and acceleration terms, a novel grey forecasting model with full-order time power terms (abbreviated as FOTP-GM(1,1)) is proposed. Firstly, two forms of sequence functions of the restored values are brought forward based respectively on whitenization method and connotation method. Then, Four forecasting properties are put forward to demonstrate that FOTP-GM(1,1) is a more general model with higher accuracy and adaptability than traditional models. Then a visual comparison method is introduced to facilitate selection of a more reasonable structure from all possible structures of the FOTP-GM(1,1) model. To verify its feasibility and efficiency, performance comparisons and suitability analyses are given by 2 examples. The first example shows that the simulative accuracy given by connotation method is higher than that by whitenization method, and it confirms that NDGM(1,1) and SAIGM(1,1) models are all special cases of the FOTP-GM(1,1) model. Then by quantitative analysis and visual comparison, the second example shows that FOTP-GM(1,1) model has better adaptability and broader universality. In the last, FOTP-GM(1,1) model is employed to forecast the potential total production volume of hydropower, nuclear power and wind power from 2017 to 2021 in China. Thus practicality of the proposed model is tested.

Graphical abstract

The paper proposed a novel adaptive intelligent grey forecasting model (FOTP-GM(1,1)), which has alterable structure that can be changed automatically according to actual application. It shows high forecasting precision and can simulate accurately three kinds of raw data sequences: one is homogeneous exponential sequence; another is non-homogeneous exponential sequence with constant, velocity and acceleration terms; the third is regressive linear sequence. The practicality and effectiveness was verified by comparison with other first-order models with one variable such as GM(1,1), NGM(1,1,k), DGM(1,1), NDGM, SAIGM, and GM(1,1,tα). Then the proposed model has been employed to predict China’s total production volume of hydropower, nuclear power and wind power successfully.

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Introduction

Accurate forecasting is the foundation and precondition of scientific decision making. In order to find the rules or patterns from the complicated phenomena, grey information system theory is proposed by Deng (1982), which is characterized by high efficient acquisition of information for system with poor information and uncertain, incomplete structures. Its goal is to discover the law of data evolution by the mining of unknown information (Deng, 2002). This theory has found its impacts on many categories such as grey incidence analysis, grey clustering, grey forecasting and grey control (Liu & Lin, 2010). As an important branch, the grey forecasting theory which takes GM(1,1) model as the center has been widely investigated and studied by many scholars in theoretical and practical fields. A series of grey forecasting models were put forward gradually and put into practical application in wide range, showing high forecasting precision and robust performance, mainly including such 5 domains as time series prediction, calamity prediction, seasonal forecasting, topological forecasting and systematic forecasting (Xie & Liu, 2009).

As the basis of the grey system of forecasting theory, GM(1,1) model is the first-order single variable grey forecasting model, i.e.x(0)(i)+az(1)(i)=b (Liu & Lin, 2010). In order to improve the accuracy of the GM(1,1) model, entire modeling process has been studies in the past 20 years. These studies mainly focus on three aspects, including optimization of parameters, improvement of model structure, selection of computing method.

In order to improve the forecasting accuracy of the model, parameter optimization of the model is required usually, mainly including 3 aspects: preprocessing of original data sequence X(0), choice of iterative datum x(0)(1) and optimization of background value Z(1) (Liu & Lin, 2006). Each of them has effluence on prediction accuracy to some extent based on fixed structure of the known forecasting model.

In fact, economy, society and ecosystem can be regarded as the generalized energy system, accumulation and release of energy in which usually conform to exponential law (Liu, Yang, & Forrest, 2017). According to GM(1,1), the forecasting sequence can be obtained as followsx̂(0)(k+1)=(1-ea)x(0)(1)-bae-ak,kN.Obviously, it is a exponential sequence, so it can achieve better fitness with relatively high accuracy. However, the development tendency of real application system cannot strictly keep to exponential law, but actually goes through three stages including incubation stage, uniform velocity phase and acceleration stage to another equilibrium state (Guo, Xiao, & Jeffrey, 2015). Three possible variation trends of some parameters for a given system are shown in Fig. 1. For this reason, errors are unavoidable for GM(1,1) model to fit non-homogeneous exponential sequence. In order to improve predictive accuracy, many derived grey forecasting models have been proposed gradually. However, none of them can achieve satisfactory forecasting accuracy. Why this is happening will be investigated in Section 2.

There are two methods often used now, which are connotation method and whitenization method. According to information coverage principle of grey derivative, the discrete grey model can be transformed into continuous whitenization differential equation. Then the time response sequence of the model would be gotten by solving the whitenization differential equation. For this reason, it is called whitenization method. As we know that the solution of GM(1,1) is gotten by whitenization method. But the skip from discreteness to continuity can result in precision loss. In order to eliminate the inherent error induced by whitenization method, the connotation method is provided to work out the time response sequence by direct solution of original discrete model without any transformation. As is known that SAIGM(1,1) model (Zeng, Meng, & Tong, 2016) and NDGM(1,1) model (Xie, Liu, Yang, & Yuan, 2013) are based on this computing method.

In this work, a more general grey forecasting model with full order time power terms and alterable structure is proposed. It is an unified representation of GM(1,1) and its derived models with one variable such as DGM(1,1), NDGM(1,1), NGM(1,1,k), and it also possesses higher forecasting precision than linear regression GM(1,1) and GM(1,1,tα). Then the proposed model is employed to predict China’s total production volume of hydropower, nuclear power and wind power successfully.

Section snippets

Literature review

In the past 20 years, three parameters are often optimized to improve the predictive accuracy of the given grey model, including original data sequence X(0), iterative datum x(0) and background value Z(1).

  • (i)

    Preprocessing of original data sequence X(0) According to axiom In accordance with information (Liu, 1991), the original data sequence X(0) is the basis of information representation of grey system. Consequently, it is the basis of grey forecasting theory too. The perturbation of individual

The definition of FOTP-GM(1,1) model

Definition 1

Assume the sequences X(0),X(1) and Z(1) have the same definitions as those in GM(1,1) model, then the equationx(0)(k)+az(1)(k)=i=1hbikh-i,kZ,k2,h1is termed as grey model with full-order time power terms (abbreviated as FOTP-GM(1,1)). And -a is called development index, bi (i = 1,2,…,h) is called grey actuating quantity and h is termed as the order of time power terms bith-i.

Definition 2

According to information coverage principle of grey derivative, the grey model in discrete form defined in Definition 1

Performance comparison and adaptability analysis

Two examples will be provided here to illustrate the advantages of the proposed FOTP-GM(1,1) model in aspects of performance, universality and adaptability by comparison with other known series of GM(1,1) models listed in Table 1.

As we have described in Section 1, there are two computing methods, namely, whitenization method and connotation method, and the solutions of DGM(1,1),SAIGM(1,1) and NDGM(1,1) models are obtained by connotation method, while solutions of GM(1,1),NGM(1,1,k) and GM(1,1,tα

Forecast of China’s total production volume of hydropower, nuclear power and wind power

In China, the 13th five-year plan for renewable energy development indicates that its government is promoting energy transformation from traditional coal-fired power to non-fossil energy. According to the plan, non-fossil energy should account for 15% and 20% in 2020 and 2030 respectively. As an important part, hydropower, nuclear power and wind power are becoming more and more important in Chinese energy structure. And its total production volume is increasing year by year dramatically, as

Conclusions

Aiming to overcome the deficiencies of the traditional prediction models, we bring forward a novel structure alterable FOTP-GM(1,1) model with attributes and advantages summarized as follows.

  • (1)

    The proposed FOTP-GM(1,1) model has full order time power terms with the basic form x̂k+az(1)(k)=i=1hbikh-i. Its order ranges from 1 to h, and its time response function of the restored values includes not only exponential term, but also constant, velocity and acceleration terms. As a matter of fact, it is

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant Numbers: 61374043, 61603392) and Nature Science Foundation of Jiangsu Province (Grant Numbers: BK20150199, BK20160275). We also thank all referees for their constructive advise which helped to improve the completeness and clarity of this paper.

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