A self-adaptive grey forecasting model and its application

TANG Xiaozhong and XIE Naiming

1.College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;2.Department of Industry and Finance, Huangshan Vocational and Technical College, Huangshan 245000, China

Abstract: GM(1,1) models have been widely used in various fields due to their high performance in time series prediction.However,some hypotheses of the existing GM(1,1) model family may reduce their prediction performance in some cases.To solve this problem,this paper proposes a self-adaptive GM(1,1)model,termed as SAGM(1,1) model,which aims to solve the defects of the existing GM (1,1) model family by deleting their modeling hypothesis.Moreover,a novel multi-parameter simultaneous optimization scheme based on firefly algorithm is proposed,the proposed multi-parameter optimization scheme adopts machine learning ideas,takes all adjustable parameters of SAGM(1,1) model as input variables,and trains it with firefly algorithm.And Sobol’ sensitivity indices are applied to study global sensitivity of SAGM(1,1) model parameters,which provides an important reference for model parameter calibration.Finally,forecasting capability of SAGM(1,1) model is illustrated by Anhui electricity consumption dataset.Results show that prediction accuracy of SAGM(1,1) model is significantly better than other models,and it is shown that the proposed approach enhances the prediction performance of GM(1,1) model significantly.

Keywords: grey forecasting model,GM(1,1) model,firefly algorithm,Sobol’ sensitivity indices,electricity consumption prediction.

1.Introduction

Grey system theory was pioneered by Deng [1] focusing on solving uncertainty problems with a small amount of data.Unlike statistical models and machine learning models, grey system theory requires very little data to uncover the intrinsic development law of an uncertain system.Over the past several decades, grey system theory has been widely applied in various fields and achieved good results [2−4].Grey system theory is composed of grey prediction, grey incidence analysis, grey control, and grey clustering, etc.[2−5], in which grey prediction is an important part and has been widely used in various fields,such as industry [6,7], agriculture [8], energy [9−11], and tourism [12−14].

GM(1,1) model, as a basic grey prediction model, has been successfully applied in energy, transportation, and other fields and achieved good results [15,16].However,as a newly born theory, GM(1,1) model still has some defects that need to be solved.To improve the prediction accuracy of GM(1,1) model, scholars have carried out a lot of creative researchs on the improvement of the GM(1,1) model in recent decades.The improved methods can be divided into several categories.

(i) Initial condition optimization.The initial value of the traditional grey model usually takesx(0)(1), which contradicts the new information priority principle.To settle this problem, scholars have conducted creative research on initial value optimization and achieved fruitful results, For example, Dang et al.took the last data item ofX(1)as the initial condition of the GM(1,1) model [17].Wang et al.proposed a novel initial value calculation method by linear weighting the first and last item ofX(1)[18].Xiong et al.took the weighted average value of all data items ofX(1)as the initial condition and constructed a new GM(1,1) model [19].Although this method considers all known information in determining the initial value, the fixed weights affect the stability of the GM(1,1)model’s predictive performance.To address this challenge, Ding proposed a new initial value construction method based on dynamic weighting coefficients [20].Zhu et al.constructed a novel initial condition optimization method by using dynamic fractional-order weighted coefficients, these studies have greatly improved the predictive capability of GM(1,1) model [21].

(ii) Background value optimization.The background value of the traditional grey model is obtained by performing an adjacent average generating operation onX(1).However, when the time series changes sharply, this background value calculation method will reduce the prediction accuracy of grey prediction model to some extent.To address this issue, a large number of background value optimization algorithms, such as linear value insert method [22,23], integration formula [24], and Simpson formula [25], have been proposed one after another over the past decades, which have significantly enhanced the predictive power of grey prediction model.

(iii) Accumulated generating operator optimization.Accumulated generating operation is an important modeling basis for grey system theory.Traditional GM(1,1)models are modeled based on first order cumulative generation series, which is contrary to the new information priority principle [26].To address this challenge, Wu et al.creatively introduced the idea of fractional order into grey modeling, which significantly improved the prediction power of the grey prediction model [26].Since then,many scholars have devoted themselves to the research of fractional order grey modeling and achieved fruitful results [27−29].

(iv) Hybrid grey models.GM(1,1) model can achieve accurate prediction when the original data sequence shows a stable increase trend.However, when the modeling sequence shows irregular fluctuation, the prediction performance of grey prediction model is poor.To address this challenge, scholars have introduced the idea of combinatorial modeling and constructed a series of hybrid grey prediction models, including Grey-Fuzzy models [30],Grey-Markov models [13−14,31], Grey-Fourier models [32,33], and other hybrid grey prediction models [34-36].These optimization techniques have greatly improved the prediction power of the GM(1,1) model.

Although the above studies improve prediction accuracy of the GM(1,1) model to some extent, these studies are all constructed on the hypothesis that some adjustable parameters are known.When original observations show great randomness, assumptions made by these studies may increase their prediction errors.To solve this problem,this paper proposes a novel self-adaptive GM(1,1) model(denoted as SAGM(1,1)) by removing the modeling assumptions of the existing GM(1,1) model family.Moreover, a novel multi-parameter simultaneous optimization scheme based on firefly optimization algorithm is designed in this paper, the new multi-parameter simultaneous optimization scheme adopts machine learning ideas,takes all the adjustable parameters of SAGM(1,1) model as input variables, and trains it with firefly algorithm.Besides, Sobol’ sensitivity indices are employed to study the global sensitivity of the proposed SAGM(1,1) model adjustable parameters, which provides an important reference for model parameter calibration.

The rest of this paper is arranged as follows: Basic theories of the GM(1,1) model are introduced in Section 2.A novel self-adaptive GM(1,1) model and its parameter optimization scheme are proposed in Section 3.In Section 4, forecasting capability of the SAGM(1,1) model is illustrated by Anhui electricity consumption dataset.Finally, Section 5 concludes this paper.

2.GM(1,1) model

GM(1,1) model, as a basic grey prediction model, has been successfully utilized in various fields and achieved good performances.The general modeling procedure of GM(1,1) model can be shown with Definitions 1−3.

Definition 1[2] SupposeX(0)=[x(0)(1),x(0)(2), ···,x(0)(n)],n≥4 is an original time series,X(1)=[x(1)(1),x(1)(2),···,x(1)(n)]) is the first-order accumulating generation sequence ofX(0), where

Definition 2[2] SupposeX(0)andX(1)as shown in Definition 1,Z(1)=[z(1)(2),z(1)(3),···,z(1)(n)] is the background value ofX(1),where

Differential equation:

is called GM(1,1) model.

Definition 3[2] Equation

is called the whitening differential equation of (2).Parametersaandbare usually calculated by least square method, and the calculation formula is defined as

where

x(1)(1)=x(0)(1),

Set initial value then time response function of (3) can be obtained.

The predicted values ofX(0)can be obtained by

3.Optimized GM(1,1) modeling approach based on firefly algorithm

In this section,we propose the SAGM(1,1) model.Then a new multi-parameter simultaneous estimation method is designed.The flow chart of SAGM(1,1) model is shown in Fig.1.

Fig.1 Flow chart of the proposed SAGM(1,1) model

3.1 Optimization scheme of the GM(1,1) model adjustable parameters

In this subsection, optimization scheme of the GM(1,1)model adjustable parameters is designed.

3.1.1 Background value

Inappropriate background value will reduce prediction accuracy of GM(1,1) model [24,37].The background value of the traditional GM(1,1) model is obtained by performing an adjacent average generating operation onX(1).However, when time series change sharply, this background value calculation method will reduce the prediction accuracy of GM(1,1) model.To address this issue,we introduce a dynamic linear interpolation method to optimize the background value.This method not only improves the prediction accuracy of the traditional grey prediction model, but also satisfies the original intention of simplicity of grey system theory, whose calculation formula is given as

3.1.2 Initial condition

The initial value of traditional grey forecasting model usually takesx(0)(1), which is contrary to the new information priority principle in grey system theory.To address this issue, we propose a new technique for determining initial value based on existing research results,and the new proposed initial value calculation method is defined as

In addition, the newly proposed initial value also satisfy β1<β2<···<βm, when parameter λ belongs to(0,1).The following formula can clearly illustrate this fact:

The time response function of (3) with the optimized initial condition can be derived.

3.1.3 Accumulated generating operator

Accumulative generation operation is an important feature that distinguishes grey modeling from other modeling methods.Traditional first-order accumulative generation operation not only contradicts the principle of new information priorities but also fails to effectively explore nonlinear components embedded in uncertainty systems,which restricts the prediction accuracy of grey forecasting models to some extent [26].In this paper, it is believed that the order of accumulation generating operator is an unknown parameter, which needs to be determined according to the characteristics of observed datasets.To further improve adaptability of GM(1,1) model, we introduce the idea of fractional order accumulative generation operation, which was first studied by Wu et al.[26].

3.2 The proposed SAGM(1,1) model

In this part, a new self-adaptive grey forecasting model is proposed, termed as SAGM(1,1) model, specific modeling steps are given in Definition 4.

Definition 4LetX(0)=[x(0)(1),x(0)(2),···,x(0)(m)](m≥4) be a non-negative original time series,γis the order of the accumulated generating operator,X(γ)=[x(γ)(1),x(γ)(2),···,x(γ)(m)] is the γ-order accumulated generating sequence (γ-AGO) ofX(0), where

Differential equation

is called SAGM(1,1) model with γ order accumulated,and its whitening equation can be expressed as

where

is called the background value of grey derivative.Parametersaγandbγcan be calculated by least-square method,its calculation formula can be given by

where

Supposex(γ)(t0)=β1x(γ)(1)+β2x(γ)(2)+···+βmx(γ)(m)is the initial condition,t0is the time input parameter, then the solution of whitening differential (11) can be obtained.

where

The simulated and predicted valuescan be obtained by performing an γth inverse accumulated generating operation (γ-IAGO) on

3.3 Parameter estimation of SAGM(1,1) model

In this subsection, a novel multi-parameter simultaneous optimization algorithm is proposed based on firefly algorithm, the new multi-parameter simultaneous optimization algorithm adopts machine learning ideas, takes all adjustable parameters of SAGM(1,1) model as input variables,and trains it with firefly algorithm.The parameter training mechanism can be expressed as

It can be seen that (15) is essentially a nonlinear constrained optimization model, and it is difficult to directly seek the exact solution by ordinary methods.Based on this, a firefly algorithm is employed to search optimal values of ε,γ,λ, andt0.

Firefly algorithm was pioneered by Yang in 2008 [38].It is a bionic optimization algorithm based on swarm intelligence.Over the past decade, firefly algorithm has been widely used in various fields due to its high computational efficiencies, simple structure, and strong searchability [39,40].The mechanism of firefly algorithm is provided below.

Assume thatXi=[xi1,xi2,···,xiM] (i=1,2,···,N) represents theith firefly in the population, whereNandMThe attractiveness between firefliesXiandXj(i≠j) can represent population size and dimension size, respectively.be calculated by

where β0is the attractiveness whenrij=0, γ denotes light absorption coefficient, andrijis the spatial distance between fireflyXiandXj, whose calculation formula can be expressed as

Suppose fireflyXiis less bright than fireflyXj, then the fireflyXiwill move to the fireflyXj, this movement can be defined as

where parametertis the number of iterations,ximandxjmare themth dimensions ofXiandXj, respectively,rand is a random number uniformly distributed in the range [0,1], and α is the step factor and α ∈[0,1].

3.4 Sobol’ sensitivity analysis

Sobol’ sensitivity analysis is a global sensitivity analysis method, which was initiated by Sobol’ in 1993 [41] and has been widely used in various fields [42,43].The core algorithm of Sobol’ indices is to decompose the total variance of the objective function into the sum of the variance of a single parameter and the variance of the interaction between multiple parameters, thereby obtaining the global sensitivity of the parameters.

Lety=f(X) be an objective function, whereX=[x1,x2,···,xm], andxk(k=1,2,···,m) is the input variable of functiony=f(X) can be expressed as the objective function.The total variance of the objective

whereV(y) stands for total variance,Viis the partial variance of the input variablexi,Vijrepresents the interaction variance between variablesxiandxj, andV1,2,···,mis the variance of interaction among multiple input variablesxi(i=1,2,···,m).Sobol’ indices can be obtained by the following equations:

The first-order Sobol’ index can be expressed as

The second-order Sobol’ index can be expressed as

The total sensitivity index of the parameterxican be defined as

4.Application

4.1 Experimental design

With the rapid development of Anhui’s economy, electricity consumption in Anhui province has increased dramatically in recent years, especially since the 12th Five-Year Plan.Statistics show that the electricity consumption in Anhui province in 2019 was 230.068 billion kW·h, an increase of 2.13 times over 2010.An efficient power supply system is an important prerequisite for social and economic development.Therefore, an accurate power demand forecast is of great significance for the government to formulate correct energy policy and power planning.In this section, Anhui’s annual electricity consumption dataset is applied to evaluate the prediction performance of the proposed SAGM(1,1) model.And the superiority of the SAGM(1,1) model is demonstrated by comparing to other competing models, including GM(1,1) model [2], the optimized GM(1,1) model(GM(1,1,x(1)(n))) [17], discrete grey forecasting model(DGM(1,1)) [44], grey system model with the fractional order accumulation (FGM(1,1)) [26], and the novel selfadaptive grey model (NSGM(1,1)) [20].Moreover, three effective model evaluation indicators including mean absolute percent error (MAPE), root mean square error(RMSE), and mean absolute error (MAE) are employed to test the prediction performance of the proposed SAGM(1,1) model, which are

wherex(0)(k) is the original observation at timek,xˆ(0)(k)is the predicted value at timek.According to the above indicators’ calculation formulas, it can be concluded that the smaller the MAPE, RMSE, and MAE values are, the higher the prediction accuracy is.

The original dataset of Anhui’s annual electricity consumption is collected from Anhui Statistical Yearbook[45−49].The data from 2010 to 2018 are used as a training set for modeling, and the data of 2019 is used as a validation set to check the prediction power of the proposed SAGM(1,1) model.The experiment is implemented by programming in python3.7 running on Windows 10 with 64 bits 2.90 GHz Intel Core i5-10400 CPU and 16.00 GB of RAM.

4.2 Sensitivity analysis for SAGM(1,1) model adjustable parameters

To identify key parameters and improve the accuracy of model parameter calibration, Sobol’ indices are used to calculate the global sensitivity of the SAGM(1,1) model adjustable parameters, the results are reported in Table 1.Table 1 shows that dynamic weighting parameter λ and time input parametert0are the two important parameters,in which time input parametert0is the most critical parameter, and its total sensitivity index is 0.66477.Moreover, there is a significant interaction between the parameters of the SAGM (1,1) model, as shown in Fig.2.Therefore, multi-parameter simultaneous optimization algorithm is an important approach to improve the accuracy of the SAGM(1,1) model parameter calibration.

Table 1 First order and total order sensitivity indices of the SAGM (1,1) model parameters

Fig.2 Second-order interaction effect of the SAGM(1,1) model parameters

4.3 Results analysis

Based on the above analysis, firefly algorithm is employed to train the adjustable parameters of SAGM(1,1)model, and the related parameters setting of the firefly algorithm are shown in Table 2.The optimization trajectory of the SAGM(1,1) model is described in Fig.3.It can be seen from Fig.3 that the firefly algorithm has high efficiency to catch the optimal solution.

Table 2 Parameters setting for firefly algorithm

Fig.3 Convergence curve for SAGM(1,1) model

The prediction results and errors of various grey models are listed in Table 3.It can be seen from Table 3 that compared with the competing models, SAGM(1,1) model has the smallest MAPE, RMSE, and MAE on both simulated period and predicted period.Therefore, the new proposed approach can significantly improve the prediction performance of GM(1,1) model.

Table 3 Comparison results of different models

5.Conclusions

Since existing GM(1,1) model family all based on assumption that some adjustable parameters are known,the assumption made by these studies may reduce their prediction performance in some cases.To address this issue, this paper proposes a novel self-adaptive GM(1,1)optimization model (denoted as SAGM(1,1)) by removing modeling assumptions of existing GM(1,1) model family.The main conclusions of this paper can be summarized.

(i) The proposed SAGM(1,1) model can effectively overcome the defects of the existing GM(1,1) model family, and significantly improve the prediction performance of the existing GM(1,1) model family in small sample time series prediction.

(ii) A new multi-parameter simultaneous optimization scheme is developed for SAGM(1,1) model.The new parameter optimization scheme adopts machine learning ideas, takes all adjustable parameters of SAGM(1,1) model as input variables, and trains it with firefly algorithm.The new scheme has greatly improved estimation accuracy of model parameters.

(iii) The global sensitivity of SAGM(1,1) model adjustable parameters are investigated by Sobol’ method.Empirical results demonstrate that there is a strong interaction effect between the parameters of SAGM (1,1)model.

Although the proposed SAGM(1,1) model provides an effective solution for small sample time series prediction,there is still room for improvement.For example, SAGM(1,1) model is a univariable grey forecasting model and does not consider the influence of external factors.Therefore, studying the self-adaptive multivariable grey forecasting model is a valuable research direction.Moreover,the proposed SAGM(1,1) model can be further investigated by being integrated with the rolling mechanism or other circumstances.