# A New Multivariable Grey Convolution Model Based on Simpson’s Rule and Its Applications

Authors: Song Ding; Ruojin Li
Journal: Complexity
 (2020)

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## Abstract

Accurate estimations can provide a solid basis for decision-making and policy-making that have experienced some kind of complication and uncertainty. Accordingly, a multivariable grey convolution model (GMC (1,n)) having correct solutions is put forward to deal with such complicated and uncertain issues, instead of the incorrect multivariable grey model (GM (1, n)). However, the conventional approach to computing background values of the GMC (1, n) model is inaccurate, and this model’s forecasting accuracy cannot be expected. Thereby, the drawback analysis of the GMC (1, n) model is conducted with mathematical reasoning, which can explain why this model is inaccurate in some applications. In order to eliminate the drawbacks, a new optimized GMC (1, n), shorted for OGMC (1, n), is proposed, whose background values are calculated based on Simpson’ rule that is able to efficiently approximate the integration of a function. Furthermore, its extended version that uses the Gaussian rule to discretize the convolution integral, abbreviated as OGMCG (1, n), is proposed to further enhance the model’s forecasting ability. In general, these two optimized models have such advantages as simplified structure, consistent forecasting performance, and satisfactory efficiency. Three empirical studies are carried out for verifying the above advantages of the optimized model, compared with the conventional GMC (1, n), GMCG (1, n), GM (1, n), and DGM (1, n) models. Results show that the new background values can effectively be calculated based on Simpson’s rule, and the optimized models significantly outperform other competing models in most cases.


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## Body

## 1. Introduction

Grey system theory has gained extensive attentions from worldwide researchers and has been successfully used in many fields with favorable outcomes since it was designed by Deng in 1982 [1–3]. This theory is capable of addressing issues characterized by uncertainty, insufficient information, and limited data points, thereby providing strong technical support for uncertain analysis [4, 5]. As the most important part of this theory, grey prediction approach and its extended versions have captured enormous attention owing to their capability to provide accurate forecasts, especially under the situation that facing sparse data and poor information [6, 7]. In contrast to the intelligent techniques and traditional statistical models, which require a large amount of data for model calibration, grey models only need limited data points to estimate the system behavior.Among the grey-model groups, theGM1,n model, of which 1 means one order of differential function and n stands for the number of total variables, gains its reputation and operates as a typical forecasting technique due to its capability to deal with the system prediction issues that are influenced by a range of relevant factors. Unfortunately, as the solutions to the whitening function for GM1,n are rough and incorrect [8], it is prone to produce large errors in applications. Thereby, since its introduction, many researchers carried out studies for further improve its prediction performance in terms of the background values [9], discrete variants [10], model structures [11], fractional model [12], and time-delayed versions [13]. Out of these, a novel grey prediction model with convolution integral, namely, GMC1,n, was designed by Tien [14] with a view to avoiding the incorrect solutions of the GM1,n model. As Tien presented, a convolution integral exists in the analytic solution to the whitening function of GM1,n, and the Trapezoid formula is used to discretize the convolution integral. Many cases have proved that the GMC1,n model outperforms the typical GM1,n. Moreover, for further adapting to the sequences having various characteristics, a range of variants derived from the GMC1,n model is put forward, such as DGDMC1,n [15–19] IGDMC1,n, FGMC1,n, NGMC1,n, GMCN1,n, and kernel-based nonlinear GMC1,n [20].In addition to these extended versions, higher accuracy grey approaches are typically obtained by reconstructing the background values, which are of great importance to the accurate parameter estimations. Essentially, the background values are used to smooth data to reduce randomness further [9] and customarily are represented by the average values of adjacent neighbor 1-AGO sequences, namely, Zi1t=0.5Xi1t+Xi1t−1,i=1,2,…. Detailed discussions on the reasonability and practicability have been carried out from diverse perspectives [1, 21, 22]. However, the rough estimations of background value are prone to give rise to the inconsistency of the grey differential and continuous equations, which results in the inaccurate forecasting performance of the target model. In order to improve the model’s performance, Wang and Hao [23] incorporate the dynamic coefficients λi∈0,1i=1,2,…,n into the structure of background values in GMC1,n, namely, the background values for the system behavior sequence Z11rp+t=λ1X11rp+t+1−λ1X11rp+t−1 and the background values for the relevant sequences Zi1t=λiXi1t+1−λiXi1t−1,i=2,3,…. Subsequently, for the purpose of further enhancing precision, various heuristic intelligent techniques are employed to determine the background value coefficient, including Particle Swarm Optimization PSO [24–26], Genetic Algorithm GA [27, 28], Ant Lion Optimizer ALO [29], and Ant Colony Algorithm ACA [30].Though the previous improvements in background values with the support of those heuristic intelligent techniques lead to the improvements of forecasting ability, they also result in the low-level, simply-repeated, and complicated computations of grey models. Meanwhile, it is troublesome and difficult for beginners or inexperienced users to choose a suitable heuristic intelligent algorithm for solving actual forecasting problems. Therefore, proposing an accurate forecasting grey model that has a simple structure and easy-understanding computations with a focus on the background values is a very valuable work. General consensus gotten from the literature was that accurate background values are regarded as a crucial requirement to ensure the reliability and practicability of theGMC1,n model [23, 24], and the forecasting precision is heavily dependent on their correct computations. Supporting this argument, one of the major contributions in this paper stems from the mathematical derivation of the actual background values, finding its differences from the estimated one in the conventional GMC1,n model. Another equally important merit of this paper is to design an optimized OGMC1,n, whose background values are reconstructed and calculated based on Simpson’s rule. Moreover, the new model’s extended version is also derived by using the Gaussian rule, instead of the Trapezoidal rule, to calculate the whitening function with convolution integral, namely, OGMCG1,n. Subsequently, three cases of diverse subjects, including tensile strength of a material, China’s gross industrial output value, and electronic waste in Washington State, are utilized for verifying the efficacy and robustness of the proposed models. Empirical results, the optimized background values, based on Simpson’s rule can act as a supplementary for the current estimation methods of grey models. Furthermore, the OGMCG1,n model having optimized background values performs best among all competitors. The detailed comparisons will be elaborated in Section 3.This work has the following structure. The research methodology is presented in Section2. In this section, the focus is on the previous procedures and shortcomings of the GMC1,n model. Subsequently, the optimized OGMC1,n and OGMCG1,n models are elaborated on. Section 3 is dedicated to three different cases for depicting the accuracy and reliability of the proposed models in comparison with a range of competitors. Conclusions and discussions are provided in the last Section 4.The nomenclature utilized in this paper is presented below and detailed explanations related to the formulas employed in each technique are given in the following sections.


## 2. Methodology

### 2.1. Overview of the GMC (1,n) Model

TheGMC1,n model has gains popularity in practical applications because of its accurate estimations and simple structures. Many researchers such as Tien [14–17], Ma et al. [31, 32], and Duman et al. [24], have elaborated the procedures for this model, which can currently be outlined below:Step 1: collect the original data. Assume thatX10=x10rp+1,x10rp+2,…,x10rp+r,r≥4 is a data sequence of a system’s characteristic variable, short for the system behavior sequence, and Xi0=xi01,xi02,…,xi0r,r≥4,i=2,3,…,n are the data sequences of relevant factors, abbreviated as the relevant series. In the series, r represents the number of entries for establishing the model, rp stands for the delayed period, and n is the total number of variables. Moreover, these two categories of series are nonnegative and equally spaced over time.Step 2: generate theX1 by using the first-order accumulated generating operation 1−AGO. Then the accumulated sequences Xi1,i=1,2,…,n of the original data Xi0i=1,2,…,n can be produced by using(1)x11rp+t=∑j=1rp+tx10rp+j,t=1,2,…,rp+r,(2)xi1t=∑j=1tx10j.Accordingly, the1−AGO sequences of the system behavior can be obtained as(3)X11=x11rp+1,x11rp+2,…,x11rp+r.And the1−AGO sequences of the influencing factors are(4)Xi1=xi11,xi12,…,xi1r,i=2,3,…,n.Step 3: build theGMC1,n model. Its first-order whitening equation is presented by(5)dX11rp+tdt+β1X11rp+t=∑i=2nβiXi1t+u,whereβ1 and u represent the system’s development coefficient and grey control parameter, respectively. Additionally, βiXi1t is known as the driving term, and βi,i=2,3,…,n are the driving coefficients corresponding to the relevant factor sequences Xi0,i=2,3,…,n, respectively.The grey derivative in equation (5) can be approximately estimated by(6)dX11rp+tdt=limΔt⟶0X11rp+t−X11rp+t−ΔtΔt=X11rp+t−X11rp+t−1=X10rp+t.In addition, the background values ofX11rp+t and X11rp+t are usually taken as(7)Z11rp+t=0.5X11rp+t+X11rp+t−1,(8)Zi1t=0.5Xi1t+Xi1t−1.However, recent studies [9, 23, 24] has pointed out that, the estimated values of the background value may be unequal to the actual values, explained in Section 2.2. Hence, based on the background values, the first-order grey differential equation can be formulated as(9)X10rp+t+β1Z11rp+t=∑i=2nβiZi1t+u.Step 4: estimate the parameters by using the Ordinary Least Squares method (OLS). Substituting the values of t into equation (9), we can obtain(10)X10rp+2+β1Z11rp+2=∑i=2nβiZi12+uX10rp+3+β1Z11rp+3=∑i=2nβiZi13+u⋮X10rp+r+β1Z11rp+r=∑i=2nβiZi1r+u.In this matrix form,Y=Bβ^, where(11)Y=X102X103⋮X10r,B=−Z11rp+2Z212⋯Zn121−Z11rp+3Z213⋯Zn131⋮⋮⋱⋮⋮−Z11rp+rZ21r⋯Zn1r1.By solving the matrixY=Bβ^, the estimated values of the parameters can be calculated:(12)β^=β^1,β^2,…,β^n,u^=BTB−1BTY.Step 5: obtain the response function for projections. As introduced above, the whitening equation of theGMC1,n model is displayed in equation (5). By substituting the parameter values β^=β^1,β^2,…,β^n,u^ into equation (5) and opting the X^11rp+1=X11rp+1 as the initial condition, the time response function can be obtained by solving the whitening equation in equation (5), which will be presented as(13)X^11rp+t=X101rp+1e−β1t−1+∫1te−β1t−τfτdτ,where(14)fτ=∑i=2nβiXi1τ+u.As Tien [14] introduced previously, the Trapezoidal rule is originally used to discretize the convolution integral in equation (13) in the traditional GMC1,n model, and subsequently, the response function for 1−AGO sequences can be determined by using the following equation:(15)X^11rp+t=X101rp+1e−β1t−1+pt−2∑ξ=2t12e−β1t−ξfξ+e−β1t−ξ+1fξ−1,where the unit step functionpt−2 in equation (15) is formulated by(16)pt−2=0,t<2,1,t≥2.Unfortunately, due to the traditionalGMC1,n model’s poor performance, the Gaussian rule gains more popularity than Trapezoidal equivalent in recent studies for discretizing the convolution integral [17, 31]. Consequently, the Gaussian rule is used in this paper as well. Then, the discrete response function can be presented below:(17)X^11rp+t=X101rp+1e−β1t−1+ut−2∑ξ=2te−β1t−ξ+1/2∗12fξ+fξ−1.where the unit step functionut−2 in equation (17) is formulated by(18)ut−2=0,t<2,1,t≥2.For convenience, theGMC1,n model whose convolution integral is discretized by using the Gaussian rule is remarked as GMCG1,n.Step 6: generate the simulative and predictive values in the original domain. Through equation (14), we will get the fitted and predicted values of the 1−AGO sequences, which normally are considered as intermediate sequences. Subsequently, the recursive function in the original domain can be given by utilizing the 1−IAGO method, which is defined as(19)X^10rp+t=X^11rp+t−X^11rp+t−1,t=2,3,…,r,r+1,…,whereX^10rp+t,t=2,3,…,r are called simulative values, and X^10rp+t,t≥r+1 are named as predictive values.As outlined previously, when theu and rp satisfy u=rp=0, the GMC1,n is reduced to the traditional GMC1,n model. Moreover, when n=1 and rp=0, the GMC1,n model becomes the original GMC1,1 model. Therefore, the GMC1,n model is extensions of the GM1,n and GM1,1 models, indicating that the GMC1,n model has wider application fields.


### 2.2. Drawback Analysis of the GMC (1,n) Model

Although the popularity ofGMC1,n has been increasing steadily, some inherent drawbacks still exist, which may generate unacceptable forecasting errors. After reviewing the procedures of the GMC1,n model, we can find that two core components of this model are the model establishment and parameter estimation, which refers to equations (5) and (9), respectively. Especially for the parameter estimation, the accuracy of the estimated parameters is the precondition of obtaining the response function in equation (5). Specifically, equation (5) is used for predicting the system behavior sequence on the basis of the known parameters β^=β^1,β^2,…,β^n,u^, which are estimated by equation (9). Hence, prior to obtaining projections, the optimal or near-optimal parameters in the GMC1,n and GMCG1,n models are essential for better forecasting precision, which are significantly influenced by the background value.To expose the inherent weakness of theGMC1,n model, the actual meaning of background value needs to be further analyzed by comparing the grey differential equation in equation (9) and whitening equation in equation (5). To this end, definite integrals of equation (5) will be given on the interval t−1,t, which is displayed as(20)∫t−1tdX11rp+τdtdτ+β1∫t−1tX11rp+τdτ=∑i=2nβi∫t−1tXi1τdτ+∫t−1tudτ,where(21)∫t−1tdX11rp+τdtdτ=X11rp+t−X11rp+t−1=X10rp+t,(22)∫t−1tudτ=u.Substituting equations (21) and (22) into equation (20), the discrete form of GMC1,n can be obtained as follows:(23)X10rp+t+β1∫t−1tX11rp+τdτ=∑i=2nβi∫t−1tXi1τdτ+u.In equation (23), the real background values in both sides of the equation are noted as Z1rp+t=∫t−1tX11rp+τdτ and Zit=∫t−1tXi1τdτ. Then, comparing equation (23) with equation (9), we find that the actual background values of ∫t−1tX11rp+t and ∫t−1tXi1t are approximately represented by the average values of adjacent neighbor sequences in equations (7) and (8) for the GMC1,n model. However, they are not always mathematically equivalent. Accordingly, the gap between the actual background values and the approximate ones produces the inaccurate parameter estimations, which exert a negative influence on the predictive performance of the whitening equation in equation (5).To address the weakness, Wang and Hao [23] put forward a new formula to calculate the background values based on the Mean Value Theorem of Integrals, which can be expressed as(24)Z11rp+t=∫t−1tX11rp+τdτ=λ1X11rp+t+1−λ1X11rp+t−1,(25)Zi1t=∫t−1tXi1τdτλi=λiXi1t+1−λiXi1t−1,where λii=1,2,…,n represents the dynamic interpolation coefficients. Obviously, equations (24) and (25) are equivalent to equations (7) and (8) when λ1=λ2=⋯λn=0.5, which infers that the optimized GMC1,n model having dynamic background values in [23] has better adaptability than the original one having fixed interpolation coefficients. To determine the dynamic interpolation coefficients, metaheuristic algorithms that have complex structures and require well-versed computer skills, such as Particle Swarm Algorithm and Genetic Algorithm, are necessary for combining with the GMC1,n model.Although the optimizedGMC1,n model in [23] provides an effective solution to avoid the deviations between the actual and estimated background values, potential difficulties in using this new model may increasingly arise, especially for beginners, due to the introduction of the complex optimization algorithms. Furthermore, these metaheuristic algorithms may not generate the satisfactory parameter estimations because they can fall into the local optima stagnation, consequently failing to find the actual global optimum.Therefore, to effectively and accurately estimate the parameters in theGMC1,n model, an alternative method based on Simpson’s rule is proposed for accurately measuring the background values in this paper. This algorithm, elaborated on in Section 2.3, has merits in simple structure and is easy to understand for beginners. Furthermore, it can also significantly improve the accuracy of the GMC1,n and GMCG1,n models, which will be demonstrated by several experiments in Section 3.


### 2.3. A New Optimized GMC (1,n) Model Based on Simpson’s Rule

In order to overcome the shortcomings of the above approaches to estimating background values, mentioned in Section2.2, a new method based on Simpson’s rule is employed to calculate the background value in the GMC1,n model. The flowchart of the new optimized GMC1,n model is displayed in Figure 1. Furthermore, the background values in [14, 23] are also presented in Figure 1 in comparison with the new model.Figure 1
Flowchart of the GMC (1,n) model with different approaches to estimating background values.Simpson’s rule, named after the English mathematician, Thomas Simpson (1710–1761), is a numerical approach to approximate the integration of a function. This method is particularly useful when integration is difficult or even impossible to do by utilizing standard techniques. The most commonly used function of Simpson’s rule is provided as(26)∫abfτdτ=b−a6fa+4fa+b2+fb.As introduced in Section2.2, the actual background value can be estimated by computing the integration ∫t−1tX11rp+t and ∫t−1tXi1t, instead of the average values of adjacent neighbor sequences in equations (7) and (8). In order to be consistent with the forms of Simpson’s rule in equation (26), the grey differential equation that is used for parameter estimation can be obtained by considering the definite integrals of equation (5) on the interval t−1,t+1, which is presented as(27)∫t−1t+1dX11rp+τdtdτ+β1∫t−1t+1X11rp+τdτ=∑i=2nβi∫t−1t+1Xi1τdτ+∫t−1t+1udτ,where(28)∫t−1t+1dX11rp+τdtdτ=X11rp+t+1−X11rp+t−1,(29)∫t−1t+1udτ=2u.Hence, substituting equations (28) and (29) into equation (27), the grey differential equation is mathematically equivalent to(30)X11rp+t+1−X11rp+t−1+β1∫t−1t+1X11rp+τdτ=∑i=2nβi∫t−1t+1Xi1τdτ+2u.According to Simpson’s rule in equation (26), we can get(31)∫t−1t+1X11rp+τdτ=13X11rp+t−1+4X11rp+t−1+X11rp+t+1,(32)∫t−1t+1Xi1rp+τdτ=13Xi1t−1+4Xi1t+Xi1t+1.Substituting and simplifying, equation (30) mathematically equals to(33)X11rp+t+1−X11rp+t−1+β13X11rp+t−1+4X11rp+t−1+X11rp+t+1=∑i=2nβi3Xi1t−1+4Xi1t+Xi1t+1+2u.Simplifying, we can get this:(34)−β13X11rp+t−1+4X11rp+t−1+X11rp+t+1+∑i=2nβi3Xi1t−1+4Xi1t+Xi1t+1+2u=X10rp+t+1+X10rp+t,t=2,3,…,r−1.Assume that(35)mt,1=−13X11rp+t−1+4X11rp+t−1+X11rp+t+1,mt,i=13Xi1t−1+4Xi1t+Xi1t+1,i=2,3,…,n,dt=X10rp+t+1+X10rp+t.Subsequently, substituting equation (35) and the values of t into equation (34), we can obtain(36)β1m2,1+β2m2,2+β3m2,3+⋯+βnm2,n+2u=d2β1m3,1+β2m3,2+β3m3,3+⋯+βnm3,n+2u=d3⋯⋯⋯⋯β1mr−1,1+β2mr−1,2+β3mr−1,3+⋯+βnmr−1,n+2u=dr−1.Equation (36) can be written into the matrix form M=Dβ^, where(37)M=d2d3⋮dr−1,D=m2,1m2,2⋯m2,n2m3,1m3,2⋯m3,32⋮⋮⋱⋮⋮mr−1,1mr−1,2⋯mr−1,n2.Solving this above matrix, the estimated values of parameters will be formulated:(38)β^=β^1,β^2,…,β^n,u^=DTD−1DTM.Similar to theGMC1,n and GMCG1,n models, we can also achieve two optimized models by using the Trapezoidal and Gaussian rule to discretize the convolution integral in equation (13). Specifically, substituting the parameter values β^=β^1,β^2,…,β^n,u^ into equations (14) and (17), the discrete response function can be obtained:(39)X^11rp+t=X10rp+1e−β1t−1+ut−2∑ξ=2te−β1t−ξ+1/2∗12fξ+fξ−1,(40)X^11rp+t=X10rp+1e−β1t−1+ut−2∑ξ=2t12e−β1t−ξfξ+e−β1t−ξ+1fξ−1,Where the unit step functionut−2 and function ft in equations (39) and (40) are formulated by(41)ut−2=0,t<2,1,t≥2,ft=∑i=2nβiXi1t+u.For convenience, these two models in (39) and (40) are remarked as OGMC1,n and OGMCG1,n models, respectively. Subsequently, by utilizing the 1−IAGO method, the recursive function in the original domain is defined as(42)X^10rp+t=X^11rp+t−X^11rp+t−1,t=2,3,…,r,r+1,…,WhereX^10rp+t,t=2,3,…,r are called simulative values, and X^10rp+t,t≥r+1 are named as predictive values.Based on Simpson’s rule, the background values of theGMC1,n model can be accurately calculated, which is quite different from those in [14, 23]. Moreover, Figure 1 provides the detailed procedures of the optimized model, which is helpful for beginners to use this model.


### 2.4. Performance Evaluation

In addition to the modeling procedures of theGMC1,n model, the measuring indicators are also essential for evaluating the model’s performance. In this study, three different indices will be adopted to examine the performance of competing models, including APE (Absolute Percentage Error), MAPE (Mean Absolute Percentage Error), and RMSE (Root Mean Squared Error).The algorithms of the three indices are calculated using the following equations:(43)APE=x^0t−x0tx0t×100%,t=2,3,…,n,n+1,…,MAPE=1n−1∑t=2nAPEt,RMSE=1n−1∑t=2nx^0t−x0t2,where x0t is the actual value at time t, and the x^0t is the corresponding simulative or predictive value at time t.


## 2.1. Overview of the GMC (1,n) Model

TheGMC1,n model has gains popularity in practical applications because of its accurate estimations and simple structures. Many researchers such as Tien [14–17], Ma et al. [31, 32], and Duman et al. [24], have elaborated the procedures for this model, which can currently be outlined below:Step 1: collect the original data. Assume thatX10=x10rp+1,x10rp+2,…,x10rp+r,r≥4 is a data sequence of a system’s characteristic variable, short for the system behavior sequence, and Xi0=xi01,xi02,…,xi0r,r≥4,i=2,3,…,n are the data sequences of relevant factors, abbreviated as the relevant series. In the series, r represents the number of entries for establishing the model, rp stands for the delayed period, and n is the total number of variables. Moreover, these two categories of series are nonnegative and equally spaced over time.Step 2: generate theX1 by using the first-order accumulated generating operation 1−AGO. Then the accumulated sequences Xi1,i=1,2,…,n of the original data Xi0i=1,2,…,n can be produced by using(1)x11rp+t=∑j=1rp+tx10rp+j,t=1,2,…,rp+r,(2)xi1t=∑j=1tx10j.Accordingly, the1−AGO sequences of the system behavior can be obtained as(3)X11=x11rp+1,x11rp+2,…,x11rp+r.And the1−AGO sequences of the influencing factors are(4)Xi1=xi11,xi12,…,xi1r,i=2,3,…,n.Step 3: build theGMC1,n model. Its first-order whitening equation is presented by(5)dX11rp+tdt+β1X11rp+t=∑i=2nβiXi1t+u,whereβ1 and u represent the system’s development coefficient and grey control parameter, respectively. Additionally, βiXi1t is known as the driving term, and βi,i=2,3,…,n are the driving coefficients corresponding to the relevant factor sequences Xi0,i=2,3,…,n, respectively.The grey derivative in equation (5) can be approximately estimated by(6)dX11rp+tdt=limΔt⟶0X11rp+t−X11rp+t−ΔtΔt=X11rp+t−X11rp+t−1=X10rp+t.In addition, the background values ofX11rp+t and X11rp+t are usually taken as(7)Z11rp+t=0.5X11rp+t+X11rp+t−1,(8)Zi1t=0.5Xi1t+Xi1t−1.However, recent studies [9, 23, 24] has pointed out that, the estimated values of the background value may be unequal to the actual values, explained in Section 2.2. Hence, based on the background values, the first-order grey differential equation can be formulated as(9)X10rp+t+β1Z11rp+t=∑i=2nβiZi1t+u.Step 4: estimate the parameters by using the Ordinary Least Squares method (OLS). Substituting the values of t into equation (9), we can obtain(10)X10rp+2+β1Z11rp+2=∑i=2nβiZi12+uX10rp+3+β1Z11rp+3=∑i=2nβiZi13+u⋮X10rp+r+β1Z11rp+r=∑i=2nβiZi1r+u.In this matrix form,Y=Bβ^, where(11)Y=X102X103⋮X10r,B=−Z11rp+2Z212⋯Zn121−Z11rp+3Z213⋯Zn131⋮⋮⋱⋮⋮−Z11rp+rZ21r⋯Zn1r1.By solving the matrixY=Bβ^, the estimated values of the parameters can be calculated:(12)β^=β^1,β^2,…,β^n,u^=BTB−1BTY.Step 5: obtain the response function for projections. As introduced above, the whitening equation of theGMC1,n model is displayed in equation (5). By substituting the parameter values β^=β^1,β^2,…,β^n,u^ into equation (5) and opting the X^11rp+1=X11rp+1 as the initial condition, the time response function can be obtained by solving the whitening equation in equation (5), which will be presented as(13)X^11rp+t=X101rp+1e−β1t−1+∫1te−β1t−τfτdτ,where(14)fτ=∑i=2nβiXi1τ+u.As Tien [14] introduced previously, the Trapezoidal rule is originally used to discretize the convolution integral in equation (13) in the traditional GMC1,n model, and subsequently, the response function for 1−AGO sequences can be determined by using the following equation:(15)X^11rp+t=X101rp+1e−β1t−1+pt−2∑ξ=2t12e−β1t−ξfξ+e−β1t−ξ+1fξ−1,where the unit step functionpt−2 in equation (15) is formulated by(16)pt−2=0,t<2,1,t≥2.Unfortunately, due to the traditionalGMC1,n model’s poor performance, the Gaussian rule gains more popularity than Trapezoidal equivalent in recent studies for discretizing the convolution integral [17, 31]. Consequently, the Gaussian rule is used in this paper as well. Then, the discrete response function can be presented below:(17)X^11rp+t=X101rp+1e−β1t−1+ut−2∑ξ=2te−β1t−ξ+1/2∗12fξ+fξ−1.where the unit step functionut−2 in equation (17) is formulated by(18)ut−2=0,t<2,1,t≥2.For convenience, theGMC1,n model whose convolution integral is discretized by using the Gaussian rule is remarked as GMCG1,n.Step 6: generate the simulative and predictive values in the original domain. Through equation (14), we will get the fitted and predicted values of the 1−AGO sequences, which normally are considered as intermediate sequences. Subsequently, the recursive function in the original domain can be given by utilizing the 1−IAGO method, which is defined as(19)X^10rp+t=X^11rp+t−X^11rp+t−1,t=2,3,…,r,r+1,…,whereX^10rp+t,t=2,3,…,r are called simulative values, and X^10rp+t,t≥r+1 are named as predictive values.As outlined previously, when theu and rp satisfy u=rp=0, the GMC1,n is reduced to the traditional GMC1,n model. Moreover, when n=1 and rp=0, the GMC1,n model becomes the original GMC1,1 model. Therefore, the GMC1,n model is extensions of the GM1,n and GM1,1 models, indicating that the GMC1,n model has wider application fields.


## 2.2. Drawback Analysis of the GMC (1,n) Model

Although the popularity ofGMC1,n has been increasing steadily, some inherent drawbacks still exist, which may generate unacceptable forecasting errors. After reviewing the procedures of the GMC1,n model, we can find that two core components of this model are the model establishment and parameter estimation, which refers to equations (5) and (9), respectively. Especially for the parameter estimation, the accuracy of the estimated parameters is the precondition of obtaining the response function in equation (5). Specifically, equation (5) is used for predicting the system behavior sequence on the basis of the known parameters β^=β^1,β^2,…,β^n,u^, which are estimated by equation (9). Hence, prior to obtaining projections, the optimal or near-optimal parameters in the GMC1,n and GMCG1,n models are essential for better forecasting precision, which are significantly influenced by the background value.To expose the inherent weakness of theGMC1,n model, the actual meaning of background value needs to be further analyzed by comparing the grey differential equation in equation (9) and whitening equation in equation (5). To this end, definite integrals of equation (5) will be given on the interval t−1,t, which is displayed as(20)∫t−1tdX11rp+τdtdτ+β1∫t−1tX11rp+τdτ=∑i=2nβi∫t−1tXi1τdτ+∫t−1tudτ,where(21)∫t−1tdX11rp+τdtdτ=X11rp+t−X11rp+t−1=X10rp+t,(22)∫t−1tudτ=u.Substituting equations (21) and (22) into equation (20), the discrete form of GMC1,n can be obtained as follows:(23)X10rp+t+β1∫t−1tX11rp+τdτ=∑i=2nβi∫t−1tXi1τdτ+u.In equation (23), the real background values in both sides of the equation are noted as Z1rp+t=∫t−1tX11rp+τdτ and Zit=∫t−1tXi1τdτ. Then, comparing equation (23) with equation (9), we find that the actual background values of ∫t−1tX11rp+t and ∫t−1tXi1t are approximately represented by the average values of adjacent neighbor sequences in equations (7) and (8) for the GMC1,n model. However, they are not always mathematically equivalent. Accordingly, the gap between the actual background values and the approximate ones produces the inaccurate parameter estimations, which exert a negative influence on the predictive performance of the whitening equation in equation (5).To address the weakness, Wang and Hao [23] put forward a new formula to calculate the background values based on the Mean Value Theorem of Integrals, which can be expressed as(24)Z11rp+t=∫t−1tX11rp+τdτ=λ1X11rp+t+1−λ1X11rp+t−1,(25)Zi1t=∫t−1tXi1τdτλi=λiXi1t+1−λiXi1t−1,where λii=1,2,…,n represents the dynamic interpolation coefficients. Obviously, equations (24) and (25) are equivalent to equations (7) and (8) when λ1=λ2=⋯λn=0.5, which infers that the optimized GMC1,n model having dynamic background values in [23] has better adaptability than the original one having fixed interpolation coefficients. To determine the dynamic interpolation coefficients, metaheuristic algorithms that have complex structures and require well-versed computer skills, such as Particle Swarm Algorithm and Genetic Algorithm, are necessary for combining with the GMC1,n model.Although the optimizedGMC1,n model in [23] provides an effective solution to avoid the deviations between the actual and estimated background values, potential difficulties in using this new model may increasingly arise, especially for beginners, due to the introduction of the complex optimization algorithms. Furthermore, these metaheuristic algorithms may not generate the satisfactory parameter estimations because they can fall into the local optima stagnation, consequently failing to find the actual global optimum.Therefore, to effectively and accurately estimate the parameters in theGMC1,n model, an alternative method based on Simpson’s rule is proposed for accurately measuring the background values in this paper. This algorithm, elaborated on in Section 2.3, has merits in simple structure and is easy to understand for beginners. Furthermore, it can also significantly improve the accuracy of the GMC1,n and GMCG1,n models, which will be demonstrated by several experiments in Section 3.


## 2.3. A New Optimized GMC (1,n) Model Based on Simpson’s Rule

In order to overcome the shortcomings of the above approaches to estimating background values, mentioned in Section2.2, a new method based on Simpson’s rule is employed to calculate the background value in the GMC1,n model. The flowchart of the new optimized GMC1,n model is displayed in Figure 1. Furthermore, the background values in [14, 23] are also presented in Figure 1 in comparison with the new model.Figure 1
Flowchart of the GMC (1,n) model with different approaches to estimating background values.Simpson’s rule, named after the English mathematician, Thomas Simpson (1710–1761), is a numerical approach to approximate the integration of a function. This method is particularly useful when integration is difficult or even impossible to do by utilizing standard techniques. The most commonly used function of Simpson’s rule is provided as(26)∫abfτdτ=b−a6fa+4fa+b2+fb.As introduced in Section2.2, the actual background value can be estimated by computing the integration ∫t−1tX11rp+t and ∫t−1tXi1t, instead of the average values of adjacent neighbor sequences in equations (7) and (8). In order to be consistent with the forms of Simpson’s rule in equation (26), the grey differential equation that is used for parameter estimation can be obtained by considering the definite integrals of equation (5) on the interval t−1,t+1, which is presented as(27)∫t−1t+1dX11rp+τdtdτ+β1∫t−1t+1X11rp+τdτ=∑i=2nβi∫t−1t+1Xi1τdτ+∫t−1t+1udτ,where(28)∫t−1t+1dX11rp+τdtdτ=X11rp+t+1−X11rp+t−1,(29)∫t−1t+1udτ=2u.Hence, substituting equations (28) and (29) into equation (27), the grey differential equation is mathematically equivalent to(30)X11rp+t+1−X11rp+t−1+β1∫t−1t+1X11rp+τdτ=∑i=2nβi∫t−1t+1Xi1τdτ+2u.According to Simpson’s rule in equation (26), we can get(31)∫t−1t+1X11rp+τdτ=13X11rp+t−1+4X11rp+t−1+X11rp+t+1,(32)∫t−1t+1Xi1rp+τdτ=13Xi1t−1+4Xi1t+Xi1t+1.Substituting and simplifying, equation (30) mathematically equals to(33)X11rp+t+1−X11rp+t−1+β13X11rp+t−1+4X11rp+t−1+X11rp+t+1=∑i=2nβi3Xi1t−1+4Xi1t+Xi1t+1+2u.Simplifying, we can get this:(34)−β13X11rp+t−1+4X11rp+t−1+X11rp+t+1+∑i=2nβi3Xi1t−1+4Xi1t+Xi1t+1+2u=X10rp+t+1+X10rp+t,t=2,3,…,r−1.Assume that(35)mt,1=−13X11rp+t−1+4X11rp+t−1+X11rp+t+1,mt,i=13Xi1t−1+4Xi1t+Xi1t+1,i=2,3,…,n,dt=X10rp+t+1+X10rp+t.Subsequently, substituting equation (35) and the values of t into equation (34), we can obtain(36)β1m2,1+β2m2,2+β3m2,3+⋯+βnm2,n+2u=d2β1m3,1+β2m3,2+β3m3,3+⋯+βnm3,n+2u=d3⋯⋯⋯⋯β1mr−1,1+β2mr−1,2+β3mr−1,3+⋯+βnmr−1,n+2u=dr−1.Equation (36) can be written into the matrix form M=Dβ^, where(37)M=d2d3⋮dr−1,D=m2,1m2,2⋯m2,n2m3,1m3,2⋯m3,32⋮⋮⋱⋮⋮mr−1,1mr−1,2⋯mr−1,n2.Solving this above matrix, the estimated values of parameters will be formulated:(38)β^=β^1,β^2,…,β^n,u^=DTD−1DTM.Similar to theGMC1,n and GMCG1,n models, we can also achieve two optimized models by using the Trapezoidal and Gaussian rule to discretize the convolution integral in equation (13). Specifically, substituting the parameter values β^=β^1,β^2,…,β^n,u^ into equations (14) and (17), the discrete response function can be obtained:(39)X^11rp+t=X10rp+1e−β1t−1+ut−2∑ξ=2te−β1t−ξ+1/2∗12fξ+fξ−1,(40)X^11rp+t=X10rp+1e−β1t−1+ut−2∑ξ=2t12e−β1t−ξfξ+e−β1t−ξ+1fξ−1,Where the unit step functionut−2 and function ft in equations (39) and (40) are formulated by(41)ut−2=0,t<2,1,t≥2,ft=∑i=2nβiXi1t+u.For convenience, these two models in (39) and (40) are remarked as OGMC1,n and OGMCG1,n models, respectively. Subsequently, by utilizing the 1−IAGO method, the recursive function in the original domain is defined as(42)X^10rp+t=X^11rp+t−X^11rp+t−1,t=2,3,…,r,r+1,…,WhereX^10rp+t,t=2,3,…,r are called simulative values, and X^10rp+t,t≥r+1 are named as predictive values.Based on Simpson’s rule, the background values of theGMC1,n model can be accurately calculated, which is quite different from those in [14, 23]. Moreover, Figure 1 provides the detailed procedures of the optimized model, which is helpful for beginners to use this model.


## 2.4. Performance Evaluation

In addition to the modeling procedures of theGMC1,n model, the measuring indicators are also essential for evaluating the model’s performance. In this study, three different indices will be adopted to examine the performance of competing models, including APE (Absolute Percentage Error), MAPE (Mean Absolute Percentage Error), and RMSE (Root Mean Squared Error).The algorithms of the three indices are calculated using the following equations:(43)APE=x^0t−x0tx0t×100%,t=2,3,…,n,n+1,…,MAPE=1n−1∑t=2nAPEt,RMSE=1n−1∑t=2nx^0t−x0t2,where x0t is the actual value at time t, and the x^0t is the corresponding simulative or predictive value at time t.


## 3. Experiments and Analysis

In this section, for the sake of demonstrating the practicability and efficacy of the new optimized model, three empirical studies are carried out in comparison with traditional multivariable grey prediction models.


### 3.1. Case One: Predicting the Tensile Strength of a Material

The multivariable grey prediction models, such asGM1,2 and GMC1,2, can be used for the indirect measurement of the tensile strength of a material [14], the efficacy of the OGMC1,2 and OGMCG1,2 models can be revealed by comparing the experimental results of predicting tensile strength of a material in comparison with the GMC1,2, GMCG1,2, GM1,2, and DGM1,2 models. The original data are presented in Table 1, which were collected from the experimental observations in Table 117 of Samuel [33]. In addition, the tensile strength represents the system behavior sequence X10, and the Brinell hardness works as the relevant series X20. Moreover, six data points of X10 and X20 are utilized for model calibration, and the remaining four observations of X10 are used for checking the predictive performance, namely, setting parameters n=2,r=6, rp=0, and rf=4 [33].Table 1
Original data in case one.
i (temperature)
Tensile strengthX10 (MPa)
Brinell hardnessX20 (HBW)
1 (400F∘)
897
514
2 (500F∘)
897
495
3 (600F∘)
890
444
4 (700F∘)
876
401
5 (800F∘)
848
352
6 (900F∘)
814
293
7 (1000F∘)
779
269
8 (1100F∘)
738
235
9 (1200F∘)
669
201
10 (1300F∘)
600
187As introduced in Sections2.1 and 2.3, the procedures and flowchart of GMC1,2 and its extensions are presented, and the procedures of GM1,2 and DGM1,2 are similar to those of GMC1,2. Thus, according to these procedures, the estimated parameter-values of six competing models are given in Table 2.Table 2
Estimated parameter values of six competing models in case one.
Parameters
GMC
1,2
OGMC
1,2
GMC
G
1,2
OGMC
G
1,2
GM
1,2
DGM
1,2
β
^
1
0.14378
0.15387
0.14378
0.15387
0.84617
−0.86519
β
^
2
0.26139
0.28354
0.26139
0.28354
1.93150
0.25951
u
^
891.16435
885.67276
891.16435
885.67276
—
755.74471TakingOGMC1,2 as an example for explanations, we can obtain the parameter values β^=β^1,β^2,u^=0.15387,0.28354,885.67276 after substituting the original data into equations (35) and (36). Subsequently, substituting the parameter values into equation (39), we will have the discrete response function:(44)X^11t=X101e−0.15387t−1+ut−2∑ξ=2te−0.15387t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=0.28354X21t+885.67276.By using the time response function in equation (44), the forecasted results and the values of three indices measuring accuracy will be obtained, which are shown in Table 3. It is obvious that four GMC1,2 and its extensions are superior to the traditional GM1,2 and DGM1,2 whose MAPE values reach 24.94% and 55.91%, respectively. The reason for this phenomenon is that GM1,2 has an incorrect time response function and DGM1,2 has narrow application fields. In contrast, the GMC1,2 and GMCG1,2 models are able to provide accurate forecasts because their modeling values with convolution integral are exact theoretically [14]. However, the OGMC1,2 and OGMCG1,2 models seem to obtain a litter better forecasting performance owing to smaller MAPE values, 2.48 and 2.52, respectively. Furthermore, the RMSE values of these two models are much smaller than the other four competitors, inferring good stability and reliability of the optimized background values based on Simpson’s rule.Table 3
The forecasted results and the values of three indices measuring accuracy by six competing models in case one.
Original values
779
738
669
600
MAPE
RMSE
GMC
1,2
Forecasted values
774.67
732.08
686.91
642.11
2.76
23.17
Error (%)
0.56
0.80
2.68
7.02
OGMC
1,2
Forecasted values
771.98
727.90
681.15
634.95
2.48
19.50
Error (%)
0.90
1.37
1.82
5.83
GMC
G
1,2
Forecasted values
771.88
729.81
685.10
640.54
2.80
22.48
Error (%)
0.91
1.11
2.41
6.76
OGMC
G
1,2
Forecasted values
768.88
725.39
679.18
633.25
2.52
19.17
Error (%)
1.30
1.71
1.52
5.54
GM
1,2
Forecasted values
650.10
554.29
467.39
430.85
24.94
172.93
Error (%)
16.55
24.89
30.14
28.19
DGM
1,2
Forecasted values
377.57
327.26
282.54
247.85
55.91
388.34
Error (%)
51.53
55.66
57.77
58.69In general, theGMC1,2, GMCG1,2, OGMC1,2, and OGMCG1,2 models share almost the similar precision in predicting the tensile strength, and they are equally applicable for further estimations. Additionally, the optimized background values based on Simpson’s rule can effectively contribute to the improvements of forecasting precision.


### 3.2. Case Two: Estimating the Gross Industrial Output Value

In addition to the numerical example from Tien’s study [14], another actual case concerning forecasting China’s industrial output value, collected from Ma and Liu [31], is selected to test the effectiveness and practicability of the proposed models. The raw observations, listed in Table 4, are collected from the National Bureau of Statistics of China, who published correct, official, and complete data. Apparently, the industrial output value is the system behavior sequence, marked as X10. Then, the total current assets and fix assets represent the relevant series, noted as X20 and X30. The GMC1,3,GMCG1,3, OGMC1,3, OGMCG1,3, GM1,3, and DGM1,3 are employed for forecasting China’s industrial output value with parameter setting m=3, r=10, rp=0, and rf=3. In other words, ten data points of the three variables are used for estimating parameters, and the last three observations of the system behavior series are selected for testing the predictive performance.Table 4
Original data in case two (unit: 104 yuan) [31].
i (year)
Industrial output values
Total current assets
Fix assets
X
1
0
X
2
0
X
3
0
1999
72707.04
49630.23
71847.09
2000
85673.66
54338.15
78646.30
2001
95448.98
57804.97
86293.10
2002
110776.48
63468.46
93887.95
2003
142271.22
76163.74
105557.09
2004
201722.19
97183.74
125761.85
2005
251619.50
111031.41
143143.63
2006
316588.96
132310.12
168850.20
2007
405177.13
163259.62
198739.27
2008
507284.89
195681.75
245352.80
2009
548311.42
223038.68
278541.09
2010
698590.42
279227.32
334839.41
2011
844268.79
327778.65
386086.72Taking theOGMC1,3 model as an example for explanations, we can obtain the parameter values β^=β^1,β^2,β^3,u^=0.53849,5.36851,−2.79249,36418.02856 by utilizing equations (27)–(36). Subsequently, substituting the parameter values into equation (39), the discrete response function will be given:(45)X^11t=X101e−0.53849t−1+ut−2∑ξ=2te−0.53849t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=0.5.36851X21t−2.79249X21t+36418.02856.Additionally, the parameter values of the other five competing models are also can be estimated, which is listed in Table5.Table 5
Estimated parameter values of six competing models in case two.
Parameters
GMC
1,3
OGMC
1,3
GMC
G
1,3
OGMC
G
1,3
GM
1,3
DGM
1,3
β
^
1
0.56111
0.53849
0.56111
0.53849
0.62073
−0.60170
β
^
2
5.79185
5.36851
5.79185
5.36851
5.01229
3.72981
β
^
3
−3.07901
−2.79249
−3.07901
−2.79249
−2.51896
−1.96757
u
^
68074.74553
36418.02856
68074.74553
36418.02856
—
22097.40372Subsequently, by using the discrete response function, the forecasted results and values of the evaluation indicators will be obtained, which are displayed in Table6. As this table illustrates, similar conclusions can be achieved: the GMC1,3,GMCG1,3, OGMC1,3, and OGMCG1,3 models significantly outperform the GM1,3 and DGM1,3 models. Moreover, the MAPE values of the GM1,3 and DGM1,3 models are 26.52% and 46.22%, which means these two models are unable to forecast future values of China’s industrial output values due to their bad performances. On the other hand, the grey prediction models with convolution integral are able to provide very competitive results in terms of improved accuracy.Table 6
The forecasted results and the values of three indices measuring accuracy by six competing models in case two.
Original values
548311.42
698590.42
844268.79
MAPE
RMSE
GMC
1,3
Forecasted values
626326.04
774366.43
963930.75
13.08
93358.22
Error (%)
14.23
10.85
14.17
OGMC
1,3
Forecasted values
622548.58
767770.55
953445.29
12.12
86055.32
Error (%)
13.54
9.90
12.93
GMC
G
1,3
Forecasted values
589395.96
721699.65
901242.19
5.85
42692.48
Error (%)
7.49
3.31
6.75
OGMC
G
1,3
Forecasted values
588046.17
718898.95
895337.97
5.40
39155.02
Error (%)
7.25
2.91
6.05
GM
1,3
Forecasted values
673916.59
898040.07
1081472.75
26.52
193065.22
Error (%)
22.91
28.55
28.10
DGM
1,3
Forecasted values
283772.70
382603.22
462874.97
46.22
324186.11
Error (%)
48.25
45.23
45.17As anticipated, with its 5.4%MAPE values and 39155.02RMSE values, the OGMCG1,3 model provides the highest prediction accuracy followed by GMCG1,3, OGMCG1,3, and GMC1,3, respectively. The reasons for such an excellent performance of this optimized model are mainly provided from two perspectives. On one hand, the Gaussian rule that discretizing the convolution interval is generally more accurate than the Trapezoid rule, which can also be verified by other studies, such as Ma and Liu [31], Tien [15], and Wang [18]. On the other hand, the improvements of the background values based on Simpson’s rule are equally important to enhancing its forecasting capability, because the optimized background values are crucial requirement to ensure the reliable and accurate parameter estimations, explained in Section 2.2. Similar findings can also be obtained from the comparison of the GMC1,3 and OGMC1,3 models, of which the accuracy rate is heavily dependent on the optimization of background values.Generally, theOGMCG1,3 model obtains the highest forecasting ability leading itself as the optimal technique for the projections of China’s industrial output value. Moreover, Simpson’s rule can be employed for calculating the background values with high precision, which is essential for parameter estimations.


### 3.3. Case Three: Forecasting Electronic Waste in Washington State

The empirical results of cases one and two demonstrate the superior forecasting performance of theOGMC1,n and OGMCG1,n models having optimized background values based on Simpson’s rule, in comparison with other competitors. Furthermore, the OGMCG1,n model achieves a litter higher accuracy than the OGMC1,n model. Therefore, in addition to the above two examples coming from the published papers, another new case concerning electronic waste is introduced to verify the proposed model’s efficacy and reliability in the following subsections.


#### 3.3.1. Data Description and Model Calibration

As introduced in earlier work [34, 35], e-waste has increasingly become one of the major contributors of municipal solid waste (MSW), whose commonly used influencing factors are the population density and median household income. In this paper, the original data sets of e-waste from 2003 to 2015 are available from the government website (ecology.wa.gov), whereas data since 2016 are unavailable. In addition, observations of these two factors are collected from the Washington State Office of Financial Management website [36, 37]. All the available data of these three variables are presented in Table 7. Moreover, in Table 7, data sets from 2003 to 2012 are used for establishing six models, whereas observations from 2013 to 2015 are employed for comparison. In the GMC (1, n) and its extensions, the parameters are used as m=3, r=10,rp=0, and rf=3.Table 7
E-waste, population density, and median household income in case three.
i (year)
E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2003
18108.19
92.07
46967.25
2004
27341.56
93.30
49585.35
2005
35887.90
94.66
50004.20
2006
46126.41
96.48
53522.31
2007
53737.51
98.06
56141.32
2008
62071.46
99.31
57857.52
2009
69246.27
100.27
55458.05
2010
68777.91
101.19
54888.45
2011
69673.02
101.84
55500.26
2012
73851.24
102.59
56443.76
2013
65894.78
103.56
57283.94
2014
67822.93
104.85
60153.00
2015
72103.41
106.26
63439.20All computations are executed on MATLAB 2016a and the parameter estimations are given in Table8. Then, we take the OGMC1,3 model as an example for explaining the main procedures. By using equations (1)–(4) and (27)–(36), we can obtain the estimated parameters, namely, β^=β^1,β^2,β^3,u^=0.31662,−420.77848,1.19678,9020.93243. Furthermore, the discrete response function will be displayed below:(46)X^11t=X101e−0.31662t−1+ut−2∑ξ=2te−0.31662t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=−420.77848X21t1.19678X21t+9020.93243. Subsequently, the visual comparison of original and forecasted OGMC1,3 values is shown in Table 9.Table 8
Estimated parameter values of six competing models in case three.
Parameters
GMC
1,3
OGMC
1,3
GMC
G
1,3
OGMC
G
1,3
GM
1,3
DGM
1,3
β
^
1
0.30047
0.31662
0.30047
0.31662
0.27732
−0.76644
β
^
2
−455.74524
−420.77848
−455.74524
−420.77848
−273.98465
−405.07032
β
^
3
1.24313
1.19678
1.24313
1.19678
0.88929
1.06417
u
^
10694.07636
9020.93243
10694.07636
9020.93243
—
3963.42238Table 9
The forecasted results and the values of three indices measuring accuracy by six competing models in case three.
Original values
65894.78
67822.93
72103.41
MAPE
RMSE
GMC
1,3
Forecasted values
74781.44
77760.78
82791.00
14.32
9865.06
Error (%)
13.49
14.65
14.82
OGMC
1,3
Forecasted values
74387.88
77235.75
82033.66
13.51
9297.73
Error (%)
12.89
13.88
13.77
GMC
G
1,3
Forecasted values
73784.29
76581.01
81414.67
12.60
8672.71
Error (%)
11.97
12.91
12.91
OGMC
G
1,3
Forecasted values
73306.88
75968.74
80567.28
11.67
8019.37
Error (%)
12.01
12.01
11.74
GM
1,3
Forecasted values
90944.46
97409.07
105286.45
42.55
29461.50
Error (%)
38.01
43.62
46.02
DGM
1,3
Forecasted values
18986.34
21522.63
24452.89
68.51
46956.34
Error (%)
71.19
68.27
66.09


#### 3.3.2. Comparative Analysis of the Forecasting Performance by Six Competing Models

As seen in Table9, notice that GM1,3 and GM1,3 produce inaccurate results with 42.55% and 68.51%MAPE values, respectively. Hence, these two models fail to predict the e-waste amount. In contrast, the GMC1,3 and GMCG1,3 models, as expected, provide obvious improvements in the forecasting precision. The reason for this situation is that the modeling values by using conventual integral in these two models are theoretically correct, while those in the GM1,3 model are wrong [8]. In addition, the Gaussian rule performs a little better than the Trapezoid rule because the GMCG1,3 model achieves higher predicting accuracy.Compared with the previousGMC1,3 and GMCG1,3 models, the OGMC1,3 and OGMCG1,3 models that have new optimized background values based on Simpson’s rule are able to produce much more accurate predictions in terms of both MAPE and RMSE values. Thus, Simpson’s rule can work as an efficient alternative technique for calculating background values, which has good capability to enhance prediction accuracy. Moreover, the OGMCG1,3 model achieves the highest predicting accuracy with 11.67%MAPE values and 8019.37RMSE values. Therefore, the e-waste amount in Washington State from 2016 to 2020 is predicted by the superior model, namely, OGMCG1,3.


#### 3.3.3. Future Projections of e-Waste via OGMCG (1, 3)

The future e-waste estimations can only be forecasted under the premise of known future values of the population density and median household income. For the future values of the population density, the Washington State Office of Financial Management has provided the trends on the website [38], which is given in Table 10. For the median household income in Table 10, its observations from 2016 to 2017 are available from the above website [36], and its future estimations from 2018 to 2020 are predicted by using the GM1,1 model [39], which is a reliable and practical technique for univariate predictions.Table 10
Estimated e-waste, population density, and median household income in case three from 2016 to 2020.
i (year)
Estimated E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2016
80377.91
108.10
65500.29
2017
85503.45
110.00
68550.43
2018
88941.10
111.77
66894.16
2019
90493.07
113.46
68298.47
2020
92405.41
115.11
69732.25Initially, data sets of the median household income from 2003 to 2017 are used for model calibration. Then we will obtain the parametersa,b=−0.02078,48518.01491, and the corresponding time response function is given below:(47)X^11t+1=2.3823×106×e0.02078t−2.3353×106.Subsequently, estimated values from 2018 to 2020 are given in Table10. By the way, the MAPE value in the in-sample period from 2003 to 2017 reaches 3.14%, inferring excellent forecasting capability of the GM1,1 model.Once we have the values of these two relevant factors from 2016 to 2020, the future projections of the e-waste amount can be forecasted by using theOGMCG1,3 model. following equations (1)–(4) and (27)–(36), the forecasts from 2016 to 2020 will be calculated in Table 10.Furthermore, by using theOGMCG1,3 model, the visual comparison of the real observations and predicted values from 2003 to 2020 are presented in Figure 2. As it can be observed from Figure 2, the predicted values fit the real observations well in the in-sample period, and further projections are likely to remain increasing from 2016 to 2020. Thus, the optimized OGMCG1,3 can act as an appropriate tool for e-waste forecast in this paper.Figure 2
Visual comparison of the real observations and predicted values from 2003 to 2020.


## 3.1. Case One: Predicting the Tensile Strength of a Material

The multivariable grey prediction models, such asGM1,2 and GMC1,2, can be used for the indirect measurement of the tensile strength of a material [14], the efficacy of the OGMC1,2 and OGMCG1,2 models can be revealed by comparing the experimental results of predicting tensile strength of a material in comparison with the GMC1,2, GMCG1,2, GM1,2, and DGM1,2 models. The original data are presented in Table 1, which were collected from the experimental observations in Table 117 of Samuel [33]. In addition, the tensile strength represents the system behavior sequence X10, and the Brinell hardness works as the relevant series X20. Moreover, six data points of X10 and X20 are utilized for model calibration, and the remaining four observations of X10 are used for checking the predictive performance, namely, setting parameters n=2,r=6, rp=0, and rf=4 [33].Table 1
Original data in case one.
i (temperature)
Tensile strengthX10 (MPa)
Brinell hardnessX20 (HBW)
1 (400F∘)
897
514
2 (500F∘)
897
495
3 (600F∘)
890
444
4 (700F∘)
876
401
5 (800F∘)
848
352
6 (900F∘)
814
293
7 (1000F∘)
779
269
8 (1100F∘)
738
235
9 (1200F∘)
669
201
10 (1300F∘)
600
187As introduced in Sections2.1 and 2.3, the procedures and flowchart of GMC1,2 and its extensions are presented, and the procedures of GM1,2 and DGM1,2 are similar to those of GMC1,2. Thus, according to these procedures, the estimated parameter-values of six competing models are given in Table 2.Table 2
Estimated parameter values of six competing models in case one.
Parameters
GMC
1,2
OGMC
1,2
GMC
G
1,2
OGMC
G
1,2
GM
1,2
DGM
1,2
β
^
1
0.14378
0.15387
0.14378
0.15387
0.84617
−0.86519
β
^
2
0.26139
0.28354
0.26139
0.28354
1.93150
0.25951
u
^
891.16435
885.67276
891.16435
885.67276
—
755.74471TakingOGMC1,2 as an example for explanations, we can obtain the parameter values β^=β^1,β^2,u^=0.15387,0.28354,885.67276 after substituting the original data into equations (35) and (36). Subsequently, substituting the parameter values into equation (39), we will have the discrete response function:(44)X^11t=X101e−0.15387t−1+ut−2∑ξ=2te−0.15387t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=0.28354X21t+885.67276.By using the time response function in equation (44), the forecasted results and the values of three indices measuring accuracy will be obtained, which are shown in Table 3. It is obvious that four GMC1,2 and its extensions are superior to the traditional GM1,2 and DGM1,2 whose MAPE values reach 24.94% and 55.91%, respectively. The reason for this phenomenon is that GM1,2 has an incorrect time response function and DGM1,2 has narrow application fields. In contrast, the GMC1,2 and GMCG1,2 models are able to provide accurate forecasts because their modeling values with convolution integral are exact theoretically [14]. However, the OGMC1,2 and OGMCG1,2 models seem to obtain a litter better forecasting performance owing to smaller MAPE values, 2.48 and 2.52, respectively. Furthermore, the RMSE values of these two models are much smaller than the other four competitors, inferring good stability and reliability of the optimized background values based on Simpson’s rule.Table 3
The forecasted results and the values of three indices measuring accuracy by six competing models in case one.
Original values
779
738
669
600
MAPE
RMSE
GMC
1,2
Forecasted values
774.67
732.08
686.91
642.11
2.76
23.17
Error (%)
0.56
0.80
2.68
7.02
OGMC
1,2
Forecasted values
771.98
727.90
681.15
634.95
2.48
19.50
Error (%)
0.90
1.37
1.82
5.83
GMC
G
1,2
Forecasted values
771.88
729.81
685.10
640.54
2.80
22.48
Error (%)
0.91
1.11
2.41
6.76
OGMC
G
1,2
Forecasted values
768.88
725.39
679.18
633.25
2.52
19.17
Error (%)
1.30
1.71
1.52
5.54
GM
1,2
Forecasted values
650.10
554.29
467.39
430.85
24.94
172.93
Error (%)
16.55
24.89
30.14
28.19
DGM
1,2
Forecasted values
377.57
327.26
282.54
247.85
55.91
388.34
Error (%)
51.53
55.66
57.77
58.69In general, theGMC1,2, GMCG1,2, OGMC1,2, and OGMCG1,2 models share almost the similar precision in predicting the tensile strength, and they are equally applicable for further estimations. Additionally, the optimized background values based on Simpson’s rule can effectively contribute to the improvements of forecasting precision.


## 3.2. Case Two: Estimating the Gross Industrial Output Value

In addition to the numerical example from Tien’s study [14], another actual case concerning forecasting China’s industrial output value, collected from Ma and Liu [31], is selected to test the effectiveness and practicability of the proposed models. The raw observations, listed in Table 4, are collected from the National Bureau of Statistics of China, who published correct, official, and complete data. Apparently, the industrial output value is the system behavior sequence, marked as X10. Then, the total current assets and fix assets represent the relevant series, noted as X20 and X30. The GMC1,3,GMCG1,3, OGMC1,3, OGMCG1,3, GM1,3, and DGM1,3 are employed for forecasting China’s industrial output value with parameter setting m=3, r=10, rp=0, and rf=3. In other words, ten data points of the three variables are used for estimating parameters, and the last three observations of the system behavior series are selected for testing the predictive performance.Table 4
Original data in case two (unit: 104 yuan) [31].
i (year)
Industrial output values
Total current assets
Fix assets
X
1
0
X
2
0
X
3
0
1999
72707.04
49630.23
71847.09
2000
85673.66
54338.15
78646.30
2001
95448.98
57804.97
86293.10
2002
110776.48
63468.46
93887.95
2003
142271.22
76163.74
105557.09
2004
201722.19
97183.74
125761.85
2005
251619.50
111031.41
143143.63
2006
316588.96
132310.12
168850.20
2007
405177.13
163259.62
198739.27
2008
507284.89
195681.75
245352.80
2009
548311.42
223038.68
278541.09
2010
698590.42
279227.32
334839.41
2011
844268.79
327778.65
386086.72Taking theOGMC1,3 model as an example for explanations, we can obtain the parameter values β^=β^1,β^2,β^3,u^=0.53849,5.36851,−2.79249,36418.02856 by utilizing equations (27)–(36). Subsequently, substituting the parameter values into equation (39), the discrete response function will be given:(45)X^11t=X101e−0.53849t−1+ut−2∑ξ=2te−0.53849t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=0.5.36851X21t−2.79249X21t+36418.02856.Additionally, the parameter values of the other five competing models are also can be estimated, which is listed in Table5.Table 5
Estimated parameter values of six competing models in case two.
Parameters
GMC
1,3
OGMC
1,3
GMC
G
1,3
OGMC
G
1,3
GM
1,3
DGM
1,3
β
^
1
0.56111
0.53849
0.56111
0.53849
0.62073
−0.60170
β
^
2
5.79185
5.36851
5.79185
5.36851
5.01229
3.72981
β
^
3
−3.07901
−2.79249
−3.07901
−2.79249
−2.51896
−1.96757
u
^
68074.74553
36418.02856
68074.74553
36418.02856
—
22097.40372Subsequently, by using the discrete response function, the forecasted results and values of the evaluation indicators will be obtained, which are displayed in Table6. As this table illustrates, similar conclusions can be achieved: the GMC1,3,GMCG1,3, OGMC1,3, and OGMCG1,3 models significantly outperform the GM1,3 and DGM1,3 models. Moreover, the MAPE values of the GM1,3 and DGM1,3 models are 26.52% and 46.22%, which means these two models are unable to forecast future values of China’s industrial output values due to their bad performances. On the other hand, the grey prediction models with convolution integral are able to provide very competitive results in terms of improved accuracy.Table 6
The forecasted results and the values of three indices measuring accuracy by six competing models in case two.
Original values
548311.42
698590.42
844268.79
MAPE
RMSE
GMC
1,3
Forecasted values
626326.04
774366.43
963930.75
13.08
93358.22
Error (%)
14.23
10.85
14.17
OGMC
1,3
Forecasted values
622548.58
767770.55
953445.29
12.12
86055.32
Error (%)
13.54
9.90
12.93
GMC
G
1,3
Forecasted values
589395.96
721699.65
901242.19
5.85
42692.48
Error (%)
7.49
3.31
6.75
OGMC
G
1,3
Forecasted values
588046.17
718898.95
895337.97
5.40
39155.02
Error (%)
7.25
2.91
6.05
GM
1,3
Forecasted values
673916.59
898040.07
1081472.75
26.52
193065.22
Error (%)
22.91
28.55
28.10
DGM
1,3
Forecasted values
283772.70
382603.22
462874.97
46.22
324186.11
Error (%)
48.25
45.23
45.17As anticipated, with its 5.4%MAPE values and 39155.02RMSE values, the OGMCG1,3 model provides the highest prediction accuracy followed by GMCG1,3, OGMCG1,3, and GMC1,3, respectively. The reasons for such an excellent performance of this optimized model are mainly provided from two perspectives. On one hand, the Gaussian rule that discretizing the convolution interval is generally more accurate than the Trapezoid rule, which can also be verified by other studies, such as Ma and Liu [31], Tien [15], and Wang [18]. On the other hand, the improvements of the background values based on Simpson’s rule are equally important to enhancing its forecasting capability, because the optimized background values are crucial requirement to ensure the reliable and accurate parameter estimations, explained in Section 2.2. Similar findings can also be obtained from the comparison of the GMC1,3 and OGMC1,3 models, of which the accuracy rate is heavily dependent on the optimization of background values.Generally, theOGMCG1,3 model obtains the highest forecasting ability leading itself as the optimal technique for the projections of China’s industrial output value. Moreover, Simpson’s rule can be employed for calculating the background values with high precision, which is essential for parameter estimations.


## 3.3. Case Three: Forecasting Electronic Waste in Washington State

The empirical results of cases one and two demonstrate the superior forecasting performance of theOGMC1,n and OGMCG1,n models having optimized background values based on Simpson’s rule, in comparison with other competitors. Furthermore, the OGMCG1,n model achieves a litter higher accuracy than the OGMC1,n model. Therefore, in addition to the above two examples coming from the published papers, another new case concerning electronic waste is introduced to verify the proposed model’s efficacy and reliability in the following subsections.


### 3.3.1. Data Description and Model Calibration

As introduced in earlier work [34, 35], e-waste has increasingly become one of the major contributors of municipal solid waste (MSW), whose commonly used influencing factors are the population density and median household income. In this paper, the original data sets of e-waste from 2003 to 2015 are available from the government website (ecology.wa.gov), whereas data since 2016 are unavailable. In addition, observations of these two factors are collected from the Washington State Office of Financial Management website [36, 37]. All the available data of these three variables are presented in Table 7. Moreover, in Table 7, data sets from 2003 to 2012 are used for establishing six models, whereas observations from 2013 to 2015 are employed for comparison. In the GMC (1, n) and its extensions, the parameters are used as m=3, r=10,rp=0, and rf=3.Table 7
E-waste, population density, and median household income in case three.
i (year)
E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2003
18108.19
92.07
46967.25
2004
27341.56
93.30
49585.35
2005
35887.90
94.66
50004.20
2006
46126.41
96.48
53522.31
2007
53737.51
98.06
56141.32
2008
62071.46
99.31
57857.52
2009
69246.27
100.27
55458.05
2010
68777.91
101.19
54888.45
2011
69673.02
101.84
55500.26
2012
73851.24
102.59
56443.76
2013
65894.78
103.56
57283.94
2014
67822.93
104.85
60153.00
2015
72103.41
106.26
63439.20All computations are executed on MATLAB 2016a and the parameter estimations are given in Table8. Then, we take the OGMC1,3 model as an example for explaining the main procedures. By using equations (1)–(4) and (27)–(36), we can obtain the estimated parameters, namely, β^=β^1,β^2,β^3,u^=0.31662,−420.77848,1.19678,9020.93243. Furthermore, the discrete response function will be displayed below:(46)X^11t=X101e−0.31662t−1+ut−2∑ξ=2te−0.31662t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=−420.77848X21t1.19678X21t+9020.93243. Subsequently, the visual comparison of original and forecasted OGMC1,3 values is shown in Table 9.Table 8
Estimated parameter values of six competing models in case three.
Parameters
GMC
1,3
OGMC
1,3
GMC
G
1,3
OGMC
G
1,3
GM
1,3
DGM
1,3
β
^
1
0.30047
0.31662
0.30047
0.31662
0.27732
−0.76644
β
^
2
−455.74524
−420.77848
−455.74524
−420.77848
−273.98465
−405.07032
β
^
3
1.24313
1.19678
1.24313
1.19678
0.88929
1.06417
u
^
10694.07636
9020.93243
10694.07636
9020.93243
—
3963.42238Table 9
The forecasted results and the values of three indices measuring accuracy by six competing models in case three.
Original values
65894.78
67822.93
72103.41
MAPE
RMSE
GMC
1,3
Forecasted values
74781.44
77760.78
82791.00
14.32
9865.06
Error (%)
13.49
14.65
14.82
OGMC
1,3
Forecasted values
74387.88
77235.75
82033.66
13.51
9297.73
Error (%)
12.89
13.88
13.77
GMC
G
1,3
Forecasted values
73784.29
76581.01
81414.67
12.60
8672.71
Error (%)
11.97
12.91
12.91
OGMC
G
1,3
Forecasted values
73306.88
75968.74
80567.28
11.67
8019.37
Error (%)
12.01
12.01
11.74
GM
1,3
Forecasted values
90944.46
97409.07
105286.45
42.55
29461.50
Error (%)
38.01
43.62
46.02
DGM
1,3
Forecasted values
18986.34
21522.63
24452.89
68.51
46956.34
Error (%)
71.19
68.27
66.09


### 3.3.2. Comparative Analysis of the Forecasting Performance by Six Competing Models

As seen in Table9, notice that GM1,3 and GM1,3 produce inaccurate results with 42.55% and 68.51%MAPE values, respectively. Hence, these two models fail to predict the e-waste amount. In contrast, the GMC1,3 and GMCG1,3 models, as expected, provide obvious improvements in the forecasting precision. The reason for this situation is that the modeling values by using conventual integral in these two models are theoretically correct, while those in the GM1,3 model are wrong [8]. In addition, the Gaussian rule performs a little better than the Trapezoid rule because the GMCG1,3 model achieves higher predicting accuracy.Compared with the previousGMC1,3 and GMCG1,3 models, the OGMC1,3 and OGMCG1,3 models that have new optimized background values based on Simpson’s rule are able to produce much more accurate predictions in terms of both MAPE and RMSE values. Thus, Simpson’s rule can work as an efficient alternative technique for calculating background values, which has good capability to enhance prediction accuracy. Moreover, the OGMCG1,3 model achieves the highest predicting accuracy with 11.67%MAPE values and 8019.37RMSE values. Therefore, the e-waste amount in Washington State from 2016 to 2020 is predicted by the superior model, namely, OGMCG1,3.


### 3.3.3. Future Projections of e-Waste via OGMCG (1, 3)

The future e-waste estimations can only be forecasted under the premise of known future values of the population density and median household income. For the future values of the population density, the Washington State Office of Financial Management has provided the trends on the website [38], which is given in Table 10. For the median household income in Table 10, its observations from 2016 to 2017 are available from the above website [36], and its future estimations from 2018 to 2020 are predicted by using the GM1,1 model [39], which is a reliable and practical technique for univariate predictions.Table 10
Estimated e-waste, population density, and median household income in case three from 2016 to 2020.
i (year)
Estimated E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2016
80377.91
108.10
65500.29
2017
85503.45
110.00
68550.43
2018
88941.10
111.77
66894.16
2019
90493.07
113.46
68298.47
2020
92405.41
115.11
69732.25Initially, data sets of the median household income from 2003 to 2017 are used for model calibration. Then we will obtain the parametersa,b=−0.02078,48518.01491, and the corresponding time response function is given below:(47)X^11t+1=2.3823×106×e0.02078t−2.3353×106.Subsequently, estimated values from 2018 to 2020 are given in Table10. By the way, the MAPE value in the in-sample period from 2003 to 2017 reaches 3.14%, inferring excellent forecasting capability of the GM1,1 model.Once we have the values of these two relevant factors from 2016 to 2020, the future projections of the e-waste amount can be forecasted by using theOGMCG1,3 model. following equations (1)–(4) and (27)–(36), the forecasts from 2016 to 2020 will be calculated in Table 10.Furthermore, by using theOGMCG1,3 model, the visual comparison of the real observations and predicted values from 2003 to 2020 are presented in Figure 2. As it can be observed from Figure 2, the predicted values fit the real observations well in the in-sample period, and further projections are likely to remain increasing from 2016 to 2020. Thus, the optimized OGMCG1,3 can act as an appropriate tool for e-waste forecast in this paper.Figure 2
Visual comparison of the real observations and predicted values from 2003 to 2020.


## 3.3.1. Data Description and Model Calibration

As introduced in earlier work [34, 35], e-waste has increasingly become one of the major contributors of municipal solid waste (MSW), whose commonly used influencing factors are the population density and median household income. In this paper, the original data sets of e-waste from 2003 to 2015 are available from the government website (ecology.wa.gov), whereas data since 2016 are unavailable. In addition, observations of these two factors are collected from the Washington State Office of Financial Management website [36, 37]. All the available data of these three variables are presented in Table 7. Moreover, in Table 7, data sets from 2003 to 2012 are used for establishing six models, whereas observations from 2013 to 2015 are employed for comparison. In the GMC (1, n) and its extensions, the parameters are used as m=3, r=10,rp=0, and rf=3.Table 7
E-waste, population density, and median household income in case three.
i (year)
E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2003
18108.19
92.07
46967.25
2004
27341.56
93.30
49585.35
2005
35887.90
94.66
50004.20
2006
46126.41
96.48
53522.31
2007
53737.51
98.06
56141.32
2008
62071.46
99.31
57857.52
2009
69246.27
100.27
55458.05
2010
68777.91
101.19
54888.45
2011
69673.02
101.84
55500.26
2012
73851.24
102.59
56443.76
2013
65894.78
103.56
57283.94
2014
67822.93
104.85
60153.00
2015
72103.41
106.26
63439.20All computations are executed on MATLAB 2016a and the parameter estimations are given in Table8. Then, we take the OGMC1,3 model as an example for explaining the main procedures. By using equations (1)–(4) and (27)–(36), we can obtain the estimated parameters, namely, β^=β^1,β^2,β^3,u^=0.31662,−420.77848,1.19678,9020.93243. Furthermore, the discrete response function will be displayed below:(46)X^11t=X101e−0.31662t−1+ut−2∑ξ=2te−0.31662t−ξ+1/2∗12fξ+fξ−1,where ut−2=0,t<2,1,t≥2, and ft=−420.77848X21t1.19678X21t+9020.93243. Subsequently, the visual comparison of original and forecasted OGMC1,3 values is shown in Table 9.Table 8
Estimated parameter values of six competing models in case three.
Parameters
GMC
1,3
OGMC
1,3
GMC
G
1,3
OGMC
G
1,3
GM
1,3
DGM
1,3
β
^
1
0.30047
0.31662
0.30047
0.31662
0.27732
−0.76644
β
^
2
−455.74524
−420.77848
−455.74524
−420.77848
−273.98465
−405.07032
β
^
3
1.24313
1.19678
1.24313
1.19678
0.88929
1.06417
u
^
10694.07636
9020.93243
10694.07636
9020.93243
—
3963.42238Table 9
The forecasted results and the values of three indices measuring accuracy by six competing models in case three.
Original values
65894.78
67822.93
72103.41
MAPE
RMSE
GMC
1,3
Forecasted values
74781.44
77760.78
82791.00
14.32
9865.06
Error (%)
13.49
14.65
14.82
OGMC
1,3
Forecasted values
74387.88
77235.75
82033.66
13.51
9297.73
Error (%)
12.89
13.88
13.77
GMC
G
1,3
Forecasted values
73784.29
76581.01
81414.67
12.60
8672.71
Error (%)
11.97
12.91
12.91
OGMC
G
1,3
Forecasted values
73306.88
75968.74
80567.28
11.67
8019.37
Error (%)
12.01
12.01
11.74
GM
1,3
Forecasted values
90944.46
97409.07
105286.45
42.55
29461.50
Error (%)
38.01
43.62
46.02
DGM
1,3
Forecasted values
18986.34
21522.63
24452.89
68.51
46956.34
Error (%)
71.19
68.27
66.09


## 3.3.2. Comparative Analysis of the Forecasting Performance by Six Competing Models

As seen in Table9, notice that GM1,3 and GM1,3 produce inaccurate results with 42.55% and 68.51%MAPE values, respectively. Hence, these two models fail to predict the e-waste amount. In contrast, the GMC1,3 and GMCG1,3 models, as expected, provide obvious improvements in the forecasting precision. The reason for this situation is that the modeling values by using conventual integral in these two models are theoretically correct, while those in the GM1,3 model are wrong [8]. In addition, the Gaussian rule performs a little better than the Trapezoid rule because the GMCG1,3 model achieves higher predicting accuracy.Compared with the previousGMC1,3 and GMCG1,3 models, the OGMC1,3 and OGMCG1,3 models that have new optimized background values based on Simpson’s rule are able to produce much more accurate predictions in terms of both MAPE and RMSE values. Thus, Simpson’s rule can work as an efficient alternative technique for calculating background values, which has good capability to enhance prediction accuracy. Moreover, the OGMCG1,3 model achieves the highest predicting accuracy with 11.67%MAPE values and 8019.37RMSE values. Therefore, the e-waste amount in Washington State from 2016 to 2020 is predicted by the superior model, namely, OGMCG1,3.


## 3.3.3. Future Projections of e-Waste via OGMCG (1, 3)

The future e-waste estimations can only be forecasted under the premise of known future values of the population density and median household income. For the future values of the population density, the Washington State Office of Financial Management has provided the trends on the website [38], which is given in Table 10. For the median household income in Table 10, its observations from 2016 to 2017 are available from the above website [36], and its future estimations from 2018 to 2020 are predicted by using the GM1,1 model [39], which is a reliable and practical technique for univariate predictions.Table 10
Estimated e-waste, population density, and median household income in case three from 2016 to 2020.
i (year)
Estimated E-waste
Population density
Median household
X
1
0 (tons)
X
2
0 (people/se mi)
IncomeX30 ($)
2016
80377.91
108.10
65500.29
2017
85503.45
110.00
68550.43
2018
88941.10
111.77
66894.16
2019
90493.07
113.46
68298.47
2020
92405.41
115.11
69732.25Initially, data sets of the median household income from 2003 to 2017 are used for model calibration. Then we will obtain the parametersa,b=−0.02078,48518.01491, and the corresponding time response function is given below:(47)X^11t+1=2.3823×106×e0.02078t−2.3353×106.Subsequently, estimated values from 2018 to 2020 are given in Table10. By the way, the MAPE value in the in-sample period from 2003 to 2017 reaches 3.14%, inferring excellent forecasting capability of the GM1,1 model.Once we have the values of these two relevant factors from 2016 to 2020, the future projections of the e-waste amount can be forecasted by using theOGMCG1,3 model. following equations (1)–(4) and (27)–(36), the forecasts from 2016 to 2020 will be calculated in Table 10.Furthermore, by using theOGMCG1,3 model, the visual comparison of the real observations and predicted values from 2003 to 2020 are presented in Figure 2. As it can be observed from Figure 2, the predicted values fit the real observations well in the in-sample period, and further projections are likely to remain increasing from 2016 to 2020. Thus, the optimized OGMCG1,3 can act as an appropriate tool for e-waste forecast in this paper.Figure 2
Visual comparison of the real observations and predicted values from 2003 to 2020.


## 4. Conclusions and Discussions

In summary, accurate future estimations play an essential role for decision-makers in framing and implementing sensible plans and policies. Thus, a grey prediction model with convolution integral, shorted forGMC1,n, is introduced alongside with its improved variants GMCG1,n. These two models have been applied to solve various prediction issues by many researchers because of their correct solutions to the whitening function. However, inaccurate methods to calculate the background values may incur large deviations of the parameter estimations, which are essential for accurate future projections. Therefore, mathematical analyses of the gap between the actual and estimated background values are discussed in detail. Moreover, Simpson’s rule is introduced to compute the background values. Subsequently, the OGMC1,n and OGMCG1,n models are designed by utilizing the optimized background values. Then, a comparative analysis via utilizing the same data points is run to analyze the performance of the proposed models in the three case studies. Through the discussion and analysis of the experimental studies, several conclusions can be drawn:(i)
The new proposedOGMC1,n and OGMCG1,n models generally outperform the previous GM1,n, DGM1,n, GMC1,n, and GMCG1,n models because of the optimized background values. Moreover, the OGMCG1,n model is normally superior to OGMC1,n due to the accurate Gaussian solution.(ii)
The optimized background values based on Simpson’s rule can significantly enhance the forecasting capability of the previousGMC1,n models. Additionally, the reconstructed background values have merits in the simplified structure, high reliability, and wide applicability, which have been proven in three cases. Most importantly, the new method can improve the accuracy of model calibration avoiding the unsatisfactory errors generated from the discrete function to the continuous one.(iii)
TheOGMCG1,n model rises as a more viable option for future projections of e-waste amount in Washington State owing to its accurate and reliable performances in three cases. Moreover, the future e-waste amount remains increasing for the upcoming years.A limitation of this paper is caused by the linear structure of the proposed model, which may produce errors when dealing with the nonlinear sequences. Therefore, future work will be carried out by proposing a nonlinearOGMC1,n model with optimized background values based on Simpson’s rule.



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*Source: 4564653-2020-02-27.xml*