# A Novel Robust Grey Model and Its Application

Authors: Fei Lan; Jiayang Kong; Riquan Yao; Cong Wei
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Previous studies paid attention to improving the predicted capability of the classical grey model, but its robustness is still unclear and not explored, in particular when these exhibit outliers in the time series, which is due to measurement error, uncorrected record, and censored date. In this study, we proposed a novel robust grey model. The novel robust grey model adopts the median regression method to address these problems caused by outliers, which provides the robust parameters. The analytical expression for the time response function and the forecasting values is derived by the grey system technique and mathematical tool. With annual observational data of Chinese electricity demand, we examine the fitness capability of the novel robust grey model, by comparing it with the classical grey model. Also, we adopt the bootstrapping test to further illustrate the sensitivity for the new robust grey model when there are outliers in the time series. To our knowledge, it is the first to introduce the bootstrapping test to the literature related to the grey model and to focus on the robustness of the grey model. The computational results suggest that the new robust grey model has higher precision than the classical grey model, but it is also very robust to outliers, whose accuracy and robustness are better than the classical grey model. Finally, we apply the novel grey model to forecast the future values in Chinese electricity demand during the year 2022 to 2025. This new model proposed in this study estimates that the Chinese electricity demand would continue to increase after the year of 2022, arriving at 10.446×105 million kW·h in the year of 2025 approximately.


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## Body

## 1. Introduction

The grey theory, pioneered by Professor Deng [1], is an approach to model the social and natural phenomenon based on a small sample and little information [2–8]. As a critical component of the grey theory, the grey model has been widely applied in many fields, including energy and economic fields [9–18]. For instance, Wang et al. grouped data and used the grey model to predict power production [19]. Zeng and Li developed an intelligent grey model and applied it to forecast natural gas demand in China [20]. Ding et al. optimized the initial values for the grey model and adopted a rolling approach to predict the electricity consumption in China [21]. Generally speaking, GM (1, 1) is short for the grey model, according to the convention of the literature on the grey system, with the first and second numbers representing the first order and univariate, respectively.The integer-order accumulative generating operation in the entire grey prediction fields, usually abbreviated as I-AGO, is the primary operation [22]. This makes these nonnegative data points better fit the exponential growth rule, revealing the inherent law of things hidden in the raw data [23]. Accordingly, many researchers used the integer-order accumulative generation operation in the grey model, particularly the first-order accumulative generation operation. However, some scholars point out that using the integer-order accumulative generating operation reduces the flexibility in the time-series transformation in real applications, limiting the accuracy of prediction. In many engineering problems, it is not tolerant for the low forecasting ability of the model. Thus, to make the transformation of time series more flexible, Wu et al. presented the fractional-order accumulative generating operation, based on the gamma function, and gave the evidence that the fractional-order accumulative generating operation is the direct generalization of the integer-order accumulative generating operation [24]. More importantly, the result from the numerical simulation using several types of time series exhibited that the grey model, adopting the fractional-order accumulative generating operation, had a better performance on prediction than the traditional grey model that used the integer-order accumulative generating operation. After that, the novel grey model is fast used by researchers in distinguished disciplines. For example, Wu et al. used it to forecast the emissions in carbon dioxide for BRICS economies [25].Meanwhile, the traditional grey model is still developing. In order to improve its accuracy of prediction, some scholars tried to extend the linear grey model to the nonlinear grey model. For example, Jiang et al. introduced a nonhomogeneous exponential form into the traditional grey model and established the grey Verhulst model [26]. Chen et al. generalized the grey Verhulst model and proposed the grey Bernoulli model [27]. Additionally, some scholars optimized the traditional grey model by introducing intelligent algorithms from computer science and game theory. For instance, Zhou et al. implemented particle swarm optimization to improve the background values [28]. Hsin and Chen applied the Nash equilibrium solution to estimate the parameters of the grey model [29]. Ma et al. provided the kernel-based grey model and discussed its applications in complex nonlinear time-series forecasting [30].However, in the traditional grey model, the structural parameters are estimated using the mean regression method. This could lead to poor robustness due to outliers and consequently make the predicted accuracy of model low. In order to overcome the drawback and boost the robustness of model, we introduce the median regression method into the traditional grey model, which estimates these parameters based on minimizing the mean absolute error. Our main contributions of the novel robust grey model are summarized as follows:(1)
The weakness of the traditional grey model, particularly, the robustness of the structural parameters, is discussed.(2)
The novel robust grey model, abbreviated as RGM (1, 1), is studied(3)
The three empirical examples from distinguished fields are used to check the robustness and accuracy of modelThe rest of this study is organized as follows. Section2 demonstrates the classical grey model and provides the details of the novel robust grey model. Section 3 discusses the accuracy and robustness of the proposed grey model. An application of prediction for future energy consumption in China is given in Section 4. Section 5 is the conclusion.


## 2. Methods

Three sections on the methodology, which is applied in this study, are introduced here. Section2.1 focuses on the classical grey model, GM (1, 1). The establishment of the novel robust grey model, abbreviated as RGM (1, 1), is provided in Section 2.2. Section 2.3 presents some statistical approaches to test the accuracy and robustness of the model.


### 2.1. The Classical Grey Model (GM (1, 1))

Professor Deng puts forward the typical grey model, which serves as a useful tool to deal with small sample and little information, particularly in uncertain issues. Compared with other forecasting model, this model has some remarkable advantages. For example, the grey model does not need too many observations, and only using three data points we could form an effective prediction model with relatively high accuracy. More importantly, the process of modeling for the grey model is open and apparent, which reduces the possibility of artificial fit that improves the credibility of the grey model. The detailed procedure of the classical grey model is provided as follows.Definition 1.
Given a raw datasetX0, it is a nonnegative time series, which could be obtained from some sources in practice. X1 is transformed from the original series by the first-order accumulative generation operation. Z1 is considered as the smoothing series using the contiguous mean generation operation [1]:(1)X0=x01,x02,x03,…,x0n−1,x0n,X1=x11,x12,x13,…,x1n−1,x1n,where x1k is notated with x1k=∑i=1kx0k, for k=1,2,3,…,n−1,n.(2)Z1=z12,z13,…,z1n−1,z1n,where z1k is notated with z1k=x1k+x1k−1/2, for k=2,3,…,n−1,n.Lemma 1.
The grey differential equation is defined below [1]:(3)x0k+αz1k=β,fork=2,3,…,n−1,n.The equation above also is called the basic form of the classical grey model, GM (1, 1).Remark 1.
The mean regression solution of the structural parameters ofα and β could be estimated as follows:(4)α^mean,β^meanT=BTB−1BTY,where B=−z121−z131−z141⋮⋮−z1n−11−z1n1 and Y=x02x03x04⋮x0n−1x0nRemark 2.
The derivative of the first-order accumulative generation series, that is,x1k, is expressed typically below:(5)dx1kdt=limΔt⟶0x1t+Δt−x1tΔt.
If the interval of the time is set as one unit, that is,Δt=1, then(6)dx1kdt≈x1t−x1t−1=x0k.
We substitute the estimates ofα and β, that is, α^mean,β^mean into the whitening differential:(7)dx1kdt+α^meanx1k=β^mean.Remark 3.
According to the whitening equation above for the classical grey model, GM (1, 1), we solve the first-order differential equation and obtain the analytical expression of the time response function as follows:(8)x1k+1=x11−b^meana^meane−a^meank+b^meana^mean,fork=2,3,…,n,where x11=x01.
Therefore, the restored values, obtained by the first-order reducing generation operation, are as follows:(9)x0k+1=x1k+1−x1k,fork=2,3,…,n.


### 2.2. The Robust Grey Model (RGM (1, 1))

The classical grey model above estimates the structural parameters using the mean regression. This makes the final established forecasting model very sensitive to outliers. Thus, the robustness of the classical grey model is problematic. In order to reduce the influence due to outliers from several sources in practice, we propose a novel robust grey model, where we solve the structural parameters by median regression. Compared with the classical grey model, the new model has many advantages. For instance, the median regression is closely analogous to the intelligent algorithm to find the optimal solution to the structural parameters. Also, it is easy for researchers to understand what the machine is doing.Definition 2.
Each point,x0k, in the series is a random variable whose distribution function is Fx=Px0k≤x. Therefore, the median of the random variable x0k is expressed below:(10)Q0.5x=arginfx∈ℜ;Fx≥0.5.
The inverse of the distribution function isFx0k−10.5=infx0k∈ℜ;Fx≥0.5.
Thus, we have(11)Q0.5x0k=Fx0k−10.5.Definition 3.
The random variable,x0k, has a cumulative distribution function; that is, Fx0k|z1k=zx|z, on the given condition, z1k=z.
Therefore, the median ofx0k|z1k=z is defined below:(12)Q0.5x0k|z1k=z=arginfx0k∈ℜ;Fx|z≥0.5.Definition 4.
We assume that these two random variablesx0k,z1k satisfy the regression x0k=β+α−z1k+εk. The error εk is an unknown independent identically distribution. Thus, the median of x is defined as follows:(13)α0.5−z1k=argminx0k∈ℜEρ0.5x0k−x^0k|z1k=z.
The loss function for the median regression is below:(14)R=∑k=2nρ0.5x0k−x^0k.
We letΔx^0k=x0k−x^0k, and ρ0.5x0k−x^0k is expressed as follows:(15)ρ0.5Δx^0k=Δx^0k0.5⋅1Δx^0k≥0−0.5⋅1Δx^0k≥0.
In equation (15), 1⋅ represents the indicator function. The loss function is essential in establishing the novel robust grey model because it gives equal weight to the values with a positive error and a negative error, reducing the sensitivity of the model to outliers.Remark 4.
According to Definitions2 to 4, the probability density function of the random variable x0k is defined as fx . Considering the loss function, ρ0.5⋅, as the weight, the expectation of the absolute deviation, that is,(16)Eρ0.5Δx^0k=∫−∞x^0k0.5⋅x−x^0kfxdx+∫x^0k+∞0.5⋅x−x^0kfxdx,would approach to the minimum when and only when x^0k=x0.50k.Proof.
We substitute equation (15) into Eρ0.5x0k−x^0k and obtain equation (14). In order to determine the fitted value x^0k at the median, we let Eρ0.5x0k−x^0k=0. Therefore, equation (16) is simplified as(17)−0.5⋅∫−∞x^0kdFx−0.5∫x^0k+∞dFx=Fx^0k−0.5=0.
By replacing the theoretical distribution,Fx, with the empirical distribution, we obtain(18)∫ρ0.5x−x^0kdFx=∑k=1nρ0.5x0k−x^0kn.
The above equation reaches the minimal whenx^0k=x0.50k. Thus, the structural parameters could be obtained using the median regression. That is,(19)minα^0.5,β^0.5∑i=2nx0k≥x^0k0.5⋅x0k−x^0k+∑i=2nx0kx^0k0.5⋅x0k−x^0k,where x^0k=β^0.5+α^0.5−z1k.
However, before going ahead, one thing needing particular attention is that the objective function, equation (19), could not be solved by traditional means such as the Lagrange multiplier approach to obtain an analytical expression. Thus, a numerical iteration method is applied here to approximate the optimal solution for these structural parameters.Lemma 2.
Based on the grey differential equation, equation (3), we use the median regression to obtain the estimates of the structural parameters α^0.5,β^0.5. It is different from the mean regression used in the classical grey model to calculate the optimal parameters α^mean,β^mean.Remark 5.
Based on the whitening equation, that is, equation (4), we solve the first-order differential equation and obtain the analytical expression of the time response function as follows:(20)x1k+1=x11−b^0.5a^0.5e−a^0.5k+b^0.5a^0.5,fork=2,3,…,n,where x11=x01.
Consequently, the restored values could be obtained from the first-order reducing generation operation.
In sum, the modeling method for the novel robust grey model is different from the classical grey model. The former adopts the median regression to calculate the structural parameters, while the latter applies the mean regression to estimate the structural parameters. The median regression technique could reduce the influence due to outliers, while the mean regression technique is heavily dependent on the conditional expectation of the random variable,x0k, whose shape would have a significant change when some data points change a little. Therefore, the prediction model based on the median regression would be more robust than that of the mean regression. It is common to see the heavy-tailed distribution in many fields, and it probably exhibits some highly influential points. Thus, we believe that the novel robust grey model would perform well in these fields.


### 2.3. Model Testing

The evaluation of the model performance should be based on two aspects, that is, the accuracy and the robustness.In order to assess the accuracy of prediction, we employ three statistical indices in this study, including the absolute percentage error, the mean absolute percentage error, and the correlation coefficient. A smaller number represents a better forecasting ability for the first two indicators. For the last index, a larger value represents a high model performance. The details of the calculation of these statistics are provided as follows.The absolute percentage error is(21)APE=εkx0k.The mean absolute percentage error is(22)MAPE=∑i=1nεk/x0k.The correlation coefficient is(23)r=covX0,X^0varX0varX^0.In addition, to check the robustness of the novel robust grey model, we perform a series of randomization tests, which will be discussed in detail in Section3.


## 2.1. The Classical Grey Model (GM (1, 1))

Professor Deng puts forward the typical grey model, which serves as a useful tool to deal with small sample and little information, particularly in uncertain issues. Compared with other forecasting model, this model has some remarkable advantages. For example, the grey model does not need too many observations, and only using three data points we could form an effective prediction model with relatively high accuracy. More importantly, the process of modeling for the grey model is open and apparent, which reduces the possibility of artificial fit that improves the credibility of the grey model. The detailed procedure of the classical grey model is provided as follows.Definition 1.
Given a raw datasetX0, it is a nonnegative time series, which could be obtained from some sources in practice. X1 is transformed from the original series by the first-order accumulative generation operation. Z1 is considered as the smoothing series using the contiguous mean generation operation [1]:(1)X0=x01,x02,x03,…,x0n−1,x0n,X1=x11,x12,x13,…,x1n−1,x1n,where x1k is notated with x1k=∑i=1kx0k, for k=1,2,3,…,n−1,n.(2)Z1=z12,z13,…,z1n−1,z1n,where z1k is notated with z1k=x1k+x1k−1/2, for k=2,3,…,n−1,n.Lemma 1.
The grey differential equation is defined below [1]:(3)x0k+αz1k=β,fork=2,3,…,n−1,n.The equation above also is called the basic form of the classical grey model, GM (1, 1).Remark 1.
The mean regression solution of the structural parameters ofα and β could be estimated as follows:(4)α^mean,β^meanT=BTB−1BTY,where B=−z121−z131−z141⋮⋮−z1n−11−z1n1 and Y=x02x03x04⋮x0n−1x0nRemark 2.
The derivative of the first-order accumulative generation series, that is,x1k, is expressed typically below:(5)dx1kdt=limΔt⟶0x1t+Δt−x1tΔt.
If the interval of the time is set as one unit, that is,Δt=1, then(6)dx1kdt≈x1t−x1t−1=x0k.
We substitute the estimates ofα and β, that is, α^mean,β^mean into the whitening differential:(7)dx1kdt+α^meanx1k=β^mean.Remark 3.
According to the whitening equation above for the classical grey model, GM (1, 1), we solve the first-order differential equation and obtain the analytical expression of the time response function as follows:(8)x1k+1=x11−b^meana^meane−a^meank+b^meana^mean,fork=2,3,…,n,where x11=x01.
Therefore, the restored values, obtained by the first-order reducing generation operation, are as follows:(9)x0k+1=x1k+1−x1k,fork=2,3,…,n.


## 2.2. The Robust Grey Model (RGM (1, 1))

The classical grey model above estimates the structural parameters using the mean regression. This makes the final established forecasting model very sensitive to outliers. Thus, the robustness of the classical grey model is problematic. In order to reduce the influence due to outliers from several sources in practice, we propose a novel robust grey model, where we solve the structural parameters by median regression. Compared with the classical grey model, the new model has many advantages. For instance, the median regression is closely analogous to the intelligent algorithm to find the optimal solution to the structural parameters. Also, it is easy for researchers to understand what the machine is doing.Definition 2.
Each point,x0k, in the series is a random variable whose distribution function is Fx=Px0k≤x. Therefore, the median of the random variable x0k is expressed below:(10)Q0.5x=arginfx∈ℜ;Fx≥0.5.
The inverse of the distribution function isFx0k−10.5=infx0k∈ℜ;Fx≥0.5.
Thus, we have(11)Q0.5x0k=Fx0k−10.5.Definition 3.
The random variable,x0k, has a cumulative distribution function; that is, Fx0k|z1k=zx|z, on the given condition, z1k=z.
Therefore, the median ofx0k|z1k=z is defined below:(12)Q0.5x0k|z1k=z=arginfx0k∈ℜ;Fx|z≥0.5.Definition 4.
We assume that these two random variablesx0k,z1k satisfy the regression x0k=β+α−z1k+εk. The error εk is an unknown independent identically distribution. Thus, the median of x is defined as follows:(13)α0.5−z1k=argminx0k∈ℜEρ0.5x0k−x^0k|z1k=z.
The loss function for the median regression is below:(14)R=∑k=2nρ0.5x0k−x^0k.
We letΔx^0k=x0k−x^0k, and ρ0.5x0k−x^0k is expressed as follows:(15)ρ0.5Δx^0k=Δx^0k0.5⋅1Δx^0k≥0−0.5⋅1Δx^0k≥0.
In equation (15), 1⋅ represents the indicator function. The loss function is essential in establishing the novel robust grey model because it gives equal weight to the values with a positive error and a negative error, reducing the sensitivity of the model to outliers.Remark 4.
According to Definitions2 to 4, the probability density function of the random variable x0k is defined as fx . Considering the loss function, ρ0.5⋅, as the weight, the expectation of the absolute deviation, that is,(16)Eρ0.5Δx^0k=∫−∞x^0k0.5⋅x−x^0kfxdx+∫x^0k+∞0.5⋅x−x^0kfxdx,would approach to the minimum when and only when x^0k=x0.50k.Proof.
We substitute equation (15) into Eρ0.5x0k−x^0k and obtain equation (14). In order to determine the fitted value x^0k at the median, we let Eρ0.5x0k−x^0k=0. Therefore, equation (16) is simplified as(17)−0.5⋅∫−∞x^0kdFx−0.5∫x^0k+∞dFx=Fx^0k−0.5=0.
By replacing the theoretical distribution,Fx, with the empirical distribution, we obtain(18)∫ρ0.5x−x^0kdFx=∑k=1nρ0.5x0k−x^0kn.
The above equation reaches the minimal whenx^0k=x0.50k. Thus, the structural parameters could be obtained using the median regression. That is,(19)minα^0.5,β^0.5∑i=2nx0k≥x^0k0.5⋅x0k−x^0k+∑i=2nx0kx^0k0.5⋅x0k−x^0k,where x^0k=β^0.5+α^0.5−z1k.
However, before going ahead, one thing needing particular attention is that the objective function, equation (19), could not be solved by traditional means such as the Lagrange multiplier approach to obtain an analytical expression. Thus, a numerical iteration method is applied here to approximate the optimal solution for these structural parameters.Lemma 2.
Based on the grey differential equation, equation (3), we use the median regression to obtain the estimates of the structural parameters α^0.5,β^0.5. It is different from the mean regression used in the classical grey model to calculate the optimal parameters α^mean,β^mean.Remark 5.
Based on the whitening equation, that is, equation (4), we solve the first-order differential equation and obtain the analytical expression of the time response function as follows:(20)x1k+1=x11−b^0.5a^0.5e−a^0.5k+b^0.5a^0.5,fork=2,3,…,n,where x11=x01.
Consequently, the restored values could be obtained from the first-order reducing generation operation.
In sum, the modeling method for the novel robust grey model is different from the classical grey model. The former adopts the median regression to calculate the structural parameters, while the latter applies the mean regression to estimate the structural parameters. The median regression technique could reduce the influence due to outliers, while the mean regression technique is heavily dependent on the conditional expectation of the random variable,x0k, whose shape would have a significant change when some data points change a little. Therefore, the prediction model based on the median regression would be more robust than that of the mean regression. It is common to see the heavy-tailed distribution in many fields, and it probably exhibits some highly influential points. Thus, we believe that the novel robust grey model would perform well in these fields.


## 2.3. Model Testing

The evaluation of the model performance should be based on two aspects, that is, the accuracy and the robustness.In order to assess the accuracy of prediction, we employ three statistical indices in this study, including the absolute percentage error, the mean absolute percentage error, and the correlation coefficient. A smaller number represents a better forecasting ability for the first two indicators. For the last index, a larger value represents a high model performance. The details of the calculation of these statistics are provided as follows.The absolute percentage error is(21)APE=εkx0k.The mean absolute percentage error is(22)MAPE=∑i=1nεk/x0k.The correlation coefficient is(23)r=covX0,X^0varX0varX^0.In addition, to check the robustness of the novel robust grey model, we perform a series of randomization tests, which will be discussed in detail in Section3.


## 3. Robustness Check

In this section, we intend to test the robustness and accuracy of the novel robust grey model developed in Section2 called RGM(1, 1). The time series used for these tests is the series of electricity demand in China between 2011 and 2019. Throughout, we compare the new model with the classical model, GM (1, 1).


### 3.1. The Robustness with an Outlier in the Sample Period

Given that the value in the year 2015 is an outlier, which is far larger than other values in the sample period, Table1 presents the results from both models. In columns (1) and (2), we report the fitted values and absolute percentage error for the classical grey model, that is, GM (1, 1), while in columns (3) and (4), we propose those obtained from the novel robust grey model, that is, RGM (1, 1).Table 1
The accuracy with an outlier in the sample period.
YearActual valuesGM (1, 1)RGM (1, 1)Fitted valuesAPE (%)Fitted valuesAPE (%)(1)(2)(3)(4)20114.700————20124.9765.90818.7305.1733.95920135.4206.14213.3215.4290.16620145.7826.38510.4295.6971.470201510.8016.63838.5435.97844.65320166.1206.90112.7616.2732.50020176.5917.1748.8546.5820.13720187.1507.4584.3086.9073.39920197.4867.7543.5807.2483.179Mean absolute percentage error12.2806.607Correlation coefficient0.5000.500For the data in 2015 that is an outlier, we observe that the classical grey model has a smaller absolute percentage error than the novel robust grey model. In other years, however, all the absolute percentage errors of RGM (1, 1) are much lower than those of GM (1, 1). That is to say, when there exists an outlier in the sample series, to make the fitted value possibly approach to the actual value that is an outlier, the classical grey model achieves it at the cost of reducing the accuracy of other predicted points.At the bottom of Table1, we provide the mean absolute percentage error and the correlation coefficient used to describe these two models’ total performance. We find that the mean absolute percentage error in the classical grey model is twice as much as that in the novel robust model, even though both models have the same correlation coefficient, suggesting that RGM (1 1) has a relatively high prediction accuracy and is not sensitive to the outlier.


### 3.2. The Bootstrapping Test for Robustness

In the Section3.1, we use a special case, where we replace the actual value in 2015 with an outlier, to demonstrate the robustness and the accuracy of the new proposed grey model, RGM(1, 1). Here, we adopt an alternative to test the robustness, that is, the bootstrapping approach. In detail, first, we replace the actual value for observation in the particular year with a random number to produce a new series with an outlier. The random number is generated from the interval between the sample series maximum and minimum. Then, we use the classical grey model and the novel robust grey model to fit the new series, respectively, and calculate the mean absolute percentage errors for these two models. Finally, we repeat this procedure 1000 times.Figures1 and 2 plot the results for the bootstrapping test. In Figure 1, we report the distribution of the mean absolute percentage errors across 1000 times for the classical grey model, GM (1,1), while in Figure 2, we report that, for the novel robust grey model, it is RGM (1, 1). We divide Figure 1, as well as Figure 2, into 9 panels. In Panel A, we replace the actual value in the year 2011 with outliers. In Panel B, we replace the actual value in 2012 with outliers. In Panel C, we replace the actual value in the year 2013 with outliers. In Panel D, we replace the actual value in the year 2014 with outliers. In Panel E, we replace the actual value in the year 2015 with outliers. In Panel F, we replace the actual value in the year 2016 with outliers. In Panel G, we replace the actual value in the year 2017 with outliers. In Panel H, we replace the actual value in the year 2018 with outliers. In Panel K, we replace the actual value in the year 2019 with outliers.Figure 1
The bootstrapping test for the robustness of the classical grey model, GM (1, 1).Figure 2
The bootstrapping test for the robustness of the novel proposed grey model, RGM (1, 1).Compared with the classical grey model, RGM (1, 1) has a smaller average across the 1000 mean absolute percentage errors. For example, in Panel A of Figure1, all the mean absolute percentage errors for GM (1, 1) are larger than 10 percentage points. In Panel A of Figure 2, however, each mean absolute percentage error for RGM (1, 1) is lower than 7 percentage points. Comparing other panels in Figure 1 and those in Figure 2, we find an analogous situation to the above.In addition, we find that, in the year 2015 that is the middle point for the sample period, the intervals of 1000 mean absolute percentage errors for both models are very similar to each other, even though we have seen that the mean absolute percentage error is smaller for RGM (1, 1) than for GM (1, 1). In other words, outliers would have a relatively low effect on the robustness and the accuracy of model when it occurs at the middle point during the sample period.


### 3.3. The Accuracy without Outliers

So far, we have investigated the robustness of the novel robust grey model, which is developed in this study. We find evidence that RGM (1, 1) is robust to outliers. Also, we see that the novel robust grey model has a better prediction performance than the classical grey model when there is an outlier in the sample. Of course, we would want to know the accuracy of RGM (1, 1) without outliers in the sample.Table2 exhibits these results. In Table 2, we use the original series of electricity demand in China from 2011 to 2019. The first column reports the fitted values for GM (1, 1). The second column presents the absolute percentage error for GM (1, 1). The third column proposes the fitted values for RGM (1, 1). The fourth column provides the absolute percentage error for RGM (1, 1).Table 2
The accuracy without outliers in the sample period.
YearActual valuesGM (1, 1)RGM (1, 1)Fitted valuesAPE (%)Fitted valuesAPE (%)(1)(2)(3)(4)20114.700————20124.9765.0160.8044.9750.02020135.4205.3082.0665.2732.71220145.7825.6162.8715.5893.33820155.8015.9422.4315.9242.12020166.1206.2872.7296.2802.61420176.5916.6520.9266.6560.98620187.1507.0381.5667.0551.32920197.4867.4460.5347.4790.094Mean absolute percentage error1.5471.468Correlation coefficient0.9920.992We find that, although not all absolute percentage errors are lower in the novel robust grey model than in the classical grey model, RGM (1, 1) has a smaller mean absolute percentage error than GM (1, 1). At the same time, the correlation coefficients for both models are the same. This suggests that the novel robust grey model developed in this study has a high ability of prediction than the classical grey model.Taken together, RGM (1, 1) not only is robust to outliers but also has a higher accuracy of prediction than the classical grey model.


## 3.1. The Robustness with an Outlier in the Sample Period

Given that the value in the year 2015 is an outlier, which is far larger than other values in the sample period, Table1 presents the results from both models. In columns (1) and (2), we report the fitted values and absolute percentage error for the classical grey model, that is, GM (1, 1), while in columns (3) and (4), we propose those obtained from the novel robust grey model, that is, RGM (1, 1).Table 1
The accuracy with an outlier in the sample period.
YearActual valuesGM (1, 1)RGM (1, 1)Fitted valuesAPE (%)Fitted valuesAPE (%)(1)(2)(3)(4)20114.700————20124.9765.90818.7305.1733.95920135.4206.14213.3215.4290.16620145.7826.38510.4295.6971.470201510.8016.63838.5435.97844.65320166.1206.90112.7616.2732.50020176.5917.1748.8546.5820.13720187.1507.4584.3086.9073.39920197.4867.7543.5807.2483.179Mean absolute percentage error12.2806.607Correlation coefficient0.5000.500For the data in 2015 that is an outlier, we observe that the classical grey model has a smaller absolute percentage error than the novel robust grey model. In other years, however, all the absolute percentage errors of RGM (1, 1) are much lower than those of GM (1, 1). That is to say, when there exists an outlier in the sample series, to make the fitted value possibly approach to the actual value that is an outlier, the classical grey model achieves it at the cost of reducing the accuracy of other predicted points.At the bottom of Table1, we provide the mean absolute percentage error and the correlation coefficient used to describe these two models’ total performance. We find that the mean absolute percentage error in the classical grey model is twice as much as that in the novel robust model, even though both models have the same correlation coefficient, suggesting that RGM (1 1) has a relatively high prediction accuracy and is not sensitive to the outlier.


## 3.2. The Bootstrapping Test for Robustness

In the Section3.1, we use a special case, where we replace the actual value in 2015 with an outlier, to demonstrate the robustness and the accuracy of the new proposed grey model, RGM(1, 1). Here, we adopt an alternative to test the robustness, that is, the bootstrapping approach. In detail, first, we replace the actual value for observation in the particular year with a random number to produce a new series with an outlier. The random number is generated from the interval between the sample series maximum and minimum. Then, we use the classical grey model and the novel robust grey model to fit the new series, respectively, and calculate the mean absolute percentage errors for these two models. Finally, we repeat this procedure 1000 times.Figures1 and 2 plot the results for the bootstrapping test. In Figure 1, we report the distribution of the mean absolute percentage errors across 1000 times for the classical grey model, GM (1,1), while in Figure 2, we report that, for the novel robust grey model, it is RGM (1, 1). We divide Figure 1, as well as Figure 2, into 9 panels. In Panel A, we replace the actual value in the year 2011 with outliers. In Panel B, we replace the actual value in 2012 with outliers. In Panel C, we replace the actual value in the year 2013 with outliers. In Panel D, we replace the actual value in the year 2014 with outliers. In Panel E, we replace the actual value in the year 2015 with outliers. In Panel F, we replace the actual value in the year 2016 with outliers. In Panel G, we replace the actual value in the year 2017 with outliers. In Panel H, we replace the actual value in the year 2018 with outliers. In Panel K, we replace the actual value in the year 2019 with outliers.Figure 1
The bootstrapping test for the robustness of the classical grey model, GM (1, 1).Figure 2
The bootstrapping test for the robustness of the novel proposed grey model, RGM (1, 1).Compared with the classical grey model, RGM (1, 1) has a smaller average across the 1000 mean absolute percentage errors. For example, in Panel A of Figure1, all the mean absolute percentage errors for GM (1, 1) are larger than 10 percentage points. In Panel A of Figure 2, however, each mean absolute percentage error for RGM (1, 1) is lower than 7 percentage points. Comparing other panels in Figure 1 and those in Figure 2, we find an analogous situation to the above.In addition, we find that, in the year 2015 that is the middle point for the sample period, the intervals of 1000 mean absolute percentage errors for both models are very similar to each other, even though we have seen that the mean absolute percentage error is smaller for RGM (1, 1) than for GM (1, 1). In other words, outliers would have a relatively low effect on the robustness and the accuracy of model when it occurs at the middle point during the sample period.


## 3.3. The Accuracy without Outliers

So far, we have investigated the robustness of the novel robust grey model, which is developed in this study. We find evidence that RGM (1, 1) is robust to outliers. Also, we see that the novel robust grey model has a better prediction performance than the classical grey model when there is an outlier in the sample. Of course, we would want to know the accuracy of RGM (1, 1) without outliers in the sample.Table2 exhibits these results. In Table 2, we use the original series of electricity demand in China from 2011 to 2019. The first column reports the fitted values for GM (1, 1). The second column presents the absolute percentage error for GM (1, 1). The third column proposes the fitted values for RGM (1, 1). The fourth column provides the absolute percentage error for RGM (1, 1).Table 2
The accuracy without outliers in the sample period.
YearActual valuesGM (1, 1)RGM (1, 1)Fitted valuesAPE (%)Fitted valuesAPE (%)(1)(2)(3)(4)20114.700————20124.9765.0160.8044.9750.02020135.4205.3082.0665.2732.71220145.7825.6162.8715.5893.33820155.8015.9422.4315.9242.12020166.1206.2872.7296.2802.61420176.5916.6520.9266.6560.98620187.1507.0381.5667.0551.32920197.4867.4460.5347.4790.094Mean absolute percentage error1.5471.468Correlation coefficient0.9920.992We find that, although not all absolute percentage errors are lower in the novel robust grey model than in the classical grey model, RGM (1, 1) has a smaller mean absolute percentage error than GM (1, 1). At the same time, the correlation coefficients for both models are the same. This suggests that the novel robust grey model developed in this study has a high ability of prediction than the classical grey model.Taken together, RGM (1, 1) not only is robust to outliers but also has a higher accuracy of prediction than the classical grey model.


## 4. Prediction

In Section3, we check the robustness of the novel robust grey model, that is, RGM (1, 1). We find that the novel grey model is more robust to outliers. In addition, we find its mean absolute percentage error for in-sample is smaller than the classical grey model, that is, GM (1, 1), suggesting that the capability of fitness in the sample for the former is better than for the latter. In this section, we apply the novel robust grey model to forecast the future electricity demand in China. We also explore whether the out-of-sample prediction capability of the novel robust grey model is better than that of the classical grey model. Table 3 presents the results.Table 3
Prediction of Chinese electricity demand during the period 2022 to 2025.
YearActual valuesGM (1, 1)RGM (1, 1)Fitted valuesAPE (%)Fitted valuesAPE (%)(1)(2)(3)(4)20228.8178.82020239.3289.33220249.8699.873202510.44210.446Mean absolute percentage error for out-of-sample0.7270.721Correlation coefficient for out-of-sample0.9990.999In the first column of Table3, we report the forecasting values obtained from the classical grey model. This model estimates that the future electricity demand in China would continue to increase, which might arrive at 10.442 × 105 million kW·h in the year of 2025. In the third column of Table 3, we report the forecasting values obtained from the novel robust grey model. This model estimates that the Chinese electricity demand would be 8.820 × 105 million kW·h in the year of 2022, 9.332 × 105 million kW·h in the year of 2023, 9.873 × 105 million kW·h in the year of 2024, and 10.446 × 105 million kW·h in the year of 2025.In the bottom of Table3, we report mean absolute percentage errors of out-of-sample during the year from 2019 to 2021, as well as correlation coefficients between the original time series and the forecasting time series, for the two model, respectively. We could see that these two correlation coefficients are the same, while the novel robust grey model has a smaller mean absolute percentage error than the classical grey model, indicating that the capability of prediction in the former is better than that in the latter.


## 5. Conclusion

We develop a novel grey model by introducing the median regression to estimate the structural parameters, abbreviated as RGM (1, 1). We compare the novel robust grey model with the classical grey model, GM (1, 1). We find that RGM (1, 1) is not sensitive to outliers. Also, we see that the accuracy of the novel robust grey model is improved compared to the classical grey model. Future research could further combine the median regression and other optimization methods to boost the robustness and accuracy of the grey model.



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*Source: 3704027-2022-05-14.xml*