A new grey quadratic polynomial model and its application in the COVID-19 in China

This paper develops a new grey prediction model with quadratic polynomial term. Analytical expressions of the time response function and the restored values of the new model are derived by using grey model technique and mathematical tools. With observations of the confirmed cases, the death cases and the recovered cases from COVID-19 in China at the early stage, the proposed forecasting model is developed. The computational results demonstrate that the new model has higher precision than the other existing prediction models, which show the grey model has high accuracy in the forecasting of COVID-19.

www.nature.com/scientificreports/ and non-homogeneous exponential series, or even some fluctuating series. (2) The analytical solution of time response function and the matrix expression of system parameters are also determined by grey technique. (3) The proposed newly model is a general grey forecasting model, and the GM(1,1) model, the NGM(1,1,k) model and the NGM(1,1,k,c) model are all special cases of the proposed model. Moreover, the feasibility of the new model is verified through two examples. (4) The new model is used to study the confirmed cases, the death cases and the recovered cases of COVID-19 in China at the early stage, and results illustrate that the new model has higher precision than other forecasting models. The rest of this paper is arranged as follows. Section 2 discusses the existing grey forecasting models. The details of the grey prediction model with quadratic polynomial term is given in Sect. 3. Section 4 provides some numerical examples. Applications are studied in the Sect. 5. Conclusions are placed in the last section.

Some existing grey forecasting models
This section provides a brief overview of some grey forecasting models which will used in the application section. They are the classical grey model GM (1,1), the discrete grey model DGM (1,1), the non-homogeneous grey model NGM (1,1,k,c) and the grey Verhulst model GVM (1,1). For concise, we only provide the whitening equation, the time response function and the restored values of them.
(1) The GM(1,1) model The classical grey model GM (1,1) is the core of the grey forecasting theory. Since been putted forward, it has been widely applied in various fields including energy, economy and education. The whitening equation of GM(1,1) model is given by The time response function and the restored values are (2) The DGM(1,1) model The discrete grey forecasting model DGM(1,1) is initially provided by Xie and Liu 28,29 , the mathematical expression is and the recursive function is given by  34 , which is able to simulate and predict original observations with an inverted U shape or a signal peak feature. The whitening equation of GVM(1,1) model is Further, the time response function and the restored values are (1) dx (1)

The grey model with quadratic polynomial term
This section discusses the grey model with quadratic polynomial term which is abbreviated as GMQP(1,1) model in the present paper. We first provide the definition of the accumulated and inverse accumulated generation operators, and then discuss the new model GMQP(1,1) along with some properties.
Accumulated and inverse accumulated generation operator. Definition 1 (Accumulated generation operator) First, we assume the original non-negative sequence is where the relationship is given by The operator A is named as the first-order accumulated genera- It follows from the definition 1 and definition 2 that the inverse accumulated generation operator is the inverse operation of the accumulated generation operator.
The grey quadratic polynomial model. Definition 3 Assume X (0) and X (1) are stated in definition 1, then the whitening differential equation of the grey model with quadratic polynomial term is defined as.
where a is the development coefficient, and bt 2 + ct + d is the grey action quantity. Obviously, when system parameter b = 0 in Eq. (12), the GMQP(1,1) model degenerates to the NGM(1,1,k,c) model.
where Proof Employing the mathematical induction considering k = 2,3,…,n into Theorem 1, we obtain that.
Converting the above equation system into the matrix form, we can get . . .
. . .   www.nature.com/scientificreports/ Assume a raw sequence X (0) = x (0) (1), x (0) (2), · · · , x (0) (m), x (0) (m + 1), . . . , x (0) (n) where a subsequence composed of the first m entries of raw sequence X (0) is applied to develop the newly proposed model, and simulation sequence is X (0) S = x (0) (1),x (0) (2), · · · ,x (0) (m) . We utilize the grey forecasting model to forecast the left n-m steps data, and the prediction sequence is X The error sequence of the simulation sequence X (0) S and the prediction sequence X (0) F are, respectively, ε S and ε F , which are given as follows Here the absolute percentage error (APE), the absolute error (MAE), the mean squares error (MSE), the mean absolute percentage error (MAPE), the root mean square percentage error (RMSPE), the index of agreement (IA) and the correlation coefficient (R) are provided below.
• The absolute percentage error

Validation of the GMQP(1,1) model
To validation of the feasibility of the new model, this section gives two numerical example where datasets are collected from published papers.

Example 1
In this example, data are all collected from Table 2 in Ref 35 . where the total energy consumption in China (unit: 10000tce). These data are used to build the GM(1,1) model, the DGM(1,1) model, the NGM(1,1,k,c) model, the GVM(1,1) model and the GMQP(1,1) model. The numerical results of these grey forecasting models are displayed in the following Tables 2, 3 and 4.  Tables 2, 3, and 4 that the new model has better performance measures than other grey forecasting models in the energy consumption of China, which show that the new structure of GMQP(1,1) model can improve the precision of grey model.

Example 2
In this example, the raw data of the electricity consumption of China are collected from Table 2 in Ref. 36    www.nature.com/scientificreports/ It follows from example 1 and example 2 that the new grey model has best performance measures, which shows the new grey models with a more flexible structure can be a good way of improving the accuracy of model.

Applications in the COVID-19 of China
In this section, we will use different grey forecasting models and the polynomial regression to study the confirmed cases, the death cases and the recovered cases from COVID-19 in China, which are the classical continuous grey model GM (1,1), the discrete grey model DGM(1,1), the non-homogeneous grey model NGM(1,1,k,c), the nonlinear grey Verhulst model GVM(1,1), the polynomial regression (PR) and the grey model with quadratic polynomial term GMQP(1,1). Moreover, the structure of the applications in the COVID-19 of China is shown in Fig. 2. Table 3. The APEs of these forecasting models in the energy consumption of China.   www.nature.com/scientificreports/ www.nature.com/scientificreports/ The confirmed cases from COVID-19 of China. In this subsection, we apply forecasting models to study the confirmed cases from COVID-19 of China. The raw data, starting 2020-01-21 to 2020-02-06, are collected from the website: http:// www. nhc. gov. cn, and displayed in the following Table 8 and Fig. 3. With these raw data, we can deduce the mathematical expressions of different grey model. Here we take the GMQP(1,1) model as an example to details show the modelling procedures.
Step 1 pre-process the raw data. It follows from Step 2 System parameter estimation. From theorem 2, and the values of X (0) and X (1) , we calculate the matrix B and the matrix Y which are given by With the help of the Eq. (17), we can obtain the values of system parameters as  440  571  830  1287  1975  2744  4515  5974  7711  9692  11791  14380 (21), respectively. Therefore, we can compute the simulation and prediction values of the confirmed cases of COVID-19 of China. By a similar argument to the other grey forecasting models which are provided below.
The GM(1,1) model. We can obtain system parameters a = -0.2441, b = 1116.9454 of the GM(1,1) model by the least squares method. And then the mathematical expression is given by.

The death cases from COVID-19 of China. This subsection discusses the death cases from COVID-19
of China by employing grey models. The raw data are collected from the website: http:// www. nhc. gov. cn, and displayed in the following Tables 12, 13, 14 and Fig. 5. The first 14 observations are used to build models, and the left three observation is used to test. Similar argument is applied to derive system parameters of each model, and then the mathematical expressions are given below.

Conclusion
This paper studied the grey forecasting model with quadratic polynomial term, and applied it to the confirmed cases, the death cases and the recovered cases from COVID-19 of China at the early stage. By using the grey technique and some mathematical derivations, the grey basic form, the time response function and the restored values are all systematically analyzed. With raw datasets of COVID-19 in China, we compute the simulation and fitting values by different forecasting models. It follows from the computational results, we can observed the new model has higher precision than other models. This also implied that our generalized model has applicable value in the COVID-19.
In this work, the GMQP(1,1) model is an univariate grey forecasting model and some factors such as social isolation and lockdown, vaccines, active treatment cannot be considered. In addition, the integer order accumulating generated operation is used to preprocess the raw data. It is generally known that the fractional order accumulating generated operation or the new information priority to preprocess raw data can get more accurate results. Thus in the future, we will continuous consider such a model with other accumulating generated operator including new information priority, fractional accumulating generated operator. Further, other multivariate grey forecasting models can be constructed to study the COVID-19.