The application of a novel grey model in the prediction of China’s aging population

The dramatic increase of the aging population is a current and primary livelihood issue in China. It is very important to accurately predict the aging population for policy-makers such as governments and insurance companies. To explore the future development trend of China’s aging population, a novel grey prediction FPTGM(1,1,α) model is established in this paper. The data from 2003 to 2012 are used to build the model, and the data from 2013 to 2018 are used to assess the modeling accuracy. The results show that the fitting and forecasting accuracy of the proposed FPTGM(1,1,α) model are higher than those of other models. This indicates that the novel grey model is more suitable for predicting the aging population in China. Combined with the idea of metabolism, the FPTGM(1,1,α) model is applied to predict China’s aging population from 2019 to 2042. Finally, some reasonable suggestions for dealing with the aging population are put forward to the government and other decision-makers.


Current status of population aging research
Population aging is an important social problem facing China. With the rapid development of our economy and society, people's living standards and medical standards have been steadily improved, life expectancy has been gradually extended, and the mortality rate has been greatly reduced, resulting in an increase in the proportion of the elderly population. According to international standards, if a country or region has more than 7 percent of its population over the age of 65, the country or region's aging problem is more prominent. According to this judgment, China entered an aging society in 2000. Due to China's strict family planning policy and the reduction of family fertility intention, the birth rate dropped sharply, which exacerbated the process of population aging in China. The proportion of the elderly population in the total population in China has increased sharply ( Fig. 1(1)). In addition, from the perspective of the world, the proportion of the elderly population in the world has also increased sharply ( Fig. 1(2)), indicating that the aging problem in China has been quite serious.
How to deal with the aging problem is the common concern of demography, economics and management. Facing the increasingly severe problem of aging in China, accurate prediction of the number of elderly population is of great significance to correctly understand the trend of aging population in China and its social and economic consequences, and provides data and theoretical support for the government to formulate relevant population policies. A scientific and reasonable prediction of population aging can accelerate the development of elderly care service industry, the implementation of national strategies and the formulation of policies. How to improve the accuracy and precision of prediction model is always the focus of research.
The population prediction began with the study of mortality rate proposed by John Graunt (Li 2016 (Hansen 1989) conducted further research on population aging in 1898, and found out the relationship between socio-economic development and population aging. In recent years, many scholars have continued to study the aging population by using different methods. The main methods used are stochastic prediction methods, time sequence prediction models, neural network prediction models, autoregressive integrated moving average models, grey prediction models, etc. For instance, Li et al. (2009) put forward the stochastic forecast of population aging in China, and used the scaled model for error to quantify the uncertainty attached to the population predictions. Thenuwara et al. (2019) applied a structural vector autoregressive framework to study whether population aging will precipitate a dramatic decline in real house prices in Australia. Jacobsen et al. (2020) utilized an artificial neural network model to study the prediction factors of attrition based on longitudinal population aging. Chang et al. (2019) used ARIMA models to find that the increase of over 65-year-old adults in Taiwan will reach up to a new high in the next decade. Duan (2019) used the grey forecasting model to predict the aging population in Chongqing in the next decade and concluded that the degree of aging continues to increase. Among these models, the grey prediction model is widely used due to its high precision and simple operation process.

Research status of grey model
The grey prediction model was first proposed by professor Deng (1982). It takes the ''small sample'' and ''incomplete information'' uncertain system as the research object. Based on the accumulation and regeneration of some known information, it extracts effective information and realizes the prediction and description of the whole system change trend. Its purpose is to predict the evolution law of system influenced by uncertain factors. It is widely used because of its easy-to-understand modeling mechanism and high predicted precision. The GM(1,1) model is the most basic and important model in grey prediction theory, and it is also the most used model in grey system model. Its basic rationale is that the random original sequence is accumulated according to time, and then the root obtained by the first-order linear differential equation is used to approximate the data trend law of the accumulated sequence. It has a strong fitting ability to the time sequence with grey index law. In fact, economic, ecological and other systems can be regarded as a generalized energy system, and most of the accumulation and release of energy has an exponential law (Deng 1982). Therefore, the GM(1,1) model has a wide application background. According to people's actual needs, the GM(1,1) model has been widely used in various fields such as energy (Hamzacebi and Es 2014;Shaikh et al. 2017;Wu et al. 2018;Ma et al. 2019a;Wang et al. 2020;Hu 2020), finance (Zhang et al. 2010;Zhang and Chen 2014), ecology and environment (Hao et al. 2006;Yu et al. 2018), etc. In a recent study, the grey prediction model began to be used to predict the aging population (Duan 2019;Chen et al. 2009;Bao et al. 2015). However, as a nascent model, GM(1,1) model not be used for effectively forecasting the homogeneous exponential sequence due to the span from the difference equation to the differential equation. In order to solve this problem, Xie et al. (2009) proposed a discrete GM(1,1) model (DGM(1,1) model) which can fully fit the homogeneous exponential sequence. Subsequently, aiming at onhomogeneous index sequence and polynomial time-vary sequence, Xie et al. (2013aXie et al. ( , 2013b) also proposed NDGM(1,1) model and DGM (1,1,m) model respectively. Based on Xie et al. research, Luo et al. (2019) proposed a polynomial discrete GM(1,1) model (DGMP(1,N) model).
To improve the forecasting accuracy of GM(1,1), Cui et al. (2010) proposed the NGM(1,1,k) model. Qian et al. (2012) proposed the GM(1,1,t a ) model based on NGM(1,1,k) model. To enhance the adaptability of GM(1,1), based on first-order accumulation generation, Wu et al. (2013) introduce fractional-order accumulation generation into the grey model and proposed fraction-order GM(1,1) model. Considering the data characteristics of small samples and increasing trend from China's annual aging population data presented Fig. 1(2), based on the research foundation of the above-mentioned various grey models, a novel grey prediction model with fractional-order time power term (abbreviated as FPTGM(1,1,a) model) is proposed in the present study. The proposed model has structural compatibility and adjustability of super parameter, and it can adapt to small sample time series modeling with different changing characteristics. Meanwhile, combined with the idea of metabolism, the proposed FPTGM(1,1,a) model is applied to the prediction of China's aging population.

Innovative points and main contribution
The innovations of this study include establishing a novel FPTGM(1,1,a) model, calculating super parameters of the proposed FPTGM(1,1,a) model by using intelligent algorithm, predicting China's population aging trend, and putting forward suggestions coping with population aging.
The contributions of this study are mainly reflected in the following aspects: First, the proposed FPTGM(1,1,a) model with structural compatibility and parameter adjustability is constructed based on fractional-order accumulation thought and fractional-order time power term. Second, the derivation of intrinsic parameters of the FPTGM(1,1,a) model are given based on the least square method, and the solution of super parameters for the FPTGM(1,1,a) model are presented based on nonlinear programming and quantum genetic algorithm. Third, based on the idea of metabolism, China's population aging trend is predicted by using the FPTGM(1,1,a) model better than other six prediction models. Lastly, good forecasting results provide good data support for the government to formulate relevant policies, and can better cope with the aging population problem.

Organization
The research status of China's population aging and prediction model is expounded as above. The remainder of this paper is organized as follows. A novel FPTGM(1,1,a) model is introduced in Sect. 3, and the solution to the FPTGM(1,1,a) model is presented in Sect. 4. The application of the FPTGM(1,1,a) model in predicting the aging of China's population is described in Sect. 5, including comparisons with other prediction models. The empirical results are analyzed and discussed and several suggestions are put forward to solve the aging population problem in Sect. 6.
3 The proposal of FPTGM(1,1,a) model In this section, we present the FPTGM(1,1,a) model, which is a new type of grey prediction model with fractionalorder polynomial time power term, including the principles and the computational steps.

The r-order accumulation generation operation
The fractional-order accumulation generation operation is an effective measure to improve the grey prediction model. Choosing the appropriate cumulative order can improve the prediction accuracy of the grey prediction model to some extent. The fractional-order accumulation generation operation is defined as follows: Definition 1 (Wu et al. 2013). Assume that X ð0Þ ¼ fx ð0Þ ð1Þ; x ð0Þ ð2Þ; Á Á Á ; x ð0Þ ðnÞg is a non-negative original sequence, X ðrÞ ¼ fx ðrÞ ð1Þ; x ðrÞ ð2Þ; . . .; x ðrÞ ðnÞg; r 2 R þ is the r-order accumulated generating sequence of X ð0Þ , where where and r is called the fractional-order. And r should be taken in the interval [0, 1]. The larger r is, the larger the weight of old data from r-order accumulated generating sequence X ðrÞ is; the smaller r is, the smaller the weight of old data from rorder accumulated generating sequence X ðrÞ is. Tuning r can change the weights of old and new data from r-order accumulated generating sequence X ðrÞ .
x ð0Þ ðkÞ ¼ ½x ðrÞ ðkÞ ðÀrÞ ¼ The r-order accumulated generating operation is an important part of the grey prediction model.

The expression of the FPTGM(1,1,a) model
This differential equation is called the whitening differential equation of the FPTGM(1,1,a) model. Where ½i is the largest integer not exceeding i, for example, ½2:4 ¼ 2.
the solution of this optimization problem can be written as follows.
The derivation of Eq. (9) is presented as follows. Bring t ¼ 2; 3; . . .; n into Eq. (5) to get the following equations Equation (11) is a set of estimates of Y ¼ Bû to the parameters a; b; c 0 ; . . .; c ½a and then the error sequence e ¼ Y À Bû can be obtained. Let ½x ðrÀ1Þ ðtÞþaz ðrÞ ðtÞ À b À t aÀ½a c 0 À t aÀ½aþ1 c 1 À Á Á Á À t a c ½a 2 then the parameters a; b; c 0 ; . . .; c ½a which makes the S minimum should satisfy Equation (12) can be equivalently transformed into Equation (13) can be expressed in matrix form as follows Because the matrix B and Y in Eq. (14) satisfy.
The derivation process is completed.

The solution method of the FPTGM(1,1,a) model
Because the expression of FPTGM(1,1,a) model is complex, the corresponding solution process is also more complicated. In this section, we will explain the solution of the FPTGM(1,1,a) model in detail. Firstly, the nonlinear programming model for solving FPTGM(1,1,a) model is introduced. Secondly, the quantum genetic algorithm (QGA) (Roy et al. 2014) for solving the nonlinear programming model is introduced. Finally, how to solve the FPTGM(1,1,a) model will be discussed systematically.

Establishment of nonlinear programming model
The FPTGM(1,1,a) model is determined based on the unknown parameters r; a, and the determination of r and a is not easy. Therefore, we establish a nonlinear programming problem to solve the prediction results of the FPTGM (1,1,a) model. The purpose is to minimize the error of the FPTGM(1,1,a) model by changing the values of r and a. In this paper, we choose the mean absolute percentage error (MAPE) as the criteria to evaluate the validation error of the proposed model, and then the mathematical formulation of the optimization problem can be written as below.
The formula for nonlinear programming with equality constraints in nature for optimization problems (12) is often solved by intelligent optimizers or heuristic algorithms (Ma et al. 2019b). In this paper, QGA is used to solve the nonlinear programming model, and then the model prediction value is obtained. According to the literature (Wei et al. 2018), the power index number of the grey prediction model cannot exceed 3, so a 2 ½0; 3Þ. To facilitate readers to understand the solution process of the proposed FPTGM(1,1,a) . . .
The application of a novel grey model in the prediction of China's aging population 12507

Introduction of quantum genetic algorithm
Quantum genetic algorithm (QGA) (Roy et al. 2014) uses multi-state gene quantum bit coding and universal quantum revolving gate operation, introduces dynamic adjusting rotation angle mechanism and quantum crossover, which is more versatile and more efficient than genetic algorithm. QGA mainly consists of the following basic parts.
(1) Initialize the population. Based on initializing the population, several chromosome codes and qubit rates are randomly generated. QGA is based on the concept of quantum bits, using binary coding in QGA, QGA stores and expresses a gene with one or more qubits, and then the qubit expression gene constitutes a chromosome. A chromosome with a qubit encoded q can be expressed as where q t i is the chromosome of the j-th individual in generation t, and n is the number of qubits contained in the chromosome ja i j 2 þ jb i j 2 ¼ 1; i ¼ 1; 2; . . .; n, m is the number of chromosomal genes. In the following example calculation, the qubit encoding ða; bÞ of each individual in the population is initialized to ð1 ffiffi ffi 2 p ; 1 ffiffi ffi 2 p Þ, which means it probability of all possible states of a chromosome expression is the same. The parameters determined in this paper are the cumulative order and the power exponent, so m ¼ 2. The quantum number contained in each chromosome is set to n ¼ 20, the initial population size is set to 40.
(2) Fitness function: Appropriate evaluation of each chromosome in the contemporary population, and evaluation of the quality of the chromosome by adaptability. In this paper, the MAPE is used as the fitness function to evaluate the pros and cons of the chromosome. (3) Iterative stop criterion: It sets the maximum genetic algebra T as the basis for the iteration stop. Set the maximum genetic algebra T ¼ 200 in the calculation, and stop the genetic update when the genetic algebra t [ T. Table 1 The algorithm for finding the optimal parameters of the FPTGM(1,1,a) model where Dh is the rotation angle. The qubit rotation update operation can be expressed aŝ where ½a i ; b i T is the i-order qubit of the chromosome; Dh i corresponds to the rotation angle at this time, and Dh i ¼ 0:001p is setting in the calculation of the following example.

Application
In this section, the FPTGM(1,1,a) model will be used to forecast the trend of population aging in China based on optimal structure parameter obtained from QGA.

Data sources
The raw data of the population aging (10 4 /people) of China is collected from the official website National Bureau of Statistics of China shown in

Model fitting and forecasting result
In the forecasting process of China's aging population, the optimal power exponent a = 1.59346 and the fractionalorder r = 0.99805 of the FPTGM(1,1,a) model are obtained by QGA (Figs. 3, 4). In order to illustrate the superiority of the proposed FPTGM(1,1,a) model in simulation and prediction, BPNN model (Hong et al. 2021), LSTM model , GM(1,1) model (Deng 1982), fractional-order GM(1,1) model (abbreviated as GM r (1,1)) (Wu et al. 2015), discrete GM(1,1) model (abbreviated as DGM (1,1) From Table 4 we can see that the fitting MAPE of the FPTGM(1,1,a) model is 0.36%, and the forecasting MAPE is 2.29%. Among the five models, the fitting and forecasting MAPE of the FPTGM(1,1,a) model is the smallest, which shows that the FPTGM(1,1,a) model is suitable for predicting the number of elderly people in China. It is noteworthy that the DGM(1,1) model, GM(1,1) model and FPGM(1,1,a) model with relatively high prediction accuracy are all very high when predicting the first four data, while the latter prediction results are relatively poor. This is why we do not continue to predict backward. If we only pursue the quantity of predictions and do not pursue precision, then the prediction is meaningless. Based on this, if we want to predict population more accurately, we can introduce the idea of metabolism to make repeated predictions, so that we can get more and more accurate predictions.

The FPTGM(1,1,a) model based on metabolic thought
The idea of metabolism is to add the latest prediction results to the original data sequence as a new prediction data source, and to delete the old information in the The application of a novel grey model in the prediction of China's aging population 12511 original time series, but it is necessary to ensure that the modeling data sequence is equal dimension. In Sect. 5.3 we find that long-term predictions can cause large deviations between predictions and real values. But in real life, long-term prediction of data is an urgent need. In order to get as much accurate prediction data as possible, this paper introduces the metabolic ideas proposed in the literature (Chen et al. 2009) into the FPTGM(1,1,a) model.
Step 2 After getting the predicted sequencê x ð0Þ ðn þ 1Þ;x ð0Þ ðn þ 2Þ;x ð0Þ ðn þ 3Þ;x ð0Þ ðn þ 4Þ, the four latest values are participated in the modeling series and the first four data points are eliminated from the modeling series, and then the next four data points are predicted again.
Step 3 Repeat step 2 to predict the next set of four data points.
Step 4 Loop through the above steps until the amount of data you get meets your needs. In order to facilitate readers' understanding, the schematic diagram for modeling process based on metabolism thought is shown in Fig. 7.

Predicted result of China's aging population from 2019 to 2042
Because the FPTGM(1,1,a) model shows the best predictive performance in the comparison described in Sect. 5.3, the novel model combined with metabolic thought is  Table 5, The visualization predicted results of China's aging population and its annual growth rate from 2019 to 2042 are plotted in Fig. 8. It can be seen from Table 5 and Fig. 8 that China's aging population will continue to grow rapidly in the next 24 years. Furthermore, the growth rate of the aging population still is increasing year by year. By 2040, China's aging population will exceed 1 billion, which means that China will be called an aging country, which will bring disastrous consequences.  Year Prediction 2 0 1 9 2 0 2 0 2 0 2 1 2 0 2 2 2 0 2 3 2 0 2 4 2 0 2 5 2 0 2 6 2 0 2 7 2 0 2 8 2 0 2 9 2 0 3 0 2 0 3 1 2 0 3 2 2 0 3 3 2 0 3 4 2 0 3 5 2 0 3 6 2 0 3 7 2 0 3 8 2 Annual growth rate of aging population (%) Fig. 8 The trend of China's aging population predicted and its annual growth rate from 2019 to 2042 The application of a novel grey model in the prediction of China's aging population 12513 6 Discussions and suggestions

Discussions
In order to predict the trend of aging population in China, a novel grey model (FPTGM(1,1,a) model) was proposed in this study. The FPTGM(1,1,a) model is a grey prediction model with better predictive performances, as can be seen by comparing the fitting and forecasting results of those models described in Sect. 5.3. In order to obtain accurate prediction values, metabolic thought was introduced into the FPTGM(1,1,a) model in this study. The FPTGM(1,1,a) model combined with metabolic thought is used to predict the number of China's aging population from 2019 to 2042 (Table 5 and Fig. 8).
Based on the data characteristics of China's aging population, the data from 2003 to 2012 is used to build fitting models, and the data from 2013 to 2018 is used to test forecasting accuracy of models. In order to confirm the feasibility and effectiveness of the proposed FPTGM(1,1,a) model, the several other prediction models (including: BPNN model, LSTM model, GM(1,1) model, GMr(1,1) model, DGM(1,1) model and NGM(1,1,k) model) were also used for comparison in terms of fitting and forecasting accuracy. The performance of the models were judged by the indicators APE and MAPE. The results showed that the proposed FPTGM(1,1,a) model was 0.36% in fitting MAPE and was 2.29% in forecasting MAPE. Composed with the other models, the proposed FPTGM(1,1,a) model has minimum fitting and forecasting MAPE. That is, it had the best fitting and forecasting performance among these models selected in this study. This may be due to small sample data series limits the application of machine learning model and deep learning model (including BPNN and LSTM model). In addition, the proposed FPTGM(1,1,a) model with super parameters and structure compatibility can adapt to time series with different changing characteristics by replacing its own super parameters. Therefore, the novel grey prediction FPTGM(1,1,a) model with excellent performance in the small sample environment is used to complete the prediction task aging population in China from 2019 to 2042. The prediction result showed that the aging population will be 139,116,600 in 2042, indicating that the aging degree of China's population will continue to increase in the future, and measures must be taken in advance to solve this problem.

Suggestions
This paper proposes a novel grey model (FPTGM(1,1,a) model) to predict the aging population in China. The FPTGM(1,1,a) model is a grey prediction model with strong predictive power, as can be seen by comparing the fitting and forecasting results of those models described in Sect. 5.3. In order to obtain a large number and accurate prediction values, this paper introduces metabolic thought into the FPTGM(1,1,a) model. The FPTGM(1,1,a) model combined with metabolic thought is used to predict the number of China's aging population from 2019 to 2042 (Table 5 and Fig. 8). The forecast results show that China's aging population will continue to grow and there is no slight relief, which is a serious problem. In order to deal with this problem, the following suggestions for this phenomenon are put forward to restrain the population of China's aging population in this paper.
(1) In controlling the birth rate, plans should be put forward to adjust the family planning policy. The introduction of education subsidies, maternity subsidies and a series of supporting measures to further enhance the willingness of residents to bear children, at the same time, gradually relax the ''second child'' policy, moderately increase the fertility rate of women. Specifically, if both spouses are single children, they can have a second child, or adjust the policy of one child in rural areas and allow the peasant family to give birth to a second child. This modest adjustment increases the total fertility rate of women and will effectively alleviate the aging of the population. It should be noted that there should be a certain time interval between the first child and the second child. It is allowed to have two children and strictly control the birth of three children. Try to delay the time between the first and second births, and do not excessively increase the pressure of family and social upbringing. (2) In view of the aging of the population caused by the extension of life, a flexible retirement system should be proposed. Whether to extend the retirement age should depend on the needs of the unit and the physical condition of the individual. It is encouraged to postpone receiving endowment insurance. At the same time, enterprises, institutions and government departments vigorously adjust the difference in pension benefits and reduce the unreasonable income gap. The index of receiving pensions should be adjusted by following the different price indices of each year to improve the level of public welfare protection for the people. The supervision should be strengthened appropriately and abnormal early retirement should be controlled strictly. The adoption of this flexible retirement system can make full use of human resources, reduce the social burden, and increase the income of pension funds, reduce expenditures, and solve the problems caused by population aging without affecting employment. (3) Improving the pension system while promoting the development of the aging industry. From the way of old-age security, in a long period of time, we should still give priority to family support, supplemented by social support. At the same time, it is necessary to increase the proportion of social support, and gradually transition to social support, supplemented by family support. But no matter what stage, the family is always the main carrier of the aged. Therefore, on the one hand, we should make great efforts to improve the economic security system, medical security system, care service system and management system for the aged, increase infrastructure services, especially specialized nursing institutions, community care and nursing centers, and strengthen the training and supervision system of nursing staff to effectively meet the challenges brought about by the aging population. On the other hand, it is necessary to use the rare huge market for the elderly to promote the development of the aging industry, to provide a growth point for economic development, to further improve the level of prosperity, and to lay the economic foundation for dealing with the more severe challenges of aging. (4) As for the legislation of the endowment insurance system, the legislation mechanism should be established and perfected as soon as possible, especially in rural areas. On the one hand, grassroots cadres should publicize the role of endowment insurance for the elderly, so that they can change the traditional concept of ''raising children for old age''. Specifically, if their first child is a girl and their second child is also a girl, the pressure on the family to have a son is increased, and it is difficult for the children to enjoy a better educational environment, then pension insurance is very important for them. On the other hand, the endowment insurance legislation mechanism will enhance people's credibility of the insurance, and they will not worry about whether they can enjoy the insurance benefits after their retirement. For some elderly people who have no children to support, they must go to the nursing home, and the insurance money is directly paid to the nursing home according to law.