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Prediction of Population Ageing in Guangdong Province Based on Metabolic GM-ARIMA Combined Model
DOI: 10.12677/sa.2024.132038, PDF, HTML, XML, 下载: 94  浏览: 174

Abstract: Based on the demographic data of Guangdong Province from 2000 to 2022, the article uses the combined weight coefficient method to construct a metabolic GM(1, 1)-ARIMA model to forecast its future demographic changes. In order to make up for the disadvantages of the ARIMA model with high sample requirement and fitting more reflective of a linear trend, and to overcome the inoperability of traditional grey prediction in medium and long-term prediction and the problem of prediction deviation caused by exponential explosive growth, firstly, the combined prediction model based on metabolism GM(1, 1)-ARIMA was constructed by using the method of least squares, MAPE and combined weight coefficients, and then, the three models of TIC, MAPE and RMSE were introduced to assess the accuracy of different combinatorial models, and finally the combinatorial model constructed using the combined weight coefficient method was selected for fitting prediction. The prediction results show that the combined model is 0.32% more accurate than the single model, which is of reference value. It also shows that the age structure of Guangdong’s population is relatively young, but it has entered the initial oligocephalic society, and will inevitably enter the deep aging and super-aging society in the future on a large scale and at a high speed, which needs to be paid great attention to.

1. 引言

2. 方法

2.1. ARIMA模型

ARIMA模型全称为自回归差分移动平均模型，其基本思想是试图通过数据的自相关性和差分的方式，提取出隐藏在数据背后的时间序列模式，然后用这些模式来预测未来的数据 [7] 。主要表达式为：

 (1)

2.2. 新陈代谢GM(1, 1)模型

2.3. 组合模型构建

1969年，由Bates等人提出组合预测模型方案，即先从各种单项预测模型中提取出有效的系统信息，找到一种准则或方式，使这些不同的预测模型进行合理有效地组合，并选择合适的权系数进行加权，最终得到最合理的组合预测模型。在进行某具体问题预测前，需要对该问题涉及到的数据进行采集，即 $Y=\left({y}_{1},{y}_{2},\cdots ,{y}_{n}\right)$ 。在本题中，本文选取两个预测模型ARIMA和新陈代谢GM(1, 1)模型，在t时刻的预测值分别记为 $\left({\phi }_{1}\left(t\right),{\phi }_{2}\left(t\right)\right)$ ，预测对应的权重为 $\lambda =\left({\lambda }_{1},{\lambda }_{2}\right)$ ，满足条件 ${\lambda }_{1}+{\lambda }_{2}=1$ ，最终得到组合预测模型表达式为 $\phi \left(t\right)={\lambda }_{1}{\phi }_{1}\left(t\right)+{\lambda }_{2}{\phi }_{2}\left(t\right)$

2.3.1. 最小二乘法确定权数

$Q={\sum }_{i=1}^{n}|{e}_{t}^{2}|={\sum }_{i=1}^{n}|{\beta }_{1}{e}_{1}^{2}\left(t\right)+{\beta }_{2}{e}_{2}^{2}\left(t\right)|$ (2)

$\begin{array}{l}\mathrm{min}\text{\hspace{0.17em}}Q={\sum }_{i=1}^{n}|{\beta }_{1}{e}_{1}^{2}\left(t\right)+{\beta }_{2}{e}_{2}^{2}\left(t\right)|\\ \text{st}.\text{\hspace{0.17em}}{\beta }_{1}+{\beta }_{2}=1,\text{\hspace{0.17em}}{\beta }_{1}\ge 0,\text{\hspace{0.17em}}{\beta }_{2}\ge 0\end{array}$ (3)

${\phi }^{\left(1\right)}\left(t\right)={\beta }_{1}{\phi }_{1}\left(t\right)+{\beta }_{2}{\phi }_{2}\left(t\right)$ (4)

2.3.2. MAPE法确定权数

$\text{MAPE}=\frac{1}{n}{\sum }_{i=1}^{n}|\frac{\phi \left(t\right)-x\left(t\right)}{x\left(t\right)}\ast 100\text{%}|\text{\hspace{0.17em}},\text{\hspace{0.17em}}t=1,2,\cdots ,n$ (5)

${\alpha }_{i}=\frac{{d}_{i}}{{d}_{1}+{d}_{2}},\text{\hspace{0.17em}}i=1,2$ (6)

${\phi }^{\left(2\right)}\left(t\right)={\alpha }_{1}{\phi }_{1}\left(t\right)+{\alpha }_{2}{\phi }_{2}\left(t\right)$ (7)

2.3.3. 组合权重系数

${\lambda }_{i}=\frac{{\alpha }_{i}{\beta }_{i}}{{\alpha }_{1}{\beta }_{1}+{\alpha }_{2}{\beta }_{2}},\text{\hspace{0.17em}}i=1,2$ (8)

${\phi }^{\left(3\right)}\left(t\right)={\lambda }_{1}{\phi }_{1}\left(t\right)+{\lambda }_{2}{\phi }_{2}\left(t\right)$ (9)

Table 1. Combined model weight allocation table

2.4. 模型评价函数

$\begin{array}{l}\text{TIC}=\frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{\left(\phi \left(t\right)-x\left(t\right)\right)}^{2}}}{\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}\phi {\left(t\right)}^{2}}+\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}x{\left(t\right)}^{2}}}\\ \text{RMSE}=\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{\left(\phi \left(t\right)-x\left(t\right)\right)}^{2}}\\ \text{MAPE}=\frac{1}{n}{\sum }_{i=1}^{n}|\frac{\phi \left(t\right)-x\left(t\right)}{x\left(t\right)}\ast 100\text{%}|\end{array}$ (10)

3. 广东省人口老龄化现状

3.1. 65岁及以上老年人口创历史新高

Figure 1. Changes in the population aged 65 and over in Guangdong Province, 2010~2022

3.2. 不同城市人口老龄化情况存在差异

Figure 2. Population structure of Guangdong Province in 2020 (Seventh Population Census)

3.3. 老年人口抚养比持续攀升

Table 2. Population structure of Guangdong Province, 2010~2022

4. 组合预测模型应用研究

4.1. ARIMA模型人口老龄化预测

Figure 3. Raw data (left) and first order difference (right) timing diagrams

Table 4. Minimum information standard (MIS)

Table 5. Parameter estimation

$\left(1-B\right)\phi \left(t\right)=29.98539+\left(1-0.13688B\right){\epsilon }_{t}$ (11)

Table 6. Individual model predicted fitted values

4.2. 新陈代谢GM(1, 1)模型人口老龄化预测

4.3. 组合模型人口老龄化预测

4.3.1. 权重构建

Figure 4. Comparison of individual model accuracy

${\phi }^{\left(1\right)}\left(t\right)=0.466{\phi }_{1}\left(t\right)+0.543{\phi }_{2}\left(t\right)$ (12)

${\phi }^{\left(2\right)}\left(t\right)=0.5912{\phi }_{1}\left(t\right)+0.4088{\phi }_{2}\left(t\right)$ (13)

${\phi }^{\left(3\right)}\left(t\right)=0.5538{\phi }_{1}\left(t\right)+0.4462{\phi }_{2}\left(t\right)$ (14)

4.3.2. 不同组合模型结果对比

Table 7. Combined model predictions

Figure 5. Comparison of errors of different combination models

Table 8. Combined model evaluation comparison

4.3.3. 组合权重组合模型预测分析

Table 9. Projections of the population structure of Guangdong Province in the next eight years

5. 结论

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