## 10.4 自上而下的方法

### 平均历史比例

$p_j=\frac{1}{T}\sum_{t=1}^{T}\frac{y_{j,t}}{{y_t}}$ 对于 $$j=1,\dots,m$$。每个比例 $$p_j$$ 反映了底层序列 $$y_{j,t}$$$$t=1,\dots,T$$ 时相对于总计 $$y_t$$ 的历史平均值。

### 历史平均数的比例

$p_j={\sum_{t=1}^{T}\frac{y_{j,t}}{T}}\Big/{\sum_{t=1}^{T}\frac{y_t}{T}}$ 对于 $$j=1,\dots,m$$。每个比例 $$p_j$$ 反映了底层序列 $$y_{j,t}$$$$t=1,\dots,T$$ 时相对于总计 $$y_t$$ 的平均值的历史平均值。

### 预测比例

• $$\hat{y}_{\text{A},h}^{(1)}=\hat{y}_{\text{B},h}^{(1)}=\hat{y}_{h}= \tilde{y}_{h}$$
• $$\hat{y}_{\text{AA},h}^{(1)}=\hat{y}_{\text{AB},h}^{(1)}=\hat{y}_{\text{AC},h}^{(1)}= \hat{y}_{\text{A},h}$$
• $$\hat{y}_{\text{AA},h}^{(2)}=\hat{y}_{\text{AB},h}^{(2)}= \hat{y}_{\text{AC},h}^{(2)}=\hat{y}_{\text{BA},h}^{(2)}= \hat{y}_{\text{BB},h}^{(2)}=\hat{y}_{h}= \tilde{y}_{h}$$
• $$\Shat{AA}{h}{1} = \Shat{AB}{h}{1}= \Shat{AC}{h}{1}= \yhat{AA}{h}+\yhat{AB}{h}+\yhat{AC}{h}$$
• $$\Shat{AA}{h}{2} = \Shat{AB}{h}{2}= \Shat{AC}{h}{2}= \Shat{A}{h}{1} = \Shat{B}{h}{1}= \hat{S}_{h}= \yhat{A}{h}+\yhat{B}{h}$$

### 参考文献

Athanasopoulos, G., Ahmed, R. A., & Hyndman, R. J. (2009). Hierarchical forecasts for Australian domestic tourism. International Journal of Forecasting, 25, 146–166. [DOI]
Gross, C. W., & Sohl, J. E. (1990). Disaggregation methods to expedite product line forecasting. Journal of Forecasting, 9, 233–254. [DOI]
Hyndman, R. J., Ahmed, R. A., Athanasopoulos, G., & Shang, H. L. (2011). Optimal combination forecasts for hierarchical time series. Computational Statistics and Data Analysis, 55(9), 2579–2589. [DOI]