时间序列分析——预测

时间序列预测

时间序列通常包含这些组成部分：线性趋势(Trend)，季节变化(Seasonality)，循环变化(Cycle)，不规则变化(Irregularity)

多步预测的五种策略

可分为单步预测（one-step-ahead）和多步预测（muti-step-ahead）

多步预测的五种策略：

• recursive (or iterated) strategy
  
• direct strategy
  
• combination of both the recursive and direct strategies, called DirREC
  
• the Multi-Input Multi-Output (MIMO) strategy
  
• DirMO strategy
  

常用指标

$\hat{y}$ 为预测值， $y$ 为实际值， $N$ 为预测数：

1. MSE
   $MSE = \frac{1}{N}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2$

2. RMSE
   $RMSE = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2}$

3. NMSE
   $NMSE = \frac{1}{N\cdot \sigma_y^2}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2$

4. MAE
   $MAE = \frac{1}{N}\sum_{i=1}^{N}|\hat{y}_i-y_i|$

5. MAPE
   $MAPE = \frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_i-y_i}{y_i}\right|$

6. sMAPE
   symmetric mean absolute percentage error
   $sMAPE = \frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_i-y_i}{(\hat{y}_i+y_i)/2}\right|$

7. MASE
   mean absolute scaled error
   $MASE = \frac{\frac{1}{h}\sum_{i=1}^h|\hat{y}_i - y_i|}{\frac{1}{l-p}\sum_{i=p+1-l}^0|\hat{y}_i - y_{i-p}|}$其中 $l$ 是训练集样本数， $h$ 是预测长度， $p$ 是季节长度