时间序列分析——预测

文章目录

• 时间序列预测
• 多步预测的五种策略
• 常用指标

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

$\hat{y}$y^​ 为预测值， $y$y 为实际值，$N$N 为预测数：

1. MSE
   $MSE = \frac{1}{N}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2$MSE=N1​i=1∑N​(y^​i​−yi​)2

2. RMSE
   $RMSE = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2}$RMSE=N1​i=1∑N​(y^​i​−yi​)2​

3. NMSE
   $NMSE = \frac{1}{N\cdot \sigma_y^2}\sum_{i=1}^{N}(\hat{y}_i-y_i)^2$NMSE=N⋅σy2​1​i=1∑N​(y^​i​−yi​)2

4. MAE
   $MAE = \frac{1}{N}\sum_{i=1}^{N}|\hat{y}_i-y_i|$MAE=N1​i=1∑N​∣y^​i​−yi​∣

5. MAPE
   $MAPE = \frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_i-y_i}{y_i}\right|$MAPE=N1​i=1∑N​∣∣∣∣​yi​y^​i​−yi​​∣∣∣∣​

6. sMAPE
   symmetric mean absolute percentage error
   $sMAPE = \frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_i-y_i}{(\hat{y}_i+y_i)/2}\right|$sMAPE=N1​i=1∑N​∣∣∣∣​(y^​i​+yi​)/2y^​i​−yi​​∣∣∣∣​

7. MASE
   mean absolute scaled error
   $MASE = \frac{\frac{1}{h}\sum_{i=1}^h|\hat{y}_i - y_i|}{\frac{1}{l-p}\sum_{i=p+1-l}^0|\hat{y}_i - y_{i-p}|}$MASE=l−p1​∑i=p+1−l0​∣y^​i​−yi−p​∣h1​∑i=1h​∣y^​i​−yi​∣​其中 $l$l 是训练集样本数，$h$h 是预测长度，$p$p 是季节长度