SARIMA时间序列分析：三大模型对比

# 参考代码
d(x,1,12)
d(x,1,12)  ma(1) sma(12)
d(x,1,12)  ar(1) sma(12)
expand 1976M1 1977M12

6.1.1时序图

$$X_t=T_t\times S_t\times I_t$$

6.1.2 $趋势效应T_t$

6.1.3 $季节效应S_t$

6.1.4指数平滑法

$X_t=(893.128+1.583t)S_j$

6.1.5 SARIMA模型建立

6.1.5.1 时序图差分$d(x,1,12)$

6.1.5.2 $ARIMA(0,1,1)\times (0,1,1{)}_{12}$

$${\nabla }_{12}\nabla X_t=(1-0.234B)\times (1-0.704B{)}^{12}{ϵ}_t$$

6.1.5.3 $ARIMA(1,1,0)\times (0,1,1{)}_{12}$

$${\nabla }_{12}\nabla X_t=\frac{1-0.670B^{12}}{1+0.234B}{ϵ}_t$$

6.1.6拟合精度比较 $$RMSE$$

$$X_t＝T_t\times S_t\times I_t$$
$$\widehat{X_t}＝T_t\times S_t＝\frac{X_t}{I_t}$$
$$MRSE=\sqrt{\frac{{\sum }_{i=1}^m{\left(X_i-{\hat{X}}_i\right)}^2}{m}}=\sqrt{\frac{{\sum }_{i=1}^m{\left(X_i-X_i/I_i\right)}^2}{m}}$$

模型	$RMSE$
$X_t=(893.128+1.583t)S_j$	7.665
${\nabla }_{12}\nabla X_t=(1-0.234B)\times (1-0.704B{)}^{12}{ϵ}_t$	3.215
${\nabla }_{12}\nabla X_t=\frac{1-0.670B^{12}}{1+0.234B}{ϵ}_t$	3.215