加载数据并估算模型。

Load the airline data set.

加载（'data_airline.mat'）y = log（数据）;t =长度（y）;mdl = arima（'Constant'，0，'D'，1，“季节性”，12，...“马拉格”，1，'SMALags'，，，，12); EstMdl = estimate(Mdl,y);

ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e -111差异0.0012634 0.00012395 10.193 2.1406E -24

检查残差是否正常。

拟合模型的一个假设是创新遵循高斯分布。推断残差，并检查它们是否正常。

res =推断（estmdl，y）;stres = res/sqrt（estmdl.variance）;图子图（1,2,1）qqplot（stres）x = -4：.05：4;[f，xi] = ksdenty（stres）;子图（1,2,2）图（xi，f，'k'，，，，'行宽'，，，，2); holdonplot(x,normpdf(x),'r-'，，，，'行宽'，，，，2) legend(“标准残差”，，，，'Standard Normal') hold离开

分位数量式图（QQ-plot）和内核密度估计没有明显违反正态性假设。

检查残差是否有自相关。

确认残差不相关。查看标准化残差的样品自相关函数（ACF）和部分自相关函数（PACF）图。

图子图（2,1,1）Autocorr（Stres）子图（2,1,2）Parcorr（Stres）

[h,p] = lbqtest(stres,'lags'，[5,10,15]，，'dof'，[3,8,13]）

h =1x3 logical array0 0 0

p =1×30.1842 0.3835 0.7321

示例ACF和PACF情节显示无显著Autocorrelation. More formally, conduct a Ljung-Box Q-test at lags 5, 10, and 15, with degrees of freedom 3, 8, and 13, respectively. The degrees of freedom account for the two estimated moving average coefficients.

Ljung-box Q检验确认了样品ACF和PACF结果。未拒绝所有自相关的无效假设，即与测试滞后的零相同等于零（h = 0) for any of the three lags.

检查预测性能。

Use a holdout sample to compute the predictive MSE of the model. Use the first 100 observations to estimate the model, and then forecast the next 44 periods.

y1 = y(1:100); y2 = y(101:end); Mdl1 = estimate(Mdl,y1);

ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.35674 0.089461 -3.9876 6.6739e-05 SMA{12} -0.63319 0.098744 -6.4124 1.4326E -100方差0.0013285 0.00015882 8.365 6.013E -17

yF1 = forecast(Mdl1,44,'y0'，y1）;pmse =平均值（（y2-yf1）。^2）

PMSE = 0.0069

图图（Y2，'r'，，，，'行宽'，2）保持on情节（yf1，'K--'，，，，'行宽'，，，，1.5) xlim([0,44]) title('Prediction Error'） 传奇（'观测到的'，，，，'预报'，，，，'Location'，，，，'西北') hold离开

The predictive ability of the model is quite good. You can optionally compare the PMSE for this model with the PMSE for a competing model to help with model selection.