我试图了解是否可以改进 ARIMA 模型。这是我的数据集（sales）：
28.35, 51.89, 37.26, 48.22, 30.93, 43.54, 35.3, 59.45, 49.41, 65.61, 36.59, 51.25, 31.42, 53.16, 39.41, 64.45, 43.94, 79.36, 52.93, 74.99, 55.03, 86.93, 41.69, 62.77, 41.29, 59.95, 40.07, 66.13, 47.15, 85.12, 74.44, 76.42, 49.17, 82.66, 49.88, 70.98, 52.83, 75.85, 61.4, 85.2, 61.99, 90.68, 48.05, 74.2, 41.7, 68, 46.41, 82.23, 62.18, 88.65, 65.21, 100.9, 46.63, 83.53, 56.57, 108.87, 51.01, 80.15, 57.03, 87.91, 62.41, 96.11, 71.41, 82.08, 62.5, 88.52, 60.53, 100.15, 67.74, 111.88, 74.64, 138.64, 97.88, 153.88, 111.34, 176.4, 67.57, 111.95, 72.36, 118.85, 82.19, 136.88, 84.95, 160.58, 64.13, 111.32, 64.65, 113.82, 74.75, 118.76, 86.28, 166.36, 71.82, 119.83, 67.64, 116.17, 77.83, 130.64, 95.23, 149.84, 115.97, 189.69, 96.35, 137.51, 82.04, 139.19, 70.68, 135.22, 69.84, 105.7, 65.47, 111.47, 63.71, 108.23, 66.81, 117.96, 86.82, 141.74, 71.97, 122.65, 89.35, 133.97, 110.07, 159.18, 117.4, 196.9, 167.69, 244.75, 85.43, 135.54, 70.51, 118.3, 78.83, 139.85, 108.57, 162.66, 139.03, 203.72, 94.37, 135.92, 80.35, 128.63, 90.2, 157.56, 112.91, 177.07, 147.28, 221.67, 90.86, 142.66, 93.96, 157.89, 121.5, 200.35, 140.08, 306.36, 187.86, 171.39, 113.52, 174.2, 108.89, 170.53, 121.49, 193.65, 148.72, 210.61, 168.46, 250.4, 213.54, 181.78, 126.56, 190.46, 137.85, 226.25, 148.68, 235.04, 170.39, 275.04, 106.68, 163.24, 109.15, 186.46, 129.33, 156.18, 91.03, 159.87, 119.43, 164.51, 92.84, 145, 87.02, 156.55, 92.76, 140.93, 102.72, 143.41, 92.11, 159.72, 96.44, 156.98,
151.38, 221.12, 174.89, 242.53, 117.66, 163.44, 111.25, 169.58, 103.27, 163.09, 105.62, 186.64, 124.75, 145.65, 108.31, 165.3, 101.91, 156.55, 101.72, 147.11, 106.25, 185.68, 146.83, 192.05, 101.46, 153.65, 105.91, 170.1, 97.07, 165.05, 106.06, 167.25, 102.68, 197.21, 99.19, 169.58, 106.66, 196.44, 103.46, 165.62, 108.77, 188.32, 117.03, 241.48, 171.6, 189.78, 110.79, 166.22, 116.14, 229.75, 144.17, 205.75, 137.51, 216.51, 111.98, 186.34, 138.92, 218.35, 172.29, 271.53, 143.24, 272.35, 274.9, 232.97, 238, 234.88, 172.19, 260.82, 143.12, 217.38, 136.56, 209.91, 144.57, 253.58, 171.79, 264.78, 189.01, 298.97, 231.23, 315.29, 198.05, 318.52, 183.21, 232.33, 161.4, 261.82, 145.56, 218.09, 140.13, 215, 154.87, 293.88, 164.71, 256.85, 192.69, 306.87, 255.16, 382.27, 298.13, 438.22, 183.88, 279.56, 217.82, 371.55, 269.81, 383.89, 211.72, 330.02, 217.97, 312.64, 227.47, 329.25, 238.65, 363.8, 280.39, 453.38, 363.84, 486.65, 647.67, 534.41, 219.69, 292.16, 209.73, 336.33, 226.43, 336.23, 249.48, 359.84, 188.05, 307.73, 231.67, 330.43, 252.22, 379.3, 293.54, 413.67, 384.64, 515.86, 482.36, 438.12
sales = as.ts(sales, frequency=2)
每周两个数据。在这个时间序列中，我看到了季节性和趋势。如我错了请纠正我。经过对数变换以稳定方差：
sales.transformed <- log(sales)
给出的模型auto.arima是：
fit <- auto.arima(sales.transformed, seasonal=TRUE)
summary(fit)
Series: sales.transformed
ARIMA(3,1,1) with drift
Coefficients:
ar1 ar2 ar3 ma1 drift
0.2340 0.6310 -0.4893 -0.8913 0.0064
s.e. 0.0541 0.0471 0.0485 0.0407 0.0018
sigma^2 estimated as 0.03276: log likelihood=96.98
AIC=-181.96 AICc=-181.71 BIC=-159.01
但是残差的行为不像白噪声：
res <- residuals(fit)
Acf(res, main="ACF of residuals")
然后我尝试了一个季节性的 ARIMA 模型。
fit <- Arima(sales.transformed, order=c(3,1,1),
seasonal=list(order=c(3,0,1), period=2))
情况正在好转：
Series: sales.transformed
ARIMA(3,1,1)(2,0,1)[2]
Coefficients:
ar1 ar2 ar3 ma1 sar1 sar2 sma1
-1.3075 0.2398 0.5474 0.9721 0.0648 -0.0803 -0.7839
s.e. 0.0630 0.1221 0.1010 0.0401 0.0848 0.0657 0.0606
sigma^2 estimated as 0.02667: log likelihood=131.05
AIC=-246.09 AICc=-245.65 BIC=-215.48
现在情况更好了。
ARIMA(5,1,0) with drift
Box Cox transformation: lambda= 0
Coefficients:
ar1 ar2 ar3 ar4 ar5 drift
-0.4224 0.0143 -0.2956 -0.0849 -0.3238 0.0015
s.e. 0.0516 0.0563 0.0541 0.0564 0.0516 0.0009
sigma^2 estimated as 0.001257: log likelihood=649.72
AIC=-1285.44 AICc=-1285.1 BIC=-1258.65
鉴于我的时间序列，我不知道该模型是否可以改进，或者这是否是 ARIMA 可以做的最好的事情。我可能应该尝试使用自回归混合模型并添加新的预测变量......
任何建议将不胜感激。