动态谐波回归
R 中的预测
Rob J. Hyndman
Professor of Statistics at Monash University
动态谐波回归
动态谐波回归
$m =$ 季节周期
任何周期函数在足够大的 K 下可用若干 sin/cos 项近似
回归系数：$\alpha_k$ 与 $\gamma_k$
$e_t$ 可建模为非季节 ARIMA 过程
假设季节模式不变
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 1),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 1, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 2),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 2, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 3),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 3, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 4),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 4, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 5),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 5, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
示例：澳大利亚咖啡馆支出
fit <- auto.arima(cafe, xreg = fourier(cafe, K = 6),
seasonal = FALSE, lambda = 0)
fit %>% forecast(xreg = fourier(cafe, K = 6, h = 24)) %>%
autoplot() + ylim(1.6, 5.1)
动态谐波回归
- 也可加入其他预测变量：$x_{t,1},...,x_{t,r}$
- 选择 K 以最小化 $AIC_c$
- K 不得超过 m/2
- 对周度、日度和更高频数据尤为有用
