Spring 2022 STAT 443
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Spring 2022 STAT 443
• Refer to Theorem 3.6.3 [Wold decomposition Theorem] for the actual theory of fore- casting
Theorem 1.1 (Wold decomposition Theorem) Any stationary process can be written as the sum of an MA(8) process and an independent deterministic process, where a deterministic process is any process whose complete realisation is a determnis- tic function of a finite number of its values.
Refer to Page 84 & 86 of Notes.
Forecasting under AR(1) Xt = ϕXt一1 + Zt , |
{Zt u WN (0, σ2 ) |
• h-step ahead forecast: Pred(Xn+h|Xn , . . . , X1 ) = ϕh Xn Update fit$model$T • h-step ahead forecasting error:
Pred(Xn+h|Xn , . . . , X1 ) = σ 2 Update fit$sigma2 |
The following are some useful functions in R:
• arima.sim: Simulate from an ARIMA model
arima.sim(n , model = list(ar= c(),ma=c()), sd)
– n: number of observations
– model: A list which gives AR and MA coefficients
– sd: standard deviation for error term
• arima: Fit an ARIMA model to a univariate time series
arima(x, order = c(p,d,q), include.mean = FALSE)
– x: a univariate time series
– order: The three integer components (p, d, q) are the AR order, the degree of differencing, and the MA order.
– include.mean : Should the ARMA model include a mean/intercept term? The default is TRUE for undifferenced series
• auto.arima: Fit best ARIMA model to univariate time series auto.arima(x)
• predict: Forecast from models fitted by arima predict(object, n.ahead = h)
– object: The result of an arima fit
– n.ahead: The number of steps ahead for which prediction is required.
• HoltWinters: Fit HoltWinters model to univariate time series
HoltWinters(x, alpha = NULL, beta = NULL, gamma = NULL)
– x: a univariate time series
– alpha: Used for exponential smoothing
– beta: Used for trend estimation. If set to FALSE, the function will do exponential smoothing.
– gamma: Used for the seasonal component. If set to FALSE, an non-seasonal model is fitted.
2022-07-14