ARIMA模型的计算公式？

Question

在某些情况下，我必须手动进行预测，这意味着使用模型的公式。对于 AR(p) 模型，这很容易。但是对于ARIMA模型(p, d, q)，d>=1，我就有点吃力了。以下示例，我使用模型 AR(2) 进行计算。我有 1990 年到 2010 年的系列，我需要 2011 年的预测：

> a<-c(198,150,120,84,150,136,80,128,160,132,144,234,300,312,400,468,420,500,650,612,516)
> series<-ts(a,frequency=1,start=c(1990))
> fit<-Arima(series,c(2,0,0),method="ML")
> fit
Series: series 
ARIMA(2,0,0) with non-zero mean 

Coefficients:
         ar1      ar2  intercept
      1.1923  -0.2881   305.3748
s.e.  0.2174   0.2346   111.5251

sigma^2 estimated as 3727:  log likelihood=-117.2
AIC=242.4   AICc=244.9   BIC=246.58

收到模型AR(2)的计算公式：

y[t]=305.3748+1.1923*y[t-1]-0.2881*y[t-2]

我预测 2011 年：

y[2011] = 305.3748+1.1923*y[2010]-0.2881*y[2009] 
        = 305.3748+1.1923*516-0.2881*612
        = 744

然而，当我拟合 ARIMA 模型 (2,1,0) 时：

> fit2<-Arima(series,c(2,1,0),method="ML")
> fit2
Series: series 
ARIMA(2,1,0)                    

Coefficients:
         ar1      ar2
      0.2561  -0.3494
s.e.  0.2196   0.2117

sigma^2 estimated as 3489:  log likelihood=-110.1
AIC=226.2   AICc=227.7   BIC=229.19

不知道d = 1时公式怎么写？还有一个问题，为什么当我用函数forecast()做预测时，结果和我用公式计算的结果不一样？

> forecast(fit,h=1)
     Point Forecast    Lo 80    Hi 80  Lo 95    Hi 95
2011       468.1754 389.9369 546.4138 348.52 587.8308

Answer 1

ARIMA(2,1,0) 只是意味着你

区分系列，即将series替换为diff(series)，
然后应用 ARIMA(2,0,0) 的公式，但应用于 diff(series)、
最终 "re-integrate" 结果 cumsum。

函数 fcst_ar2 包含 ARIMA(2,0,0) 的公式，fcst_diff_ar2 ARIMA(2,1,0) 的公式：

library(forecast)

#--------------------------------------------------------

fcst_ar2 <- function( coef, series, horizon )
{
  y <- as.vector(series) - coef[3]

  for ( i in 1:horizon ) {
    y <- c( y, coef[2:1] %*% tail(y,2) )
  }

  return( y + coef[3] )
}

#--------------------------------------------------------

fcst_diff_ar2 <- function( coef, series, horizon )
{
  y <- as.vector(series)

  return( cumsum( c(y[1], fcst_ar2( c(coef,0), diff(y), horizon ) ) ) )
}

#========================================================
# Example:

a<-c(198,150,120,84,150,136,80,128,160,132,144,234,300,312,400,468,420,500,650,612,516)
series<-ts(a,frequency=1,start=c(1990))
fit<-Arima(series,c(2,0,0),method="ML")
fit2<-Arima(series,c(2,1,0),method="ML")

#--------------------------------------------------------
fcst_ar2( coef(fit), series, 3 )
#  [1] 198.0000 150.0000 120.0000  84.0000 150.0000 136.0000  80.0000 128.0000
#  [9] 160.0000 132.0000 144.0000 234.0000 300.0000 312.0000 400.0000 468.0000
# [17] 420.0000 500.0000 650.0000 612.0000 516.0000 468.1754 438.8091 417.5725

forecast( fit, 3 )
#      Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
# 2011       468.1754 389.9369 546.4138 348.5200 587.8308
# 2012       438.8091 317.0562 560.5621 252.6041 625.0142
# 2013       417.5725 266.9412 568.2038 187.2018 647.9432

#--------------------------------------------------------
fcst_diff_ar2( coef(fit2), series, 3 )
#  [1] 198.0000 150.0000 120.0000  84.0000 150.0000 136.0000  80.0000 128.0000
#  [9] 160.0000 132.0000 144.0000 234.0000 300.0000 312.0000 400.0000 468.0000
# [17] 420.0000 500.0000 650.0000 612.0000 516.0000 504.6897 535.3388 547.1413

forecast( fit2, 3 )
#      Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
# 2011       504.6897 428.9867 580.3927 388.9119 620.4675
# 2012       535.3388 413.7918 656.8858 349.4487 721.2289
# 2013       547.1413 405.0444 689.2383 329.8228 764.4599

ARIMA模型的计算公式？

Calculation formula of ARIMA model?

r

time-series

forecasting

autoregressive-models