R code

GM <- function(x,adv=0) {
  x0 <- x
  k = length(x0) 
  # AGO
  x1 = cumsum(x0)
  # construct matrix B & Y
  B = cbind(-0.5*(x1[-k]+x1[-1]),rep(1,times=k-1))
  Y = x0[-1]
  # compute BtB...
  BtB = t(B)%*%B
  BtB.inv = solve(BtB)
  BtY = t(B)%*%Y
  
  # estimate
  alpha = BtB.inv%*%BtY
  # estimate
  predict <- function(k) {
    y = (x0[1] - alpha[2]/alpha[1])*exp(-alpha[1]*k)+alpha[2]/alpha[1]
    return(y)
  }
  pre <- sapply(X=0:(k-1),FUN=predict)
  prediff <- c(pre[1],diff(pre))
  
  # Accuracy test
  # Residual
  error <- x0-prediff
  
  # Residual mean square error      
  s2 = sqrt(sum((abs(error)-mean(abs(error)))^2/(length(error)-1)))

  # Monotone sequence mean square error
  s1 = sqrt(sum((x0-mean(x0))^2)/(length(x0)-1))
  # Variance ratio
  C = s2/s1
  # Small residual error probability
  p = sum((abs(abs(error)-mean(abs(error))))<0.6745*s1)/length(error)
  
  # RE
  RE = numeric(length = length(x0))
  for (i in 1:length(x0)) {
    RE[i] = (abs(x0[i]-prediff[i])/x0[i])
  }
  ARE = mean(RE)
  result <- list(actual = x0,
                 fit = prediff,
                 a = alpha[1],
                 b =alpha[2],
                 C = C,
                 P = p,
                 RE = RE,
                 ARE = ARE)
  
  # estimate: k+adv
  if (adv > 0) {
    pre.adv = numeric(adv)
    for (i in 1:adv) {
      pre.adv[i] <- predict(k-1+i)-predict(k-2+i)
    }
    result$predict <- pre.adv
  }
  class(result) <- 'GM1.1'
  return(result)
}
# 案例
x = c(1,2,3,4,5,6,7,8)
GM(x,2)  # 向后预测两个，默认为零