使用矢量化生成时间序列

Question

我想生成一个包含50个值的AR（2）过程（3阶自回归过程）：

其中：Y（t）是时间序列，E（t）是误差项。

从标准正态分布中随机选取误差项值，我们假设如下：

作为一名前Java程序员和相对较新的R转换器，循环的使用对我来说非常明显，下面的代码就是我如何去做。但是，我一直在尝试转向矢量化范例，并一直在寻找一种方法来执行这项任务的矢量化方式＆＃39;。非常感谢你的帮助。

set.seed(123) # Reproducible
u <- rnorm(50) # Vector of randomly generated numbers for the error term
Y <- vector(length = 50) # Vector that will contain the generated time series values
a1 <- 0.25 # alpha 1
a2 <- 1.17 # alpha 2
# Based on the assumptions made, we can derive the values for the first 2 time periods:
Y[1] <- a1*0 + a2*0 + u[1]
Y[2] <- a1*Y[1] + a2*0 + u[2]

# The following "loop" generates the remaining values
i <- 3
while(i <= 50){
  Y[i] <- a1 * Y[i-1] + a2 * Y[i-2] + u[i]
  i <- i+1
}

Y
 [1]   -0.5604756   -0.3702964    0.8103777   -0.1601440    1.0373937    1.7870450    2.1214280    1.3561384    2.1342525    1.6745831
[11]    4.1398030    3.3540268    6.0828477    5.5556059    7.9499921   10.2744701   12.3679588   13.1465026   18.4584934   19.5232400
[21]   25.4094235   28.9765717   35.9471640   42.1604887   51.9732648   60.6343946   76.8051055   90.2968912  111.2980593  134.7256925
[31]  164.3266167  198.4156429  242.7611779  293.7147301  358.2808418  433.9050850  528.2187738  639.6617312  777.6254355  942.4301133
[41] 1144.7345808 1388.6189605 1685.2288034 2048.1603406 2484.9657471 3016.4659267 3661.1235209 4444.0793591 5395.3143244 6548.3180622