问题描述
我正在模拟一个一维对称随机游走过程:
I'm simulating a one-dimensional and symmetric random walk procedure:
y[t] = y[t-1] + epsilon[t]
其中白噪声由 epsilon[t] ~ N(0,1)
在时间段 t
中表示.在这个过程中没有漂移.
where white noise is denoted by epsilon[t] ~ N(0,1)
in time period t
. There is no drift in this procedure.
另外,RW 是对称的,因为 Pr(y[i] = +1) = Pr(y[i] = -1) = 0.5
.
Also, RW is symmetric, because Pr(y[i] = +1) = Pr(y[i] = -1) = 0.5
.
这是我在 R 中的代码:
Here's my code in R:
set.seed(1)
t=1000
epsilon=sample(c(-1,1), t, replace = 1)
y<-c()
y[1]<-0
for (i in 2:t) {
y[i]<-y[i-1]+epsilon[i]
}
par(mfrow=c(1,2))
plot(1:t, y, type="l", main="Random walk")
outcomes <- sapply(1:1000, function(i) cumsum(y[i]))
hist(outcomes)
我想模拟 1000 个不同的 y[i,t]
系列(i=1,...,1000; t=1,...,1000
).(之后我会在t=3
,t=5y[1]=0
)的概率代码> 和 t=10
.)
I would like to simulate 1000 different y[i,t]
series (i=1,...,1000; t=1,...,1000
). (After that, I will check the probability of getting back to the origin (y[1]=0
) at t=3
, t=5
and t=10
.)
哪个函数可以让我用 y[t]
随机游走时间序列进行这种重复?
Which function would allow me to do this kind of repetition with y[t]
random walk time-series?
推荐答案
由于y[t] = y[0] + sum epsilon[i]
,其中sum
取自 i=1
到 i=t
,序列 y[t]
可以立即计算,例如使用 R cumsum
函数.重复系列 T=10³ 次就很简单了:
Since y[t] = y[0] + sum epsilon[i]
, where the sum
is taken from i=1
to i=t
, the sequence y[t]
can be computed at once, using for instance R cumsum
function. Repeating the series T=10³ times is then straightforward:
N=T=1e3
y=t(apply(matrix(sample(c(-1,1),N*T,rep=TRUE),ncol=T),1,cumsum))
因为 y
的每一行都是模拟的随机游走序列.
since each row of y
is then a simulated random walk series.
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