如何在R中重复1000次此随机游走模拟？

Question

I'm simulating a one-dimensional and symmetric random walk procedure:

$$y_t=y_{t-1}+\varepsilon_t$$

where white noise is denoted by $\\varepsilon_t \\sim N(0,1)$ in time period $t$ . There is no drift in this procedure.

Also, RW is symmetric, because $Pr(y_i=+1)=Pr(y_i=-1)=0.5$.

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)

I would like to simulate 1000 different $y_{it}$ 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 $t=10$.

Which function does allow me to do this kind of repetition with $y_t$ random walk time-series?

Answer 1

Try the following:

length_of_time_series <- 1000
num_replications <- 1000

errors <- matrix(rnorm(length_of_time_series*num_replications),ncol=num_replications)
rw <- apply(errors, 2, cumsum)

This creates 1000 random walks simultaneously by first defining a matrix filled with white noise error terms drawn from a standard normal distribution, and in the second step I calculate the cumulative sums, which should correspond to your random walk, assuming that y_0=0 .

Note that I have ignored that your errors are either -1 or 1, since I am not sure that this is what you intended. Having said that, you can adjust the above code easily with the following line to create errors that are either 1 or -1:

 errors2 <- ifelse(errors > 0, 1, -1)

If you really want to go the way of doing it repeatedly as opposed to doing it simultaneously, you can define a function that returns the random walk, and use replicate . Note that you should not hardcode the seed inside the function to avoid replicating the same random walk all the time.