Generate random data based on correlation matrix for multiple timesteps in R

Question

I would like to simulate data for some cases (eg nPerson=1000 obversations) at some consecutive timesteps (eg ts = 3) for N intercorrelated variables (eg N=5).

The simulation should be based on a correlation matrix (corrMat, nrows=nPerson,.ncols = N). corrMat should be identical for all timesteps.

I already found out that the MASS package has a function to create random data fitting the constraints given by corrMat.

t1 <- mvrnorm(nPerson,mu=rep(0, N),Sigma=corrMat,empirical=T)

Now I would like to simulate t2 as a function of t1 and corrMat. The data of t2 therefore should correlate according to corrMat and they should also have same variance as the variables of t1.

One important constrained: for the intial values corrMat[i,i] = 1, for consequtive timesteps it should be posible, that corrMat[i,i] < 1, because each variable is depending on itsself a timestep before, but a perfect correlation is notintended.

Maybe there is a variance decomposition of the correlation matrix, that calculates an error variance for each of the n variables at the next time step, so that one could calculate the values at timestep t+1 as sum of the weighted correlations of the variables at timestep t and then adding a random error,distributed according to the error variance (with mean of error = 0) that replicates the correlation matrix again at t+1.

Assuming normal errors:

getRand <- function (range) {
  return (rnorm(1,mean=0, sd=range)  )
}

That the (very simplified) code for the i-th variable x_i:

x_i[t+1] = 0 
for (j:1..N) {
  x_i[t+1] = x_i[t+1] + corrMat[i,j]  * x_j[t] 
}
x_i[t+1] = x_i[t+1] + getRand(sdErr)

So the question would be more specific: how to calculate sdErr?

For simplification I try to assume, that the variance for all variables should be 1.

Thank you for any hint, how to get one step further!

Answer 1

I will do a mathematical formulation of the problem to stats.stackexchange.com, as mikeck suggested to discuss details of the correlation problems more in depth.

I still am interested in finding a geneal formula to calculate sdErr to use it in the calculation of x_i[t+1].

But meanwhile I found a useful practical solution to the specific question "how to calculate sdErr?" without a formula for sdErr:

(1) simply calculate all variables WITHOUT errors (according to the equation above).

(2) calculate variances of the new variables

(3) calculate (for each i) differences var(x_i[t]) - var(x_i[t+1]) = sdErr ^ 2 So this sdErr can be added to each variable for each new observation. This should lead to observations at t+1 which at least have the same variances as the observations in t.

Details concercing the question, if the model definition is adequate, will be part of another post.