I have been simulating a lot of data lately  with various covariance (correlation) structures, and one that I have been using is the autocorrelation (or autoregressive) structure, where there is a “lag” between variables. The matrix is a v-dimension matrix of the form

$$\begin{bmatrix} 1 & \rho & \rho^2 & \dots & \rho^{v-1}\\ \rho & 1& \ddots & \dots & \rho^{v-2}\\ \vdots & \ddots & \ddots & \ddots & \vdots\\ \rho^{v-2} & \dots & \ddots & \ddots & \rho\\ \rho^{v-1} & \rho^{v-2} & \dots & \rho & 1 \end{bmatrix}$$,

where \(\rho \in [-1, 1]\) is the lag. Notice that the lag decays to 0 as v increases.

My goal was to make the construction of such a matrix simple and easy in R.  The method that I used explored a function I have not used yet in R called “lower.tri” for the lower triangular part of the matrix.  The upper triangular part is referenced with “upper.tri.”

My code is as follows:

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autocorr.mat <- function(p = 100, rho = 0.9) {
    mat <- diag(p)
    autocorr <- sapply(seq_len(p), function(i) rho^i)
    mat[lower.tri(mat)] <- autocorr[unlist(sapply(seq.int(p-1, 1), function(i) seq_len(i)))]
    mat[upper.tri(mat)] <- autocorr[unlist(sapply(seq_len(p - 1), function(i) seq.int(i, 1)))]
    return(mat)
}

I really liked it because I feel that it is simple, but then I found Professor Peter Dalgaard’s method, which I have slightly modified. It is far better than mine, easy to understand, and slick. Oh so slick. Here it is:

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autocorr.mat <- function(p = 100, rho = 0.9) {
    mat <- diag(p)
    return(rho^abs(row(mat)-col(mat)))
}

Professor Dalgaard’s method puts mine to shame. It is quite obvious how to do it once it is seen, but I certainly wasn’t thinking along those lines.