Generate a simulated covariance matrix

Question

I am working on a problem where one of the input variables is the level of covariance between the entries in a particular matrix. To simulate this problem in Mathematica, I will need to feed my code some simulated covariance matrices.

I will, therefore, need to generate these pretend covariance matrices for dummy data for a range of specified mean levels of covariance.

Is there a way to do this in Mathematica?

Answer 1

I have previously used the following two helper functions to generate the format of covariance and correlation matrices:

covariancematrix[n_] :=
  Table[
    σ[i] σ[j] ρ[i, j]^(1 - KroneckerDelta[i, j]),
    {i, 1, n, 1}, {j, 1, n, 1}
  ] /. {ρ[i_, j_] :> ρ[j, i] /; i > j}

correlationmatrix[n_] :=
  Table[
    σ[i] σ[j] ρ[i, j]^(1 - KroneckerDelta[i, j]),
    {i, 1, n, 1}, {j, 1, n, 1}
  ] / Product[σ[i], {i, 1, n, 1}] /. {ρ[i_, j_] :> ρ[j, i] /; i > j}

I could then use the following for instance:

covariancematrix[4]

(* Out:
{{σ[1]^2, ρ[1, 2] σ[1] σ[2], ρ[1, 3] σ[1] σ[3], ρ[1, 4] σ[1] σ[4]},
 {ρ[1, 2] σ[1] σ[2], σ[2]^2, ρ[2, 3] σ[2] σ[3], ρ[2, 4] σ[2] σ[4]}, 
 {ρ[1, 3] σ[1] σ[3], ρ[2, 3] σ[2] σ[3], σ[3]^2, ρ[3, 4] σ[3] σ[4]},
 {ρ[1, 4] σ[1] σ[4], ρ[2, 4] σ[2] σ[4], ρ[3, 4] σ[3] σ[4], σ[4]^2}}
*)

and I can then substitute appropriate values for standard deviations and correlation coefficients:

covariancematrix[4];
% /. {σ[1] -> 0.1, σ[2] -> 0.1, σ[3] -> 1000, σ[4] -> 100000};
% /. {ρ[1, 2] -> 0, ρ[1, 3] -> 0, ρ[1, 4] -> 0, 
      ρ[2, 3] -> 0.4, ρ[2, 4] -> 0.4, ρ[3, 4] -> 0.7}

(* Out:
{{0.01, 0., 0., 0.},
 {0., 0.01, 40., 4000.},
 {0., 40., 1000000, 7.*10^7},
 {0., 4000., 7.*10^7, 10000000000}}
*)

You can check that the matrices thus generated are positive definite, as required:

PositiveDefiniteMatrixQ[%]
(* Out: True *)