MvNormal Error with Symmetric & Positive Semi-Definite Matrix - matlab

Question

The summary of my problem is that I am trying to replicate the Matlab function:
mvnrnd(mu', sigma, 200)
into Julia using:
rand( MvNormal(mu, sigma), 200)'
and the result is a 200 x 7 matrix, essentially generating 200 random return time series data.
Matlab works, Julia doesn't.
My input matrices are:
mu = [0.15; 0.03; 0.06; 0.04; 0.1; 0.02; 0.12]
sigma = [0.0035 -0.0038 0.0020 0.0017 -0.0006 -0.0028 0.0009;
-0.0038 0.0046 -0.0011 0.0001 0.0003 0.0054 -0.0024;
0.0020 -0.0011 0.0041 0.0068 -0.0004 0.0047 -0.0036;
0.0017 0.0001 0.0068 0.0125 0.0002 0.0109 -0.0078;
-0.0006 0.0003 -0.0004 0.0002 0.0025 -0.0004 -0.0007;
-0.0028 0.0054 0.0047 0.0109 -0.0004 0.0159 -0.0093;
0.0009 -0.0024 -0.0036 -0.0078 -0.0007 -0.0093 0.0061]
Using Distributions.jl, running the line:
MvNormal(sigma)
Produces the error:
ERROR: LoadError: Base.LinAlg.PosDefException(4)
The matrix sigma is symmetrical but only positive semi-definite:
issym(sigma) #symmetrical
> true
isposdef(sigma) #positive definite
> false
using LinearOperators
check_positive_definite(sigma) #check for positive (semi-)definite
> true
Matlab produces the same results for these tests however Matlab is able to generate the 200x7 random return sample matrix.
Could someone advise as to what I could do to get it working in Julia? Or where the issue lies?
Thanks.

Answer 1

The issue is that the covariance matrix is indefinite. See
julia> eigvals(sigma)
7-element Array{Float64,1}:
-3.52259e-5
-2.42008e-5
2.35508e-7
7.08269e-5
0.00290538
0.0118957
0.0343873
so it is not a covariance matrix. This might have happened because of rounding so if you have access to unrounded data you can try that instead. I just tried and I also got an error in Matlab. However, in contrast to Julia, Matlab does allow the matrix to be positive semidefinite.
A way to make this work is to add a diagonal matrix to the original matrix and then input that to MvNormal. I.e.
julia> MvNormal(randn(7), sigma - minimum(eigvals(Symmetric(sigma)))*I)
Distributions.MvNormal{PDMats.PDMat{Float64,Array{Float64,2}},Array{Float64,1}}(
dim: 7
μ: [0.889004,-0.768551,1.78569,0.130445,0.589029,0.529418,-0.258474]
Σ: 7x7 Array{Float64,2}:
0.00353523 -0.0038 0.002 0.0017 -0.0006 -0.0028 0.0009
-0.0038 0.00463523 -0.0011 0.0001 0.0003 0.0054 -0.0024
0.002 -0.0011 0.00413523 0.0068 -0.0004 0.0047 -0.0036
0.0017 0.0001 0.0068 0.0125352 0.0002 0.0109 -0.0078
-0.0006 0.0003 -0.0004 0.0002 0.00253523 -0.0004 -0.0007
-0.0028 0.0054 0.0047 0.0109 -0.0004 0.0159352 -0.0093
0.0009 -0.0024 -0.0036 -0.0078 -0.0007 -0.0093 0.00613523
)
The "covariance" matrix is of course not the same anymore, but it is very close.