Can't get a positive definite variance matrix when very small eigen values

Question

To run a Canonical correspondence analysis (cca package ade4) I need a positive definite variance matrix. (Which in theory is always the case) but:

matrix(c(2,59,4,7,10,0,7,0,0,0,475,18714,4070,97,298,0,1,0,17,7,4,1,4,18,36),nrow=5)
> a
     [,1] [,2]  [,3] [,4] [,5]
[1,]    2    0   475    0    4
[2,]   59    7 18714    1    1
[3,]    4    0  4070    0    4
[4,]    7    0    97   17   18
[5,]   10    0   298    7   36

> eigen(var(a))
$values
[1]  6.380066e+07  1.973658e+02  3.551492e+01  1.033096e+01
[5] -1.377693e-09

The last eigen value is -1.377693e-09 which is < 0. But the theorical value is > 0.
I can't run the function if one of the eigen value is < 0

I really don't know how to fix this without changing the code of the function cca()

Thanks for help

Answer 1

You can change the input, just a little bit, to make the matrix positive definite.

If you have the variance matrix, you can truncate the eigenvalues:

correct_variance <- function(V, minimum_eigenvalue = 0) {
  V <- ( V + t(V) ) / 2
  e <- eigen(V)
  e$vectors %*% diag(pmax(minimum_eigenvalue,e$values)) %*% t(e$vectors)
}
v <- correct_variance( var(a) )
eigen(v)$values
# [1] 6.380066e+07 1.973658e+02 3.551492e+01 1.033096e+01 1.326768e-08

Using the singular value decomposition, you can do the same thing directly with a.

truncate_singular_values <- function(a, minimum = 0) { 
  s <- svd(a)
  s$u %*% diag( ifelse( s$d > minimum, s$d, minimum ) ) %*% t(s$v)
}
svd(a)$d
# [1] 1.916001e+04 4.435562e+01 1.196984e+01 8.822299e+00 1.035624e-01
eigen(var( truncate_singular_values(a,.2) ))$values
# [1] 6.380066e+07 1.973680e+02 3.551494e+01 1.033452e+01 6.079487e-09

However, this changes your matrix a by up to 0.1, which is a lot (I suspect it is that high because the matrix a is square: as a result, one of the eigenvalues of var(a) is exactly 0.)

b <- truncate_singular_values(a,.2)
max( abs(b-a) )
# [1] 0.09410187

We can actually do better simply by adding some noise.

b <- a + 1e-6*runif(length(a),-1,1)  # Repeat if needed
eigen(var(b))$values
# [1] 6.380066e+07 1.973658e+02 3.551492e+01 1.033096e+01 2.492604e-09