Iman Conover (Excel / VBA)

Thu, 14 May 2026 10:41:00 +0100

“When your work speaks for itself, don’t interrupt.” [Henry J. Kaiser]

Abstract

If you need to generate correlated random numbers, the Iman Conover approach is a good method which is to be preferred to the Cholesky decomposition.

In 1982 Iman and Conover published their original article (external link!) “A distribution-free approach to inducing rank correlation among input variables”.

Rick Wicklin wrote in 2021 on (external link!) “Simulate correlated variables by using the Iman-Conover transformation”. His article includes a SAS implementation of the Iman Conover approach.

Cholesky (VBA)

Thu, 14 May 2026 02:46:00 +0100

Abstract

You can generate correlated random variables easily with a Cholesky (pronounce “koleski”) decomposition. I present a simple example here. For the better Iman Conover approach look here.

Appendix – Cholesky and RandCorr Code

Please read my Disclaimer.

Function Cholesky(vA As Variant) As Variant
'I suggest to use the Cholesky decomposition just for purposes of demonstration.
'Better options are (in this order): tred2, tqli, eigsrt from Numerical Recipes.
'SVD also works but is computationally more expensive by far since it does not
'make use of symmetry.
'(Thanks to my former colleague Glen R.)
'Source (EN): https://www.sulprobil.de/cholesky_en/
'Source (DE): https://www.berndplumhoff.de/cholesky_de/
'Bernd Plumhoff 02-Nov-2024 PB V1.1
Dim d As Double
Dim i As Long, j As Long, k As Long, n As Long
With Application.WorksheetFunction
On Error Resume Next
vA = .Transpose(.Transpose(vA))
On Error GoTo 0
n = UBound(vA, 1)
If n <> UBound(vA, 2) Then
  Cholesky = CVErr(xlErrRef)
  Exit Function
End If

ReDim dR(1 To n, 1 To n) As Double 'Zeroing all elements
For j = 1 To n
  d = 0#
  For k = 1 To j - 1
    d = d + dR(j, k) * dR(j, k)
  Next k
  dR(j, j) = vA(j, j) - d
  If dR(j, j) > 0# Then
    dR(j, j) = Sqr(dR(j, j))
    For i = j + 1 To n
      d = 0#
      For k = 1 To j - 1
        d = d + dR(i, k) * dR(j, k)
      Next k
      dR(i, j) = (vA(i, j) - d) / dR(j, j)
    Next i
  Else
    'Cannot continue with usual Cholesky
    'Fill this column with zeros. Idea: Glen R.
    For i = j To n
      dR(i, j) = 0#
    Next i
  End If
Next j
Cholesky = dR
End With
End Function

Function RandCorr(n As Long, vVarCovar As Variant) As Variant
'Returns Ubound(vVarCovar,1) correlated random number vectors of length n.
'vVarCovar is a square matrix containing the variance/covariance matrix.
'Please notice that you will only get a "proxy" correlation, not an exact one.
'Bernd Plumhoff 06-Nov-2009 PB V0.2
Dim vA As Variant
Dim d As Double
Dim i As Long, j As Long, k As Long, m As Long

With Application.WorksheetFunction
vA = .Transpose(.Transpose(vVarCovar))
m = UBound(vA, 1)
If m <> UBound(vA, 2) Then
  RandCorr = CVErr(xlErrRef)
  Exit Function
End If

ReDim Db(1 To m, 1 To m) As Double
For j = 1 To m
  d = 0#
  For k = 1 To j - 1
    d = d + Db(j, k) * Db(j, k)
  Next k
  Db(j, j) = vA(j, j) - d
  If Db(j, j) <= 0 Then
    RandCorr = CVErr(xlErrNum)
    Exit Function
  End If
  Db(j, j) = Sqr(Db(j, j))
  
  For i = j + 1 To m
    d = 0#
    For k = 1 To j - 1
      d = d + Db(i, k) * Db(j, k)
    Next k
    Db(i, j) = (vA(i, j) - d) / Db(j, j)
  Next i
Next j

ReDim vR(1 To n, 1 To m) As Variant
For i = 1 To n
  For j = 1 To m
    vR(i, j) = .Norm_S_Inv(Rnd())
  Next j
Next i
vR = .MMult(vR, Db)
RandCorr = vR
End With
End Function

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Please read my Disclaimer.

Generate Correlated Random Numbers on Sulprobil

Iman Conover (Excel / VBA)

Abstract

Cholesky (VBA)

Abstract

Appendix – Cholesky and RandCorr Code

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