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Re: (Q) transformation of R^n simplex to R^(n-1)
Posted:
Jul 15, 1996 8:17 AM
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Please check my contribution to the thread :
"Uniform distribution over polyhedron"
I understand you want to get rid of the dependent constraints over the simplex.
I suppose also that you want to cover the simplex uniformly under
transformation
mapping.
You can do it in two ways.
1) you immerse the simplex into R^n and use n identical variables with
independent constraints. That is
X_i = log(U_i) 0 < U_i < 1
and
x_i = X_i/\sum X_i
2) If you insist on having n-1 variables instead of n, you can do the following
transformation: (for 3 dimensions for example) you can use
x_3 = 1-sqrt(U_2)
x_2 = (1-U_1)*sqrt(U_2)
x_1 = U_1*sqrt(U_2)
This generalizes to higher dimension involving U_i^{1/i} 0 < U_i < 1
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