Regularization of the AVO inverse problem by means of a ...

APPENDIX A
Multivariate t distribution and
Regularizations
A.1 Multivariate t distribution
The general central multivariate t distribution is given by,
P (x i |µ, Ψ, ν) =
Γ( ν+p
2 )|Ψ|−1/2
(πν) p/2Γ( ν 1
2 )[1 + ν (xi − µ) T Ψ−1 (xi − µ)] (ν+p)
2
, (A.1)
The model x i is a p dimensional vector whose elements are in the range −∞ < xj < ∞
for j = 1, 2, 3 , ..., p which are correlated and having a p dimensional vector center, µ. The
degree of correlation is represented by a p×p dimensional scale matrix, Ψ. The parameter ν
is the degree of freedom. For N independent samples (x 1 , x 2 , x 3 ,.., x N ), the joint probability
density distribution is the product of each density which can be written as
where
P (X|µ, Ψ, ν) = Po(Ψ, p, ν)
N
i=0
Po(Ψ, p, ν) = po =
1
[1 + 1
ν (xi − µ) T Ψ −1 (xi − µ)] (ν+p)
2
N
i=0
83
Γ( ν+p
2 )|Ψ|−1/2
(πν) p/2 Γ( ν
2 )
. (A.2)