Quando uma variável aleatória multivariada (X1,X2,…,Xn) possui uma matriz de covariância não-regenerada C=(γij)=(Cov(Xi,Xj)) , o conjunto de todas as combinações lineares reais de Xi forma um n -dimensional espaço vectorial real com base E=(X1,X2,…,Xn) e um produto interno não degenerado, dado por
⟨Xi,Xj⟩=γij .
Sua base dupla em relação a esse produto interno , , é definida exclusivamente pelas relaçõesE∗=(X∗1,X∗2,…,X∗n)
⟨X∗i,Xj⟩=δij ,
the Kronecker delta (equal to 1 when i=j and 0 otherwise).
The dual basis is of interest here because the partial correlation of Xi and Xj is obtained as the correlation between the part of Xi that is left after projecting it into the space spanned by all the other vectors (let's simply call it its "residual", Xi∘) and the comparable part of Xj, its residual Xj∘. Yet X∗i is a vector that is orthogonal to all vectors besides Xi and has positive inner product with Xi whence Xi∘ must be some non-negative multiple of X∗i, and likewise for Xj. Let us therefore write
Xi∘=λiX∗i, Xj∘=λjX∗j
for positive real numbers λi and λj.
The partial correlation is the normalized dot product of the residuals, which is unchanged by rescaling:
ρij∘=⟨Xi∘,Xj∘⟩⟨Xi∘,Xi∘⟩⟨Xj∘,Xj∘⟩−−−−−−−−−−−−−−−−√=λiλj⟨X∗i,X∗j⟩λ2i⟨X∗i,X∗i⟩λ2j⟨X∗j,X∗j⟩−−−−−−−−−−−−−−−−−−√=⟨X∗i,X∗j⟩⟨X∗i,X∗i⟩⟨X∗j,X∗j⟩−−−−−−−−−−−−−−√ .
(In either case the partial correlation will be zero whenever the residuals are orthogonal, whether or not they are nonzero.)
We need to find the inner products of dual basis elements. To this end, expand the dual basis elements in terms of the original basis E:
X∗i=∑j=1nβijXj .
Then by definition
δik=⟨X∗i,Xk⟩=∑j=1nβij⟨Xj,Xk⟩=∑j=1nβijγjk .
In matrix notation with I=(δij) the identity matrix and B=(βij) the change-of-basis matrix, this states
I=BC .
That is, B=C−1, which is exactly what the Wikipedia article is asserting. The previous formula for the partial correlation gives
ρij⋅=βijβiiβjj−−−−−√=C−1ijC−1iiC−1jj−−−−−−√ .