Triangular decomposition of a covariance matrix
Decomposes a symmetric positive definite matrix with a variant of the Cholesky decomposition.
triDec(Sigma)
Sigma |
a symmetric positive definite matrix. |
Any symmetric positive definite matrix Sigma can be decomposed as Sigma = B %*% Delta %*% t(B) where B is upper triangular with ones along the main diagonal and Delta is diagonal. If Sigma is a covariance matrix, the concentration matrix is Σ^{-1} = A^T Δ^{-1} A where A = B^{-1} is the matrix of the regression coefficients (with the sign changed) of a system of linear recursive regression equations with independent residuals. In the equations each variable i is regressed on the variables i+1, …, d. The elements on the diagonal of Δ are the partial variances.
A |
a square upper triangular matrix of the same order as
|
B |
the inverse of |
Delta |
a vector containing the diagonal values of Δ. |
Giovanni M. Marchetti
Cox, D. R. \& Wermuth, N. (1996). Multivariate dependencies. London: Chapman \& Hall.
## Triangular decomposition of a covariance matrix
B <- matrix(c(1, -2, 0, 1,
0, 1, 0, 1,
0, 0, 1, 0,
0, 0, 0, 1), 4, 4, byrow=TRUE)
B
D <- diag(c(3, 1, 2, 1))
S <- B %*% D %*% t(B)
triDec(S)
solve(B)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.