triDec: Triangular decomposition of a covariance matrix
In ggm: Graphical Markov Models with Mixed Graphs

Description Usage Arguments Details Value Author(s) References See Also Examples

Decomposes a symmetric positive definite matrix with a variant of the Cholesky decomposition.

1	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 `Sigma` with ones on the diagonal.
`B`	the inverse of `A`, another triangular matrix with unit diagonal.
`Delta`	a vector containing the diagonal values of Δ.

Giovanni M. Marchetti

Cox, D. R. \& Wermuth, N. (1996). Multivariate dependencies. London: Chapman \& Hall.

chol

## 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)