R: Triangular decomposition of a covariance matrix

triDec {ggm} | R Documentation

## Triangular decomposition of a covariance matrix

### Description

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

### Usage

    triDec(Sigma)

### Arguments

`Sigma` | a symmetric positive definite matrix.

### Details

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.

### Value

`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.

### Author(s)

Giovanni M. Marchetti

### References

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

### See Also

`chol`

### Examples

    ## 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)

[Package ggm version 2.3 Index]