dlm (version 1.1-6.1)

dlmSvd2var: Compute a nonnegative definite matrix from its Singular Value Decomposition

Description

The function computes a nonnegative definite matrix from its Singular Value Decomposition.

Usage

dlmSvd2var(u, d)

Value

The function returns a nonnegative definite matrix, reconstructed from its SVD, or a list of such matrices (see details above).

Arguments

u

a square matrix, or a list of square matrices for a vectorized usage.

d

a vector, or a matrix for a vectorized usage.

Author

Giovanni Petris

Details

The SVD of a nonnegative definite \(n\) by \(n\) square matrix \(x\) can be written as \(u d^2 u'\), where \(u\) is an \(n\) by \(n\) orthogonal matrix and \(d\) is a diagonal matrix. For a single matrix, the function returns just \(u d^2 u'\). Note that the argument d is a vector containing the diagonal elements of \(d\). For a vectorized usage, u is a list of square matrices, and d is a matrix. The returned value in this case is a list of matrices, with the element \(i\) being u[[i]] %*% diag(d[i,]^2) %*% t(u[[i]]).

References

Horn and Johnson, Matrix analysis, Cambridge University Press (1985)

Examples

x <- matrix(rnorm(16),4,4)
x <- crossprod(x)
tmp <- La.svd(x)
all.equal(dlmSvd2var(tmp$u, sqrt(tmp$d)), x)
## Vectorized usage
x <- dlmFilter(Nile, dlmModPoly(1, dV=15099, dW=1469))
x$se <- sqrt(unlist(dlmSvd2var(x$U.C, x$D.C)))
## Level with 50% probability interval
plot(Nile, lty=2)
lines(dropFirst(x$m), col="blue")
lines(dropFirst(x$m - .67*x$se), lty=3, col="blue")
lines(dropFirst(x$m + .67*x$se), lty=3, col="blue")