| nearPD {Matrix} | R Documentation |
Computes the nearest positive definite matrix to an approximate one, typically a correlation or variance-covariance matrix.
nearPD(x, corr = FALSE, keepDiag = FALSE, do2eigen = TRUE,
only.values = FALSE,
eig.tol = 1e-06, conv.tol = 1e-07, posd.tol = 1e-08,
maxit = 100, trace = FALSE)
x |
numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. |
corr |
logical indicating if the matrix should be a correlation matrix. |
keepDiag |
logical, generalizing corr: if TRUE, the
resulting matrix should have the same diagonal
(diag(x)) as the input matrix. |
do2eigen |
logical indicating if a
posdefify() eigen step should be applied to
the result of the Higham algorithm. |
only.values |
logical; if TRUE, the result is just the
vector of eigen values of the approximating matrix. |
eig.tol |
defines relative positiveness of eigenvalues compared to largest one, λ_1. Eigen values λ_k are treated as if zero when λ_k / λ_1 <= eig.tol. |
conv.tol |
convergence tolerance for Higham algorithm. |
posd.tol |
tolerance for enforcing positive definiteness (in the
final posdefify step when do2eigen is TRUE). |
maxit |
maximum number of iterations allowed. |
trace |
logical or integer specifying if convergence monitoring should be traced. |
This implements the algorithm of Higham (2002), and then forces
positive definiteness using code from
posdefify. The algorithm of Knol DL and ten
Berge (1989) (not implemented here) is more general in (1) that it
allows constraints to fix some rows (and columns) of the matrix and (2)
to force the smallest eigenvalue to have a certain value.
Note that setting corr = TRUE just sets diag(.) <- 1
within the algorithm.
If only.values = TRUE, a numeric vector of eigen values of the
approximating matrix;
Otherwise, as by default, an S3 object of class
"nearPD", basically a list with components
mat |
a matrix of class dpoMatrix, the
computed positive-definite matrix. |
eigenvalues |
numeric vector of eigen values of mat. |
corr |
logical, just the argument corr. |
normF |
the Frobenius norm (norm(x-X, "F")) of the
difference between the original and the resulting matrix. |
iterations |
number of iterations needed. |
converged |
logical indicating if iterations converged. |
Jens Oehlschlaegel donated a first version. Subsequent changes by the Matrix package authors.
Cheng, Sheung Hun and Higham, Nick (1998) A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization; SIAM J. Matrix Anal. Appl., 19, 1097–1110.
Knol DL, ten Berge JMF (1989) Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika 54, 53–61.
Higham, Nick (2002) Computing the nearest correlation matrix - a problem from finance; IMA Journal of Numerical Analysis 22, 329–343.
A first version of this (with non-optional corr=TRUE)
has been available as nearcor(); and
more simple versions with a similar purpose
posdefify(), both from package sfsmisc.
set.seed(27)
m <- matrix(round(rnorm(25),2), 5, 5)
m <- m + t(m)
diag(m) <- pmax(0, diag(m)) + 1
(m <- round(cov2cor(m), 2))
str(near.m <- nearPD(m, trace = TRUE))
round(near.m$mat, 2)
norm(m - near.m$mat) # 1.102
if(require("sfsmisc")) {
m2 <- posdefify(m) # a simpler approach
norm(m - m2) # 1.185, i.e., slightly "less near"
}
round(nearPD(m, only.values=TRUE), 9)
## A longer example, extended from Jens' original,
## showing the effects of some of the options:
pr <- Matrix(c(1, 0.477, 0.644, 0.478, 0.651, 0.826,
0.477, 1, 0.516, 0.233, 0.682, 0.75,
0.644, 0.516, 1, 0.599, 0.581, 0.742,
0.478, 0.233, 0.599, 1, 0.741, 0.8,
0.651, 0.682, 0.581, 0.741, 1, 0.798,
0.826, 0.75, 0.742, 0.8, 0.798, 1),
nrow = 6, ncol = 6)
nc. <- nearPD(pr, conv.tol = 1e-7) # default
nc.$iterations # 2
nc.1 <- nearPD(pr, conv.tol = 1e-7, corr = TRUE)
nc.1$iterations # 11 (!)
ncr <- nearPD(pr, conv.tol = 1e-15)
str(ncr)# 3 iterations
ncr.1 <- nearPD(pr, conv.tol = 1e-15, corr = TRUE)
ncr.1 $ iterations # 27 !
## But indeed, the 'corr = TRUE' constraint did ensure a better solution;
## cov2cor() does not just fix it up equivalently :
norm(pr - cov2cor(ncr$mat)) # = 0.09994
norm(pr - ncr.1$mat) # = 0.08746