Nearest Positive Definite Matrix
Compute the nearest positive definite matrix to an approximate one, typically a correlation or variance-covariance matrix.
nearPD(x, corr = FALSE, keepDiag = FALSE, base.matrix = FALSE, do2eigen = TRUE, doSym = FALSE, doDykstra = TRUE, only.values = FALSE, ensureSymmetry = !isSymmetric(x), eig.tol = 1e-06, conv.tol = 1e-07, posd.tol = 1e-08, maxit = 100, conv.norm.type = "I", trace = FALSE)
x: numeric approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. If x is not symmetric (and ensureSymmetry is not false), symmpart(x) is used.
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.
base.matrix: logical indicating if the resulting mat
component should be a base matrix or (by default) a Matrix of class dpoMatrix.
do2eigen: logical indicating if a posdefify() eigen step should be applied to the result of the Higham algorithm.
doSym: logical indicating if X <- (X + t(X))/2 should be done, after X <- tcrossprod(Qd, Q); some doubt if this is necessary.
doDykstra: logical indicating if Dykstra's correction should be used; true by default. If false, the algorithm is basically the direct fixpoint iteration .
only.values: logical; if TRUE, the result is just the vector of eigenvalues of the approximating matrix.
ensureSymmetry: logical; by default, symmpart(x)
is used whenever isSymmetric(x) is not true. The user can explicitly set this to TRUE or FALSE, saving the symmetry test. Beware however that setting it FALSE
for an a symmetric input x, is typically nonsense!
eig.tol: defines relative positiveness of eigenvalues compared to largest one, . Eigenvalues are treated as if zero when .
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.
conv.norm.type: convergence norm type (norm(*, type)) used for Higham algorithm. The default is "I"
(infinity), for reasons of speed (and back compatibility); using "F" is more in line with Higham's proposal.
trace: logical or integer specifying if convergence monitoring should be traced.
This implements the algorithm of Higham (2002), and then (if do2eigen is true) forces positive definiteness using code from posdefify. The algorithm of Knol and ten Berge (1989) (not implemented here) is more general in that it allows constraints to (1) fix some rows (and columns) of the matrix and (2) force the smallest eigenvalue to have a certain value.
Note that setting corr = TRUE just sets diag(.) <- 1
within the algorithm.
Higham (2002) uses Dykstra's correction, but the version by Jens did not use it (accidentally), and still gave reasonable results; this simplification, now only used if doDykstra = FALSE, was active in nearPD() up to Matrix version 0.999375-40.
If only.values = TRUE, a numeric vector of eigenvalues 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 eigenvalues 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.
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.
Jens donated a first version. Subsequent changes by the Matrix package authors.
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.
## Higham(2002), p.334f - simple example A <- matrix(1, 3,3); A[1,3] <- A[3,1] <- 0 n.A <- nearPD(A, corr=TRUE, do2eigen=FALSE) n.A[c("mat", "normF")] n.A.m <- nearPD(A, corr=TRUE, do2eigen=FALSE, base.matrix=TRUE)$mat stopifnot(exprs = { #=-------------- all.equal(n.A$mat[1,2], 0.760689917) all.equal(n.A$normF, 0.52779033, tolerance=1e-9) all.equal(n.A.m, unname(as.matrix(n.A$mat)), tolerance = 1e-15)# seen rel.d.= 1.46e-16 }) 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 / 1.08 if(requireNamespace("sfsmisc")) { m2 <- sfsmisc::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 / 12 (!) ncr <- nearPD(pr, conv.tol = 1e-15) str(ncr)# still 2 iterations ncr.1 <- nearPD(pr, conv.tol = 1e-15, corr = TRUE) ncr.1 $ iterations # 27 / 30 ! ncF <- nearPD(pr, conv.tol = 1e-15, conv.norm = "F") stopifnot(all.equal(ncr, ncF))# norm type does not matter at all in this example ## 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 / 0.08805 ### 3) a real data example from a 'systemfit' model (3 eq.): (load(system.file("external", "symW.rda", package="Matrix"))) # "symW" dim(symW) # 24 x 24 class(symW)# "dsCMatrix": sparse symmetric if(dev.interactive()) image(symW) EV <- eigen(symW, only=TRUE)$values summary(EV) ## looking more closely {EV sorted decreasingly}: tail(EV)# all 6 are negative EV2 <- eigen(sWpos <- nearPD(symW)$mat, only=TRUE)$values stopifnot(EV2 > 0) if(requireNamespace("sfsmisc")) { plot(pmax(1e-3,EV), EV2, type="o", log="xy", xaxt="n", yaxt="n") for(side in 1:2) sfsmisc::eaxis(side) } else plot(pmax(1e-3,EV), EV2, type="o", log="xy") abline(0, 1, col="red3", lty=2)
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