C++ implementation of Matrix's nearPD

With `ensureSymmetry` it makes sure it is symmetric by applying 0.5*(t(x) + x) before using lotriNearPD

lotriNearPD(
  x,
  keepDiag = FALSE,
  do2eigen = TRUE,
  doDykstra = TRUE,
  only.values = FALSE,
  ensureSymmetry = !isSymmetric(x),
  eig.tol = 1e-06,
  conv.tol = 1e-07,
  posd.tol = 1e-08,
  maxit = 100L,
  trace = FALSE
)

Arguments

x: numeric \(n \times n\) 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.
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()` (like in the package `sfsmisc`) eigen step should be applied to the result of the Higham algorithm
doDykstra: logical indicating if Dykstra's correction should be used; true by default. If false, the algorithm is basically the direct fixpoint iteration \(Y_k = P_U(P_S(Y_{k-1}))\).
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 asymmetric input x, is typically nonsense!
eig.tol: defines relative positiveness of eigenvalues compared to largest one, \(\lambda_1\). Eigenvalues \(\lambda_k\) are treated as if zero when \(\lambda_k / \lambda_1 \le eig.tol\).
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.

Value

unlike the matrix package, this simply returns the nearest positive definite matrix

Details

This implements the algorithm of Higham (2002), and then (if do2eigen is true) forces positive definiteness using code from `sfsmisc::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 Oehlschlägel 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.

References

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.

Author

Jens Oehlschlägel donated a first version to Matrix. Subsequent changes by the Matrix package authors, later modifications to C++ by Matthew Fidler

Examples


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))
#>       [,1]  [,2]  [,3]  [,4]  [,5]
#> [1,]  1.00  0.65 -0.46 -1.15 -0.76
#> [2,]  0.65  1.00  0.58  0.50 -0.90
#> [3,] -0.46  0.58  1.00 -0.45 -0.32
#> [4,] -1.15  0.50 -0.45  1.00  0.25
#> [5,] -0.76 -0.90 -0.32  0.25  1.00

near.m <- lotriNearPD(m)
round(near.m, 2)
#>       [,1]  [,2]  [,3]  [,4]  [,5]
#> [1,]  1.31  0.41 -0.24 -0.85 -0.75
#> [2,]  0.41  1.19  0.41  0.27 -0.91
#> [3,] -0.24  0.41  1.15 -0.24 -0.32
#> [4,] -0.85  0.27 -0.24  1.28  0.26
#> [5,] -0.75 -0.91 -0.32  0.26  1.00
norm(m - near.m) # 1.102 / 1.08
#> [1] 1.079735

round(lotriNearPD(m, only.values=TRUE), 9)
#> [1] 2.800681404 1.831722441 1.229003616 0.076994641 0.000000028

## 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  <- lotriNearPD(pr)