R: Nearest Positive Definite Matrix

nearPD {Matrix} | R Documentation

## Nearest Positive Definite Matrix

### Description

Compute the nearest positive definite matrix to an approximate one, typically a correlation or variance-covariance matrix.

### Usage

    nearPD(x, corr = FALSE, keepDiag = 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)

### Arguments

`x` | numeric n * 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.
`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.
`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 Y(k) = P_U(P_S(Y(k-1))).
`only.values` | logical; if `TRUE`, the result is just the vector of eigen values 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, 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.
`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.

### Details

This implements the algorithm of Higham (2002), and then (if `do2eigen` is true) 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.

Higham (2002) uses Dykstra's correction, but the version by Jens Oehlschlaegel did not use it (accidentally), and has still lead to good results; this simplification, now only via `doDykstra = FALSE`, was active in `nearPD()` upto Matrix version 0.999375-40.

### Value

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.

### Author(s)

Jens Oehlschlaegel donated a first version. Subsequent changes by the Matrix package authors.

### 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.

### See Also

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.

### Examples

     ## 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")]
     stopifnot(all.equal(n.A$mat[1,2], 0.760689917),
    	   all.equal(n.A$normF, 0.52779033, tolerance=1e-9) )

     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(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 / 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(require("sfsmisc")) {
           plot(pmax(1e-3,EV), EV2, type="o", log="xy", xaxt="n",yaxt="n")
           eaxis(1); eaxis(2)
    } else plot(pmax(1e-3,EV), EV2, type="o", log="xy")
    abline(0,1, col="red3",lty=2)

[Package Matrix version 1.1-4 Index]