is.CNSD: Check for Conditional Negative Semi-Definiteness

Description

This function checks whether a symmetric matrix is Conditionally Negative Semi-Definite (CNSD). Note that this function does not check whether the matrix is actually symmetric.

Usage

is.CNSD(X, method = "alg1", tol = 1e-08)

Value

boolean, which is TRUE if X is CNSD

Arguments

X

a symmetric matrix

method

a string, specifiying the method to be used. "alg1" is based on algorithm 1 in Ikramov and Savel'eva (2000). "alg2" is based on theorem 3.2 in Ikramov and Savel'eva (2000). "eucl" is based on Glunt (1990).

tol

torelance value. Eigenvalues between -tol and tol are assumed to be zero.

Symmetric, CNSD matrices are, e.g., euclidean distance matrices, whiche are required to produce Positive Semi-Definite correlation or kernel matrices. Such matrices are used in models like Kriging or Support Vector Machines.

References

Ikramov, K. and Savel'eva, N. Conditionally definite matrices, Journal of Mathematical Sciences, Kluwer Academic Publishers-Plenum Publishers, 2000, 98, 1-50

Glunt, W.; Hayden, T. L.; Hong, S. and Wells, J. An alternating projection algorithm for computing the nearest Euclidean distance matrix, SIAM Journal on Matrix Analysis and Applications, SIAM, 1990, 11, 589-600

Examples

Run this code

# The following permutations will produce
# a non-CNSD distance matrix with Insert distance
# and a CNSD distance matrix with Hamming distance
x <- list(c(2,1,4,3),c(2,4,3,1),c(4,2,1,3),c(4,3,2,1),c(1,4,3,2))
D <- distanceMatrix(x,distancePermutationInsert)
is.CNSD(D,"alg1")
is.CNSD(D,"alg2")
is.CNSD(D,"eucl")
D <- distanceMatrix(x,distancePermutationHamming)
is.CNSD(D,"alg1")
is.CNSD(D,"alg2")
is.CNSD(D,"eucl")