Test if a Distance Matrix Is Euclidean

Checks whether a distance matrix corresponds to Euclidean distances.

Usage

isEuclid(distmat, tol = 0.0000001)

Arguments

distmat

an object of class dist.

tol

numeric tolerance for detecting negative eigenvalues, relative to the largest absolute eigenvalue.

Value

a logical scalar. Returns TRUE if the distance matrix is (approximately) Euclidean, otherwise FALSE.

Details

The test is based on the eigenvalues of the double-centered squared distance matrix \(B = -\frac{1}{2} J D^2 J\). A distance matrix is Euclidean if and only if \(B\) is positive semi-definite, i.e., all eigenvalues are non-negative (within numerical tolerance).

The tolerance is applied relative to the largest absolute eigenvalue, so that the test is invariant to rescaling of the distances. Note that this holds in both directions: the comparison below uses max(abs(lambda)) without an absolute floor, so shrinking all distances by a constant factor cannot turn a non-Euclidean matrix into a Euclidean one.

The returned logical value carries additional diagnostic information as attributes:

• eigenvalues: Eigenvalues of the centered matrix

• minEigenvalue: Smallest eigenvalue

• tol: Tolerance used for the test

See also

Other data.predicate: flags(), isDichotomous(), isLowCardinality(), isNumeric(), isWholeLike(), isZero(), nUnique()

Examples

d <- dist(matrix(rnorm(20), ncol = 2))
res <- isEuclid(d)
res
#> [1] TRUE
#> attr(,"eigenvalues")
#>  [1]  1.851127e+01  6.672260e+00  1.684038e-15  1.082417e-15  9.048295e-16
#>  [6] -1.331686e-17 -4.259570e-16 -1.486064e-15 -2.322664e-15 -3.071728e-15
#> attr(,"minEigenvalue")
#> [1] -3.071728e-15
#> attr(,"tol")
#> [1] 1e-07

# Access diagnostics
attr(res, "eigenvalues")
#>  [1]  1.851127e+01  6.672260e+00  1.684038e-15  1.082417e-15  9.048295e-16
#>  [6] -1.331686e-17 -4.259570e-16 -1.486064e-15 -2.322664e-15 -3.071728e-15
attr(res, "minEigenvalue")
#> [1] -3.071728e-15