stats (version 3.3.2)

mahalanobis: Mahalanobis Distance

Description

Returns the squared Mahalanobis distance of all rows in x and the vector \(\mu\) = center with respect to \(\Sigma\) = cov. This is (for vector x) defined as $$D^2 = (x - \mu)' \Sigma^{-1} (x - \mu)$$

Usage

mahalanobis(x, center, cov, inverted = FALSE, ...)

Arguments

x

vector or matrix of data with, say, \(p\) columns.

center

mean vector of the distribution or second data vector of length \(p\) or recyclable to that length. If set to FALSE, the centering step is skipped.

cov

covariance matrix (\(p \times p\)) of the distribution.

inverted

logical. If TRUE, cov is supposed to contain the inverse of the covariance matrix.

...

passed to solve for computing the inverse of the covariance matrix (if inverted is false).

See Also

cov, var

Examples

require(graphics)

ma <- cbind(1:6, 1:3)
(S <-  var(ma))
mahalanobis(c(0, 0), 1:2, S)

x <- matrix(rnorm(100*3), ncol = 3)
stopifnot(mahalanobis(x, 0, diag(ncol(x))) == rowSums(x*x))
        ##- Here, D^2 = usual squared Euclidean distances

Sx <- cov(x)
D2 <- mahalanobis(x, colMeans(x), Sx)
plot(density(D2, bw = 0.5),
     main="Squared Mahalanobis distances, n=100, p=3") ; rug(D2)
qqplot(qchisq(ppoints(100), df = 3), D2,
       main = expression("Q-Q plot of Mahalanobis" * ~D^2 *
                         " vs. quantiles of" * ~ chi[3]^2))
abline(0, 1, col = 'gray')