mahalanobis {stats}

R Documentation

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.
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=.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')

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