mahalanobis
Mahalanobis Distance
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
$$D^2 = (x - \mu)' \Sigma^{-1} (x - \mu)$$
- Keywords
- multivariate
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 ($p x p$) of the distribution.
- inverted
- logical. If
TRUE,covis supposed to contain the inverse of the covariance matrix. - ...
- passed to
solvefor computing the inverse of the covariance matrix (ifinvertedis false).
See Also
Examples
library(stats)
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')
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