Mahalanobis distance in R

Question

I have found the mahalanobis.dist function in package StatMatch (http://cran.r-project.org/web/packages/StatMatch/StatMatch.pdf) but it isn't doing exactly what I want. It seems to be calculating the mahalanobis distance from each observation in data.y to each observation in data.x

I would like to calculate the mahalanobis distance of one observation in data.y to all observations in data.x. Basically calculate a mahalanobis distance of one point to a "cloud" of points if that makes sense. Kind of getting at the idea of the probability of an observation being part of another group of observations

This person (http://people.revoledu.com/kardi/tutorial/Similarity/MahalanobisDistance.html) seems to be doing this and I've tried to replicate his process in R but it is failing when I get to the bottom part of the equation:

mahaldist = sqrt((inversepooledcov %*% t(meandiffmatrix)) %*% meandiffmatrix)

All the code I am working with is here:

a = rbind(c(2,2), c(2,5), c(6,5),c(7,3))

colnames(a) = c('x', 'y')

b = rbind(c(6,5),c(3,4))

colnames(b) = c('x', 'y')

acov = cov(a)
bcov = cov(b)

meandiff1 = mean(a[,1]) - mean(b[,1])

meandiff2 = mean(a[,2]) - mean(b[,2])

meandiffmatrix = rbind(c(meandiff1,meandiff2))

totaldata = dim(a)[1] + dim(b)[1]

pooledcov = (dim(a)[1]/totaldata * acov) + (dim(b)[1]/totaldata * bcov)

inversepooledcov = solve(pooledcov)

mahaldist = sqrt((inversepooledcov %*% t(meandiffmatrix)) %*% meandiffmatrix)

Answer 1

How about using the mahalanobis function in the stats package:

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

Answer 2

I've been trying this out from the same website that you looked at and then stumbled upon this question. I managed to get the script to work, But I get a different result.

#WORKING EXAMPLE
#MAHALANOBIS DIST OF TWO MATRICES

#define matrix
mat1<-matrix(data=c(2,2,6,7,4,6,5,4,2,1,2,5,5,3,7,4,3,6,5,3),nrow=10)
mat2<-matrix(data=c(6,7,8,5,5,5,4,7,6,4),nrow=5)
#center data
mat1.1<-scale(mat1,center=T,scale=F)
mat2.1<-scale(mat2,center=T,scale=F)
#cov matrix
mat1.2<-cov(mat1.1,method="pearson")
mat2.2<-cov(mat2.1,method="pearson")
n1<-nrow(mat1)
n2<-nrow(mat2)
n3<-n1+n2
#pooled matrix
mat3<-((n1/n3)*mat1.2) + ((n2/n3)*mat2.2)
#inverse pooled matrix
mat4<-solve(mat3)
#mean diff
mat5<-as.matrix((colMeans(mat1)-colMeans(mat2)))
#multiply
mat6<-t(mat5) %*% mat4
#multiply
sqrt(mat6 %*% mat5)

I think the function mahalanobis() is used to compute mahalanobis distances between individuals (rows) in one matrix. The function pairwise.mahalanobis() from package(HDMD) can compare two or more matrices and give mahalanobis distances between the matrices.

Answer 3

Your output before taking the square root is :

inversepooledcov %*% t(meandiffmatrix) %*% meandiffmatrix
          [,1]        [,2]
x -0.004349227 -0.01304768
y  0.114529639  0.34358892

I think you can'take the square root of negative numbers number, so you have NAN for negative elements:

 sqrt(inversepooledcov %*% t(meandiffmatrix) %*% meandiffmatrix)
       [,1]      [,2]
x       NaN       NaN
y 0.3384223 0.5861646

Warning message:
In sqrt(inversepooledcov %*% t(meandiffmatrix) %*% meandiffmatrix) :
  NaNs produced

Answer 4

You can wrap the function stats::mahalanobis as bellow to output a mahalanobis distance matrix (pairwise mahalanobis distances):

# x - data frame
# cx - covariance matrix; if not provided, 
#      it will be estimated from the data
mah <- function(x, cx = NULL) {
  if(is.null(cx)) cx <- cov(x)
  out <- lapply(1:nrow(x), function(i) {
    mahalanobis(x = x, 
                center = do.call("c", x[i, ]),
                cov = cx)
  })
  return(as.dist(do.call("rbind", out)))
}

Then, you can cluster your data and plot it, for example:

# Dummy data
x <- data.frame(X = c(rnorm(10, 0), rnorm(10, 5)), 
                Y = c(rnorm(10, 0), rnorm(10, 7)), 
                Z = c(rnorm(10, 0), rnorm(10, 12)))
rownames(x) <- LETTERS[1:20]
plot(x, pch = LETTERS[1:20])

# Comute the mahalanobis distance matrix
d <- mah(x)
d

# Cluster and plot
hc <- hclust(d)
plot(hc)

Answer 5

There a very easy way to do it using R Package "biotools". In this case you will get a Squared Distance Mahalanobis Matrix.

#Manly (2004, p.65-66)

x1 <- c(131.37, 132.37, 134.47, 135.50, 136.17)
x2 <- c(133.60, 132.70, 133.80, 132.30, 130.33)
x3 <- c(99.17, 99.07, 96.03, 94.53, 93.50)
x4 <- c(50.53, 50.23, 50.57, 51.97, 51.37)

#size (n x p) #Means 
x <- cbind(x1, x2, x3, x4) 

#size (p x p) #Variances and Covariances
Cov <- matrix(c(21.112,0.038,0.078,2.01, 0.038,23.486,5.2,2.844, 
        0.078,5.2,24.18,1.134, 2.01,2.844,1.134,10.154), 4, 4)

library(biotools)
Mahalanobis_Distance<-D2.dist(x, Cov)
print(Mahalanobis_Distance)