Mahalanobis distance of each pair of observations

Question

I am trying to compute the Mahalanobis distance between each observations of a dataset dat, where each row is an observation and each column is a variable. Such distance is defined as:

I wrote a function that does it, but I feel like it is slow. Is there any better way to compute this in R ?

To generate some data to test the function:

generateData <- function(nObs, nVar){
  library(MASS)
  mvrnorm(n=nObs, rep(0,nVar), diag(nVar))
  }

This is the function I have written so far. They both work and for my data (800 obs and 90 variables), it takes approximatively 30 and 33 seconds for the method = "forLoop" and method = "apply", respectively.

mhbd_calc2 <- function(dat, method) { #Method is either "forLoop" or "apply"
  dat <- as.matrix(na.omit(dat))
  nObs <- nrow(dat)
  mhbd <- matrix(nrow=nObs,ncol = nObs)
  cv_mat_inv = solve(var(dat))

  distMH = function(x){  #Mahalanobis distance function
    diff = dat[x[1],]-dat[x[2],]
    diff %*% cv_mat_inv %*% diff
  }

  if(method=="forLoop")
  {
    for (i in 1:nObs){
      for(j in 1:i){
        mhbd[i,j] <- distMH(c(i,j))
      }
    }
  }
  if(method=="apply")
  {
    mhbd[lower.tri(mhbd)] = apply(combn(nrow(dat),2),2, distMH)
  }
  result = sqrt(mhbd)
  colnames(result)=rownames(dat)
  rownames(result)=rownames(dat)
  return(as.dist(result))
}

NB: I tried using outer() but it was even slower (60seconds)

Answer 1

You need some mathematical knowledge.

1. Do a Cholesky factorization of empirical covariance, then standardize your observations;
2. use dist to compute Euclidean distance on transformed observations.

---

dist.maha <- function (dat) {
  X <- as.matrix(na.omit(dat))  ## ensure a valid matrix
  V <- cov(X)  ## empirical covariance; positive definite
  L <- t(chol(V))  ## lower triangular factor
  stdX <- t(forwardsolve(L, t(X)))  ## standardization
  dist(stdX)  ## use `dist`
  }

Example

set.seed(0)
x <- matrix(rnorm(6 * 3), 6, 3)

dist.maha(x)
#         1        2        3        4        5
#2 2.362109                                    
#3 1.725084 1.495655                           
#4 2.959946 2.715641 2.690788                  
#5 3.044610 1.218184 1.531026 2.717390         
#6 2.740958 1.694767 2.877993 2.978265 2.794879

The result agrees with your mhbd_calc2.