# Is Mahalanobis distance equivalent to the Euclidean one on the PCA-rotated data?

I've been led to believe (see here and here) that Mahalanobis distance is the same as the Euclidean distance on the PCA-rotated data. In other words, taking multivariate normal data $X$, the Mahalanobis distance of all of the $x$'s from any given point (say $\mathbf{0}$) should be the same as the Euclidean distance of the entries of $X^{rot}$ from $\mathbf{0}$, where $X^{rot}$ is the product of the data and the PCA rotation matrix.

1. Is this true?

My code below is suggesting to me that it is not. In particular, it looks like the variance of the Mahalanobis distance around the PCA-Euclidean distance is increasing in the magnitude of the PCA-Euclidean distance. Is this a coding error, or a feature of the universe? Does it have to do with imprecision in an estimate of something? Something that gets squared?

N=1000
cr = runif(1,min=-1,max=1)
A = matrix(c(1,cr,cr,1),2)
e<-mvrnorm(n = N,rep(0,2),A)
mx = apply(e, 2, mean)
sx = apply(e, 2, sd)
e = t(apply(e,1,function(X){(X-mx)/sx}))
plot(e[,1],e[,2])
dum<-rep(0,2)
md = mahalanobis(e,dum,cov(e))

pc = prcomp(e,center=F,scale=F)
d<-as.matrix(dist(rbind(dum,pc$x),method='euclidean',diag=F)) d<-d[1,2:ncol(d)] plot(d,md^.5) abline(0,1)  2. If the answer to the above is true, can one use the PCA-rotated Euclidean distance as a stand-in for the Mahalanobis distance when$p>n$? If not, is there a similar metric that captures multivariate distance, scaled by correlation, and for which distributional results exist to allow the calculation of the probability of an observation? EDIT I've run a few simulations to investigate the equivalence of MD and SED on scaled/rotated data over a gradient of n and p. As I mentioned previously, I'm interested in the probability of an observation. I am hoping to find a good way to get the probability of an observation being part of a multivariate normal distribution, but for which I've got$n<p$data to estimate the distribution. See the code below. It looks like the PCA-scaled/rotated SED is slightly biased estimator of the MD, with a fair amount of variance that seems to stop increasing when$p=N$. f = function(N=1000,n,p){ a = runif(p^2,-1,1) a = matrix(a,p) S = t(a)%*%a x = mvrnorm(N,rep(0,p),S) mx = apply(x, 2, mean) sx = apply(x, 2, sd) x = t(apply(x,1,function(X){(X-mx)/sx})) Ss = solve(cov(x)) x = x[sample(1:N,n,replace=F),] md = mahalanobis(x,rep(0,p),Ss,inverted=T) prMD<-pchisq(md,df = p) pc = prcomp(x,center=F,scale=F) d<-mahalanobis(scale(pc$x),rep(0,ncol(pc$x)),diag(rep(1,ncol(pc$x))))
prPCA<-pchisq(d,df = min(p,n))#N is the number of PCs where N<P
return(data.frame(prbias = as.numeric(mean(prMD - prPCA)), prvariance = as.numeric(mean((prMD - prPCA)^2))))
}

grid = data.frame(n=100,p=2:200)
grid$prvariance <-grid$prbias <-NA

for (i in 1:nrow(grid)){
o = f(n=grid[i,]$n,p=grid[i,]$p)
grid[i,3:4]<-o
}

par(mfrow=c(1,2))
with(grid, plot(p,prbias))
abline(v=100)
m = lm(prbias~p,data=grid)
abline(m,col='red',lty=2)
with(grid, plot(p,prvariance))
abline(v=100) Two questions: 1. Any criticism of what I'm finding in these simulations? 2. Can anyone formalize what I'm finding with an analytical expression for the bias and the variance as functions of n and p? I'd accept an answer that does this.

• From the very formula of Mahalanobis distance it follows that it is equal to Euclidean distance when the covariance matrix is identity matrix (or, to extend without loss of key - proportional to identity). Unless data are exactly spherical the covariances between their PCs are diagonal, not identity, matrix. – ttnphns Aug 10 '15 at 18:09
• To note, PCA doesn't just rotate the data it also scales it differently in different directions. Scaling has an effect on distance measurements. – conjectures Aug 10 '15 at 18:49
• Squared Mahalanobis distances between the data points are exactly proportional to the weighted squared Euclidean distances computed on the principal components of the data. The weight being 1/eigenvalue of the component. (And same hold also when we speak about distances between points and centroid, instead of point-point distances) This weighting is what compensates for the differences between Mahalanobis and Euclidean which I have touched in my comment. – ttnphns Aug 10 '15 at 19:23
• @ttnphns That could be an answer for the first part of my question. I'm still thinking about the second part however -- whether it is a good stand-in for mahalanobis when $P>N$. I'm thinking however that the curse of dimensionality might start becoming pretty severe. – generic_user Aug 10 '15 at 19:43
• You can search this site Mahalanobis singular which might yield answers to your second question – ttnphns Aug 10 '15 at 19:50

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.
Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. Indeed, think about Euclidean distance in the PCA-transformed data; when $n<p$, some of the eigenvalues of covariance matrix are zero and the corresponding PCs have zero variance (all the data points are projected to zero). It is therefore impossible to standardize these PCs, as it is impossible to divide by zero. Mahalanobis distance cannot be defined "in these directions".