Solved – How is “Orthogonal distance” computed

Given a $n$ by $p$ matrix $pmb X$ the SVD decomposition of $pmb X$ is:

$$text{svd}((pmb X-bar{x})/sqrt{n-1})=pmb{UDV}'$$

(I will denote $pmb V_k$ the matrix formed of the first $k$ columns of $pmb V$ and $pmb D_k$ the diagonal matrix formed of the first $k$ rows and columns of $pmb D$)

The SVD decomposition divides the total variance of $pmb X$ unto two mutually orthogonal components:

• The variance of the projection of $(pmb X-bar{x})$ on the space spanned by the first $k$ singular vectors of $(pmb X-bar{x})$:

$$sqrt{(pmb X-bar{x})'pmb{V_kD_k^{-1}V_k'}(pmb X-bar{x})}$$

• The variance of the projection of $(pmb X-bar{x})$ on the space orthogonal to the first $k$ singular vectors of $(pmb X-bar{x})$:

$$sqrt{(pmb X-bar{x})'(pmb{I_k}-pmb{V_kV_k'})(pmb X-bar{x})}$$

(where $pmb I_k$ is the rank $k$ identity matrix) which is also equivalent to:

$$||pmb X-bar{x}-(pmb X-bar{x})pmb{V_k^{}V_k'}||$$.

In R

n<-100 p<-20 k<-5 x<-matrix(rnorm(n*p),nc=p) #your data matrix  #the orthogonal distances:  data_centered<-sweep(x,2,colMeans(x),FUN="-") loadings<-svd(data_centered/sqrt(nrow(data_centered)-1),nu=0)$v[,1:k] orthDist<-data_centered-data_centered%*%loadings%*%t(loadings) orthDist<-sqrt(rowSums(orthDist*orthDist)) 

you will find a more complete code to compute OD (and SD, the statistical distances on the space spanned by the loading matrix) in the rrcov package:

library(rrcov) rrcov:::.distances 

the function is not documented, but the slot [email protected] therein are the orthogonal distances.