Euclidean distance.

d(x,y)=i=1n(xiyi)2

An alternative is correlation-based distance which considers two observations to be similar if their features are highly correlated

d(x,y)=2m[1r(x,y)]

This dissimilarity measure is equivalent to the Euclidean distance when the data are standardized to have mean 0 and standard deviation 1 .

Analytically show the equivalence of these two dissimilarity measures.