Normalized Euclidean Distance versus cross correlation?

Question

Normalized Euclidean Distance and Normalized Cross - Correlation can both be used as a metric of distance between vectors.

What is the difference between these metrics? It seems to me that they are the same, although I have not seen this explicitly stated in any textbook or literature.

thank you.

Answer 1

These two metrics are not the same.

The normalized Euclidean distance is the distance between two normalized vectors that have been normalized to length one. If the vectors are identical then the distance is 0, if the vectors point in opposite directions the distance is 2, and if the vectors are orthogonal (perpendicular) the distance is sqrt(2). It is a positive definite scalar value between 0 and 2.

The normalized cross-correlation is the dot product between the two normalized vectors. If the vectors are identical, then the correlation is 1, if the vectors point in opposite directions the correlation is -1, and if the vectors are orthogonal (perpendicular) the correlation is 0. It is a scalar value between -1 and 1.

This all comes with the understanding that in time-series analysis the cross-correlation is a measure of similarity of two series as a function of the lag of one relative to the other.