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Recently I have started looking for the definition of normalized Euclidean distance between two real vectors u and v. So far, I have discovered two apparently unrelated definitions:

http://en.wikipedia.org/wiki/Mahalanobis_distance

and

http://reference.wolfram.com/language/ref/NormalizedSquaredEuclideanDistance.html

I am familiar with the context of the Wikipedia definition. However, I am yet to discover any context for the Wolfram.com definition:

NormalizedSquaredEuclideanDistance[u,v] is equivalent to 1/2*Norm[(u-Mean[u])-(v-Mean[v])]^2/(Norm[u-Mean[u]]^2+Norm[v-Mean[v]]^2)

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I simplified the above expression to the following:

NormalizedSquaredEuclideanDistance[u,v] = 0.5*Var(u - v)/[Var(u) + Var(v)]

(where Var(x) denotes the variance of x)

The intuitive meaning of this definition is not very clear. Any help on this will be appreciated.

Thanks and regards,

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The normalized squared euclidean distance gives the squared distance between two vectors where there lengths have been scaled to have unit norm. This is helpful when the direction of the vector is meaningful but the magnitude is not. It's not related to Mahalanobis distance.

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  • $\begingroup$ Thanks a lot, Aaron. Please search the string "normalized Euclidean distance" in the Wikipedia page en.wikipedia.org/wiki/Mahalanobis_distance and let me know if the definition given there is wrong. $\endgroup$ – PTDS Feb 4 '15 at 7:47
  • $\begingroup$ I think one desirable property of NED is that it should always lie between 0 and 1. The Wolfram.com definition has this property. The Wikipedia definition, squared and divided by N [also replace the SD s_i in the denominator by the RANGE of the i-th component of the vector], also has this property. If I am not mistaken, your definition does not have this property, right? $\endgroup$ – PTDS Feb 5 '15 at 20:56

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