Warning: super-technical post ahead, but I have made this point in oral discussions at least a few times, so I thought I would write it up. It is a trivial algebraic manipulation, but because “ℓ₁ norm” sounds too mathy for ecologists while “Bray-curtis” sounds too ecological and ad-hoc for mathematically minded people, it’s good to see that it’s the same thing on normalized data.

Assuming you have two feature vectors, Xᵢ, Xⱼ, if they have been normalized to sum to 1, then the Bray-Curtis dissimilarity is just their ℓ₁ distance, aka Manhattan distance (times ½, which is a natural normalization so that the result is between zero and one).

Very useful. There’s a sign error in the last two derivation step, I think. Should be a minus sign between the terms of the absolute values…

You are right! fixed.

Hi Luis, Very interesting post! Do you have a useful reference for this?

No, not really.