Bray-Curtis dissimilarity on relative abundance data is the Manhattan distance (aka L₁ distance)

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).

This is the Wikipedia definition of the Bray-Curtis dissimiliarity (there are a few other, equivalent, definitions around, but we’ll use this one):

BC = 1 – 2 Cᵢⱼ/(Sᵢ + Sⱼ), where Cᵢⱼ =Σₖmin(Xᵢₖ, Xⱼₖ)  and Sᵢ = ΣₖXᵢₖ.

While the Manhattan distance is D₁ = Σₖ|Xᵢₖ – Xⱼₖ|

We are assuming that they sum to 1, so Sᵢ=Sⱼ=1. Thus,

BC = 1 – Σₖmin(Xᵢₖ, Xⱼₖ)

Now, this still does not look like the Manhattan distance (D₁, above). But for any a and b, it holds that

min(a,b) = (a + b)/2 – |a – b|/2

(this is easiest to see graphically:  start at the midpoint, (a+b)/2 and subtract half of the difference, |a-b|/2).

Thus, BC = 1 -Σₖmin(Xᵢₖ, Xⱼₖ) = 1 – Σₖ{(Xᵢₖ + Xⱼₖ)/2 – |Xᵢₖ – Xⱼₖ|/2}

Because, we assumed that the vectors were normalized,  Σₖ(Xᵢₖ + Xⱼₖ)/2 =(ΣₖXᵢₖ +ΣₖXⱼₖ)/2 = 1, so

BC = 1 – 1 + Σₖ|Xᵢₖ – Xⱼₖ|/2 = D₁/2.

6 thoughts on “Bray-Curtis dissimilarity on relative abundance data is the Manhattan distance (aka L₁ distance)

  1. 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…

  2. Hi, thanks for your post.

    I have a matrix where the rows are sampleIDs and columns are taxonomic abundances. The rows sum to 100. If I understand your post correctly, to get Bray Curtis, I need to divide each number by 100, so the rows sum to 1.

    Then each row represents a vector as your Xs.

    The I need to calculate the Manhatten distance between all vectors (rows) I have and the multiply the resulting distance metric by 0.5. The I have the BC distance, is that correct?

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