Hi @yuqing,

Welcome to the forum!

It has to do with the way weight is calculated. In unweighted UniFrac, we calculate distance as \frac{\cap \textrm{ branches}}{\cup \textrm{ branches}} (Shared history / total history). So, this value is always between 0 and 1.

Weighted UniFrac is the sum of the the branch length by the absolute difference in count fractions \sum_{i}^{n}{b_{i} \mid \frac{A_{i}}{A_{T}} - \frac {B_{i}}{B_{T}} \mid } where b_{i} is the branch length, A_{i} is the total number of sequences in sample A along the length of the branch, A_{T} is the total number of sequences in sample A, B_{i} are the sequences in B along the branch and B_{T} the total sequences in B. But, because of this scaling, the distance can easily exceed one. There's also a rescaling factor that can be used to adjust for the fact that a longer branch length may correlate to more difference... I think it's probably easier for you to read the original paper than for me to try and dance around the math.

I do want to note that there's no requirements distances be between 0 and 1, though. I'm about to go to work, which is a distance of about 8 metro stops. I could theoretically rescale that as the fraction of the subway if that was somehow more meaningful here, but I can also just represent it as 8 subway stops. The issue is the relative scale, not the quantity. The one thing I *can't* do is compare my distance of 8 subway stops with someone else who's scaled their metro distance, because then it's not the same thing!

Best,

Justine