How to relate isometric log ratio [ILR] transformed compositional data to original attributes?

I am having trouble grasping the concept of isometric log ratio transformation and how to interpret the resulting transformation. To my understanding, the ILR represents log ratios between left and right subtrees going down the dendrogram of the basis? Since the input (n,m) matrix is projected into a (n,m-1) space there doesn’t seem to be a direct way to associate the transformed data with the original attributes? I am implemnting it via and there an inverse ilr function that projects the data back into the relative abundance but I don’t understand how to interpret the transformed data. I have a few questions:

(1) Know which columns represent which attributes or nodes?
(2) Why exactly do you end up with m-1 attributes?
(3) If you had X which was (n=sample rows,m= attribute columns) where the data was converted to relative abundance so each sample sums to 1, could you calculate the ILR with respect to axis=0 if you wanted to create a pairwise correlation matrix of the attributes? shape = (m,m)
(4) Is there a way to directly relate the transformed data with specific attributes in the original?

  1. Have you been able to run the gneiss tutorial?

There are also a few similar posts on how to interpret balances, which may help with figuring out how to interpret these results

  1. You end up with m-1 attributes since we are dealing with proportions, which live in a m-1 dimensional space. The ilr is a direct mapping from the m-simplex to m-1 real space.

Another way to think of it is the ilr transform computes the transformation between the proportions in the tips and log ratios in the ancestral states in the tree. If there are m tips, there are exactly m-1 internal nodes.

  1. No. The ilr tranform will not give pairwise correlations, that is a very different discussion. I would look at tools such as SPEIC-EASI to answer those sorts of questions

  2. If I am understanding this question correctly, the answer is yes.
    Each balance just represents two groups of features (in the context of phylogenetics, you can think of a balance as an ancestral state in the phylogenetic tree.)


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