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I just need a very short summary of what the standard way to deal with compositional data is.

I've skimmed pages in a 500-page long book on the topic, and I didn't really gather much. I would like to know the general idea before I delve into the details.

By compositional data, I mean response variables $p_1, ..., p_k$ with $\sum p_i = \mathcal{S}$ for some fixed value $\mathcal{S}$.

So far, I gather that we consider the "simplex" of ALL such response values. This is a $k-1$ dimensional space. On this, it seems a geometry can be defined with some nice properties. We also consider transformations that take a value in the simplex, and returns some coefficients.

And then what? What's the point of those coefficients? Is that choice arbitrary? What will we do with those coefficients to help us analyze the data?

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Lecture Notes on Compositional Data Analysis These lecture notes by Pawlowsky-Glahn, Egozcue, and Tolosana-Delgado are excellent. There are a few errors in them but it is a much shorter (≈80 pages) read than the full textbook that came out of these notes.

Even shorter: Compositional Data Analysis in a Nutshell (2 pages).

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