An invariant from category theory solves a problem in mathematical ecology [pdf]
maths.ed.ac.uk
maths.ed.ac.uk
"The starting point is the connection between diversity and entropy. We will discover:
• how Shannon entropy, originally defined for communications engineering, can also be understood through biological diversity (Chapter 2);
• how deformations of Shannon entropy express a spectrum of viewpoints on the meaning of biodiversity (Chapter 4);
• how these deformations provably provide the only reasonable abundance-based measures of diversity (Chapter 7);
• how to derive such results from characterization theorems for the power means, of which we prove several, some new (Chapters 5 and 9).
Complementing the classical techniques of these proofs is a large-scale categorical programme, which has produced both new mathematics and new measures of diversity now used in scientific applications. For example, we will find: [...]"
"The question of how to quantify diversity is far more mathematically profound than is generally appreciated. This book makes the case that the theory of diversity measurement is fertile soil for new mathematics, just as much as the neighbouring but far more thoroughly worked field of information theory"
I'm not too sure where the category theoretical stuff enters though. They mention that metric spaces have a magnitude, but their end result looks more like a channel capacity (with the confusion matrix being the probability to confuse one species with another). Which, you know, makes sense, if you've got 'N' signals but they're so easily confused with one another that you can only send 'n' signals worth of data then your channels are not too diverse.
They do mention that this is equivalent to some modified version of the category theoretical heuristic, but is that really interesting? The link to Euler characteristic is intriguing, but from the way they end up at their final definition I'm not sure if metric spaces are really the natural context to talk about these things. It almost feels like they've stepped over an enriched category that would provide a more natural fit.
So, I guess, (\R_{>= 0}, >=, +, 0) is like, a monoidal category with + as the monoidal operation?
So like, for x,y,z in the metric space, the
well, from hom(x,y) and hom(y,z) I guess the idea is there is a designated composition morphism
from hom(x,y) monoidalProduct hom(y,z) to hom(x,z)
which is specifically,
hom(x,y)+hom(y,z) >= hom(x,z)
(I said designated, but there is only the one, which is just the fact above.)
I.e. d(x,y)+d(y,z) >= d(x,z)
(Note: I didn’t manage to “just guess” this. I’ve seen it before, and was thinking it through as part of remembering how the idea worked. I am commenting this to both check my understanding in case I’m wrong, and to (assuming I’m remembering the idea correctly) provide an elaboration on what you said for anyone who might want more detail.)
This works, but it's not quite what you want in most cases. There's a lot of stuff that requires you to enrich over a closed category, so instead we define `Hom(a,b)` to be `max(b - a, 0)` (which you can very roughly think of as replacing the mere proposition `a < b` with its "witnesses"). See https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf for more.
At least what they seem to be doing has little to do with metrics, and a lot more to do with probability distributions.
-edit- by the way, since it may not be obvious to everyone, the reason why an ecologist might care bout evenness is because extremely rare species are often not very important to the wider community. From an ecological function perspective, there is very little difference between my above example of the 99%/1% community and a community that is 100% species A. So an community with two, equally populous species might have more functional diversity than a community with one very abundant species and several more, very rare species.
(Less emphasis of the category theory, and more attention to the basic math behind entropy-like diversity measures)
(TLDR) There are two important aspects:
1. Generalize diversity measures to when the categories are not fully distinct (as assumed for the Shannon entropy calculation) but have similarities parametrized in a "Z"-matrix here.
2. A parameter "q" to represent whether you value a category highly (for diversity purposes) even when it has only a single distinct example highly (q --> 0) or whether you value it highly only when it has many many examples (q --> infinity)
* With Z = identity matrix (categories completely distinct) they show how different values of q reproduce different measures that have been considered before. (nicely summarized in a table)
* The generalization (parameterized by Z) when the categories can/do have some overlap is very elegant, and feels like an important step forward. (especially how it makes the measure robust to how we partition a bunch of examples into distinct categories, so long as we keep track of the similarity between the categories)
* The paper also summarizes a bunch of sensible properties that we would want any diversity measure to satisfy (like the one I just mentioned above).
Fun stuff!
The presentation gives us a whole range of views on biodiversity, making it clear that we need to think about both common and rare species when we're figuring out diversity. It hints that this magnitude idea might give us a more detailed picture of diversity by considering how important different species are relative to each other. Then the presentation gets into the weeds of category theory, chatting about enriched categories and size-like invariants. It talks about stuff like monoidal categories, V-enriched categories, and linear categories. These ideas are brought up as tools to help us understand and calculate magnitude.
The presentation also explores how these category theory ideas relate to metric spaces. It suggests that getting a handle on this relationship could give us even more insight into how to max out diversity. Lastly, the presentation brings up the Euler characteristic, which is a concept from algebraic topology. It hints that the Euler characteristic and magnitude are pretty tight, and understanding this connection could give us even more insight into how to max out diversity.
So, to wrap it up, the presentation is a thorough look at how category theory ideas, especially magnitude, can be used to tackle problems in mathematical ecology. It suggests that these ideas can give us a more detailed understanding of diversity and might even help us figure out how to get the most diversity.
References:
My personal approach to magnitude is not based on category theory but rather based on weightings of a metric space. If your metric satisfies certain properties, you can obtain a measure of the 'effective number of points' of a metric space. This is particularly relevant when looking at the metric space from different scales---zooming in gives you a lot of disconnected points, while zooming out gives you clusters. Magnitude then captures the changes in the number of points in a principled manner.
Some questions and thoughts.
It seems that the result could vary based on how you construct the similarity matrix Z, e.g. is it purely taxonomic? or does it try to account for the ecological roles that a species is playing in the community, etc.
A seeming limitation is that the optimization works only for a fixed set of n species. While it is useful for managing existing communities, it means that there is still a question of whether larger n is strictly better, and leaves open questions of how to deal with transient or migratory members (if the community is spatially bound).
The answer I think, is that it depends on how the similarity matrix is constructed. If every species is fully dissimilar then increasing n is always a good thing. If you use niche space to construct it and new species do not some add or enter new niches so they overlap with others, then they will be close to another species in the matrix and increasing n will not have much impact. On the other hand if you use a purely taxonomic approach then you wind up balancing the number of birds and mammals regardless of niche.
It is not clear to me whether it is possible to construct a similarity matrix that can account for the interaction between n, the carrying capacity of the ecosystem, and the number of available niches (or the ability of species to create new niches). By analogy if you have a stream (sunlight) powering water wheels, how many wheels and how many levels of gears (layers in the ecosystem) can be added, created, and/or sustained? At what point does adding an additional species mean that either two species are forced to be close together in the similarity matrix or both their populations must shrink in size because they must compete for the same energy sources?
Does the model sometimes produce impractical results, e.g. that it is good to have a single member of a sexually reproducing species (this is probably an orthogonal concern and you would want to scale to real population sizes such that the minimum corresponded to the smallest viable a self sustaining population)?
Is there evidence that maximizing diversity using this measure actually produces more robust and stable ecologies?
Let's take the mammoth - it's basically a hairy elephant and the genetic material that could make one is 99+% present in Asian elephants ( possibly 100%, just not all in the same individual ) today.
A bit of selective breeding and you could probably produce one quite quickly [1]
ie the true measure of diversity is the pool of genetic diversity out there - rather than some arbitrary classifications.
[1] Look at the variety in shape and form of domestic dogs - they all came from the same stock a mere 20-40K years ago.
http://www.loujost.com/Statistics%20and%20Physics/Diversity%...
The first explains the problem of measuring ecological diversity, and the problem with old, traditional measures like Shannon entropy or Gini-Simpson index. The second introduces the species similarity matrix Z and the viewpoint parameter q.
https://www.maths.ed.ac.uk/~tl/riken/
Now you know how to measure the diversity of a given community.
But if you are given the number of species and their similarity matrix Z, and the viewpoint parameter q, if you get to design a community (decide the relative abundance of each species), how would you maximize the diversity? Which distribution of relative abundances will maximize the measured diversity? The posted talk gives the answer, and the surprising result that the answer does not depend on q.