Large metacommunities

This page is about measuring diversity over data that is large enough for the shape of the calculation to matter - hundreds of thousands of subcommunities, thousands of types, or both. A gridded landscape is the usual reason: a 1 km grid over a country is a few hundred thousand cells, and a species list for it can run to thousands.

Nothing here is a different API. It is the same diversity, subdiv and metadiv as everywhere else, used in the order that keeps the intermediate results small.

What is actually big

Two things scale with the size of your data, and everything else is small beside them.

Your abundance matrix, which is types × subcommunities. That is yours, and it has to exist.

The ordinariness, Z * p, which is the same shape. When there is no similarity between types - UniqueTypes, the default - this is free, because the ordinariness is the abundances and the package does not copy them. With a similarity matrix it is a second array of the same size, computed once and cached on the metacommunity.

Everything else is proportional to the answer rather than to the data. The individual diversities of a measure are computed on demand rather than stored, so building a measure over a metacommunity you have already measured allocates nothing at all; a subcommunity-level result is one row per subcommunity, at about 88 bytes a row.

So for nt types and np subcommunities, in double precision:

	bytes
abundances	`8 nt np`
ordinariness, without similarity	0
ordinariness, with similarity	`8 nt np`
the similarity matrix itself	`8 nt^2`
a `subdiv` result, per measure per order	`88 np`
an `inddiv` result, per measure per order	`88 nt np`

A worked case: 2000 species on a 1 km grid over a box containing the UK is roughly 10^6 cells, so 16 GB of abundances, or 32 GB with a 2000 × 2000 similarity matrix - while a subcommunity-level result for one measure at one order is about 88 MB.

`inddiv` is the one that does not scale

inddiv returns a row per type per subcommunity, not per subcommunity. At the size above that is 2 * 10^9 rows, which is not a table anyone wants. subdiv and metadiv are the ones that scale; reach for inddiv on a subset, or on data small enough to look at.

Build the metacommunity once

A Metacommunity caches the three things every measure needs - the ordinariness, the subcommunity weights and the metacommunity ordinariness - on first use. Each is a full pass over the abundance matrix, so the first measure pays for them and every later one does not.

The corollary is that a metacommunity is not a live view of your data. It computes from what it was given and never invalidates, so build it when the abundances are final, and build a fresh one rather than editing an existing one.

Ask for everything at once

diversity takes a collection of levels, a collection of measures and one or more orders, and returns them in a single table:

julia> using Diversity, Diversity.ShortNames
julia> pop = rand(20, 500); pop ./= sum(pop);
julia> mc = Metacommunity(pop)Metacommunity{Float64, Matrix{Float64}, Matrix{Float64}, UniqueTypes, Subcommunities} with 20 species in 500 subcommunities measuring Unique diversity.

Species names:
1, 2, 3...19, 20

Subcommunity names:
1, 2, 3...499, 500
julia> levels = [subcommunityDiversity, metacommunityDiversity];
julia> result = diversity(levels, [ᾱ, ρ̄, Γ], mc, [0, 1, 2]);
julia> size(result)(4509, 8)

The arithmetic is the same as calling the wrappers one at a time - the power means still have to be taken. What changes is how much is built on the way: one table rather than six that you then have to join. Measured over 200 types and 200,000 subcommunities, for exactly that call:

	time	memory
`diversity(levels, [ᾱ, ρ̄, Γ], mc, [0, 1, 2])`	2035 ms	165 MiB
the six wrapper calls, then `vcat`	2133 ms	275 MiB

Do not build a table you are only going to write out

For a run whose output is destined for a file, a DataFrame is an expensive intermediate. Every result function takes an optional first argument naming what to return, so the columns can be handed to a writer without one:

using CSV, Tables
CSV.write("diversity.csv",
          subdiv(Tables.columntable, NormalisedAlpha(mc), [0, 1, 2]))

This is worth doing at scale because most of a result is repetition. Of the eight columns, five hold one value repeated for every row and two cycle a short list of names; only diversity is data. They are held as rules rather than arrays, and a column table keeps them that way, so CSV.write reads each column once and never materialises it. Asking for a DataFrame - the default - materialises them deliberately, so that what you get back is an ordinary mutable table.

Which sinks get their own treatment

The first argument is resolved through Tables.materializer, and only types that define a method for it - DataFrame among them - are constructed directly. Anything else falls back to Tables.columntable, which is a perfectly good answer and is what you want here, but it does mean naming a writer rather than a table type will not do what it looks like it does. Write the file with CSV.write or Arrow.write and pass them a column table, as above.

Empty subcommunities are nearly free

A subcommunity with no individuals in it - a sea cell in a species grid, a cell excluded from a landscape simulation - has no defined diversity, and its rows come back as NaN:

julia> sparse = zeros(4, 5);
julia> sparse[:, [2, 4]] .= [1 4; 2 3; 3 2; 4 1];
julia> sparse ./= sum(sparse);
julia> subdiv(NormalisedAlpha(Metacommunity(sparse)), 1).diversity5-element Vector{Float64}:
 NaN
   3.5961154666243225
 NaN
   3.5961154666243225
 NaN

Such subcommunities are recognised from the weights the metacommunity has already computed rather than by inspecting their abundances, so they cost almost nothing: over 100,000 subcommunities of which 27% were occupied, they accounted for 16% of the run before this was so, and now they account for essentially none of it.

They also do not affect anything else. Removing them entirely gives identical answers for every remaining subcommunity and for the metacommunity as a whole, at every order and with or without similarity - abundances are relative to the whole metacommunity, so zeroes do not move the total; a subcommunity's measures depend only on its own composition and on the metacommunity; and a power mean ignores zero-weight entries. So you may keep a fixed grid and let cells fall in and out of use without the measures noticing, or drop the empty cells and get the same numbers over a smaller matrix - whichever suits the data you have.