Proper use of scaling on axes from ordinations (like MDS and PCA)?

[kevbonham]

I wonder - could this be related to the eigenvalue ratio, as suggested in the F1000 paper?

If it is, it’s not straightforward. Using my real life example, I took some subsets of the data and calculated the aspect ratio with the highest correlation (maxc), what that correlation was (maxcor) and the ratio of eigenvalues. Here’s what I found:

function maxcor(mds, dm; dim1=1, dim2=2, rng = 0.1:0.01:3)
    proj = projection(mds)
    x, y = (proj[:,i] for i in [dim1, dim2])

    e1, e2 = eigvals(mds)[[dim1, dim2]]
    
    res = [getcor(i, dm, x, y) for i in rng]
    (mc, i) = findmax(res)
    return (maxc = rng[i], maxcor=mc, eigratio=e2/e1)
end


function ploteigs(dm; dims=50, points=200)
    c = Float64[]
    er = Float64[]
    cor = Float64[]
    
    for _ in 1:points
        picks = rand(1:size(dm, 1), dims)
        subdm = dm[picks, picks]
        mds = fit(MDS, subdm, distances=true)

        for i in 1:5
            mc = maxcor(mds, subdm, dim1=1, dim2=i)
            push!(c, mc[:maxc])
            push!(er, mc[:eigratio])
            push!(cor, mc[:maxcor])
        end
    end
    
    println(length(c))
    scatter(c, er, zcolor=cor)
end

ploteigs(dm)