Compute AUC without needing specific thresholds

[ericphanson]

Currently, Lighthouse asks for a set of input thresholds (e.g. 0.0:0.01:1.0), then computes binary stats for each threshold, then forms corresponding fpr/tpr curves, then computes the area underneath those curves w/ the trapezoid rule.

I came across An Improved Model Selection Heuristic for AUC today which among other things, pointed out that you can compute the AUC without reference to specific thresholds by sorting the soft-labels of all instances, and then doing a particular kind of count:

I think this is a nice algorithm we should consider using, so that the accuracy of the AUC estimate is not influenced by the granularity of the thresholds chosen.

[ericphanson]

Some (minimally-tested) code to compute the ROC AUC in this way (plain) along with the "scored" ROC AUC they discuss in that paper:

"""
     ROC_AUC(positive_scores, negative_scores)

- positive_scores: predictions for the "positive" class in binary classification (e.g. class 2)
- negative_scores: predictions for the "negative" class in binary classification (e.g. class 1)

"""
function ROC_AUC(positive_scores, negative_scores)
    m = length(positive_scores)
    n = length(negative_scores)

    T = promote_type(eltype(positive_scores), eltype(negative_scores))
    table = @NamedTuple{score::T, label::String}[]
    sizehint!(table, m+n)

    append!(table, ((; score, label="positive") for score in positive_scores))
    append!(table, ((; score, label="negative") for score in negative_scores))
    sort!(table, by = x -> x.score, rev=true)

    scored_AOC = zero(T)
    scored_AUC = zero(T)
    scored_r = zero(T)
    scored_c = zero(T)
    c = 0
    AUC = zero(T)
    for row in table
        if row.label == "positive"
            scored_c += row.score
            scored_AOC += scored_r
            c += 1
        else
            scored_r += row.score
            scored_AUC += scored_c
            AUC += c
        end
    end
    R⁻ = (m*scored_r - scored_AOC)/(m*n)
    R⁺ = scored_AUC/(m*n)

    scored = R⁺ - R⁻ 
    plain = AUC / (m*n)
    return (; scored, plain)
end

# Example 1 from the paper - agrees!
using Test

M1 = ROC_AUC([1.0, 0.7, 0.6],  [0.5, 0.4, 0.0])
M2 = ROC_AUC([1.0, 0.9, 0.5],  [0.6, 0.2, 0.0])

@test M1 == (scored = 0.46666666666666656, plain = 1.0)
@test M2 == (scored = 0.5444444444444443, plain = 0.8888888888888888)