Weight candidate predictions by cross-validated risk, on the simplex

Z must hold CROSS-VALIDATED predictions; in-sample predictions make super learner pick the most overfit candidate and nothing in the arithmetic can detect it. Weights are restricted to the simplex. Optimisation is Frank-Wolfe with a FIXED step schedule 2/(t + 2).

Usage

Superlrn(Z, Y, iters = 500)

Arguments

Z

Cross-validated predictions, one column per candidate.

Y

Outcome.

iters

Frank-Wolfe steps (fixed budget).

Value

List with weights, risk, sl_risk, discrete_risk, discrete_index, fitted, beats_discrete, n, K.

Details

Formula: Psi_SL(W) = sum_k alpha_k Psi_k(W), alphahat = argmin over the simplex of sum_i (Y_i - (Z alpha)_i)^2

References

van der Laan, Polley & Hubbard (2007), Super Learner, UC Berkeley Division of Biostatistics Working Paper 222; the construction as given in Polley's dissertation, Super Learner, UC Berkeley (escholarship qt4qn0067v).

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Superlrn(V, V)
#> $weights
#> [1] 1
#> 
#> $risk
#> [1] 0
#> 
#> $sl_risk
#> [1] 0
#> 
#> $discrete_risk
#> [1] 0
#> 
#> $discrete_index
#> [1] 1
#> 
#> $fitted
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $beats_discrete
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $K
#> [1] 1
#> 
#> $method
#> [1] "Super learner: simplex-constrained ensemble of candidates"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"