Concept

G optimality — where it appears

The design criterion that minimises the largest prediction variance anywhere in the region the experiment covers. It is equivalent to D-optimality by the general equivalence theorem, which is one of the few places two criteria written for different purposes coincide exactly.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The variance touches p and never crosses it. d(x) = f(x)′M⁻¹f(x) along the diagonal of a square region, for the D-optimal measure. The line at 6 is the number of parameters in the model. Kiefer and Wolfowitz's theorem says a design is D-optimal exactly when the largest d anywhere in the region is p — not approximately, equals — so the optimal curve is tangent to that line at its support points and below it everywhere else. Here the largest value anywhere on a 41×41 grid is 6.000000000.

The theorem that says when to stop

A search that maximises the volume of the information has no way of knowing it has finished, because nothing tells it what the maximum is. Kiefer and Wolfowitz's equality does — a design is D-optimal exactly when the worst prediction anywhere in the region equals the number of parameters, which is 6.000000000059 here, gated at machine precision.

optimality · Equivalence

Four criteria on 5 designs, on a square region. Each design scored as an efficiency — its value over the best attainable — so four criteria in as many different units sit on one scale where 1 is the optimum. Rows are ordered by D. D picks 13-run exchange; A picks face-centred composite; G picks 13-run exchange; I picks face-centred composite. Every design has been scaled to just fit the region first, because a design run at settings the region does not contain is not a competitor on it. The disagreement is the point: the letter is a choice, and it is almost never reported as one.

Four letters and two camps

D, A, G and I are four ways of turning one matrix into one number, and they do not agree. The design that wins on D is the worst thing here on I. And the same two designs swap places entirely when the region changes from a square to a disc — on all four criteria at once.

optimality · Criterion

What visiting fewer settings costs, 6 parameters. Carathéodory's bound puts the support of an optimal measure between 6 and 21. 6 settings: D-efficiency 88.90%, G-efficiency 57.18%, 0 degrees of freedom for lack of fit; 7 settings: D-efficiency 94.54%, G-efficiency 61.22%, 1 degrees of freedom for lack of fit; 8 settings: D-efficiency 95.99%, G-efficiency 64.60%, 2 degrees of freedom for lack of fit; 9 settings: D-efficiency 97.40%, G-efficiency 82.76%, 3 degrees of freedom for lack of fit. The saturated design has none, and buying the first one costs about five points of efficiency to get back.

How many places a design goes

Carathéodory's bound puts an optimal design's support between six and twenty-one settings, and every design in this field that can fit the model visits exactly nine. The count is not a choice anybody makes, it decides how many degrees of freedom are left to check the model with, and the first spare setting costs six points of efficiency to get back.

optimality · Equivalence

Named alongside it

The objects these essays reach for when they reach for this one.

D-optimality Experimental design Information matrix Optimal design Prediction variance Central composite design Design measure A-optimality Degrees of freedom Equivalence theorem Lack-of-fit Response-surface

All concepts