An exact 13-run design, found rather than looked up

The exact 13-run design a point-exchange search returns, over the same 121 candidate settings. It puts 9 distinct settings on the page and repeats some of them, which is what an integer number of runs can do in place of a weight. Its D-efficiency against the optimal measure is 99.77%.

The largest value over the square of a mixture of the two pieces — weight t on the radial piece and the rest on the cross piece — on a 201 × 201 grid, against t. At t = 0 it is 0.500, at the edge midpoints; at t = 1 it is 0.333, at the centre and the corners; it falls to exactly 0.2 at t = 0.6 and nowhere else — 0.250 at t = 0.5 and 0.233 at t = 0.7.

Exact weights: 0.20000, 0.20000, 0.20000, 0.40000, 0.40000 and 1.40000 — the minimum, 1⁄5, attained three times. The subgradient search's weights: 0.19999, 0.19999, 0.20214, 0.40180, 0.40242 and 1.40423 — two at the minimum and the third, the interaction's, at 0.20214, because its corners carry a little more than 1⁄20 each.

The gap between 1⁄5 and the best smallest eigenvalue the subgradient search has found, at 10, 30, 100, 300, 1,000, 3,000, 10,000, 40,000 iterations: 5.28e-2, 1.93e-2, 8.88e-3, 3.11e-3, 6.94e-4, 1.57e-4, 5.14e-5, 1.34e-5. At forty thousand the search is within 0.0067% of the optimum. The derivative-based search stops at 0.0744, 37.2% of it.

For each design, its smallest eigenvalue divided by the certified optimum of 1⁄5, and the lower bound that comes from the design alone — its smallest eigenvalue over the largest value of the certificate built from its own minimum eigenvector. D-optimal measure: efficiency 49.6%, bound 17.2%; A-optimal measure: efficiency 82.6%, bound 34.1%; I-optimal measure: efficiency 87.7%, bound 36.7%; the search's E design: efficiency 100.0%, bound 40.0%; 3² factorial, equal weights: efficiency 55.6%, bound 21.0%; face-centred composite, 5 centres: efficiency 76.9%, bound 30.8%; rotatable composite, scaled in: efficiency 38.5%, bound 7.7%.

The exact 13-run design a point-exchange search returns, over the same 121 candidate settings. It puts 9 distinct settings on the page and repeats some of them, which is what an integer number of runs can do in place of a weight. Its D-efficiency against the optimal measure is 99.77%.

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.

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.

The D-efficiency of the best N-run design at each size, against the optimal measure. It is not a rising curve. 13 runs reaches 99.77% and 14 falls to 99.44%, because the optimal weights are real numbers and N runs is an integer approximation to them, so how good a design can be depends on how well N divides. A Wald interval behaves the same way: a larger sample sometimes makes its coverage worse, for exactly this reason.

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.

D-optimal measure: 9 distinct settings, 3 degrees of freedom for lack of fit; A-optimal measure: 9 distinct settings, 3 degrees of freedom for lack of fit; I-optimal measure: 9 distinct settings, 3 degrees of freedom for lack of fit; 3² factorial: 9 distinct settings, 3 degrees of freedom for lack of fit; face-centred composite: 9 distinct settings, 3 degrees of freedom for lack of fit; rotatable composite: 9 distinct settings, 3 degrees of freedom for lack of fit; factorial with centres: 5 distinct settings, -1 degrees of freedom for lack of fit. Carathéodory's bound allows anything from 6 to 21 and every design that can fit the model lands on the same number.

a 2² factorial has already been run and 4 further runs are to be placed. The stationarity condition is no longer max d = p; it is max d = (p − λ·tr(M⁻¹M_fixed))/(1 − λ) with λ = 0.5000 the share of runs already spent, which is 6.0000 here. The search reaches 6.000000000 against it, and the largest value anywhere on a 41×41 grid is 6.000000000.

a 2² factorial has been run. Placing further runs at the optimum computed as though it had not is 84.74% efficient at 2 added, 91.97% efficient at 4 added, 96.75% efficient at 8 added, 98.89% efficient at 16 added. Repeating the design already run is not a design at all — it visits too few distinct settings to fit the model, at any number of repetitions.

the D-optimal design's smallest eigenvalue is 0.09927, attained once; the A-optimal design's smallest eigenvalue is 0.16516, attained once; the I-optimal design's smallest eigenvalue is 0.17541, attained once; the E-optimal design's smallest eigenvalue is 0.19999, attained 2 times. A criterion that reads the smallest eigenvalue has no derivative where that eigenvalue is repeated, and the E-optimal design is exactly there.

Maximising the smallest eigenvalue of the information matrix. A subgradient ascent reaches 0.199987 with 2 eigenvalues at the minimum. A multiplicative update built on a derivative — the algorithm this field's other three criteria use — reaches 0.074421, which is 37.2% of it. The D-optimal design, which was not chosen for this, reaches 0.099268.