Normal Distribution
nothing is called 95% until it has been counted
Concept

Response-surface — where it appears

A fitted model of how a response varies across a region of factor settings, usually a quadratic in two or more factors. Its stationary point can be a maximum, a minimum or a saddle, and which one it is decides whether the fitted optimum is anywhere worth running.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Where a D-optimal design puts its runs. The D-optimal measure over 121 candidate settings on a square region. It keeps 9 of them and discards the rest, and the 9 it keeps are the settings a catalogue would have offered without any of this arithmetic. What the search adds is the weights: 0.1458, 0.0962, 0.0802, which nine equal runs cannot express.

A design is a number

A standard design is taken from a catalogue and then measured. Turn the arithmetic round and a design becomes the answer to an optimisation — and over 121 candidate settings the search keeps nine of them, which are exactly the nine a catalogue would have offered, at weights nine equal runs cannot express.

optimality · Criterion 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 Where a Ds-optimal design puts its runs. The Ds-optimal measure over 121 candidate settings on a square region. It keeps 9 of them and discards the rest, and the 9 it keeps are the settings a catalogue would have offered without any of this arithmetic. What the search adds is the weights: 0.2500, 0.1250, 0.0625, which nine equal runs cannot express.

The two terms anybody wanted

D-optimality estimates all six parameters of a quadratic as precisely as possible. Nobody wants that. An experimenter looking for a maximum wants the two curvature terms, and the design that gives them is not the D-optimal one — it is a quarter of the runs at the centre, exactly, and the D-optimal design is 75.3% efficient for the question that was actually asked.

criteria · Criterion What the fit calls the shape, against what it is. One eigenvalue held at −3 and the other swept from −2 to 2, so the truth is a maximum on the left and a saddle on the right and the change happens at exactly zero. At an eigenvalue of −0.25 — a genuine maximum — the fit reports a saddle on 26.4% of studies; at +0.25 — a genuine saddle — it reports a maximum on 25.1%. The standard error of a squared coefficient under this design is 0.3791, and the region of confusion is about that wide either side of zero.

The sign the curvature has

A fitted surface reports a maximum, a minimum or a saddle, and the report is a comparison of two estimated eigenvalues against zero. At a true second eigenvalue of −0.25 the fit calls a genuine maximum a saddle on 26.4% of studies, and at +0.25 it calls a genuine saddle a maximum on 25.1%.

surface · Optimum What the main effects cost the rest of the model. The compound optimum of α·log Ds(main effects) + (1 − α)·log D on an 11-by-11 grid of the square, at 14 weights from α = 0 to 0.999, placed by its D-efficiency against the D-optimal design and its main-effect efficiency against the factorial. The D-optimal design (α = 0) gives the main effects 74.3% of what the factorial gives. At α = 0.5 the design keeps 94.6% of D and gives 85.6%; at 0.9, 64.9% and 96.8%; at 0.99, 31.3% and 99.7%. The factorial itself (open, at the left edge) gives the main effects everything and the model nothing: its D-efficiency is zero. The design optimal for the two quadratic terms (red) gives the main effects 50.0% at a D-efficiency of 83.6%.

A subset the model cannot use

Nominate the two main effects of a quadratic response surface as the parameters of interest and the optimal design is the 2² factorial — four corners, a quarter each — which leaves the full model's information matrix at rank four of six and the curvature with nothing at all. That is why the field's algorithm would not converge on it. Approached through a compound criterion instead, the trade is explicit: 96.8% of the main effects' best information costs the model 35% of its D-efficiency, and the last three points cost nine tenths of what is left about the curvature.

criteria · Criterion The ridge, when the fitted optimum is outside the region. One fitted surface. Its stationary point is at a radius of 2.289 and the fit calls the shape a maximum. The ridge is the best setting at each radius, found by the Lagrange condition (B̂ − μI)x = −ĝ/2; the fitted response rises along it from 59.93 at the centre to 62.10 at the edge. The true optimum is at (0.4, 0.3).

When the best setting is outside the region

On a flat surface at twice the noise the fitted optimum lands outside the experimental region on 24.9% of studies and more than three coded units out on 11.8%. The answer is a ridge — the best setting at each radius, with a closed form — and the two obvious rules for using it turn out to be within four per cent of each other.

surface · Optimum What a confirmation run at the chosen setting would find. The true optimum is worth 62.348. At σ = 2 the fit predicts 62.679 at the setting it recommends and the truth there is 61.821 — a gap of 0.858, which is 0.72 of the prediction's own standard error. The setting itself gives up 0.527 against the best available.

The run that confirms it

The setting a response-surface analysis recommends was chosen because the fitted surface was highest there, so the height the fit predicts at it is a maximum over a random field. At twice the noise the fit predicts 0.858 more than is there — 0.72 of the prediction's own standard error — and the gap is not noise, it is the selection.

surface · Optimum The E-optimality certificate along the axis x₂ = 0. The certificate — a fifth of the squared distance of the point's squared radius from one, plus a fifth of the squared difference of its two squared coordinates — along the axis x₂ = 0, with its two pieces drawn alone at full weight: the radial piece, a third of the squared distance from the unit circle in squared radius, which reaches 0.333 here, and the cross piece, half the squared difference of the squared coordinates, which reaches 0.500. The certificate's largest value on the cut is 0.2000, reached at x = -1.0, 0.0, 1.0; the smallest eigenvalue of the nine-point design is 0.2.

A certificate for the worst direction

E-optimality was the one design criterion here whose answer was claimed rather than proved. It has a certificate after all, and on the square it fits on one line — (1⁄5)(1 − x₁² − x₂²)² + (1⁄5)(x₁² − x₂²)², which never exceeds 1⁄5 and touches it at exactly the nine points of the 3² grid. That proves the design with 2⁄5 of its weight at the centre, 1⁄10 at each edge midpoint and 1⁄20 at each corner is E-optimal on the whole square, and it is built from two directions mixed three to two, either of which alone fails.

optimality · Equivalence Seven ways to report the height at the setting a response-surface analysis recommends, at noise 2. Bias and root mean squared error of the reported height about the true height at the chosen setting, over 2,000 studies with a central composite design of thirteen runs, or the same design run twice. 13 runs, the fit's own height: bias 0.884, error 1.537. 13 runs, optimism bootstrapped away: bias 0.183, error 1.396. One confirmation run alone: bias -0.005, error 2.005. Corrected fit and one confirmation run: bias 0.056, error 1.121. 26 runs, the fit's own height: bias 0.475, error 0.998. 26 runs, optimism bootstrapped away: bias 0.050, error 0.924. 26 runs split: choose on 13, estimate on 13: bias -0.014, error 1.213.

The height at the chosen setting

The height a response-surface fit predicts at the setting it recommends reads high, because the setting was chosen where the fit was highest. The obvious repair — choose on some runs, estimate on the rest — is impossible in the usual thirteen-run design, which has nine distinct settings where two separate fits need twelve. A parametric bootstrap of the optimism removes four fifths of it at no cost in runs, 0.884 down to 0.183 at twice the noise. With the design run twice, correcting the full fit beats splitting it: an error of 0.924 against 1.213, at a better setting.

surface · Optimum

Named alongside it

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

Experimental designCentral composite designD-optimalityInformation matrixOptimal designPrediction varianceCurvatureDesign measureEquivalence theoremStationary pointEigenvalueCanonical analysis

All concepts