Optimum — the series · Normal Distribution

Twenty walks up the same hill, σ = 2. Each walk fits a plane to the same four-corner factorial, takes its gradient as a direction, and steps along it until a run comes in below the one before. The true optimum is the cross. 80% of the walks stop before the best point on their own path — not because the direction was wrong, but because one noisy run is enough to stop them, and the direction error costs only 3.9% of the available gain.

Walking up the gradient

The fitted gradient is wrong by an angle with a closed form, σ/(|β|√N), and what that angle costs is its squared cosine — twelve per cent at twenty degrees. What costs a third of the gain is not the direction at all. It is deciding where to stop.

part 1 · surface

Where the maximum is, from 15 runs. One dataset, one fitted quadratic, and two answers to "where is the best setting". The delta method reports 0.80 ± 0.46, a finite interval it will report whatever the data does. Fieller's set is 0.49 to 1.76, because the curvature here has t = -4.04. The true optimum is at 0.75.

The optimum is a ratio, and its interval is sometimes the whole line

The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.

part 2 · surface

One curve is a binomial coefficient and the other is a line. The number of subsets a maximin over this dictionary would have to score, against the number the exchange algorithm actually scores. At three functions the walk is 2,024 subsets and is the honest answer; at eight it is 735,471 and the exchange algorithm has looked at 421. The warrant for the second curve is the four sizes where both exist and agree, which is a weak warrant — it says the algorithm has not yet been wrong, not that it cannot be — and it is the only one available past the point the first curve leaves the page.

Where the enumeration stops

A maximin over an eight-function dictionary is a walk over seventy subsets. Over twenty-four it is 735,471 at eight functions, and the exchange algorithm that replaces the walk scores 421. What licenses the second curve is four sizes where both exist and agree, which is a weaker warrant than it looks.

part 3 · product

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%.

part 4 · surface

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.

part 5 · surface

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.

part 6 · surface