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

Ridge analysis — where it appears

Tracing the best setting of a fitted response surface at each distance from the centre, rather than reporting the single stationary point. It is what replaces a recommendation when the stationary point lies outside the region an experiment can reach, and it prices each step outwards.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

Named alongside it

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

Experimental designPrediction varianceResponse-surfaceStationary pointCanonical analysisCentral composite designConstrained optimisationCurvatureEigenvalueForecast intervalMonte CarloSaddle point

All concepts