Stationary point — where it appears
Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
Where the derivative is zero
The delta method reads a standard error off a tangent line, and at a flat point the tangent says the spread is zero. The interval built on it for a squared mean covers 99.991% there and 85.978% one and a half standard errors away, with nearly every miss on the same side — and the law it should have used is a χ², not a normal.
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%.
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.
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.
A flat point with more than one direction
At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.
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.
Testing for a flat point first
Test whether the gradient is zero, and if the test does not reject, read the interval off the second-order law that holds at a stationary point. For a squared mean that two-stage rule covers about 50% at its worst whatever the pretest's level — 49.72% with a 5% test, where the tangent interval it was meant to rescue never falls below 85.98% — because its worst sits exactly where the true distance equals the test's critical value, half the samples are declared flat there, and the second-order interval cannot reach the truth from any of them.
Named alongside it
The objects these essays reach for when they reach for this one.
Saddle pointCentral composite designConfidence intervalCoverageDelta methodEigenvalueExperimental designMonte CarloResponse-surfaceChi-squareClosed formCurvature