Concept

Canonical analysis — where it appears

Rotating a fitted quadratic surface to the eigenvectors of its quadratic form, so it becomes a sum of independent squared terms. The signs of the eigenvalues then say whether the stationary point is a maximum, a minimum or a saddle, and their sizes say how steeply the surface falls away in each direction.

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

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

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

Named alongside it

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

Curvature Eigenvalue Experimental design Response-surface Saddle point Stationary point Central composite design Constrained optimisation Least squares Prediction variance Ridge analysis

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