Canonical reduction of second-order fitted models subject to linear restrictions

Canonical reduction of second-order response surfaces is a useful technique for finding the form and shape of surfaces and often for discovering redundancies that enable the surface to be expressible in a simpler form with fewer canonical predictor variables than there are original predictor variables. Canonical reduction of models subject to linear restrictions has received little attention, possibly due to the apparent difficulty of performing it. An important special application is when the predictor variables are mixture ingredients that must sum to a constant; other linear restrictions may also be encountered in such problems. A possible difficulty in interpretation is that the stationary point may fall outside the permissible restricted space. Here, techniques for performing such a canonical reduction are given, and two mixture examples in the literature are re-examined, and canonically reduced, to illustrate what canonical reduction can and cannot provide.