9.3.6 Quadratic Remappings With Gradient-Hesian Approximations
A quadratic remapping can be constructed with a gradient-Hessian approximation. Constructed about the point 1|0μ = 0E(1R), the remapping has form
[9.29]
Quadratic remappings should generally not be constructed in this manner. Because they are based upon first and second partial derivatives, the approximations are too localized.
9.3.7 Interpolation and the Method of Least Squares
To construct a quadratic approximation that is good over a larger region of values for 1R, we may apply ordinary interpolation or ordinary least squares to fit a quadratic remapping. To do so, we
- select a set of realizations {1r[1], 1r[2], … , 1r[l]} for 1R,
- value 1p[k] = θ(1r[k]) for each, and
- either interpolate or apply least squares to the points (1r[k],1p[k]).
Since ordinary least squares is a generalization of ordinary interpolation (see Exercise 2.16), we consider both approaches simultaneously. Suppose the form of quadratic polynomial we wish to fit has m unique potentially nonzero components ci, j, bi, and a. With interpolation, the number l of realizations 1r[k] equals m. With least squares, l exceeds m.
If the dimensionality of 1R is large, l will also be large. Since we must perform l valuations 1p[k] = θ(1r [k]), it is desirable that θ be as easy to value as possible. It may be advisable to precede the remapping with a holdings remapping that simplifies θ.