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

1. select a set of realizations {1r[1], 1r[2], … , 1r[l]} for 1R,
2. value 1p[k] = θ(1r[k]) for each, and
3. 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.