On smoothness terms in multidimensional Whittaker graduation

Abstract

Multidimensional Whittaker graduations are obtained by minimizing a linear combination of a measure of lack-of-fit between the graduated and ungraduated values, and measures of lack-of-smoothness of the graduated values across each dimension. For convenience we shall use the terminology ‘smoothness’ rather than ‘lack-of-smoothness’ when referring to the terms included in the objective function. Standard smoothness terms are sums of squares of differences, with respect to a single variable, of the graduated values. Consequently, such terms may be thought of as measuring lack-of-smoothness in directions that are parallel to the various axes. Additional smoothness of the graduated values may be obtained by including smoothness terms that are based on mixed differences. In this paper a method is developed for determining which smoothness terms to include in the objective function. This analysis assumes the graduator has pre-specified a polynomial model which represents the graduated values under ideal or ultimate smoothness. A second problem of determining formulas for these smoothness terms is also solved. Whittaker graduation is most efficiently done when the objective function is expressed in terms of matrices. Smoothness terms may then be written as quadratic forms in the graduated values. Lowrie (1993) explains how to construct the matrices of these quadratic forms; however, a basic formula for these matrices has not been found. In this paper formulas for these matrices are developed in terms of Kronecker products. An extension to nonstandard smoothness terms is also discussed.