Piecewise linear approximation of multivariate functions

To approximate functions of a single variable by using linear interpolation is routine in empirical studies. Here, we consider approximating functions of several variables in a similar piecewise linear manner. We focus on the nontrivial part of this technique, which is that of choosing the appropriate “pieces” for the piecewise linear approximation. Precisely, we seek to identify the best interpolants to use at a point of interpolation. This is not an issue for functions of a single variable, since the linear ordering of the number line leaves us with no choice. For functions of several variables, we propose several simple tools to help uncover undesirable choices. The techniques presented are useful in the empirical study of quantitatively complex functional relationships whose qualitative behavior is nevertheless known and simple. Response time relationships parametrized by workloads in computer performance modeling often fall into this category, and an actual bivariate function of this type is used to motivate the development.