Curve Fitting to Step Functions

Abstract

The methods presented in this paper provide an improved technique for fitting a continuous function to discontinuous step functions so often evolving from empirical studies such as estimation of demand and supply functions by linear programming. Curve fitting to step functions is analyzed using the criterion of minimizaton of the integral of squared distances between the curve being fitted and a step function. The criterion is a limiting form of least squares fit where all points on the step function are treated as observations. Derivations are given for equations specifying the minimum integral fit for polynomials, a criterion of fit analogous to the coefficient of multiple determination in least squares fitting, and the analogue of the standard error of estimate in least squares. STEP FUNCTIONS frequently arise in empirical investigations when theoretical considerations suggest a continuous relation. By a step function we mean a function such that the range is divided into a finite number of intervals with the dependent variable constant on a given interval. Graphically, the function appears as a series of steps (Figure 1). The results of linear programming models, in which a price or resource has been permitted to vary, often yield step functions.' Economic engineering studies may also generate empirical measures in the form of step functions.2 Sometimes the discontinuous function may be consistent with theory, but the convenience afforded by a continuous relationship makes it desirable to obtain a continuous approximation.