Least-Squares Fitting of a Polynomial Constrained to be Either Non-Negative, Non-Decreasing or Convex

Abstract

An algorithm is provided to fit the (unique) least-squares polynomialY(x) tondata points (x, y) subject to a constraint that it be either non-negative, non-positive, non-decreasing, non-increasing, convex or concave at all pointsxin a closed intervalI.Some generalization to a general linear model can be handled, and certain alternative constraints can also be dealt with. The problem is close to the classical problem of Quadratic Programming (Q.P.), and some of the known results in Q.P. may be used in fitting the curve. The algorithm terminates in at most three steps. The number of steps depends on the degree of the polynomialY(x) and the data. When no more than two steps are required, the algorithm is very easy to carry out but when three steps are necessary, it becomes more difficult. A useful statistical application is the fitting of a polynomial to growth data. In that case, we might like the fitted curve to be non-decreasing over the range of interest.