Shape-Constrained Regression using Sum of Squares Polynomials
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We consider the problem of fitting a polynomial to a set of data points, each data point consisting of a feature vector and a response variable. In contrast to standard least-squares polynomial regression, we require that the polynomial regressor satisfy shape constraints, such as monotonicity with respect to a variable, Lipschitz-continuity, or convexity over a region. Constraints of this type appear quite frequently in a number of areas including economics, operations research, and pricing. We show how to use semidefinite programming to obtain polynomial regressors that have these properties. We further show that, under some assumptions on the generation of the data points, the regressors obtained are consistent estimators of the underlying shape-constrained function that maps the feature vectors to the responses. We apply our methodology to the US KLEMS dataset to estimate production of a sector as a function of capital, energy, labor, materials, and services. We observe that it outperforms the more traditional approach (which consists in modelling the production curves as Cobb-Douglas functions) on 50 out of the 65 industries listed in the KLEMS database.
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