Least Squares Data Fitting Subject to Decreasing Marginal Returns
Let data of a univariate process be given. If the data are related by a sigmoid curve, but the sigmoid property has been lost due to the errors of the measuring process, then the least sum of squares change to the data that provides nonnegative third divided differences is proposed. The method is highly suitable for estimating points on a sigmoid curve of unspecified parametric form subject to increasing marginal returns or subject to diminishing marginal returns. It is a structured quadratic programming calculation, which is solved very efficiently by a special least squares algorithm that takes into account the form of the constraints. Some numerical results illustrate the method on a variety of data sets. Moreover, two applications of the method on real economic data demonstrate its modeling capability. The first one concerns renewable energy consumption data, which exhibit a sigmoid pattern. The second one concerns technological substitutions among the PDP computers to the VAX computers.
KeywordsApproximation Decreasing marginal returns Divided difference Least squares data fitting Optimization Renewable energy consumption Sigmoid Technological substitution
This work was partially supported by the University of Athens under Research Grant 11105.
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