Robust approximation of uncertain functions where adaptation is impossible

The paper is concerned with the approximation of functions with an environmental parameter that is difficult or impossible to adapt to. Approximation with respect to the ordinary least-squares criterion provides a good overall approximation but at the cost of large approximation errors for some values of the independent variables. An alternative training method using the risk-averting training criterion is proposed that provides robust function approximation. The method adaptively adjusts the sensitivity index of the risk-averting criterion to tune to the effects of the unobservable parameter, when the measurement noises are negligible or unbiased. Numerical examples are presented illustrating the efficacy of the proposed adaptive risk-averting training method.