Multifactor B-spline mixed models in designed experiments for the engine mapping problem

In practical applications, a response surface may have a shape that is too complicated to be adequately described by a low-order polynomial over the region where predictions are required. The use of polynomial spline regression models in which the effects of one or more factors are modeled using B-spline basis functions allows greater model flexibility and can lead to more accurate predictions than traditional polynomial models. Multifactor B-spline regression models are described that have useful properties for designing experiments, including the advantage that linear model theory applies, An application of the models is discussed in a case study from the automotive industry on modeling torque in engine mapping using a two-stage hierarchical model, A B-spline model with known knots is used at the second stage to represent each of the response features of engineering interest that are carried forward from the first-stage modeling. The model is fitted over a constrained design region using data obtained from an experiment that employs a V-optimal design generated by a standard search algorithm using a tailored candidate list. Validation runs are used to assess the accuracy of predictions and to show that the approach is effective.