Optimal design of experiments for functional linear models with dynamic factors

In this work, we build optimal experimental designs for precise estimation of the functional coefficient of linear regression model where both the response and the factors are continuous functions. After obtaining the variance-covariance matrix of the estimator of the functional coefficient which minimizes the integrated sum of square of errors, we extend the classical definition of optimal design to this estimator, and we provide the expression of the A-optimal and of the D-optimal designs. Examples of optimal designs for dynamic experimental factors are then computed through a suitable algorithm, and we discuss different scenarios in terms of the set of basis functions used for their representation. Finally, we present an illustrative example inspired by a real application in a pharmaceutical manufacturing process, to illustrate the feasibility and the advantages of our methodology.