Optimal knots for smoothing splines and adaptive smoothing splines in generalized linear models: Design approach

Smoothing splines play a crucial role as a prevalent tool for estimating unknown functions in non parametric regression. In fitting functions with varying roughness, adaptive smoothing splines provide the flexibility to adjust the smoothing parameter adaptively with the changes over the predictor variable within specific intervals, facilitating accommodation for changing roughness. Despite its intuitive appeal and high flexibility, the implementation of this estimator poses challenges, especially in the absence of specified knot numbers, knot locations, and certain simplifying assumptions on the smoothing parameter. D-optimal designs are considered to specify the knot locations in this models. The prior information from data as the prior curvature knowledge can be useful for this optimization problem. For this purpose, the smoothing parameter in this optimization problem can be considered as a function based on the prior curvature. We apply this method to responses from the exponential family of distributions. The mathematical expression of the estimator of the curve in a generalized linear mixed model enables us to derive an information matrix. The derived information matrix and hence the optimum design will depend on the values of the model's unknown parameters. In such situations, numerical techniques should be considered for evaluating and optimizing this criterion.