An additive penalty P-Spline approach to derivative estimation

P-Splines are commonly used for derivative estimation where a non-linear relationship exists between the response and explanatory variables. However, questions about the error of these estimates have arisen. Incorporating an extra penalty term in a P-Spline model is proposed as an improvement when derivative estimation is of primary concern. This additive penalty approach to derivative estimation is shown to improve on the P-Spline estimates based on the results of a simulation study to compare the performance when estimating the first and second derivatives of six simulated functions. A method for generating variability bands for P-Spline derivative estimates with and without an additive penalty is given. The proposed additive penalty variability bands are shown to behave better than their single penalty counterpart. Motivating examples in environmental and sports science are used to demonstrate the need for accurate derivative estimates and the benefit of using an additional penalty term to this end.