An n-th Order Generalization of the Activity Measure for Continuous Systems

Abstract

In this work we present an analytical expression that generalizes the definition of activity measure in continuous time signals. We define the activity of order n and show that it allows to estimate the number of sections of polinomials up to order n that are needed to represent that signal with certain accuracy. We apply this concept to obtain a lower bound for the number of steps performed by quantization–based integration algorithms in the simulation of ordinary differential equations. We performed a practical analysis over a first order example system, computing the activity of order n and comparing it with the number of steps required integration methods of different orders. We corroborated the theoretical predictions, which indicate that the activity measure can be used as a reference for assessing the suitability of different algorithms depending on how close they perform in comparison with the theoretical lower bound. Finally, a discussion is provided which indicates that further research is needed in order to test the results presented in this work in the context of stiff systems.