Numerical Differentiation by Integrated Series Expansion (NDBISE) in the Context of Ordinary Differential Equation Estimation Problems

Parameter or model estimation of ordinary differential equations (ODE) nowadays frequently involves the numerical calculation of derivatives from noisy data. This study presents a novel differentiation method (NDBISE) for such calculations. The method was benchmarked against 57 differential equations and compared to five numerical differentiation methods: central finite differences, numerical derivatives using the Friedrichs mollifier, weak differentiation using this mollifier, first order polynomial approximation and spectral derivatives. For the latter method, a novel criterion is presented that allows the determination of the low pass filter parameter. The hyperparameters of all these methods are optimized in order to get a reasonable comparison. The resilience against larger noise or fewer data points per time interval is examined. It turns out that the novel method is overall superior to the other methods. The derivative for the 42 real world data points of the Hudson bay lynx hare data (years 1900-1920) is also calculated. The results match the derivative of a curve fit to the data points astonishingly close. Using a Savitsky-Golay filter, the method can be leveraged, to calculate second and third order derivatives, so that the results are close to the theoretically expected outcome. The paper references a GitHub project that contains code and an application manual to reproduce all figures of the result section from ODE state data.