Numerical Differentiation by Integration (NDBI) in the Context of Ordinary Differential Equation Estimation Problems

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