Parameter certainty quantification in nonlinear models

Estimating model parameters from experimental data is a common practice across various research fields. For nonlinear models, the parameters are estimated using an optimization algorithm that minimizes an objective function. Assessing the certainty of these parameter estimates is crucial to address questions such as “what is the probability the estimation error is smaller than 5%?”, “is our experiment sensitive enough to estimate all parameters?”, and “how much can we change each parameter while still fitting the data accurately?”. Typically, the certainty levels are quantified using a linear approximation of the model. However, we show that in models that are highly nonlinear in their parameters or in the presence of large experimental errors, this method fails to capture the certainty levels accurately. To address these limitations, we present an alternative method based on the Hessian approximation of the objective function. We show that this method captures the certainty levels more accurately and can be derived geometrically. We demonstrate the efficacy of our approach through a case study involving a nonlinear hyperelastic material constitutive model and an application on a nonlinear model for the conductivity of electrolyte solutions. Despite its higher computational cost, we recommend adopting the Hessian approximation when accurate certainty levels are required in highly nonlinear models.