The uncertainty analysis in linear and nonlinear regression revisited: application to concrete strength estimation

Regression is a common technique in engineering when physical laws are unknown. Practitioners usually look for a unique set of true parameters that optimally explain the observed data. This is, for instance, the case in concrete strength estimation where engineers have been looking for an universal law to estimate this magnitude. We show that this approach is incorrect if the uncertainty of the regression problem is not properly taken into account. The uncertainty analysis of linear regression problems is revisited providing an analytical expression for the direction of maximum uncertainty where most of the models are sampled when partial information is used. We also analyse the case of 1D nonlinear regression models (exponential and potential models) and the multivariate case. We show a simple way of sampling the posterior distribution of the model parameters by performing least-squares of different data bags (bootstrap), introducing the percentile curves for the concrete strength estimation, comparing the results obtained from the linearized and nonlinear bootstrap procedures in the case of the nonlinear regression models. The methodology introduced in this paper constitutes a robust and simple way of assessing the intrincic uncertainty of these well-known parameter identification problems and adopting more robust decisions.