Publication Details
Abstract
Turning measurements into a quantitative law involves two linked problems: choosing the structure of the functional relationship and estimating (reconstructing) its parameters together with their uncertainty. In practice the two are often treated separately, which leads to over-fitted empirical formulas, parameters that cannot be determined from the available data, and unstable solutions of ill-posed inverse problems. This paper proposes and tests an integrated workflow that combines information-theoretic structure selection (AICc, BIC, Akaike weights), cross-validation and residual diagnostics with nonlinear least-squares estimation by the Levenberg–Marquardt method, local and bootstrap uncertainty quantification, identifiability analysis based on the conditioning of the Fisher information matrix, and Tikhonov regularisation for linear ill-posed problems. The workflow is evaluated in four controlled numerical experiments with known ground truth: selection among six candidate laws, a Monte Carlo study of estimator accuracy versus noise level and experimental design, reconstruction of the parameters of a damped oscillator from a noisy trajectory, and deconvolution of a Gaussian-blurred signal. The exponential law was recovered with an Akaike weight of 0.947 although a cubic polynomial fitted almost equally well (R2 = 0.9925 vs 0.9931). At 5 % noise, the relative RMSE of the parameters stayed below 4.2 %, while shortening the observation window from 12 to 3 units increased the condition number of JTJ from 127 to 1.27·104 and the error of the asymptote from 3.2 % to 37.8 %. Oscillator parameters were recovered to within 0.3 % (ω) and 1.9 % (ζ) from all six starting points; Tikhonov regularisation reduced the reconstruction error of an operator with condition number 3.8·1011 from 6·108 to 3.4 %. The results show that the experimental design, not the fitting algorithm, is usually the limiting factor.