Curve-fitting overlapped bands: Quantification and improvement of curve-fitting robustness in the presence of errors in the model and in the data

Estimation of the band parameters of overlapped bands often relies on curve-fitting. It has been demonstrated that curve-fitting provides the maximum likelihood estimation of band parameters under a series of assumptions. One of these assumptions is that the curve-fitting model is correct and any error in the data is random. Under real conditions, we have to acknowledge the unavoidable presence of errors in the model and systematic errors in the data. Here, we derive an expression for the estimation of how these errors affect the quality of the parameters obtained from curve-fitting. In addition, we derive theoretical expressions to quantify the extent to which different methods can improve the curve-fitting robustness to these errors. The methods considered are: (i) deterministic and (ii) probabilistic constraints in the band parameters, (iii) curve-fitting band-narrowed data, and (iv) building a more accurate model. The theoretical expressions obtained are tested in the curve-fitting of a synthetic noisy spectrum with either baseline or band shape errors, and in the curve-fitting of the experimental infrared amide I band of the membrane protein bacteriorhodopsin.