Box plots: A simple graphical tool for visualizing overfitting in peak fitting as demonstrated with X-ray photoelectron spectroscopy data

While peak fitting of spectra/data is frequently performed in science, recent reports suggest that the quality of peak fitting in the scientific literature is often inadequate. Here, we describe a new statistical tool for determining the quality of fitting protocols, illustrating this capability with X-ray photoelectron spectroscopy (XPS) data. This tool, box plots of random starting conditions and their results, helps identify local minima in the multidimensional fit space of the fit parameters. Ideally, there should be a single global minimum for a fitting protocol such that different, reasonable starting conditions lead to the same result. To determine whether a fit space contains multiple local minima, a series of reasonable starting conditions is randomly chosen for the fit. If the boxes in the box plot of the peak areas of these multiple fits are narrow, the different possibilities converge to a single global minimum. Conversely, if the boxes are wide, multiple local minima are present. This method is related to the mathematical concept of 'disproof by contradiction'. Our approach is demonstrated with four- and ten-component fits to a moderately complex C 1s XPS narrow scan. The results from our analysis compare favorably to those of traditional Monte Carlo analyses and uniqueness plots, where box plots are also applied to the Monte Carlo results, and each of these statistical tools performs a different function/probes a fit space/protocol differently.