Precision comparison of intensity ratios and area ratios in spectral analysis

Abstract

Abstract The long-debated question in analytical chemistry of which of the area ratio or the intensity ratio is the more precise has yielded no definitive analytical conclusion. To address this issue theoretically, we derived analytical solutions for the lower limits of estimation precision for spectral parameters, including the intensity ratio and area ratio, based on the Cramér–Rao lower bound (CRLB) framework for a Gaussian spectrum. The precisions of spectral parameter estimations from the analytical solutions were consistent with results obtained from Monte Carlo simulations. Our theoretical and simulation results revealed that the precision of estimating the area ratio surpassed that of the intensity ratio by a factor of $$\sqrt{2}$$ 2 . Additionally, our experimental results aligned well with both theoretical predictions and simulation outcomes, further validating our approach. This increased precision of the area ratio is due to negative covariance between intensity and bandwidth, rather than the area containing more intensity information, as often misinterpreted. Consequently, and quite counter intuitively, prior bandwidth and intensity related information does not improve the area ratio precision: it worsens it. The analytical solution we derived represents the fundamental limits of spectral parameter measurement precision. Thus, it can be used as an alternative method for estimating the minimum error when experimental measurement uncertainty cannot be determined.

Keywords

• Cramér–Rao lower bound
• Fisher information
• Monte Carlo simulation
• Intensity ratio
• Area ratio
• Variance–covariance matrix