Hagiwara Yuuki, Kuwatani Tatsu
Research Institute for Marine Geodynamics, Japan Agency for Marine-Earth Science and Technology, Yokosuka, 237-0061, Japan.
Sci Rep. 2024 Oct 15;14(1):22898. doi: 10.1038/s41598-024-71653-3.
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 . 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.
分析化学中关于面积比和强度比哪个更精确这一长期争论的问题，尚未得出明确的分析结论。为了从理论上解决这个问题，我们基于高斯光谱的克拉美 - 罗下界（CRLB）框架，推导了光谱参数估计精度下限的解析解，包括强度比和面积比。从解析解得到的光谱参数估计精度与蒙特卡罗模拟结果一致。我们的理论和模拟结果表明，估计面积比的精度比强度比高出 倍。此外，我们的实验结果与理论预测和模拟结果都非常吻合，进一步验证了我们的方法。面积比精度的提高是由于强度和带宽之间的负协方差，而不是像通常误解的那样面积包含更多的强度信息。因此，非常违反直觉的是，先前的带宽和强度相关信息并不能提高面积比精度，反而会使其变差。我们推导的解析解代表了光谱参数测量精度的基本极限。因此，当无法确定实验测量不确定度时，它可以用作估计最小误差的替代方法。
