5:15 PM - 7:15 PM
[MZZ42-P03] Theoretical Framework for Analytical Precision: Guidelines for Efficient High-Precision Analysis Based on Fisher Information
Keywords:Cramér–Rao lower bound, Fisher information, Analytical chemistry, Signal processing, Precision, Metrology
Introduction
Signal data analysis forms the foundation of advancements in science and engineering. The precise estimation of peak parameters, such as intensity, position, bandwidth, and area, is not merely a technical objective but a crucial element underpinning discoveries. To achieve efficient high-precision analysis, it is essential to comprehensively understand the physical principles governing the estimation of these parameters and to establish a theoretical framework that connects ‘analytical precision" with ‘instrumental performance." This framework enables researchers to quantitatively evaluate the improvement in precision achieved by upgrading instruments. Thus, the goal of this study is to support decision-making for precision optimization and provide a foundation for effectively utilizing research funds.
Methods
Analytical solutions for the precision of peak parameter estimation were derived within the framework of the Cramér–Rao inequality (Cramér, 1946; Rao, 1945) and Fisher information (Fisher, 1922). This study focused on signal profiles commonly observed in analytical chemistry, specifically Gaussian and Lorentzian profiles. For noise, the assumption was made that Poisson noise dominates under conditions of sufficiently strong signals. The cost function applied was the weighted sum of squared residuals, which is universally used in curve fitting software (Wojdyr, 2010). By specifying these three conditions—signal profile, noise, and cost function—it becomes possible to derive analytical solutions for precision under various scenarios.
The precision of Lorentzian peak parameter estimation (Hagiwara and Kuwatani, 2025) derived under these conditions was compared directly with that of Gaussian profiles (Hagiwara and Kuwatani, 2024). This enabled a comprehensive investigation into the principles determining the precision of parameter estimation. To confirm the validity of the analytical solutions, Monte Carlo simulations and numerical calculations were conducted. Finally, the parameters in the analytical solutions were expressed in terms of instrumental performance, thereby establishing a theoretical framework that links "analytical precision" with "instrumental performance."
Results
The main findings are as follows: Figures illustrate where intensity, position, and bandwidth information is distributed in the signal data for Gaussian and Lorentzian profiles. Notably, information is not always concentrated in regions of strong signal. Furthermore, the regions rich in information vary depending on the parameter—for example, intensity information is concentrated at the peak center, while position information is abundant around the full width at half maximum (FWHM). Under identical conditions for intensity and bandwidth, the precision of position and position-difference estimation is approximately {4√(ln2/π)}0.5 ~ 1.37 times better for Gaussian profiles than for Lorentzian profiles. This is attributed to the steeper gradients and lower tail intensities of Gaussian profiles, which increase the information content related to peak position. Lorentzian profiles exhibit a strong negative correlation between intensity and bandwidth compared to Gaussian profiles. This correlation implies that higher intensity corresponds to narrower bandwidths and vice versa, making it challenging to estimate intensity and bandwidth independently with high precision. Additionally, since bandwidth information is concentrated in the tails (as shown in the figures), missing or noisy data in these regions significantly reduces precision. For Lorentzian profiles, the relative standard deviation of area ratios is √3 times smaller than that of intensity ratios, making area ratios a more precise estimator. In profiles with strong negative intensity-bandwidth correlations and large peak areas (like Lorentzian profiles), area ratios tend to outperform intensity ratios in precision. By expressing factors such as the x-axis spacing of signal data (Hagiwara et al., 2023), signal intensity, and bandwidth (Liu and Berg, 2012) in terms of instrumental parameters, a theoretical framework linking analytical precision with instrumental performance was established. Numerical calculations demonstrated how upgrades to instrument components (e.g., spectrometer focal length, diffraction grating constants, detector pixel size, slit width) improve the precision of δ13C measurements of CO2 using Raman spectroscopy.
Signal data analysis forms the foundation of advancements in science and engineering. The precise estimation of peak parameters, such as intensity, position, bandwidth, and area, is not merely a technical objective but a crucial element underpinning discoveries. To achieve efficient high-precision analysis, it is essential to comprehensively understand the physical principles governing the estimation of these parameters and to establish a theoretical framework that connects ‘analytical precision" with ‘instrumental performance." This framework enables researchers to quantitatively evaluate the improvement in precision achieved by upgrading instruments. Thus, the goal of this study is to support decision-making for precision optimization and provide a foundation for effectively utilizing research funds.
Methods
Analytical solutions for the precision of peak parameter estimation were derived within the framework of the Cramér–Rao inequality (Cramér, 1946; Rao, 1945) and Fisher information (Fisher, 1922). This study focused on signal profiles commonly observed in analytical chemistry, specifically Gaussian and Lorentzian profiles. For noise, the assumption was made that Poisson noise dominates under conditions of sufficiently strong signals. The cost function applied was the weighted sum of squared residuals, which is universally used in curve fitting software (Wojdyr, 2010). By specifying these three conditions—signal profile, noise, and cost function—it becomes possible to derive analytical solutions for precision under various scenarios.
The precision of Lorentzian peak parameter estimation (Hagiwara and Kuwatani, 2025) derived under these conditions was compared directly with that of Gaussian profiles (Hagiwara and Kuwatani, 2024). This enabled a comprehensive investigation into the principles determining the precision of parameter estimation. To confirm the validity of the analytical solutions, Monte Carlo simulations and numerical calculations were conducted. Finally, the parameters in the analytical solutions were expressed in terms of instrumental performance, thereby establishing a theoretical framework that links "analytical precision" with "instrumental performance."
Results
The main findings are as follows: Figures illustrate where intensity, position, and bandwidth information is distributed in the signal data for Gaussian and Lorentzian profiles. Notably, information is not always concentrated in regions of strong signal. Furthermore, the regions rich in information vary depending on the parameter—for example, intensity information is concentrated at the peak center, while position information is abundant around the full width at half maximum (FWHM). Under identical conditions for intensity and bandwidth, the precision of position and position-difference estimation is approximately {4√(ln2/π)}0.5 ~ 1.37 times better for Gaussian profiles than for Lorentzian profiles. This is attributed to the steeper gradients and lower tail intensities of Gaussian profiles, which increase the information content related to peak position. Lorentzian profiles exhibit a strong negative correlation between intensity and bandwidth compared to Gaussian profiles. This correlation implies that higher intensity corresponds to narrower bandwidths and vice versa, making it challenging to estimate intensity and bandwidth independently with high precision. Additionally, since bandwidth information is concentrated in the tails (as shown in the figures), missing or noisy data in these regions significantly reduces precision. For Lorentzian profiles, the relative standard deviation of area ratios is √3 times smaller than that of intensity ratios, making area ratios a more precise estimator. In profiles with strong negative intensity-bandwidth correlations and large peak areas (like Lorentzian profiles), area ratios tend to outperform intensity ratios in precision. By expressing factors such as the x-axis spacing of signal data (Hagiwara et al., 2023), signal intensity, and bandwidth (Liu and Berg, 2012) in terms of instrumental parameters, a theoretical framework linking analytical precision with instrumental performance was established. Numerical calculations demonstrated how upgrades to instrument components (e.g., spectrometer focal length, diffraction grating constants, detector pixel size, slit width) improve the precision of δ13C measurements of CO2 using Raman spectroscopy.