2:15 PM - 2:30 PM
[SCG45-03] How to improve the measurement precision of spectral position, width, intensity, and area: Derivation of Cramér–Rao lower bound and benchmarking via Monte Carlo simulations
Keywords:Cramér–Rao lower bound, Fisher information, ICP-MS, chromatography, Raman spectroscopy
Introduction
Spectral properties extracted from spectra, such as peak position (ωc), bandwidth (Γ), intensity (I), area (A), wavenumber difference (Δω), ratio of intensities (RI), and ratio of areas (RA), have been used in various chemical analyses (e.g., Raman, Brillouin, XRD, FTIR, NMR, EDS, ICP-MS, Chromatography) as an indicator of the physicochemical properties of a substance. Although achieving the ultimate precision of estimation is an important goal in any analysis, the following fundamental questions regarding the precision of estimation of spectral properties remain unanswered:
1) How is the precision of the estimation of spectral characteristics expressed analytically in the shot noise limit?
2) Which is the more precise, the area ratio or the intensity ratio (or area or intensity)?
3) In the shot noise limit, to what extent does pixel binning, which treats multiple adjacent pixels as one large pixel, improve the precision of estimating spectral characteristics?
4) To what extent does upgrading the instrument performance (e.g., detector or spectrometer) improve the estimation precision of spectral characteristics?
Methods
We conducted theoretical investigations and Monte Carlo simulations to address these questions. For spectra with Gaussian profiles, we derived analytical solutions for lower bounds on the precision of estimating spectral characteristics in the shot noise limit based on the Cramér-Rao lower bound (CRLB) and the Fisher information framework. For the Monte Carlo simulation, a total of 8100 spectra with two gaussian profiles sufficiently far apart were generated under various conditions. To extract spectral properties from the generated spectra, a python nonlinear fitting routine (scipy.optimize.curve_fit) based on the Levenberg-Marquardt algorithm was used. Finally, we compared the precision of the estimates of spectral characteristics obtained from theory and simulation.
Results
The following results were obtained for each of the above problems 1)-4).
1) The analytical solutions obtained were consistent with the results of the Monte Carlo simulation. As intuition suggests, the relative standard deviations of all estimators are proportional to "the inverse of the square root of the intensity (1/I)1/2. The relative standard deviations of bandwidth, intensity, area, and their ratios are all proportional to {Δx/Γ}1/2, where Δx is the distance between the data points that make up the spectrum. On the other hand, the relative standard deviations of peak positions and wavenumber differences are both proportional to {Γ×Δx}1/2. Therefore, under the condition that Δx/I = const., the intensity and bandwidth can be estimated more precisely for broad peaks, while the peak position can be estimated more precisely for narrow peaks.
2) In analytical chemistry, the debate over the greater precision between intensity ratio and area ratio has been longstanding (Kipiniak, 1981). Experimental research presents varied findings: in some instances, the intensity ratio is more precise (Dyson, 1999), in others, the area ratio holds more precision (Pauls et al., 1986), and sometimes, no significant difference in precision is observed between the two (Park et al., 1987). Our results indicate that the precision of the area ratio (or area) estimation is √2 times better than that of the intensity ratio (or intensity) estimation (see figure below). The reason for the higher precision of the area ratio (or area) is due to the negative covariance between the intensity and the bandwidth. As a result, contrary to intuition, a priori information related to intensity and bandwidth does not improve the precision of area ratio (or area), but rather worsens it.
3) At the shot noise limit, it has been shown that pixel binning neither improves nor worsens the estimation precision of all spectral characteristics.
4) By integrating the obtained CRLB with spectroscopic knowledge, we established a theoretical foundation linking ‘estimation precision of spectral characteristics‘ and ‘instrument performance‘. This theoretical framework allows us to quantitatively evaluate the limits of precision achievable with a particular analytical instrument and the degree of improvement in precision that can be achieved by upgrading the instrument. We used this platform to numerically calculate the degree to which upgrading instrument performance (focal length of the spectrometer, grating constant, detector pixel size, and slit width) improves the precision of δ13C measurements for CO2 using Raman spectroscopy.
Spectral properties extracted from spectra, such as peak position (ωc), bandwidth (Γ), intensity (I), area (A), wavenumber difference (Δω), ratio of intensities (RI), and ratio of areas (RA), have been used in various chemical analyses (e.g., Raman, Brillouin, XRD, FTIR, NMR, EDS, ICP-MS, Chromatography) as an indicator of the physicochemical properties of a substance. Although achieving the ultimate precision of estimation is an important goal in any analysis, the following fundamental questions regarding the precision of estimation of spectral properties remain unanswered:
1) How is the precision of the estimation of spectral characteristics expressed analytically in the shot noise limit?
2) Which is the more precise, the area ratio or the intensity ratio (or area or intensity)?
3) In the shot noise limit, to what extent does pixel binning, which treats multiple adjacent pixels as one large pixel, improve the precision of estimating spectral characteristics?
4) To what extent does upgrading the instrument performance (e.g., detector or spectrometer) improve the estimation precision of spectral characteristics?
Methods
We conducted theoretical investigations and Monte Carlo simulations to address these questions. For spectra with Gaussian profiles, we derived analytical solutions for lower bounds on the precision of estimating spectral characteristics in the shot noise limit based on the Cramér-Rao lower bound (CRLB) and the Fisher information framework. For the Monte Carlo simulation, a total of 8100 spectra with two gaussian profiles sufficiently far apart were generated under various conditions. To extract spectral properties from the generated spectra, a python nonlinear fitting routine (scipy.optimize.curve_fit) based on the Levenberg-Marquardt algorithm was used. Finally, we compared the precision of the estimates of spectral characteristics obtained from theory and simulation.
Results
The following results were obtained for each of the above problems 1)-4).
1) The analytical solutions obtained were consistent with the results of the Monte Carlo simulation. As intuition suggests, the relative standard deviations of all estimators are proportional to "the inverse of the square root of the intensity (1/I)1/2. The relative standard deviations of bandwidth, intensity, area, and their ratios are all proportional to {Δx/Γ}1/2, where Δx is the distance between the data points that make up the spectrum. On the other hand, the relative standard deviations of peak positions and wavenumber differences are both proportional to {Γ×Δx}1/2. Therefore, under the condition that Δx/I = const., the intensity and bandwidth can be estimated more precisely for broad peaks, while the peak position can be estimated more precisely for narrow peaks.
2) In analytical chemistry, the debate over the greater precision between intensity ratio and area ratio has been longstanding (Kipiniak, 1981). Experimental research presents varied findings: in some instances, the intensity ratio is more precise (Dyson, 1999), in others, the area ratio holds more precision (Pauls et al., 1986), and sometimes, no significant difference in precision is observed between the two (Park et al., 1987). Our results indicate that the precision of the area ratio (or area) estimation is √2 times better than that of the intensity ratio (or intensity) estimation (see figure below). The reason for the higher precision of the area ratio (or area) is due to the negative covariance between the intensity and the bandwidth. As a result, contrary to intuition, a priori information related to intensity and bandwidth does not improve the precision of area ratio (or area), but rather worsens it.
3) At the shot noise limit, it has been shown that pixel binning neither improves nor worsens the estimation precision of all spectral characteristics.
4) By integrating the obtained CRLB with spectroscopic knowledge, we established a theoretical foundation linking ‘estimation precision of spectral characteristics‘ and ‘instrument performance‘. This theoretical framework allows us to quantitatively evaluate the limits of precision achievable with a particular analytical instrument and the degree of improvement in precision that can be achieved by upgrading the instrument. We used this platform to numerically calculate the degree to which upgrading instrument performance (focal length of the spectrometer, grating constant, detector pixel size, and slit width) improves the precision of δ13C measurements for CO2 using Raman spectroscopy.