Precision in Peak Parameter Estimation for the Pseudo-Voigt Profile: A Novel Optimization Approach for High-Precision Analysis via Mixing Parameter Control

Abstract

High-precision measurement of peak parameters such as intensity (I), peak position (ω _c ), full width at half-maximum (Γ), and area (A) is pivotally important for advancing scientific research. Achieving high-precision requires elucidating the physical principles governing measurement precision and establishing guidelines for optimizing analytical conditions. Although the pseudo-Voigt profile is a widely used line-shape model, the underlying principles governing the precision of its parameter estimation remained unclear. For this study, we developed a model to quantify the parameter estimation precision under arbitrary conditions by integrating theoretical analysis, numerical calculations, and Monte Carlo simulations. Our quantification results indicate that when the mixing parameter (η) is fixed, the precision of I, Γ, and A is proportional to {Δx/ΓI}^0.5, whereas the precision of ω _c is proportional to {ΓΔx/I}^0.5, where Δx denotes the sampling interval. Furthermore, the analytical precision exhibits η-dependence: for I and Γ, when the profile becomes more Lorentzian, the absolute value of the covariance between Γ and η as well as between I and η increases, thereby degrading their estimation precision. This finding suggests that in addition to conventional methods such as improving the signal-to-noise ratio and reducing sampling interval, appropriately controlling η can be an effective strategy for optimizing precision. For instance, if broadening effects (e.g., instrumental or Doppler broadening) are deliberately introduced to tune η from 1 to 0, then this alone improves Γ estimation precision by a factor of 3.7, equivalent to a 14-fold increase in signal intensity. Furthermore, when the effect of increased Γ due to broadening is considered, even greater improvements in precision can be achieved. Overall, our model provides a foundational framework for research on peak parameter estimation. It serves as an alternative approach to error estimation when experimental evaluation is challenging and as a quantitative tool for assessing precision gain from instrument upgrades.

Across a broad range of fields, peak parameters extracted from signal data, such as intensity (I), peak position (ω _c ), full width at half-maximum (Γ), and area (A), serve as key indices for evaluating the physicochemical properties of materials. These parameters must be estimated with the highest possible precision to identify subtle phenomena that are difficult to discern and to facilitate more rapid analyses with minimal sample quantities. These demands underscore the importance and challenges of achieving precise estimation in various fields and analytical techniques.

In practice, observed line profiles result from the simultaneous contributions of multiple factors, including Doppler broadening caused by thermal motion, instrumental broadening, crystal size or lattice strain broadening, , and natural, pressure, or power broadening. , Consequently, neither a purely Gaussian profile nor a purely Lorentzian profile describes these profiles adequately; instead, the Voigt profile, defined as the convolution of the two functions, emerges as the most appropriate model. − However, because the Voigt profile lacks a closed-form analytical expression, its use in data analysis and curve fitting incurs high computational costs.

To address this issue, the pseudo-Voigt profile (eq ) has been employed as a computationally efficient approximation that retains the essential characteristics of the original Voigt profile. − Numerous studies have specifically examined the accuracy of the approximation of the pseudo-Voigt profile and development of methods to estimate the original physical parameters more accurately from the extracted parameters. ,− However, to achieve high-precision measurements efficiently, it is imperative not only to assess the approximation accuracy but also to elucidate the underlying physical principles that define the limits of peak parameter estimation precision, thereby providing guidelines for optimizing analytical instruments and measurement conditions. To date, no study has investigated the physical principles that establish these limits for the pseudo-Voigt profile.

Earlier studies of the precision of peak parameter estimation have predominantly examined Gaussian profiles − and Lorentzian profiles. − However, most of these studies used assumed conditions in which the peaks have low signal-to-noise ratios (SNR) and the noise follows the Gaussian noise limit. Consequently, their analyses are not applicable to high-precision measurement conditions, where the SNR is high and Poisson noise predominates. Under the Poisson noise limit, noise is intensity-dependent and is more complex. Studies of precision under this limit are scarce, even for Gaussian and Lorentzian profiles individually. ,, Moreover, the precision of area estimation is influenced by the covariance between intensity and bandwidth (as well as with η in the pseudo-Voigt profile), yet few studies have considered this factor, even for the well-studied Gaussian profiles. ,−

Accordingly, this study has three main objectives: (1) to elucidate the physical principles which define the limits of precision in estimating each peak parameter of the pseudo-Voigt profile under the Poisson noise limit, (2) to clarify the effect of optimizing the mixing parameter η on high-precision analysis, alongside conventional strategies such as increasing the SNR and using a finer sampling interval, thereby proposing new guidelines for precision enhancement, and (3) to provide a definitive answer to the longstanding question in analytical chemistry of whether intensity ratios or area ratios serve as more precise indicators. To achieve these objectives, we have developed a model to quantify the estimation precision of each peak parameter, in addition to their ratios and differences, for the pseudo-Voigt profile under the Poisson noise limit, integratively employing theoretical analyses based on the Cramér–Rao inequality and Fisher information, numerical calculations, and Monte Carlo simulations. For this study, we use general peak parameters such as I, Γ, the sampling interval (Δx), and η to establish a methodology that is independent of any specific analytical technique or field. Furthermore, we take advantage of a unique opportunity to compare the estimation precision of peak parameters for the pseudo-Voigt profile directly with those for Gaussian and Lorentzian profiles, thereby elucidating how profile shape differences affect the parameter estimation precision.

The model for peak parameter estimation precision derived in this study is expected to serve as a foundation for extensive research on analytical precision, with various applications including the following: (1) providing an alternative method for estimating errors when experimental error determination is challenging because of limited data; ,, (2) enabling the quantitative assessment of precision improvements resulting from instrumentation upgrades by integrating our model with analyses of the relationship between peak parameters and analytical device performance; (3) providing guidelines for system improvement by determining whether analytical precision has reached its theoretical limit or is constrained by external factors, thereby identifying bottlenecks in precision enhancement; and (4) serving as a benchmark for evaluating novel analytical methods.

Theory

Analytical expressions for the Fisher information matrix are derived under the following seven assumptions: (1) the noise variance equals the signal intensity at all data points. (2) The data sampling interval (Δx) remains constant throughout the analysis region. (3) The peak intensity is zero outside the sampling range. (4) The profile is sampled finely enough for summation to be approximated by integration. (5) The baseline is constant. (6) Peaks do not mutually interfere. (7) Noise at each data point is statistically independent. It can be approximated by a Gaussian probability density function because of the high mean intensity. This approximation is justified by the central limit theorem, which allows the Poisson noise distribution to approximate a Gaussian distribution when the signal intensity is high.

The pseudo-Voigt profile is expressed as a function of intensity (I), peak position (ω _c ), full width at half-maximum (Γ), and the mixing parameter (η) by the following equation

$${\gamma }^{\mathrm{P}\mathrm{V}}({\omega }_i;\mathit{\theta ̂})=I[(1-\eta )\exp \{-4\ln (2){\left(\frac{{\omega }_i-{\omega }_c}{\mathrm{\Gamma }}\right)}^2\}+\eta \frac{1}{1+{4\left(\frac{{\omega }_i-{\omega }_c}{\mathrm{\Gamma }}\right)}^2}]$$ 1

Here, the parameter vector is defined as θ = (θ₁, θ₂, θ₃, θ₄) = (I, ω _c , Γ, and η). The unit of I is not a digital count but an absolute physical unit that reflects the characteristics of Poisson noise (e.g., photons or electrons). Mixing parameter η represents the contribution of the Lorentzian component, where η = 0 corresponds to a purely Gaussian profile and η = 1 corresponds to a purely Lorentzian profile.

Next, the lower bound of the variance–covariance matrix $\mathbb{C}$ is constrained using the inverse of the Fisher information matrix $\mathbb{F}$ , , as expressed by the Cramér–Rao inequality: , $\mathbb{C}\ge {\mathbb{F}}^{-1}$ . This inequality implies a fundamental limit to the precision with which the model parameters can be estimated. The Fisher information matrix is defined as one-half of the Hessian matrix $\mathbb{H}$ , with its elements given as

$$F_{kl}=\frac{1}{2}{H}_{kl}=\frac{1}{2}\frac{{\partial }^2{\chi }^2}{\partial {\theta }_k\partial {\theta }_l}=\sum _i(\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_k}\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_l}+\frac{{\partial }^2\gamma ({\omega }_i;\theta )}{\partial {\theta }_k\partial {\theta }_l}{R}_i){\sigma }_i^{‐2}$$ 2

The quantity χ² is defined as the weighted sum of squared residuals: $\sum _i^n{\{[\gamma ({\omega }_i;\mathrm{\theta ̂})-{\gamma }_i]/{\sigma }_i\}}^2$ ) where the residual at the i-th data point is given as R _i (= $\gamma ({\omega }_i;\mathrm{\theta ̂})-{\gamma }_i$ ). Here, σ _i represents the total noise variance at each data point. For the pseudo-Voigt profile under the Poisson noise limit, it is given as ${{\sigma }_i}^2=\gamma ({\omega }_i;\mathrm{\theta ̂})$ . Because the second term in eq is negligible, the matrix elements of $\mathbb{F}$ can be approximated as $F_{kl}\approx \sum _i\left(\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_k}\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_l}\right){\sigma }_i^{-2}$ . − Furthermore, the summation in eq can be approximated as

$$F_{kl}\approx \frac{1}{\mathrm{\Delta }x}{\int }_{-\infty }^{\infty }\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_k}\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_l}{\sigma }_i^{-2}d{\omega }_i$$ 3

From eqs –, the elements of the Fisher information matrix F _kl for the pseudo-Voigt profile are given as

$$F_{11}^{\mathrm{P}\mathrm{V}}=\frac{\mathrm{\Gamma }}{2I\mathrm{\Delta }x}[\sqrt{\frac{\pi }{\ln 2}}(1-\eta )+\pi \eta ]$$ 4

$$F_{12}^{\mathrm{P}\mathrm{V}}=F_{23}^{\mathrm{P}\mathrm{V}}=F_{24}^{\mathrm{P}\mathrm{V}}=0$$ 5

$$F_{13}^{\mathrm{P}\mathrm{V}}=\frac{1}{2\mathrm{\Delta }x}[\sqrt{\frac{\pi }{\ln 2}}(1-\eta )+\pi \eta ]$$ 6

$$F_{14}^{\mathrm{P}\mathrm{V}}=\frac{\mathrm{\Gamma }}{2\mathrm{\Delta }x}[\pi -\sqrt{\frac{\pi }{\ln 2}}]$$ 7

$$F_{22}^{\mathrm{P}\mathrm{V}}=\frac{64I}{\mathrm{\Gamma }\mathrm{\Delta }x}{\int }_{-\infty }^{\infty }{\left(\frac{{\omega }_i-{\omega }_c}{\mathrm{\Gamma }}\right)}^2\frac{{K({\omega }_i)}^2}{(1-\eta )G({\omega }_i)+\eta L({\omega }_i)}d{\omega }_i$$ 8

$$F_{33}^{\mathrm{P}\mathrm{V}}=\frac{64I}{\mathrm{\Gamma }\mathrm{\Delta }x}{\int }_{-\infty }^{\infty }{\left(\frac{{\omega }_i-{\omega }_c}{\mathrm{\Gamma }}\right)}^4\frac{{K({\omega }_i)}^2}{(1-\eta )G({\omega }_i)+\eta L({\omega }_i)}d{\omega }_i$$ 9

$$F_{34}^{\mathrm{P}\mathrm{V}}=\frac{8I}{\mathrm{\Gamma }\mathrm{\Delta }x}{\int }_{-\infty }^{\infty }{\left(\frac{{\omega }_i-{\omega }_c}{\mathrm{\Gamma }}\right)}^2\frac{K({\omega }_i)[L({\omega }_i)-G({\omega }_i)]}{(1-\eta )G({\omega }_i)+\eta L({\omega }_i)}d{\omega }_i$$ 10

$$F_{44}^{\mathrm{P}\mathrm{V}}=\frac{\mathrm{\Gamma }I}{\mathrm{\Delta }x}{\int }_{-\infty }^{\infty }\frac{{[L({\omega }_i)-G({\omega }_i)]}^2}{(1-\eta )G({\omega }_i)+\eta L({\omega }_i)}d{\omega }_i$$ 11

Here, the functions are defined as G(ω _i ) = exp­[−4ln(2)­{(ω _i – ω _c )/Γ}²], L(ω _i ) = 1/[1 + 4­{(ω _i – ω _c )/Γ}²], and K(ω _i ) = [(1−η) ln(2)G(ω _i ) + ηL(ω _i )²]. Superscripts PV, G, and L respectively denote that the variable is associated with the pseudo-Voigt, Gaussian, and Lorentzian profile. In eqs –, the integrals include a numerator involving the squared combination of G(ω _i ) and L(ω _i ), whereas the denominator consists of their weighted sum, (1 – η)G(ω _i ) + ηL(ω _i ). As a result, these integrals cannot be expressed in terms of their elementary functions. Consequently, only the matrix elements F ₁₁ , F ₁₂ , F ₁₃ , F ₁₄ , F ₂₃ , and F ₂₄ have closed-form analytical expressions. As a result, the Fisher information matrix for the estimated parameter vector, θ̂, in the pseudo-Voigt profile, denoted by ${\mathbb{F}}^{\mathrm{P}\mathrm{V}}$ , is given as

$${\mathbb{F}}^{\mathrm{P}\mathrm{V}}=\left(\begin{array}{llll}\frac{\mathrm{\Gamma }[\sqrt{\frac{\pi }{\ln 2}}(1-\eta )+\pi \eta ]}{2I\mathrm{\Delta }x}& 0& \frac{[\sqrt{\frac{\pi }{\ln 2}}(1-\eta )+\pi \eta ]}{2\mathrm{\Delta }x}& \frac{\mathrm{\Gamma }\left[\pi -\sqrt{\frac{\pi }{\ln 2}}\right]}{2\mathrm{\Delta }x}\\ 0& F_{22}^{\mathrm{P}\mathrm{V}}& 0& 0\\ \frac{[\sqrt{\frac{\pi }{\ln 2}}(1-\eta )+\pi \eta ]}{2\mathrm{\Delta }x}& 0& F_{33}^{\mathrm{P}\mathrm{V}}& F_{34}^{\mathrm{P}\mathrm{V}}\\ \frac{\mathrm{\Gamma }\left[\pi -\sqrt{\frac{\pi }{\ln 2}}\right]}{2\mathrm{\Delta }x}& 0& F_{34}^{\mathrm{P}\mathrm{V}}& F_{44}^{\mathrm{P}\mathrm{V}}\end{array}\right)$$ 12

Because the Fisher information matrix is symmetric by definition, the matrix ${\mathbb{F}}^{\mathrm{P}\mathrm{V}}$ is also symmetric. For the Gaussian and Lorentzian profiles, the Fisher information matrices ${\mathbb{F}}^G$ and ${\mathbb{F}}^L$ are 3 × 3 matrices. All their elements can be expressed analytically as presented below ,

$${\mathbb{F}}^G=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}\left(\begin{array}{ccc}\frac{\mathrm{\Gamma }}{I\mathrm{\Delta }x}& 0& \frac{1}{\mathrm{\Delta }x}\\ 0& 8\ln 2\frac{I}{\mathrm{\Gamma }\mathrm{\Delta }x}& 0\\ \frac{1}{\mathrm{\Delta }x}& 0& \frac{3I}{\mathrm{\Gamma }\mathrm{\Delta }x}\end{array}\right)$$ 13

$${\mathbb{F}}^L=\frac{\pi }{2}\left(\begin{array}{ccc}\frac{\mathrm{\Gamma }}{I\mathrm{\Delta }x}& 0& \frac{1}{\mathrm{\Delta }x}\\ 0& \frac{2I}{\mathrm{\Gamma }\mathrm{\Delta }x}& 0\\ \frac{1}{\mathrm{\Delta }x}& 0& \frac{3I}{2\mathrm{\Gamma }\mathrm{\Delta }x}\end{array}\right)$$ 14

It is noteworthy that, for the Gaussian profile, the calculations of ${\mathbb{F}}^G$ , ${\mathbb{C}}^G$ , and ${\sigma }_{{\theta }_k}^{*G}$ were performed after correcting the coefficient error in the F ₃₃ term reported by Hagiwara and Kuwatani (details are provided in another report of our work). Figure presents the values of the matrix elements F _kl calculated for the Gaussian, Lorentzian, and pseudo-Voigt profiles based on eqs –. Theoretical values for the Gaussian and Lorentzian profiles (eqs and ) are in agreement with results of our numerical calculations (Figures S1 and S2).

1.

Dependence of the Fisher information matrix elements F _kl on the mixing parameter (η). Green circles and pink squares respectively represent the values of the Fisher information matrix elements F _kl for the Gaussian and Lorentzian profiles, computed under the conditions I = 1000, Γ = 10, and Δx = 0.1 according to eqs and . Gray dashed lines show the analytically derived values of F _kl for the pseudo-Voigt profile (specifically F ₁₁, F ₁₂, F ₁₃, F ₁₄, F ₂₃, and F ₂₄). Light blue crosses denote the numerically computed F _kl for the pseudo-Voigt profile, together with their corresponding fitted curves. All numerical calculations were performed with I = 1000, ω _c = 0, Γ = 10, Δx = 0.1, and η of 0.001–0.999 (Table S1).

Next, using eqs – and the relation $\mathbb{C}$ $={\mathbb{F}}^{-1}$ , we compute the variance–covariance matrix $\mathbb{C}$ . For the pseudo-Voigt profile, only C ₁₂ = C ₂₃ = C ₂₄ = 0 can be expressed analytically. By contrast, all elements of the covariance matrix for the Gaussian and Lorentzian profiles can be expressed in closed form as ,

$${\mathbb{C}}^G=\sqrt{\frac{\ln 2}{\pi }}\left(\begin{array}{ccc}\frac{3I\mathrm{\Delta }x}{\mathrm{\Gamma }}& 0& -2\mathrm{\Delta }x\\ 0& \frac{1}{4\ln 2}\frac{\mathrm{\Gamma }\mathrm{\Delta }x}{I}& 0\\ -2\mathrm{\Delta }x& 0& \frac{\mathrm{\Gamma }\mathrm{\Delta }x}{I}\end{array}\right)$$ 15

$${\mathbb{C}}^L=\frac{2}{\pi }\left(\begin{array}{lll}\frac{3I\mathrm{\Delta }x}{\mathrm{\Gamma }}& 0& -2\mathrm{\Delta }x\\ 0& \frac{\mathrm{\Gamma }\mathrm{\Delta }x}{2I}& 0\\ -2\mathrm{\Delta }x& 0& \frac{2\mathrm{\Gamma }\mathrm{\Delta }x}{I}\end{array}\right)$$ 16

Figure presents the covariance matrix elements C _kl for the Gaussian and Lorentzian profiles, as calculated from eqs and . The theoretical values are in agreement with results of our numerical calculations (Figures S3 and S4).

2.

Dependence of the variance–covariance matrix elements C _kl on the mixing parameter (η). Green circles and pink squares respectively display the values of the variance–covariance matrix elements C _kl for the Gaussian and Lorentzian profiles, calculated under the conditions I = 1000, Γ = 10, and Δx = 0.1 based on eqs and . Light blue crosses represent the numerically computed C _kl for the pseudo-Voigt profile, along with their fitted curves. Each element C _kl was obtained as the inverse of the Fisher information matrix (i.e., $\mathbb{C}={\mathbb{F}}^{-\mathbf{1}}$ ). Calculations were performed under the same conditions as those of Figure , with η of 0.001–0.999 (Table S1).

We then note that the standard deviation of each peak parameter σ_θk is given as $\sqrt{C_{\mathrm{k}\mathrm{k}}}$ . Using standard error propagation, we compute the relative or absolute standard deviations for I, ω _c , Γ, and η, as well as for the intensity ratio (R _I = I _w /I _s ), peak position difference (Δω = |ω _w – ω _s |), full width at half maximum (fwhm) ratio (R _Γ = Γ _w /Γ _s ), and η ratio (Rη = η _w /η _s ). Here, subscripts w and s stand for the weaker and stronger peaks. Because an analytical solution for C _kl is not available for the pseudo-Voigt profile, the standard deviations cannot be expressed in terms of elementary functions. In contrast, for the Gaussian and Lorentzian profiles, the standard deviations can be expressed analytically, as presented below.

For the Gaussian profile ,

$${\sigma }_I^{*G}=\frac{{\sigma }_I^G}{I}={\left\{3\sqrt{\frac{\ln 2}{\pi }}\frac{\mathrm{\Delta }x}{I\mathrm{\Gamma }}\right\}}^{1/2}$$ 17

$${\sigma }_{{\omega }_c}^G={\left\{\frac{1}{4\sqrt{\pi \ln 2}}\frac{\mathrm{\Gamma }\mathrm{\Delta }x}{I}\right\}}^{1/2}$$ 18

$${\sigma }_{\mathrm{\Gamma }}^{*G}=\frac{{\sigma }_{\mathrm{\Gamma }}^G}{\mathrm{\Gamma }}={\left\{\sqrt{\frac{\ln 2}{\pi }}\frac{\mathrm{\Delta }x}{I\mathrm{\Gamma }}\right\}}^{1/2}$$ 19

$${\sigma }_{R_I}^{*G}=\frac{{\sigma }_{R_I}^G}{R_I}={\left\{3\sqrt{\frac{\ln 2}{\pi }}\frac{1}{I_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 20

$${\sigma }_{\mathrm{\Delta }\omega }^G={\left\{\frac{1}{4\sqrt{\pi \ln 2}}\frac{1}{I_w}({\mathrm{\Gamma }}_w\mathrm{\Delta }{x}_w+R_I{\mathrm{\Gamma }}_s\mathrm{\Delta }{x}_s)\right\}}^{1/2}$$ 21

$${\sigma }_{R_{\mathrm{\Gamma }}}^{*G}=\frac{{\sigma }_{R_{\mathrm{\Gamma }}}^G}{R_{\mathrm{\Gamma }}}={\left\{\sqrt{\frac{\ln 2}{\pi }}\frac{1}{I_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 22

For the Lorentzian profile

$${\sigma }_I^{*L}=\frac{{\sigma }_I^L}{I}={\left\{\frac{6\mathrm{\Delta }x}{\pi I\mathrm{\Gamma }}\right\}}^{1/2}$$ 23

$${\sigma }_{{\omega }_c}^L={\left\{\frac{\mathrm{\Gamma }\mathrm{\Delta }x}{\pi I}\right\}}^{1/2}$$ 24

$${\sigma }_{\mathrm{\Gamma }}^{*L}=\frac{{\sigma }_{\mathrm{\Gamma }}^L}{\mathrm{\Gamma }}={\left\{\frac{4\mathrm{\Delta }x}{\pi I\mathrm{\Gamma }}\right\}}^{1/2}$$ 25

$${\sigma }_{R_I}^{*L}=\frac{{\sigma }_{R_I}^L}{R_I}={\left\{\frac{6}{{\pi I}_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 26

$${\sigma }_{\mathrm{\Delta }\omega }^L={\left\{\frac{1}{\pi I_w}({\mathrm{\Gamma }}_w\mathrm{\Delta }{x}_w+R_I{\mathrm{\Gamma }}_s\mathrm{\Delta }{x}_s)\right\}}^{1/2}$$ 27

$${\sigma }_{R_{\mathrm{\Gamma }}}^{*L}=\frac{{\sigma }_{R_{\mathrm{\Gamma }}}^L}{R_{\mathrm{\Gamma }}}={\left\{\frac{4}{{\pi I}_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 28

Here, the asterisk denotes the relative standard deviation.

Next, we derive the standard deviation of the area A ^PV. Area is a function of I, Γ, and η. It is given as

$$A^{\mathrm{P}\mathrm{V}}=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}(1-\eta )I\mathrm{\Gamma }+\frac{\pi }{2}\eta I\mathrm{\Gamma }$$ 29

For this derivation, the integration range is assumed to be infinite. However, for practical analyses, the integration range is finite; it might affect the computed peak area. Because A ^PV is proportional to the product IΓη, its precision must account for the covariances among these parameters. The variance of A ^PV, denoted ${{\sigma }_A^{\mathrm{P}\mathrm{V}}}^2$ , is given as the product of the covariance matrix ${\mathbb{C}}^{\mathrm{P}\mathrm{V}}$ and the gradient of A ^PV with respect to θ̂ as

$${{\sigma }_A^{\mathrm{P}\mathrm{V}}}^2=\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\mathrm{\theta ̂}}^T}{\mathbb{C}}^{\mathrm{P}\mathrm{V}}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\theta ̂}}$$ 30

with $\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\mathrm{\theta ̂}}^T}$ = $(\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I},\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\omega }_c},\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }},\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }{)}^T$ . Thus, the variance is expressed as

$${{\sigma }_A^{\mathrm{P}\mathrm{V}}}^2={\left(\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I}\right)}^2{{\sigma }_I^{\mathrm{P}\mathrm{V}}}^2+{\left(\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}\right)}^2{{\sigma }_{\mathrm{\Gamma }}^{\mathrm{P}\mathrm{V}}}^2+{\left(\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }\right)}^2{{\sigma }_{\eta }^{\mathrm{P}\mathrm{V}}}^2+2\sum _{i<j}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\theta }_i}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\theta }_j}\mathrm{C}\mathrm{o}\mathrm{v}({\theta }_i,{\theta }_j)$$ 31

The partial derivatives of A ^PV with respect to I, Γ, and η are

$$\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I}=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}(1-\eta )\mathrm{\Gamma }+\frac{\pi }{2}\eta \mathrm{\Gamma }$$ 32

$$\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}(1-\eta )I+\frac{\pi }{2}\eta I$$ 33

$$\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }=I\mathrm{\Gamma }[\frac{\pi }{2}-\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}]$$ 34

Furthermore, the covariance contribution in eq is

$$\sum _{i<j}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\theta }_i}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial {\theta }_j}\mathrm{C}\mathrm{o}\mathrm{v}({\theta }_i,{\theta }_j)=\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}{C}_{13}^{\mathrm{P}\mathrm{V}}+\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }{C}_{14}^{\mathrm{P}\mathrm{V}}+\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }{C}_{34}^{\mathrm{P}\mathrm{V}}$$ 35

In eq the first and third terms are negative, whereas the second term is positive (Figure S5). Because eq includes terms that cannot be expressed analytically, the standard deviation of the area for the pseudo-Voigt profile cannot be represented in the closed form. In contrast, the analytical solutions for the relative standard deviations of the area and the area ratio for the Gaussian and Lorentzian profiles are

$${\sigma }_A^{*G}=\frac{{\sigma }_A^G}{A^G}={\left\{2\sqrt{\frac{\ln 2}{\pi }}\frac{\mathrm{\Delta }x}{I\mathrm{\Gamma }}\right\}}^{1/2}$$ 36

$${\sigma }_A^{*L}=\frac{{\sigma }_A^L}{A^L}={\left\{\frac{2}{\pi }\frac{\mathrm{\Delta }x}{I\mathrm{\Gamma }}\right\}}^{1/2}$$ 37

$${\sigma }_{R_A}^{*G}=\frac{{\sigma }_{R_A}^G}{R_A}={\left\{2\sqrt{\frac{\ln 2}{\pi }}\frac{1}{I_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 38

$${\sigma }_{R_A}^{*L}=\frac{{\sigma }_{R_A}^L}{R_A}={\left\{\frac{2}{{\pi I}_w}(\frac{\mathrm{\Delta }{x}_w}{{\mathrm{\Gamma }}_w}+R_I\frac{\mathrm{\Delta }{x}_s}{{\mathrm{\Gamma }}_s})\right\}}^{1/2}$$ 39

From the equations presented above, the analytical expressions for the relative standard deviations of R _I , R _Γ, and R _A for the Gaussian and Lorentzian profiles are identical, apart from constant coefficients. Consequently, the ratios are ,

$${\sigma }_{R_I}^{*G}:{\sigma }_{R_{\mathrm{\Gamma }}}^{*G}:{\sigma }_{R_A}^{*G}=\sqrt{3}:1:\sqrt{2}$$ 40

$${\sigma }_{R_I}^{*L}:{\sigma }_{R_{\mathrm{\Gamma }}}^{*L}:{\sigma }_{R_A}^{*L}=\sqrt{3}:\sqrt{2}:1$$ 41

Figure presents the relative and absolute standard deviations computed based on eqs –.

3.

(a–e, g–k) Light blue crosses denote the standard deviations obtained from the diagonal elements of the numerically computed variance–covariance matrix, with corresponding fitted approximations shown as light blue curves. Green circles and pink squares respectively denote the theoretical standard deviations ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{G}}$ and ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{L}}$ for the Gaussian and Lorentzian profiles, calculated from eqs – and – under the conditions I = 1000, Γ = 10, and Δx = 0.1. (f, l) show the dependence of the relative standard deviation ratios (σ _{R I} /σ _{R A} , σ _{R Γ} /σ _{R A} , and σ _{R I} /σ _{R Γ} ) on the mixing parameter η. Filled circles and squares correspond to theoretical values at η = 0 and η = 1, respectively, as derived from eqs and . Numerical and theoretical results were obtained under I = 1000, ω _c = 0, Γ = 10, Δx = 0.1, with η varying from 0.0001 to 0.9999 (Table S1). Open symbols represent results from Monte Carlo simulations, with corresponding data in Table S2. Error bars show the 95% confidence intervals computed using eqs S15 and S16.

The precision of peak parameter estimation for Gaussian profiles under the Poisson noise limit was derived earlier by Hagen et al. and by Hagiwara and Kuwatani. However, their formulations differ in both the coefficients and the variable terms. This discrepancy likely arises from differences in the definition of Poisson noise: Hagen et al. defined the variance as σ _i = Δx ${\gamma }^G({\omega }_i;\mathit{\theta ̂})$ , whereas Hagiwara and Kuwatani used ${\gamma }^G({\omega }_i;\mathit{\theta ̂})$ . As a result, the analytical expression for precision derived by Hagen et al. lacks a Δx term, implying that the sampling interval does not influence precision. This conclusion contradicts both the qualitative reasoning and our simulation results. In contrast, the precision of peak parameter estimation for Lorentzian profiles under the Poisson noise limit has only been derived by Hagiwara and Kuwatani. Hagiwara and Kuwatani verified the consistency of eqs – and – using Monte Carlo simulations, whereas for this study, we further confirm their agreement with numerical calculations (Figures S2 and S4). No report of an earlier study has described the examination of the precision of peak parameter estimation for the pseudo-Voigt profile. Therefore, this study offers a unique opportunity for systematic comparison of the estimation precision of peak parameters for pseudo-Voigt, Gaussian, and Lorentzian profiles.

Methods

Numerical Calculation

For verification of the consistency between the analytical solutions and the numerical results, and for numerical evaluation of the matrix elements of the pseudo-Voigt profile that cannot be expressed in closed form, we performed numerical calculations of the Fisher information matrix $\mathbb{F}$ and its inverse, the covariance matrix $\mathbb{C}$ . These computations were implemented by using the Python programming language. The matrix elements F _kl were computed according to eq and S1–S4 as the integral: $\left(\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_k}\frac{\partial \gamma ({\omega }_i;\theta )}{\partial {\theta }_l}\right){\sigma }_i^{-2}$ , where integration was performed numerically over the interval [−50,000, 50,000] using the function scipy.integrate.quad. The calculations were conducted under the conditions of I = 1000, ω _c = 0, Γ = 10, and Δx = 0.1, with η varied from 0.0001 to 0.9999, resulting in a total of 42 distinct parameter sets for ${\mathbb{F}}^{\mathrm{P}\mathrm{V}}$ . In addition, the inverse of $\mathbb{F}$ was computed via the numpy.linalg.inv function to obtain $\mathbb{C}$ , thereby evaluating the lower bound of the precision in parameter estimation as stipulated by the Cramér–Rao inequality. The relative standard deviations of the area and area ratio were calculated by combining the obtained variance–covariance matrix with eqs –.

The results obtained for ${\mathbb{F}}^{\mathrm{P}\mathrm{V}}$ and ${\mathbb{C}}^{\mathrm{P}\mathrm{V}}$ are depicted in Figures and , where they are represented by light blue cross symbols and corresponding approximation curves. The standard deviation for each peak parameter, computed as ${\sigma }_{{\theta }_k}=\sqrt{C_{\mathrm{k}\mathrm{k}}}$ , is presented in Figure with light blue cross symbols and fitted curves. All numerical data for $\mathbb{F}$ , $\mathbb{C}$ , and σ_θk are presented in Table S1.

Monte Carlo Simulation

To validate the precision of peak parameter estimates obtained from theoretical and numerical calculations, we performed Monte Carlo simulations. For this approach, we generated synthetic signal data and conducted curve fitting to extract the peak parameters. The standard deviations were then computed from the dispersion of the fitted parameter values.

In the simulation, two pseudo-Voigt profiles were generated with peak positions set as ω _c, w = 4000 and ω _c, s = 12,000. The expected intensities of the two peaks were fixed respectively as I _w = 1000 and I _s = 10,000. A sampling interval of Δx = 0.1 and a bandwidth of Γ = 10 were used, whereas η was varied in increments of 0.1 from 0.1 to 0.9, yielding data under nine different conditions. Under these simulation conditions, the separation ratio Δω/Γ exceeded 800 in every case. For the Lorentzian function, the intensity at a distance of 800Γ from the peak center decreases to approximately 4 × 10^–7 of the peak intensity. For the Gaussian function, the intensity at a distance of 3Γ drops to about 1 × 10^–11 of the peak intensity, indicating that interference between the peaks is negligible.

For each condition, 999 data sets were generated. Then peak parameter extraction was performed using Python’s nonlinear fitting library (scipy.optimize.curve_fit). To facilitate rapid convergence to the global minimum, known parameter values were used as initial estimates. A cost function minimizing the weighted sum of the squared residuals was employed. This approach is analogous to those used in widely adopted curve-fitting software such as Fityk and GRAMS/AI (Thermo Fisher Scientific Inc.). The convergence tolerances (xtol = gtol = ftol) were initially set to 1 × 10^–15. If convergence was not achieved, then the tolerances were relaxed incrementally by 2 orders of magnitude. The trust region reflective algorithm was adopted as the optimization method, with the maximum number of iterations set to 10,000.

A total of 8991 signal data sets yielded extracted peak parameters, which are presented in Table S2. The mean values and standard deviations of the peak parameters across the nine conditions are presented in Table S3.

Results

Figure a–j displays the η dependence of the Fisher information matrix elements F _kl as obtained from both theoretical derivations and numerical calculations. For F ₁₁ , F ₁₂ , F ₁₃ , F ₁₄ , F ₂₃ , and F ₂₄ , all elements with available analytical solutions, and the theoretical and numerical results are in excellent agreement, confirming the high reliability of both methods. Moreover, the values of F ₁₁ , F ₁₂ , F ₁₃ , and F ₂₃ can be expressed as a weighted sum of the Gaussian and Lorentzian components, i.e., F _kl = (1 – η)F _kl + ηF _kl . In contrast, F ₂₂ and F ₃₃ cannot be expressed in this form. However, in the limits as η → 0 and η → 1, F _kl converges respectively to F _kl and F _kl . Additionally, F ₃₄ and F ₄₄ increase exponentially in the vicinity of η ≈ 0, indicating that as η → 0, the information content for η is high, allowing it to be estimated with high precision.

Figure a–j presents a comparison of the variance–covariance matrix elements C _kl obtained from theory and numerical calculations. It is noteworthy that for C ₁₁ , C ₁₃ , and C ₃₃ , the values converge to those of the Gaussian profile, C _kl , in the limit as η → 0. In contrast, for η → 1, the absolute values satisfy |C _kl | > |C _kl |, which indicates that the effect of mixing Gaussian and Lorentzian components on the estimation precision is asymmetric with respect to η. Furthermore, although C ₂₂ does not correspond exactly to a weighted sum of C ₂₂ and C ₂₂ , it converges to C ₂₂ and C ₂₂ , respectively, in the limits as η → 0 and η → 1. As with the pure Gaussian and Lorentzian cases, C ₂₂ is given simply as the reciprocal of F ₂₂ . , With the exception of C ₁₂ , C ₂₃ , and C ₂₄ , which are identically zero, no element of the covariance matrix ${\mathbb{C}}^{\mathrm{P}\mathrm{V}}$ can be expressed as a weighted sum of C _kl and C _kl .

Figure presents a comparison of the estimation precision of the peak parameters, as found using theoretical analysis, numerical calculations, and Monte Carlo simulations. For all parameters, the numerical and simulation results agree very well. Moreover, for all parameters (with the exception of those pertaining to η), the relative standard deviation ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ for the pseudo-Voigt profile converges to that of the Gaussian profile, ${\sigma }_{{\theta }_k}^{*G}$ , in the limit as η → 0. However, in the limit as η → 1, ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ matches ${\sigma }_{{\theta }_k}^{*L}$ only for ω _c , Δω, A, and R _A . For these parameters, ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ can be reasonably approximated using a weighted sum of ${\sigma }_{{\theta }_k}^{*G}$ and ${\sigma }_{{\theta }_k}^{*L}$ (i.e., ${\sigma }_{{\theta }_k}^{*G+\mathrm{L}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=(1-\eta ){\sigma }_{{\theta }_k}^{*G}+{\eta \sigma }_{{\theta }_k}^{*L}$ , as shown in Figure b,e,h,k). In contrast, for I, R _I , Γ, and R _Γ, in the limit as η → 1, ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ is up to 2.26 times larger (for Γ and R _Γ; Figure c,i) and 1.22 times larger (for I and R _I ; Figure a,g) than the values predicted by the weighted sum approximation. The estimation precision for η itself exhibits marked asymmetry with respect to its value, approaching near-zero precision as η → 0 (Figure d). Moreover, for the Gaussian and Lorentzian profiles, the precision values of R _A (or A) are superior to those for R _I (or I), respectively, by factors of $\sqrt{3/2}$ and $\sqrt{3}$ (eqs and ). In the case of the pseudo-Voigt profile, in the limit as η → 1, the ratios ${\sigma }_{R_I}^{*\mathrm{P}\mathrm{V}}/{\sigma }_{R_A}^{*\mathrm{P}\mathrm{V}}$ and σ* _I /σ* _A approach 2.11, indicating that the precision of R _A (or A) is up to 2.11 times better than that of R _I (or I) (Figures f,l). This finding implies that using R _A (or A) instead of R _I (or I) allows for precision equivalent to that obtained with a signal that is up to 4.45 times stronger.

Figure presents the relative error distributions of the parameters obtained from Monte Carlo simulations alongside the standard deviation ellipses computed numerically from eqs –, demonstrating excellent agreement between the two approaches. It is noteworthy that a negative correlation between I and Γ is observed, which is consistent with the negative value of C ₁₃ (Figure c). In other words, as I increases, Γ tends to decrease; as I decreases, Γ tends to increase. This tendency is more pronounced for the Lorentzian profile than for the Gaussian profile. In fact, it becomes even stronger for the pseudo-Voigt profile at η > 0.42 (Figures c and b). In the pseudo-Voigt profile, as η → 1, C ₁₃ decreases further below that of the Lorentzian profile (Figure c), thereby leading to a stronger correlation between I and Γ (Figure b). The observed positive correlation between I and η (Figure c) is consistent with the positive value of C ₁₄ (Figure d), indicating that as I increases, η also increases. This finding suggests that in the pseudo-Voigt profile, a larger η corresponds to a stronger contribution from the Lorentzian component, which, in turn, results in a higher peak intensity. By contrast, the negative correlation between Γ and η (Figure f) corresponds to the negative value of C ₃₄ (Figure i) and indicates that as η increases, Γ tends to decrease. Finally, no significant correlation is observed among I and ω _c , ω _c and Γ, or ω _c and η, which is consistent with the fact that C ₁₂, C ₂₃, and C ₂₄ are all zero (Figure a,d,e).

4.

Correlation between relative errors of peak parameters in the pseudo-Voigt profile: (a) intensity vs. peak position, (b) intensity vs. fwhm, (c) intensity vs. mixing parameter, (d) peak position vs. fwhm, (e) peak position vs. mixing parameter, and (f) fwhm vs. mixing parameter. Blue symbols represent 999 data points obtained from simulated signal data under the conditions I = 1000, ω _c = 4000, Γ = 10, Δx = 0.1, and η = 0.5. Black solid, dashed, and dotted lines respectively show covariance ellipses corresponding to 1σ, 2σ, and 3σ levels, computed from the Fisher information matrix under these conditions. Green and pink dotted lines respectively show the 3σ covariance ellipses for the Gaussian and Lorentzian profiles, calculated under I = 1000, ω _c = 0, Γ = 10, and Δx = 0.1. Simulation data are presented in Table S2.

Discussion

Empirical Expression for Peak Parameter Estimation Precision in the Pseudo-Voigt Profile

For Gaussian and Lorentzian profiles, the estimation precision of peak parameters can be derived analytically as functions of I, Γ, and Δx (eqs –). However, for the pseudo-Voigt profile, elements F ₂₂, F ₃₃, F ₃₄, and F ₄₄ cannot be expressed in terms of elementary functions (eqs –). Consequently, an analytical solution for the standard deviation is not available. For this subsection, we therefore derive an empirical expression for the estimation precision of peak parameters in the pseudo-Voigt profile.

Because the pseudo-Voigt profile is a weighted sum of Gaussian and Lorentzian functions (eq ), one of the most straightforward methods for expressing the estimation precision of peak parameters is to approximate it as a weighted sum of the analytical solutions for the precision of the Gaussian and Lorentzian components (i.e., ${\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=(1-\eta ){\sigma }_{{\theta }_k}^{*G}+{\eta \sigma }_{{\theta }_k}^{*L}$ ). Indeed, for ω _c , Δω, A, and R _A , it has been confirmed that ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ can be reasonably approximated by ${\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ (Figure b,e,h,k). Because ${\sigma }_{{\theta }_k}^{*G}$ and ${\sigma }_{{\theta }_k}^{*L}$ are known (eqs –), an empirical formula for estimating ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ under arbitrary conditions of I, Γ, Δx, and η is obtainable if the ratio between ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ and ${\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ can be modeled. To ascertain the relation between I, Γ, Δx, η, and ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ , we performed numerical calculations of ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ over a total of 237,600 conditions with I = 10³–10⁶, Γ = 1–10, Δx = 0.01–1.0, and η = 0.001–0.999. The results were compared with ${\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ (Table S4 presents related details). These calculations were performed for values of Γ/Δx ranging from 1 to 1000, corresponding to cases where between 1 and 1000 data points are included within the fwhm: a range that covers many practical scenarios in chemical analysis.

Figure presents that, for all peak parameters, the ratio ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}/{\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ depends solely on η: it is independent of Γ and Δx. To model the η dependence of this ratio, we performed curve fitting using various candidate models, including first–24th order polynomials, exponential functions, logarithmic functions, rational functions, and combinations of polynomials with other functions. Particularly, numerical calculations demonstrated that for I and Γ, the model function must satisfy f _{θ k} (0) = 1. For ω _c and A, it must satisfy both g _{θ k} (0) = 1 and g _{θ k} (1) = 1. Also, for η, it must satisfy h _{θ k} (0) = 0. Consequently, we selected model functions that fulfill these boundary constraints. To avoid convergence to local minima, the fitting procedure was repeated with random initial guesses for the parameters. The model was selected by comparing the residual sum of squares, Akaike’s information criterion, Bayesian information criterion, and 5-fold cross-validation results for all candidate models. Detailed information related to the evaluated model functions, the model selection procedure, and the fitting results is provided in the Supporting Information and Table S5. As a result, for I and Γ, the function defined by eq was chosen, whereas for ω _c and A, the function defined by eq was adopted, as presented below.

$$f_{{\theta }_k}(\eta )=1+\sum _{i=1}^8(p_i{\eta }^i)+p_9\ln (1+p_{10}\eta )$$ 42

$$g_{{\theta }_k}(\eta )=1+\eta (\eta -1)[\sum _{i=1}^8(p_i{\eta }^{i-1})+p_9\ln (1+p_{10}\eta )]$$ 43

The fitted parameters p _i are presented in Table . Using eqs and , the empirical formulas for ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ as functions of I, Γ, Δx, and η are given as

$${\sigma }_I^{*\mathrm{P}\mathrm{V}}=f_I(\eta )[(1-\eta ){\sigma }_I^{*G}+{\eta \sigma }_I^{*L}]$$ 44

$${\sigma }_{{\omega }_c}^{\mathrm{P}\mathrm{V}}=g_{{\omega }_c}(\eta )[(1-\eta ){\sigma }_{{\omega }_c}^G+{\eta \sigma }_{{\omega }_c}^L]$$ 45

$${\sigma }_{\mathrm{\Gamma }}^{*\mathrm{P}\mathrm{V}}=f_{\mathrm{\Gamma }}(\eta )[(1-\eta ){\sigma }_{\mathrm{\Gamma }}^{*G}+{\eta \sigma }_{\mathrm{\Gamma }}^{*L}]$$ 46

$${\sigma }_A^{*\mathrm{P}\mathrm{V}}=g_A(\eta )[(1-\eta ){\sigma }_A^{*G}+{\eta \sigma }_A^{*L}]$$ 47

5.

Dependence of the normalized standard deviation of peak parameters in the pseudo-Voigt profile’s peak parameters on the mixing parameter η, together with the best-fit model. The normalized standard deviation is defined as the ratio ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathbf{k}}}^{*\mathbf{P}\mathbf{V}}/{\mathit{\sigma }}_{{\mathit{\theta }}_{\mathbf{k}}}^{*\mathbf{G}+\mathbf{L}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{a}\mathbf{r}}$ , where the predicted standard deviation based on the weighted sum of the Gaussian and Lorentzian contributions is ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{G}+\mathit{L}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{a}\mathbf{r}}=(\mathbf{1}-\mathit{\eta }){\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{G}}+{\mathit{\eta }\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{L}}$ . (a–d) respectively present results for (a) intensity, (b) peak position, (c) full width at half-maximum (fwhm), and (d) area. Blue symbols are values of ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathbf{P}\mathbf{V}}/{\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{k}}}^{*\mathit{G}+\mathit{L}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{a}\mathbf{r}}$ obtained from numerical calculations performed under 237,600 conditions spanning the parameter ranges I = 10³–10⁶, Γ = 1–10, Δx = 0.01–1.0, and η = 0.001–0.999 (Table S4). The best-fit model is described by eqs and , with corresponding model parameters presented in Table .

1. Fitting Parameters for Eqs. – and .

	f I (η)	gω(η)	fΓ(η)	g A (η)	fη(η)
p 1	0.18048	–0.65277	1.60950	0.06110	4.34923
p 2	–0.78572	–0.71219	–5.36278	0.04935	–16.23451
p 3	3.84928	2.23240	23.01545	0.04500	75.01915
p 4	–10.29070	–6.01379	–60.20845	–0.26126	–204.52516
p 5	17.18979	10.75665	98.70767	0.76342	344.59331
p 6	–16.91906	–11.78860	–96.21326	–1.19016	–342.02286
p 7	8.97672	7.11471	50.90463	0.93835	184.20357
p 8	–2.00891	–1.80230	–11.32548	–0.29406	–41.63627
p 9	0.00477	0.13412	0.02385	–0.28274	0.04816
p 10	308.75319	383.23065	275.06340	0.23319	1857.31525

For these equations, the maximum relative errors, defined as 100 × $\underset{1\le i\le n}{\max }|{{\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}}_i-\hat{{{\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}}_i}|/{{\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}}_i$ , are 0.26%, 0.05%, 0.89%, and 0.002%, respectively, for eqs –, which is sufficiently precise for practical purposes. By combining eqs – with standard error propagation, we can calculate the η-dependence of the standard deviations for I, ω _c , Γ, and A, as well as for R _I , Δω, R _Γ, and R _A (Figure ).

6.

Dependence of standard deviations of various parameters on Γ/Δx or ΓΔx in the pseudo-Voigt profile. Panels show the standard deviations of intensity, peak position, bandwidth, area, intensity ratio, peak position difference, bandwidth ratio, and area ratio as predicted using the empirical equations (eqs –). Green and pink dashed lines respectively show theoretical solutions for pure Gaussian and pure Lorentzian cases (eqs – and –). Results are presented for four values of the mixing parameter: η = 0.01, 0.33, 0.66, and 0.99.

Because η is a parameter that is not present in the Gaussian or Lorentzian profiles, its standard deviation cannot be expressed as a weighted sum of ${\sigma }_{{\theta }_k}^{*G}$ and ${\sigma }_{{\theta }_k}^{*L}$ as in eqs –. However, based on the same 237,600 data sets, we determined empirically that the standard deviation of η, σ_η, is proportional to (Δx/ΓI)^0.5, similar to I, Γ, and A. To account for the effect of η on σ_η, we tested several models, which revealed that the function h _η(η) defined by the following equation performed best.

$$h_{\eta }(\eta )=\sum _{i=1}^8(p_i{\eta }^i)+p_9\ln (1+p_{10}\eta )$$ 48

Consequently, we modeled σ_η as

$${\sigma }_{\eta }^{\mathrm{P}\mathrm{V}}=h_{\eta }(\eta ){\left\{\frac{\mathrm{\Delta }x}{I\mathrm{\Gamma }}\right\}}^{1/2}$$ 49

The parameters for h _η(η) obtained from the fitting are presented in Table . As shown in Figure , the numerical results for σ_η agree extremely well with those in eq . The maximum relative error was 8.6% at η = 0.001 and 0.7% for η ≥ 0.01, demonstrating that the model explains the numerical results with practically sufficient precision.

7.

Comparison of the standard deviations of the mixing parameter (η) for the pseudo-Voigt profile obtained from the empirical formula (eq ) and numerical calculations. Results were obtained under five conditions with η = 0.01, 0.25, 0.5, 0.75, and 0.99. Cross symbols denote σ _η values obtained from numerical calculations under the conditions I = 1000, Γ = 1–10, Δx = 0.01–1.0, and η = 0.001–0.999 (Table S4).

The deviation between ${\sigma }_{{\omega }_{\mathrm{c}}}^{\mathrm{P}\mathrm{V}}$ and ${\sigma }_{{\omega }_{\mathrm{c}}}^{\mathrm{G}+\mathrm{L}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ reaches a maximum of 2.7% at η ≈ 0.33, whereas that between σ* _A and σ* _A reaches a maximum of 1.4% at η ≈ 0.48 (Figure b,d). Because uncertainties are generally reported to one or two significant digits, these minor deviations are practically negligible. Consequently, for ω _c , Δω, A, and R _A , it is acceptable to approximate ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ by ${\sigma }_{{\theta }_k}^{*G+L\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}$ . In contrast, for I (or R _I ), in the limit η → 1, σ* _I is up to 22% larger than σ* _I (Figure a). Moreover, the uncertainties predicted by the weighted-sum model are up to ten times worse than those for ω _c or A. Even so, because uncertainties are typically reported to one or two significant digits, an increase by a factor of approximately 1.22 is generally negligible. Therefore, the estimation precision for I and R _I can also be approximated as ${\sigma }_{{\theta }_k}^{*\mathrm{P}\mathrm{V}}$ ∼ σ* _I without any important practical issue. By contrast, the deviation between σ*_Γ and σ*_Γ reaches a maximum of 126% in the limit as η → 1 (Figures c and c), implying that the signal intensity required to achieve a given target precision could be underestimated by as much as a factor of 2.26² (∼5.11). Consequently, we recommend using eq instead of the simple weighted-sum model for estimating σ*_Γ .

Guidelines for Instrument and Measurement Condition Optimization in High-Precision Analysis

This paper presents a novel approach that extends conventional strategies for optimizing instrumentation and measurement conditions for the high-precision estimation of peak parameters I, ω _c , Γ, and A. Traditionally, under the Poisson noise limit for Gaussian and Lorentzian profiles, it has been considered that a wide bandwidth, fine sampling interval, and high signal intensity, i.e., minimizing Δx/ΓI, are key factors for improving the precision of I, Γ, and A estimation. , From the present study, we have demonstrated that this principle also applies to the pseudo-Voigt profile when the shape parameter η remains constant, as shown by eqs , , and and Figure . Furthermore, when the condition Δx _w /Γ _w ≫R _I Δx _s /Γ _s is satisfied, the precision of the estimated intensity ratio, fwhm ratio, and area ratio is governed primarily by two factors: the number of sampling intervals spanning the fwhm of the weaker peak (quantified by Γ _w /Δx _w ) and the intensity of the weaker peak (I _w ). Consequently, achieving high precision requires minimizing the Δx _w /Γ _w I _w . However, if the bandwidth of the stronger peak is exceptionally narrow, the sampling interval for the stronger peak is coarse, or the intensity ratio approaches unity, then the characteristics of the stronger peak might also strongly influence the precision of R _I , R _Γ, and R _A estimations.

Similarly to the optimization of I, Γ, and A, for Gaussian and Lorentzian profiles, the precision of ω _c is enhanced by refining the sampling interval and increasing the signal intensity. However, in contrast, a narrower bandwidth is preferable for ω _c estimation. , For our study, we demonstrated further that, for the pseudo-Voigt profile, minimizing ΓΔx/I is effective for reducing ${\sigma }_{{\omega }_c}^{\mathrm{P}\mathrm{V}}$ (eq ; Figure ). Moreover, regarding the precision of the peak position difference Δω, if the condition Γ _w Δx _w ≫R _I Γ _s Δx _s is met, then minimizing Γ _w Δx _w /I _w is effective for enhancing precision. However, if the bandwidth of the stronger peak is exceptionally wide, its sampling interval is coarse, or the intensity ratio is nearly 1, then the characteristics of the stronger peak might strongly affect the precision of the Δω estimation.

A central finding of this study is that even when I, Γ, and Δx are held constant, the precision of parameter estimation varies significantly with η (Figure ). Particularly as η approaches 1, as the profile becomes increasingly Lorentzian, the precision of both the Γ and I estimation deteriorates considerably (Figure a,c). Traditional approaches have primarily emphasized improvement of the signal-to-noise ratio and refining the sampling interval to enhance precision. Our results suggest that appropriate control of η provides a novel strategy for precision improvement.

Parameters Γ and η in the signal data are influenced by various broadening effects. For instance, when Doppler or instrumental broadening predominates, the profile tends to be Gaussian. , In contrast, when natural, pressure, or power broadening is dominant, the profile becomes more Lorentzian. , These effects can be modulated by adjusting the sample’s temperature and pressure conditions , as well as the analytical settings (e.g., slit width, grating constant, focal length of spectrometer, and detector pixel size). , Consequently, deliberately inducing broadening that yields a more Gaussian-like profile is particularly effective at enhancing the precision of I and Γ estimations (Figure a,c).

Specifically, changing η from 1 to 0 improves the precision of Γ estimation by up to 3.7-fold, an effect that is equivalent to increasing the signal intensity by approximately 14-fold. Similarly, the precision of both intensity and intensity ratio estimations improves by a factor of 1.4 when η is altered from 1 to 0, which is comparable to doubling the signal intensity. Moreover, because the precision of I and Γ estimation is proportional to (Δx/ΓI)^0.5 (eqs , , and ), an increase in Γ because of broadening also enhances precision. For example, if Γ doubles, then the precision improves further by a factor of $\sqrt{2}$ , which is equivalent to the improvement achieved by doubling the signal intensity.

In many analytical applications, such as comparison of the concentration or isotope ratios between two substances, the primary goal is to achieve a high-precision estimation of peak parameters rather than determine their absolute values directly. In these cases, any broadening-induced modifications to peak shape are of little consequence because calibration using standard samples ensures that the absolute parameter values remain reliable for comparative purposes.

Nonetheless, these high-precision measurement strategies have inherent limitations. For instance, in practical analyses, adjusting η is not always feasible due to methodological constraints and analyte-specific behaviors. In addition, factors such as fluctuations in I caused by changes in analytical conditions and the impact of increased Γ on peak interference must be addressed comprehensively to maintain precision. Despite these limitations, this study’s insights reveal that controlling η beyond conventional methods such as enhancing the signal-to-noise ratio and refining the sampling interval offers a deeper understanding of the impact of analytical conditions on precision. Ultimately, this approach provides effective guidelines for optimizing analytical systems.

Finally, based on the theoretical framework developed by previous studies, , we briefly demonstrate how our precision models can inform the design and optimization of common grating-based spectrometers, such as those used in Raman, fluorescence, and absorption spectroscopy, in which dispersed light is detected by an area detector. In a system characterized by focal length f, grating constant k, and half the angle between the incident and diffracted beams (α/2), the reciprocal dispersion (RD) is given by

$$\mathrm{R}\mathrm{D}=\frac{\cos (\alpha /2+\mathrm{\Phi })\times k}{f}$$ 50

where the grating angle Φ for first-order diffraction is

$$\mathrm{\Phi }={\sin }^{-1}\left[\frac{\lambda }{2k\cos \alpha /2}\right]$$ 51

Here, λ denotes the wavelength. The sampling interval Δx is then expressed as

$$\mathrm{\Delta }x=w_{\det }\times \mathrm{R}\mathrm{D}$$ 52

where w _det denotes the pixel width of the detector. The effective fwhm, Γ, of a peak can be approximated as the convolution of the true line width Γ_Nat, the instrumental spectral resolution Γ_Spec, and additional broadening effects Γ_Other, including excitation laser line width, Doppler broadening, and natural-, pressure-, or power-induced broadening

$$\mathrm{\Gamma }=\sqrt{{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{a}\mathrm{t}}^2+{\mathrm{\Gamma }}_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}^2+{\mathrm{\Gamma }}_{\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}^2}$$ 53

The instrumental contribution Γ_Spec is expressed as

$${\mathrm{\Gamma }}_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}=\frac{k{b}_{\mathrm{i}\mathrm{m}\mathrm{g}}}{f}\cos \{{\sin }^{-1}\left[\frac{\lambda }{2k\cos (\alpha /2)}\right]+\frac{\alpha }{2}\}$$ 54

where b _img is the width of the slit image on the detector, determined by the entrance slit width b _ent and the optical configuration. By substituting these expressions for Δx and Γ into our precision models (eqs – and ), we provide a method for quantitatively predicting how instrumental changes, such as pixel size, slit width, grating groove density, and focal length, affect the precision of I, ω _c , Γ, and A. While the above example assumes a grating-based spectrometer with area detection, the procedure is equally applicable to other analytical systems: one needs only to express Δx and Γ as functions of the instrument’s performance parameters and substitute them into our models to derive a direct relationship between measurement precision and hardware specifications. For a concrete illustration of how variations in focal length, pixel size, grating groove density, and slit width influence the precision of area ratios in Gaussian and Lorentzian profiles, see Figure of Hagiwara and Kuwatani. These relationships are particularly valuable for guiding instrument design and upgrades to meet specific precision requirements as well as for optimizing research budgets by identifying cost-effective performance enhancements.

Which is More Precise for the Pseudo-Voigt Profile: The Intensity Ratio or Area Ratio?

The question of whether the intensity ratio (R _I ) or the area ratio (R _A ) yields higher estimation precision has been a longstanding debate in analytical chemistry because this choice directly influences researchers’ decision-making regarding the precise estimation of target variables during data analysis. ,− Because of various factors, including noise and instrumental differences, distortions in signal data, and large uncertainties associated with variance estimation, earlier experimental studies undertaken to address this issue faced challenges in drawing definitive conclusions. To minimize the effects of experimental uncertainties and to provide a more definitive answer, a combination of theoretical analysis, numerical calculations, and Monte Carlo simulations is highly effective.

Using a theoretical approach, we demonstrated that, for Gaussian and Lorentzian profiles, the relative standard deviations of the intensity ratio and area ratio are given respectively as ${\sigma }_{R_I}^{*G}/{\sigma }_{R_A}^{*G}=\sqrt{3/2}$ and ${\sigma }_{R_I}^{*L}/{\sigma }_{R_A}^{*L}=\sqrt{3}$ (eqs and ), confirming that the estimation precision of the area ratio is superior. In practical terms, using the area ratio instead of the intensity ratio is equivalent to achieving the same precision under signal conditions that are 1.5 and 3 times stronger. For this study, we extended this analysis to the pseudo-Voigt profile and showed that in the limit as η → 1, the ratio ${\sigma }_{R_I}^{*\mathrm{P}\mathrm{V}}/{\sigma }_{R_A}^{*\mathrm{P}\mathrm{V}}$ to 2.11 suggests that the precision advantage of the area ratio is even greater, corresponding to a precision equivalent to that achieved under signal conditions up to 4.45 times stronger (Figure l). In this subsection, we discuss (1) why the estimation precision of R _A is superior to that of R _I and (2) why, as η approaches 1, the ratio ${\sigma }_{R_I}^{*G}/{\sigma }_{R_A}^{*G}$ exceeds the values $\sqrt{3/2}$ and $\sqrt{3}$ observed respectively for Gaussian and Lorentzian profiles.

Earlier studies , have demonstrated that, for Gaussian and Lorentzian profiles, the superior precision of the area ratio over the intensity ratio is attributable to the negative covariance between I and Γ. Specifically, when I increases, Γ tends to decrease and vice versa. Therefore, the product IΓ, which is proportional to the area, remains approximately constant (Figure b). However, this negative covariance also means that the uncertainty in one parameter amplifies the uncertainty in the other, making it difficult to estimate I or Γ independently with high precision. This interplay among the peak parameters ultimately results in R _A being estimated more precisely than R _I .

For the pseudo-Voigt profile, the influence of η must also be considered. In the limit of η → 0, the correlations between I and η and between Γ and η vanish (Figure d,i), meaning that η does not affect the estimation of other parameters. Consequently, the estimation precision of the pseudo-Voigt profile converges to that of the Gaussian profile in this limit. The ratio ${\sigma }_{R_I}^{*\mathrm{P}\mathrm{V}}/{\sigma }_{R_A}^{*\mathrm{P}\mathrm{V}}$ asymptotically approaches $\sqrt{3/2}$ . Conversely, as η approaches 1, the correlation between I and η increases monotonically (Figure d), whereas the correlation between Γ and η decreases monotonically (Figure i). Therefore, near η = 1, η begins to influence the estimation of both I and Γ considerably. Particularly in the limit as η → 1, the correlation coefficient between Γ and η (i.e., $C_{13}/\sqrt{C_{11}{C}_{33}}$ ) reaches −0.90, indicating a very strong negative correlation that degrades the precision of independently estimating Γ (Figures c,f and c). Moreover, the correlation coefficient between I and η (i.e., $C_{14}/\sqrt{C_{11}{C}_{44}}$ ) reaches 0.57 in this limit, signifying a moderate correlation that also adversely affects the precision of I estimation (Figures a,c and a). As a consequence, the precision of the intensity ratio worsens, causing the ratio ${\sigma }_{R_I}^{*\mathrm{P}\mathrm{V}}/{\sigma }_{R_A}^{*\mathrm{P}\mathrm{V}}$ to increase. In contrast, the estimation precision of the area and area ratio remains unaffected because the negative covariances between I and Γ and between Γ and η tend to preserve the product IΓη (i.e., the area remains nearly constant). In fact, when comparing the contributions of individual terms in eq to the precision of the area, the covariance terms $\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial I}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}{C}_{13}^{\mathrm{P}\mathrm{V}}$ and $\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \mathrm{\Gamma }}\frac{\partial A^{\mathrm{P}\mathrm{V}}}{\partial \eta }{C}_{34}^{\mathrm{P}\mathrm{V}}$ contribute greatly to the improvement in precision (Figure S5). Therefore, to maximize the estimation precision of the target variable, it is imperative not only to control the mixing parameter η but also to judiciously select the peak parameters.

Conclusions

For this study, we employed a multifaceted approach combining theoretical analysis, numerical calculations, and Monte Carlo simulations to elucidate the physical principles governing the estimation precision of pseudo-Voigt profile parameters: intensity (I), peak position (ω _c ), full width at half-maximum (Γ), mixing parameter (η), and area (A). We developed a model that expresses estimation precision in terms of I, Γ, η, and the sampling interval (Δx), providing a comprehensive framework that bridges theoretical precision limits with practical experiment parameters.

Our analysis revealed that under constant η, the precision of I, Γ, and A improves as the ratio Δx/ΓI is minimized, whereas the precision of ω _c is enhanced by reducing ΓΔx/I. Furthermore, our investigation into the η-dependence of peak parameter estimations demonstrated that decreasing η from 1 to 0 improves the estimation precision of Γ by up to 3.7-fold: an effect equivalent to increasing the signal intensity by approximately 14-fold. Similarly, the precision of the I estimation increases by a factor of 1.4, which is comparable to the effect of doubling the signal intensity.

These findings underscore that beyond conventional strategies such as enhancing the signal-to-noise ratio and optimizing the sampling interval, controlling the mixing parameter η offers a useful approach to precision optimization. For instance, inducing controlled Doppler or instrument-induced broadening to shift the profile toward a more Gaussian shape (i.e., reducing η) can simultaneously improve the precision of both I and Γ. Furthermore, because the estimation precision of I and Γ is proportional to (Δx/ΓI)^0.5, any increase in Γ caused by such broadening further enhances the precision. This strategy provides practical guidelines for designing and optimizing analytical instruments and experiment conditions.

In summary, findings from our study not only deepen the understanding of the interplay between analytical parameters and estimation precision but also pave the way for novel optimization strategies in high-precision quantitative analysis. The precision model derived herein is anticipated to serve as a foundational tool for further research related to measurement precision, providing an alternative method for error estimation in cases where direct experimental error quantification is challenging and offering a means for quantitative assessment of improvements in precision resulting from instrument upgrades.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsmeasuresciau.5c00030.

• Detailed numerical data obtained from computational analysis and Monte Carlo simulations (XLSX)

• Details of the Fisher information matrix derivation, confidence interval estimation, model fitting constraints, and evaluation metrics are provided, along with supplementary figures illustrating how the Fisher information matrix elements and variance-covariance matrix elements depend on profile type, sampling interval, and bandwidth. (PDF)

The authors declare no competing financial interest.