| Title: | Conducting the Peak Fitting Based on the EM Algorithm |
|---|---|
| Description: | The peak fitting of spectral data is performed by using the frame work of EM algorithm. We adapted the EM algorithm for the peak fitting of spectral data set by considering the weight of the intensity corresponding to the measurement energy steps (Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019, 2021 and 2023) <doi:10.1080/14686996.2019.1620123>, <doi:10.1080/27660400.2021.1899449> <doi:10.1080/27660400.2022.2159753>. The package efficiently estimates the parameters of Gaussian mixture model during iterative calculation between E-step and M-step, and the parameters are converged to a local optimal solution. This package can support the investigation of peak shift with two advantages: (1) a large amount of data can be processed at high speed; and (2) stable and automatic calculation can be easily performed. |
| Authors: | Tarojiro Matsumura [aut, cre] |
| Maintainer: | Tarojiro Matsumura <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.3.1 |
| Built: | 2024-10-31 16:33:22 UTC |
| Source: | https://github.com/cran/EMpeaksR |
Visualization of the result of spect_em_dsgmm().
show_dsgmm_curve(spect_em_dsgmm_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init)show_dsgmm_curve(spect_em_dsgmm_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init)
spect_em_dsgmm_res |
data set obtained by spect_em_dsgmm() |
x |
measurement steps |
y |
intensity |
mix_ratio_init |
initial values of the mixture ratio of the components |
mu_init |
initial values of the mean of the components |
sigma_init |
initial values of the standard deviation of the components |
alpha_init |
initial values of the asymmetric parameter of the components |
eta_init |
initial values of the mixing ratio of Gauss and Lorentz distribution |
Perform a visualization of fitting curve estimated by Doniach-Sunjic-Gauss mixture model.
Show the fitting curve and variation of the parameters.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Doniach-Sunjic-Gauss mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(20, 50, 80) true_sigma <- c(3, 3, 3) true_alpha <- c(0.1, 0.3, 0.1) true_eta <- c(0.4, 0.6, 0.1) true_mix_ratio <- rep(1/3, 3) degree <- 4 #trancated Doniach-Sunjic-Gauss truncated_dsg <- function(x, mu, sigma, alpha, eta) { ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma)) / sum( ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma))) } y <- c(true_mix_ratio[1]*truncated_dsg(x = x, mu = true_mu[1], sigma = true_sigma[1], alpha = true_alpha[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_dsg(x = x, mu = true_mu[2], sigma = true_sigma[2], alpha = true_alpha[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_dsg(x = x, mu = true_mu[3], sigma = true_sigma[3], alpha = true_alpha[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(4, 3, 2) alpha_init <- c(0.3, 0.2, 0.4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_DSG_res <- spect_em_dsgmm(x = x, y = y, mu = mu_init, sigma = sigma_init, alpha = alpha_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_dsgmm_curve(SP_ECM_DSG_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init) #Showing the result of spect_em_dsgmm() print(cbind(c(mu_init), c(sigma_init), c(alpha_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_DSG_res$mu, SP_ECM_DSG_res$sigma, SP_ECM_DSG_res$alpha, SP_ECM_DSG_res$eta, SP_ECM_DSG_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_alpha, true_eta, true_mix_ratio))#generating the synthetic spectral data based on three component Doniach-Sunjic-Gauss mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(20, 50, 80) true_sigma <- c(3, 3, 3) true_alpha <- c(0.1, 0.3, 0.1) true_eta <- c(0.4, 0.6, 0.1) true_mix_ratio <- rep(1/3, 3) degree <- 4 #trancated Doniach-Sunjic-Gauss truncated_dsg <- function(x, mu, sigma, alpha, eta) { ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma)) / sum( ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma))) } y <- c(true_mix_ratio[1]*truncated_dsg(x = x, mu = true_mu[1], sigma = true_sigma[1], alpha = true_alpha[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_dsg(x = x, mu = true_mu[2], sigma = true_sigma[2], alpha = true_alpha[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_dsg(x = x, mu = true_mu[3], sigma = true_sigma[3], alpha = true_alpha[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(4, 3, 2) alpha_init <- c(0.3, 0.2, 0.4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_DSG_res <- spect_em_dsgmm(x = x, y = y, mu = mu_init, sigma = sigma_init, alpha = alpha_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_dsgmm_curve(SP_ECM_DSG_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init) #Showing the result of spect_em_dsgmm() print(cbind(c(mu_init), c(sigma_init), c(alpha_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_DSG_res$mu, SP_ECM_DSG_res$sigma, SP_ECM_DSG_res$alpha, SP_ECM_DSG_res$eta, SP_ECM_DSG_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_alpha, true_eta, true_mix_ratio))
Visualization of the result of spect_em_gmm().
show_gmm_curve(spect_em_gmm_res, x, y, mix_ratio_init, mu_init, sigma_init)show_gmm_curve(spect_em_gmm_res, x, y, mix_ratio_init, mu_init, sigma_init)
spect_em_gmm_res |
data set obtained by spect_em_gmm() |
x |
measurement steps |
y |
intensity |
mix_ratio_init |
initial values of the mixture ratio of the components |
mu_init |
initial values of the mean of the components |
sigma_init |
initial values of the standard deviation of the components |
Perform a visualization of fitting curve estimated by Gaussian mixture model.
Show the fitting curve and variation of the parameters.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
#generating the synthetic spectral data based on three component Gausian mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 y <- c(true_mix_ratio[1] * dnorm(x = x, mean = true_mu[1], sd = true_sigma[1])*10^degree + true_mix_ratio[2] * dnorm(x = x, mean = true_mu[2], sd = true_sigma[2])*10^degree + true_mix_ratio[3] * dnorm(x = x, mean = true_mu[3], sd = true_sigma[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) #Coducting calculation SP_EM_G_res <- spect_em_gmm(x, y, mu = mu_init, sigma = sigma_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_gmm_curve(SP_EM_G_res, x, y, mix_ratio_init, mu_init, sigma_init) #Showing the result of spect_em_gmm() print(cbind(c(mu_init), c(sigma_init), c(mix_ratio_init))) print(cbind(SP_EM_G_res$mu, SP_EM_G_res$sigma, SP_EM_G_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_mix_ratio))#generating the synthetic spectral data based on three component Gausian mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 y <- c(true_mix_ratio[1] * dnorm(x = x, mean = true_mu[1], sd = true_sigma[1])*10^degree + true_mix_ratio[2] * dnorm(x = x, mean = true_mu[2], sd = true_sigma[2])*10^degree + true_mix_ratio[3] * dnorm(x = x, mean = true_mu[3], sd = true_sigma[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) #Coducting calculation SP_EM_G_res <- spect_em_gmm(x, y, mu = mu_init, sigma = sigma_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_gmm_curve(SP_EM_G_res, x, y, mix_ratio_init, mu_init, sigma_init) #Showing the result of spect_em_gmm() print(cbind(c(mu_init), c(sigma_init), c(mix_ratio_init))) print(cbind(SP_EM_G_res$mu, SP_EM_G_res$sigma, SP_EM_G_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_mix_ratio))
Visualization of the result of spect_em_lmm().
show_lmm_curve(spect_em_lmm_res, x, y, mix_ratio_init, mu_init, gam_init)show_lmm_curve(spect_em_lmm_res, x, y, mix_ratio_init, mu_init, gam_init)
spect_em_lmm_res |
data set obtained by spect_em_lmm() |
x |
measurement steps |
y |
intensity |
mix_ratio_init |
initial values of the mixture ratio of the components |
mu_init |
initial values of the mean of the components |
gam_init |
initial values of the scale parameter of the components |
Perform a visualization of fitting curve estimated by Lorentz mixture model.
Show the fitting curve and variation of the parameters.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Lorentz mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_gam <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Lorentz distribution dCauchy <- function(x, mu, gam) { (dcauchy(x, mu, gam)) / sum(dcauchy(x, mu, gam)) } y <- c(true_mix_ratio[1] * dCauchy(x = x, mu = true_mu[1], gam = true_gam[1])*10^degree + true_mix_ratio[2] * dCauchy(x = x, mu = true_mu[2], gam = true_gam[2])*10^degree + true_mix_ratio[3] * dCauchy(x = x, mu = true_mu[3], gam = true_gam[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) gam_init <- c(2, 5, 4) #Coducting calculation SP_ECM_L_res <- spect_em_lmm(x, y, mu = mu_init, gam = gam_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_lmm_curve(SP_ECM_L_res, x, y, mix_ratio_init, mu_init, gam_init) #Showing the result of spect_em_lmm() print(cbind(c(mu_init), c(gam_init), c(mix_ratio_init))) print(cbind(SP_ECM_L_res$mu, SP_ECM_L_res$gam, SP_ECM_L_res$mix_ratio)) print(cbind(true_mu, true_gam, true_mix_ratio))#generating the synthetic spectral data based on three component Lorentz mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_gam <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Lorentz distribution dCauchy <- function(x, mu, gam) { (dcauchy(x, mu, gam)) / sum(dcauchy(x, mu, gam)) } y <- c(true_mix_ratio[1] * dCauchy(x = x, mu = true_mu[1], gam = true_gam[1])*10^degree + true_mix_ratio[2] * dCauchy(x = x, mu = true_mu[2], gam = true_gam[2])*10^degree + true_mix_ratio[3] * dCauchy(x = x, mu = true_mu[3], gam = true_gam[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) gam_init <- c(2, 5, 4) #Coducting calculation SP_ECM_L_res <- spect_em_lmm(x, y, mu = mu_init, gam = gam_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_lmm_curve(SP_ECM_L_res, x, y, mix_ratio_init, mu_init, gam_init) #Showing the result of spect_em_lmm() print(cbind(c(mu_init), c(gam_init), c(mix_ratio_init))) print(cbind(SP_ECM_L_res$mu, SP_ECM_L_res$gam, SP_ECM_L_res$mix_ratio)) print(cbind(true_mu, true_gam, true_mix_ratio))
Visualization of the result of spect_em_pvmm().
show_pvmm_curve(spect_em_pvmm_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init)show_pvmm_curve(spect_em_pvmm_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init)
spect_em_pvmm_res |
data set obtained by spect_em_pvmm() |
x |
measurement steps |
y |
intensity |
mix_ratio_init |
initial values of the mixture ratio of the components |
mu_init |
initial values of the mean of the components |
sigma_init |
initial values of the standard deviation of the components |
eta_init |
initial values of the mixing ratio of Gauss and Lorentz distribution |
Perform a visualization of fitting curve estimated by Pseudo-Voigt mixture model.
Show the fitting curve and variation of the parameters.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_eta <- c(0.3, 0.8, 0.5) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_PV_res <- spect_em_pvmm(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_curve(SP_ECM_PV_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init) #Showing the result of spect_em_pvmm() print(cbind(c(mu_init), c(sigma_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_PV_res$mu, SP_ECM_PV_res$sigma, SP_ECM_PV_res$eta, SP_ECM_PV_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio))#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_eta <- c(0.3, 0.8, 0.5) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_PV_res <- spect_em_pvmm(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_curve(SP_ECM_PV_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init) #Showing the result of spect_em_pvmm() print(cbind(c(mu_init), c(sigma_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_PV_res$mu, SP_ECM_PV_res$sigma, SP_ECM_PV_res$eta, SP_ECM_PV_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio))
Visualization of the result of spect_em_pvmm_lback().
show_pvmm_lback_curve(spect_em_pvmm_lback_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init, x_lower, x_upper)show_pvmm_lback_curve(spect_em_pvmm_lback_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init, x_lower, x_upper)
spect_em_pvmm_lback_res |
data set obtained by spect_em_pvmm_lback() |
x |
measurement steps |
y |
intensity |
mu_init |
initial values of the mean of the components |
sigma_init |
initial values of the standard deviation of the components |
eta_init |
initial values of the mixing ratio of Gauss and Lorentz distribution |
mix_ratio_init |
initial values of the mixture ratio of the components |
x_lower |
lower limit of the measurement steps. Default is a minimum of x |
x_upper |
upper limit of the measurement steps. Default is a maximum of x |
Perform a visualization of fitting curve estimated by pseudo-Voigt mixture model with a linear background.
Show the fitting curve and variation of the parameters.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2023). High-throughput XPS spectrum modeling with autonomous background subtraction for 3 d 5/2 peak mapping of SnS. Science and Technology of Advanced Materials: Methods, 3(1), 2159753.
#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) K <- 3 true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- c(0.5/3, 0.5/3, 0.5/3, 0.5) true_eta <- c(0.4, 0.6, 0.1) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree + true_mix_ratio[4]*(c(500*x + 15000) / sum(500*x + 15000))*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values mu_init <- c(30, 40, 60) sigma_init <- c(4, 4, 4) mix_ratio_init <- rep(1/(length(mu_init)+3), length(mu_init)+3) eta_init <- c(1, 1, 1) #Coducting calculation SP_ECM_PV_LBACK_res <- spect_em_pvmm_lback(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, x_lower = min(x), x_upper = max(x), conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_lback_curve(spect_em_pvmm_lback_res = SP_ECM_PV_LBACK_res, x = x, y = y, mix_ratio_init = mix_ratio_init, mu_init = mu_init, sigma_init = sigma_init, eta_init = eta_init, x_lower = min(x), x_upper = max(x)) #Showing the result of spect_em_pvmm_lback() print(cbind(SP_ECM_PV_LBACK_res$mu, SP_ECM_PV_LBACK_res$sigma, SP_ECM_PV_LBACK_res$eta, SP_ECM_PV_LBACK_res$mix_ratio[1:K])) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio[1:K]))#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) K <- 3 true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- c(0.5/3, 0.5/3, 0.5/3, 0.5) true_eta <- c(0.4, 0.6, 0.1) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree + true_mix_ratio[4]*(c(500*x + 15000) / sum(500*x + 15000))*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values mu_init <- c(30, 40, 60) sigma_init <- c(4, 4, 4) mix_ratio_init <- rep(1/(length(mu_init)+3), length(mu_init)+3) eta_init <- c(1, 1, 1) #Coducting calculation SP_ECM_PV_LBACK_res <- spect_em_pvmm_lback(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, x_lower = min(x), x_upper = max(x), conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_lback_curve(spect_em_pvmm_lback_res = SP_ECM_PV_LBACK_res, x = x, y = y, mix_ratio_init = mix_ratio_init, mu_init = mu_init, sigma_init = sigma_init, eta_init = eta_init, x_lower = min(x), x_upper = max(x)) #Showing the result of spect_em_pvmm_lback() print(cbind(SP_ECM_PV_LBACK_res$mu, SP_ECM_PV_LBACK_res$sigma, SP_ECM_PV_LBACK_res$eta, SP_ECM_PV_LBACK_res$mix_ratio[1:K])) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio[1:K]))
Perform a peak fitting based on the spectrum adapted ECM algorithm by Doniach-Sunjic-Gauss mixture model.
spect_em_dsgmm(x, y, mu, sigma, alpha, eta, mix_ratio, conv.cri, maxit)spect_em_dsgmm(x, y, mu, sigma, alpha, eta, mix_ratio, conv.cri, maxit)
x |
measurement steps |
y |
intensity |
mu |
mean of the components |
sigma |
standard deviation of the components |
alpha |
asymmetric parameter of the component |
eta |
mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
mixture ratio of the components |
conv.cri |
criterion of the convergence |
maxit |
maximum number of the iteration |
Peak fitting is conducted by spectrum adapted ECM algorithm.
mu |
estimated mean of the components |
sigma |
estimated standard deviation of the components |
alpha |
estimated asymmetric parameter of the components |
eta |
estimated mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
estimated mixture ratio of the components |
it |
number of the iteration to reach the convergence |
LL |
variation of the weighted log likelihood values |
MU |
variation of mu |
SIGMA |
variation of sigma |
ALPHA |
variation of alpha |
ETA |
variation of beta |
MIX_RATIO |
variation of mix_ratio |
W_K |
decomposed component of the spectral data |
convergence |
message for the convergence in the calculation |
cal_time |
calculation time to complete the peak fitting. Unit is seconds |
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Doniach-Sunjic-Gauss mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(20, 50, 80) true_sigma <- c(3, 3, 3) true_alpha <- c(0.1, 0.3, 0.1) true_eta <- c(0.4, 0.6, 0.1) true_mix_ratio <- rep(1/3, 3) degree <- 4 #trancated Doniach-Sunjic-Gauss truncated_dsg <- function(x, mu, sigma, alpha, eta) { ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma)) / sum( ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma))) } y <- c(true_mix_ratio[1]*truncated_dsg(x = x, mu = true_mu[1], sigma = true_sigma[1], alpha = true_alpha[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_dsg(x = x, mu = true_mu[2], sigma = true_sigma[2], alpha = true_alpha[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_dsg(x = x, mu = true_mu[3], sigma = true_sigma[3], alpha = true_alpha[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(4, 3, 2) alpha_init <- c(0.3, 0.2, 0.4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_DSG_res <- spect_em_dsgmm(x = x, y = y, mu = mu_init, sigma = sigma_init, alpha = alpha_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_dsgmm_curve(SP_ECM_DSG_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init) #Showing the result of spect_em_dsgmm() print(cbind(c(mu_init), c(sigma_init), c(alpha_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_DSG_res$mu, SP_ECM_DSG_res$sigma, SP_ECM_DSG_res$alpha, SP_ECM_DSG_res$eta, SP_ECM_DSG_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_alpha, true_eta, true_mix_ratio))#generating the synthetic spectral data based on three component Doniach-Sunjic-Gauss mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(20, 50, 80) true_sigma <- c(3, 3, 3) true_alpha <- c(0.1, 0.3, 0.1) true_eta <- c(0.4, 0.6, 0.1) true_mix_ratio <- rep(1/3, 3) degree <- 4 #trancated Doniach-Sunjic-Gauss truncated_dsg <- function(x, mu, sigma, alpha, eta) { ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma)) / sum( ((eta*(((gamma(1-alpha)) / ((x-mu)^2+(sqrt(2*log(2))*sigma)^2)^((1-alpha)/2)) * cos((pi*alpha/2)+(1-alpha)*atan((x-mu) / (sqrt(2*log(2))*sigma))))) + (1-eta)*dnorm(x, mu, sigma))) } y <- c(true_mix_ratio[1]*truncated_dsg(x = x, mu = true_mu[1], sigma = true_sigma[1], alpha = true_alpha[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_dsg(x = x, mu = true_mu[2], sigma = true_sigma[2], alpha = true_alpha[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_dsg(x = x, mu = true_mu[3], sigma = true_sigma[3], alpha = true_alpha[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(4, 3, 2) alpha_init <- c(0.3, 0.2, 0.4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_DSG_res <- spect_em_dsgmm(x = x, y = y, mu = mu_init, sigma = sigma_init, alpha = alpha_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_dsgmm_curve(SP_ECM_DSG_res, x, y, mix_ratio_init, mu_init, sigma_init, alpha_init, eta_init) #Showing the result of spect_em_dsgmm() print(cbind(c(mu_init), c(sigma_init), c(alpha_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_DSG_res$mu, SP_ECM_DSG_res$sigma, SP_ECM_DSG_res$alpha, SP_ECM_DSG_res$eta, SP_ECM_DSG_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_alpha, true_eta, true_mix_ratio))
Perform a peak fitting based on the spectrum adapted EM algorithm by Gaussian mixture model.
spect_em_gmm(x, y, mu, sigma, mix_ratio, conv.cri, maxit)spect_em_gmm(x, y, mu, sigma, mix_ratio, conv.cri, maxit)
x |
measurement steps |
y |
intensity |
mu |
mean of the components |
sigma |
standard deviation of the components |
mix_ratio |
mixture ratio of the components |
conv.cri |
criterion of the convergence |
maxit |
maximum number of the iteration |
Peak fitting is conducted by spectrum adapted EM algorithm.
mu |
estimated mean of the components |
sigma |
estimated standard deviation of the components |
mix_ratio |
estimated mixture ratio of the components |
it |
number of the iteration to reach the convergence |
LL |
variation of the weighted log likelihood values |
MU |
variation of mu |
SIGMA |
variation of sigma |
MIX_RATIO |
variation of mix_ratio |
W_K |
decomposed component of the spectral data |
convergence |
message for the convergence in the calculation |
cal_time |
calculation time to complete the peak fitting. Unit is seconds |
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
#generating the synthetic spectral data based on three component Gausian mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 y <- c(true_mix_ratio[1] * dnorm(x = x, mean = true_mu[1], sd = true_sigma[1])*10^degree + true_mix_ratio[2] * dnorm(x = x, mean = true_mu[2], sd = true_sigma[2])*10^degree + true_mix_ratio[3] * dnorm(x = x, mean = true_mu[3], sd = true_sigma[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) #Coducting calculation SP_EM_G_res <- spect_em_gmm(x, y, mu = mu_init, sigma = sigma_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_gmm_curve(SP_EM_G_res, x, y, mix_ratio_init, mu_init, sigma_init) #Showing the result of spect_em_gmm() print(cbind(c(mu_init), c(sigma_init), c(mix_ratio_init))) print(cbind(SP_EM_G_res$mu, SP_EM_G_res$sigma, SP_EM_G_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_mix_ratio))#generating the synthetic spectral data based on three component Gausian mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 y <- c(true_mix_ratio[1] * dnorm(x = x, mean = true_mu[1], sd = true_sigma[1])*10^degree + true_mix_ratio[2] * dnorm(x = x, mean = true_mu[2], sd = true_sigma[2])*10^degree + true_mix_ratio[3] * dnorm(x = x, mean = true_mu[3], sd = true_sigma[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) #Coducting calculation SP_EM_G_res <- spect_em_gmm(x, y, mu = mu_init, sigma = sigma_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_gmm_curve(SP_EM_G_res, x, y, mix_ratio_init, mu_init, sigma_init) #Showing the result of spect_em_gmm() print(cbind(c(mu_init), c(sigma_init), c(mix_ratio_init))) print(cbind(SP_EM_G_res$mu, SP_EM_G_res$sigma, SP_EM_G_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_mix_ratio))
Perform a peak fitting based on the spectrum adapted ECM algorithm by Lorentz mixture model.
spect_em_lmm(x, y, mu, gam, mix_ratio, conv.cri, maxit)spect_em_lmm(x, y, mu, gam, mix_ratio, conv.cri, maxit)
x |
measurement steps |
y |
intensity |
mu |
mean of the components |
gam |
scale parameter of the components |
mix_ratio |
mixture ratio of the components |
conv.cri |
criterion of the convergence |
maxit |
maximum number of the iteration |
Peak fitting is conducted by spectrum adapted ECM algorithm.
mu |
estimated mean of the components |
gam |
estimated scale parameter of the components |
mix_ratio |
estimated mixture ratio of the components |
it |
number of the iteration to reach the convergence |
LL |
variation of the weighted log likelihood values |
MU |
variation of mu |
GAM |
variation of gam |
MIX_RATIO |
variation of mix_ratio |
W_K |
decomposed component of the spectral data |
convergence |
message for the convergence in the calculation |
cal_time |
calculation time to complete the peak fitting. Unit is seconds |
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Lorentz mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_gam <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Lorentz distribution dCauchy <- function(x, mu, gam) { (dcauchy(x, mu, gam)) / sum(dcauchy(x, mu, gam)) } y <- c(true_mix_ratio[1] * dCauchy(x = x, mu = true_mu[1], gam = true_gam[1])*10^degree + true_mix_ratio[2] * dCauchy(x = x, mu = true_mu[2], gam = true_gam[2])*10^degree + true_mix_ratio[3] * dCauchy(x = x, mu = true_mu[3], gam = true_gam[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) gam_init <- c(2, 5, 4) #Coducting calculation SP_ECM_L_res <- spect_em_lmm(x, y, mu = mu_init, gam = gam_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_lmm_curve(SP_ECM_L_res, x, y, mix_ratio_init, mu_init, gam_init) #Showing the result of spect_em_lmm() print(cbind(c(mu_init), c(gam_init), c(mix_ratio_init))) print(cbind(SP_ECM_L_res$mu, SP_ECM_L_res$gam, SP_ECM_L_res$mix_ratio)) print(cbind(true_mu, true_gam, true_mix_ratio))#generating the synthetic spectral data based on three component Lorentz mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_gam <- c(3, 3, 3) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Lorentz distribution dCauchy <- function(x, mu, gam) { (dcauchy(x, mu, gam)) / sum(dcauchy(x, mu, gam)) } y <- c(true_mix_ratio[1] * dCauchy(x = x, mu = true_mu[1], gam = true_gam[1])*10^degree + true_mix_ratio[2] * dCauchy(x = x, mu = true_mu[2], gam = true_gam[2])*10^degree + true_mix_ratio[3] * dCauchy(x = x, mu = true_mu[3], gam = true_gam[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) gam_init <- c(2, 5, 4) #Coducting calculation SP_ECM_L_res <- spect_em_lmm(x, y, mu = mu_init, gam = gam_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_lmm_curve(SP_ECM_L_res, x, y, mix_ratio_init, mu_init, gam_init) #Showing the result of spect_em_lmm() print(cbind(c(mu_init), c(gam_init), c(mix_ratio_init))) print(cbind(SP_ECM_L_res$mu, SP_ECM_L_res$gam, SP_ECM_L_res$mix_ratio)) print(cbind(true_mu, true_gam, true_mix_ratio))
Perform a peak fitting based on the spectrum adapted ECM algorithm by Pseudo-Voigt mixture model.
spect_em_pvmm(x, y, mu, sigma, eta, mix_ratio, conv.cri, maxit)spect_em_pvmm(x, y, mu, sigma, eta, mix_ratio, conv.cri, maxit)
x |
measurement steps |
y |
intensity |
mu |
mean of the components |
sigma |
standard deviation of the components |
eta |
mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
mixture ratio of the components |
conv.cri |
criterion of the convergence |
maxit |
maximum number of the iteration |
Peak fitting is conducted by spectrum adapted ECM algorithm.
mu |
estimated mean of the components |
sigma |
estimated standard deviation of the components |
eta |
estimated mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
estimated mixture ratio of the components |
it |
number of the iteration to reach the convergence |
LL |
variation of the weighted log likelihood values |
MU |
variation of mu |
SIGMA |
variation of sigma |
ETA |
variation of beta |
MIX_RATIO |
variation of mix_ratio |
W_K |
decomposed component of the spectral data |
convergence |
message for the convergence in the calculation |
cal_time |
calculation time to complete the peak fitting. Unit is seconds |
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_eta <- c(0.3, 0.8, 0.5) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_PV_res <- spect_em_pvmm(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_curve(SP_ECM_PV_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init) #Showing the result of spect_em_pvmm() print(cbind(c(mu_init), c(sigma_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_PV_res$mu, SP_ECM_PV_res$sigma, SP_ECM_PV_res$eta, SP_ECM_PV_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio))#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_eta <- c(0.3, 0.8, 0.5) true_mix_ratio <- rep(1/3, 3) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values K <- 3 mix_ratio_init <- c(0.2, 0.4, 0.4) mu_init <- c(20, 40, 70) sigma_init <- c(2, 5, 4) eta_init <- c(0.5, 0.4, 0.3) #Coducting calculation SP_ECM_PV_res <- spect_em_pvmm(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_curve(SP_ECM_PV_res, x, y, mix_ratio_init, mu_init, sigma_init, eta_init) #Showing the result of spect_em_pvmm() print(cbind(c(mu_init), c(sigma_init), c(eta_init), c(mix_ratio_init))) print(cbind(SP_ECM_PV_res$mu, SP_ECM_PV_res$sigma, SP_ECM_PV_res$eta, SP_ECM_PV_res$mix_ratio)) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio))
Perform a peak fitting based on the spectrum adapted ECM algorithm by pseudo-Voigt mixture model with a linear background.
spect_em_pvmm_lback(x, y, mu, sigma, eta, mix_ratio, x_lower, x_upper, conv.cri, maxit)spect_em_pvmm_lback(x, y, mu, sigma, eta, mix_ratio, x_lower, x_upper, conv.cri, maxit)
x |
measurement steps |
y |
intensity |
mu |
mean of the components |
sigma |
standard deviation of the components |
eta |
mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
mixture ratio of the components |
x_lower |
lower limit of the measurement steps. Default is a minimum of x |
x_upper |
upper limit of the measurement steps. Default is a maximum of x |
conv.cri |
criterion of the convergence |
maxit |
maximum number of the iteration |
Peak fitting is conducted by spectrum adapted ECM algorithm.
mu |
estimated mean of the components |
sigma |
estimated standard deviation of the components |
eta |
estimated mixing ratio of Gauss and Lorentz distribution |
mix_ratio |
estimated mixture ratio of the components |
it |
number of the iteration to reach the convergence |
LL |
variation of the weighted log likelihood values |
MU |
variation of mu |
SIGMA |
variation of sigma |
ETA |
variation of beta |
MIX_RATIO |
variation of mix_ratio |
W_K |
decomposed component of the spectral data |
convergence |
message for the convergence in the calculation |
cal_time |
calculation time to complete the peak fitting. Unit is seconds |
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2019). Spectrum adapted expectation-maximization algorithm for high-throughput peak shift analysis. Science and technology of advanced materials, 20(1), 733-745.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2021). Spectrum adapted expectation-conditional maximization algorithm for extending high–throughput peak separation method in XPS analysis. Science and Technology of Advanced Materials: Methods, 1(1), 45-55.
Matsumura, T., Nagamura, N., Akaho, S., Nagata, K., & Ando, Y. (2023). High-throughput XPS spectrum modeling with autonomous background subtraction for 3 d 5/2 peak mapping of SnS. Science and Technology of Advanced Materials: Methods, 3(1), 2159753.
#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) K <- 3 true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- c(0.5/3, 0.5/3, 0.5/3, 0.5) true_eta <- c(0.4, 0.6, 0.1) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree + true_mix_ratio[4]*(c(500*x + 15000) / sum(500*x + 15000))*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values mu_init <- c(30, 40, 60) sigma_init <- c(4, 4, 4) mix_ratio_init <- rep(1/(length(mu_init)+3), length(mu_init)+3) eta_init <- c(1, 1, 1) #Coducting calculation SP_ECM_PV_LBACK_res <- spect_em_pvmm_lback(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, x_lower = min(x), x_upper = max(x), conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_lback_curve(spect_em_pvmm_lback_res = SP_ECM_PV_LBACK_res, x = x, y = y, mix_ratio_init = mix_ratio_init, mu_init = mu_init, sigma_init = sigma_init, eta_init = eta_init, x_lower = min(x), x_upper = max(x)) #Showing the result of spect_em_pvmm_lback() print(cbind(SP_ECM_PV_LBACK_res$mu, SP_ECM_PV_LBACK_res$sigma, SP_ECM_PV_LBACK_res$eta, SP_ECM_PV_LBACK_res$mix_ratio[1:K])) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio[1:K]))#generating the synthetic spectral data based on three component Pseudo-Voigt mixture model. x <- seq(0, 100, by = 0.5) K <- 3 true_mu <- c(35, 50, 65) true_sigma <- c(3, 3, 3) true_mix_ratio <- c(0.5/3, 0.5/3, 0.5/3, 0.5) true_eta <- c(0.4, 0.6, 0.1) degree <- 4 #Normalized Pseudo-Voigt distribution truncated_pv <- function(x, mu, sigma, eta) { (eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) / sum(eta*dcauchy(x, mu, sqrt(2*log(2))*sigma) + (1-eta)*dnorm(x, mu, sigma)) } y <- c(true_mix_ratio[1]*truncated_pv(x = x, mu = true_mu[1], sigma = true_sigma[1], eta = true_eta[1])*10^degree + true_mix_ratio[2]*truncated_pv(x = x, mu = true_mu[2], sigma = true_sigma[2], eta = true_eta[2])*10^degree + true_mix_ratio[3]*truncated_pv(x = x, mu = true_mu[3], sigma = true_sigma[3], eta = true_eta[3])*10^degree + true_mix_ratio[4]*(c(500*x + 15000) / sum(500*x + 15000))*10^degree) plot(y~x, main = "genrated synthetic spectral data") #Peak fitting by EMpeaksR #Initial values mu_init <- c(30, 40, 60) sigma_init <- c(4, 4, 4) mix_ratio_init <- rep(1/(length(mu_init)+3), length(mu_init)+3) eta_init <- c(1, 1, 1) #Coducting calculation SP_ECM_PV_LBACK_res <- spect_em_pvmm_lback(x = x, y = y, mu = mu_init, sigma = sigma_init, eta = eta_init, mix_ratio = mix_ratio_init, x_lower = min(x), x_upper = max(x), conv.cri = 1e-2, maxit = 2000) #Plot fitting curve and trace plot of parameters show_pvmm_lback_curve(spect_em_pvmm_lback_res = SP_ECM_PV_LBACK_res, x = x, y = y, mix_ratio_init = mix_ratio_init, mu_init = mu_init, sigma_init = sigma_init, eta_init = eta_init, x_lower = min(x), x_upper = max(x)) #Showing the result of spect_em_pvmm_lback() print(cbind(SP_ECM_PV_LBACK_res$mu, SP_ECM_PV_LBACK_res$sigma, SP_ECM_PV_LBACK_res$eta, SP_ECM_PV_LBACK_res$mix_ratio[1:K])) print(cbind(true_mu, true_sigma, true_eta, true_mix_ratio[1:K]))