Methanol example

Matt Moores, Benjamin Moskowitz and Karen Faulds

2019-04-29

There are 3 R functions available in the package serrsBayes for model-based, statistical analysis of spectroscopy:

This vignette will provide recommendations for choosing which function to use for a given purpose. As an example, let’s say I wanted to fit a Raman spectrum of methanol. With a quick Google search, I find that there should be 4 peaks located at wavenumbers 1033 (C-O stretching), 1106 (CH3 rocking), 1149 (CH3 bending), and 1448 cm\(^{-1}\). However, we know that these peaks can shift slightly due to differences in temperature, pressure, spectrometer calibration, etc. My sample is in the liquid phase, so these peaks will not be exactly Gaussian or Lorentzian, but somewhere in the middle. On the other hand, I don’t want to wait for hours for SMC to finish running.

The first point to emphasise is that fitSpectraMCMC should not be used for real data. It exists mainly to illustrate the differences between MCMC and SMC algorithms, so is only of interest to computational statisticians.

fitSpectraSMC is the fastest method, but can also be difficult to tune. You need to choose either Lorentzian or Gaussian broadening functions. You also need to use fixed peak locations and the results are sensitive to the choice of priors$bl.smooth, however if you manage to get it working for your data then there is no real reason to use fitVoigtPeaksSMC.

The main differences in fitVoigtPeaksSMC are as follows:

Raw Data

The first step is always to read in the spectrum and plot it. Normally I would use the R package hyperSpec for this, but the data that we are going to use is included in the R package serrsBayes:

library(serrsBayes)
data("methanol", package = "serrsBayes")
wavenumbers <- methanol$wavenumbers
spectra <- methanol$spectra

peakLocations <- c(1033, 1106, 1149, 1448)
nPK <- length(peakLocations)
pkIdx <- numeric(nPK)
for (i in 1:nPK) {
  pkIdx[i] <- which.min(wavenumbers < peakLocations[i])
}
nWL <- length(wavenumbers)
plot(wavenumbers, spectra[1,], type='l', col=4,
     xlab=expression(paste("Raman shift (cm"^{-1}, ")")), ylab="Intensity (a.u.)", main="Observed Raman spectrum for methanol")
points(peakLocations, spectra[1,pkIdx], cex=2, col=2)
text(peakLocations + c(100,20,40,0), spectra[1,pkIdx] + c(0,700,400,700), labels=peakLocations)

fitSpectraSMC

We can use exactly the same priors as given in the package README:

lPriors <- list(scale.mu=log(11.6) - (0.4^2)/2, scale.sd=0.4, bl.smooth=10^11, bl.knots=50,
                 beta.mu=5000, beta.sd=5000, noise.sd=200, noise.nu=4)
tm <- system.time(result <- fitSpectraSMC(wavenumbers, spectra, peakLocations, lPriors))
print(paste(tm["elapsed"]/60, "minutes"))
## [1] "2.84936666666667 minutes"

We can see there is a problem with the model fit:

samp.idx <- sample.int(length(result$weights), 200, prob=result$weights)
plot(wavenumbers, spectra[1,], type='l', lwd=3,
     xlab=expression(paste("Raman shift (cm"^{-1}, ")")), ylab="Intensity (a.u.)", main="Fitted model with Lorentzian peaks")
for (pt in samp.idx) {
  bl.est <- result$basis %*% result$alpha[,1,pt]
  lines(wavenumbers, bl.est, col="#C3000009")
  lines(wavenumbers, bl.est + result$expFn[pt,], col="#00C30009")
}

Adjusting the priors is not going to fix this. The underlying issue is that the peak location of 1448 cm\(^{-1}\) doesn’t match our observed data. Even a small difference in the location of one of the peaks can adversely effect the entire model fit. This is the kind of issue that the function fitVoigtPeaksSMC is designed to handle.

fitVoigtPeaksSMC

The priors here are very similar to what we used for TAMRA in the other vignette. However, we don’t need as many bl.knots because we have a narrower range of wavenumbers. Instead, we will use bl.knots = 50, the same as what we used for fitSpectraSMC above.

lPriors2 <- list(loc.mu=peakLocations, loc.sd=rep(50,nPK), scaG.mu=log(16.47) - (0.34^2)/2,
                 scaG.sd=0.34, scaL.mu=log(25.27) - (0.4^2)/2, scaL.sd=0.4, noise.nu=5,
                 noise.sd=50, bl.smooth=1, bl.knots=50)
data("result2", package = "serrsBayes")
if(!exists("result2")) {
  tm2 <- system.time(result2 <- fitVoigtPeaksSMC(wavenumbers, spectra, lPriors2, npart=3000))
  result2$time <- tm2
  save(result2, file="Figure 2/result2.rda")
}

fitVoigtPeaksSMC takes more than twice as long to fit the model, due to estimating the additional parameters. However, we obtain a much better fit to the observed spectrum:

print(paste(result2$time["elapsed"]/60, "minutes"))
## [1] "7.21733333333347 minutes"
samp.idx <- 1:nrow(result2$location)
plot(wavenumbers, spectra[1,], type='l', lwd=3,
     xlab=expression(paste("Raman shift (cm"^{-1}, ")")), ylab="Intensity (a.u.)", main="Fitted model with Voigt peaks")
samp.mat <- resid.mat <- matrix(0,nrow=length(samp.idx), ncol=nWL)
samp.sigi <- samp.lambda <- numeric(length=nrow(samp.mat))
for (pt in 1:length(samp.idx)) {
  k <- samp.idx[pt]
  samp.mat[pt,] <- mixedVoigt(result2$location[k,], result2$scale_G[k,],
                              result2$scale_L[k,], result2$beta[k,], wavenumbers)
  samp.sigi[pt] <- result2$sigma[k]
  samp.lambda[pt] <- result2$lambda[k]
  
  Obsi <- spectra[1,] - samp.mat[pt,]
  g0_Cal <- length(Obsi) * samp.lambda[pt] * result2$priors$bl.precision
  gi_Cal <- crossprod(result2$priors$bl.basis) + g0_Cal
  mi_Cal <- as.vector(solve(gi_Cal, crossprod(result2$priors$bl.basis, Obsi)))

  bl.est <- result2$priors$bl.basis %*% mi_Cal # smoothed residuals = estimated basline
  lines(wavenumbers, bl.est, col="#C3000009")
  lines(wavenumbers, bl.est + samp.mat[pt,], col="#00C30009")
  resid.mat[pt,] <- Obsi - bl.est[,1]
}

The model fit could be further improved. For example, it appears that there is a shoulder peak around 1550 cm\(^{-1}\). However, we can see that the posterior distribution for the fourth peak location is very far away from the theoretical value of 1448:

for (j in 1:4) {
  hist(result2$location[,j], main=paste("Peak",j),
       xlab=expression(paste("Peak location (cm"^{-1}, ")")),
       freq=FALSE, xlim=range(result2$location[,j], peakLocations[j]))
  lines(density(result2$location[,j]), col=4, lty=2, lwd=3)
  abline(v=peakLocations[j], col=2, lty=3, lwd=3)
}

Posteriors for the peak locations.Posteriors for the peak locations.Posteriors for the peak locations.Posteriors for the peak locations.

This explains why the results from fitSpectraSMC were so poor. You really need to have very precise information about all of the peak locations in order for that function to perform well. Otherwise, fitVoigtPeaksSMC will usually give much better fit to the observed spectrum.