Sequential Monte Carlo and stochastic processes in spectroscopy

Abstract

In this thesis, we introduce stochastic processes and Bayesian inference methods with a particular emphasis on spectroscopic applications. First, we apply sequential Monte Carlo samplers for estimating the modulating error function and line shape parameters that are relevant for coherent anti-Stokes Raman scattering spectroscopy. Second, we model posterior distributions of deconvoluted spectroscopic data as log-Gaussian Cox processes. Furthermore, we estimate the number of line shapes and their locations via sampling local maxima of the log-intensity of the log-Gaussian Cox process. Third, we propose nested sequential Monte Carlo samplers as an inference tool for mixtures of Gaussian process experts. The nested sequential Monte Carlo samplers allow for unbiased and more efficient inference in terms of Kullback–Leibler divergence while being thoroughly amenable to parallelisation. Lastly, we propose a new type of stochastic process, the log-Gaussian gamma process. We generate synthetic coherent anti-Stokes Raman scattering spectra with a combination of Gaussian and log-Gaussian gamma processes. We use the synthetic spectra to train a partially-Bayesian neural network for the inverse problem of estimating an underlying Raman spectrum from coherent anti-Stokes Raman scattering spectra.

The Bayesian paradigm offers explicit and robust uncertainty quantification, an aspect that has seen little attention in spectroscopic analysis. All in all, we propose Bayesian statistical methods for spectroscopic analysis while extending existing statistics literature with the proposal of nested sequential Monte Carlo samplers for mixtures of Gaussian process experts and log-Gaussian gamma processes.