Bayesian Inference for NMR Spectroscopy with Applications to Chemical Quantification

Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic properties of atomic nuclei to discover the structure, reaction state and chemical environment of molecules. We propose a probabilistic generative model and inference procedures for NMR spectroscopy. Specically, we use a weighted sum of trigonometric functions undergoing exponential decay to model free induction decay (FID) signals. We discuss the challenges in estimating the components of this general model { amplitudes, phase shifts, frequencies, decay rates, and noise variances { and oer practical solutions. We compare with conventional Fourier transform spectroscopy for estimating the relative concentrations of chemicals in a mixture, using synthetic and experimentally acquired FID signals. We nd the proposed model is particularly robust to low signal to noise ratios (SNR), and overlapping peaks in the Fourier transform of the FID, enabling accurate predictions (e.g., 1% sensitivity at low SNR) which are not possible with conventional spectroscopy (5% sensitivity). 1. Introduction. Nuclear magnetic resonance (NMR) spectroscopy has greatly advanced our understanding of molecular properties, and is now widespread in analytical chemistry. The theory of nuclear magnetic resonance postulates that protons and neutrons behave like gyroscopes that spin about their axes, generating their own small magnetic elds. These concepts were rst described by Rabi et al. (1939), for which Isidor Rabi was awarded the 1944 Nobel prize in physics. Later, Bloch, Hanson and Packard (1946) and Purcell, Torrey and Pound (1946) showed how NMR could be used to understand the structure of molecules in liquids and solids, for which they shared the Nobel prize in physics in 1952. Richard Ernst then won the 1991 Nobel prize in chemistry for developing Fourier transform NMR spectroscopy 1 , which led to the prevalence of NMR as an analytic technique. NMR spectroscopy is well suited to studying both organic and inorganic molecules, including proteins, and other biochemical species (Barrett et al., 2013), and is routinely used to identify the structure of unknown chemical species or the composition of mixtures. NMR spectroscopy is quantitative, chemically specic and non-invasive and therefore