Bayesian analysis of time-domain magnetic resonance signals

Abstract

Although the discrete Fourier transform remains the dominant means of processing NMR data (l-4), other methods of analyzing time-domain signals exist (5), and there has been recent interest among magnetic resonance scientists in applying alternative analysis techniques in an effort to improve signal-to-noise and resolution of the resulting frequency-domain spectrum. Recently, Bretthorst (6) and Jaynes ( 7) introduced a novel approach to the general time-domain signal analysis problem utilizing the techniques of Bayesian probability theory. Their approach is particularly suited to the class of time-domain signals characteristic of pulsed magnetic resonance spectroscopy, namely, a sum of decaying sinusoids. An especially attractive feature of Bayesian spectrum analysis lies in its ability to integrate out of the parameteroptimization-search procedure many of the parameters that define the model (socalled “nuisance” parameters), This greatly reduces the complexity of the optimization-search process. These nuisance parameters, such as sinusoid amplitude and phase, are then readily estimated, if needed, once the primary model parameters (e.g., frequencies and decay rates) are found via standard optimization-search algorithms. In this communication we demonstrate the application of Bayesian spectrum analysis to a time-domain (Bloch decay) 13C NMR signal from a standard ASTM reference sample of 1,4-dioxane in benzenede. Generally, a substantial amount of prior information is available regarding the “true” signal resulting from an NMR experiment. Making use of this information ought to improve our results. However, simply taking the Fourier transform of the data affords no way to take it into account. This information can be incorporated advantageously into the time-domain analysis, yielding a more powerful parameter determination. In the Bayesian spectrum-analysis and parameter-estimation technique, one analyzes the data in terms of some model which expresses the prior information. The data are fitted to the model using probability theory to obtain the “best” estimated parameters. The residuals are then reviewed to see if there is any coherent characteristic that has not been accounted for in the model. If there is, the model is updated and the entire process repeated until all coherent characteristics are removed from the residuals, that is, until the data accurately map onto the model. The Bayesian analysis gives a simple and elegant interpretation to the model-fitting problem and places the discrete Fourier transform in a new light. When fitting data to a model, the data may be thought of as a vector in an N-dimensional vector space