Bayesian wavelet regression on curves with application to a spectroscopic calibration problem

Motivated by calibration problems in near-infrared (NIR) spectroscopy we co nsider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced point s and there may be multiple predictands. We tackle this regression problem by calculating the wavelet transforms of the discretized curves. then apply ing a Bayesian variable selection method using mixture priors to the multiv ariate regression of predictands on wavelet coefficients. Far prediction pu rposes, we average over a set of likely models. Applied to a particular pro blem in NIR spectroscopy, this approach was able to find subsets of the wav elet coefficients with overall better predictive performance than the more usual approaches. In the application, the available predictors are measurem ents of the NIR reflectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths. The aim is to predict the composition (i.e., the fat, flour, sugar, and water content) of the dough pieces using the spectral var iables. Thus we have a multivariate regression of four predictands on 256 p redictors with quite high intercorrelation among the predictors. A training set of 39 samples is available to fit this regression. Applying a wavelet transform replaces the 256 measurements on each spectrum with 256 wavelet c oefficients that carry the same information. The variable selection method could use subsets of these coefficients that gave good predictions for all four compositional variables on a separate test set of samples. Selecting i n the wavelet domain rather than from the original spectral variables is ap pealing in this application, because a single wavelet coefficient can carry information from a band of wavelengths in the original spectrum. This band can be narrow or wide, depending on the scale of the wavelet selected.