Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Abstract

Motivated by calibration problems in near-infrared (N IR) spectroscopy, we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands. We tackle this regression problem by calculating the wavelet transforms of the discretized curves, then applying a Bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coef cients. For prediction purposes, we average over a set of likely models. Applied to a particular problem in N IR spectroscopy, this approach was able to nd subsets of the wavelet coef cients with overall better predictive performance than the more usual approaches. In the application, the available predictors are measurements of the N IR re ectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths. The aim is to predict the composition (i.e., the fat, our, sugar, and water content) of the dough pieces using the spectral variables. Thus we have a multivariate regression of four predictands on 256 predictors with quite high intercorrelation among the predictors. A training set of 39 samples is available to t this regression. Applying a wavelet transform replaces the 256 measurements on each spectrum with 256 wavelet coef cients that carry the same information. The variable selection method could use subsets of these coef cients that gave good predictions for all four compositional variables on a separate test set of samples. Selecting in the wavelet domain rather than from the original spectral variables is appealing in this application, because a single wavelet coef cient 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.

Sandwich algorithms for Bayesian variable selection

“…We consider the biscuit dough dataset, which was previously analyzed by Brown et al [1] and more recently by [11]. The dataset was obtained from a near-infrared (NIR) spectroscopy experiment used to analyze the composition of biscuit dough pieces, and it is available as part of the R package ppls [17].…”

“…Email addresses: joyee-ghosh@uiowa.edu (Joyee Ghosh), aixin-tan@uiowa.edu (Aixin Tan) 1 The authors contributed equally. 2 Supplementary material includes details of algorithms, statement and proof of a lemma, and theoretical comparisons of sandwich algorithms in some examples.…”

Bayesian linear regression and variable selection for spectroscopic calibration

“…The advantage of introducing Bayesian methodology into neural networks was reviewed [10] through the application to the calibration of near infrared spectroscopy. As early advocates of Bayesian approach, Brown and co-workers proposed a number of calibration models that demonstrated promising results, such as Bayesian variable selection methods for calibration [21,22] and wavelet regression [23]. One salient feature of Brown's work is that Markov chain Monte Carlo (MCMC) simulation is employed for the inference of the model parameters.…”

Combination of Models Obtained by Regression in the Wavelet Domain

“…Models are then built by applying regression techniques to the wavelet coefficients resulting from the bank filter decomposition [4,6,23]. In many applications, such a procedure led to improvements in the prediction performance of the resulting models [24].…”