A probabilistic class-modelling method based on prediction bands for functional spectral data: Methodological approach and application to near-infrared spectroscopy

Abstract

Class-modelling methods aim to predict the conformity of new unknown samples with a single target class, using statistical decision rules built exclusively with objects of that class. This article introduces a novel class-modelling method for spectral data. The method uses the concept of β%-prediction band for functional data to classify spectra. The band is defined by an upper and a lower limiting spectra which delimit critical trajectories for β% of future spectra of the target class. It is constructed in three main steps: firstly, a naïve bootstrap sample of calibration spectra is projected onto a parsimonious principal component (PC) basis and their scores are estimated. The posterior predictive distribution of the scores on each PC is estimated using a Bayesian zero-mean normal model. This procedure is repeated on naïve bootstrap estimations of the PCs to obtain the predictive distribution of the scores. These enable to account for all modelling uncertainties including the random deviation of scores from their zero-mean on each PC, uncertainty in the variance of scores (eigenvalue) on each PC, and uncertainty in the PC estimations. Secondly, the predicted scores are back-transformed to the original signal scale to obtain the predictive distribution of future spectra. Thirdly, the predicted spectra are ranked to select the β% most central ones as typical set, whose ranges of variation are used to construct the simultaneous limits of the band. Once the band is constructed, reconstructions of future unknown test spectra by bootstrap PC models are projected onto it, and the extent to which they overlap with it is used to decide their acceptance or rejection. The statistical properties and classification performances of the proposed prediction band are evaluated on real near-infrared datasets and compared to the well-known soft-independent modelling of class analogy (SIMCA) model. The results of the evaluation provide evidence that the proposed prediction band possesses satisfactory predictive performances. It even outperforms the SIMCA while offering attractive advantages like risk-management and straightforward physical interpretability of outlyingness patterns of tested spectra.

Keywords: Bayesian chemometrics; Class-modelling; Depth statistic; Functional data analysis; Multivariate data analysis; Prediction band; Spectral predictive distribution.