The trilinear constraint adapted to solve data with strong patterns of outlying observations or missing values

Abstract

The possibility to perform trilinear decompositions of data sets has the clear advantage of providing unique solutions. Excitation-emission fluorescence matrices (EEM) are the best known paradigm of chemical measurements providing a trilinear structure associated with the configuration of excitation, emission and sample modes. Chemometric tools, such as Parallel Factor Analysis (PARAFAC) and Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS) with trilinear constraint, assist in solving the mixture analysis problem by exploiting the trilinear behavior of the EEM measurements. However, the spectroscopic nature of EEM measurements makes that no emission signal can be recorded below the current excitation wavelength, generating a strong and systematic pattern of outlier (zero observations) in EEM data that challenges the classical analysis by MCR-ALS or PARAFAC. Several approaches have been proposed to deal with this problem, such as the identification of outlying values below the excitation wavelength and, thus, the use of data imputation in PARAFAC, but they show severe limitations when systematic outlying data patterns occur. In this paper, we propose a new implementation of the trilinear constraint in MCR-ALS algorithm to cope with EEM measurements where a strongly patterned of outlying data is present. This approach preserves the trilinear property and does not require any data imputation step to replace the outlying observations. Its performance is tested on simulated data, controlled pharmaceutical mixtures and hyperspectral images of a plant tissue (HSI). It should be noted that the approach proposed is applicable to EEM data, where a systematic pattern of outlying observations exist, but can be generalized to the treatment of any trilinear data set with a strong pattern of missing values.