Multivariate resolution of flattened absorption spectra with weighted least squares; weight selection and rotational ambiguity

Abstract

Multivariate curve resolution (MCR) of absorption spectra is now a ubiquitously used tool. However, MCR methods, which use ordinary least squares (OLS) approach, assume that the measurement uncertainties are unbiased and homoscedastic. This is not true for absorption measurements, in which uncertainty variance and bias both increase as the true absorbance increases. The bias produces a well-known flattening/saturation of the peaks at high optical densities, which makes the data nonlinear and unsuitable for OLS-based MCR analysis. This problem can be reduced by using weighted least squares (WLS). In the present paper, the ability of WLS-based MCR to handle simulated and real datasets with realistic optical noise and flattening was assessed. Three weighting schemes were tested: OLS (unity weights), weights based on the maximum likelihood principle (MLP) and the physics of absorption measurement, and weights based on empirical cutoff (zero weights for saturated data points). The abilities of MCR to recover the true profiles and to evaluate rotational ambiguity of the solutions were compared for the 3 weighting schemes. MLP-and cutoff-based WLS-MCR produced better resolution of flattened data than OLS, but the success of the extension to strongly flattened spectra depended on data structure. MLP-based MCR was general and stable, while cutoff-based MCR was more sensitive to the data but could recover unbiased profiles. Generally, the use of WLS can expand MCR functionality to the analysis of flattened spectra. The specifics of finding WLS bilinear solutions and approaches to migrate factorbased MCR methods from OLS to WLS are also discussed. KEYWORDS absorption spectroscopy, multivariate curve resolution, peak flattening, rotational ambiguity, weighted least squares | INTRODUCTIONA combination of optical spectroscopy with multivariate curve resolution (MCR) is a mature and well-developed chemometric approach. 1-5 The rapidity and inexpensiveness of optical spectroscopy and its adaptability for many scientific and practical tasks are naturally complemented by the abilities of MCR to extract contributions of individual sources of spectral variance from strongly overlapped and correlated data. Multivariate curve resolution has been widely used for resolving peaks in chromatography, studying kinetics and equilibriums, etc. 2-5 Recently, adaptations of MCR for solving calibration tasks have also been developed. 6 These methods possess a second-order advantage, ie, the possibility to quantify components in the presence of unknown interferences.Generally, MCR assumes that the measured response x is a bilinear combination of several properties. In 2-way MCR the data matrix X can be written in compact matrix form:where C 0 is an m × k left factor matrix, S 0 is an n × k right factor matrix, and k is the number of factors. In practical