Methods for systematic investigation of measurement error covariance matrices

Abstract

Several recently developed methods for multivariate data analysis allow the incorporation of prior information about the measurement error structure into the analysis to permit better model estimation and prediction. This error structure is described in the form of the measurement error covariance matrix, which defines the complex relationship between measurement uncertainty at one channel and those at other channels and/or for other samples. In this work, a systematic approach for characterizing the measurement error covariance matrix for a particular experimental or instrumental environment is presented. This approach involves a number of strategies that include visualization of covariance and correlation matrices, bilinear modelling through principal components analysis (PCA) and target-testing factor analysis, trilinear modelling through PARAFAC, and refinement of models to include interaction terms and independent errors. The primary goals of this characterization are to obtain a better understanding of the factors contributing to measurement error and to develop parametric models for error covariance that do not rely on extensive replication. To illustrate this approach, four experimental data sets are employed: (1) UV-visible absorbance data, (2) near-infrared (NIR) reflectance data, (3) fluorescence emission data, and (4) short-wave NIR (SW-NIR) absorbance data from a kinetics experiment. For both the UV-visible and SW-NIR spectra, the main contribution to the error structure is a constant offset term that appears to have a dependence on the reciprocal of the wavelength. The NIR reflectance spectra are dominated by constant and multiplicative offset noise that have a strong interaction. The fluorescence data is affected by independent shot-noise with a variance proportional to the magnitude of the spectrum, as well as by correlated offset noise with two largely independent terms, one which is fixed in magnitude and the other which depends on the square root of the spectrum. It was also found that the SW-NIR data exhibits a strong correlation among samples.