Abstract
Many methods have been developed to allow for consideration of measurement errors during multivariate data analyses. The incorporation of the error structure into the analytical framework, usually described in terms of the covariance matrix of measurement errors, can provide better model estimation and prediction. However, little effort has been made to evaluate the effects of heteroscedastic measurement uncertainties on multivariate analyses when the covariance matrix of measurement errors changes with the measurement conditions. For this reason, the present work describes a new numerical procedure for analyses of heteroscedastic systems (heteroscedastic principal component regression or H-PCR) that takes into consideration the variations of the covariance matrix of measurement fluctuations. In order to illustrate the proposed approach, near infrared (NIR) spectra of xylene and toluene mixtures were measured at different temperatures and stirring velocities and the obtained data were used to build calibration models with different multivariate techniques, including H-PCR. Modeling of available xylene–toluene NIR data revealed that H-PCR can be used successfully for calibration purposes and that the principal directions obtained with the proposed approach can be quite different from the ones calculated through standard PCR, when heteroscedasticity is disregarded explicitly.
Highlights
Multivariate calibration methods constitute indispensable tools for solving many “realworld” problems [1]
It has long been recognized that measurement errors are inherent components of experimental measurements, traditional multivariate calibration methods, such as principal components analysis (PCA), principal components regression (PCR), partial least squares (PLS) and parallel factor analysis (PARAFAC), implicitly assume the occurrence of independent and identically distributed measurement Gaussian errors [2]
Modeling of available xylene–toluene near infrared (NIR) data revealed that heteroscedastic principal component regression (H-PCR) can be used successfully for calibration purposes and that the principal directions obtained with the proposed approach can be quite different from the ones calculated through standard PCR, when heteroscedasticity is disregarded explicitly
Summary
Multivariate calibration methods constitute indispensable tools for solving many “realworld” problems [1]. It has long been recognized that measurement errors are inherent components of experimental measurements, traditional multivariate calibration methods, such as principal components analysis (PCA), principal components regression (PCR), partial least squares (PLS) and parallel factor analysis (PARAFAC), implicitly assume the occurrence of independent and identically distributed measurement Gaussian errors [2]. PCA and PCR can possibly be regarded as the most popular and most powerful chemometric tools for process monitoring and quantitative analyses [3,4,5,6]. According to the PCA technique, the number of variables of the problem can be reduced through suitable linear combinations, so that variable combinations concentrate the highest possible variance of the available data [8,9]. The objective of PCA is generally twofold: to determine the best calibration model with the smallest number of variables [5]
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