Interval estimation in multivariate curve resolution by exploiting the principles of error propagation in linear least squares

Abstract Interval estimation in Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS) is a challenging problem. Several algorithms including Bayesian, Monte-Carlo, bootstrap and jackknife resampling approaches have been proposed previously to address this problem in MCR. In the present contribution, constructing the confidence intervals (CIs) in MCR-ALS resolved profiles using the principles of error propagation in linear least squares (LS) parameter estimation is proposed. In MCR-ALS, every set of profiles in the row subspace (scores) can be considered as coefficients for estimating their counterparts in the columns subspace (loadings) and vice versa. Therefore, it should be possible to calculate the CIs for these coefficients (scores or loadings) using the principles of error propagation in LS. The confidence intervals of the coefficients in linear regression are directly related to the inverse of the matrix of sums of squares and sums of cross products of independent variables and to the standard deviation of the residuals. This idea has been successfully applied in this work for the construction of the CIs of the resolved profiles in the MCR-ALS algorithm. The proposed approach is named ‘Confidence Intervals based on Least Squares’ (CILS). The weighted version of this approach has also been implemented and named as CIWLS. The latter method can be used for handling datasets with a known type of error structure. The performances of the CILS and CIWLS approaches are evaluated in this work for the calculation of the CIs for several simulated three component LC-MS and LC-DAD datasets, with different homo- and heteroscedastic noise levels. The CIs obtained by CILS and CIWLS are compared with those obtained by the Monte-Carlo noise addition method. The empirical coverage probabilities of the CIs are also computed to check if the calculated CIs achieve the nominal confidence level. The average standard errors (ASE) are used as measures of the level of uncertainty (precision) in the recovered profiles. The results obtained in this work revealed that the CIs obtained by the CILS method are in agreement with those obtained by the Monte-Carlo approach. The main advantage of the CILS method is that the calculations are faster and require less computation time. Finally, the performance of the CIWLS algorithm was assessed in the analysis of a real environmental dataset for the source apportionment of particulate matter (PM) in air samples collected in Northern Spain. The results obtained by the CIWLS method were in agreement with those previously reported for this real dataset and the CIs of the contribution and composition profiles of the PM10 sources were properly estimated.