Nicolaas (Klaas) M. Faber

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

This paper gives an introduction to multivariate calibration from a chemometrics perspective and reviews the various proposals to generalize the well-established univariate methodology to the multivariate domain. Univariate calibration leads to relatively simple models with a sound statistical underpinning. The associated uncertainty estimation and figures of merit are thoroughly covered in several official documents. However, univariate model predictions for unknown samples are only reliable if the signal is sufficiently selective for the analyte of interest. By contrast, multivariate calibration methods may produce valid predictions also from highly unselective data. A case in point is quantification from near-infrared (NIR) spectra. With the ever-increasing sophistication of analytical instruments inevitably comes a suite of multivariate calibration methods, each with its own underlying assumptions and statistical properties. As a result, uncertainty estimation and figures of merit for multivariate calibration methods has become a subject of active research, especially in the field of chemometrics.

Recent developments in CANDECOMP/PARAFAC algorithms: a critical review

Several recently proposed algorithms for fitting the PARAFAC model are investigated and compared to more established alternatives. Alternating least squares (ALS), direct trilinear decomposition (DTLD), alternating trilinear decomposition (ATLD), self-weighted alternating trilinear decomposition (SWATLD), pseudo alternating least squares (PALS), alternating coupled vectors resolution (ACOVER), alternating slice-wise diagonalization (ASD) and alternating coupled matrices resolution (ACOMAR) are compared on both simulated and real data. For the recent algorithms, only unconstrained three-way models can be fitted. In contrast, for example, ALS allows modeling of higher-order data, as well as incorporating constraints on the parameters and handling of missing data. Nevertheless, for three-way data, the newer algorithms are interesting alternatives. It is found that the ALS estimated models are generally of a better quality than any of the alternatives even when overfactoring the model, but it is also found that ALS is significantly slower. Based on the results (in particular the poor performance of DTLD), it is advised that (a slightly modified) ASD may be a good alternative to ALS when a faster algorithm is desired.

Net analyte signal calculation for multivariate calibration

Abstract A unifying framework for calibration and prediction in multivariate calibration is shown based on the concept of the net analyte signal (NAS). From this perspective, the calibration step can be regarded as the calculation of a net sensitivity vector, whose length is the amount of net signal when the value of the property of interest (e.g. analyte concentration) is equal to unity. The prediction step can be interpreted as projecting a measured spectrum onto the direction of the net sensitivity vector. The length of the projected spectrum divided by the length of the net sensitivity vector is the predicted value of the property of interest. This framework, which is equivalent to the univariate calibration approach, is used for critically revising different definitions of NAS and their calculation methods. The framework is particularized for the classical least squares (CLS), principal component regression (PLS) and partial least-squares (PCR) regression models.

Standard error of prediction for multiway PLS: 1. Background and a simulation study

While a multitude of expressions has been proposed for calculating sample-specific standard errors of prediction when using partial least squares (PLS) regression for the calibration of first-order data, potential generalisations to multiway data are lacking to date. We have examined the adequacy of two approximate expressions when using unfold- or tri-PLS for the calibration of second-order data. The first expression is derived under the assumption that the errors in the predictor variables are homoscedastic, i.e., of constant variance. In contrast, the second expression is designed to also work in the heteroscedastic case. The adequacy of the approximations is tested using extensive Monte Carlo simulations while the practical utility is demonstrated in Part 2 of this series.

Limit of detection estimator for second-order bilinear calibration

A new approach is developed for estimating the limit of detection in second-order bilinear calibration with the generalized rank annihilation method (GRAM). The proposed estimator is based on recently derived expressions for prediction variance and bias. It follows the latest IUPAC recommendations in the sense that it concisely accounts for the probabilities of committing both types I and II errors, i.e. false positive and false negative declarations, respectively. The estimator has been extensively validated with simulated data, yielding promising results.

Estimating the uncertainty in estimates of root mean square error of prediction: application to determining the size of an adequate test set in multivariate calibration

Abstract Root mean square error of prediction (RMSEP) is widely used as a criterion for judging the performance of a multivariate calibration model; often it is even the sole criterion. Two methods are discussed for estimating the uncertainty in estimates of RMSEP. One method follows from the approximate sampling distribution of mean square error of prediction (MSEP) while the other one is based on performing error propagation, which is a distribution-free approach. The results from a small Monte Carlo simulation study suggest that, provided that extreme outliers are removed from the test set, MSEP estimates are approximately proportional to a χ 2 random variable with n degrees of freedom, where n is the number of samples in the test set. It is detailed how this knowledge can be used to determine the size of an adequate test set. The advantages over the guideline issued by the American Society for Testing and Materials (ASTM) are discussed. The expression derived by the method of error propagation is shown to systematically overestimate the true uncertainty. A correction factor is introduced to ensure approximate correct behaviour. A close agreement is found between the uncertainties calculated using the two complementary methods. The consequences of using a too small test set are illustrated on a practical data set.

Uncertainty estimation for multivariate regression coefficients

Five methods are compared for assessing the uncertainty in multivariate regression coefficients, namely, an approximate variance expression and four resampling methods (jack-knife, bootstrapping objects, bootstrapping residuals, and noise addition). The comparison is carried out for simulated as well as real near-infrared data. The calibration methods considered are ordinary least squares (simulated data), partial least squares regression, and principal component regression (real data). The results suggest that the approximate variance expression is a viable alternative to resampling.

Detection limits in classical multivariate calibration models

Abstract This work presents a new approach for calculating multivariate detection limits for the commonly used classical or direct calibration models. The derived estimator, which is in accordance with latest IUPAC recommendations, accounts for the different sources of error related to the calibration and prediction steps. Since the multivariate detection limit for a given analyte is influenced by the presence of other components in the sample, a different detection limit is calculated for each analyte and analysed sample, at the chosen significance levels α and β . The proposed methodology has been experimentally validated by determining four pesticides in water using a FIA method with diode-array detection. The results compare favourably with the ones obtained using previously proposed estimators.

Sample-specific standard error of prediction for partial least squares regression

The development of an adequate expression for sample-specific standard error of prediction for partial least squares regression is a major trend in chemometrics literature. This article focuses on three generally applicable expressions, namely one recommended by the American Society for Testing and Materials (ASTM), one implemented in Unscrambler software and a simplification derived under the errors-in-variables (EIV) model. Results obtained for a near-infrared data set taken from the literature demonstrate that the EIV expression works best.

Second-order bilinear calibration: the effects of vectorising the data matrices of the calibration set

In a groundbreaking paper, Linder and Sundberg [Chemometr. Intell. Lab. Syst. 42 (1998) 159] developed a statistical framework for the calibration of second-order bilinear data. Within this framework, they formulated three different predictor construction methods [J. Chemom. 16 (2002) 12], namely the so-called naive method, the bilinear least squares (BLLS) method, and a refined version of the latter that takes account of the calibration uncertainty. Elsewhere [J. Chemom. 15 (2001) 743], a close relationship is established between the naive method and the generalized rank annihilation method (GRAM) by comparing expressions for prediction variance. Here it is proved that the BLLS method can be interpreted to work with vectorised data matrices, which establishes an algebraic relationship with so-called unfold partial least squares (PLS) and unfold principal component regression (PCR). It is detailed how these results enable quantifying the effects of vectorising bilinear second-order data matrices on analytical figures of merit and variance inflation factors.