Linear calibrations in chromatography: The incorrect use of ordinary least squares for determinations at low levels,and the need to redefine the limit of quantification with this regression model

Ordinary least squares is widely applied as the standard regression method for analytical calibrations, and it is usually accepted that this regression method can be used for quantification starting at the limit of quantification. However, it requires calibration being homoscedastic and this is not common. Different calibrations have been evaluated to assess whether ordinary least squares is adequate to quantify estimates at low levels. All calibrations evaluated were linear and heteroscedastic. Despite acceptable values for precision at limit of quantification levels were obtained, ordinary least squares fitting resulted in significant and unacceptable bias at low levels. When weighted least squares regression was applied, bias at low levels was solved and accurate estimates were obtained. With heteroscedastic calibrations, limit values determined by conventional methods are only appropriate if weighted least squares are used. A “practical limit of quantification” can be determined with ordinary least squares in heteroscedastic calibrations, which should be fixed at a minimum of 20 times the value calculated with conventional methods. Biases obtained above this “practical limit” were acceptable applying ordinary least squares and no significant differences were obtained between the estimates measured using weighted and ordinary least squares when analyzing real‐world samples.     

Statistical comparison of calibration methods in multicomponent analysis

Spectrophotometric multicomponent analysis is considerd on the basis of inverse multivariate calibration with linear methods (ordinary least squares, principal component, ridge and partial least squares regression) and with the non-linear methods ACE and the non-linear partial least squares. The performance of the different methods is compared by paired F-tests. As an estimate of the error variance the residual mean sum of squares in the analysis of variance table is used. The comparison is demonstrated for the infrared spectrometric analysis of the hydroxyl group content of brown coal measured in diffuse reflectance. Although the error variances among the calibration methods differ gradually, the differences are much less pronounced at statistical level.     

Simulation of high- and low-resolution mass spectra for assessment of calibration methods

Calibrating mixtures of residual gases in quadrupole mass spectrometry (QMS) can be difficult since low m/z ratios of molecular ions and their fragments result in overlap of signals especially in the lower mass regions. This causes problems in univariate calibration methods and encourages use of full spectral multivariate methods. Experimental assessment of regression methods has limitations since experimental sources of error can only be minimised and not entirely eliminated. A method of simulating full spectra at low and high resolution to accurate masses is described and these are then used for a calibration study of some popular linear regression methods [classical least squares regression (CLS), partial least squares (PLS), principal component regression (PCR)].     

Robust regression techniques A useful alternative for the detection of outlier data in chemical analysis

The validation of an analytical procedure means the evaluation of some performance criteria such as accuracy, sensitivity, linear range, capability of detection, selectivity, calibration curve, etc. This implies the use of different statistical methodologies, some of them related with statistical regression techniques, which may be robust or not. The presence of outlier data has a significant effect on the determination of sensitivity, linear range or capability of detection amongst others, when these figures of merit are evaluated with non-robust methodologies.In this paper some of the robust methods used for calibration in analytical chemistry are reviewed: the Huber M-estimator; the Andrews, Tukey and Welsh GM-estimators; the fuzzy estimators; the constrained M-estimators, CM; the least trimmed squares, LTS. The paper also shows that the mathematical properties of the least median squares (LMS) regression can be of great interest in the detection of outlier data in chemical analysis. A comparative analysis is made of the results obtained by applying these regression methods to synthetic and real data. There is also a review of some applications where this robust regression works in a suitable and simple way that proves very useful to secure an objective detection of outliers. The use of a robust regression is recommended in ISO 5725-5.     

New approach by Kriging models to problems in QSAR

Most models in quantitative structure and activity relationship (QSAR) research, proposed by various techniques such as ordinary least squares regression, principal components regression, partial least squares regression, and multivariate adaptive regression splines, involve a linear parametric part and a random error part. The random errors in those models are assumed to be independently identical distributed. However, the independence assumption is not reasonable in many cases. Some dependence among errors should be considered just like Kriging. It has been successfully used in computer experiments for modeling. The aim of this paper is to apply Kriging models to QSAR. Our experiments show that the Kriging models can significantly improve the performances of the models obtained by many existing methods.     

化学计量学在我国光度分析中的进展

本文介绍了化学计量学在我国光度分析中的研究应用及某些进展。这些方法可用于消除干扰,改善选择性,实现无机、有机及药物多组分混合物以及复方制剂的同时测定。     

Comparison of regression techniques for linear calibration

A recently presented regression technique for linear calibration, which is based on a variance component model for univariate quantitative measurement data, is compared with the conventional and far spread regression techniques ordinary least squares regression and weighted least squares regression. The associated statistical models and estimations are represented. Its application is demonstrated at some practical examples. With consideration of special variation causes, like matrix influence or the influence of several operating conditions on the measurement response, it can be shown that the application of the variance component model is an advantage.     

High-speed chromatographic analysis of high-fructose corn syrup for process monitoring

High-speed chromatography is coupled with numerical methods for analyzing unresolved chromatograms and applied to a process analysis of high-fructose corn syrup. A column selection process is demonstrated where a minimum amount of resolution is sacrificed in order to decrease analysis time from over 5 min to 25 sec. Two data analysis methods, linear least squares regression and the sequential chromatogram ratio technique coupled with sequential suppression, are compared for their ability to quantitate the poorly resolved chromatograms. Both methods fit pure component analyte chromatograms, collected on a computer, to a sample chromatogram with unknown concentrations of each analyte. For a high-fructose corn syrup sample with a nominal fructose concentration of 55%, linear least squares analysis gave a fructose concentration percentage of 57.2 +/- 0.9%. The sequential chromatogram ratio algorithm gave a fructose concentration percentage of 57.9 +/- 0.7%.     

多元光度测定病态系统的岭回归估计

考察了含多种酚类的有机混合体系的同时测定，发现对于这类化学与光谱性质皆十分相近的混合物体系，最小二乘法和卡尔曼滤波方法都不能给出令人满意的结果，甚至出现负值，而脾岭回归估计方法对此类病态体系进行分析，所得结果明显优于最小二乘法与卡尔曼滤波方法。     

Halide ion-selective electrode array calibration

The calibration of several ions (Cl(-), Br(-), F(-) and OH(-)) measured with an ion selective electrodes (ISE) array has been carried out in the presence of interferents using an experimental design and multivariate calibration methods. Partial least squares regression and principal component regression do not seem to improve the test set prediction compared to multivariate linear regression. In the case of very slight or no interference on the ISE, each ion can be determined using the corresponding ISE and univariate calibration methods, but the use of multivariate methods does not lead to worse results.     

Robust online algorithm for adaptive linear regression parameter estimation and prediction

In this article, we focus on adaptive linear regression methods and propose a new technique. The article begins with a review of the online passive aggressive algorithm (OPAA), an adaptive linear regression algorithm from the machine learning literature. We highlight the strengths and weaknesses of OPAA and compare it with other popular adaptive regression techniques such as moving window and recursive least squares, recursive partial least squares, and just‐in‐time or locally weighted regression. Modifications to OPAA are proposed to make it more robust and better suited for industrial soft‐sensor applications. The new algorithm is called smoothed passive aggressive algorithm (SPAA), and like OPAA, it follows a cautious parameter update strategy but is more robust. The trade‐off between SPAA's computation complexity and accuracy can be easily controlled by manipulating just two tuning parameters. We also demonstrate that the SPAA framework is quite flexible and a number of variants are easily formulated. Application of SPAA to estimate the time‐varying parameters of a numerically simulated autoregressive with exogenous terms (ARX) model and to predict the Reid vapor pressure of the bottoms flow from an industrial column demonstrates its superior performance over OPAA and comparable performance with the other popular algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

化学计量学在过程分析中的应用及进展

评述了化学计量学方法在生产过程分析中各个方面 ,如过程优化、过程模拟、仪器及仪器校正、过程监测等方面的应用 ,并展望了化学计量学在过程分析中的应用前景     

Non-linear calibration models for near infrared spectroscopy

Different calibration techniques are available for spectroscopic applications that show nonlinear behavior. This comprehensive comparative study presents a comparison of different nonlinear calibration techniques: kernel PLS (KPLS), support vector machines (SVM), least-squares SVM (LS-SVM), relevance vector machines (RVM), Gaussian process regression (GPR), artificial neural network (ANN), and Bayesian ANN (BANN). In this comparison, partial least squares (PLS) regression is used as a linear benchmark, while the relationship of the methods is considered in terms of traditional calibration by ridge regression (RR). The performance of the different methods is demonstrated by their practical applications using three real-life near infrared (NIR) data sets. Different aspects of the various approaches including computational time, model interpretability, potential over-fitting using the non-linear models on linear problems, robustness to small or medium sample sets, and robustness to pre-processing, are discussed. The results suggest that GPR and BANN are powerful and promising methods for handling linear as well as nonlinear systems, even when the data sets are moderately small. The LS-SVM is also attractive due to its good predictive performance for both linear and nonlinear calibrations.     

Multivariate versus univariate calibration for nonlinear chemiluminescence data: Application to chromium determination by luminol-hydrogen peroxide reaction

Multivariate calibration is tested as an alternative to model chromium(III) concentration versus chemiluminescence registers obtained from luminol-hydrogen peroxide reaction. The multivariate calibration approaches included have been: conventional linear methods (principal component regression (PCR) and partial least squares (PLS)), nonlinear methods (nonlinear variants and variants of locally weighted regression) and linear methods combined with variable selection performed in the original or in the transformed data (stepwise multiple linear regression procedure). Both the direct and inverse univariate approaches have been also tested.

The use of a double logarithmic transformation previous to the linear regression has been also evaluated. A new double logarithmic transformation previous to the linear regression is proposed in order to avoid the effect of the noise in the calibration model. Pre-processing, optimization and prediction ability of the multivariate calibration models has been studied at nine different experimental conditions including batch and FIA measurements. Box-plots, PCA and cluster analysis have been employed to test the prediction ability of the different models tested. Nonlinear PCR and nonlinear PLS provide the best results. Real samples have been analyzed and compared with the reference method. The results confirm the successful use of the proposed methodology.     

Envirometrics. Part I: Modeling of water salinity and air quality data

Envirometrics utilises advanced mathematical, statistical and information tools to extract information. Two typical environmental data sets are analysed using MVATOB (Multi Variate Analysis TOol Box). The first data set corresponds to the variable river salinity. Least median squares (LMS) detected the outliers whereas linear least squares (LLS) could not detect and remove the outliers. The second data set consists of daily readings of air quality values. Outliers are detected by LMS and unbiased regression coefficients are estimated by multi-linear regression (MLR). As explanatory variables are not independent, principal component regression (PCR) and partial least squares regression (PLSR) are used. Both examples demonstrate the superiority of LMS over LLS.     

A robust partial least squares regression method with applications

Partial least squares (PLS) regression is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper, we show that if the sample covariance matrix is properly robustified further robustification of the linear regression steps of the PLS algorithm becomes unnecessary. The robust estimate of the covariance matrix is computed by searching for outliers in univariate projections of the data on a combination of random directions (Stahel—Donoho) and specific directions obtained by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto [1]. It is shown that this procedure is fast to apply and provides better results than other methods proposed in the literature. Its performance is illustrated by Monte Carlo and by an example, where the algorithm is able to show features of the data which were undetected by previous methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Multicomponent kinetic determinations using multivariate calibration techniques

Multivariate calibration techniques for use in multicomponent kinetic-based determinations are reviewed. Multivariate calibration is a chemometric tool that continues to grow in popularity among analytical chemists. Multicomponent kinetic methods depend on differences in rates of reactions or processes to distinguish among the components. Kinetic profiles or a combination of kinetic profiles and spectra are commonly used. Because of their ability to process large quantities of data, multivariate calibration techniques are well suited for kinetic-based determinations. The concepts and principles of multivariate calibration are discussed first. Classical least squares regression, principal component regression, partial least squares regression and artificial neural networks are the multivariate calibration techniques considered here in detail. Recent examples of the application of these techniques to multicomponent kinetic determinations are reviewed. Both single and multiwavelength kinetic data are considered.