Practical guidelines for reporting results in single- and multi-component analytical calibration: A tutorial

Practical guidelines for reporting analytical calibration results are provided. General topics, such as the number of reported significant figures and the optimization of analytical procedures, affect all calibration scenarios. In the specific case of single-component or univariate calibration, relevant issues discussed in the present Tutorial include: (1) how linearity can be assessed, (2) how to correctly estimate the limits of detection and quantitation, (2) when and how standard addition should be employed, (3) how to apply recovery studies for evaluating accuracy and precision, and (4) how average prediction errors can be compared for different analytical methodologies. For multi-component calibration procedures based on multivariate data, pertinent subjects here included are the choice of algorithms, the estimation of analytical figures of merit (detection capabilities, sensitivity, selectivity), the use of non-linear models, the consideration of the model regression coefficients for variable selection, and the application of certain mathematical pre-processing procedures such as smoothing.     

Multisignal calibration in spark- and ICP-OES

The advantages of multivariate calibration by using more than one selective emission line for one analyte are described in comparison with univariate calibration. Principle component regression (PCR) and partial least squares regression (PLS) are compared with classical univariate linear regression of single lines. Practical applications in spark- and inductively coupled plasma optical emission spectroscopy (ICP-OES) show that the sensitivity can be increased and thus a lower limit of detection be obtained by means of these multivariate techniques. Results from the investigations concerning the problem of collinearity are also discussed. Disturbed analyte lines are included in multisignal calibration. Their influence on the results of calibration is described.     

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.     

Characterization of the measurement error structure in 1D H NMR data for metabolomics studies

NMR-based metabolomics is characterized by high throughput measurements of the signal intensities of complex mixtures of metabolites in biological samples by assaying, typically, bio-fluids or tissue homogenates. The ultimate goal is to obtain relevant biological information regarding the dissimilarity in patho-physiological conditions that the samples experience. For a long time now, this information has been obtained through the analysis of measured NMR signals via multivariate statistics.NMR data are quite complex and the use of such multivariate statistical methods as principal components analysis (PCA) for their analysis assumes that the data are multivariate normal with errors that are identical, independent and normally distributed (i.e. iid normal). There is a consensus that these assumptions are not always true for these data and, thus, several methods have been devised to transform the data or weight them prior to analysis by PCA. The structure of NMR measurement noise, or the extent to which violations of error homoscedasticity affect PCA results have neither been characterized nor investigated.A comprehensive characterization of measurement uncertainties in NMR based metabolomics was achieved in this work using an experiment designed to capture contributions of several sources of error to the total variance in the measurements. The noise structure was found to be heteroscedastic and highly correlated with spectral characteristics that are similar to the mean of the spectra and their standard deviation. A model was subsequently developed that potentially allows errors in NMR measurements to be accurately estimated without the need for extensive replication.     

Generalized error-dependent prediction uncertainty in multivariate calibration

Most of the current expressions used to calculate figures of merit in multivariate calibration have been derived assuming independent and identically distributed (iid) measurement errors. However, it is well known that this condition is not always valid for real data sets, where the existence of many external factors can lead to correlated and/or heteroscedastic noise structures. In this report, the influence of the deviations from the classical iid paradigm is analyzed in the context of error propagation theory. New expressions have been derived to calculate sample dependent prediction standard errors under different scenarios. These expressions allow for a quantitative study of the influence of the different sources of instrumental error affecting the system under analysis. Significant differences are observed when the prediction error is estimated in each of the studied scenarios using the most popular first-order multivariate algorithms, under both simulated and experimental conditions.     

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)].     

Construction of stable multivariate calibration models using unsupervised segmented principal component regression

In multivariate spectral calibration by principal component regression (PCR), the principal components (PCs) are calculated from the response data measured at all employed instrument channels; however some channels are redundant and their responses do not possess useful information. Thus, the extracted PCs possess mixed information from both useful and redundant channels. In this work, we propose a segmentation approach based on unsupervised pattern recognition to identify the most informative spectral region and then to construct a stable multivariate calibration model by PCR. In this method, the instrument channels are clustered into different segments via Kohonen self‐organization map. The spectral data of each segment are then subjected to PCA and the derived PCs are used as input variables for an inverse least square (ILS) regression model employing stepwise selection of the informative PCs. The proposed method was evaluated by the analysis of four simulated and six experimental data sets. It was found that our proposed method can model the above data sets with prediction errors lower than conventional partial least squares (PLS) and PCR methods. In addition, the prediction ability of our method was better than the previously reported models for these data sets. Copyright © 2011 John Wiley & Sons, Ltd.     

Multivariate calibration leverages and spectral F‐ratios via the filter factor representation

Diagnostics are fundamental to multivariate calibration (MC). Two common diagnostics are leverages and spectral F‐ratios and these have been formulated for many MC methods such as partial least square (PLS), principal component regression (PCR) and classical least squares (CLS). While these are some of the most common methods of calibration in analytical chemistry, ridge regression is also common place and yet spectral F‐ratios have not been developed for it. Noting that ridge regression is a form of Tikhonov regularization (TR) and using the unifying filter factor representation for MC, this paper develops the filter factor form of leverages and spectral F‐ratios. The approach is applied to a spectral data set to demonstrate computational speed‐up advantages and ease of implementation for the filter factor representation. Copyright © 2010 John Wiley & Sons, Ltd.     

Daubechies小波主成分回归法机理及算法研究

将小波变换与主成分回归相结合,提出一种新型多元校正算法---小波基主成分回归法。理论分析和仿真实验表明,该法可更有效地去除噪声,提取有用信息。将其用于氯霉素及甲硝唑实际药物体系分析,与主成分回归法(PCR)相比,得到的回收率总平均相对误差由1.70%下降到0.90%。此外,通过将统计判据和小波多尺度分析相结合,发展了一种新的因子数判定方法。理论和实验研究表明,该法比传统因子数判定法具有更高的可靠性。     

Transfer of multivariate regression models between high‐resolution NMR instruments: application to authenticity control of sunflower lecithin

In recent years the number of spectroscopic studies utilizing multivariate techniques and involving different laboratories has been dramatically increased. In this paper the protocol for calibration transfer of partial least square regression model between high‐resolution nuclear magnetic resonance (NMR) spectrometers of different frequencies and equipped with different probes was established. As the test system previously published quantitative model to predict the concentration of blended soy species in sunflower lecithin was used. For multivariate modelling piecewise direct standardization (PDS), direct standardization, and hybrid calibration were employed. PDS showed the best performance for estimating lecithin falsification regarding its vegetable origin resulting in a significant decrease in root mean square error of prediction from 5.0 to 7.3% without standardization to 2.9–3.2% for PDS. Acceptable calibration transfer model was obtained by direct standardization, but this standardization approach introduces unfavourable noise to the spectral data. Hybrid calibration is least recommended for high‐resolution NMR data. The sensitivity of instrument transfer methods with respect to the type of spectrometer, the number of samples and the subset selection was also discussed. The study showed the necessity of applying a proper standardization procedure in cases when multivariate model has to be applied to the spectra recorded on a secondary NMR spectrometer even with the same magnetic field strength. Copyright © 2016 John Wiley & Sons, Ltd.     

Net analyte signal-based simultaneous determination of antazoline and naphazoline using wavelength region selection by experimental design-neural networks

Net analyte signal (NAS)-based multivariate calibration methods were employed for simultaneous determination of anthazoline and naphazoline. The NAS vectors calculated from the absorbance data of the drugs mixture were used as input for classical least squares (CLS), principal component and partial least squares regression PCR and PLS methods. A wavelength selection strategy was used to find the best wavelength region for each drug separately. As a new procedure, we proposed an experimental design-neural network strategy for wavelength region optimization. By use of a full factorial design method, some different wavelength regions were selected by taking into account different spectral parameters including the starting wavelength, the ending wavelength and the wavelength interval. The performance of all the multivariate calibration methods, in all selected wavelength regions for both drugs, was evaluated by calculating a fitness function based on the root mean square error of calibration and validation. A three-layered feed-forward artificial neural network (ANN) model with back-propagation learning algorithm was employed to model the nonlinear relationship between the spectral parameters and fitness of each regression method. From the resulted ANN models, the spectral regions in which lowest fitness could be obtained were chosen. Comparison of the results revealed that the net NAS-PLS resulted in lower prediction error than the other models. The proposed NAS-based calibration method was successfully applied to the simultaneous analyses of anthazoline and naphazoline in a commercial eye drop sample.     

Sensitive analytical method for simultaneous analysis of some vasoconstrictors with highly overlapped analytical signals

Multivariate calibration method is a powerful mathematical tool that can be applied in analytical chemistry when the analytical signals are highly overlapped. The method with regression by partial least squares is proposed for the simultaneous spectrophotometric determination of adrenergic vasoconstrictors in decongestive solution containing two active components: phenyleprine hydrochloride and trimazoline hydrochloride. These sympathomimetic agents are that frequently associated in pharmaceutical formulations against the common cold. The proposed method, which is, simple and rapid, offers the advantages of sensitivity and wide range of determinations without the need for extraction of the vasoconstrictors. In order to minimize the optimal factors necessary to obtain the calibration matrix by multivariate calibration, different parameters were evaluated. The adequate selection of the spectral regions proved to be important on the number of factors. In order to simultaneously quantify both hydrochlorides among excipients, the spectral region between 250 and 290 nm was selected. A recovery for the vasoconstrictor was 98–101%. The developed method was applied to assay of two decongestive pharmaceutical preparations.     

Comparative study of net analyte signal-based methods and partial least squares for the simultaneous determination of amoxycillin and clavulanic acid by stopped-flow kinetic analysis

A comparative study about advantages and limitations of net analyte signal (NAS)-based methods (NBMs) and partial least squares (PLS) calibration in kinetic analysis has been performed. The different multivariate calibration methods were applied to the determination of binary mixtures of amoxycillin and clavulanic acid, by stopped-flow kinetic analysis. The reactions of oxidation of these compounds with cerium(IV), in sulphuric acid medium, were monitored by following the changes on the fluorescence of the oxidation products, in stopped-flow mode. The differences on the kinetic profiles obtained at λ_ex=256 nm and λ_em=351 nm, were used to determine mixtures of both compounds by multivariate calibration of the kinetic data, using PLS-1, a modification of hybrid linear analysis (HLA) and net analyte pre-processing combined with classical least squares (NAP/CLS) methods. The NBMs allowed the selection of optimal time data regions by calculating the minimum error indicator function (EIF), improving the results and making NBMs very convenient for the analysis. In addition, the use of the net analyte signal concept allows the calculation of the analytical figures of merit, limit of detection (LOD), sensitivity and selectivity, for each component.     

Application of Multivariate Calibration Techniques to HPLC Data for Quantitative Analysis of a Binary Mixture of Hydrochlorathiazide and Losartan in Tablets

New multivariate approaches have been applied to high-performance liquid chromatography (HPLC) with multiwavelength photodiode-array (PDA) detection. Multivariate calibration techniques such as partial least squares (PLS), principal component regression (PCR), classical least squares (CLS), and inverse least squares (ILS) was subjected to HPLC data for simultaneous quantitative analysis of synthetic binary mixtures and a commercial tablet formulation containing hydrochlorothiazide (HCT) and losartan potassium (LST). The combined use of HPLC and multivariate calibrations has been denoted HPLC–CLS, HPLC–ILS, HPLC–PCR, and HPLC–PLS. Successful chromatographic separation of the two active compounds and enalapril maleate, used as internal standard (IS), was accomplished by means of a 4.6 mm i.d. × 250 mm, 5 m particle, Waters Symmetry C₁₈ reversed-phase column and a mobile phase consisting of 60:40 acetate buffer (0.2 M, pH 4.8)–acetonitrile (v/v, 60:40). HPLC data based on the ratio of analyte peak areas to IS peak area were obtained by PDA detection at five-wavelengths (250, 255, 260, 265, and 270 nm). The HPLC–CLS, HPLC–ILS, HPLC–PCR, and HPLC–PLS calibration plots for hydrochlorothiazide and losartan potassium were constructed separately by using the peak-area ratios corresponding to the concentrations of each active compound. The HPLC multivariate calibrations obtained were tested for different synthetic mixtures containing HCT and LST in the presence of the IS. These multivariate chromatographic methods were also applied to a commercial pharmaceutical dosage form containing HCT and LST. The results obtained from the multivariate calibrations were compared with those obtained by use of another, classical HPLC method using single-wavelength detection.Revised: 29 September 2004 and 4 January 2005     

小波变换-主成分回归分光光度法同时测定邻苯二酚、间苯二酚和对苯二酚三组分

根据小波变换具有将信号分频的特点,本文提出了将小波变换与主成分回归(PCR)相结合的一种多元校正算法。该法能更有效地去除噪声,提取有用信息,并将其用于分析邻苯二酚、间苯二酚、对苯二酚三组分体系。实验结果表明,本法比直接用主成分回归处理效果好,得到的平均相对误差从2．24％降低到1．19％。     

Butene concentration prediction in ethylene/propylene/1-butene terpolymers by FT-IR spectroscopy through multivariate statistical analysis and artificial neural networks

This paper reports the development of calibration models for quality control in the production of ethylene/propylene/1-butene terpolymers by the use of multivariate tools and FT-IR spectroscopy.1-Butene concentration prediction is achieved in terpolymers by coupling FT-IR spectroscopy to multivariate regression tools. A dataset of 26 terpolymers (14 coming from a constrained experimental design for mixtures, plus 12 terpolymers used for external validation) was analysed by FT-IR spectroscopy. An internal method of “Polimeri Europa” plant, based on ¹³C NMR spectroscopy is used to determine the percentage of 1-butene in the samples. Then, different multivariate tools are used for 1-butene concentration prediction based on the FT-IR spectra recorded. Different multivariate calibration methods were explored: principal component regression (PCR), partial least squares (PLS), stepwise OLS regression (SWR) and artificial neural networks (ANNs). The model obtained by back-propagation neural networks turned out to be the best one. The performances of the BP-ANN model were further improved by variable selection procedures based on the calculation of the first derivative of the network.The proposed approach allows the monitoring in real time of the polymer synthesis and the estimation of the characteristics of the product attainable from the concentration of 1-butene.     

Simultaneous spectrophotometric determination of phenanthridine, phenanthridinone and phenanthridine N-oxide using multivariate calibration methods

The multivariate calibration methods, partial least squares (PLS) and principle component regression (PCR) have been used to determine phenanthridine, phenanthridinone and phenanthridine N-oxide in spiked human plasma samples. Resolution of binary and ternary mixtures of analytes with minimum sample pre-treatment and without analyte separation has been successfully achieved analyzing the UV spectral data. The net analyte signal (NAS) concept was also used to calculate multivariate analytical figures of merit such as limit of detection, selectivity and sensitivity. The simultaneous determination of three analytes was possible by PLS and PCR processing of sample absorbance in the 210–355 nm region. Good recoveries were obtained for both synthetic mixtures and spiked human plasma samples.     

Partial least-squares regression on design variables as an alternative to analysis of variance

The partial least-squares (PLS) algorithm has become popular for explorative multivariate data analysis and for multivariate calibration. The same PLS algorithm can also be used for confirmatory data analysis. The discussion is limited to analysis of a single response variable. A close correspondence of PLS1 regression to classical analysis of variance (ANOVA) is demonstrated. The design of an experiment is described in terms of discrete design variables for main effects and simple interactions (dummy variables). These are used as regressors X = (x₁, x₂,…,) for modelling the response variable of the experiment, y. As in conventional use of PLS1 regression, the algorithm gives a concentrated model or diagram of the most important, y-relevant variability types in the X-data. In the present case, this gives the combination of design variables that models the variations in y. A simple plot of the resulting factor loadings immediately reveals the important design variables. Statistical tests and confidence regions in the PLS solution give additional safeguards against interpretation of spurious effects. The method is applied to two data sets. One concerns assessment of personal preference for blackcurrent juice, studied in a 2⁵ factorial experiment; these data are also studied with missing values and as fractional factorials. The other ceoncers spectrophotometric absorbance-based colour assessments of pigment in strawberry jam in a 3-factor design with 2, 2 and 3 levels in the respective factors.     

Simultaneous Polarographic Determination of Lead (II) and TIN (II) by Multivariate Calibration

ABSTRACT

The polarographic waves of lead (II) and tin (II) overlap due to their similar reductive potentials and it is difficult to determine these two components simultaneously without a pre-separation. In this paper, differential pulse polarography (DPP) combined with multivariate calibration approaches, such as classical least squares (CLS), principal component regression (PCR) and partial least squares (PLS), were successfully applied to the resolution of overlapping polarographic waves of these two components in the concentration range of 0.05-3.50 mg 1^?1. Satisfactory quantitative results were obtained.     

Improvement of multivariate calibration techniques applied to 1-to-N component mixtures through an optimized experimental design

This work describes a novel experimental design aimed at building a calibration set constituted by samples containing a different number of components. The algorithm performs a reiteration process to maintain the number of samples at the lower value as possible and to ensure an homogeneous presence of all the concentration levels. The mixture design was applied to a drug system composed by one-to-four components in different combination. The resolution of the system was performed by three multivariate UV spectrophotometric methods utilizing principal component regression (PCR) and partial last squares (PLS1 and PLS2) algorithms. The calibration set was composed by 61 references on four concentration levels, including 15 samples for each quaternary, ternary and binary composition and 16 one-component samples. The calibration models were optimized through a careful selection of number of factors and wavelength zones, in such a way as to remove interferences from instrumental noise and excipients present in the pharmaceutical formulations. The prediction power of the regression models were verified and compared by analysis of an external prediction set. The models were finally used to assay pharmaceutical specialities containing the studied drugs in one-to-four formulations.