Multivariate Calibration,an Overview

Abstract

Numerous methods of multivariate calibration methods exist with ridge regression, principal component regression, and partial least squares being the most popular methods in analytical chemistry. This mini‐review overviews multivariate calibration and provides a common theme with respect to the bias/variance tradeoff (harmony) and the harmony/parsimony tradeoff for model selection. Other multivariate calibration considerations are briefly reviewed. A few applications are noted.     

Supervised principal components: a new method for multivariate spectral analysis

The supervised principal components (SPC) method was proposed by Bair and Tibshirani for statistics regression problems where the number of variables greatly exceeds the number of samples. This case is extremely common in multivariate spectral analysis. The objective of this research is to apply SPC to near‐infrared and Raman spectral calibration. SPC is similar to traditional principal components analysis except that it selects the most significant part of wavelength from the high‐dimensional spectral data, which can reduce the risk of overfitting and the effect of collinearity in modeling according to a semi‐supervised strategy. In this study, four conventional regression methods, including principal component regression, partial least squares regression, ridge regression, and support vector regression, were compared with SPC. Three evaluation criteria, coefficient of determination (R²), external correlation coefficient (Q²), and root mean square error of prediction, were calculated to evaluate the performance of each algorithm on both near‐infrared and Raman datasets. The comparison results illustrated that the SPC model had a desirable ability of regression and prediction. We believe that this method might be an alternative method for multivariate spectral analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Near Infrared Spectral Similarity Combined with Variable Selection Method in the Quality Control of Flos Lonicerae: A Preliminary Study

This paper indicates the possibility to use near infrared (NIR) spectral similarity as a rapid method to estimate the quality of Flos Lonicerae. Variable selection together with modelling techniques is utilized to select representative variables that are used to calculate the similarity. NIR is used to build calibration models to predict the bacteriostatic activity of Flos Lonicerae. For the determination of the bacteriostatic activity, the in vitro experiment is used. Models are built for the Gram‐positive bacteria and also for the Gram‐negative bacteria. A genetic algorithm combined with partial least squares regression (GA‐PLS) is used to perform the calibration. The results of GA‐PLS models are compared to interval partial least squares (iPLS) models, full‐spectrum PLS and full‐spectrum principal component regression (PCR) models. Then, the variables in the two GA‐PLS models are combined and then used to calculate the NIR spectral similarity of samples. The similarity based on the characteristic variables and full spectrum is used for evaluating the fingerprints of Flos Lonicerae, respectively. The results show that the combination of variable selection method, modelling techniques and similarity analysis might be a powerful tool for quality control of traditional Chinese medicine (TCM).     

Simultaneous Determination of Eight Bioactive Components of Radix Dipsaci by Near-infrared Spectroscopy

A new method was developed using Fourier transform near-infrared spectroscopy and high-performance liquid chromatography with diode array detection for the identification and determination of eight major compounds in crude and sweated Radix Dipsaci. Partial least square regression was selected for the analysis. Multiplicative scatter correction, first derivative, and a Savitzky–Golay filter were used for the spectral pretreatment of the crude material, while standard normal variation, first derivative, and the Savitzky–Golay filter were used for the sweated samples. The correlation coefficients of the calibration models were above 0.99 and the root mean square error of calibration, the root mean square error of prediction, and root mean square error of cross-validation were under 0.63. The developed models were used to analyze unknown crude and sweated Radix Dipsaci with satisfactory results. The established methods were rapid, simple, nondestructive, and useful for quality control of Radix Dipsaci.     

人工神经网络紫外光度法同时测定甲基苯甲醛的三种同分异构体及其与偏最小二乘法的比较

同分异构体的同时测定一直是分析化学领域的热点和难点问题,本文将化学计量学中的多元校正方法,如偏最小二乘法和人工神经网络法与紫外分光光度法相结合,同时测定了紫外吸收光谱严重重叠的甲基苯甲醛的三种同分异构体混合体系中各组分的含量。确定了测定的最佳波长范围为230～304 nm;测得48个混合标样的吸光度值用于建立模型,其中,邻、间、对甲基苯甲醛的浓度范围分别为6.0～15.0、7.0～16.0和8.0～19.0 μg·mL-1。7个模拟样品作为监测集用于检验所建立模型的预测性能。本文还讨论了三种组分浓度比例对所建立模型预测性能的影响并确定了可以准确测定的浓度比例范围。所建立的方法用于模拟样品的测定,其回收率在84.00%与109.60%之间。与偏最小二乘法的测定结果比较,经成对t检验表明,两种方法对邻、间甲基苯甲醛测定结果无显著性差异;而对甲基苯甲醛的测定,人工神经网络法的测定结果优于偏最小二乘法。     

Prediction of physical–chemical properties of crude oils by 1H NMR analysis of neat samples and chemometrics

In this work, we report the feasibility study to predict the properties of neat crude oil samples from 300‐MHz NMR spectral data and partial least squares (PLS) regression models. The study was carried out on 64 crude oil samples obtained from 28 different extraction fields and aims at developing a rapid and reliable method for characterizing the crude oil in a fast and cost‐effective way. The main properties generally employed for evaluating crudes' quality and behavior during refining were measured and used for calibration and testing of the PLS models. Among these, the UOP characterization factor K (K_UOP) used to classify crude oils in terms of composition, density (D), total acidity number (TAN), sulfur content (S), and true boiling point (TBP) distillation yields were investigated. Test set validation with an independent set of data was used to evaluate model performance on the basis of standard error of prediction (SEP) statistics. Model performances are particularly good for K_UOP factor, TAN, and TPB distillation yields, whose standard error of calibration and SEP values match the analytical method precision, while the results obtained for D and S are less accurate but still useful for predictions. Furthermore, a strategy that reduces spectral data preprocessing and sample preparation procedures has been adopted. The models developed with such an ample crude oil set demonstrate that this methodology can be applied with success to modern refining process requirements. Copyright © 2012 John Wiley & Sons, Ltd.     

QSAR modeling based on the bias/variance compromise: a harmonious

Modeling quantitative structure–activity relationships (QSAR) is considered with an emphasis on prediction. An abundance of methods are available to develop such models. Using a harmonious approach that balances the bias and variance of predictions, the best calibration models are identified relative to the bias and variance criteria used. Criteria utilized to determine the adequacy of models are the root mean square error of calibration (RMSEC) and validation (RMSEV), respective R ² values, and the norm of the regression vector. QSAR data from the literature are used to demonstrate concepts. For these data sets and criteria used, it is suggested that models obtained by ridge regression (RR) are more harmonious and parsimonious than models obtained by partial least squares (PLS) and principal component regression (PCR) when the data is mean-centered. The most harmonious RR models have the best bias/variance tradeoff reflected by the smallest RMSEC, RMSEV, and regression vector norms and the largest calibration and validation R ² values. The most parsimonious RR models have the smallest effective rank.     

Interrelationships between generalized Tikhonov regularization,generalized net analyte signal,and generalized least squares for desensitizing a multivariate calibration to interferences

Orthogonal pre‐processing (orthogonal projection) of spectral data is a common approach to generate analyte‐specific information for use in multivariate calibration. The goal of this pre‐processing is to remove from each spectrum the respective sample interferent contributions (spectral interferences from overlap, scatter, noise, etc.). Two approaches to accomplish orthogonal pre‐processing are net analyte signal (NAS) and generalized least squares (GLS). Developed in this paper is the mathematical relationship between NAS and GLS. It is also realized that orthogonal NAS pre‐processing can remove too much analyte signal and that the degree of interferent correction can be regulated. Similar to GLS, the degree of correction is accomplished by using a regularization (tuning) parameter to form generalized NAS (GNAS). Also developed in this paper is an alternative to GNAS and GLS based on generalized Tikhonov regularization (GTR). The mathematical relationships between GTR, GNAS, and GLS are derived. A result is the ability to express the model vector as the sum of two contributions: the orthogonal NAS contribution and a non‐NAS contribution from the interferent components. Thus, rather than the usual situation of sequentially pre‐processing data by either GNAS or GLS followed by model building with the pre‐processed data, the methods of GTR, GNAS, and GLS are expressed as direct computations of model vectors allowing concurrent pre‐processing and model building to occur. Simultaneous pre‐processing and model forming are shown to be natural to the GTR process. Two near‐infrared spectroscopic data sets are studied to compare the theoretical relationships between GTR, GNAS, and GLS. One data set covers basic calibration, and the other data set is for calibration maintenance. Filter factor representation is key to developing the interprocess relationships. Copyright © 2013 John Wiley & Sons, Ltd.     

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)].     

Development of variance‐filtered instrumental transfer methods for high‐resolution NMR spectroscopy

The introduction of a variance‐filter to both direct standardization (DS) and piece‐wise direct standardization (PDS) instrumental transfer methods for the analysis of NMR spectral data is described. The variance‐filter modification allows for the identification of regions in the NMR spectra that are not adequately represented by the limited number of transfer calibration samples used during the calculation of the instrument‐to‐instrument transfer matrix. For these spectral frequencies, the corresponding portion of the transfer matrix is replaced by identity (or scaled identity) prior to the secondary instrumental data sets being transferred to the target instrument response. The spectral matching performance of the variance‐filtered instrumental transfer method as applied to high‐resolution ¹H NMR spectra is presented along with possible uses and limitations. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of Multivariate Calibration Using a Tikhonov Regularization Approach and the Generalized Pair‐Correlation Method with Nonlinear Data

Abstract

In order to reduce data nonlinearity and overfitting with the multivariate calibration model y=Xb, a modified Tikhonov regularization (TR) algorithm is evaluated for selecting key variables from an X augmented with extra columns that contain the original measured variables (x _ij ) as squared terms (x _ij ²) and other orders. The TR approach simultaneously develops the multivariate calibration model. The new generalized pair‐correlation method (GPCM) is also studied for variable selection followed by partial least squares (PLS) for multivariate calibration. Results from synthetic spectral data are compared when using the modified TR approach, GPCM, and PLS without variable selection. The GPCM usually performs slightly better than the TR approach for tabulated bias and variance measures and in some cases, at a sacrifice to parsimony. The method of PLS without variable selection performs the worst. By using synthetic spectral data sets, how the methods work could be studied. Thus, results from this study will aid investigators of real spectral data sets exhibiting nonlinear behavior.     

Regression models with process variables and parallel blocks of raw material measurements

The topic of this paper is regression models based on designed experiments, where additional spectroscopic measurements are also available. This particular case describes a situation with two spectral blocks with no natural order: The blocks are parallel. Three methods are described, which combine least squares regression of the design variables with PCA or PLS on the spectra. The methods properties are explored in two simulation studies based on real experiments. The results show that the methods are equal when it comes to prediction, but interpretability varies. One of the methods, LS‐ParPLS, is especially interesting when it comes to interpretability because it splits the spectral information into two parts; information that is common in both blocks and information that is unique for each block. Copyright © 2008 John Wiley & Sons, Ltd.     

Reliability and uncertainty in the estimation of p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> by least squares nonlinear regression analysis of multiwavelength spectrophotometric pH titration data

When drugs are poorly soluble then, instead of the potentiometric determination of dissociation constants, pH-spectrophotometric titration can be used along with nonlinear regression of the absorbance response surface data. Generally, regression models are extremely useful for extracting the essential features from a multiwavelength set of data. Regression diagnostics represent procedures for examining the regression triplet (data, model, method) in order to check (a) the data quality for a proposed model; (b) the model quality for a given set of data; and (c) that all of the assumptions used for least squares hold. In the interactive, PC-assisted diagnosis of data, models and estimation methods, the examination of data quality involves the detection of influential points, outliers and high leverages, that cause many problems when regression fitting the absorbance response hyperplane. All graphically oriented techniques are suitable for the rapid estimation of influential points. The reliability of the dissociation constants for the acid drug silybin may be proven with goodness-of-fit tests of the multiwavelength spectrophotometric pH-titration data. The uncertainty in the measurement of the pK _a of a weak acid obtained by the least squares nonlinear regression analysis of absorption spectra is calculated. The procedure takes into account the drift in pH measurement, the drift in spectral measurement, and all of the drifts in analytical operations, as well as the relative importance of each source of uncertainty. The most important source of uncertainty in the experimental set-up for the example is the uncertainty in the pH measurement. The influences of various sources of uncertainty on the accuracy and precision are discussed using the example of the mixed dissociation constants of silybin, obtained using the SQUAD(84) and SPECFIT/32 regression programs.     

Rapid Determination of Metabolites in Bio‐fluid Samples by Raman Spectroscopy and Optimum Combinations of Chemometric Methods

The application of Raman spectroscopic techniques combined with multivariate chemometrics signal processing promise new means for the rapid multidimensional analysis of metabolites non‐destructively, with little or no sample preparation and little sensitivity to water. However, Rayleigh scattering, fluorescence and uncontrolled variance present substantial challenges for the accurate quantitative analysis of metabolites at physiological levels in biologically varying samples. Effective strategies include the application of chemometrics pretreatments for reducing Raman spectral interference. However, the arbitrary application of individual or combined pretreatment procedures can significantly alter the outcome of a measurement, thereby complicating spectral analysis. This paper evaluates and compares six signal pretreatment methods for correcting the baseline variances, together with three variable selection methods for eliminating uninformative variables, all within the context of multivariate calibration models based on partial least squares (PLS) regression. Raman spectra of 90 artificial bio‐fluid samples with eight urine metabolites at near‐physiological concentrations were used to test these models. The combination of multiplicative scatter correction (MSC), continuous wavelet transform (CWT), randomization test (RT) and PLS modeling presented the best performance for all the metabolites. The correlation coefficient (R) between predicted and prepared concentration reached as high as 0.96.     

Determination of galactosamine impurities in heparin samples by multivariate regression analysis of their <Superscript>1</Superscript>H NMR spectra

Heparin, a widely used anticoagulant primarily extracted from animal sources, contains varying amounts of galactosamine impurities. Currently, the United States Pharmacopeia (USP) monograph for heparin purity specifies that the weight percent of galactosamine (%Gal) may not exceed 1%. In the present study, multivariate regression (MVR) analysis of ¹H NMR spectral data obtained from heparin samples was employed to build quantitative models for the prediction of %Gal. MVR analysis was conducted using four separate methods: multiple linear regression, ridge regression, partial least squares regression, and support vector regression (SVR). Genetic algorithms and stepwise selection methods were applied for variable selection. In each case, two separate prediction models were constructed: a global model based on dataset A which contained the full range (0–10%) of galactosamine in the samples and a local model based on the subset dataset B for which the galactosamine level (0–2%) spanned the 1% USP limit. All four regression methods performed equally well for dataset A with low prediction errors under optimal conditions, whereas SVR was clearly superior among the four methods for dataset B. The results from this study show that ¹H NMR spectroscopy, already a USP requirement for the screening of contaminants in heparin, may offer utility as a rapid method for quantitative determination of %Gal in heparin samples when used in conjunction with MVR approaches.     

Scope of partial least‐squares regression applied to the enantiomeric composition determination of ketoprofen from strongly overlapped chromatographic profiles

Valuable quantitative information could be obtained from strongly overlapped chromatographic profiles of two enantiomers by using proper chemometric methods. Complete separation profiles where the peaks are fully resolved are difficult to achieve in chiral separation methods, and this becomes a particularly severe problem in case that the analyst needs to measure the chiral purity, i.e., when one of the enantiomers is present in the sample in very low concentrations. In this report, we explore the scope of a multivariate chemometric technique based on unfolded partial least‐squares regression, as a mathematical tool to solve this quite frequent difficulty. This technique was applied to obtain quantitative results from partially overlapped chromatographic profiles of R‐ and S‐ketoprofen, with different values of enantioresolution factors (from 0.81 down to less than 0.2 resolution units), and also at several different S:R enantiomeric ratios. Enantiomeric purity below 1% was determined with excellent precision even from almost completely overlapped signals. All these assays were tested on the most demanding condition, i.e., when the minor peak elutes immediately after the main peak. The results were validated using univariate calibration of completely resolved profiles and the method applied to the determination of enantiomeric purity of commercial pharmaceuticals.     

The Strongest Brønsted Acid: Protonation of Alkanes by H(CHB11F11) at Room Temperature

What is the strongest acid? Can a simple Brønsted acid be prepared that can protonate an alkane at room temperature? Can that acid be free of the complicating effects of added Lewis acids that are typical of common, difficult‐to‐handle superacid mixtures? The carborane superacid H(CHB₁₁F₁₁) is that acid. It is an extremely moisture‐sensitive solid, prepared by treatment of anhydrous HCl with [Et₃Si? H? SiEt₃][CHB₁₁F₁₁]. It adds H₂O to form [H₃O][CHB₁₁F₁₁] and benzene to form the benzenium ion salt [C₆H₇][CHB₁₁F₁₁]. It reacts with butane to form a crystalline tBu⁺ salt and with n‐hexane to form an isolable hexyl carbocation salt. Carbocations, which are thus no longer transient intermediates, react with NaH either by hydride addition to re‐form an alkane or by deprotonation to form an alkene and H₂. By protonating alkanes at room temperature, the reactivity of H(CHB₁₁F₁₁) opens up new opportunities for the easier study of acid‐catalyzed hydrocarbon reforming.     

Bayesian linear regression and variable selection for spectroscopic calibration

This paper presents a Bayesian approach to the development of spectroscopic calibration models. By formulating the linear regression in a probabilistic framework, a Bayesian linear regression model is derived, and a specific optimization method, i.e. Bayesian evidence approximation, is utilized to estimate the model “hyper-parameters”. The relation of the proposed approach to the calibration models in the literature is discussed, including ridge regression and Gaussian process model. The Bayesian model may be modified for the calibration of multivariate response variables. Furthermore, a variable selection strategy is implemented within the Bayesian framework, the motivation being that the predictive performance may be improved by selecting a subset of the most informative spectral variables. The Bayesian calibration models are applied to two spectroscopic data sets, and they demonstrate improved prediction results in comparison with the benchmark method of partial least squares.     

Using the L1 norm to select basis set vectors for multivariate calibration and calibration updating

With projection based calibration approaches, such as partial least squares (PLS) and principal component regression (PCR), the calibration space is spanned by respective basis vectors (latent vectors). Up to rank k basis vectors are formed where k ≤ min(m,n) with m and n denoting the number of calibration samples and measured variables. The user needs to decide how many and which respective basis vectors (tuning parameters). To avoid the second issue, basis vectors are selected top‐down starting with the first and sequentially adding until model criteria are satisfied. Ridge regression (RR) avoids the issues by using the full set of basis vectors. Another approach is to select a subset from the total available. The presented work develops a process based on the L₁ vector norm to select basis vectors. Specifically, the L₁ norm is used to select singular value decomposition (SVD) basis set vectors for PCR (LPCR). Because PCR, PLS, RR, and others can be expressed as linear combination of the SVD basis vectors, the focus is on selection and comparison using the SVD basis set. Results based on respective tuning parameter selections and weights applied to the SVD basis vectors for LPCR, top‐down PCR, correlation PCR (CPCR), PLS, and RR are compared for calibration and calibration updating using spectroscopic data sets. The methods are found to predict equivalently. In particular, the L₁ norm produces similar results to those obtained by the well‐studied CPCR process. Thus, the new method provides a different theoretical framework than CPCR for selecting basis vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

Simultaneous quantification of both triterpenoid and steroidal saponins in various Yunnan Baiyao preparations using HPLC–UV and HPLC–MS

Yunnan Baiyao (YNBY) is one of the best known traditional Chinese medicines. Saponins are considered to be its active components. In this study, an HPLC method was first developed for the simultaneous quantitative analysis of thirteen saponins, including five triterpenoid saponins and eight steroidal saponins, in a series of YNBY preparations, i. e., powder, capsules, aerosol, toothpaste, plaster, and adhesive bandage. The pre‐treatment methods for each dosage form were investigated and optimized. The HPLC separation was performed on a Shim‐pack C₁₈ reversed‐phase column in gradient mode with UV detection at 203 nm. All calibration curves showed good linear regression (r² ? 0.9981) within the test ranges. Precisions and repeatabilities of the methods were better than 4.22 and 4.78%, respectively. Recoveries were better than 90.5%, even in the analysis of the least abundant saponins in a complex YNBY plaster. HPLC–ESI‐TOF/MS was used for definite identification of compounds in the preparations. This proposed method was successfully applied to quantify the 13 bioactive constituents in 27 commercial samples to evaluate the quality of YNBY preparations. The overall results demonstrate that this method is simple, reliable, and suitable for the quality control of YNBY. Furthermore, the retention behavior of these saponins in reversed‐phase chromatography is described.