Linear Multivariate Calibration Research Articles

A novel multiple linear multivariate NIR calibration model‐based strategy for in‐line monitoring of continuous mixing

The capability of near infra‐red (NIR) spectroscopy to predict many different variables, such as concentration and humidity, has been demonstrated in many published works. Several of those articles have been in the subject of real time prediction of continuous operations. However, those demonstrations have been for narrow ranges of the variables, especially for powder concentration, which could present a nonlinear behavior of the NIR absorbance as a function of the entire range of concentration. This work developed a novel strategy to predict the entire range of powder concentration using multiple linear NIR calibration models. The root mean standard error of prediction and relative standard deviation (RSD) parameters were used to establish the number of the multiple linear calibration models; other statistical features were used to establish the correct prediction. It was found that a minimum number of linear partial least squares (PLS) calibration models were necessary to accurately predict the range from 0 to 100% w/w. This technique could also be used with other nonlinear behaviors. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3123–3132, 2014

A novel algorithm for linear multivariate calibration based on the mixed model of samples

We present a novel algorithm for linear multivariate calibration that can generate good prediction results. This is accomplished by the idea of that testing samples are mixed by the calibration samples in proper proportion. The algorithm is based on the mixed model of samples and is therefore called MMS algorithm. With both theoretical support and analysis of two data sets, it is demonstrated that MMS algorithm produces lower prediction errors than partial least squares (PLS2) model, has similar prediction performance to PLS1. In the anti-interference test of background, MMS algorithm performs better than PLS2. At the condition of the lack of some component information, MMS algorithm shows better robustness than PLS2.

Application of random forest regression to spectral multivariate calibration

The performance of the random forest (RF) algorithm on the spectroscopic data was studied and compared by bootstrap aggregating of classification and regression trees (bagging CART), partial least squares (PLS) and nonlinear support vector machine (SVM) algorithms. The performances of these algorithms were investigated on four real data sets; these data sets were: (1) UV-Visible spectra of two cardiovascular drugs (hydrochlorothiazide and valsartan); (2) visible spectra of copper, cobalt and nickel complexes with 4-(2-pyridylazo) resorcinol (PAR) as chromogenic reagent; (3) near infrared spectra of corn samples, and (4) near infrared diffuse transmission spectra of pharmaceutical tablets. Results indicate that besides its comparable accuracy and mathematical simplicity, it is computationally fast and robust to noise. Therefore, RF is a useful tool for regression studies and has potential for modeling linear and nonlinear multivariate calibration.

Potentially antioxidant compounds indicated from Mallotus and Phyllanthus species fingerprints

The genera of Mallotus and Phyllanthus contain several species that are commonly used as traditional medicines in oriental countries. Some species show interesting pharmaceutical activities, such as an antioxidant activity. To produce clinically useful medicines or food supplements (nutraceuticals) from these herbs, the species should be identified and a thorough quality control should be implemented. Nowadays, the integration of chromatographic and chemometric approaches allows a high-throughput identification and activity prediction of medicinal plants. In this study, Principal Component Analysis (PCA) and Hierarchical Cluster Analysis (HCA) were applied and compared to distinguish Mallotus and Phyllanthus species. Moreover, peaks from their chromatographic fingerprints, which were responsible for their antioxidant activity were assigned. For the latter purpose, the relevant information was extracted from the chromatographic fingerprints using linear multivariate calibration techniques, i.e., Partial Least Squares (PLS) and Orthogonal Projections to Latent Structures (O-PLS). Results reveal that exploratory analysis using PCA shows somewhat diverging clustering tendencies between Mallotus and Phyllanthus samples than HCA. However, both approaches mainly confirm each other. Concerning the multivariate calibration techniques, both PLS and O-PLS models demonstrate good predictive abilities. By comparing the regression coefficients of the models with the chromatographic fingerprints, the peaks that are potentially responsible for the antioxidant activity of the extracts could be confirmed.

Quantitative Analysis of Powder Mixtures by Raman Spectrometry: the influence of particle size and its correction

Particle size distribution and compactness have significant confounding effects on Raman signals of powder mixtures, which cannot be effectively modeled or corrected by traditional multivariate linear calibration methods such as partial least-squares (PLS), and therefore greatly deteriorate the predictive abilities of Raman calibration models for powder mixtures. The ability to obtain directly quantitative information from Raman signals of powder mixtures with varying particle size distribution and compactness is, therefore, of considerable interest. In this study, an advanced quantitative Raman calibration model was developed to explicitly account for the confounding effects of particle size distribution and compactness on Raman signals of powder mixtures. Under the theoretical guidance of the proposed Raman calibration model, an advanced dual calibration strategy was adopted to separate the Raman contributions caused by the changes in mass fractions of the constituents in powder mixtures from those induced by the variations in the physical properties of samples, and hence achieve accurate quantitative determination for powder mixture samples. The proposed Raman calibration model was applied to the quantitative analysis of backscatter Raman measurements of a proof-of-concept model system of powder mixtures consisting of barium nitrate and potassium chromate. The average relative prediction error of prediction obtained by the proposed Raman calibration model was less than one-third of the corresponding value of the best performing PLS model for mass fractions of barium nitrate in powder mixtures with variations in particle size distribution, as well as compactness.

Polynomial multivariate least-squares regression for modeling nonlinear data applied to in-depth characterization of chromatographic resolution

Well-established, linear multivariate calibration methods such as multivariate least-squares regression (MLR), principal component regression (PCR), or partial least squares (PLS) have two limitations: (i) measured data must be linearly related to the response variables and (ii) predictor variables xn = 1, …, N cannot be coupled to each other. For evaluation of nonlinear data, however, these restrictions need to be overcome and thus polynomial multivariate least-squares regression (PMLR or “response surfaces”) has been introduced here. PMLR is based on multivariate least squares but incorporates all combinations of predictor variables up to a user-selected polynomial order (e.g., including u or v = 0). Because of the inclusion of such coupled terms and their powers, PMLR models are better adapted to model nonlinear data and can help to enhance the prediction step's accuracy and precision. PMLR has been based on MLR because it facilitates—unlike PCR or PLS—a physical and chemical interpretation of the predictors. Hence, the origins and the relevance of nonlinear and/or coupled predictors can be investigated. The details of the PMLR algorithm and its implementation are presented along with a method for model optimization utilizing gradients of response surfaces. Newly developed PMLR models up to quintic order have been applied to predict a chromatograph's peak resolution as a function of six-instrument parameters. It has been demonstrated that PMLR is better capable than MLR and PCR to describe these nonlinear and coupled instrument parameters. In addition, the novel software tool has been utilized for model optimization to determine instrument parameters, which result in the best chromatographic resolution. Copyright © 2011 John Wiley & Sons, Ltd.

Comparison of various chemometric approaches for large near infrared spectroscopic data of feed and feed products

In the present study, different multivariate regression techniques have been applied to two large near-infrared data sets of feed and feed ingredients in order to fulfil the regulations and laws that exist about the chemical composition of these products. The aim of this paper was to compare the performances of different linear and nonlinear multivariate calibration techniques: PLS, ANN and LS-SVM. The results obtained show that ANN and LS-SVM are very powerful methods for non-linearity but LS-SVM can also perform quite well in the case of linear models. Using LS-SVM an improvement of the RMS for independent test sets of 10% is obtained in average compared to ANN and of 24% compared to PLS.

Calibration of Multiplexed Fiber-Optic Spectroscopy

Large-scale commercial bioprocesses that manufacture biopharmaceutical products such as monoclonal antibodies generally involve multiple bioreactors operated in parallel. Spectra recorded during in situ monitoring of multiple bioreactors by multiplexed fiber-optic spectroscopies contain not only spectral information of the chemical constituents but also contributions resulting from differences in the optical properties of the probes. Spectra with variations induced by probe differences cannot be efficiently modeled by the commonly used multivariate linear calibration models or effectively removed by popular empirical preprocessing methods. In this study, for the first time, a calibration model is proposed for the analysis of complex spectral data sets arising from multiplexed probes. In the proposed calibration model, the spectral variations introduced by probe differences are explicitly modeled by introducing a multiplicative parameter for each optical probe, and then their detrimental effects are effectively mitigated through a "dual calibration" strategy. The performance of the proposed multiplex calibration model has been tested on two multiplexed spectral data sets (i.e., MIR data of ternary mixtures and NIR data of bioprocesses). Experimental results suggest that the proposed calibration model can effectively mitigate the detrimental effects of probe differences and hence provide much more accurate predictions than commonly used multivariate linear calibration models (such as PLS) with and without empirical data preprocessing methods such as orthogonal signal correction, standard normal variate, or multiplicative signal correction.

Wavelength selection-based nonlinear calibration for transcutaneous blood glucose sensing using Raman spectroscopy.

While Raman spectroscopy provides a powerful tool for noninvasive and real time diagnostics of biological samples, its translation to the clinical setting has been impeded by the lack of robustness of spectroscopic calibration models and the size and cumbersome nature of conventional laboratory Raman systems. Linear multivariate calibration models employing full spectrum analysis are often misled by spurious correlations, such as system drift and covariations among constituents. In addition, such calibration schemes are prone to overfitting, especially in the presence of external interferences that may create nonlinearities in the spectra-concentration relationship. To address both of these issues we incorporate residue error plot-based wavelength selection and nonlinear support vector regression (SVR). Wavelength selection is used to eliminate uninformative regions of the spectrum, while SVR is used to model the curved effects such as those created by tissue turbidity and temperature fluctuations. Using glucose detection in tissue phantoms as a representative example, we show that even a substantial reduction in the number of wavelengths analyzed using SVR lead to calibration models of equivalent prediction accuracy as linear full spectrum analysis. Further, with clinical datasets obtained from human subject studies, we also demonstrate the prospective applicability of the selected wavelength subsets without sacrificing prediction accuracy, which has extensive implications for calibration maintenance and transfer. Additionally, such wavelength selection could substantially reduce the collection time of serial Raman acquisition systems. Given the reduced footprint of serial Raman systems in relation to conventional dispersive Raman spectrometers, we anticipate that the incorporation of wavelength selection in such hardware designs will enhance the possibility of miniaturized clinical systems for disease diagnosis in the near future.

Simultaneous kinetic determination of 3-hydroxybutyrate and 3-hydroxyvalerate in biopolymer degradation processes

A new kinetic method is proposed for the simultaneous determination of 3-hydroxybutyrate (3HB) and 3-hydroxyvalerate (3HV) based on the different rate of the 3-hydroxybutyrate dehydrogenase-catalysed reactions of these compounds with coenzyme NAD +. A flow injection system with two reactors of immobilised 3-hydroxybutyrate dehydrogenase and dual detection is used. The concentrations of NADH produced after two different reaction times are measured by fluorometry or spectrophotometry and multivariate linear calibration is applied for quantification. Concentrations of 3HB and 3HV between 1 × 10 −6 and 1 × 10 −4 M can be determined at an average sampling frequency of 20 h −1. In contrast to usual methods, the proposed here makes possible the discrimination of 3HB and 3HV without previous separation so that usual extraction with chlorinated solvents and/or chromatographic separation is not required. The method is of interest in a wide variety of fields concerning PHAs, as it can provide information on the degradation rate and mechanism, composition and structure of these polymers. Its applicability has been proved through the determination of 3HB and 3HV in the digests of some chemically degraded commercial PHAs.

Potential antioxidant compounds in Mallotus species fingerprints. Part I: Indication, using linear multivariate calibration techniques

Some Mallotus species are used in traditional medicine in Vietnam and China. Some also show interesting activities, such as antioxidant and cytotoxic ones. Combining fingerprint technology with data-handling techniques allows indicating the peaks potentially responsible for given activities. In this study it is aspired to indicate from chromatographic fingerprints the peaks potentially responsible for the antioxidant activity of several Mallotus species. Relevant information was extracted using linear multivariate calibration techniques, both before and after alignment of the fingerprints with correlation optimized warping (COW). From the studied techniques, stepwise multiple linear regression is least recommended as it made an inadequate variable selection. Principal component regression theoretically can take largely varying variables uncorrelated to the antioxidant activity into account. However, in practice in the actual case study this problem was limited. These problems in principle do not occur using partial least squares (PLS) models. Of the tested PLS methods, orthogonal projections to latent structures was preferred because of its simplicity, reproducibility, reduced model complexity and improved interpretability of the regression coefficients, yielding a clearer view on the individual contribution of the compounds. Furthermore, reducing analysis times from 60 min to 35 and 22.5 min resulted in the same main compounds, indicated responsible for the antioxidant activity. Models built after alignment by COW did not result in additional information.

Potential antioxidant compounds in Mallotus species fingerprints. Part I: Indication, using linear multivariate calibration techniques

Some Mallotus species are used in traditional medicine in Vietnam and China. Some also show interesting activities, such as antioxidant and cytotoxic ones. Combining fingerprint technology with data-handling techniques allows indicating the peaks potentially responsible for given activities. In this study it is aspired to indicate from chromatographic fingerprints the peaks potentially responsible for the antioxidant activity of several Mallotus species. Relevant information was extracted using linear multivariate calibration techniques, both before and after alignment of the fingerprints with correlation optimized warping (COW). From the studied techniques, Stepwise Multiple Linear Regression is least recommended as it made an inadequate variable selection. Principal Component Regression theoretically can take largely varying variables uncorrelated to the antioxidant activity into account. However, in practice in the actual case study this problem was limited. These problems in principle do not occur using Partial Least Squares (PLS) models. Of the tested PLS methods, Orthogonal Projections to Latent Structures was preferred because of its simplicity, reproducibility, reduced model complexity and improved interpretability of the regression coefficients, yielding a clearer view on the individual contribution of the compounds. Furthermore, reducing analysis times from 60 min to 35 and 22.5 min resulted in the same main compounds, indicated responsible for the antioxidant activity. Models built after alignment by COW did not result in additional information.

Support vector regression for functional data in multivariate calibration problems

Quantitative analyses involving instrumental signals, such as chromatograms, NIR, and MIR spectra have been successfully applied nowadays for the solution of important chemical tasks. Multivariate calibration is very useful for such purposes and the commonly used methods in chemometrics consider each sample spectrum as a sequence of discrete data points. An alternative way to analyze spectral data is to consider each sample as a function, in which a functional data is obtained. Concerning regression, some linear and nonparametric regression methods have been generalized to functional data. This paper proposes the use of the recently introduced method, support vector regression for functional data (FDA-SVR) for the solution of linear and nonlinear multivariate calibration problems. Three different spectral datasets were analyzed and a comparative study was carried out to test its performance with respect to some traditional calibration methods used in chemometrics such as PLS, SVR and LS-SVR. The satisfactory results obtained with FDA-SVR suggest that it can be an effective and promising tool for multivariate calibration tasks.

Relevance vector machines for multivariate calibration purposes

AbstractThe introduction of support vector regression (SVR) and least square support vector machines (LS‐SVM) methods for regression purposes in the field of chemometrics has provided advantageous alternatives to the existing linear and nonlinear multivariate calibration (MVC) approaches. Relevance vector machines (RVMs) claim the advantages attributed to all the SVM‐based methods over many other regression methods. Additionally, it also exhibits advantages over the standard SVM‐based ones since: it is not necessary to estimate the error/margin trade‐off parameter C and the insensitivity parameter ε in regression tasks, it is applicable to arbitrary basis functions, the algorithm gives probability estimates seamlessly and offer, additionally, excellent sparseness capabilities, which can result in a simple and robust model for the estimation of different properties. This paper presents the use of RVMs as a nonlinear MVC method capable of dealing with ill‐posed problems. To study its behavior, three different chemometric benchmark datasets are considered, including both linear and non‐linear solutions. RVM was compared with other calibration approaches reported in the literature. Although RVM performance is comparable with the best results obtained by LS‐SVM, the final model achieved is sparser, so the prediction process is faster. Taking into account the other advantages attributed to RVMs, it can be concluded that this technique can be seen as a very promising option to solve nonlinear problems in MVC. Copyright © 2008 John Wiley & Sons, Ltd.

Exploration of linear multivariate calibration techniques to predict the total antioxidant capacity of green tea from chromatographic fingerprints

Nowadays fingerprinting is a generally applied technique for the identification and quality assessment of herbal products. In this study it was aimed to predict a quantitative property, the antioxidant capacity of green tea, from chromatographic fingerprints. Different linear multivariate calibration techniques, commonly applied on spectral data, were explored and compared. When the chromatograms were appropriately pretreated, all tested techniques were able to predict the total antioxidant capacity with a precision comparable to that of the reference method (Trolox equivalent antioxidant capacity assay). Stepwise multiple linear regression (MLR) however is less recommended because of inadequate variable selection. Principal components regression (PCR) also seems less preferable, because large variations not correlated with the total antioxidant capacity were also included in the model. This problem does not occur with partial least squares (PLS) models. Of all tested PLS methods, orthogonal projections to latent structures (O-PLS) was preferred because of its simplicity, reproducibility, good interpretability of the compounds’ contribution to the antioxidant capacity and its good predictive and describing abilities.

Multi-elemental analysis of solidified mineral melt samples by Laser-Induced Breakdown Spectroscopy coupled with a linear multivariate calibration

Laser-Induced Breakdown Spectroscopy (LIBS) has been successfully applied for multi-elemental analysis of solidified mineral melt samples containing several oxides present in various concentrations. The plasma was generated using a Nd:YAG laser and the spectra were acquired using an Echelle spectrometer, coupled to an ICCD detector, which covers a spectral range from 200 to 780 nm. Using a set of 19 calibration samples, we first established univariate calibration curves for the major elements (Al, Fe, Mg, Ca, Ti and Si). We found out that the presence of matrix effects makes such a model, traditionally used in LIBS, not satisfying for quantitative analysis of such samples. Indeed, no sufficiently linear trends can be extracted from the calibration curves for the elements of interest considering all the samples. Instead, a much more robust calibration approach was obtained by considering a multivariate model. The matrix effects are then taken into account by correcting the spectroscopic signals emitted by a given species due the presence of the others ones. More specifically, we established here a calibration model using a 2nd order polynomial linear multivariate inverse regression. The capability of this approach was then checked using a 2nd set of samples with an unknown composition. A good agreement was observed between the analysis provided by X-ray fluorescence (XRF) and the LIBS measurements coupled to the multivariate model for the unknown samples.

The application of direct orthogonal signal correction for linear and non-linear multivariate calibration

Traditionally, the direct orthogonal signal correction (DOSC) is always used together with a latent variable method such as partial least square (PLS) or principal component regression (PCR), to build a linear calibration model. In this study, PLS and least square support vector machine (LSSVM) were used to develop the linear and non-linear relation between spectra and components, respectively. DOSC was used to preprocess the input data, and the effect of DOSC pretreatment on linear and non-linear calibration model was investigated. The experiment was performed with three data sets. The first one was the acousto-optic tunable filter near infrared (AOTF-NIR) spectra of apples, the second one was the temperature-induced spectra of a ternary mixture of ethanol, water and 2-propanol, and the third one was the NIR spectra of corn. For all of the applications, the relation between spectra and components can be clearly observed in the spectra plot or the score plot after DOSC pretreatment. DOSC improved the predictive ability of PLS model. However, DOSC removed useful non-linear information that was related to components, thus, was not able to improve the performance of LSSVM model. DOSC pretreatment seems to be not suitable for non-linear calibration.

Multicomponent Determination of Organophosphorus Pesticides in Grain Samples by Linear Sweep Stripping Voltammetry and Multivariate Calibration

Organophosphorus pesticides, such as parathion methyl (PTM), fenitrothion (FT), parathion (PT), and isocarbophos (ICP), have sensitive but overlapped voltammetric peaks with peak potentials −309, −364, −317, and −480 mV, respectively, in Britton‐Robinson buffer of pH 4.8 by application of linear sweep stripping voltammetry (LSSV). In this work, two multivariate calibration methods, partial least squares (both PLS‐1 and PLS‐2), and principal component regression (PCR), were applied to quantitatively resolve the overlapping voltammogram of the mixtures of these four pesticides. The prediction results obtained from a set of independent test samples showed that PLS‐1 method performed better prediction ability than PLS‐2 and PCR methods. The proposed method was successfully applied to the determination of these four pesticides in grain samples after a pre‐extraction step with a solvent of acetone.

Application of SIC (simple interval calculation) for object status classification and outlier detection—comparison with regression approach

AbstractWe introduce a novel approach termed simple interval calculation (SIC) for classification of object status in linear multivariate calibration (MVC) and other data analytical contexts. SIC is a method that directly constructs an interval estimator for the predicted response. SIC is based on the single assumption that all errors involved in MVC are limited. We present the theory of the SIC method and explain its realization by linear programming techniques. The primary SIC consequence is a radically new object classification that can be interpreted using a two‐dimensional object status plot (OSP), ‘SIC residual vs SIC leverage’. These two new measures of prediction quality are introduced in the traditional chemometric MVC context. Simple straight demarcations divide the OSP into areas which quantitatively discriminate all objects involved in modeling and prediction into four different types: boundary samples, which are the significant objects (for generating the entire data structure) within the training subset; insiders, which are samples that comply with the model; outsiders, which are samples that have large prediction errors; and finally outliers, which are those samples that cannot be predicted at all with respect to a given model. We also present detailed comparisons of the new SIC approach with traditional chemometric methods applied for MVC, classification and outlier detection. These comparisons employ four real‐world data sets, selected for their particular complexities, which serve as showcases of SIC application on intricate training and test set data structures. Copyright © 2005 John Wiley & Sons, Ltd.

On estimation in multivariate linear calibration with elliptical errors

The estimation problem in multivariate linear calibration with elliptical errors is considered under a loss function which can be derived from the Kullback-Leibler distance. First, we discuss the problem under normal errors and give unbiased estimate of risk of an alternative estimator by means of the Stein and Stein-Haff identities for multivariate normal distribution. From the unbiased estimate of risk, it is shown that a shrinkage estimator improves on the classical estimator under the loss function. Furthermore, from the extended Stein and Stein-Haff identities for our elliptically contoured distribution, the above result under normal errors is extended to the estimation problem under elliptical errors. We show that the shrinkage estimator obtained under normal models is better than the classical estimator under elliptical errors with the above loss function and hence we establish the robustness of the above shrinkage estimator.