Calibration by orthogonal and common least squares — Theoretical and practical aspects

Consequences resulting from a three-dimensional calibration model introduced in [5] are investigated. Accordingly, there exists a different statistical background for the calibration, the analytical evaluation and the validation step. If the errors of the concentration values are not negligible compared with the errors of the measured values, orthogonal calibration models have to be used instead of the common Gaussian least squares (GLS). Four different approximation models of orthogonal least squares, Wald's approximation (WA), Mandel's approximation (MA), Geometrical mean (GM), and Principal component estimation (PC) are investigated and compared with each other and with GLS by simulations and by real analytical applications. From the simulations it can be seen that GLS is affected by bias in the estimates of both slope and intercept in the case of increasing concentration error. On the other hand, the orthogonal models estimate the calibration parameter better. The best fit is obtained by Wald's approximation. It is shown by simulations and real analytical calibration problems that orthogonal calibration has to be used in all cases in which the concentration errors cannot be neglected compared to the errors of the measured values. This is in particular relevant in recovery experiments for validation by means of comparison of methods. In such cases orthogonal least squares methods have always to be applied where the use of WA is recommended. The situation is different in the case of ordinary calibration experiments. The examples considered show small existing differences between the classical GLS and the orthogonal procedures. In doubtful cases both GLS and WA should be computed where the latter should be used if significant differences appear.     

Method comparison on estimation for an unknown specimen in a heteroscedastic controlled calibration model

In analytical chemistry applications, statistical calibration models are commonly used to estimate the true value of an unknown specimen. In this article, we consider a heteroscedastic controlled calibration model in which both dependent and independent variables are subject to heteroscedastic measurement errors. The main task of using this model is to estimate the true value of an unknown regressor (independent variable) under the condition that a set of observations on its corresponding response (dependent variable) is available. We introduce four estimation methods to the problem of interest, including generalized least squares (GLS), modified least squares, corrected score, and expectation maximization‐based (EM‐based) methods. Furthermore, an interval estimation based on an asymptotic method is also derived. We compare their performance through detailed simulation studies. In consequence, GLS and EM‐based methods are recommended in practical use. A real data example is given to illustrate the application of the calibration model. Copyright © 2013 John Wiley & Sons, Ltd.     

Spectrophotometric determination of metal ions in electroplating solutions in the presence of EDTA with the aid of multivariate calibration and artificial neural networks

Metal ions such as Co(II), Ni(II), Cu(II), Fe(III) and Cr(III), which are commonly present in electroplating baths at high concentrations, were analysed simultaneously by a spectrophotometric method modified by the inclusion of the ethylenediaminetetraacetate (EDTA) solution as a chromogenic reagent. The prediction of the metal ion concentrations was facilitated by the use of an orthogonal array design to build a calibration data set consisting of absorption spectra collected in the 370-760 nm range from solution mixtures containing the five metal ions earlier. With the aid of this data set, calibration models were built based on 10 different chemometrics methods such as classical least squares (CLS), principal component regression (PCR), partial least squares (PLS), artificial neural networks (ANN) and others. These were tested with the use of a validation data set constructed from synthetic solutions of the five metal ions. The analytical performance of these chemometrics methods were characterized by relative prediction errors and recoveries (%). On the basis of these results, the computational methods were ranked according to their performances using the multi-criteria decision making procedures preference ranking organization method for enrichment evaluation (PROMETHEE) and geometrical analysis for interactive aid (GAIA). PLS and PCR models applied to the spectral data matrix that used the first derivative pre-treatment were the preferred methods. They together with ANN-radial basis function (RBF) and PLS were applied for analysis of results from some typical industrial samples analysed by the EDTA-spectrophotometric method described. DPLS, DPCR and the ANN-RBF chemometrics methods performed particularly well especially when compared with some target values provided by industry.     

Multivariate near infrared spectroscopy models for predicting methanol and water content in biodiesel

The transesterification of vegetable oils, animal fats or waste oils with an alcohol (such as methanol) in the presence of a homogeneous catalyst (sodium hydroxide or methoxyde) is commonly used to produce biodiesel. The quality control of the final product is an important issue and near infrared (NIR) spectroscopy recently appears as an appealing alternative to the conventional analytical methods. The use of NIR spectroscopy for this purpose first involves the development of calibration models to relate the near infrared spectrum of biodiesel with the analytical data. The type of pre-processing technique applied to the data prior to the development of calibration may greatly influence the performance of the model. This work analyses the effect of some commonly used pre-processing techniques applied prior to partial least squares (PLS) and principal components regressions (PCR) in the quality of the calibration models developed to relate the near infrared spectrum of biodiesel and its content of methanol and water. The results confirm the importance of testing various pre-processing techniques. For the water content, the smaller validation and prediction errors were obtained by a combination of a second order Savitsky-Golay derivative followed by mean centring prior to PLS and PCR, whereas for methanol calibration the best results were obtained with a first order Savitsky-Golay derivative plus mean centring followed by the orthogonal signal correction.     

Selecting the optimum number of partial least squares components for the calibration of attenuated total reflectance-mid-infrared spectra of undesigned kerosene samples

Selecting the correct dimensionality is critical for obtaining partial least squares (PLS) regression models with good predictive ability. Although calibration and validation sets are best established using experimental designs, industrial laboratories cannot afford such an approach. Typically, samples are collected in an (formally) undesigned way, spread over time and their measurements are included in routine measurement processes. This makes it hard to evaluate PLS model dimensionality. In this paper, classical criteria (leave-one-out cross-validation and adjusted Wold's criterion) are compared to recently proposed alternatives (smoothed PLS-PoLiSh and a randomization test) to seek out the optimum dimensionality of PLS models. Kerosene (jet fuel) samples were measured by attenuated total reflectance-mid-IR spectrometry and their spectra where used to predict eight important properties determined using reference methods that are time-consuming and prone to analytical errors. The alternative methods were shown to give reliable dimensionality predictions when compared to external validation. By contrast, the simpler methods seemed to be largely affected by the largest changes in the modeling capabilities of the first components.     

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.     

Infrared Spectroscopy as Alternative to Wet Chemical Analysis to Characterize Eucalyptus globulus Pulps and Predict Their Ethanol Yield for a Simultaneous Saccharification and Fermentation Process

Bioethanol can be obtained from wood by simultaneous enzymatic saccharification and fermentation step (SSF). However, for enzymatic process to be effective, a pretreatment is needed to break the wood structure and to remove lignin to expose the carbohydrates components. Evaluation of these processes requires characterization of the materials generated in the different stages. The traditional analytical methods of wood, pretreated materials (pulps), monosaccharides in the hydrolyzated pulps, and ethanol involve laborious and destructive methodologies. This, together with the high cost of enzymes and the possibility to obtain low ethanol yields from some pulps, makes it suitable to have rapid, nondestructive, less expensive, and quantitative methods to monitoring the processes to obtain ethanol from wood. In this work, infrared spectroscopy (IR) accompanied with multivariate analysis is used to characterize chemically organosolv pretreated Eucalyptus globulus pulps (glucans, lignin, and hemicellulosic sugars), as well as to predict the ethanol yield after a SSF process. Mid (4,000–400 cm^?1) and near-infrared (12,500–4,000 cm^?1) spectra of pulps were used in order to obtain calibration models through of partial least squares regression (PLS). The obtained multivariate models were validated by cross validation and by external validation. Mid-infrared (mid-IR)/NIR PLS models to quantify ethanol concentration were also compared with a mathematical approach to predict ethanol yield estimated from the chemical composition of the pulps determined by wet chemical methods (discrete chemical data). Results show the high ability of the infrared spectra in both regions, mid-IR and NIR, to calibrate and predict the ethanol yield and the chemical components of pulps, with low values of standard calibration and validation errors (root mean square error of calibration, root mean square error of validation (RMSEV), and root mean square error of prediction), high correlation between predicted and measured by the reference methods values (R ² between 0.789 and 0.997), and adequate values of the ratio between the standard deviation of the reference methods and the standard errors of infrared PLS models relative performance determinant (RPD) (greater than 3 for majority of the models). Use of IR for ethanol quantification showed similar and even better results to the obtained with the discrete chemical data, especially in the case of mid-IR models, where ethanol concentration can be estimated with a RMSEV equal to 1.9 g?L^?1. These results could facilitate the analysis of high number of samples required in the evaluation and optimization of the processes.     

Near-infrared reflectance spectroscopy and multivariate calibration techniques applied to modelling the crude protein, fibre and fat content in rapeseed meal

Near-infrared reflectance spectroscopy (NIRS) is often applied when a rapid quantification of major components in feed is required. This technique is preferred over the other analytical techniques due to the relatively few requirements concerning sample preparations, high efficiency and low costs of the analysis. In this study, NIRS was used to control the content of crude protein, fat and fibre in extracted rapeseed meal which was produced in the local industrial crushing plant. For modelling the NIR data, the partial least squares approach (PLS) was used. The satisfactory prediction errors were equal to 1.12, 0.13 and 0.45 (expressed in percentages referring to dry mass) for crude protein, fat and fibre content, respectively. To point out the key spectral regions which are important for modelling, uninformative variable elimination PLS, PLS with jackknife-based variable elimination, PLS with bootstrap-based variable elimination and the orthogonal partial least squares approach were compared for the data studied. They enabled an easier interpretation of the calibration models in terms of absorption bands and led to similar predictions for test samples compared to the initial models.     

基于粒子群算法的最小二乘支持向量机在红花提取液近红外定量分析中的应用

提出一种基于粒子群算法的最小二乘支持向量机(PSO-LS-SVM)方法,用于建立红花提取过程关键质控指标的定量分析模型.近红外光谱数据经波段选择、预处理和主成分分析(降维)后,利用粒子群优化(PSO)算法对最小二乘支持向量机算法中的参数进行优化,然后使用最优参数建立固含量和羟基红花黄色素A(HSYA)浓度的定量校正模型.将校正结果与偏最小二乘法回归(PLSR)和BP神经网络(BP-ANN)比较,并将所建的3个模型用于红花提取过程未知样本的预测.结果表明,BP-ANN校正结果优于PSO-LS-SVM和PLSR,但是对验证集和未知样品集的预测能力较差,而PSO-LS-SVM和PLSR模型的校正、验证结果相近,相关系数均大于0.987,RMSEC和RMSEP值相近且小于0.074,RPD值均大于6.26,RSEP均小于5.70％.对于未知样品集,pSO-LS-SVM模型的RPD值大于8.06,RMSEP和RSEP值分别小于0.07％和5.84％,较BP-ANN和PLSR模型更低.本研究所建立的PSO-LS-SVM模型表现出较好的模型稳定性和预测精度,具有一定的实践意义和应用价值,可推广用于红花提取过程的近红外光谱定量分析.     

Problems of calibration in trace, in situ-micro and surface analysis

Summary A systematic survey will be given on different strategies of calibration in dependence on given analytical and statistical conditions, particularly on several procedures of least squares regression (ordinary, orthogonal, unweighted and weighted LSR), of robust regression, addition methods and multicomponent calibration. In this connection calibration by means of latent variables (principal component regression PCR, partial least squares PLS) will be dealt with. The special conditions in the case of microanalysis and surface analysis will be considered under practical analytical as well as chemometrical aspects. Problems of homogeneity, representativness of samples and sample regions will be treated.