Relation between prediction errors of inverse and classical calibration

The formulae for prediction errors of inverse and classical calibration derived by Centner, Massart and de Jong in the Fresenius’ Journal of Analytical Chemistry (1998) 361 : 2–9 are reconsidered. All calculations assume univariate calibration by ordinary least squares regression applied to an infinite number of data pairs. Inverse calibration gives rise to an error variance which is smaller by a certain factor than that of classical calibration. This factor amounts to unity plus the ratio of the variances of the measurement errors and of the responses used for the calibration. The root mean squared error of prediction is also smaller for inverse than for classical calibration, namely by the square root of this factor. A prediction error calculated in that way agrees well with a result obtained by Monte Carlo simulations. Received: 23 December 1999 / Revised: 14 February 2000 / Accepted: 15 February 2000     

A new and consistent parameter for measuring the quality of multivariate analytical methods: Generalized analytical sensitivity

Generalized analytical sensitivity (γ) is proposed as a new figure of merit, which can be estimated from a multivariate calibration data set. It can be confidently applied to compare different calibration methodologies, and helps to solve literature inconsistencies on the relationship between classical sensitivity and prediction error. In contrast to the classical plain sensitivity, γ incorporates the noise properties in its definition, and its inverse is well correlated with root mean square errors of prediction in the presence of general noise structures. The proposal is supported by studying simulated and experimental first-order multivariate calibration systems with various models, namely multiple linear regression, principal component regression (PCR) and maximum likelihood PCR (MLPCR). The simulations included instrumental noise of different types: independently and identically distributed (iid), correlated (pink) and proportional noise, while the experimental data carried noise which is clearly non-iid.     

New Calibration Model: Combining Integrated Calibration Method and H-point Standard Addition Method to Detect and Avoid Interference Effects

A new calibration methodology based on the combination of integrated calibration method (ICM) and the H-point standard addition method (HPSAM) is presented. It allows the diagnosis and correction of errors caused in an analytical system by different kinds of interference effects. Six calibration solutions consisting of mixtures of sample, diluent, and one standard are prepared in accordance with the ICM principle to integrate the external calibration method with the standard addition method and thereby to detect and eliminate proportional interferences. Absorbance increments chosen according to the HPSAM principle are proposed to correct the errors caused by additive interferences. A set of as many as six apparent estimations of analyte concentration in a single calibration procedure is calculated for validating accuracy. As a consequence, doing calibration by the ICM-HPSAM method, it is possible to obtain the final analytical results with considerably improved accuracy. The determination of calcium in several different water samples (containing amounts between 4.9 and 127?mg?L^?1) with Arsenazo III has been chosen as an example because it is biased if the errors are not diagnosed and corrected. The results are characterized by small (not higher than 8%) relative error (RE), and good precision (RSD values smaller than 6%).     

Enhancing calibration models for non-invasive near-infrared spectroscopical blood glucose determination

Partial least-squares regression (PLS) and radial basis function (RBF) networks are used to compute calibration models for non-invasive blood glucose determination by NIR diffuse reflectance spectroscopy. A model computation shows that even extremely small deviations of the spectra induce increased prediction errors. Since the spectral contribution of blood glucose is much smaller than deviations resulting from the non-invasive measuring process a method based on Pearson’s correlation coefficient can be used for evaluating the quality of the recorded spectra during the prediction step. Another method is based on the leverage values from the hat matrix of the RBF network. Both methods lead to a significant decrease in prediction error.     

Inverse vs. classical calibration for small data sets

In classical calibration, the statistically uncertain variable y is regressed on the error-free variable x for a number of known samples, and the results are used to estimate the x value (x0) for an unknown sample from its measured y value (y0). It has long been known that inverse calibration--regression of x on y for the same data--is more efficient in its prediction of x0 from y0 than the seemingly more appropriate classical procedure, over large ranges of the controlled variable x. In the present work, theoretical expressions and Monte Carlo calculations are used to illustrate that the comparison favors the inverse procedure even more for small calibration data sets than for the large sets that have been emphasized in previous studies.     

Evolving factor analysis-based method for correcting monitoring delay in different batch runs for use with PLS: On-line monitoring of a transesterification reaction by ATR-FTIR

In this work, the base-catalyzed transesterification of soybean oil with ethanol was monitored on-line using mid-infrared spectroscopy (MIRS) and the yield of fatty acid ethyl esters (biodiesel) was obtained by (1)H NMR spectroscopy. The MIRS monitoring carried out for 12min, was performed using a cylindrical internal reflectance cell of PbSe in the range of 3707-814cm(-1) with eight co-added scans. Two different data treatment strategies were used: in the first, the models were built using the raw data and in the other, evolving factor analysis (EFA) was used to overcome the sensor time delay due to the on-line analysis, producing significantly better results. In addition, models based on partial least squares (PLS) using three batches for calibration and another for validation were compared with models with just one batch for calibration and three for validation. The models were compared between each other using root mean square error of prediction (RMSEP) and root mean square prediction difference (RMSPD), obtaining relative errors under 3%.     

应用遗传算法和PLS的近红外光谱预测玉米中淀粉含量的研究

以普通玉米籽粒为试验材料,在应用遗传算法结合偏最小二乘回归法对近红外光谱数据进行特征波长选择的基础上,应用偏最小二乘回归法建立了特征波长测定玉米籽粒中淀粉含量的校正模型．试验结果表明,基于11个特征波长所建立的校正模型,其校正误差（RMSEC）、交叉检验误差（RMSECV）和预测误差（RMSEP）分别为0.30％、0.35％和0.27％,校正数据集和独立的检验数据集的预测值与实际测定值之间的相关系数分别达到0.9279和0.9390,与全光谱数据所建立的预测模型相比,在预测精度上均有所改善,表明应用遗传算法和PLS进行光谱特征选择,能获得更简单和更好的模型,为玉米籽粒中淀粉含量的近红外测定和红外光谱数据的处理提供了新的方法与途径．     

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.     

Determination of Iridoids in Gentiana rigescens by Infrared Spectroscopy and Multivariate Analysis

Gentiana rigescens is a famous herbal medicine in China for treatment of convulsion, rheumatism, and jaundice. Here, the infrared determination of gentiopicroside, swertiamarin, sweroside, and loganic acid in G. rigescens from different areas and varieties was presented for the first time. Reference information for the iridoids were obtained by high-performance liquid chromatography. Partial least squares was used to characterize the relationship between spectra matrix and concentration vector for the determination of the analytes. For determination of gentiopicroside, the appropriate performance of partial least squares model was acquired with coefficient of determination of calibration and coefficient of determination of prediction values of 0.965 and 0.868. The root mean square error of estimation (RMSEE), root mean square error of cross validation (RMSECV), root mean square error of prediction (RMSEP), and residual predictive deviation (RPD) values were 2.612, 5.292, 5.239?mg g^?1, and 2.701, respectively, based on the first derivative and multiplicative scatter correction. For determination of the total iridoids, the best results were obtained using the coefficient of determination of calibration and coefficient of determination of prediction of 0.943 and 0.834, RMSEE, RMSECV, RMSEP and RPD of 3.896, 7.536, 6.543?mg g^?1 and 2.438, respectively, based on the first derivative. Both models were reliable and robust. The results demonstrated that infrared spectroscopy provided a rapid, low-cost tool to monitor the quality of G. rigescens by the determination of the iridoids.     

近红外光谱法同时测定通天口服液中天麻素和芍药苷的含量

本文应用近红外光谱结合偏最小二乘法建立了同时测定通天口服液中天麻素与芍药苷含量的方法。以高效液相色谱(HPLC)法测定通天口服液样品中天麻素和芍药苷的化学参考值,随机抽取60个样本作校正集,20个样本作预测集。用偏最小二乘法(PLS)将校正集样本的近红外光谱与相应样本的天麻素和芍药苷含量分别相关联建立模型。结果表明,天麻素和芍药苷校正模型的决定系数分别为96.28%、94.55%,模型的交叉验证均方差分别为0.0336、0.00908,预测集的决定系数分别为94.23%、92.86%,预测集均方差分别为0.0453、0.00839。同时还做了模型的精密度实验,该方法能用于大批量样品的快速分析。     

Resolution optimisation in micellar electrokinetic chromatography using empirical models

Theoretical and empirical models can be used to model the migration or separation characteristics in micellar electrokinetic chromatography in order to optimise the resolution. In this paper only empirical models were used, because it is easier and more straightforward to obtain these models. Several empirical approaches for the optimisation of the resolution were compared in order to determine which response should be modelled preferably. The use of models of the effective mobility in combination with average plate numbers proved to be the most suitable approach to optimisation of the resolution, because the relative prediction errors of the models of the effective mobility were a factor of 2-4 smaller than the relative prediction errors of the models of the apparent mobility. Moreover for the least separated peak pair the resolutions based on the models of the apparent and effective mobility showed relative prediction errors that were approximately a factor of 2 smaller than the relative prediction errors of the resolutions based on the models of the resolution and separation factor. The predictions of the separation factor based on the different models generally showed lower prediction errors than the predictions of the corresponding resolutions. Although the relative prediction errors were large, particularly for closely migrating compounds, the empirical approach will probably lead to the optimum separation buffer composition.     

Genetic algorithms applied to the selection of factors in principal component regression

Using principal component regression (PCR) as a multivariate calibration tool, always brings up the question what subset of factors, i.e. principal components (PCs) gives the best calibration model. Normally factor selection is based on deterministic methods like top–down procedures, forward–backward-stepwise variable selection or correlated principal component regression (CPCR). In contrast to this, we applied a stochastic method, i.e. a genetic algorithm (GA) for factor selection in this paper. A new kind of fitness function was applied which combined the prediction error of the calibration and an independent validation set. The performance of eigenvalue and correlation ranking was compared. A general statistical criterion for judging the significance of differences between individual calibration models is introduced. In this context it could be shown that for the uncertainties of the standard deviations representing the prediction errors a very simple approximation formula holds which only includes the number of standards. For the current applications it is shown that the GA gives a result very close to CPCR-solutions.     

Determination of water in lubricating oils by mid- and near-infrared spectroscopy

Mid-infrared (MIR) and near-infrared (NIR) spectroscopy were used to determine water in lubricating oils with high additive contents that introduce large errors in determinations by the Karl-Fischer and hydride methods. MIR spectra were obtained in the attenuated total reflectance (ATR) mode and exhibited water specific band absorption in the region 3100–3700cm^–1, which facilitated calibration. Multivariate (partial least-squares regression, PLSR) and univariate calibration (based on peak height and band area as independent variables) were tested. Both led to errors of prediction less than 5%. NIRS determinations rely on absorbance and first-derivative spectra, in addition to two different types of multivariate calibration,viz. inverse multiple linear regression (MLR) and partial least-squares regression (PLSR). Both approaches gave similar results, with errors of prediction less than 2%.For none of the proposed approaches any sample pretreatment for recording spectra is required.     

Determination of API Gravity and Total and Basic Nitrogen Content by Mid- and Near-Infrared Spectroscopy in Crude Oil with Multivariate Regression and Variable Selection Tools

Abstract

American Petroleum Institute (API) gravity is an important parameter in the crude oil industry and the nitrogen compounds are related to the toxic effects of the oil in refineries and the environment. In this paper, 194 crude oil samples with API gravities ranging from 11.4 to 57.5 were used for the purpose of estimating the physicochemical properties: API gravity, total nitrogen content (TNC) and basic nitrogen content (BNC). Initially, infrared spectra in the mid and near regions (MIR and NIR) were collected, then full-spectral partial least squares (PLS) and the orthogonal projections to latent structures (OPLS) chemometric models were developed and validated, as well as models using interval PLS (iPLS), synergy interval PLS (siPLS) and competitive adaptive reweighted sampling PLS (CARSPLS) as variable selection tools. For API gravity and TNC, the best calibration technique is the NIR CARSPLS with a root mean square error of prediction (RMSEP) values of 0.9 and 0.0275?wt%, respectively. For BNC, the best technique is MIR siPLS with a prediction error of 0.0134?wt%. The results were validated based on the evaluation of the figures of merit, a statistical evaluation of the accuracy, characterization of the systematic error and measurement for errors in the residues. The results were satisfactory considering the high variability of the data and the diversity of the samples, demonstrating suitable applicability for practical analysis.     

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.     

Determination of low concentrations by spectrophotometry: Uranium-thiocyanate system

Errors in the determination of low concentrations by spectrophotometry are investigated with the uranium-thiocyanate system as an example. The reagent blank has significant absorption and measurements are made at 375 nm instead of the λ_max. The error in the intercept of the calibration curve is an important factor in such measurements and the errors involved in the estimation of 1 μg/ml (normal working range 4–40μg/ml) are studied. It is shown that both random and systematic errors associated with the intercept are responsible for observed errors. The two types of errors are resolved by ANOVA (analysis of variance). The error in the measurement of a single value is estimated and compared with measured values for different calibration ranges. It is seen that two factors predominantly influence the error in the measured concentration — the variance from regression and the closeness of the measured value to the mean of the calibration range.     

径向基函数网络在近红外人体无创伤血糖浓度检测基础研究中的应用

在近红外无创伤血糖浓度检测的基础研究中，对于多组分的混合物的分析，常因光谱与样品浓度之间呈现非线性响应，使得基于线性模型的校正方法失效。本文讨论了非线性校正方法径向基函数神经网络( RBFN )的有效性，并与线性校正方法中的主成分分析和偏最小二乘法作了对比研究。验证实验所用样品为①葡萄糖水溶液②包含牛血红蛋白和白蛋白的葡萄糖水溶液，结果表明：在①实验中PLS模型和RBFN预测标准偏差分别为8．2、8．9；在②实验中分别为15．6、8．8。可见在样品组分增多时，RBFN算法较线性PLS方法建立的模型预测能力强。     

Improved mass accuracy for higher mass peptides by using SWIFT excitation for MALDI-FTICR mass spectrometry

Stepwise-external calibration has previously been shown to produce sub part-per-million (ppm) mass accuracy for the MALDI-FTICR/MS analyses of peptides up to m/z 2500. The present work extends these results to ions up to m/z 4000. Mass measurement errors for ions of higher mass-to-charge are larger than for ions below m/z 2500 when using conventional chirp excitation to detect ions. Mass accuracy obtained by using stored waveform inverse Fourier transform (SWIFT) excitation was evaluated and compared with chirp excitation. Analysis of measurement errors reveals that SWIFT excitation provides smaller deviations from the calibration equation and better mass accuracy than chirp excitation for a wide mass range and for widely varying ion populations.     

Determination of polyphenols in oats by near-infrared spectroscopy (NIRS) and two-dimensional correlation spectroscopy

Two-dimensional correlation spectroscopy (2DCOS) and near-infrared spectroscopy (NIRS) were used to determine the polyphenol content in oat grain. A partial least squares (PLS) algorithm was used to perform the calibration. A total of 116 representative oat samples from four locations in China were prepared and the corresponding near-infrared spectra were measured. Two-dimensional correlation spectroscopy was employed to select wavelength bands for the PLS regression model for the polyphenol determination. The number of PLS components and intervals was optimized according to the coefficients of determination (R²) and root mean square error of cross validation (RMSECV) in the calibration set. The performance of the final model was evaluated using the correlation coefficient (R) and the root mean square error of validation (RMSEV) in the prediction set. The results showed the band corresponding to the optimal calibration model was between 1350 and 1848?nm and the optimal spectral preprocessing combination was second derivative with second smoothing. The optimal regression model was obtained with an R² of 0.8954 and an RMSECV of 0.06651 in the calibration set and R of 0.9614 and RMSEV of 0.04573 in the prediction set. These measurements reveal the calibration model had qualified predictive accuracy. The results demonstrated that the 2DCOS with PLS was a simple and rapid method for the quantitative determination of polyphenols in oats.