Robust regression techniques A useful alternative for the detection of outlier data in chemical analysis

The validation of an analytical procedure means the evaluation of some performance criteria such as accuracy, sensitivity, linear range, capability of detection, selectivity, calibration curve, etc. This implies the use of different statistical methodologies, some of them related with statistical regression techniques, which may be robust or not. The presence of outlier data has a significant effect on the determination of sensitivity, linear range or capability of detection amongst others, when these figures of merit are evaluated with non-robust methodologies.In this paper some of the robust methods used for calibration in analytical chemistry are reviewed: the Huber M-estimator; the Andrews, Tukey and Welsh GM-estimators; the fuzzy estimators; the constrained M-estimators, CM; the least trimmed squares, LTS. The paper also shows that the mathematical properties of the least median squares (LMS) regression can be of great interest in the detection of outlier data in chemical analysis. A comparative analysis is made of the results obtained by applying these regression methods to synthetic and real data. There is also a review of some applications where this robust regression works in a suitable and simple way that proves very useful to secure an objective detection of outliers. The use of a robust regression is recommended in ISO 5725-5.     

Beyond linear least squares regression

Regression is a collection of statistical methods that are used to study relationships among predictor and response variables. In addition to the most popular linear model, solved by least squares, several other techniques have found an application in analytical chemistry. Biased methods such as stepwise regression, ridge regression, principal components regression, and partial least squares regression are especially useful in cases of poorly or underdetermined systems with collinearity. When structural and/or distributional assumptions associated with linear least squares are violated, nonlinear regression, robust regression or generalized least squares estimators may offer potential remedies.     

Fuzzy robust estimation of central location

The problem of a new robust algorithm for estimation of central location has been described in a mathematically simpler way using the fuzzy sets theory. It was compared with ordinary mean estimator and other robust estimators - median, 5% trimmed mean and Huber-, Tukey-, Hampel-, and Andrews-type M-estimators. The performance of Fuzzy 1-means algorithm (FM) proposed is demonstrated by applying it to different data sets from published literature and has been shown to exceed that of conventional ordinary mean estimator and equals or often exceeds that of the most robust estimators.     

Direct determination of antibacterial norfloxacin in urine by isopotential fluorimetry

A novel and simple method is presented for the determination of norfloxacin in human urine by matrix isopotential synchronous fluorescence spectrometry (MISF). This method is useful for the determination of compounds in samples with unknown background fluorescence, such as norfloxacin in urine, without the need for tedious sample pre-treatment. The method was performed in ethanol-water medium (10% v/v), at an apparent pH of 4.8 provided by adding sodium acetate-acetic acid buffer solution. In the determination of norfloxacin in urine the fluorescent intensity varied linearly with its concentration from 20 to 200 ng/mL. An exhaustive statistical analysis has been developed to all calibration graphs, this treatment includes robust regression such as least median of squares, which also detects outliers and leverage points. The overall least-squares regression has been applied to find the more exact straight line that fits the experimental data. The error propagation has been considered to calculate the detection limit and the repeatability of the method. The method shows very low detection limits with acceptable recoveries and precisions. The applicability of the proposed method was illustrated in the determination of norfloxacin in human urine sample without sample pre-treatment.     

Spatial sign preprocessing: a simple way to impart moderate robustness to multivariate estimators

The spatial sign is a multivariate extension of the concept of sign. Recently multivariate estimators of covariance structures based on spatial signs have been examined by various authors. These new estimators are found to be robust to outlying observations. From a computational point of view, estimators based on spatial sign are very easy to implement as they boil down to a transformation of the data to their spatial signs, from which the classical estimator is then computed. Hence, one can also consider the transformation to spatial signs to be a preprocessing technique, which ensures that the calibration procedure as a whole is robust. In this paper, we examine the special case of spatial sign preprocessing in combination with partial least squares regression as the latter technique is frequently applied in the context of chemical data analysis. In a simulation study, we compare the performance of the spatial sign transformation to nontransformed data as well as to two robust counterparts of partial least squares regression. It turns out that the spatial sign transform is fairly efficient but has some undesirable bias properties. The method is applied to a recently published data set in the field of quantitative structure-activity relationships, where it is seen to perform equally well as the previously described best linear model for these data.     

Independent component analysis and its applications in signal processing for analytical chemistry

Independent component analysis (ICA) is a statistical method the goal of which is to find a linear representation of non-Gaussian data so that the components are statistically independent, or as independent as possible. In an ICA procedure, the estimated independent components (ICs) are identical to or highly correlated to the spectral profiles of the chemical components in mixtures under certain circumstances, so the latent variables obtained are chemically interpretable and useful for qualitative analysis of mixtures without prior information about the sources or reference materials, and the calculated demixing matrix is useful for simultaneous determination of polycomponents in mixtures. We review commonly used ICA algorithms and recent ICA applications in signal processing for qualitative and quantitative analysis. Furthermore, we also review the preprocessing method for ICA applications and the robustness of different ICA algorithms, and we give the empirical criterion for selection of ICA algorithms in signal processing for analytical chemistry.     

Quantitative characterization of changes in dynamical behavior for single-particle tracking studies

Single-particle tracking experiments have been used widely to study the heterogeneity of a sample. Segments with dissimilar diffusive behaviors are associated with different intermediate states, usually by visual inspection of the tracking trace. A likelihood-based, systematic approach is presented to remove this incertitude. Maximum likelihood estimators are derived for the determination of diffusion coefficients. A likelihood ratio test is applied to the localization of the changes in them. Simulations suggest that the proposed procedure is statistically robust and is able to quantitatively recover time-dependent changes in diffusion coefficients even in the presence of large measurement noise.     

通用模拟退火用于稳健多元分析校正

模拟退火是一种全局优化算法，具有跨越局部最优点的机制，最小一乘是一种较常用的最小二乘更为稳健的优化准则，更适用于可能偏离正态分布的实际数据集，本文探讨了用最小一乘为准则并利用模拟退火方法同时测定多组分体系的可能性。应用于2－3组分药物体系分析，获得了满意的结果，本文还探讨了改变步长提高模拟退火算法优化精度的方法。     

Robust regression methods to calibrate a continuous flow analyzer in the colorimetric analysis of inorganic phosphorus in seawater

The presence of a salinity effect in the automated analysis of phosphorus in seawater performed by a phosphomolybdate complex can cause the inaccurate estimation of the true amount, because the calibration procedure performed by least-squares regression does not take into account the different analytical response of samples and standard solutions. The error arising from this salt effect is corrected by calibrating the autoanalyzer with robust regression methods (RRMs), which allow definition of the range of salinity in which the analytical response of the autoanalyzer is only dependent on the concentration of marine nutrients. The RRMs are especially helpful in the analysis of oligotrophic samples where the salinity effect is particularly relevant.     

The effect of the choice of method for determining assigned value on the assessment of performance in melting point analysis

The determination of melting point is a fundamental test in the Pharmaceutical industry, since it is one of the simplest techniques for the identification of a chemical substance. The melting point provides information on both identity and purity of a chemical substance and for that reason is a key test in the PHARMASSURE proficiency testing (PT) scheme. The PT scheme assesses participant??s determination of melting point, using chemicals of high purity and basing the assigned value on the robust consensus mean (median). In recent rounds, melting point reference standards have been provided as the test material and a reference value used as the assigned value for PT assessment. Comparison of the PT results over a number of rounds, using test materials with a wide range of melting points, shows the overall performance of the participant group is worse in rounds where a reference material and associated reference assigned value are used for performance assessment. When participants were assessed against the reference assigned value, a positive bias was observed in the participant??s results. Detailed information regarding the methodology used demonstrated that the majority of participants use the same analytical method, EU.Ph.2.2.14 (Council of Europe, Strasbourg, 2011), for the determination of melting point although this procedure allows flexibility in key methodological parameters, such as heating ramp rate, which may fail to ensure consistent performance across the group of participant laboratories.     

How to estimate moments and quantiles of environmental data sets with non-detected observations? A case study on volatile organic compounds in marine water samples

Concentrations of 27 priority volatile organic compounds were measured in water samples of the North Sea and Scheldt estuary during a 3-year monitoring study. Despite the use of a sensitive analytical method, a number of data were censored. That is, some concentrations were below the decision limit or critical level defined by IUPAC. To characterize the observed measurement results, an attempt was made to identify an appropriate procedure to compute summary statistics for the censored data sets. Several parametric and robust parametric approaches based on the maximum likelihood principle and probability-plot regression method were evaluated for the estimation of the mean, standard deviation, median and interquartile range using three uncensored analytes (1,1,2-trichloroethane, tetrachloroethene and o-xylene) from the monitoring survey. Performance was assessed by artificially censoring the observed concentrations and estimating moments and quantiles at each censoring level. Results showed that methods with the least distributional assumptions, such as the robust bias-corrected restricted maximum likelihood method, perform best for estimating the mean and standard deviation, while both parametric and robust parametric techniques can be used for quantiles. Hence, summary statistics could be estimated with little bias (5-10%) up to 80% of censoring for the data sets employed in this study.     

Mass spectrometric analysis for carboxylic acids as viable markers of petroleum hydrocarbon biodegradation

The characterization of metabolites, which are considered markers of bacterial degradation of hydrocarbons, is gaining in importance. Over the years, carboxylic acids have served as useful indicators of aerobic and anaerobic hydrocarbon biodegradation. This interest has been accompanied by the extensive and robust development of analytical methods for monitoring, untargeted identification, and specific and sensitive determination of carboxylic acids in complex matrices. This review discusses critically the state-of-the-art of mass spectrometry as a reliable analytical technique to identify and quantify carboxylic acid metabolites. Attention is paid to sample pre-treatment, selective group pre-concentration, and gas and liquid chromatography preceding mass spectrometry to alleviate matrix effects and ionization discrimination. Recent specific applications of mass spectrometry in monitoring carboxylic acids for assessing hydrocarbon biodegradation are reviewed. Presently, no single technique is sufficient for holistic profiling of carboxylic acids. The direct characterization of carboxylic acids by mass spectrometry is the ultimate goal but despite recent significant developments, challenges persist.     

Use of characteristic functions derived from proficiency testing data to evaluate measurement uncertainties

Interlaboratory comparisons show that reproducibility standard deviations are dependent on the concentration of the analyte. Many attempts have been made to model this. In this paper, ??characteristic?? functions are used for modelling the concentration dependence of the reproducibility standard deviations based on data from proficiency tests for water analysis. The characteristics of the resulting functions can be used for the estimation of measurement uncertainties at different concentration levels. These functions are especially useful to determine the concentration levels below which absolute uncertainties tend to be constant and above which the relative uncertainties are more constant. By comparing the characteristic functions of different analytical procedures for the determination of the same analyte, the performance of these procedures under routine application can be compared. Finally, these functions may be used to get an indication on the average quality of analytical result in a specific field to be used by regulators in order to formulate requirements in the legislation that are in accordance with current measurement quality.     

Robust linear regression taking into account errors in the predictor and response variables

We developed a robust regression technique that is a generalization of the least median of squares (LMS) technique to the field in which the errors in both the predictor and the response variables are taken into account. This simple generalization is limited in the sense that the resulting straight line is found by using only two points from the initial data set. In this way a simulation step is added by using the Monte Carlo method to generate the best robust regression line. We call this new technique 'bivariate least median of squares' (BLMS), following the notation of the LMS method. We checked the robustness of the new regression technique by calculating its breakdown point, which was 50%. This confirms the robustness of the BLMS regression line. In order to show its applicability to the chemical field we tested it on simulated data sets and real data sets with outliers. The BLMS robust regression line was not affected by many types of outlying points in the data sets.     

Interpolation and extrapolation problems of multivariate regression in analytical chemistry: benchmarking the robustness on near-infrared (NIR) spectroscopy data

Modern analytical chemistry of industrial products is in need of rapid, robust, and cheap analytical methods to continuously monitor product quality parameters. For this reason, spectroscopic methods are often used to control the quality of industrial products in an on-line/in-line regime. Vibrational spectroscopy, including mid-infrared (MIR), Raman, and near-infrared (NIR), is one of the best ways to obtain information about the chemical structures and the quality coefficients of multicomponent mixtures. Together with chemometric algorithms and multivariate data analysis (MDA) methods, which were especially created for the analysis of complicated, noisy, and overlapping signals, NIR spectroscopy shows great results in terms of its accuracy, including classical prediction error, RMSEP. However, it is unclear whether the combined NIR + MDA methods are capable of dealing with much more complex interpolation or extrapolation problems that are inevitably present in real-world applications. In the current study, we try to make a rather general comparison of linear, such as partial least squares or projection to latent structures (PLS); "quasi-nonlinear", such as the polynomial version of PLS (Poly-PLS); and intrinsically non-linear, such as artificial neural networks (ANNs), support vector regression (SVR), and least-squares support vector machines (LS-SVM/LSSVM), regression methods in terms of their robustness. As a measure of robustness, we will try to estimate their accuracy when solving interpolation and extrapolation problems. Petroleum and biofuel (biodiesel) systems were chosen as representative examples of real-world samples. Six very different chemical systems that differed in complexity, composition, structure, and properties were studied; these systems were gasoline, ethanol-gasoline biofuel, diesel fuel, aromatic solutions of petroleum macromolecules, petroleum resins in benzene, and biodiesel. Eighteen different sample sets were used in total. General conclusions are made about the applicability of ANN- and SVM-based regression tools in the modern analytical chemistry. The effectiveness of different multivariate algorithms is different when going from classical accuracy to robustness. Neural networks, which are capable of producing very accurate results with respect to classical RMSEP, are not able to solve interpolation problems or, especially, extrapolation problems. The chemometric methods that are based on the support vector machine (SVM) ideology are capable of solving both classical regression and interpolation/extrapolation tasks.     

A quick, convenient and economical method for the reliable determination of methylglyoxal in millimolar concentrations: the N-acetyl-l-cysteine assay

The determination of methylglyoxal (MG) concentrations in vivo is gaining increasing importance as high levels of MG are linked to various health impairments including complications of diabetes. In order to standardize the measurements of MG in body fluids, it is necessary to precisely determine the concentration of MG stock solutions used as analytical standards. The "gold standard" method for the determination of MG concentration in the millimolar range is an enzyme-catalyzed endpoint assay based on the glyoxalase I catalyzed formation of S-lactoylglutathione. However, as this assay used purified glyoxalase I enzyme, it is quite expensive. Another method uses a derivation reaction with 2,4-dinitrophenylhydrazine, but this substance is explosive and needs special handling and storage. In addition, precipitation of the product methylglyoxal-bis-2,4-dinitrophenylhydrozone during the reaction limits the reliability of this method. In this study, we have evaluated a new method of MG determination based on the previously published fast reaction between MG and N-acetyl-L-cysteine at room temperature which yields an easily detectable condensation product, N-α-acetyl-S-(1-hydroxy-2-oxo-prop-1-yl)cysteine. When comparing these three different assays for the measurement of MG concentrations, we find that the N-acetyl-L-cysteine assay is the most favorable, providing an economical and robust assay without the need for the use of hazardous or expensive reagents.     

偏最小二乘回归用于近红外光谱分析的稳健策略

偏最小二乘法（PLS）在近红外光谱（NIR）定量分析中占有重要地位,但预测结果往往容易受到样本分组和奇异样本等因素的影响,稳健性不强。多模型PLS (EPLS）方法在模型稳健性上得到提高,然而它无法识别样本中存在的奇异样本。为了同时提高模型的预测准确性和稳健性,本文提出了一种根据取样概率重新取样的多模型PLS方法,称为稳健共识PLS（RE-PLS）方法。该方法通过迭代赋权偏最小二乘法(IRPLS)计算样本回归残差得到每个校正集样本的取样概率,然后根据样本的取样概率来选择训练子集建立多个PLS模型,最后将所有PLS模型的预测结果平均作为最终预测结果。该方法用于两种不同植物样品的近红外光谱建模,并与传统的PLS及EPLS方法进行比较。结果表明该方法可以有效的避免校正集中奇异样本对模型的影响,同时可以提高预测精确度和稳健性。对于含有较多奇异样本的,复杂近红外光谱烟草实际样本,利用简单PLS或者EPLS方法建模预测效果不是很理想,而RE-PLS凭借其独特优势则有望在这种复杂光谱定量分析中得到广泛的应用。     

Laser Ablation Inductively Coupled Plasma Mass Spectrometry for Determination of Trace Elements in Geological Glasses

 Laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) was used as a powerful multielement analytical method for trace analysis of geological glasses which are useful as reference materials for geochemical in-situ microanalytical work. The quantification of the analytical results was carried out using the BCR-2G and NIST 612 glass standard reference material (SRM). The experimentally determined relative sensitivity coefficients (RSC) for both SRMs vary between 0.2 and 3 for most of the elements, with increasing mass an increasing of relative sensitivity coefficients was observed. The relative standard deviation (RSD) for determination of trace element concentration of most elements (N=3) are between 2 and 10%. The determination of trace elements in various geological glasses by LA-ICP-MS yielded a good agreement with the reference values and those results of other trace analytical methods. Received October 15, 1999. Revision April 14, 2000.     

Estimation of Performance Characteristics of an Analytical Method Using the Data Set Of The Calibration Experiment

Abstract

A model to calculate the analytical sensitivity, limit of detection, limit of determination and precision of a method of instrumental analysis through a data set obtained by calibration experiment using the statistical analysis of linear regression, is proposed. This model has been applied to spectrophotometric, spectrofluorimetric and chromatographic methods. The values obtained are independent of the instrument used and can be applied as a criterion of comparison between different methods proposed for the same analyte. Also, these characteristics have been calculated using the IUPAC suggested model and both results have been compared.     

Comparison of IFCC and NGSP methods for determination of glycated haemoglobin using advanced regression techniques

The objective of this work was a correct statistical comparison of two assay methods for determination of glycated haemoglobin HbA1c. The immunoturbidimetric determination of HbA1c was performed in two ways: using an automatic analyser Hitachi 912, calibrated according to the IFCC reference system (International Federation of Clinical Chemistry and Laboratory Medicine) and using an analyser Advia 1200 and the NGSP reference system (National Glycohaemoglobin Standardization Program). For statistical comparison of these two analytical methods several advanced regression methods were used, which respect random errors of both compared methods. Specifically, Deming regression with and without weights, orthogonal regression, and Passing-Bablok regression were employed. The results demonstrate that the investigated analytical assay methods do not correspond to each other. The summarized results indicate usefulness of better harmonisation of two existing reference systems.