Linear calibrations in chromatography: The incorrect use of ordinary least squares for determinations at low levels,and the need to redefine the limit of quantification with this regression model

Ordinary least squares is widely applied as the standard regression method for analytical calibrations, and it is usually accepted that this regression method can be used for quantification starting at the limit of quantification. However, it requires calibration being homoscedastic and this is not common. Different calibrations have been evaluated to assess whether ordinary least squares is adequate to quantify estimates at low levels. All calibrations evaluated were linear and heteroscedastic. Despite acceptable values for precision at limit of quantification levels were obtained, ordinary least squares fitting resulted in significant and unacceptable bias at low levels. When weighted least squares regression was applied, bias at low levels was solved and accurate estimates were obtained. With heteroscedastic calibrations, limit values determined by conventional methods are only appropriate if weighted least squares are used. A “practical limit of quantification” can be determined with ordinary least squares in heteroscedastic calibrations, which should be fixed at a minimum of 20 times the value calculated with conventional methods. Biases obtained above this “practical limit” were acceptable applying ordinary least squares and no significant differences were obtained between the estimates measured using weighted and ordinary least squares when analyzing real‐world samples.     

Comparison of approaches for parameter estimation on stochastic models: Generic least squares versus specialized approaches

Parameter estimation for models with intrinsic stochasticity poses specific challenges that do not exist for deterministic models. Therefore, specialized numerical methods for parameter estimation in stochastic models have been developed. Here, we study whether dedicated algorithms for stochastic models are indeed superior to the naive approach of applying the readily available least squares algorithm designed for deterministic models.We compare the performance of the recently developed multiple shooting for stochastic systems (MSS) method designed for parameter estimation in stochastic models, a stochastic differential equations based Bayesian approach and a chemical master equation based techniques with the least squares approach for parameter estimation in models of ordinary differential equations (ODE). As test data, 1000 realizations of the stochastic models are simulated. For each realization an estimation is performed with each method, resulting in 1000 estimates for each approach. These are compared with respect to their deviation to the true parameter and, for the genetic toggle switch, also their ability to reproduce the symmetry of the switching behavior. Results are shown for different set of parameter values of a genetic toggle switch leading to symmetric and asymmetric switching behavior as well as an immigration-death and a susceptible-infected-recovered model. This comparison shows that it is important to choose a parameter estimation technique that can treat intrinsic stochasticity and that the specific choice of this algorithm shows only minor performance differences.     

Non-linear multiparameter estimation in spectrophotometry: Part I. The selection of close-to-linear model functions with easy convergence using least squares algorithms

Least squares optimization offers an objective function which, compared to other computational strategies, is favourable for non-linear models. Generally, iterative procedures using approximations of the objective function are used to find the global minimum and its parameter set. Convergence is a crucial aspect in these calculations. In this respect it is not always realized that for a specific model many model functions can usually be raised, each exhibiting a different degree of non-linearity and, on substitution in the objective function, an associated ease of convergence. For good convergence, a close-to-linear model function with a low intrinsic non-linearity should be selected. The remaining parameter non-linearity can be decreased by change of parameters. The measures of non-linearity are elucidated geometrically in the sample space in relation to possible non-Gaussian behaviour and bias of the estimated parameters.     

An assisted approach of the global optimization for the experimental conditions in capillary electrophoresis

Experimental conditions have effect on the separation of capillary electrophoresis (CE) directly. In this work, a set of index to describe the separation in CE was established properly. Based on a combination of genetic algorithm and least square support vector machine, an assisted approach of global optimization for experimental conditions was proposed for the first time, and it was applied to the separation of four synthetic compounds by CE in nonaqueous system. Under the optimum conditions obtained by this approach, the result of the experiment was satisfactory and proved that this novel approach was effective. Furthermore, we investigated the most important conditions that mainly affect the separation effectiveness of CE by partial least squares regression analysis. Because of the generalization of this new approach proposed, it can be applied to the optimization of other experimental processes.     

Simultaneous spectrophotometric determination of p-benzoquinone and chloranil after microcrystalline naphthalene extraction by using genetic algorithm-based wavelength selection-partial least squares regression.

Microcrystalline naphthalene extraction has been used for the preconcentration of p-benzoquinone and tetrachloro-p-benzoquinone (chloranil), after their reaction by aniline, and later simultaneous spectrophotometric analysis by genetic algorithm-partial least squares (GA-PLS) calibration. The chemical variables affecting the analytical performance of the methodology were studied and optimized. Under the optimum conditions i.e., [aniline] = 0.05 M and [naphthalene] = 2.2% (w/v), preconcentration of 25 ml of sample solution permitted the detection of 0.32 and 0.23 microg ml(-1) for p-benzoquinone and chloranil, respectively. The predictive abilities of partial least squares regression (PLS) and genetic algorithm-partial least squares regression (GA-PLS) were examined for simultaneous determination of two quinones. The GA-PLS shows superiority over other PLS methods due to the wavelength selection in PLS calibration using a genetic algorithm without loss of prediction capacity, provides useful information about the chemical system.     

Effective GA approach for a direct evaluation of reaction kinetic within EPDM accelerated sulphur crosslinking

A direct genetic algorithm (GA) approach with kinetic base, to provide effective numerical estimates of vulcanization level for EPDM cross-linked with accelerated sulphur is presented. The model requires a preliminary characterization of rubber through standard rheometer tests. A recently presented kinetic exponential model is used as starting point to develop the algorithm proposed. In such a model, three kinetic constants have to be determined by means of a non-linear least-squares curve fitting. The approach proposed circumvents a sometimes inefficient and not convergent non-linear data fitting, disregarding at a first attempt reversion and finding the local minimum of a suitable two-variable error function, to have an estimate of the first two kinetic constants. A comparison between present GA approach and traditional gradient based algorithms is discussed. The last constant, representing reversion is again evaluated through a minimization performed on a single variable error function. The applicability of the approach is immediate and makes the model extremely appealing when fast and reliable estimates of crosslinking density of cured EPDM are required. To show the capabilities of the approach proposed, a comprehensive comparison with both available experimental data and results obtained numerically with a least square exponential model for a real compound at different temperatures is provided.     

On the use of genetic algorithms for response surface modeling in high-performance liquid chromatography and their combination with the Microsoft Solver

Four genetic algorithms--the classical, Haupt's, Brunetti's and a modification of the classical algorithm suggested in the present paper--are examined when they are used for the modeling of response surfaces in high-performance liquid chromatography (HPLC). We found that the best results are obtained from our modification and the worst by Haupt's algorithm. The classical genetic algorithm gives satisfactory results, better than those of Brunetti's algorithm. We also ascertained that all genetic algorithms may get stuck in a local minimum other than the global one, except for our modification, which can be considered to approach a global method. Finally, the time needed for the optimization of a genetic algorithm and the combination of a genetic algorithm with a non-linear least-squares routine are considered and discussed.     

Multivariate calibration of spectral data using dual-domain regression analysis

To date, few efforts have been made to take simultaneous advantage of the local nature of spectral data in both the time and frequency domains in a single regression model. We describe here the use of a novel chemometrics algorithm using the wavelet transform. We call the algorithm dual-domain regression, as the regression step defines a weighted model in the time-domain based on the contributions of parallel, frequency-domain models made from wavelet coefficients reflecting different scales. In principle, any regression method can be used, and implementation of the algorithm using partial least squares regression and principal component regression are reported here. The performance of the models produced from the algorithm is generally superior to that of regular partial least squares (PLS) or principal component regression (PCR) models applied to data restricted to a single domain. Dual-domain PLS and PCR algorithms are applied to near infrared (NIR) spectral datasets of Cargill corn samples and sets of spectra collected on batch chemical reactions run in different reactors to illustrate the improved robustness of the modeling.     

Kinetic evaluation of non-isothermal thermoanalytical curves in the case of independent thermal reactions

A least squares curve-fitting method was developed for the following thermo-analytical problem: “Find the kinetic parameters and the unknown initial amounts of the reactants from non-isothermal thermoanalytical curves in the case of two or more independent or quasi-independent thermal reactions”. From a numerical point of view this problem differs from the non-linear least squares techniques used in other areas of reaction kinetics. The special difficulties which have arisen in such calculations were eliminated by parameter transformations and by separating the linear and non-linear parts of the problem. The method can be applied at any T(t) functions. Thermo-analytical curves differing in temperature program can be evaluated simultaneously.     

组合偏最小二乘回归方法在近红外光谱定量分析中的应用

针对近红外光谱数据局部效应显著,变量个数多,彼此间常存在严重的复共线性,并多与样品组分含量呈非线性关系,构建一种组合非线性偏最小二乘回归(E-S-QPLSR)方法。它采用无重复采样技术(subag-ging),从训练样本中生成若干子样,然后每个子样通过二次多项式偏最小二乘回归(QPLSR),建立其子模型,并实现对训练样本因变量的定量预测,再将它们交由线性PLS算法用于计算各子模型的组合权系数。将该法应用于80个玉米样品的水组分含量与其近红外光谱的定量关系建模,效果良好,显示出很强的学习能力,所建模型的预报性能也优于其它方法。     

Local classification: Locally weighted–partial least squares-discriminant analysis (LW–PLS-DA)

The possibility of devising a simple, flexible and accurate non-linear classification method, by extending the locally weighted partial least squares (LW–PLS) approach to the cases where the algorithm is used in a discriminant way (partial least squares discriminant analysis, PLS-DA), is presented. In particular, to assess which category an unknown sample belongs to, the proposed algorithm operates by identifying which training objects are most similar to the one to be predicted and building a PLS-DA model using these calibration samples only. Moreover, the influence of the selected training samples on the local model can be further modulated by adopting a not uniform distance-based weighting scheme which allows the farthest calibration objects to have less impact than the closest ones.     

Statistical comparison of calibration methods in multicomponent analysis

Spectrophotometric multicomponent analysis is considerd on the basis of inverse multivariate calibration with linear methods (ordinary least squares, principal component, ridge and partial least squares regression) and with the non-linear methods ACE and the non-linear partial least squares. The performance of the different methods is compared by paired F-tests. As an estimate of the error variance the residual mean sum of squares in the analysis of variance table is used. The comparison is demonstrated for the infrared spectrometric analysis of the hydroxyl group content of brown coal measured in diffuse reflectance. Although the error variances among the calibration methods differ gradually, the differences are much less pronounced at statistical level.     

Comparison of robust regression methods based on least-median and adaptive kalman filtering approaches applied to linear calibration data

Five algorithms for data analysis are evaluated for their abilities to discriminate against outliers in small data sets (4–10 points). These methods included least-squares regression, the least absolute -deviation method, the least median of squares method, and two techniques based on an adaptive Kalman filter. For data sets consisting of 4–9 points with one outlier, the average errors in the estimation of the slope were found to be 18.9 % by least-squares, 17.7% by the least absolute deviation method, 0.5% by the least median of squares algorithm, 9.1% by an adaptive Kalman filter algorithm, and 0.9% by a zero-lag adaptive Kalman filter algorithm. Based on these results, the conclusion is that the zero-lag adaptive Kalman filter and the least median of squares approaches are best suited for the detection of outliers in small calibration data sets.     

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。     

Robustness properties of a robust partial least squares regression method

The presence of multicollinearity in regression data is no exception in real life examples. Instead of applying ordinary regression methods, biased regression techniques such as principal component regression and ridge regression have been developed to cope with such datasets. In this paper, we consider partial least squares (PLS) regression by means of the SIMPLS algorithm. Because the SIMPLS algorithm is based on the empirical variance-covariance matrix of the data and on least squares regression, outliers have a damaging effect on the estimates. To reduce this pernicious effect of outliers, we propose to replace the empirical variance-covariance matrix in SIMPLS by a robust covariance estimator. We derive the influence function of the resulting PLS weight vectors and the regression estimates, and conclude that they will be bounded if the robust covariance estimator has a bounded influence function. Also the breakdown value is inherited from the robust estimator. We illustrate the results using the MCD estimator and the reweighted MCD estimator (RMCD) for low-dimensional datasets. Also some empirical properties are provided for a high-dimensional dataset.     

构建支持向量机-偏最小二乘法为药物构效关系建模

为研究药物构效关系积累样本数据的过程中,需为小样本建模。此时较易造成过拟合,影响模型的预测性能和稳定性。为此可用偏最小二乘(PLS)法从样本数据中成对地提取最优成分,消除自变量间的复共线性,并有效的降维,然后应用最小二乘支持向量机对成对成分进行非线性回归,并以基于误差修正的策略调整,使之更有效地表达自、因变量间的非线性关系。由此构建为EB-LSSVM-PLS算法,所建模型的预报精度高,稳定性良好。将其应用于新型黄烷酮类衍生物的QSAR建模,效果令人满意,其泛化性能优于其它方法。     

Open-path FT-IR spectrometry: is completely unattended operation feasible?

Most protocols used for open-path Fourier transform infrared spectrometry (OP/FT-IR) require that spectra be measured at a resolution of 1 cm(-1) and that the concentrations of the analytes be calculated by classical least squares regression (CLS). These specifications were largely developed for monitoring light molecules with easily resolvable rotational fine structure. For most volatile organic compounds in air, the rotational fine structure is not resolvable and better accuracy can be obtained when the spectrum is measured at lower resolution (typically 8 cm(-1)), provided that the algorithm used for quantification is partial least squares regression (PLS). By measuring the spectrum at low resolution, the need for a liquid-nitrogen-cooled mercury cadmium telluride detector is reduced and a pyroelectric detector operating at ambient temperature can be used instead. By using PLS rather than CLS, spectral features due to water vapor do not have to be compensated and a short-path background spectrum can be used, greatly simplifying field measurements.     

Chemometrics-assisted spectrophotometric methods for simultaneous determination and complexation study of Fe(III), Al(III) and V(V) with morin in micellar media

Evolutionary factor analysis (EFA) and rank annihilation factor analysis (RAFA) were applied to resolve the two-way equilibrium spectrophotometric data belonging to the complexes of Fe(III), Al(III) and V(V) with morin (3,5,7,20,40-penta hydroxy flavone) as chelating agent in triton X-100 micellar media. Then, partial least square regression combined with genetic algorithm for wavelength selection (GA-PLS) was used for simultaneous determination of the metal ions. The parameters controlling behavior of the system were investigated and optimum conditions were selected. The predictive abilities of partial least squares regression (PLS) and genetic algorithm-partial least squares regression (GA-PLS) were examined in simultaneous determination of ternary mixtures of metal ions over the concentration range of 17.0-170.0ngml(-1), 25.0-180.0ngml(-1) and 40.0-325.0ngml(-1) for Fe(III), Al(III) and V(V), respectively. The relative standard errors for prediction of the ions in synthetic mixtures were lower than 5% and the mean recoveries in the tap water spiked samples were 104.2 and 101.7% for PLS and GA-PLS, respectively.