A comparison of methods for estimating prediction intervals in NIR spectroscopy: Size matters

In this article we demonstrate that, when evaluating a method for determining prediction intervals, interval size matters more than coverage because the latter can be fixed at a chosen confidence level with good reliability. To achieve the desired coverage, we employ a simple non-parametric method to compute prediction intervals by calibrating estimated prediction errors, and we extend the basic method with a continuum correction to deal with small data sets. In our experiments on a collection of several NIR data sets, we evaluate several existing methods of computing prediction intervals for partial least-squares (PLS) regression. Our results show that, when coverage is fixed at a chosen confidence level, and the number of PLS components is selected to minimize squared error of point estimates, interval estimation based on the classic ordinary least-squares method produces the narrowest intervals, outperforming the U-deviation method and linearization, regardless of the confidence level that is chosen.     

Validation of analytical methods using a regression procedure

The evaluation and validation of analytical methods and instruments require comparison studies using sample material for testing accuracy and precision. In analytical chemistry, the commonly accepted means of analyzing data from method comparison studies is least-squares regression analysis, a model which has limitations. In this paper, the results from ordinary least-squares and many other regression approaches recommended in the literature were compared with a new regression procedure that takes into account the errors in both variables (methods). After a discussion of the properties of the regression procedure, recommendations are given for carrying out a method comparison study using informational analysis of variance. The efficiency of the regression procedure proposed is demonstrated by applying it to different data sets from published literature.     

Least-squares regression of adsorption equilibrium data: comparing the options

Experimental and simulated adsorption equilibrium data were analyzed by different methods of least-squares regression. The methods used were linear regression, nonlinear regression, and orthogonal distance regression. The results of the regression analysis of the experimental data showed that the different regression methods produced different estimates of the adsorption isotherm parameters, and consequently, different conclusions about the surface properties of the adsorbent and the mechanism of adsorption. A Langmuir-type simulated data set was calculated and several levels of random error were added to the data set. The results of regression analysis of the simulated data set showed that orthogonal distance regression gives the most accurate and efficient estimates of the isotherm parameters. Nonlinear regression and one form of the linearized Langmuir isotherm also gave accurate estimates, but only at low levels of random error.     

A new fuzzy regression algorithm

A new fuzzy regression algorithm is described and compared with conventional ordinary and weighted least-squares and robust regression methods. The application of these different methods to relevant data sets proves that the performance of the procedure described in this paper exceeds that of the ordinary least-squares method and equals and often exceeds that of weighted or robust methods, including the two fuzzy methods proposed previously (Otto, M.; Bandemer, H., Chemom. Intell. Lab. Syst. 1986, 1, 71. Hu, Y.; Smeyers-Verbeke, J.; Massart, D. L. Chemom. Intell. Lab. Syst. 1990, 8, 143). Moreover, we emphasize the effectiveness and the generality of the two new criteria proposed in this paper for diagnosing the linearity of calibration lines in analytical chemistry.     

Maximum likelihood multivariate calibration

Two new approaches to multivariate calibration are described that, for the first time, allow information on measurement uncertainties to be included in the calibration process in a statistically meaningful way. The new methods, referred to as maximum likelihood principal components regression (MLPCR) and maximum likelihood latent root regression (MLLRR), are based on principles of maximum likelihood parameter estimation. MLPCR and MLLRR are generalizations of principal components regression (PCR), which has been widely used in chemistry, and latent root regression (LRR), which has been virtually ignored in this field. Both of the new methods are based on decomposition of the calibration data matrix by maximum likelihood principal component analysis (MLPCA), which has been recently described (Wentzell, P. D.; et al. J. Chemom., in press). By using estimates of the measurement error variance, MLPCR and MLLRR are able to extract the optimum amount of information from each measurement and, thereby, exhibit superior performance over conventional multivariate calibration methods such as PCR and partial least-squares regression (PLS) when there is a nonuniform error structure. The new techniques reduce to PCR and LRR when assumptions of uniform noise are valid. Comparisons of MLPCR, MLLRR, PCR, and PLS are carried out using simulated and experimental data sets consisting of three-component mixtures. In all cases of nonuniform errors examined, the predictive ability of the maximum likelihood methods is superior to that of PCR and PLS, with PLS performing somewhat better than PCR. MLLRR generally performed better than MLPCR, but in most cases the improvement was marginal. The differences between PCR and MLPCR are elucidated by examining the multivariate sensitivity of the two methods.     

Logistic 组稀疏回归模型的 Bayes 建模及变分推断

在工程应用中,如数据挖掘、成本预测以及风险预测等,Logistic 回归是一类十分重要的预测方法．当前,大部分 Logistic 回归方法都是基于优化准则而设计,这类回归方法具有参数调试过程繁琐、模型解释性差、估计子没有置信区间等缺点．本文从 Bayes 概率角度研究 Logistic 组稀疏性回归的建模与推断问题．具体来说,首先利用高斯-方差混合公式提出 Logistic 组稀疏回归的 Bayes 概率模型;其次,通过变分 Bayes 方法设计出一个高效的推断算法．在模拟数据上的实验结果表明,本文所提出的方法具有较好的预测性能．     

New robust bilinear least squares method for the analysis of spectral-pH matrix data

A new second-order multivariate method has been developed for the analysis of spectral-pH matrix data, based on a bilinear least-squares (BLLS) model achieving the second-order advantage and handling multiple calibration standards. A simulated Monte Carlo study of synthetic absorbance-pH data allowed comparison of the newly proposed BLLS methodology with constrained parallel factor analysis (PARAFAC) and with the combination multivariate curve resolution-alternating least-squares (MCR-ALS) technique under different conditions of sample-to-sample pH mismatch and analyte-background ratio. The results indicate an improved prediction ability for the new method. Experimental data generated by measuring absorption spectra of several calibration standards of ascorbic acid and samples of orange juice were subjected to second-order calibration analysis with PARAFAC, MCR-ALS, and the new BLLS method. The results indicate that the latter method provides the best analytical results in regard to analyte recovery in samples of complex composition requiring strict adherence to the second-order advantage. Linear dependencies appear when multivariate data are produced by using the pH or a reaction time as one of the data dimensions, posing a challenge to classical multivariate calibration models. The presently discussed algorithm is useful for these latter systems.     

A diffusion-based compensation approach for intensity inhomogeneity correction in MRI

Intensity inhomogeneity is considered as an inherent artifact in magnetic resonance images and is prominent in high-field strength scanners. An effective and conceptually simple retrospective correction technique is introduced in this article that implements a compensation function based on spatially constrained fuzzy c-means clustering to reduce the effect of intensity inhomogeneity. Intensity compensation functions are estimated on each clustered region and are subsequently processed with an anisotropic diffusion strategy. The proposed approach does not require any parametric models or prior knowledge on the acquisition process for the intensity inhomogeneity correction. The proposed diffusion based technique was evaluated on simulated and real data sets and the results were compared with some of the prominent correction methods. The quantitative analyses in terms of coefficient of variation and coefficient of joint variation ensure the effectiveness of the proposed methodology. The experimental analyses of the results show that the proposed methodology outperforms the state-of-the-art approaches.     

Visible/near infrared-partial least-squares analysis of Brix in sugar cane juice: A test field for variable selection methods

Several variable selection algorithms were applied in order to sort informative wavelengths for building a partial least-squares (PLS) model relating visible/near infrared spectra to Brix degrees in samples of sugar cane juice. Two types of selection methods were explored. A first group was based on the PLS regression coefficients, such as the selection of coefficients significantly larger than their uncertainties, the estimation of the variable importance in projection (VIP), and uninformative variable elimination (UVE). The second group involves minimum error searches conducted through interval PLS (i-PLS), variable-size moving-window (VS-MW), genetic algorithms (GA) and particle swarm optimization (PSO). The best results were obtained using the latter two methodologies, both based on applications of natural computation. The results furnished by inspection of the spectrum of regression coefficients may be dangerous, in general, for selecting informative variables. This important fact has been confirmed by analysis of a set of simulated data mimicking the experimental sugar cane juice spectra.     

On-line process control of gradient elution liquid chromatography

The typical measure of the stability of analytical HPLC methods in the pharmaceutical laboratory is standards injected repeatedly throughout the sample sequence. To obtain improved control of the analysis and reduction of the number of standards and replicates, a novel approach to treat the analytical run as a process, where the chromatographic data is the product, is proposed. Thus, an alternative and continuous system suitability test procedure is described. This is obtained by continuous monitoring of several parameters of the chromatographic system such as pressure, temperature, and conductivity. The data are analyzed in real time with chemometrics to produce easily interpreted control charts. Gradient elution LC is extensively employed in pharmaceutical analysis. A gradient elution system is inherently dynamic due to the mobile-phase composition being changed during the chromatographic run. To handle the dynamics, suitable chemometric tools are needed. In this report, we extend the use of liquid chromatography process control (LCPC) to gradient elution LC by creating partial least-squares regression batch models of the data collected. The gradient elution LCPC system was evaluated by inducing disturbances, and it was shown to easily detect any real or simulated deviation.     

Time-alignment of bidimensional chromatograms in the presence of uncalibrated interferences using parallel factor analysis: Application to multi-component determinations using liquid-chromatography with spectrofluorimetric detection

A new methodology for the alignment of matrix chromatographic data is proposed, based on the decomposition of a three-way array composed of a test and a reference data matrix using a suitably initialized and constrained parallel factor (PARAFAC) model. It allows one to perform matrix alignment when the test data matrix contains unexpected chemical interferences, in contrast to most of the available algorithms. A series of simulated analytical systems is studied, as well as an experimental one, all having calibrated analytes and also potential interferences in the test samples, i.e., requiring the second-order advantage for successful analyte quantitation. The results show that the newly proposed method is able to properly align the different data matrix, restoring the trilinearity which is required to process the calibration and test data with second-order multivariate calibration algorithms such as PARAFAC. Recent models including unfolded partial least-squares regression (U-PLS) and N-dimensional PLS (N-PLS), combined with residual bilinearization (RBL), are also applied to both simulated and experimental data. The latter one corresponds to the determination of the polycyclic aromatic hydrocarbons benzo[b]fluoranthene and benzo[k]fluoranthene in the presence of benzo[j]fluoranthene as interference. The analytical figures of merit provided by the second-order calibration models are compared and discussed.     

Transport properties of mixtures of rarefied gases. Hydrogen–methane system

An analysis and generalization of experimental data on transport properties based on the relations of molecular-kinetic theory and three-parameter interaction potentials of the Lennard-Jones (m–6) family have been performed for the mixture of rarefied neutral gases "hydrogen–methane." Nine parameters of the potentials have been restored in joint data processing using the weight nonlinear least-squares method. Tables of reference data for viscosity, the interdiffusion coefficient, and the thermal diffusion factor have been calculated in the temperature interval 200–1500 K. Errors of reference data in the entire interval of concentrations, including those of pure components, have been evaluated using the matrix of parametric errors.     

Prediction intervals for load-sharing systems in accelerated life testing

Based on accelerated lifetime experiments, we consider the problem of constructing prediction intervals for the time point at which a given number of components of a load-sharing system fails. Our research is motivated by lab experiments with prestressed concrete beams where the tension wires fail successively. Due to an audible noise when breaking, the time points of failure could be determined exactly by acoustic measurements. Under the assumption of equal load sharing between the tension wires, we present a model for the failure times based on a birth process. We provide a model check based on a Q-Q plot including a simulated simultaneous confidence band and four simulation-free prediction methods. Three of the prediction methods are given by confidence sets where two of them are based on classical tests and the third is based on a new outlier-robust test using sign depth. The fourth method uses the implicit function theorem and the δ-method to get prediction intervals without confidence sets for the unknown parameter. We compare these methods by a leave-one-out analysis of the data on prestressed concrete beams. Moreover, a simulation study is performed to discuss advantages and drawbacks of the individual methods.     

Statistics of the slope-method estimator

The slope method has customarily been used and is still used for inversion of atmospheric optical parameters, extinction, and backscatter in homogeneous atmospheres from lidar returns. Our aim is to study the underlying statistics of the old slope method and ultimately to compare its inversion performance with that of the present-day nonlinear least-squares solution (the so-called exponential-curve fitting). The contents are twofold: First, an analytical study is conducted to characterize the bias and the mean-square-estimation error of the regression operator, which permits estimation of the optical parameters from the logarithm of the range-compensated lidar return. Second, universal plots for most short- and far-range tropospheric backscatter lidars are presented as a rule of thumb for obtaining the optimum regression interval length that yields unbiased estimates. As a result, the simple graphic basis of the slope method is still maintained, and its inversion performance improves up to that of the present-day computer-oriented exponential-curve fitting, which ends the controversy between these two algorithms.     

Framework for multivariate selectivity analysis, part I: theoretical and practical merits

A number of definitions of multivariate selectivity have been proposed in the literature. Arguably, the one that enjoys the greatest chemometric attention has been the net analyte signal (NAS) based definitions of Lorber and Zinn. Recent works have suggested that similar inference can be made for inverse least-squares calibration methods (e.g., principal components regression). However, the properties of inverse calibration methods are markedly different than classical methods, so in many practical cases involving inverse models classically derived figures of merit cannot be transparently interpreted. In Part I of this work, we discuss a selectivity framework that is theoretically consistent regardless of the calibration method. Importantly, it is also experimentally measurable, either through controlled selectivity experiments, or through analysis on opportunistically acquired sample measurements. It is statistically advantageous to use the former if such control is achievable. Selectivity is defined to be a function of the change in predicted analyte concentration that will result from a change in the concentration of an interferant, an approach consistent with traditional definitions of analytical selectivity and National Committee for Clinical Laboratory Standards recommendations for interference testing. Unlike the NAS-based definition of selectivity, the definition discussed herein is relevant to only a particular analyte-interferant pair. The theoretical and experimental aspects of this approach are illustrated with simulated data herein and in Part II of this paper, which investigates several experimental near-infrared data sets.     

Beschleunigte Zuverlässigkeitstests unter Berücksichtigung von Vertrauensbereichen

End-of-life tests (EoL-Tests) are typically associated with considerable resources. Accelerated EoL-tests aid at minimizing the required time and budget. Typically, the sample’s failure behavior is described by lifetime models such as the Arrhenius model for only constant stresses. Such models are adapted to the obtained experimental data and are then used to estimate the reliability in the field. Unfortunately, real stress profiles are time-dependent. Despite the effort of parameterizing lifetime models, distribution functions are assumed for the components which ignore the influence of the applied stress. Calculating statistical parameters, such as the reliability inevitably leads to inaccurate results. Furthermore, reliability is often determined based on limited sample sizes. Consequently, reliability prediction is subject to uncertainty and is therefore specified including a confidence interval.This paper presents a new approach which enables the prediction of operative reliability concerning time-dependent stresses with a confidence level. Based on the presented method it is possible to estimate the operative reliability and its confidence interval for transient stresses. This fundamental work uses simulative data to verify the methodology.Two methods for calculating the operative reliability function with time-depending stresses are explained: The cumulative exposure model and the model of age.The most relevant methods to determine a confidence level are introduced briefly. Finally, it is shown how the new method for calculating the operative reliability and the associated confidence intervals for lifetime models is derived. The functionality of the new method, namely the Dubi Bootstrap Simulation (DUBS) is shown by an example enhancing its applicability. The validation of the new approach is done in a separate article using a practical example.     

Multiple and simultaneous fluorophore detection using fluorescence spectrometry and partial least-squares regression with sample-specific confidence intervals

Fluorescent labeling is widely used in biological and chemical analysis, and the drive for increased throughput is stretching multiplexing capabilities to the limit. The limiting factor in multiplexed analyses is the ability to subsequently deconvolute the signals. Consequently, alternative approaches for interpreting complex data sets are required to allow individual components to be identified. Here we have investigated the application of a novel approach to multiplexed analysis that does not rely on multivariate curve resolution to achieve signal deconvolution. The approach calculates a sample-specific confidence interval for a multivariate (partial least-squares regression (PLSR)) prediction, thereby enabling the estimation of the presence or absence of each fluorophore based on the total spectral signal. This approach could potentially be applied to any multiplexed measurement system and has the advantage over the current algorithm-based methods that the requirement for resolution of spectral peaks is not central to the method. Here, PLSR was used to obtain the concentrations for up to eight dye-labeled oligonucleotides at levels of (0.6-5.3) x 10(-6) M. The sample-specific prediction intervals show good discrimination for the presence/absence of seven of the eight labeled oligonucleotides with efficiencies ranging from approximately 91 to 100%.     

Asymptotic efficiency properties of least squares in an ultrastructural model

This paper considers the estimation of both the intercept term and the slope parameter in a linear ultrastructural model using direct regression and reverse regression methods. Without assuming the errors to be normally distributed, asymptotic expressions for the bias and the variance of the estimators are derived and analyzed. The effect of departure of the errors from normality is also studied.     

Asymptotic efficiency properties of least squares in an ultrastructural model

This paper considers the estimation of both the intercept term and the slope parameter in a linear ultrastructural model using direct regression and reverse regression methods. Without assuming the errors to be normally distributed, asymptotic expressions for the bias and the variance of the estimators are derived and analyzed. The effect of departure of the errors from normality is also studied.