Selected abstracts

[A][B][C][D][E][F][G][H][I][J][K][L][M][N][O][P][Q][R][S][T][U][V][W][X][Y][Z]


A


M. Amrhein, B. Srinivasan and D. Bonvin
Calibration of spectral reaction data
Chemometrics and Intelligent Laboratory Systems, 46 (1999) 249-264

Calibration is the first step in the prediction of concentrations from spectral measurements of chemical reaction systems. It is a well-known fact that the species in the calibration set must include those in the new set. Typically, the calibration set is constructed from non-reacting mixtures of known concentrations. In this paper, it is proposed instead to use the calibration data from reacting mixtures, thereby avoiding the independent variation of possibly highly-reactive intermediates. However, for the prediction to be correct, restrictions on the initial and inlet concentrations of the new data set must be imposed. When these restrictions can not be met, calibration of data in reaction-variant form is proposed. The methodology is illustrated experimentally using an esterification reaction.

Return to General on calibration

C.M. Andersen, R. Bro and P.B. Brockhoff
Quantifying and handling errors in instrumental measurements using the measurement error theory
Journal of Chemometrics, 17 (2003) 621-629

Measurement error modelling is used for investigating the influence of measurement/sampling error on univariate predictions of water content and water-holding capacity (reference measurement) from nuclear magnetic resonance (NMR) relaxations (instrumental) measured on two gadoid fish species. This is a new way of using the measurement error theory. Reliability ratios illustrate that the models for the two fish species are influenced differently by the error. However, the error seems to influence the predictions of the two reference measures in the same way. The effect of using replicated xmeasurements is illustrated by simulated data and by NMR relaxations measured several times on each fish. The standard error of the physical determination of the reference values is lower than the standard error of the NMR measurements. In this case, lower prediction error is obtained by replicating the instrumental measurements. A new general formula is given for how to correct the least squares regression coefficient when a different number of replicated x-measurements is used for prediction than for calibration. It is shown that the correction should be applied when the number of replicates in prediction is less than the number in calibration. This can be relevant for online applications and in other situations where it is important to make fast measurements in the prediction phase.

Return to Reliability of univariate calibration

D.T. Andrews, L. Chen, P.D. Wentzell and D.C. Hamilton
Comments on the relationship between principal components analysis and weighted linear regression for bivariate data sets
Chemometrics and Intelligent Laboratory Systems, 34 (1996) 231-244

Regression and principal components analysis (PCA) are two of the most widely used techniques in chemometrics. In this paper, these methods are compared by considering their application to linear, two-dimensional data sets with a zero intercept. The need for accommodating measurement errors with these methods is addressed and various techniques to accomplish this are considered. Seven methods are examined: ordinary least squares (OLS), weighted least squares (WLS), the effective variance method (EVM), multiply weighted regression (MWR), unweighted PCA (UPCA), and two forms of weighted PCA. Additionally, five error structures in x and y are considered: homoscedastic equal, homoscedastic unequal, proportional equal, proportional unequal, and random. It is shown that for certain error structures, several of the methods are mathematically equivalent. Furthermore, it is demonstrated that all of the methods can be unified under the principle of maximum likelihood estimation, embodied in the general case by MWR. Extensive simulations show that MWR produces the most reliable parameter estimates in terms of bias and mean-squared error. Finally, implications for modeling in higher dimensions are considered.

Return to Maximum likelihood calibration

D.T. Andrews and P.D. Wentzell
Applications of maximum likelihood principal component analysis: incomplete data sets and calibration transfer
Analytica Chimica Acta, 350 (1997) 341-352

The application of a new method to the multivariate analysis of incomplete data is described. The new method, called maximum likelihood principal component analysis (MLPCA), is analogous to conventional principal component analysis (PCA), but incorporates measurement error variance information in the decomposition of multivariate data. Missing measurements can be handled in a reliable and simple manner by assigning large measurement uncertainties to them. The problem of missing data is pervasive in chemistry, and MLPCA is applied to three sets of experimental data to illustrate its utility. For exploratory data analysis, a data set from the analysis of archeological artifacts is used to show that the principal components extracted by MLPCA retain much of the original information even when a significant number of measurements are missing. Maximum likelihood projections of censored data can often preserve original clusters among the samples and can, through the propagation of error, indicate which samples are likely to be projected erroneously. To demonstrate its utility in modeling applications, MLPCA is also applied in the development of a model for chromatographic retention based on a data set which is only 80% complete. MLPCA can predict missing values and assign error estimates to these points. Finally, the problem of calibration transfer between instruments can be regarded as a missing data problem in which entire spectra are missing on the 'slave' instrument. Using NIR spectra obtained from two instruments, it is shown that spectra on the slave instrument can be predicted from a small subset of calibration transfer samples even if a different wavelength range is employed. Concentration prediction errors obtained by this approach were comparable to cross-validation errors obtained for the slave instrument when all spectra were available.

Return to Maximum likelihood calibration

B


K. Baumann, H. Albert and M. von Korff
A systematic evaluation of the benefits and hazards of variable selection in latent variable regression. Part I. Search algorithm, theory and simulations
Journal of Chemometrics, 16 (2002) 339-350

Variable selection is an extensively studied problem in chemometrics and in the area of quantitative structure-activity relationships (QSARs). Many search algorithms have been compared so far. Less well studied is the influence of different objective functions on the prediction quality of the selected models. This paper investigates the performance of different cross-validation techniques as objective function for variable selection in latent variable regression. The results are compared in terms of predictive ability, model size (number of variables) and model complexity (number of latent variables). It will be shown that leave-multiple-out cross-validation with a large percentage of data left out performs best. Since leave-multiple-out cross-validation is computationally expensive, a very efficient tabu search algorithm is introduced to lower the computational burden. The tabu search algorithm needs no user-defined operational parameters and optimizes the variable subset and the number of latent variables simultaneously.

Return to Validation

K. Baumann, M. von Korff and H. Albert
A systematic evaluation of the benefits and hazards of variable selection in latent variable regression. Part II. Practical applications
Journal of Chemometrics, 16 (2002) 351-360

Leave-multiple-out cross-validation (LMO-CV) is compared to leave-one-out cross-validation (LOO-CV) as objective function in variable selection for four real data sets. Two data sets stem from NIR spectroscopy and two from quantitative structure-activity relationships. In all four cases, LMO-CV outperforms LOO-CV with respect to prediction quality, model complexity (number of latent variables) and model size (number of variables). The number of objects left out in LMO-CV has an important effect on the final results. It controls both the number of latent variables in the final model and the prediction quality. The results of variable selection need to be validated carefully with a validation step that is independent of the variable selection. This step needs to be done because the internal figures of merit (i.e. anything that is derived from the objective function value) do not correlate well with the external predictivity of the selected models. This is most obvious for LOO-CV. LOO-CV without further constraints always shows the best internal figures of merit and the worst prediction quality.

Return to Validation

K. Baumann
Cross-validation as the objective function for variable-selection techniques
Trends in Analytical Chemistry, 22 (2003) 395-406

Different methods of cross-validation are studied for their suitability to guide variable-selection algorithms to yield highly predictive models. It is shown that the commonly applied leave-one-out cross-validation has a strong tendency to overfitting, underestimates the true prediction error, and should not be used without further constraints or further validation. Alternatives to leave-one-out cross-validation and other validation methods are presented.

Return to Validation

K. Baumann and N. Stiefl
Validation tools for variable subset regression
Journal of Computer-Aided Molecular Design, 18 (2004) 549-562

Variable selection is applied frequently in QSAR research. Since the selection process influences the characteristics of the finally chosen model, thorough validation of the selection technique is very important. Here, a validation protocol is presented briefly and two of the tools which are part of this protocol are introduced in more detail. The first tool, which is based on permutation testing, allows to assess the inflation of internal figures of merit (such as the cross-validated prediction error). The other tool, based on noise addition, can be used to determine the complexity and with it the stability of models generated by variable selection. The obtained statistical information is important in deciding whether or not to trust the predictive abilities of a specific model. The graphical output of the validation tools is easily accessible and provides a reliable impression of model performance. Among others, the tools were employed to study the influence of leave-one-out and leave-multiple-out cross-validation on model characteristics. Here, it was confirmed that leave-multiple-out cross-validation yields more stable models. To study the performance of the entire validation protocol, it was applied to eight different QSAR data sets with default settings. In all cases internal and external model performance was good, indicating that the protocol serves its purpose quite well.

Return to Validation

G. Bergmann, B. von Oepen and P. Zinn
Improvement in the definitions of sensitivity and selectivity
Analytical Chemistry, 59 (1987) 2522-2526

New definitions of sensitivity and selectivity of a multicomponent analysis are presented. An algorithm is developed that relates analytical precision to the noise of the measured spectra, provided that the corresponding sensitivities and selectivities are known. These expressions are derived from a system of linear equations by using the Gaussian rule of error propagation. The algorithm allows for prediction of the standard deviation of concentrations in multicomponent analysis. The new definitions are confirmed by simulated spectra and by IR spectrometric investigations.

Return to Analytical figures of merit

H.F.M. Boelens, W.Th. Kok, O.E. de Noord and A.K. Smilde
Performance optimization of spectroscopic process analyzers
Analytical Chemistry, 76 (2004) 2656-2663

To increase the power and the robustness of spectroscopic process analyzers, methods are needed that suppress the spectral variation that is not related to the property of interest in the process stream. An approach for the selection of a suitable method is presented. The approach uses the net analyte signal (NAS) to analyze the situation and to select methods to suppress the nonrelevant spectral variation. The empirically determined signal-to-noise of the NAS is used as a figure of merit. The advantages of the approach are (i) that the error of the reference method does not affect method selection and (ii) that only a few spectral measurements are needed. A diagnostic plot is proposed that guides the user in the evaluation of the particular suppression method. As an example, NIR spectroscopic monitoring of a mol-sieve separation process is used.

Return to Analytical figures of merit

R. Boqué, M.S. Larrechi and F.X. Rius
Multivariate detection limits with fixed probabilities of error
Chemometrics and Intelligent Laboratory Systems, 45 (1999) 397-408

In this paper, a new approach to calculate multivariate detection limits (MDL) for the commonly used inverse calibration model is discussed. The derived estimator follows the latest recommendations of the International Union of Pure and Applied Chemistry (IUPAC) concerning the detection capabilities of analytical methods. Consequently, the new approach: (a) is based on the theory of hypothesis testing and takes into account the probabilities of false positive and false negative decisions, and (b) takes into account all the different sources of error, both in calibration and prediction steps, which affect the final result. The MDL is affected by the presence of other analytes in the sample to be analysed; therefore, it has a different value for each sample to be tested and so the proposed approach attempts to find whether the concentration derived from a given response can be detected or not at the fixed probabilities of error. The estimator has been validated with and applied to real samples analysed by NIR spectroscopy.

Return to Limit of detection

R. Boqué, N.M. Faber and F.X. Rius
Detection limits in classical multivariate calibration models
Analytica Chimica Acta, 423 (2000) 41-49

This work presents a new approach for calculating multivariate detection limits for the commonly used classical or direct calibration models. The derived estimator, which is in accordance with latest IUPAC recommendations, accounts for the different sources of error related to the calibration and prediction steps. Since the multivariate detection limit for a given analyte is influenced by the presence of other components in the sample, a different detection limit is calculated for each analyte and analysed sample, at the chosen significance levels a and b. The proposed methodology has been experimentally validated by determining four pesticides in water using a FIA method with diode-array detection. The results compare favourably with the ones obtained using previously proposed estimators.

Return to Limit of detection

R. Boqué, J. Ferré, N.M. Faber and F.X. Rius
Limit of detection estimator for second-order bilinear calibration
Analytica Chimica Acta, 451 (2002) 313-321

A new approach is developed for estimating the limit of detection in second-order bilinear calibration with the generalized rank annihilation method (GRAM). The proposed estimator is based on recently derived expressions for prediction variance and bias. It follows the latest IUPAC recommendations in the sense that it concisely accounts for the probabilities of committing both type I and type II errors, i.e. false positive and false negative declarations respectively. The estimator has been extensively validated with simulated data, yielding promising results.

Return to Limit of detection

J.W.B. Braga and R.J. Poppi
Figures of merit for the determination of the polymorphic purity of carbamazepine by infrared spectroscopy and multivariate calibration
Journal of Pharmaceutical Sciences, 93 (2004) 2124-2134

Polymorphism is an important property in the quality control of pharmaceutical products. In this regard, partial least squares regression and the net analytical signal were used to build and validate a multivariate calibration model using diffuse reflectance infrared spectroscopy in the region of 900-1100 cm-1 for the determination of the polymorphic purity of carbamazepine. Physical mixtures of the polymorphs were made by weight, from 80 to 100% (w/w) form III mixed with form I. Figures of merit, such as sensitivity, analytical sensitivity, selectivity, confidence limits, precision (mean, repeatability, intermediate), accuracy, and signal-to-noise ratio were calculated. Feasible results were obtained with maximum absolute error of 2% and an average error of 0.53%, indicating that the proposed methodology can be used by the pharmaceutical industry as an alternative to the X-ray diffraction (United States Pharmacopoeia method).

Return to Validation or return to Analytical figures of merit

J.W.B. Braga and R.J. Poppi
Comparison of variance sources and confidence limits in two PLSR models for determination of the polymorphic purity of carbamazepine
Chemometrics and Intelligent Laboratory Systems, 80 (2006) 50-56

This paper presents a study of the variance sources and confidence limits in two PLSR models for the determination of the polymorphic purity of Carbamazepine, using near and mid infrared spectroscopy. The variance sources estimated and compared were reference values, instrumental responses and fit of the model. The variance of instrumental responses was estimated experimentally and theoretically, and the differences were discussed. The confidence limits at three confidence levels: 95%, 90% and 50% were determined, presenting a good agreement with the expected values. The predictive ability of the models was compared, showing that both present the same overall performances with RMSEP of 0.67% for near infrared and 0.62% for mid infrared spectroscopy. It was also verified that, for both models, the main variance source remains the error of the PLSR model.

Return to Reliability of multivariate calibration

R. Bro, N.D. Sidiropoulos and A.K. Smilde
Maximum likelihood fitting using ordinary least squares algorithms
Journal of Chemometrics, 16 (2002) 387-400

In this paper a general algorithm is provided for maximum likelihood fitting of deterministic models subject to Gaussian-distributed residual variation (including any type of non-singular covariance). By deterministic models is meant models in which no distributional assumptions are valid (or applied) on the parameters. The algorithm may also more generally be used for weighted least squares (WLS) fitting in situations where either distributional assumptions are not available or other than statistical assumptions guide the choice of loss function. The algorithm to solve the associated problem is called MILES (Maximum likelihood via Iterative Least squares EStimation). It is shown that the sought parameters can be estimated using simple least squares (LS) algorithms in an iterative fashion. The algorithm is based on iterative majorization and extends earlier work for WLS fitting of models with heteroscedastic uncorrelated residual variation. The algorithm is shown to include several current algorithms as special cases. For example, maximum likelihood principal component analysis models with and without offsets can be easily fitted with MILES. The MILES algorithm is simple and can be implemented as an outer loop in any least squares algorithm, e.g. for analysis of variance, regression, response surface modeling, etc. Several examples are provided on the use of MILES.

Return to Maximum likelihood calibration

R. Bro and C.M. Andersen
Theory of net analyte signal vectors in inverse regression
Journal of Chemometrics, 17 (2003) 646-652

The net analyte signal and the net analyte signal vector are useful measures in building and optimizing multivariate calibration models. In this paper a theory for their use in inverse regression is developed. The theory of net analyte signal was originally derived from classical least squares in spectral calibration where the responses of all pure analytes and interferents are assumed to be known. However, in chemometrics, inverse calibration models such as partial least squares regression are more abundant and several tools for calculating the net analyte signal in inverse regression models have been proposed. These methods yield different results and most do not provide results that are in accordance with the chosen calibration model. In this paper a thorough development of a calibration-specific net analyte signal vector is given. This definition turns out to be almost identical to the one recently suggested by Faber (Anal. Chem. 1998; 70: 5108-5110). A required correction of the net analyte signal in situations with negative predicted responses is also discussed.

Return to Analytical figures of merit

R. Bro, Å. Rinnan and N.M. Faber
Standard error of prediction for multilinear PLS. 2. Practical implementation in fluorescence spectroscopy
Chemometrics and Intelligent Laboratory Systems, 75 (2005) 69-76

In Part 1 of this series, a new simplified expression was derived for estimating sample-specific standard error of prediction in inverse multivariate regression. The focus was on the application of this expression in multilinear partial least squares (N-PLS) regression, but its scope is more general. In this paper, the expression is applied to a fluorescence spectroscopic calibration problem where N-PLS regression is appropriate. Guidelines are given for how to cope in practice with the main assumptions underlying the proposed methodology. The sample-specific uncertainty estimates yield coverage probabilities close to the stated nominal value. Similar results were obtained for standard (i.e., linear) PLS regression and principal component regression on data rearranged to ordinary two-way matrices. The two-way results highlight the generality of the proposed expression.

Return to Reliability of multiway calibration

C.D. Brown, L. Vega-Montoto and P.D. Wentzell
Derivative preprocessing and optimal corrections for baseline drift in multivariate calibration
Applied Spectroscopy, 54 (2000) 1055-1068

The characteristics of baseline drift are discussed from the perspective of error covariance. From this standpoint, the operation of derivative filters as preprocessing tools for multivariate calibration is explored. It is shown that convolution of derivative filter coefficients with the error covariance matrices for the data tend to reduce the contributions of correlated error, thereby reducing the presence of drift noise. This theory is corroborated by examination of experimental error covariance matrices before and after derivative preprocessing. It is proposed that maximum likelihood principal components analysis (MLPCA) is an optimal method for countering the deleterious effects of drift noise when the characteristics of that noise are known, since MLPCA uses error covariance information to perform a maximum likelihood projection of the data. In simulation and experimental studies, the performance of MLPCR and derivative-preprocessed PCR are compared to that of PCR with multivariate calibration data showing significant levels of drift. MLPCR is found to perform as well as or better than derivative PCR (with the best-suited derivative filter characteristics), provided that reasonable estimates of the drift noise characteristics are available. Recommendations are given for the use of MLPCR with poor estimates of the error covariance information.

Return to Maximum likelihood calibration

C.D. Brown
Discordance between net analyte signal theory and practical multivariate calibration
Analytical Chemistry, 76 (2004) 4364-4373

Lorber's concept of net analyte signal is reviewed in the context of classical and inverse least-squares approaches to multivariate calibration. It is shown that, in the presence of device measurement error, the classical and inverse calibration procedures have radically different theoretical prediction objectives, and the assertion that the popular inverse least-squares procedures (including partial least squares, principal components regression) approximate Lorber's net analyte signal vector in the limit is disproved. Exact theoretical expressions for the prediction error bias, variance, and mean-squared error are given under general measurement error conditions, which reinforce the very discrepant behavior between these two predictive approaches, and Lorber's net analyte signal theory. Implications for multivariate figures of merit and numerous recently proposed preprocessing treatments involving orthogonal projections are also discussed.

Return to Analytical figures of merit

C.D. Brown and T.D. Ridder
Framework for multivariate selectivity analysis, Part I: Theoretical and practical merits
Applied Spectroscopy, 59 (2005) 787-803

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.

Return to Analytical figures of merit

A.J. Burnham, J.F. MacGregor and R. Viveros
A statistical framework for multivariate latent variable regression methods based on maximum likelihood
Journal of Chemometrics, 13 (1999) 49-65

A statistical framework is developed to contrast methods used for parameter estimation for a latent variable multivariate regression (LVMR) model. This model involves two sets of variables, X and Y, both with multiple variables and sharing a common latent structure with additive random errors. The methods contrasted are partial least squares (PLS) regression, principal component regression (PCR), reduced rank regression (RRR) and canonical co-ordinate regression (CCR). The framework is based on a constrained maximum likelihood analysis of the model under assumptions of multivariate normality. The constraint is that the estimates of the latent variables are restricted to be linear functions of the X variables, which is the form of the estimates for the methods being contrasted. The resulting framework is a continuum regression that goes from RRR to PCR depending on the ratio of error variances in the X and Y spaces. PLS does not arise as a member of the continuum; however, the method does offer some insight into why PLS would work well in practice. The constrained maximum likelihood result is also compared with the unconstrained maximum likelihood analysis to investigate the impact of the constraint. The results are illustrated on a simulated example.

Return to Maximum likelihood calibration

C


R.J. Carroll and D. Ruppert
The use and misuse of orthogonal regression in linear errors-in-variables models
The American Statistician, 50 (1996) 1-6

Orthogonal regression is one of the standard linear regression methods to correct for the effects of measurement error in predictors. We argue that orthogonal regression is often misused in errors-in-variables linear regression because of a failure to account for equation errors. The typical result is to overcorrect for measurement error, that is, to overestimate the slope, because equation error is ignored. The use of orthogonal regression must include a careful assessment of equation error, and not merely the usual (often informal) estimation of the ratio of measurement error variances. There are rarer instances, for example, an example from geology discussed here, where the use of orthogonal regression without proper attention to modeling may lead to either overcorrection or undercorrection, depending on the relative sizes of the variances involved. Thus our main point, which does not seem to be widely appreciated, is that orthogonal regression, just like any measurement error analysis, requires careful modeling of error.

Return to Reliability of univariate calibration

H.R. Cederkvist, A.H. Aastveit and T. Næs
A comparison of methods for testing differences in predictive ability
Journal of Chemometrics, 19 (2005) 500-509

This paper is devoted to the comparison of methods for testing difference in prediction ability of regression-based predictors. Focus is on both parametric and non-parametric methods. Comparisons are done using the bootstrap based on residuals from real data sets. The power of the methods for different alternatives is the main tool for evaluating the properties of the methods. The main conclusion from the bootstrap simulations is that the most suitable method seems to be two-way analysis of variance of the absolute values of the prediction errors.

Return to Validation

V. Centner, D.-L. Massart, O.E. De Noord, S. De Jong, B.M. Vandeginste and C. Sterna
Elimination of uninformative variables for multivariate calibration
Analytical Chemistry, 68 (1996) 3851-3858

A new method for the elimination of uninformative variables in multivariate data sets is proposed. To achieve this, artificial (noise) variables are added and a closed form of the PLS or PCR model is obtained for the data set containing the experimental and the artificial variables. The experimental variables that do not have more importance than the artificial variables, as judged from a criterion based on the b coefficients, are eliminated. The performance of the method is evaluated on simulated data. Practical aspects are discussed on experimentally obtained near-IR data sets. It is concluded that the elimination of uninformative variables can improve predictive ability.

Return to Reliability of multivariate calibration

A.K. Conlin, E.B. Martin and A.J. Morris
Confidence limits for contribution plots
Journal of Chemometrics, 14 (2000) 725-736

An issue often raised in multivariate statistical process control, when using statistical projection-based techniques to define nominal process behaviour, is that of the assured identification of the variables causing an out-of-statistical-control signal. One approach which has been adopted is that once a change in process operating conditions has been detected, the contribution of the individual variables to the principal component scores or squared prediction error, the Q-statistic, are examined. Adopting this approach, it is important that those variables responsible for, or contributing to, the process change are clearly identifiable. In process modelling and estimation studies, confidence bounds are typically placed around the model predictions. Currently confidence bounds are not used to identify the limits of normal behaviour for the individual multivariate statistical contributions, resulting in the interpretation of the contribution plot being left to the user. This paper presents a potential solution to the definition of confidence bounds for contribution plots. The methodology is based on bootstrap estimates of the standard deviations of the loading matrix. The proposed approach is evaluated using data from a benchmark simulation of a continuous stirred tank reaction system. The preliminary results are encouraging.

Return to Reliability of principal component analysis

L.A. Currie
Nomenclature in evaluation of analytical methods including detection and quantification capabilities
Pure & Applied Chemistry, 67 (1995) 1699-1723

The IUPAC nomenclature document has been prepared to help establish a uniform and meaningful approach to terminology, notation, and formulation for performance characteristics of the Chemical Measurement Process (CMP). Following definition of the CMP and its Performance Characteristics, the document addresses fundamental quantities related to the observed response and calibration, and the complement to the calibration function: the evaluation function. Performance characteristics related to precision and accuracy comprise the heart of the document. These include measures for the means or "expected values" of the relevant chemical quantities, as well as dispersion measures such as variance and standard error. Attention is given also to important issues involving: assumptions, internal and external quality control, estimation, error propagation and uncertainty components, and bounds for systematic error. Special treatment is given to terminology and concepts underlying detection and quantification capabilities in chemical metrology, and the significance of the blank. The document concludes with a note on the important distinction between the Sampled Population and the Target Population, especially in relation to the interlaboratory environment.

Return to Limit of detection

L.A. Currie
Detection: International update, and some emerging di-lemmas involving calibration, the blank, and multiple detection decisions
Chemometrics and Intelligent Laboratory Systems, 37 (1997) 151-181

Detection and quantification capabilities represent fundamental performance characteristics of measurement processes, yet there have been decades of confusion and miscommunication regarding the underlying concepts and terminology. New, co-ordinated documents prepared by the International organization for standardization (ISO) [1] and the International union of pure and applied chemistry (IUPAC) [2] promise to alleviate this situation by providing, for the first time, a harmonized position of standards and recommendations for adoption by the international scientific community. The first section of this paper contains a brief introduction to the events leading to the ISO and IUPAC efforts. Section 2 consists of (1) a brief review of the history of 'detection limits' in chemistry, illustrating the critical need for the development of a sound and uniform system of terms and symbols; and (2) a review of the ISO-IUPAC deliberations and the ensuing harmonized position on concepts and nomenclature. Section 3 treats fundamental applications of the underlying concepts, together with a series of unresolved or 'open' questions involving: detection and quantification capabilities in the signal and concentration domains, respectively; and the link between calibration and detection and quantification limits, together with the blank-intercept dichotomy. Also included are special treatments and approximations, developed in part for the IUPAC document, involving the non-central-t, and the exact (non-normal) distribution of the estimated concentration. The final section (Section 4) introduces issues and approaches to multiple independent and multivariate detection decisions and limits, and concludes with a glimpse at some challenges involving the multivariate blank and non-monotonic calibration functions.

Return to Limit of detection

D


K. Danzer and L.A. Currie
Guidelines for calibration in analytical chemistry
Part 1. Fundamentals and single component calibration
Pure & Applied Chemistry, 70 (1998) 993-1014

This IUPAC nomenclature document has been prepared to establish a uniform and meaningful approach to terminology, notation, and formulation for calibration in analytical chemistry. In this first part, general fundamentals of calibration are presented, namely for both relationships of qualitative and quantitative variables (relations between variables characterizing certain types of analytes and measured signals in certain positions of a measured function on the one hand and between variables characterizing the amount or concentration of the chemical species and the intensities of the measured signals, on the other hand). On this basis, the fundamentals of the common single component calibration which models the relationship y = f(x) between the signal intensities y and the amounts or concentrations x of the analyte under given conditions are represented. Additional papers will be prepared dealing with extensive relationships between several signal intensities and analyte contents, namely with multivariate calibration and with optimization and experimental design.

Return to Official literature

K. Danzer, M. Otto and L.A. Currie
Guidelines for calibration in analytical chemistry
Part 2. Multispecies calibration
Pure & Applied Chemistry, 76 (2004) 1215-1225

Calibration in analytical chemistry refers to the relation between sample domain and measurement domain (signal domain) expressed by an analytical function x = fs(Q) representing a pattern of chemical species Q and their amounts or concentrations x in a given test sample on the one hand and a measured function y = f(z) that may be a spectrum, chromatogram, etc.
    Simultaneous multispecies analyses are carried out mainly by spectroscopic and chromatographic methods in a more or less selective way. For the determination of n species Qi (i = 1, 2...n), at least n signals must be measured which should be well separated in the ideal case. In analytical practice, the situation can be different.

Return to Official literature

M. de Castro, M. Galea-Rojas, H. Bolfarine and M.V. de Castilho
Detection of analytical bias when comparing two or more measuring methods
Journal of Chemometrics, 18 (2004) 431-440

The main goal of this paper is to consider maximum likelihood inference for models used in the detection of analytical bias in the comparison of two or more methods of measurement. We embrace a functional errors-in-variables regression model with an EM-type algorithm for computing maximum likelihood estimates and to obtain consistent estimators for the asymptotic variance of the maximum likelihood estimators, which seems not to be found in the literature. Wald-type statistics are proposed for testing hypotheses related to the bias of the analytical methods with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Some approaches specific for the two-methods comparison problem are not directly extendable to this more general situation. Results of simulation studies and an application to a real data set are also reported.

Return to Reliability of univariate calibration

M. de Castro, M. Galea-Rojas, H. Bolfarine and M.V. de Castilho
Local influence in regression models for the detection of analytical bias
Chemometrics and Intelligent Laboratory Systems, 83 (2006) 139-147

This work deals with errors-in-variables regression models applied in comparing two analytical methods. The chief aim of the paper is the assessment of effects of minor perturbations of data on test statistics. To accomplish this we resort to methods of local influence. The techniques provide to the analyst a valuable tool that enables to identify potential influential elements and to quantify the effects of perturbations on test statistics. An illustrative example with a real data set is also reported.

Return to Reliability of univariate calibration

F.J. del Río, J. Riu and F.X. Rius
Prediction intervals in linear regression taking into account errors on both axes
Journal of Chemometrics, 15 (2001) 773-788

This study reports the expressions for the variances in the prediction of the response and predictor variables calculated with the bivariate least squares (BLS) regression technique. This technique takes into account the errors on both axes. Our results are compared with those of a simulation process based on six different real data sets. The mean error in the results from the new expressions is between 4% and 5%. With weighted least squares, ordinary least squares, the constant variance ratio approach and orthogonal regression, on the other hand, mean errors can be as high as 85%, 277%, 637% and 1697% respectively. An important property of the prediction intervals calculated with BLS is that the results are not affected when the axes are switched.

Return to Reliability of univariate calibration

F.J. del Río Bocio, J. Riu, R. Boqué and F.X. Rius
Limits of detection in linear regression with errors in the concentration
Journal of Chemometrics, 17 (2003) 413-421

This paper discusses a method for calculating the limit of detection in analytical methods in which the calibration stage takes into account the errors in both concentrations and instrumental responses. The proposed method considers the heteroscedastic individual errors on both axes, i.e. it uses the bivariate least squares (BLS) regression method. The expressions were applied to X-ray fluorescence (XRF) and capillary electrophoresis (CE) determinations to calculate the limits of detection of nine elements in solid rocks and three anions in water respectively. The limits of detection with these expressions have been compared with those obtained with ordinary least squares (OLS) and weighted least squares (WLS). The results show that the limits of detection when the BLS procedure is used are smaller than when the other techniques are used.

Return to Reliability of univariate calibration

M.C. Denham
Prediction intervals in partial least squares
Journal of Chemometrics, 11 (1997) 39-52

Partial least squares (PLS) regression has become a popular technique within the chemometric community, particularly for dealing with calibration problems. An important aspect of calibration is the implicit requirement to predict values for future samples. The PLS predictor is non-linear with a presently unknown statistical distribution. We consider approaches for providing prediction intervals rather than point predictions based on sample reuse strategies, and, by application of an algorithm for calculating the first derivative of the PLS predictor, local linear approximation. We compare these approaches, together with a naïve approach which ignores the non-linearities induced by the PLS estimation method, using a simulated example.

Return to Reliability of multivariate calibration

M.C. Denham
Choosing the number of factors in partial least squares regression: estimating and minimizing the mean squared error of prediction
Journal of Chemometrics, 14 (2000) 351-361

We investigate a number of approaches to estimating the mean squared error of prediction (MSEP) in partial least squares (PLS) regression without resorting to external validation. Using two simulation examples based on real data, performances of the methods are evaluated in terms of their accuracy and their usefulness in determining the optimal number of factors to include in the PLS model. We find that for problems with relatively few variables, based on ignoring the effect of non-linearity in PLS regression or using a linear approximation give good estimates of MSEP, with little to choose between them. However, where linear approximation is feasible, we prefer it, since it gives estimates of MSEP which have lower bias and variance than cross-validation. In situations where there are large numbers of variables, these methods break down. In these circumstances, cross-validation and bootstrapping methods are better able to capture the changes in MSEP with the number of factors fitted and thus are more useful for identifying the optimal PLS regression model.

Return to Reliability of multivariate calibration or return to Component selection

R. DiFoggio
Examination of some misconceptions about near-infrared analysis
Applied Spectroscopy, 49 (1995) 67-75

There are some recurrent misconceptions about near-infrared (near-IR or NIR) analysis and similar chemometric techniques that have resurfaced as this technology has become more popular and widely used. These misconceptions are that (1) near-IR can never perform better than the primary reference technique from which it was calibrated, (2) near-IR models that are linear in absorbance cannot account for nonlinear effects, and (3) bias and skew adjustments are purely mathematical manipulations without physical assumptions. This paper examines how these misconceptions arise and, using real and synthetic spectra and simulated noise, presents counter examples to them. This paper also shows how one can approximate a change of scale using bias and skew adjustments to the model and lists some conditions on a model's coefficient weights that make a model insensitive to small changes in spectrometer response.

Return to General on calibration or return to Validation

R. DiFoggio
Guidelines for applying chemometrics to spectra: feasibility and error propagation
Applied Spectroscopy, 54 (2000) 94A-113A

This article concentrates on the two most basic and intertwined concepts needed for chemometric modeling of spectral data - application feasibility and error propagation. Among the topics discussed are how application feasibility depends on user requirements, how spectral noise and artifacts propagate through a model, and how lab error affects model coefficients. The approach is axiomatic, starting out with a few obvious assumptions that eventually lead to some surprising conclusions. For example, for many liquid samples, we can gauge, at a glance, the temperature sensitivity of a model from the closeness of that model's offset constant to the calibration set's average property value. For easy reference, the most important assumptions and conclusions are listed as a series of 30 different "Points" that are highlighted in the main text and that are also collected together, in condensed form, into a single table.

Return to General on calibration or return to Limit of detection

R. DiFoggio
Desensitizing models using covariance matrix transforms or counter-balanced distortions
Journal of Chemometrics, 19(2005) 203-215

This paper presents a generalization of the Lagrange multiplier equation for a regression subject to constraints. It introduces two methods for desensitizing models to anticipated spectral artifacts such as baseline variations, wavelength shift, or trace contaminants. For models derived from a covariance matrix such as multiple linear regression (MLR) and principal components regression (PCR) models, the first method shows how a covariance matrix can be desensitized to an artifact spectrum, v, by adding s2 v Kronecker.gif v to it. For models not derived from a covariance matrix, such as partial least squares (PLS) or neural network (NN) models, the second method shows how distorted copies of the original spectra can be prepared in a counter-balanced manner to achieve desensitization. Unlike earlier methods that added random distortions to spectra, these new methods never introduce any accidental correlations between the added distortions and the Y-block. The degree of desensitization is controlled by a parameter, s, for each artifact from zero (no desensitization) to infinity (complete desensitization, which is the Lagrange multiplier limit). Unlike Lagrange multipliers, these methods permit partial desensitization so we can individually vary the degree of desensitization to each artifact, which is important when desensitization to one artifact inhibits desensitization to another.

Return to General on calibration

D.L. Duewer, B.R. Kowalski and J.L. Fasching
Improving the reliability of factor analysis of chemical data by utilizing the measured analytical uncertainty
Analytical Chemistry, 48 (1976) 2002-2010

A procedure for including measured analytical uncertainty into data analysis methodology is discussed, with partlcular reference to factor analysis. The suitability of various dispersion matrices and matrix rank determlnatlon criteria for data having analytical uncertalnty Is Investigated. A crlterion useful for judging the number of factors insensitive to analytlcal uncertainty is presented. A model data structure for investigating the behavlor of factor analysis techniques In a known, controlled manner is described and analyzed. A chemically interestlng test data base having analytical uncertainty Is analyzed and compared wlth the model data. The limit to meaningful factor analysis for a portlon of the API Project 44 low resolution mass spectral data is explored.

Return to Reliability of principal component analysis

E


F


N.M. Faber, L.M.C. Buydens and G. Kateman
Standard errors in the eigenvalues of a cross-product matrix: theory and applications
Journal of Chemometrics, 7 (1993) 495-526

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by the method of error propagation. Cross-product matrices frequently arise in multivariate data analysis, especially in principal component analysis (PCA). The derived standard errors account for the variability in the data as a result of measurement noise and are therefore essentially different from the standard errors developed in multivariate statistics. Those standard errors were derived in order to account for the finite number of observations on a fixed number of variables, the so-called sampling error. They can be used for making inferences about the population eigenvalues. Making inferences about the population eigenvalues is often not the purpose of PCA in physical sciences. This is particularly true if the measurements are performed on an analytical instrument that produces two-dimensional arrays for one chemical sample: the rows and columns of such a data matrix cannot be identified with observations on variables at all. However, PCA can still be used as a general data reduction technique, but now one must consider the effect of measurement noise on the standard errors in the eigenvalues. The consequences for significance testing of the eigenvalues, as well as the usefulness for error estimates for scores and loadings of PCA, multiple linear regression (MLR) and the generalized rank annihilation method (GRAM) are discussed. Monte Carlo simulations are deployed for testing the adequacy of the derived expressions.

Return to Reliability of principal component analysis

N.M. Faber, M.J. Meinders, P. Geladi, M. Sjöström, L.M.C. Buydens and G. Kateman
Random error bias in principal component analysis. I: Derivation of theoretical predictions
Analytica Chimica Acta, 304 (1995) 257-271

Principal component analysis (PCA) or singular value decomposition (SVD) are multivariate techniques that are often used to compress large data matrices to a relevant size. Subsequent data analysis then proceeds with the model representation of the data. In this first paper expressions are derived for the prediction of the bias in the eigenvalues of PCA and singular values of SVD that results from random measurement errors in the data. Theoretical expressions for the prediction of this `random error bias' have been given in the statistics literature. These results are, however, restricted to the case that only one principal component (PC) is significant. The first objective of this paper is to extend these results to an arbitrary number of significant PCs. For the generalization Malinowski's error functions are used. A signal-to-noise ratio is defined that describes the error situation for each individual PC. This definition enhances the interpretability of the derived expressions. The adequacy of the derived expressions is tested by a limited Monte Carlo study. This finally leads to the second objective of this paper. Simulation results are always restricted to the class of data that is well represented in the study. Thus rather than giving extensive simulation results it is outlined how the validation and evaluation of theoretical predictions can proceed for a specific application in practice.

Return to Reliability of principal component analysis

N.M. Faber, M.J. Meinders, P. Geladi, M. Sjöström, L.M.C. Buydens and G. Kateman
Random error bias in principal component analysis. II: Application of theoretical predictions to multivariate problems
Analytica Chimica Acta, 304 (1995) 273-283

In the first part of this paper expressions were derived for the prediction of random error bias in the eigenvalues of principal component analysis (PCA) and the singular values of singular value decomposition (SVD). The main issues of Part I were to investigate the question whether adequate prediction of this bias is possible and to discuss how the validation and evaluation of these predictions could proceed for a specific application in practice. The main issue of this second part is to investigate how random error bias should be taken into account. This question will be treated for a number of seemingly disparate multivariate problems. For example, the construction of confidence intervals for the bias-corrected quantities will be discussed with respect to the estimation of the number of significant principal components. The consequences of random error bias for calibration and prediction with ordinary least squares (OLS), principal component regression (PCR), partial least squares (PLS) and the generalized rank annihilation method (GRAM) will also be outlined. Finally, the derived bias expressions will be compared in detail with the corresponding results for OLS and GRAM.

Return to Reliability of principal component analysis

N.M. Faber and B.R. Kowalski
Prediction error in least squares regression: further critique on the deviation used in The Unscrambler
Chemometrics and Intelligent Laboratory Systems, 34 (1996) 283-292

In a recent critique on the deviation used in the Unscrambler a correction factor was proposed that gave improved results. However, this investigation was carried out under the standard assumption of the classical regression model, i.e. the measurement errors in the response and predictor variables were neglected. In many chemical applications this assumption is too resrictive. Moreover, even in absence of measurement errors, some problems remain and, consequently, further critique on the deviation used in the Unscrambler is in order. This paper recommends an extension of a recently derived expression for prediction variance that includes all measurement errors and is valid for multivariate calibration using ordinary least squares, principal component regression and partial least squares. The theory is illustrated on practical examples taken from the literature.

Return to Reliability of multivariate calibration

N.M. Faber, A. Lorber and B.R. Kowalski
Generalized rank annihilation method: standard errors in the estimated eigenvalues if the instrumental errors are heteroscedastic and correlated
Journal of Chemometrics, 11 (1997) 95-109

The generalized rank annihilation method (GRAM) is a method for curve resolution and calibration that uses two data matrices simultaneously, i.e. one for the unknown and one for the calibration sample. The method is known to become an eigenvalue problem for which the eigenvalues are the ratios of the concentrations for the samples under scrutiny. Previously derived standard errors in the estimated eigenvalues of GRAM have very recently been shown to be based on unrealistic assumptions about the measurement errors. In this paper a systematic notation is introduced that enables the propagation of errors that are based on realistic assumptions concerning the data generating process. The error propagation will be performed in detail for the case that one data order modulates the second one. Extensions to more complicated error models are indicated.

Return to Reliability of multiway calibration

N.M. Faber and B.R. Kowalski
Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares
Journal of Chemometrics, 11 (1997) 181-238

Multivariate calibration aims to model the relation between a dependent variable, e.g. analyte concentration, and the measured independent variables, e.g. spectra, for complex mixtures. The model parameters are obtained in the form of a regression vector from calibration data by regression methods like principal component regression (PCR) or partial least squares (PLS). Subsequently, this regression vector is used to predict the dependent variable for unknown mixtures. The validation of the obtained predictions is a crucial part of the procedure, i.e. together with the point estimate an interval estimate is desired. The associated prediction intervals can be constructed from the covariance matrix of the estimated regression vector. However, currently known expressions for PCR and PLS are derived within the classical regression framework, i.e. they only take the uncertainty in the dependent variable into account. This severely limits their capability for establishing realistic prediction intervals in practical situations. In this paper expressions are derived using the method of error propagation that also account for the measurement errors in the independent variables. An exact linear relation is assumed between the dependent and independent variables. The obtained expressions are therefore valid for the classical errors-in-variables (EIV) model. In order to make the presentation reasonably self-contained relevant expressions are reviewed for the classical regression model as well as the classical EIV model, especially for ordinary least squares (OLS). The consequences for the limit of detection, wavelength selection, sample selection and local modeling are discussed. Diagnostics are proposed to determine the adequacy of the approximations used in the derivations. Finally, PCR and PLS are so-called biased regression methods. Compared with OLS they yield small variance at the expense of an increased bias. It follows that bias may be an important ingredient of the obtained predictions. Therefore considerable attention is paid to the quantification of bias and new stopping rules for model selection in PCR and PLS are proposed. The theoretical ideas are illustrated by the analysis of real data taken from the literature (classical regression model) as well as simulated data (classical EIV model).

Return to Validation or return to Reliability of multivariate calibration

N.M. Faber and B.R. Kowalski
Improved prediction error estimates for multivariate calibration by correcting for the measurement error in the reference values
Applied Spectroscopy, 51 (1997) 660-665

The validation of multivariate calibration models using measured reference values leads to a so-called apparent prediction error estimate which is systematically larger than the true prediction error. The reason for this difference is clear: the measured reference values contain an irrelevant random component, the measurement error, which can not be predicted by any model, not even the 'true' one. However, the contribution of the measurement error in the reference values to the apparent prediction error estimate is interpreted as an inadequacy of the calibration model rather than an inadequacy of the reference values themselves. This phenomenon of confounding has been pointed out recently by several researchers but no generally applicable solution was given. In this paper we propose a simple correction procedure that yields a more realistic estimate of the true prediction error. A large potential improvement over the conventional estimate is demonstrated for a variety of applications taken from the literature.

Return to Validation

N.M. Faber and B.R. Kowalski
Improved estimation of the limit of detection in multivariate calibration
Fresenius' Journal of Analytical Chemistry, 357 (1997) 789-795

The limit of detection is one of the most important performance characteristics of an analytical procedure. This paper critically examines a recently introduced expression for estimating a multivariate limit of detection. The main result is that the proposed expression is inconsistent. An application from inductively coupled plasma - optical emission spectrometry (ICP-OES) shows that the true limit of detection may be overestimated by more than 20%. In addition, the practical evaluation of the proposed expression amounts to an iterative procedure, which is unattractive from an application point of view. A modification is derived that solves both problems. The relevant expressions are discussed with respect to their interpretability in terms of experimental design.

Return to Limit of detection

N.M. Faber, A. Lorber and B.R. Kowalski
Analytical figures of merit for tensorial calibration
Journal of Chemometrics, 11 (1997) 419-461

The subject of analytical figures of merit for tensorial calibration is critically reviewed. Tensorial calibration derives it name from tensor algebra, which provides a classification of calibration methods depending on the complexity of the data that is obtained for one chemical sample. Expressions for net analyte signal, sensitivity (classical model formulation), 'inverse sensitivity' (inverse model formulation), selectivity, signal-to-noise ratio and limit of detection (in signal space) are proposed for Nth-order data (N ³ 2) that are consistent with the accepted zeroth-order definitions and previously proposed definitions for first-order data. Useful relationships between the proposed figures of merit and prediction error variance are described. A selectivity-based rule of thumb is derived to compare data analysis across orders. Central to the currently proposed framework for analytical figures of merit is the reduction of a complex data structure to the scalar net analyte signal. This allows for the construction of a univariate calibration graph (classical or inverse model), independent of the complexity of the data. Enhanced visualization and interpretation is obtained that may help to bridge the gap between Nth-order calibration and the intuitive understanding of zeroth-order data.

Return to Analytical figures of merit

N.M. Faber, D.L. Duewer, S.J. Choquette, T.L. Green and S.N. Chesler
Characterizing the uncertainty in near-infrared spectroscopic prediction of mixed-oxygenate concentrations in gasoline: sample-specific prediction intervals
Analytical Chemistry, 70 (1998) 2972-2982

Oxygenates are added to gasoline to reduce exhaust emission levels of carbon monoxide and to boost octane. The U.S. National Institute of Standards and Technology (NIST) provides eight Standard Reference Materials (SRMs) for single-oxygenates in reference gasoline. A previous study demonstrated the feasibility of non-destructively quantifying oxygenate concentration in SRM gasoline ampules using near-infrared (NIR) spectroscopy combined with multivariate calibration techniques. A drawback of this approach has been that an average prediction uncertainty, rather than a sample-specific one, is obtained. Recent developments in multivariate calibration theory for prediction error variance cure this problem. This report: (1) characterizes the significant sources of uncertainties in multivariate calibration using principal component regression and partial least-squares and (2) interprets prediction results in terms of multivariate analytical figures of merit.

Return to Reliability of multivariate calibration

N.M. Faber
Estimating the uncertainty in estimates of root mean squared error of prediction: application to determining the size of an adequate test set in multivariate calibration
Chemometrics and Intelligent Laboratory Systems, 49 (1999) 79-89

Root mean square error of prediction (RMSEP) is widely used as a criterion for judging the performance of a multivariate calibration model; often it is even the sole criterion. Two methods are discussed for estimating the uncertainty in estimates of RMSEP. One method follows from the approximate sampling distribution of mean square error of prediction (MSEP) while the other one is based on performing error propagation, which is a distribution-free approach. The results from a small Monte Carlo simulation study suggest that, provided that extreme outliers are removed from the test set, MSEP estimates are approximately proportional to a c2 random variable with n degrees of freedom, where n is the number of samples in the test set. It is detailed how this knowledge can be used to determine the size of an adequate test set. The advantages over the guideline issued by the American Society for Testing and Materials (ASTM) are discussed. The expression derived by the method of error propagation is shown to systematically overestimate the true uncertainty. A correction factor is introduced to ensure approximate correct behaviour. A close agreement is found between the uncertainties calculated using the two complementary methods. The consequences of using a too small test set are illustrated on a practical data set.

Return to Validation

N.M. Faber
A closer look at the bias-variance tradeoff in multivariate calibration
Journal of Chemometrics, 13 (1999) 185-192

Principal component regression (PCR) and partial least squares (PLS) avoid the high variance often associated with the ordinary least squares (OLS) results by allowing a small bias in the model. This paper presents a closer look at this bias-variance tradeoff by discussing three practical aspects: (1) variance increases relatively slowly with increasing model complexity, (2) bias may be zero for the optimum model, and (3) variance does not necessarily increase with increasing model complexity. While the first aspect is well known, the last two aspects are not. The second aspect implies that so-called biased regression methods do not necessarily yield biased predictions while the third aspect, which is only encountered with nonlinear estimation methods such as PLS, even contradicts the concept of bias-variance tradeoff. The possibility of having both variance and bias decreasing with increasing PLS model complexity is illustrated using a near-infrared data set published by Fearn.

Return to Reliability of multivariate calibration

N.M. Faber
Improved computation of the standard error in the regression coefficient estimates of a multivariate calibration model
Analytical Chemistry, 72 (2000) 4675-4676

A multivariate calibration model consists of regression coefficient estimates whose significance depends on the associated standard errors. A recently introduced leave-one-out (LOO) method for computing these standard errors is modified to achieve consistency with the jackknife method. The proposed modification amounts to multiplying the LOO standard errors with the factor (n-1)/n½, where n denotes the number of calibration samples. The potential improvement for realistic values of n is illustrated using a practical example.

Return to Reliability of multivariate calibration

N.M. Faber
Comparison of two recently proposed expressions for partial least squares regression prediction error
Chemometrics and Intelligent Laboratory Systems, 52 (2000) 123-134

Two recently proposed expressions for partial least squares (PLS) prediction error are compared. Using extensive Monte Carlo simulations, it is found that the expression based on the so-called errors-in-variables approach yields prediction intervals with coverage probabilities close to their nominal value, whereas the expression, which is implemented in the latest version of Unscrambler (7.0), is found to behave unsatisfactorily. The difference between the two approaches is illustrated on a real near-infrared data set taken from the literature.

Return to Reliability of multivariate calibration

N.M. Faber
Quantifying the effect of measurement errors on the uncertainty in bilinear model predictions: a small simulation study
Analytica Chimica Acta, 439 (2001) 193-201

Four methods are compared for quantifying the effect of measurement errors on the uncertainty in bilinear model predictions. These methods amount to (1) evaluating an approximate expression for prediction variance, (2) bootstrapping residuals left after fitting the data matrices using a singular value decomposition, (3) adding noise from an appropriate distribution to the original data, and (4) jack-knifing rows and columns of the data matrices. The comparison is carried out for liquid chromatography / ultraviolet data obtained from Malinowski and the models are constructed using the generalized rank annihilation method. It is found that the first three methods give highly consistent results whereas the jack-knife yields uncertainty estimates that have no clear interpretation.

Return to Reliability of multiway calibration

N.M. Faber, J. Ferré and R. Boqué
Iteratively reweighted generalized rank annihilation method. 1. Improved handling of prediction bias
Chemometrics and Intelligent Laboratory Systems, 55 (2001) 67-90

The generalized rank annihilation method (GRAM) is a method for curve resolution and calibration that uses two bilinear data matrices simultaneously, i.e., one for the unknown and one for the calibration sample. A GRAM calculation amounts to solving an eigenvalue problem for which the eigenvalues are related to the predicted analyte concentrations. Previous studies have shown that random measurement errors bring about a bias in the eigenvalues, which directly translates into prediction bias. In this paper, accurate formulas are derived that enable removing most of this bias. Two bias correction methods are investigated. While the first method directly subtracts bias from the eigenvalues obtained by the original GRAM, the second method first applies a weight to the data matrices to reduce bias. These weights are specific for the analyte of interest and must be determined iteratively from the data. Consequently, the proposed modification is called iteratively reweighted GRAM (IRGRAM). The results of Monte Carlo simulations show that both methods are effective in the sense that the standard error in the bias-corrected prediction compares favourably with the root mean squared error that accompanies the original quantity. However, IRGRAM is found to perform best because the increase of variance caused by subtracting bias is minimised. In the original formulation of GRAM only a single calibration sample is exploited. The error analysis is extended to cope with multiple calibration samples.

Return to Reliability of multiway calibration

N.M. Faber, R. Boqué and J. Ferré
Iteratively reweighted generalized rank annihilation method. 2. Least squares property and variance expressions
Chemometrics and Intelligent Laboratory Systems, 55 (2001) 91-100

The generalized rank annihilation method (GRAM) has been criticised for not having a global least squares fitting property such as the alternating least squares (ALS) method. In Part 1 of this series, we have modified GRAM by introducing a weight for the data matrices. The proposed modification is called iteratively reweighted GRAM (IRGRAM). Here, it is shown that these weights enable one to shed new light on the least squares fitting properties of GRAM and ALS. Inequalities are derived which suggest that IRGRAM compares favourably with ALS in terms of model fit to the data matrices. Although applying different weights directly affects the sums of squares explained by IRGRAM and ALS, error propagation shows that the first-order approximation to prediction variance remains unaltered when using IRGRAM. In contrast, the effect on the variance in the estimated profiles depends on the analyte under consideration. This result suggests that the amount of fitted data does not give a clear indication of the performance of bilinear calibration models.

Return to Reliability of multiway calibration

N.M. Faber and R. Bro
Standard error of prediction for multiway PLS. 1. Background and a simulation study
Chemometrics and Intelligent Laboratory Systems, 61 (2002) 133-149

While a multitude of expressions has been proposed for calculating sample-specific standard errors of prediction when using partial least squares regression (PLS) for the calibration of first-order data, potential generalisations to multiway data are lacking to date. We have examined the adequacy of two approximate expressions when using unfold- or tri-PLS for the calibration of second-order data. The first expression is derived under the assumption that the errors in the predictor variables are homoscedastic, i.e., of constant variance. In contrast, the second expression is designed to also work in the heteroscedastic case. The adequacy of the approximations is tested using extensive Monte Carlo simulations while the practical utility is demonstrated in Part 2 of this series.

Return to Reliability of multiway calibration

N.M. Faber, J. Ferré, R. Boqué and J.H. Kalivas
Second-order bilinear calibration: the effect of vectorising the data matrices of the calibration set
Chemometrics and Intelligent Laboratory Systems, 63 (2002) 107-116

In a ground-breaking paper, Linder and Sundberg developed a statistical framework for the calibration of second-order bilinear data (Chemom. Intell. Lab. Syst. 42 (1998) 159). Within this framework, they formulated three different predictor construction methods (J. Chemom. 16 (2002) 12), namely the so-called naïve method, the bilinear least squares (BLLS) method, and a refined version of the latter that takes account of the calibration uncertainty. Elsewhere (N.M. Faber, J. Chemom. 15 (2001) 743) a close relationship is established between the naïve method and the generalized rank annihilation method (GRAM) by comparing expressions for prediction variance. Here it is proved that the BLLS method can be interpreted to work with vectorised data matrices, which establishes an algebraic relationship with so-called unfold partial least squares (PLS) and unfold principal component regression (PCR). It is detailed how these results enable quantifying the effects of vectorising bilinear second-order data matrices on analytical figures of merit and variance inflation factors.

Return to Analytical figures of merit

N.M. Faber
Towards a rehabilitation of the generalized rank annihilation method (GRAM)
Analytical and Bioanalytical Chemistry, 372 (2002) 683-687

The trilinear PARAFAC model occupies a central place in multiway analysis, because the components of a data array can often be uniquely resolved. This paper compares the resolution for a large variety of methods, namely the generalized rank annihilation method (GRAM), alternating least squares (ALS), alternating trilinear decomposition (ATLD), alternating coupled vectors resolution (ACOVER), alternating slice-wise diagonalization (ASD), alternating coupled matrices resolution (ACOMAR), self-weighted alternating trilinear decomposition (SWATLD) and pseudo alternating least squares (PALS). The comparison is carried out using Monte Carlo simulations. It is shown that GRAM performs well for moderately as well as highly overlapped data. The current results strongly contrast the previously claimed superiority of the alternatives listed above.

Return to Method comparison studies

N.M. Faber
Uncertainty estimation for multivariate regression coefficients
Chemometrics and Intelligent Laboratory Systems, 64 (2002) 169-179

Five methods are compared for assessing the uncertainty in multivariate regression coefficients, namely an approximate variance expression and four resampling methods (jack-knife, bootstrapping objects, bootstrapping residuals and noise addition). The comparison is carried out for simulated as well as real near-infrared data. The calibration methods considered are ordinary least squares (simulated data), partial least squares regression and principal component regression (real data). The results suggest that the approximate variance expression is a viable alternative to resampling.

Return to Reliability of multivariate calibration

N.M. Faber, R. Bro and P.K. Hopke
Recent developments in CANDECOMP/PARAFAC algorithms: A critical review
Chemometrics and Intelligent Laboratory Systems, 65 (2003) 119-137

Several recently proposed algorithms for fitting the PARAFAC model are investigated and compared to more established alternatives. Alternating least squares (ALS), direct trilinear decomposition (DTLD), alternating trilinear decomposition (ATLD), self-weighted alternating trilinear decomposition (SWATLD), pseudo alternating least squares (PALS), alternating coupled vectors resolution (ACOVER), alternating slice-wise diagonalization (ASD) and alternating coupled matrices resolution (ACOMAR) are compared on both simulated and real data. For the recent algorithms, only unconstrained, three-way models can be fitted. In contrast, for example, ALS allows modeling of higher-order data, as well as incorporating constraints on the parameters and handling of missing data. Nevertheless, for three-way data, the newer algorithms are interesting alternatives. It is found that the ALS estimated models are generally of a better quality than any of the alternatives even when overfactoring the model, but it is also found that ALS is significantly slower. Based on the results (in particular the poor performance of DTLD), it is advised that (a slightly modified) ASD may be a good alternative to ALS when a faster algorithm is desired.

Return to Method comparison studies

N.M. Faber, X.-H. Song and P.K. Hopke
Sample-specific standard error of prediction for partial least squares regression
Trends in Analytical Chemistry, 22 (2003) 330-334

The development of an adequate expression for sample-specific standard error of prediction for partial least squares regression is a major trend in the chemometrics literature. This paper focuses on three generally applicable expressions, namely the one recommended by the American Society for Testing and Materials (ASTM), the one implemented in the Unscrambler software and a simplification derived under the errors-in-variables (EIV) model. Results obtained for a near-infrared data set taken from the literature demonstrate that the EIV expression works best.

Return to Reliability of multivariate calibration

N.M. Faber, J. Ferré, R. Boqué and J.H. Kalivas
Quantifying selectivity in spectrophotometric multicomponent analysis
Trends in Analytical Chemistry, 22 (2003) 352-361

According to the latest recommendation of the International Union of Pure and Applied Chemistry, "selectivity refers to the extent to which the method can be used to determine particular analytes in mixtures or matrices without interferences from other components of similar behavior". Owing to its prime importance as an analytical figure of merit, numerous proposals have been published on how to quantify selectivity in spectrophotometric multicomponent analysis. We show that the criterion independently developed by Lorber (Anal. Chem. 58 (1986) 1167) and Bergmann et al. (Anal. Chem. 59 (1987) 2522) is the most suitable one, because it directly relates to prediction uncertainty and allows for a consistent generalization to more complex systems of chemical analysis.

Return to Analytical figures of merit

S.Z. Fairchild and J.H. Kalivas
PCR eigenvector selection based on correlation relative standard deviations
Journal of Chemometrics, 15 (2001) 615-625

While principal component regression (PCR) is often performed with eigenvectors ordered by decreasing singular values, PCR models have been formed using other eigenvector arrangements. A common criterion for organizing eigenvectors involves absolute correlations between respective eigenvectors and the prediction property being modeled. However, correlation cut-off values for eigenvector selection are inconsistent between data sets, and additional criteria are needed such as the root mean square error of cross-validation (RMSECV). Furthermore, correlations for some selected eigenvectors are often extremely low (e.g. values of 01 have been considered acceptable) and it is difficult to justify inclusion of these eigenvectors. Relative standard deviations (RSDs) of correlations are evaluated in this paper as an alternative method of eigenvector selection. This paper reveals distinct advantages for using eigenvectors ordered by RSDs of correlations compared to eigenvectors ordered by absolute correlations. In particular, RSDs can be used to determine significant eigenvectors without resorting to additional criteria such as the RMSECV. Additionally, inspection of RSD values explains why different correlation cut-off values are obtained for different data sets as well as why correlations can be small.

Return to Component selection

J.A. Fernández Pierna, F. Wahl, O.E. de Noord and D.L. Massart
Methods for outlier detection in prediction
Chemometrics and Intelligent Laboratory Systems, 63 (2002) 27-39

If a prediction sample is different from the calibration samples, it can be considered as an outlier in prediction. In this work, two techniques, the use of the uncertainty estimation and convex hull method are studied to detect such prediction outliers. Classical techniques (Mahalanobis distance and X-residuals), potential functions and robust techniques are used for comparison. It is concluded that the combination of the convex hull and the uncertainty estimation offers a practical way for detecting outliers in prediction. By adding the potential function method, inliers can also be detected.

Return to Outlier detection

J.A. Fernández Pierna, L. Jin, F. Wahl, N.M. Faber and D.L. Massart
Estimation of partial least squares regression (PLSR) prediction uncertainty when the reference values carry a sizeable measurement error
Chemometrics and Intelligent Laboratory Systems, 65 (2003) 281-291

The prediction uncertainty is studied when using a multivariate partial least squares regression model constructed with reference values that contain a sizeable measurement error. Several approximate expressions for calculating a sample-specific standard error of prediction have been proposed in the literature. In addition, Monte Carlo simulation methods such as the bootstrap and the noise addition method can give an estimate of this uncertainty. In this paper, two approximate expressions are compared with the simulation methods for three near-infrared data sets.

Return to Reliability of multivariate calibration

J. Ferré, S.D. Brown and F.X. Rius
Improved calculation of the net analyte signal in inverse multivariate calibration
Journal of Chemometrics, 15 (2001) 537-553

The net analyte signal (NAS) is the part of the measured signal that a calibration model relates to the property of interest (e.g. analyte concentration). Accurate values of the NAS are required in multivariate calibration to calculate analytical figures of merit such as sensitivity, selectivity, signal-to-noise ratio, and limit of detection. This paper presents an improved version of the calculation method for the NAS in inverse models proposed by Lorber et al. (Anal. Chem. 1997; 69: 1620). Model coefficients and predictions calculated with the improved NAS are the same as those from the common equations of principal component regression (PCR) and partial least-squares (PLS) regression. The necessary alterations to the calculations of sensitivity, selectivity and the pseudounivariate presentation of the model are also provided.

Return to Analytical figures of merit

J. Ferré and N.M. Faber
Net analyte signal calculation for multivariate calibration
Chemometrics and Intelligent Laboratory Systems, 69 (2003) 123-136

A unifying framework for calibration and prediction in multivariate calibration is shown based on the concept of the net analyte signal (NAS). From this perspective, the calibration step can be regarded as the calculation of a net sensitivity vector, whose length is the amount of net signal when the value of the property of interest (e.g. analyte concentration) is equal to unity. The prediction step can be interpreted as projecting a measured spectrum onto the direction of the net sensitivity vector. The length of the projected spectrum divided by the length of the net sensitivity vector is the predicted value of the property of interest. This framework, which is equivalent to the univariate calibration approach, is used for critically revising different definitions of NAS and their calculation methods. The framework is particularized for the classical least squares (CLS), principal component regression (PLS) and partial least-squares (PLS) regression models.

Return to Analytical figures of merit

J. Ferré and N.M. Faber
Generalization of rank reduction problems with Wedderburn's formula
Journal of Chemometrics, 17 (2003) 603-607

In first- and second-order calibration methods based on spectroscopic data, the calculation of the space spanned by the spectra of the interferences has been an important research subject for, among many other applications, calculating the net analyte signal and obtaining figures of merit. Recently, many different calculation methods have been introduced. We show that the calculation of this space can be interpreted from a unified point of view, namely from the rank-one down-dating Wedderburn formula. This formula enables one to better understand the properties of the calculation methods currently available. A number of recently introduced signal pre-processing methods also fit into the proposed framework.

Return to Analytical figures of merit

G


M. Galea-Rojas, M.V. de Castilho, H. Bolfarine and M. de Castro
Detection of analytical bias
Analyst, 128 (2003) 1073-1081

The main object of this paper is to consider maximum likelihood estimators for models used in detection of analytical bias. We consider the regression model proposed in Ripley and Thompson (Analyst, 112, 1987, p. 377) with an EM-type algorithm for computing maximum likelihood estimators and obtain consistent estimators for the asymptotic variance of the maximum likelihood estimators, which seems not to be available in the literature. Wald type statistics are proposed for testing hypothesis related to the bias of the analytical methods with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. The main conclusion is that proposed approaches in the literature underestimate the covariance matrix of the maximum likelihood estimators. Results of simulation studies and applications to real data sets are reported to illustrate comparisons with other approaches.

Return to Reliability of univariate calibration

N.B. Gallagher, B.M. Wise and D.M. Sheen
Error analysis for estimation of trace vapor concentration-pathlength in stack plumes
Applied Spectroscopy, 57 (2003) 614-621

Near-infrared hyperspectral imaging is finding utility in remote sensing applications such as detection and quantification of chemical vapor effluents in stack plumes. Optimizing the sensing system or quantification algorithms is difficult because reference images rarely are well characterized. The present work uses a radiance model for a down-looking scene and a detailed noise model for dispersive and Fourier transform spectrometers to generate well-characterized synthetic data. These data were used with a classical least-squares-based estimator in an error analysis to obtain estimates of different sources of concentration-pathlength quantification error in the remote sensing problem. Contributions to the overall quantification error were the sum of individual error terms related to estimating the background, atmospheric corrections, plume temperature, and instrument signal-to-noise ratio. It was found that the quantification error depended strongly on errors in the background estimate and second most on instrument signal-to-noise ratio. Decreases in net analyte signal (e.g., due to low analyte absorbance or increasing the number of analytes in the plume) led to increases in the quantification error as expected. These observations have implications on instrument design and strategies for quantification. The outlined approach could be used to estimate detection limits or perform variable selection for given sensing problems.

Return to Reliability of multivariate calibration

I. García, L. Sarabia, M.C. Ortiz and J.M. Aldama
Three-way models and detection capability of a gas chromatography—mass spectrometry method for the determination of clenbuterol in several biological matrices: the 2002/657/EC European Decision
Analytica Chimica Acta, 515 (2004) 55-63

Clenbuterol has been extracted by mixed solid-phase extraction from two biological matrices (bovine hair and urine) and detected by GC/MS (selected ion monitoring (SIM) and full-SCAN modes). The analytical signal has been modelled with univariate and three-way models, namely DTLD, PARAFAC, PARAFAC2, Tucker3 and trilinear PLS. Since clenbuterol is a banned substance a comparative study of the capability of detection (CCb, X0 = 0) has been performed as a function of the sample (hair, 74 mg kg-1 and urine, 0.36 mg l-1), the mode in which the signals are monitored (SCAN, 283 mg kg-1 and SIM, 74 mg kg-1) and the statistical model (univariate, 283 mg kg-1 and trilinear PLS, 20.91 mg kg-1). The capability of detection has been calculated as stated in ISO 11843 and Decision 2002/657/EC setting in all cases the probabilities of false positive and of false negative at 0.05.
    The identification of the mass spectra must be done to confirm the presence of clenbuterol and has been carried out through PARAFAC. The correlation coefficient between the spectra estimated by PARAFAC and the library spectra is 0.96 (hair, SCAN mode) and 1.00 (hair and urine, SIM mode).
    The Decision 2002/657/EC advocates the use of independent mass fragments to identify banned compounds. These recommendations together with the effect of the number of ions registered on the capability of detection have lead us to select five uncorrelated fragments (86, 243, 262, 264 and 277) from the data set of 210 ions by hierarchical clustering of variables.

Return to Limit of detection

I. García, L. Sarabia, M.C. Ortiz and J.M. Aldama
Usefulness of D-optimal designs and multicriteria optimization in laborious analytical procedures. Application to the extraction of quinolones from eggs
Journal of Chromatography A, 1085 (2005) 190-198

An analytical method has been developed to extract ciprofloxacin and enrofloxacin from eggs. The aim of this work is to determine the experimental conditions of extraction providing high recoveries with small standard deviations. An experimental design based on the D-optimality criterion and replicated three times was built to evaluate the effect of five factors related to the extraction which is the most inaccurate stage of the procedure. This non-classical design is needed because there are several practical constraints: (i) the extraction procedure is time-consuming, quinolones are not stable and the design must be performed in a single working session. (ii) The tube capacity of the centrifuge is 6, so the number of experiments will be 6 or a multiple of 6. In the optimal experimental conditions, the extraction is performed once with 5ml of methanol. Then, fatty acids are removed with a mixture of hexane/ether. Analytes are finally separated and detected by HPLC-fluorescence without the additional step of purification by solid-phase extraction (SPE). Under these conditions, the mean recovery is 64% and 70% and the standard deviation 5% and 4% for ciprofloxacin and enrofloxacin, respectively. The capability of decision, CCa, is 3.1 and 2.8 bg kg-1 of ciprofloxacin and enrofloxacin, respectively. The capability of detection, CCb, is 7.8 and 7.0 mg kg-1 of ciprofloxacin and enrofloxacin, respectively. In both cases the probabilities of false positive, a, and of false negative, b, were fixed at 0.05.

Return to Limit of detection

P. Geladi
Some recent trends in the calibration literature
Chemometrics and Intelligent Laboratory Systems, 60 (2002) 211-224

Calibration is a part of all scientific activity and many calibration methods have been proposed and introduced over the years. Univariate calibration is quite simple but not as useful as multivariate calibration, which is complicated. Outsiders and active chemometricians alike tend to lose overview of the literature. What are the differences between the calibration methods? What should be looked at besides the choice of calibration methods? What are the best diagnostics and how should they be interpreted? Many more questions can be asked. A general introduction to some of the topics is given and supplemented by recent literature.

Return to General on calibration

P.J. Gemperline
Computation of the range of feasible solutions in self-modeling curve resolution algorithms
Analytical Chemistry, 71 (1999) 5398-5404

Self-modeling curve resolution (SMCR) describes a set of mathematical tools for estimating pure-component spectra and composition profiles from mixture spectra. The source of mixture spectra may be overlapped chromatography peaks, composition profiles from equilibrium studies, kinetic profiles from chemical reactions and batch industrial processes, depth profiles of treated surfaces, and many other types of materials and processes. Mathematical solutions are produced under the assumption that pure-component profiles and spectra should be nonnegative and composition profiles should be unimodal. In many cases, SMCR may be the only method available for resolving the composition profiles and pure-component spectra from these measurements. Under ideal circumstances, the SMCR results are accurate quantitative estimates of the true underlying profiles. Although SMCR tools are finding wide use, it is not widely known or appreciated that, in most circumstances, SMCR techniques produce a family of solutions that obey non-negatively constraints. In this paper, we present a new method for computation of the range of feasible solutions and use it to study the effect of chromatographic resolution, peak height, spectral dissimilarity, and signal-to-noise ratios on the magnitude of feasible solutions. An illustration of its use in resolving composition profiles from a batch reaction is also given.

Return to Reliability of multiway calibration

D. Giménez, D. Grasso, L. Sarabia and M.C. Ortiz
Determination of quinolones by fluorescent excitation emission
Talanta, 64 (2004) 442-451

In this work the viability of a fluorescent technique for the determination of quinolones is studied. This analytical technique allows one to analyze the effect of the increasing order of the analytical signal from a univariate calibration (zero order data) to partial least squares (PLS) calibration (first order). The comparison has been done through the figures of merit of the analytical procedure (technique and calibration) in accordance with the ISO norm and the 2002/657/EC European Decision about residuals.

Return to Limit of detection

L.A. Goodman and S.J. Haberman
The analysis of nonadditivity in two-way analysis of variance
Journal of the American Statistical Association, 85 (1990) 139-145

Singular value decompositions are used to study interactions in two-way layouts. In contrast to traditional treatments, emphasis is on description of the nature of the interactions rather than on tests of whether they do or do not exist. The first model considered assumes that the matrix of interactions has rank 1, so in a singular value decomposition of the interactions, only one singular value is not 0. Given that this model holds and interactions are present, normal approximations are obtained under conditions described for the estimates of this singular value and for the corresponding scores for row categories and column categories. Thus approximate confidence intervals can be obtained for the parameters in this singular value decomposition of interactions. In addition, under this model, approximate t and F tests can be constructed for hypotheses that assign specified values to the row and column scores in the singular value decomposition. Results are illustrated in a reanalysis of data previously analyzed by Johnson and Graybill (1972). The model that assumes that the interactions matrix is of rank 1 is also used to test the appropriateness of the scoring system that is associated with Tukey's one degree of freedom for nonadditivity. In addition, models are considered in which the singular value decomposition of the interactions has rank greater than 1, and normal approximations for parameter estimates and approximate t and F tests for hypotheses concerning the scores in the singular value decomposition are derived for these models as well.

Return to Reliability of principal component analysis

S. Gourvénec, J.A. Fernández Pierna, D.L. Massart and D.N. Rutledge
An evaluation of the PoLiSh smoothed regression and the Monte Carlo cross-validation for the determination of the complexity of a PLS model
Chemometrics and Intelligent Laboratory Systems, 68 (2003) 41-51

A crucial point of the PLS algorithm is the selection of the right number of factors or components (i.e., the determination of the optimal complexity of the system to avoid overfitting). The leave-one-out cross-validation is usually used to determine the optimal complexity of a PLS model, but in practice, it is found that often too many components are retained with this method. In this study, the Monte Carlo Cross-Validation (MCCV) and the PoLiSh smoothed regression are used and compared with the better known adjusted Wold's R criterion.

Return to Component selection

R.L. Green and J.H. Kalivas
Graphical diagnostics for regression model determinations with consideration of the bias/variance trade-off
Chemometrics and Intelligent Laboratory Systems, 60 (2002) 173-188

Estimates of model parameters (regression coefficients forming the regression vector) for a multivariate linear model have been the subject of considerable discussion. Regression diagnostics utilized in chemometrics for a multivariate linear model are often based on a single number such as the coefficient of determination, root mean square error of cross-validation, selectivity, etc. Additionally, regression diagnostics commonly applied focus on model bias and do not include variance or model complexity. This paper demonstrates that substantial information is available through a graphical study of trends in model parameters as determined by plots of regression diagnostics using bias, variance, and/or model complexity measures. Also illustrated is that by using harmonious graphics which simultaneously use bias and variance information, determination of proper model parameters without cross-validation is possible. This paper concludes with comments on the next level of regression diagnostics, including use of color, sound, and virtual reality.

Return to Component selection

M.L. Griffiths, R.P. Barbagallo and J.T. Keer
Multiple and simultaneous fluorophore detection using fluorescence spectrometry and partial least-squares regression with sample-specific confidence intervals
Analytical Chemistry, 78 (2006) 513-523

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 algorithmbased 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 dyelabeled oligonucleotides at levels of (0.6-5.3) × 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 ~91 to 100%.

Return to Reliability of multivariate calibration

M.L. Griffiths and S.L.R. Ellison
A simple numerical method of estimating the contribution of reference value uncertainties to sample-specific uncertainties in multivariate regression
Chemometrics and Intelligent Laboratory Systems, 83 (2006) 133-138

The development of a sample-specific standard error of prediction (SS-SEP) for multivariate methods has been a major topic in the chemometrics literature. The expressions are, however, dependent on the predictive algorithm used. This study proposes a general and relatively economical numerical approach to the problem independent of the prediction algorithm used that potentially provides a validation technique for less established and more complex theoretical expressions. A comparison is then made to the standard errors estimated using the EIVapproach in a PLS example. An important part of any such prediction interval is the contribution from uncertainties in reference values. The numerical method provided slightly optimistic standard errors, but these were sufficient to check the adequacy of the reference values and, for modest reference value uncertainties, for routine uncertainty estimation.

Return to Reliability of multivariate calibration

S.P. Gurden, J.A. Westerhuis, R. Bro and A.K. Smilde
A comparison of multiway regression and scaling methods
Chemometrics and Intelligent Laboratory Systems, 59 (2001) 121-136

Recent years have seen an increase in the number of regression problems for which the predictor andror response arrays have orders higher than two, i.e. multiway data. Examples are found in, e.g. industrial batch process analysis, chemical calibration using second-order instrumentation and quantitative structure-activity relationships QSAR . As these types of problems increase in complexity in terms of both the dimensions and the underlying structures of the data sets, the number of options with respect to different types of scaling and regression models also increases. In this article, three methods for multiway regression are compared: unfold partial least squares PLS , multilinear PLS and multiway covariates regression MCovR . All three methods differ either in the structural model imposed on the data or the way the model components are calculated. Three methods of scaling multiway arrays are also compared, along with the option of applying no scaling. Three data sets drawn from industrial processes are used in the comparison. The general conclusion is that the type of data and scaling used is more important than the type of regression model used in terms of predictive ability. The models do differ, however, in terms of interpretability.

Return to Method comparison studies

H


A. Herrero, S. Zamponi, R. Marassi, P. Conti, M.C. Ortiz and L.A. Sarabia
Determination of the capability of detection of a hyphenated method: application to spectroelectrochemistry
Chemometrics and Intelligent Laboratory Systems, 61 (2002) 63-74

A procedure to evaluate the capability of detection of a second-order analytical technique to determine an analyte in presence of an interferent has been proposed taking into account a and b errors in a similar way as ISO norms indicate for the univariate analytical methods. The potentiality of spectroelectrochemistry as a quantitative three-way technique of analysis has been analysed. Trilinearity of spectroelectrochemical data has been studied since it is a necessary condition to apply the Trilinear Decomposition (TLD) method. As an example, the voltabsorptometric determination of o-tolidine in presence of high concentration of ferrocyanide was chosen to test the applicability of the proposed method. In the same way the capability of discrimination has been determined. In addition, a second-order standard addition method (SOSAM) has been applied to calculate the concentration of the analyte of interest in presence of this interferent, avoiding the need to previously identify and determine the quantity of the interferent.

Return to Limit of detection

R.F. Hirsch, G.L. Wu and P.C. Tway
Reliability of factor analysis in the presence of random noise or outlying data
Chemometrics and Intelligent Laboratory Systems, 1 (1987) 265-272

The influence of random and determinate error on abstract factor analysis is investigated. A Monte Carlo study of the effect of random error in data from NMR and GC-MS experiments indicates that the eigenvalue ratio test and Malinowski's indicator function are more accurate that the imbedded error function in determining the number of significant factors or components. Outliers affect the analysis, but the level of deviation of a single outlier can be considerably larger than the random experimental error in the data before the effect is significant.

Return to Reliability of principal component analysis

J.T. Hwang and A. Ding
Prediction intervals for artificial neural networks
Journal of the American Statistical Association, 92 (1997) 748-757

The artificial neural network (ANN) is becoming a very popular model for engineering and scientific applications. Inspired by brain architecture, artificial neural networks represent a class of nonlinear models capable of learning from data. Neural networks have been applied in many areas, including pattern matching, classification, prediction, and process control. This article focuses on the construction of prediction intervals. Previous statistical theory for constructing confidence intervals for the parameters (or the weights in an ANN), is inappropriate, because the parameters are unidentifiable. We show in this article that the problem disappears in prediction. We then construct asymptotically valid prediction intervals and also show how to use the prediction intervals to choose the number of nodes in the network. We then apply the theory to an example for predicting the electrical load.

Return to Reliability of multivariate calibration

J.T. Hwang and D. Nettleton
Principal component regression with data-chosen components and related methods
Technometrics, 45 (2003) 70-79

Multiple regression with correlated explanatory variables is relevant to a broad range of problems in the physical, chemical, and engineering sciences. Chemometricians in particular have made heavy use of principal component regression and related procedures for predicting a response variable from a large number of highly correlated variables. In this article we develop a general theory for selecting principal components that yield estimates of regression coefficients with low mean squared error. Our numerical results suggest that the theory also can be used to improve partial least squares regression estimators and regression estimators based on rotated principal components. Although our work has been motivated by the statistical genetics problem of mapping quantitative trait loci, the results are applicable to any problem in which estimation of regression coefficients for correlated explanatory variables is of interest.

Return to Component selection

I


J


K


J.H. Kalivas
Optimization using variations of simulated annealing
Chemometrics and Intelligent Laboratory Systems, 15 (1992) 1-12

Simulated annealing (SA) was originally developed for determining global optima for combinatorial optimization problems, but it has been shown to be amenable to continuous functions as well. Difficulties in SA arise when determining proper operating cooling schedule variables for particular optimization problems. Generalized simulated annealing (GSA) reduces many of these complications. However, GSA usually locates near-global optima and not exact optima. If GSA is set up interactively, convergence to the exact global optimum becomes achievable. In order to attain automatic convergence to the exact global optimum, the variable step size generalized simulated annealing (VSGSA) algorithm tunes its operation variables based on data collected during optimization. This tutorial describes the basic operation principles for SA, GSA, and VSGSA, and incorporates worked examples from the literature.

Return to Numerical optimization

H.A.L. Kiers
Bootstrap confidence intervals for three-way methods
Journal of Chemometrics, 18 (2004) 22-36

Results from exploratory three-way analysis techniques such as CANDECOMP/PARAFAC and Tucker3 analysis are usually presented without giving insight into uncertainties due to sampling. Here a bootstrap procedure is proposed that produces percentile intervals for all output parameters. Special adjustments are offered for handling the non-uniqueness of the solutions. The percentile intervals indicate the instability of the sample solutions. By means of a simulation study it is demonstrated that the percentile intervals can fairly well be interpreted as confidence intervals for the output parameters.

Return to Reliability of multiway calibration

L


M. Laasonen, T. Harmia-Pulkkinen, C. Simard, M. Räsänen and H. Vuorela
Development and validation of a near-infrared method for the quantitation of caffeine in intact single tablets
Analytical Chemistry, 75 (2003) 754-760

A near-infrared spectroscopic method was developed and validated for determining the caffeine concentration of single and intact tablets in a Finnish pharmaceutical product containing 58.82% (m/m) caffeine.The spectral region of interest contained a total of 474 data points. The second derivative of Savitsky-Golay, a standard normal variate, and mean centering were used as spectral preprocessing options. The feasibility study showed nonuniformity of caffeine repartition within each tablet. Thus, spectra were recorded from both faces of the tablets, and the analysis result for a single tablet was reported as the average of both face determinations. Precision of the method was validated because the relative standard deviations from repeatability and intermediate precision tests were below 0.75% (m/m). Accuracy validation proved that the NIR results were not significantly different (P = 0.09, n = 12) from the results obtained with the reference HPLC method. The limit of quantification for caffeine was 13.7% (m/m) in the tablets. The method was found to be unaffected by NIR source replacement, but the repeatability of the results was affected if the sample holder was not placed in the correct position in the light beam. Routine NIR analysis of caffeine in tablet form was found to be more flexible and much faster than that performed with the HPLC method.

Return to Validation

P.M. Lang and J.H. Kalivas
A global perspective on multivariate methods in spectral chemical analysis
Journal of Chemometrics, 7 (1993) 153-163

This paper consists of two distinct but related parts. In the first part, a geometric theory of generalized inverses is presented and a methodology based on this theory is developed and applied to solve K-matrix and P-matrix forms of Beer's law. It is shown that most currently accepted and practiced methods for solving these forms of Beer's law are just special cases of this geometric theory of generalized inverses. In addition, this geometric theory is used to explain why the current methods work and why they fail.
    In the second part a general methodology that includes as special cases least squares, principal component regression, partial least squares 1 and 2, continuum regression plus a variety of other described and undescribed methodologies is presented and then applied to solve the P-matrix formulation of Beer's law. This general methodology, like the first, is also geometric in nature and relies on an understanding of projections.
    The main emphasis of the paper is on the perspective, which if understood, provides the proper foundation fro answering the general but extremely hard and possible unanswerable question 'what is the best method?'.

Return to General on calibration

R. Leardi
Genetic algorithms in chemometrics and chemistry: a review
Journal of Chemometrics, 15 (2001) 559-569

The use of genetic algorithms has been growing exponentially since Holland published the first papers about them. Thanks to the extraordinary increase in calculation power, nowadays it is possible to apply them to extremely complex problems. A considerable number of papers in which genetic algorithms have been applied have been published in several scientific journals. This review is of course far from being a complete summary of the application of genetic algorithms to chemical problems; its goal is to show the reader the main fields of application of this technique, together with providing a list of references on the subject.

Return to Numerical optimization

S. Ledauphin, M. Hanafi and E.M. Qannari
Simplification and signification of principal components
Chemometrics and Intelligent Laboratory Systems, 74 (2004) 277-281

Within the framework of principal component analysis (PCA), we propose a procedure of hypothesis testing to assess the signification of the principal components and the signification of the variable contributions to the determination of the principal components. If a variable contribution turns out to be nonsignificant then the loading associated with this variable is set to zero. This leads to a simplification of principal components in that sense that they can be more easily interpreted. Hypothesis testing is based on a procedure of simulation by permutations of the rows of the data matrix at hand. The interest of this procedure is illustrated using a real data set.

Return to Component selection

M.N. Leger and P.D. Wentzell
Maximum likelihood principal components regression on wavelet-compressed data
Applied Spectroscopy, 58 (2004) 855-862

Maximum likelihood principal component regression (MLPCR) is an errors-in-variables method used to accommodate measurement error information when building multivariate calibration models. A hindrance of MLPCR has been the substantial demand on computational resources sometimes made by the algorithm, especially for certain types of error structures. Operations on these large matrices are memory intensive and time consuming, especially when techniques such as cross-validation are used. This work describes the use of wavelet transforms (WT) as a data compression method for MLPCR. It is shown that the error covariance matrix in the wavelet and spectral domains are related through a two-dimensional WT. This allows the user to account for any effects of the wavelet transform on spectral and error structures. The wavelet transform can be applied to MLPCR when using either the full error covariance matrix or the smaller pooled error covariance matrix. Simulated and experimental near-infrared data sets are used to demonstrate the benefits of using wavelets with the MLPCR algorithm. In all cases, significant compression can be obtained while maintaining favorable predictive ability. Considerable time savings were also attained, with improvements ranging from a factor of 2 to a factor of 720. Using the WT-compressed data in MLPCR gave a reduction in prediction errors compared to using the raw data in MLPCR. An analogous reduction in prediction errors was not always seen when using PCR.

Return to Maximum likelihood calibration

M.N. Leger, L. Vega-Montoto and P.D. Wentzell
Methods for systematic investigation of measurement error covariance matrices
Chemometrics and Intelligent Laboratory Systems, 77 (2005) 181-205

Several recently developed methods for multivariate data analysis allow the incorporation of prior information about the measurement error structure into the analysis to permit better model estimation and prediction. This error structure is described in the form of the measurement error covariance matrix, which defines the complex relationship between measurement uncertainty at one channel and those at other channels and/or for other samples. In this work, a systematic approach for characterizing the measurement error covariance matrix for a particular experimental or instrumental environment is presented. This approach involves a number of strategies that include visualization of covariance and correlation matrices, bilinear modelling through principal components analysis (PCA) and target-testing factor analysis, trilinear modelling through PARAFAC, and refinement of models to include interaction terms and independent errors. The primary goals of this characterization are to obtain a better understanding of the factors contributing to measurement error and to develop parametric models for error covariance that do not rely on extensive replication. To illustrate this approach, four experimental data sets are employed: (1) UV-visible absorbance data, (2) near-infrared (NIR) reflectance data, (3) fluorescence emission data, and (4) short-wave NIR (SW-NIR) absorbance data from a kinetics experiment. For both the UV-visible and SW-NIR spectra, the main contribution to the error structure is a constant offset term that appears to have a dependence on the reciprocal of the wavelength. The NIR reflectance spectra are dominated by constant and multiplicative offset noise that have a strong interaction. The fluorescence data is affected by independent shot-noise with a variance proportional to the magnitude of the spectrum, as well as by correlated offset noise with two largely independent terms, one which is fixed in magnitude and the other which depends on the square root of the spectrum. It was also found that the SW-NIR data exhibits a strong correlation among samples.

Return to Maximum likelihood calibration

M. Linder and R. Sundberg
Second order calibration: bilinear least squares regression and a simple alternative
Chemometrics and Intelligent Laboratory Systems, 42 (1998) 159-178

The bilinear regression model is to be estimated from a number of specimens of known composition. We propose a simple estimator and study how it works on real and simulated data. The estimator, which we call the SVD (singular value decomposition) estimator is usually not much less efficient than bilinear least squares. The advantages of our method over bilinear least squares are that it is faster and more easily computed, its standard errors are explicit (and derived in the paper), and it has a simpler correlation structure.

Return to Reliability of multiway calibration

M. Linder and R. Sundberg
Precision of prediction in second-order calibration, with focus on bilinear regression methods
Journal of Chemometrics, 16 (2002) 12-27

We consider calibration of hyphenated instruments with particular focus on determination of the unknown concentrations of new specimens. A hyphenated instrument generates for each specimen a two-way array of data. These are assumed to depend on the concentrations through a bilinear regression model, where each constituent is characterized by a pair of profiles to be determined in the calibration. We discuss the problem of predicting the unknown concentrations in a new specimen, after calibration. We formulate three different predictor construction methods, a naive method, a least squares method, and a refined version of the latter that takes account of the calibration uncertainty. We give formulae for the uncertainty of the predictors under white noise, when calibration can be seen as precise. We refine these formulae to allow for calibration uncertainty, in particular when calibration is carried out by the bilinear least squares (BLLS) method or the singular value decomposition (SVD) method proposed by Linder and Sundberg (Chemometrics Intell. Lab. Syst. 1998; 42: 159-178). By error propagation formulae and previous results on the precision of Ahat.gif and Bhat.gif we can obtain approximate standard errors for the predicted concentrations, according to each of the two estimation methods. The performance of the predictors and the precision formulae is illustrated on both real (fluorescence) and simulated data.

Return to Reliability of multiway calibration

X. Liu and N. Sidiropoulos
Cramér-Rao lower bounds for low-rank decomposition of multidimensional arrays
IEEE Transactions on Signal Processing, 49 (2001) 2074-2086

Unlike low-rank matrix decomposition, which is generally nonunique for rank greater than one, low-rank three- and higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher-dimensional arrays. This paper develops Cramér-Rao Bound (CRB) results for low-rank decomposition of three- and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided.

Return to Reliability of multiway calibration

A. Liwo, P. Skurski, S. Oldziej, L. Lankiewicz, J. Malicka and W. Wiczk
A new approach to the resolution of the excitation-emission spectra of multicomponent systems
Computers and Chemistry, 21 (1997) 89-96

A method has been developed for the calculation of the emission, absorption, and concentration characteristics of chemical species from the fluorescence spectra recorded for a series of solutions of varying composition at different excitation wavelengths. The three-dimensional fluorescence-intensity array (whose ijkth element is the fluorescence intensity of the ith solution at the jth excitation and the kth emission wavelength) is assumed to be a trilinear product of the excitation (X), emission (Y), and concentration (C) characteristics of the emitting species. The decomposition of experimental spectra is carried out in three steps. In the first step, an eigenvector analysis of the Gauss transform of the fluorescence array is performed, whose rank is the lower bound to the rank of the matrices X, Y, and C, and thereby to the number of independently emitting chemical species present in the system. In the second step, factorization of the eigenvectors of the Gauss transform of the fluorescence array is performed, which gives approximate factor matrices X, Y and C. The obtained factors correspond to exact decomposition, if there is no error in the data. In the third step, the factors are refined by means of a nonlinear least-squares method, which is directed at minimizing the sum of the squares of the differences between the elements of the observed and calculated fluorescence array. Application of the method to the fluorescence spectrum of mixtures of anthracene and 9-methylanthracene resulted in perfect resolution of the fluorescence spectra into the component excitation and emission spectra and correct profiles of component concentrations. It is also shown that if the numbers of absorbing and emitting species are not equal, as occurs for the small beta, Greek-naphtol-small beta, Greek-naphtolate system in acid and neutral aqueous solution, unambiguous decomposition of the spectrum is impossible to carry out.

Return to Reliability of multiway calibration

R.A. Lodder and G.M. Hieftje
Quantile BEAST attacks the false-sample problem in near-infrared reflectance analysis
Applied Spectroscopy, 42 (1988) 1351-1365

The multiple linear regression approach typically need in near-infrared calibration yields equations in which any amount of reflectance at the analytical wavelengths leads to a corresponding composition value. As a result, when the sample contains a component not present in the training set, erroneous composition values can arise without any indication of error. The Quantile BEAST (Bootstrap Error-Adjusted Single-sample Technique) is described here as a method of detecting one or more "false" samples. The BEAST constructs a multidimensional form in space using the reflectance values of each training-set sample at a number of wave-lengths. New samples are then projected into this space, and a confidence test is executed to determine whether the new sample is part of the training-set form. The method is more robust than other procedures because it relies on few assumptions about the structure of the data; therefore, deviations from assumptions do not affect the results of the confidence test.

Return to Reliability of classification

R.A. Lodder and G.M. Hieftje
Detection of subpopulations in near-infrared reflectance analysis
Applied Spectroscopy, 42 (1988) 1500-1512

In typical near-infrared multivariate statistical analyses, samples with similar spectra produce points that cluster in a certain region of spectral hyperspace. These clusters can vary significantly in shape and size due to variation in sample packings, particle-size distributions, component concentrations, and drift with time. These factors, when combined with discriminate analysis using simple distance metrics, produce a test in which a result that places a particular point inside a particular cluster does not necessarily mean that the point is actually a member of the cluster. Instead, the point may be a member of a new, slightly different cluster that overlaps the first. A new cluster can be created by factors like low-level contamination or instrumental drift. An extension added to part of the BEAST (Bootstrap Error-Adjusted Single-sample Technique) can be used to set nonparametric probability-density contours inside spectral clusters as well as outside, and when multiple points begin to appear in a certain region of cluster-hyperspace the perturbation of these density contours can be detected at an assigned significance level. The detection of false samples both within and beyond 3 SDs of the center of the training set is possible with this method. This procedure is shown to be effective for contaminant levels of a few hundred ppm in an over-the-counter drug capsule, and is shown to function with as few as one or two wavelengths, suggesting its application to very simple process sensors.

Return to Reliability of classification

R.A. Lodder and G.M. Hieftje
Quantile analysis: a method for characterizing data distributions
Applied Spectroscopy, 42 (1988) 1512-1520

Analyzing distribution of data represents a common problem in chemistry. Quantile-quantile (QQ) plots provide a useful way to attack this problem. These graphs are often used in the form of the normal probability plot, determine whether the residuals from a fitting process are randomly distributed and therefore whether an assumed model fits the data at hand. By comparing the integrals of two probability density functions in a single plot, QQ plotting methods are able to capture the location, scale, and skew of a data set. This procedure provides more information to the analyst than do classical statistical methods that rely on a single test statistic for distribution comparisons.

Return to Reliability of classification

J.A. Lopes and J.C. Menezes
Industrial fermentation end-product modelling using multilinear PLS
Chemometrics and Intelligent Laboratory Systems, 68 (2003) 75-81

In this paper a trilinear version of the PLS algorithm was used to model the performance of an industrial fed-batch fermentation process. Trilinear data obtained from process operation was used to derive a model for the end-process active product ingredient (API) concentration prediction. Obtained multilinear PLS models were compared with the correspondent bilinear models. A genetic algorithm was used to select appropriate calibration sets (to reduce the influence of nominal batches). A validation coefficient of determination (QY2) of 91.4% was obtained for the multilinear PLS model after batch selection (prediction intervals were estimated using bootstrapping). Examination of the multilinear PLS model weights lead to the delimitation of a small time region (from 50 to 75 processing hours) almost exclusively responsible for the fermentation performance.

Return to Reliability of multiway calibration

A. Lorber
Error propagation and figures of merit for quantitation by solving matrix equations
Analytical Chemistry, 58 (1986) 1167-1172

Quantitation from one-dimensional data enables the simultaneous determination of all components contributing to the spectrum. However, the applicability of the procedure is limited because of the figures of merit; error propagation, signal to noise, limit of detection, precision, accuracy, sensitivity, and selectivity are not determined for each component. It is suggested that by considering the "net analyte signal", error propagation and other figures of merit are defined for each component. Net analyte signal is defined as the part of the signal that is orthogonal to the spectra of the other components. The mathematical results were applied to absorbance data of a four RNA nucleotide mixture, and it was found that it succeeds well in predicting both precision and accuracy.

Return to Analytical figures of merit

A. Lorber, A. Harel, Z. Goldbart and I.B. Brenner
Curve resolution and figures of merit estimation for determination of trace elements in geological materials by inductively coupled plasma atomic emission spectrometry
Analytical Chemistry, 59 (1987) 1260-126

In geochemical analysis using inductively coupled plasma atomic emission spectrometry (ICP-AES), spectral interferences and background enhancement in response to sample concomitants are the main cause of deterioration of the limit of detection (LOD) and inaccuracy of the determination at the trace and minor element levels. In this account, we describe the chemometric procedure of curve resolution for compensating for these sources of error. A newly developed method for calculating figures of merit is used to evaluate the correction procedure, test the statistical significance of the determined concentration, and determine LODs for each sample. The technique involves scanning the vicinity of the spectral line of the analyte. With prior knowledge of potential spectral interferences, deconvolution of the overlapped response is possible. Analytical data for a wide range of geological standard reference materials demonstrate the effectiveness of the chemometric techniques. Separation of 0.002 nm spectral coincidence, employing a 0.02 nm resolution spectrometer, is demonstrated.

Return to Analytical figures of merit

A. Lorber and B.R. Kowalski
The effect of interferences and calibration design on accuracy: implications for sensor and sample selection
Journal of Chemometrics, 2 (1988) 67-79

Methods of multivariate calibration use models that relate spectral data or sensor array responses to the concentrations of analytes. The goal is to insure that the calibration model can accurately estimate analyte concentrations in unknown samples not contained in the calibration set. The sensors or spectral channels (e.g. wavelengths) selected for incorporation in the model, as well as samples selected for the calibration step, are know to have an effect on the accuracy of analysis for unknown samples. This work provides a fundamental treatment of this effect and derives criteria for optimal selection. Additionally, a proof is given for the advantage of having more sensors and calibration samples than analytes - the overdetermined case.

Return to Analytical figures of merit

A. Lorber, N.M. Faber and B.R. Kowalski
Net analyte signal calculation in multivariate calibration
Analytical Chemistry, 69 (1997) 1620-1626

Net analyte signal plays an important role in the calculation of figures of merit for characterizing a calibration model. Until now its computation has only been feasible for the direct calibration model, which requires knowledge of pure spectra or concentrations of all contributing species in the calibration samples. An increasingly important calibration model is the inverse calibration model, which also allows for quantitation if the knowledge about the interferents is incomplete. This paper shows that net analyte signal computation is possible for the inverse calibration case. Application to the determination of protein content in wheat samples by near-infrared spectrometry is presented. Net analyte signal calculation was used to estimate selectivities (ratio of signal available for quantitation to the total measured signal). The selectivities were found to range between 0 and 2% of the measured reflectance signal. A new measure for outlier diagnosis based on the correlation of the net analyte signal to the regression coefficients vector is introduced and tested on the same data.

Return to Analytical figures of merit

M


E.R. Malinowski
Theory of error in factor analysis
Analytical Chemistry, 49 (1977) 606-612

A theory of error for abstract factor analysis (AFA) is developed. It is shown that the resulting eigenvalues can be grouped into two sets: a primary set which contains the true factors together with a mixture of error and a secondary set which consists of pure error. Removal of the secondary set from the AFA scheme leads to data improvement. Three types of errors are shown to exist: RE, real error; XE, extracted error; and IE, imbedded error. These errors are related in a pythagorean sense and can be calculated from a knowledge of the secondary eigenvalues, the size of the data matrix, and the number of factors involved. Mathematical models are used to illustrate and verify various facets of the theory.

Return to Reliability of principal component analysis

E.R. Malinowski
Determination of the number of factors and the experimental error in a data matrix
Analytical Chemistry, 49 (1977) 612-617

An imbedded error function and an indicator function, calculated solely from the eigenvalues which result from subjecting a data matrix to abstract factor analysis, are used to determine not only the number of controlling factors but also the root mean square of the experimental error without any a priori knowledge of the error. Model data are used to illustrate the behavior of these functions. The method is applied to problems of interest to chemists, involving nuclear magnetic resonance, absorption spectroscopy, mass spectra, gas-liquid chromatography, and drug activity.

Return to Reliability of principal component analysis

E.R. Malinowski
Theory of error applied to factor loadings resulting from combination target factor analysis
Analytica Chimica Acta, 122 (1980) 327-330

The theory of error for target factor analysis is used to derive a simple equation from which the root-mean-square error in the factor loadings can be calculated. The method is applied to a problem in gas-liquid chromatography and is shown to agree with errors estimated by the 'jackknife' method.

Return to Reliability of principal component analysis

R. Marbach
On Wiener filtering and the physics behind statistical modeling
Journal of Biomedical Optics, 7 (2002) 130-147

The closed-form solution of the so-called statistical multivariate calibration model is given in terms of the pure component spectral signal, the spectral noise, and the signal and noise of the reference method. The ''statistical'' calibration model is shown to be as much grounded on the physics of the pure component spectra as any of the ''physical'' models. There are no fundamental differences between the two approaches since both are merely different attempts to realize the same basic idea, viz., the spectrometric Wiener filter. The concept of the application-specific signal-to-noise ratio (SNR) is introduced, which is a combination of the two SNRs from the reference and the spectral data. Both are defined and the central importance of the latter for the assessment and development of spectroscopic instruments and methods is explained. Other statistics like the correlation coefficient, prediction error, slope deficiency, etc., are functions of the SNR. Spurious correlations and other practically important issues are discussed in quantitative terms. Most important, it is shown how to use a priori information about the pure component spectra and the spectral noise in an optimal way, thereby making the distinction between statistical and physical calibrations obsolete and combining the best of both worlds. Companies and research groups can use this article to realize significant savings in cost and time for development efforts.

Return to General on calibration

R. Marbach
A new method for multivariate calibration
Journal of Near Infrared Spectroscopy, 13 (2005) 241-254

A new method for multivariate calibration is described that combines the best features of "classical" (also called "physical" or "K-matrix") calibration and "inverse" (or "statistical" or "P-matrix") calibration. By estimating the spectral signal in the physical way and the spectral noise in the statistical way, so to speak, the prediction accuracy of the inverse model can be combined with the low cost and ease of interpretability of the classical model, including "built-in" proof of specificity of response. The cost of calibration is significantly reduced compared to today's standard practice of statistical calibration using partial least squares or principal component regression, because the need for lab-reference values is virtually eliminated. The method is demonstrated on a data set of near-infrared spectra from pharmaceutical tablets, which is available on the web (so-called Chambersburg Shoot-out 2002 data set). Another benefit is that the correct definitions of the "limits of multivariate detection" become obvious. The sensitivity of multivariate measurements is shown to be limited by the so-called "spectral noise", and the specificity is shown to be limited by potentially existing "unspecific correlations". Both limits are testable from first principles, i.e. from measurable pieces of data and without the need to perform any calibration.

Return to General on calibration

H. Mark, G.E. Ritchie, R.W. Roller, E.W. Ciurczak, C. Tso and S.A. MacDonald
Validation of a near-infrared transmission spectroscopic procedure, Part A: validation protocols
Journal of Pharmaceutical and Biomedical Analysis, 28 (2002) 251-260

It is possible to devise calibration and validation protocols that enable the ICH guidelines to conform to the specialized requirements of the NIR method of analysis. Some of the required characteristics of evaluation specified by the guidelines, such as accuracy and repeatability, can be applied directly, just as with any other analytical method. Other characteristics are adapted through the novel use of specialized statistics, or through the use of creative methods and procedures to match the recommendations of the guidelines to the unique and specialized requirements of the NIR method.

Return to Validation

H. Martens and P. Dardenne
Validation and verification of regression in small data sets
Chemometrics and Intelligent Laboratory Systems, 44 (1998) 99-121

Four different methods of using small data sets in multivariate modeling are compared w.r.t. predictive precision in the long-run. The modeling in this case concerns multivariate calibration: ŷ = f(X). The study consists of a Monte Carlo simulations within a large data base of real data; X = NIR reflectance spectra and y = protein percentage, measured in 922 whole maize plant samples. Small data set (40-120 objects) were repeatedly selected at random from the data base, each time simulating the situation of having only a small set of samples available for estimating, optimizing and assessing the calibration model. The 'true' apparent prediction error was each time controlled in the remaining data base. This was replicated 100 times in order to study the statistical performance of the four different validation methods. In each Monte Carlo replicate, the splitting of the available data sets into calibration set and test set was compared to full cross validation. The results demonstrated that removing samples from an already limited set of available samples to an independent VALIDATION TEST SET seriously reduced the predictive performance of the calibration models, and at the same time gave uncertain, systematically over-optimistic assessment of the models' predictive performance. Full CROSS VALIDATION gave improved predictive performance, and gave only slightly over-optimistic assessment of this predictive performance. Further removal of even more of the available samples for use in an independent VERIFICATION TEST SET gave in-the-long-run correct, although uncertain estimates of the predictive performance of the calibrated models, but this performance level had seriously deteriorated. Alternative verification of the model's predictive performance by the method of CROSS VERIFICATION gave results very similar to those of the cross validation. These results from real data correspond closely to previous findings for artificially simulated data. It appears that full cross validation is superior to both the use of independent validation test set and independent verification test set.

Return to Validation

H. Martens and M. Martens
Modified Jack-knife estimation of parameter uncertainty in bilinear modelling by partial least squares regression (PLSR)
Food Quality and Preference, 11 (2000) 5-16

A method for assessing the uncertainty of the individual bilinear model parameters from two-block regression modelling by multivariate partial least squares regression (PLSR) is presented. The method is based on the so-called "Jack-knife" resampling, comparing the perturbed model parameter estimates from cross-validation with the estimates from the full model. The conventional jack-knifing from ordinary least squares regression is modified in order to compensate for rotational ambiguities of bilinear modelling. The method is intended to make "do-it-yourself" multivariate data-analysis by non-statisticians more safe, in particular in cases with many collinear and noisy regressor and -regressand variables (which is very common in practice). Its use is illustrated by a real example, where the chemical and physical properties of different cocoa drinks are predicted from sensory analysis.

Return to Reliability of multivariate calibration

H. Martens, M. Høy, F. Westad, D. Folkenberg and M. Martens
Analysis of designed experiments by stabilised PLS Regression and jack-knifing
Chemometrics and Intelligent Laboratory Systems, 58 (2001) 151-170

Pragmatical, visually oriented methods for assessing and optimising bi-linear regression models are described, and applied to PLS Regression (PLSR) analysis of multi-response data from controlled experiments. The paper outlines some ways to stabilise the PLSR method to extend its range of applicability to the analysis of effects in designed experiments. Two ways of passifying unreliable variables are shown. A method for estimating the reliability of the cross-validated prediction error RMSEP is demonstrated. Some recently developed jack-knifing extensions are illustrated, for estimating the reliability of the linear and bi-linear model parameter estimates. The paper illustrates how the obtained PLSR "significance" probabilities are similar to those from conventional factorial ANOVA, but the PLSR is shown to give important additional overview plots of the main relevant structures in the multi-response data.
    The study is part of an ongoing effort to establish a cognitively simple and versatile approach to multivariate data analysis, with reliability assessment based on the data at hand, and with little need for abstract distribution theory.

Return to Reliability of multivariate calibration

N.J. Messick, J.H. Kalivas and P.M. Lang
Selectivity and related measures for nth-order data
Analytical Chemistry, 68 (1996) 1572-1579

Analytical figures of merit are often used as criteria to decide whether or not a given instrumental method is suitable for attacking an analytical problem. To date, figures of merit primarily exist for analytical instruments producing data indexed by one variable, i.e., first-order instruments and first-order data. Almost none exist for instruments that generate data indexed by two variables, i.e., second-order instruments and data, and none exist for instruments supplying data indexed by three or more variables, i.e., nth-order instruments and data. This paper develops practical mathematical tools that can be used to create several figures of merit for nth-order instrumentation, namely, selectivity, net analyte signal, and sensitivity. In particular, the paper fully develops a local selectivity measure for second-order instrumentation and tests its performance using simulated second-order data and real second-order data obtained by gas chromatography with Fourier transform infrared detection and liquid chromatography with photodiode array detection. Also included in the paper is a brief discussion on practical uses of nth-order figures of merit.

Return to Analytical figures of merit

L. Milan and J. Whittaker
Application of the parametric bootstrap to models that incorporate a singular value decomposition
Applied Statistics, 44 (1995) 31-49

Simulation is a standard technique for investigating the sampling distribution of parameter estimators. The bootstrap is a distribution-free method of assessing sampling variability based on resampling from the empirical distribution; the parametric bootstrap resamples from a fitted parametric model. However, if the parameters of the model are constrained, and the application of these constraints is a function of the realized sample, then the resampling distribution obtained from the parametric bootstrap may become badly biased and overdispersed. Here we discuss such problems in the context of estimating parameters from a bilinear model that incorporates the singular value decomposition (SVD) and in which the parameters are identified by the standard orthogonality relationships of the SVD. Possible effects of the SVD parameter identification are arbitrary changes in the sign of singular vectors, inversion of the order of singular values and rotation of the plotted co-ordinates. This paper proposes inverse transformation or 'filtering' techniques to avoid these problems. The ideas are illustrated by assessing the variability of the location of points in a principal co-ordinates diagram and in the marginal sampling distribution of singular values. An application to the analysis of a biological data set is described. In the discussion it is pointed out that several exploratory multivariate methods may benefit by using resampling with filtering.

Return to Reliability of principal component analysis

N


B. Nadler and R.R. Coifman
Partial least squares, Beer's law and the net analyte signal: statistical modeling and analysis
Journal of Chemometrics, 19 (2005) 45-54

Partial least squares (PLS) is one of the most common regression algorithms in chemistry, relating input-output samples (xi, yi) by a linear multivariate model. In this paper we analyze the PLS algorithm under a specific probabilistic model for the relation between x and y. Following Beer's law, we assume a linear mixture model in which each data sample (x, y) is a random realization from a joint probability distribution where x is the sum of k components multiplied by their respective characteristic responses, and each of these components is a random variable. We analyze PLS on this model under two idealized settings: one is the ideal case of noise-free samples and the other is the case of an infinite number of noisy training samples. In the noise-free case we prove that, as expected, the regression vector computed by PLS is, up to normalization, the net analyte signal. We prove that PLS computes this vector after at most k iterations, where k is the total number of components. In the case of an infinite training set corrupted by unstructured noise, we show that PLS computes a final regression vector which is not in general purely proportional to the net analyte signal vector, but has the important property of being optimal under a mean squared error of prediction criterion. This result can be viewed as an asymptotic optimality of PLS in the limit of a very large but finite training set.

Return to Analytical figures of merit or return to Reliability of multivariate calibration

B. Nadler and R.R. Coifman
The prediction error in CLS and PLS: the importance of feature selection prior to multivariate calibration
Journal of Chemometrics, 19 (2005) 107-118

Classical least squares (CLS) and partial least squares (PLS) are two common multivariate regression algorithms in chemometrics. This paper presents an asymptotically exact mathematical analysis of the mean squared error of prediction of CLS and PLS under the linear mixture model commonly assumed in spectroscopy. For CLS regression with a very large calibration set the root mean squared error is approximately equal to the noise per wavelength divided by the length of the net analyte signal vector. It is shown, however, that for a finite training set with n samples in p dimensions there are additional error terms that depend on s2p2/n2, where s is the noise level per co-ordinate. Therefore in the 'large p—small n' regime, common in spectroscopy, these terms can be quite large and even dominate the overall prediction error. It is demonstrated both theoretically and by simulations that dimensional reduction of the input data via their compact representation with a few features, selected for example by adaptive wavelet compression, can substantially decrease these effects and recover the asymptotic error. This analysis provides a theoretical justification for the need to perform feature selection (dimensional reduction) of the input data prior to application of multivariate regression algorithms.

Return to Reliability of multivariate calibration

M.N. Nounou, B.R. Bakshi, P.K. Goel and X. Shen
Bayesian principal component analysis
Journal of Chemometrics, 16 (2002) 576-595

Principal component analysis (PCA) is a dimensionality reduction modeling technique that transforms a set of process variables by rotating their axes of representation. Maximum likelihood PCA (MLPCA) is an extension that accounts for different noise contributions in each variable. Neither PCA nor any of its extensions utilizes external information about the model or data, such as the range or distribution of the underlying measurements. Such prior information can be extracted from measured data and can be used to greatly enhance the model accuracy. This paper develops a Bayesian PCA (BPCA) modeling algorithm that improves the accuracy of estimating the parameters and measurements by incorporating prior knowledge about the data and model. The proposed approach integrates modeling and feature extraction by simultaneously solving parameter estimation and data reconciliation optimization problems. Methods for estimating the prior parameters from available data are discussed. Furthermore, BPCA reduces to PCA or MLPCA when a uniform prior is used. Several examples illustrate the benefits of BPCA versus existing methods even when the measurements violate the assumptions about their distribution.

Return to Reliability of principal component analysis or return to Maximum likelihood calibration

O


H. Ogasawara
Standard errors of the principal component loadings for unstandardized and standardized variables
British Journal of Mathematical and Statistical Psychology, 53 (2000) 155-174

The asymptotic standard errors of the estimates of the principal component loadings for standardized variables are derived under the assumption of multivariate normality. The standard errors are obtained for the usual unrotated case where a loading matrix is orthogonal and for the cases with orthogonally or obliquely rotated components. The corresponding standard errors for unstandardized variables and the asymptotic correlations among the estimators of the parameters for the unstandardized and standardized variables are simultaneously derived, together with the standard errors for the standardized variables. Results of a simulation illustrate the accuracy of the theoretical asymptotic standard errors and correlations.

Return to Reliability of principal component analysis

H. Ogasawara
Concise formulas for the standard errors of component loading estimates
Psychometrika, 67 (2002) 289-297

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.

Return to Reliability of principal component analysis

H. Ogasawara
Asymptotic biases of the unrotated/rotated solutions in principal component analysis
British Journal of Mathematical and Statistical Psychology, 57 (2004) 353-376

Asymptotic biases of the parameter estimates in principal component analysis with substantial misspecification are derived. The solutions for unstandardized and standardized observed variables are considered with and without orthogonal and oblique rotations. The distribution of observed variables can be non-normal as long as the finite fourth-order moments of the observed variables exist. When multivariate normality holds for the observed variables, substantial reduction of the amount of computation can be achieved. Numerical examples with simulations are given, with some discussion on the tendency of the biases to reduce the absolute values of parameter estimates.

Return to Reliability of principal component analysis

H. Ogasawara
Higher-order asymptotic standard error and asymptotic expansion in principal component analysis
Communications in Statistics - Simulation and Computation, 35 (2006) 201-223

Asymptotic expansions of the distributions of the estimators of unrotated and orthogonally rotated component loadings are given under nonnormality of observed variables in principal component analysis for sample covariance and correlation matrices. The expansions include those for the Studentized statistics of the estimators with unknown standard errors. The expansions with the adjustment of the higher-order asymptotic variance of estimators are also presented with weight for partial adjustment. The formula is applied to the estimators of the contributions of unrotated/rotated components as well as their loadings, which includes eigenvalues as special cases. Simulations were performed to see the accuracy of the asymptotic moments and the higher-order standard errors in samples with finite sample sizes.

Return to Reliability of principal component analysis

A.C. Olivieri
A simple approach to uncertainty propagation in preprocessed multivariate calibration
Journal of Chemometrics, 16 (2002) 207-217

A simple approach is described to estimate the confidence limit for the concentrations predicted by multivariate calibration when preprocessing techniques such as orthogonal signal correction or net analyte calculation are applied. It involves reconstructing the unpreprocessed data using the extracted spectral factors and those employed for prediction in order to correctly estimate the sample leverage. Monte Carlo simulations carried out by adding random noise to both concentrations and analytical signals for theoretical binary mixtures are in excellent agreement with the calculations. Experimental multicomponent examples were studied by a similar Monte Carlo approach, and the obtained variances are also in agreement with the calculated values. Implications concerning the limits of detection in the latter case are also discussed.

Return to Reliability of multivariate calibration

A.C. Olivieri and N.M. Faber
Standard error for prediction in parallel factor (PARAFAC) analysis of three-way data
Chemometrics and Intelligent Laboratory Systems, 70 (2004) 75-82

A simple approach is described to calculate sample-specific standard errors for the concentrations predicted by a three-way parallel factor (PARAFAC) analysis model. It involves a first-order error propagation equation in which the correct leverages and sensitivity values are introduced. Monte Carlo simulation results obtained by adding random noise to both concentrations and instrumental signals for theoretical binary mixtures are in good agreement with the proposed approach. An experimental multicomponent example was studied by a similar Monte Carlo approach, and the obtained standard errors are also in agreement with the calculated values. Implications concerning the limit of detection are discussed.

Return to Reliability of multiway calibration

A. Olivieri, N.M. Faber, J. Ferré, R. Boqué, J.H. Kalivas and H. Mark
Guidelines for calibration in analytical chemistry
Part 3. Uncertainty estimation and figures of merit for multivariate calibration
Pure & Applied Chemistry, 78 (2006) 633-661

This paper gives an introduction to multivariate calibration from a chemometrics perspective and reviews the various proposals to generalize the well-established univariate methodology to the multivariate domain. Univariate calibration leads to relatively simple models with a sound statistical underpinning. The associated uncertainty estimation and figures of merit are thoroughly covered in several official documents. However, univariate model predictions for unknown samples are only reliable if the signal is sufficiently selective for the analyte of interest. By contrast, multivariate calibration methods may produce valid predictions also from highly unselective data. A case in point is quantification from near-infrared (NIR) spectra. With the ever-increasing sophistication of analytical instruments inevitably comes a suite of multivariate calibration methods, each with its own underlying assumptions and statistical properties. As a result, uncertainty estimation and figures of merit for multivariate calibration methods has become a subject of active research, especially in the field of chemometrics.

Return to Official literature

M.C. Ortiz, J. Arcos, J.V. Juarros, J. López-Palacios and L.A. Sarabia
Robust procedure for calibration and calculation of the detection limit of trimipramine by adsorptive stripping voltammetry at a carbon paste electrode
Analytical Chemistry, 65 (1993) 678-682

A method for the determination of trimipramine by adsorptive stripping voltammetry using a carbon paste electrode has been developed. For routine calibration and calculation of the detection Iimit, a robust regression method (least median squares) has been proposed, providing the method with adequate reliability and detectability. The methodology used in this work overcomes the experimental drawbacks arising from the need to renew the electrode surface after each measurement. The detection limits thus achieved (1.16 × 10-8 to 2.41 × 10-8 M) take into account the sensitivity of the analytical method, the nature of the analyte, and the risk of false positive and false negative results the analyst is willing to accept.

Return to Limit of detection

M.C. Ortiz, L.A. Sarabia, A. Herrero, M. S. Sánchez, M.B. Sanz, M.E. Rueda, D. Giménez and M.E. Meléndez
Capability of detection of an analytical method evaluating false positive and false negative (ISO 11843) with partial least squares
Chemometrics and Intelligent Laboratory Systems, 69 (2003) 21-33

Analytical techniques based on soft multivariate calibrations (as those which provide first and second order analytical signals necessarily are) remain outside the field of application of the ISO norms related to capability of detection. In this work, a complete solution for the problem of applying ISO norm 11843 to soft calibration (for instance, one or multi-way partial least squares (PLS)) is provided. The methodological procedure is applied to different case studies which implies different analytical techniques.

Return to Limit of detection

M.C. Ortiz, L.A. Sarabia, I. García, D. Giménez and E. Meléndez
Capability of detection and three-way data
Analytica Chimica Acta, 559 (2006) 124-136

Due to the possibility of making analytical determinations in the presence of non-modelled interferents and to identify the analyte of interest, calibrations based on scores of PARAFAC decomposition of three-way data are becoming increasingly important in routine analysis.
    Furthermore, the IUPAC and EU (European Decision 2002/657/EC) have accepted the definition given by the ISO 11843 for the capability of detection as the minimum net quantity detectable with a pre-set probability of false positive and false negative. What is more, recently our research group has generalised this definition of capability of detection, CCb, to multivariate calibrations. In practice, CCb is a good measure of the quality of the calibration because in its definition it brings together analytical sensitivity with precision in analytical determinations.
    This paper studies the effect of the pre-treatment of the sample, the signal/noise ratio and the second-order advantage on CCb when using second-order signals modelled by PARAFAC. All of them are experimental factors which influence the quality of the calibration. Analytical pre-treatment is habitual in the analysis of real samples. Specifically, we analyse the effect of the extraction phase and the clean-up of milk samples on the determination of chlortetracycline by HPLC-DAD. It is shown that it is more efficient to do the joint PARAFAC decomposition of the pure standards with the milk samples.
    Secondly, the effect of asymmetry on CCb, according to the path of the noise of the signals, is studied. Specifically, in the determination of naphthalene by excitation-emission spectroscopy, EEM, it is the emission spectrum which limits the capability of detection. It is shown that by eliminating the spectra with the poorest signal/noise ratio in this path, the capability of detection can be substantially improved.
    Thirdly, the impact on CCb when the second-order advantage is used, that is when PARAFAC calibration is used over samples with an unknown interference not modelled in the calibration step. This is important to apply a PARAFAC calibration to routine analysis in the IUPAC and European Decision framework. Specifically, in the determination of enrofloxacine in poultry feeding water through excitation-emission fluorescence CCb is evaluated when the PARAFAC is built only with calibration samples or with the calibration samples plus the test samples with uncalibrated and unknown interferent.

Return to Limit of detection

M.C. Ortiz, L.A. Sarabia and A. Herrero
Robust regression techniques. A useful alternative for the detection of outlier data in chemical analysis
Talanta, 70 (2006) 499-512

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 andWelsh 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.

Return to Limit of detection

P


P. Paatero
A weighted non-negative least squares algorithm for three-way 'PARAFAC' factor analysis
Chemometrics and Intelligent Laboratory Systems, 38 (1997) 223-242

A time-efficient algorithm PMF3 is presented for solving the three-way PARAFAC (CANDECOMP) factor analytic model. In contrast to the usual alternating least squares, the PMF3 algorithm computes changes to all three modes simultaneously. This typically leads to convergence in 40-100 iteration steps. The equations of the weighted multilinear least squares fit are given. The optional non-negativity is achieved by imposing a logarithmic penalty function. The algorithm contains a possibility for dynamical reweighting of the data during the iteration, allowing a robust analysis of outlier-containing data. The problems typical of PARAFAC models are discussed (but not solved): multiple local solutions, degenerate solutions, non-identifiable solutions. The question of how to verify the solution is discussed at length. The program PMF3 is available for 486-Pentium based PC computers.

Return to Reliability of multiway calibration

A. Phatak, P.M. Reilly and A. Penlidis
An approach to interval estimation in partial least squares regression
Analytica Chimica Acta, 277 (1993) 495-501

Although partial least squares regression (PLS) is widely used in chemometrics for quantitative spectral analysis, little is known about the distribution of the prediction error from calibration models based on PLS. As a result, we must rely on computationally intensive procedures like bootstrapping to produce confidence intervals for predictions, or, in many cases, we must do with no interval estimates at all, only point estimates. In this paper we present an approach, based on the linearization of the PLS estimator, that allows us to construct approximate confidence intervals for predictions from PLS.

Return to Reliability of multivariate calibration

Q


R


M.S. Reis and P.M. Saraiva
Integration of data uncertainty in linear regression and process optimization
AIChE Journal, 51 (2005) 3007-3019

Data uncertainties provide important information that should be taken into account along with the actual data. In fact, with the development of measurement instrumentation methods and metrology, one is very often able to rigorously specify the uncertainty associated with each measured value. The use of this piece of information, together with raw measurements, should—in principle—lead to more sound ways of performing data analysis, empirical modeling, and subsequent decision making. In this paper, we address the issues of using data uncertainty in the task of model estimation and, when it is already available, we show how the integration of measurement and actuation uncertainty can be achieved in the context of process optimization. Within the scope of the first task (model estimation), we make reference to several methods designed to take into account data uncertainties in linear multivariate regression (multivariate least squares, maximum likelihood principal component regression), and others whose potential to deal with noisy data is well known (partial least squares, principal component regression, and ridge regression), as well as modifications of previous methods that we developed, and compare their performance. MLPCR2 tends to achieve better predictive performance than all the other tested methods. The potential benefits of including measurement and actuation uncertainties in process optimization are also illustrated.

Return to Maximum likelihood calibration

M.S. Reis and P.M. Saraiva
Heteroscedastic latent variable modelling with applications to multivariate statistical process control
Chemometrics and Intelligent Laboratory Systems, 80 (2006) 57-66

We present an approach for conducting multivariate statistical process control (MSPC) in noisy environments, i.e., when the signal to noise ratio is low, and, furthermore, noise standard deviation (uncertainty) affecting each collected value can vary over time, and is assumingly known. This approach is based upon a latent variable model structure, HLV (standing for heteroscedastic latent variable model), that explicitly integrates information regarding data uncertainty. Moderate amounts of missing data can also be handled in a coherent and fully integrated way through HLV. Several examples show the added value achieved under noisy conditions by adopting such an approach and a case study illustrates its application to a real industrial context of pulp and paper product quality data analysis.

Return to Maximum likelihood calibration

G.E. Ritchie, R.W. Roller, E.W. Ciurczak, H. Mark, C. Tso and S.A. MacDonald
Validation of a near-infrared transmission spectroscopic procedure, Part B: Application to alternate content uniformity and release assay methods for pharmaceutical solid dosage forms
Journal of Pharmaceutical and Biomedical Analysis, 29 (2002) 259-271

NIR analytical methods can be validated to meet the requirements of demonstrating that it is suitable for the analysis of the metrials for which it is being used. Applying previously described protocols for NIR methods to the analysis of two types of pharmaceutical products shows that for these products, NIR is suitable as an alternate analytical method for assay and for content uniformity.

Return to Validation

J. Riu and R. Bro
Jack-knife technique for outlier detection and estimation of standard errors in PARAFAC models
Chemometrics and Intelligent Laboratory Systems, 65 (2003) 35-49

In the last years, multi-way analysis has become increasingly important because it has proved to be a valuable tool, e.g. in interpreting data provided by instrumental methods that describe the multivariate and complex reality of a given problem. Parallel factor analysis (PARAFAC) is one of the most widely used multi-way models. Despite its usefulness in many applications, up to date there is no available tool in the literature to estimate the standard errors associated with the parameter estimates. In this study, we apply the so-called jack-knife technique to PARAFAC in order to find the associated standard errors to the parameter estimates from the PARAFAC model. The jack-knife technique is also shown to be useful for detecting outliers. An example of fluorescence data (emission/excitation landscapes) is used to show the applicability of the method.

Return to Reliability of multiway calibration

B.A. Roscoe and P.K. Hopke
Error estimates for factor loadings and scores obtained with target transformation factor analysis
Analytica Chimica Acta, 132 (1981) 89-97

Methods of calculating the errors associated with the reproduction of data from the results of target-transformation factor analysis are demonstrated. Errors in the factor loadings and scores are produced by two methods: the jack-knife method, which is time-consuming, and a faster calculation procedure. The agreement shown between the two methods demonstrates the effectiveness of the calculation approach as a quick and simple method of error estimation.

Return to Reliability of principal component analysis

D.N. Rutledge, A. Barros and I. Delgadillo
PoLiSh—smoothed partial least-squares regression
Analytica Chimica Acta, 446 (2001) 281-296

Partial least-squares (PLS) regression is a very widely used technique in spectroscopy for calibration/prediction purposes. One of the most important steps in the application of the PLS regression is the determination of the correct number of dimensions to use in order to avoid over-fitting, and therefore to obtain a robust predictive model. The "structured" nature of spectroscopic signals may be used in several ways as a guide to improve the PLS models. The aim of this work is to propose a new technique for the application of PLS regression to signals (FT-IR, NMR, etc.). This technique is based on the Savitsky—Golay (SG) smoothing of the loadings weights vectors (w) obtained at each iteration step of the NIPALS procedure. This smoothing progressively "displaces" the random or quasi-random variations from earlier (most important) to later (less important) PLS latent variables. The Durbin—Watson (DW) criterion is calculated for each PLS vectors (p, w, b) at each iteration step of the smoothed NIPALS procedure in order to measure the evolution of their "noise" content. PoLiSh has been applied to simulated datasets with different noise levels and it was found that for those with noise levels higher than 10—20%, an improvement in the predictive ability of the models is observed. This technique is also important as a tool to evaluate the true dimensionality of signal matrices for complex PLS models, by comparing the DW profile of the PoLiSh vectors at different smoothing degrees with those of the unsmoothed PLS models.

Return to Component selection

S


M.B. Sanz, L.A. Sarabia, A. Herrero and M.C. Ortiz
Capability of discrimination: application to soft calibration methods
Analytica Chimica Acta, 446 (2001) 297-311

Given a nominal concentration, in order to know the behaviour of an analytical procedure in samples with similar concentration, the minimum discriminable concentration is defined as the smallest concentration of the analyte in a sample which can be distinguished, with probability 1 - β, from the nominal value. The definition generalises the concept of minimum detectable net concentration established by ISO norm 11843 which is restricted to the case in which the nominal concentration is zero. Given an analytical procedure with a well-established net detectable concentration (detection limit), it may not be possible to discriminate this same concentration when the procedure is used in samples with a much higher nominal concentration. For this reason, the discrimination capability is a criterion for the selection of an analytical procedure when it is going to be used to determine concentrations well above its detection limit. The discrimination capability is established as a hypothesis test based on the data of a calibration carried out in a range of concentrations which contains the nominal value.
    As an application, the discrimination capability has been estimated when the concentration of a sample test is obtained by means of a partial least squares (PLS) calibration. In this case, the proposed procedure is composed of three steps, the first of which consists of the soft multivariate calibration. The second step is the evaluation of the discrimination capability by a regression of the concentration found with the multivariate calibration versus the true concentration of a new set of reference samples. The standard deviation of the regression estimates the repeatability (analytical procedure and soft calibration jointly) as the concentration range analysed in the regression. In the third step, the capability of discrimination calculated is applied to a new test sample if the repeatability has not changed as the level of concentration considered.
    The procedure developed has been applied to the determination of benzaldehyde by means of differential pulse polarography (DPP), where univariate calibration cannot be applied and a PLS calibration is appropriate. The capability of discrimination has been evaluated at two different concentration ranges: from 0.10 to 1.05 μM, and from 0.0199 to 0.1740 mM, with an estimated repeatability of 3.1 × 10-2 μM and 3.2 × 10-3 mM, respectively. The "DPP + PLS" capability of discrimination performance is analysed by means of the "false noncompliance" and the "false compliance" probability in the case that the analysis method has to discriminate differences lower than 10% of the nominal concentration with sufficient warranty.

Return to Analytical figures of merit

M.B. Sanz, L.A. Sarabia, A. Herrero and M.C. Ortiz
Multivariate analytical sensitivity in the determination of selenium, copper, lead and cadmium by stripping voltammetry when using soft calibration
Analytica Chimica Acta, 489 (2003) 85-94

A new approach to the multivariate sensitivity concept based on the determination of the capability of discrimination of a method of analysis is shown. Thus the analytical sensivity is defined in this work by the analyte concentration that a analytical method is able to discriminate, which implies the estimation of the 'false noncompliance' and 'false compliance'. In this approach the estimation of the multivariate analytical sensivity is independent of scale factors and calibration models, and allows one to study the behaviour of a analytical method for several concentrations and matrix. The estimation of this parameter in the simultaneous determination of selenium, copper, lead and cadmium by stripping voltammetry when using soft calibration is carried out, showing that different multivariate analytical sensivities are obtained for each metal.

Return to Analytical figures of merit

L. Sarabia and M.C. Ortiz
DETARCHI: a program for detection limits with specified assurance probabilities and characteristic curves of detection
Trends in Analytical Chemistry, 13 (1994) 1-6

C.A. Clayton and his collaborators [Anal. Chem., 59 (1987) 2506] established the detection limit as an hypothesis test to determine the presence or absence of an analyte in a sample, with an evaluation of the probability of obtaining a false negative or false positive. The program DETARCHI implements this method and calculates the characteristic curves of detection associated with the test.

Return to Limit of detection

L.A. Sarabia, M. Cruz Ortiz, M. Julia Arcos, M. Sagrario Sánchez, A. Herrero and S. Sanllorente
Multivariate detection capability using a neural classifier for nonselective signals
Chemometrics and Intelligent Laboratory Systems, 61 (2002) 89-104

A new methodology is proposed based on a neural network to determine the detection capability of an analytical procedure, in complex matrices, with the evaluation of the probability of false detection, a, and false nondetection, b, according to the ISO norms. This methodology is designed for first or greater order signals for which there is currently no procedure with these characteristics, which makes it difficult to use these signals in analytical procedures standardized according to the ISO norm.
    The procedure consists of: (i) an experimental design suited to the increase in analyte to be detected from a threshold level; (ii) a homogenisation of the multivariate signals by a Piecewise Direct Standardization (PDS) transformation; (iii) the training of a neural network with stochastic learning, Genetic Inside Neural Network (GINN), which optimizes a and b directly.
    The procedure was applied to the polarographic determination of Tl(I)/Pb(II) mixtures and indomethacin/tenoxicam mixtures. In the first case one can assure the detection of 1 mM (threshold: 12 mM) with a and b less than 5% for both metals. While for the tenoxicam it is possible to detect less than 10% of 12 mM (threshold) with a <10% and b <5%, for indomethacin one can assure less than 10% of 86 mM (threshold) with a and b less than 5%.

Return to Limit of detection

J. Saurina, C. Leal, R. Compañó, M. Granados, M. Dolors Prat and R. Tauler
Estimation of figures of merit using univariate statistics for quantitative second-order multivariate curve resolution
Analytica Chimica Acta, 432 (2001) 241-251

In this study, a straightforward approach is proposed for determining the figures of merit in multivariate curve resolution. The method is based on the recovery of the pure response profiles of the analytes after a curve resolution procedure. Figures of merit such as limit of detection, sensitivity, precision and accuracy are estimated from these pure analyte responses using univariate statistics directly from a calibration graph, as usual in univariate calibration. Examples of the calculation of the figures of merit in the determination of triphenyltin in synthetic and natural sea water samples by using excitation-emission matrix fluorescence are given.

Return to Analytical figures of merit

S.K. Schreyer, M. Bidinosti and P.D. Wentzell
Application of maximum likelihood principal components regression to fluorescence emission spectra
Applied Spectroscopy, 56 (2002) 789-796

The application of maximum likelihood multivariate calibration methods to the fluorescence emission spectra of mixtures of acenaphtylene, naphtalene, and phenanthrene in acetonitrile is described. Maximum likelihood principal components regression (MLPCR) takes into account the measurement error structure in the spectral data in constructing the calibration model. Measurement errors for the fluorescence spectra are shown to exhibit both a heteroscedastic and correlated noise structure. MLPCR is compared with principal components regression (PCR) and partial least-squares regression (PLS). The application of MLPCR reduces the prediction errors by about a factor of two over PCR and PLS when a pooled estimate of the measurement error covariance matrix is employed. However, when only the heteroscedasticity is incorporated into MLPCR, no improvement in results is observed, indicating the importance of accounting for correlated measurement errors.

Return to Maximum likelihood calibration

M. Schuermans, I. Markovskya, P.D. Wentzell and S. Van Huffel
On the equivalence between total least squares and maximum likelihood PCA
Analytica Chimica Acta, 544 (2005) 254-267

The maximum likelihood PCA (MLPCA) method has been devised in chemometrics as a generalization of the well-known PCA method in order to derive consistent estimators in the presence of errors with known error distribution. For similar reasons, the total least squares (TLS) method has been generalized in the field of computational mathematics and engineering to maintain consistency of the parameter estimates in linear models with measurement errors of known distribution. The basic motivation for TLS is the following. Let a set of multidimensional data points (vectors) be given. How can one obtain a linear model that explains these data? The idea is to modify all data points in such a way that some norm of the modification is minimized subject to the constraint that the modified vectors satisfy a linear relation. Although the name "total least squares" appeared in the literature only 25 years ago, this method of fitting is certainly not new and has a long history in the statistical literature, where the method is known as "orthogonal regression", "errors-in-variables regression" or "measurement error modeling". The purpose of this paper is to explore the tight equivalences between MLPCA and element-wise weighted TLS (EW-TLS). Despite their seemingly different problem formulation, it is shown that both methods can be reduced to the same mathematical kernel problem, i.e. finding the closest (in a certain sense) weighted low rank matrix approximation where the weight is derived from the distribution of the errors in the data. Different solution approaches, as used in MLPCA and EW-TLS, are discussed. In particular, we will discuss the weighted low rank approximation (WLRA), the MLPCA, the EW-TLS and the generalized TLS (GTLS) problems. These four approaches tackle an equivalent weighted low rank approximation problem, but different algorithms are used to come up with the best approximation matrix.We will compare their computation times on chemical data and discuss their convergence behavior.

Return to Maximum likelihood calibration

M.B. Seasholtz and B. Kowalski
The parsimony principle applied to multivariate calibration
Analytica Chimica Acta, 277 (1993) 165-177

The general principle of parsimonious data modeling states that if two models in some way adequately model a given set of data, the one that is described by a fewer number of parameters will have better predictive ability given new data. This concept is of interest in multivariate calibration since several new non-linear modeling techniques have become available. Three such methods are neural networks, projection pursuit regression (PPR) and multivariate adaptive regression splines (MARS). These methods, while capable of modeling non-linearities, typically have very many parameters that need to be estimated during the model building phase. The biased calibration methods, principal components regression (PCR) and partial least squares (PLS) are linear methods and so may not as efficiently describe some types of non-linearities, however have comparably very few parameters to be estimated. It is therefore of interest to study the parsimony principle formally in order to understand under what circumstances the various methods are appropriate. In this paper, the mathematical theory of parsimonious data modeling is presented. The assumptions made in the theory are shown to hold for multivariate calibration methods. This theory is used to provide a procedure for selecting the most parsimonious model structure for a given calibration application.

Return to General on calibration

H. Seipel and J.H. Kalivas
Effective rank for multivariate calibration methods
Journal of Chemometrics, 18 (2004) 306-311

In order to determine the proper multivariate calibration model, it is necessary to select the number of respective basis vectors (latent vectors, factors, etc.) when using principal component regression (PCR) or partial least squares (PLS). These values are commonly referred to as the prediction rank of the model. Comparisons between PCR and PLS models for a given data set are often made with the prediction rank to determine the more parsimonious model, ignoring the fact that the values have been obtained using different basis sets. Additionally, it is not possible to use this approach for determining the prediction rank of models generated by other modeling methods such as ridge regression (RR). This paper presents measures of effective rank for a given model that can be applied to all modeling methods, thereby providing inter-model comparisons. A definition based on the regression vector norm and is compared with two alternative forms from the literature. With a proper definition of effective rank, a better assessment of degrees of freedom for statistical computations is possible. Additionally, the true nature of variable selection for improved parsimony can be properly assessed. Spectroscopic data sets are used as examples with PCR, PLS and RR.

Return to Component selection

S. Serneels, P. Lemberge and P. J. Van Espen
Sample specific prediction intervals in SIMPLS
In: M. Vilares, M. Tenenhaus, P. Coelho, V. Esposito Vinzi and A. Morineau (eds.), PLS and related methods, pp. 219-233. DECISIA, Levallois Perret, France (2003)

In this article, we propose two new differentiated algorithms for SIMPLS. These algorithms allow efficient computation of the Jacobian matrix for the regression vector, taking into account error propagation of either X, y or both X and y, the last case being better known as the errors-in-variables problem. The Jacobian matrices thus computed lead to a new way of calculating a sample-specific prediction error in SIMPLS regression.

Return to Reliability of multivariate calibration

S. Serneels, P. Lemberge and P.J. Van Espen
Calculation of PLS prediction intervals using efficient recursion relations for the Jacobian matrix
Journal of Chemometrics, 18 (2004) 76-80

Several algorithms to calculate the vector of regression coefficients and the Jacobian matrix for partial least squares regression have been published. Whereas many efficient algorithms to calculate the regression coefficients exist, algorithms to calculate the Jacobian matrix are inefficient. Here we introduce a new, efficient algorithm for the Jacobian matrix, thus making the calculation of prediction intervals via a local linearization of the PLS estimator more practicable.

Return to Reliability of multivariate calibration

S. Serneels, C. Croux and P.J. Van Espen
Influence properties of partial least squares regression
Chemometrics and Intelligent Laboratory Systems, 71 (2004) 13-20

In this paper, we compute the influence function (IF) for partial least squares (PLS) regression. Thereunto, we design two alternative algorithms, according to the PLS algorithm used. One algorithm for the computation of the influence function is based on the Helland PLS algorithm, whilst the other is compatible with SIMPLS.
    The calculation of the influence function leads to new influence diagnostic plots for PLS. An alternative to the well-known Cook's Distance (CD) plot is proposed, as well as a variant which is sample specific. Moreover, a novel estimate of prediction variance is deduced. The validity of the latter is corroborated by dint of a Monte Carlo simulation.

Return to Reliability of multivariate calibration

S. Serneels, M. Moens, P.J. Van Espen and F. Blockhuys
Identification of micro-organisms by dint of the electronic nose and trilinear partial least squares regression
Analytica Chimica Acta, 516 (2004) 1-5

Ventilator-associated pneumonia is one of the most lethal infections occurring in intensive care units of hospitals. In order to obtain a faster method of diagnosis, we proposed to apply the electronic nose to cultures of the relevant micro-organisms. This allowed to halve the time of the analysis. In the current paper, we focus on the application of some chemometrical tools which enhance the performance of the method. Trilinear partial least squares (tri-PLS) regression is used to perform calibration and is shown to produce satisfactory predictions. Sample specific prediction intervals are produced for each predicted value, which allows us to eliminate erroneous predictions. The method is applied to an external validation set and it is shown that only a single observation out of 22 is being wrongly classified, so that the method is acceptable for inclusion in the clinical routine.

Return to Reliability of multiway calibration

S. Serneels, P. Geladi, M. Moens, F. Blockhuys and P.J. Van Espen
Influence properties of trilinear partial least squares regression
Journal of Chemometrics, 19 (2005) 405-411

In this article we derive an algorithm to compute the influence function for tri-PLS1 regression. Based on the influence function, we propose the squared influence diagnostic plot to assess the influence of individual samples on calibration and prediction. We illustrate the applicability of the squared influence diagnostic plot for tri-PLS1 to two different data sets which have previously been reported in literature. Finally we note that from the influence function, a new estimate of prediction variance can be obtained.

Return to Reliability of multiway calibration

S. Serneels and P.J. Van Espen
Bootstrap confidence intervals for trilinear partial least squares regression
Analytica Chimica Acta, 544 (2005) 153-158

The boostrap is a successful technique to obtain confidence limits for estimates where it is theoretically impossible to establish an exact expression thereunto. Trilinear partial least squares regression (tri-PLS) is an estimator for which this is the case; in the current paper we thus propose to apply the bootstrap in order to obtain confidence intervals for the predictions made by tri-PLS. By dint of an extensive simulation study, we show that bootstrap confidence intervals have a desirable coverage. Finally, we apply the method to an identification problem of micro-organisms and show that from the bootstrap confidence intervals, the organisms can (up to a misclassification probability of 3.5%) correctly be identified.

Return to Reliability of multiway calibration

P. Stoica and T. Söderström
Partial least squares: a first-order analysis
Scandinavian Journal of Statistics, 25 (1998) 17-24

We compare the partial least squares (PLS) and the principal component analysis (PCA), in a general case in which the existence of a true linear regression is not assumed. We prove under mild conditions that PLS and PCA are equivalent, to within a first-order approximation, hence providing a theoretical explanation for empirical findings reported by other researchers. Next, we assume the existence of a true linear regression equation and obtain asymptotic formulas for the bias and variance of the PLS parameter estimator.

Return to Reliability of multivariate calibration or return to Method comparison studies

R. Sundberg
Interplay between chemistry and statistics, with special reference to calibration and the generalized standard addition method
Chemometrics and Intelligent Laboratory Systems, 4 (1988) 299-305

Ever since the beginning of this century problems in the field of chemistry and chemical engineering have played an important role in the development of statistics, and statistical models and methods have proved to be an efficient tool for planning of chemical experiments and in the analysis of data. After mentioning a number of mutually important statistical methods and chemical problem areas, I will discuss in more detail the interplay and lack thereof in the field of calibration. In particular I will describe and discuss the generalized standard addition method (GSAM) with a two-fold intention: to propose statistically motivated improvements and future developments in the theory of estimation and design for this method of multicomponent analysis, as well as to illustrate how a statistician looks at the topic, for the possible benefit of chemists in their future contacts with statistics and statisticians.

Return to General on calibration

R. Sundberg
Multivariate calibration - direct and indirect regression methodology
Scandinavian Journal of Statistics, 26 (1999) 161-207

This paper tries first to introduce and motivate the methodology of multivariate calibration. Next a review is given, mostly avoiding technicalities, of the somewhat messy theory of the subject. Two approaches are distinguished: the estimation approach (controlled calibration) and the prediction approach (natural calibration). Among problems discussed are the choice of estimator, the choice of confidence region, methodology for handling situations with more variables than observations, near-collinearities (with countermeasures like ridge type regression, principal components regression, partial least squares regression and continuum regression), pretreatment of data, and cross-validation vs true prediction. Examples discussed in detail concern estimation of the age of a rhinoceros from its horn lengths (low-dimensional), and nitrate prediction in waste-water from high-dimensional spectroscopic measurements.

Return to General on calibration

J.M. Sutter, J.H. Kalivas and P.M. Lang
Which principal components to utilize for principal component regression
Journal of Chemometrics, 6 (1992) 217-225

Principal components (PCs) for principal component regression (PCR) have historically been selected from the top down for a reliable predictive model. That is, the PCs are arranged in a list starting with the most informative (PC associated with the largest singular value) and proceeding to the least informative (PC associated wit the smallest singular value). PCs are then chosen starting at the top of this list. This paper discusses an alternative procedure of treating PC selection as an optimization problem. Specifically, without any regard to the ordering, the optimal subset of PCs for an acceptable predictive model is desired. Five data sets are analyzed using the conventional and alternative approaches. Two data sets are spectroscopic in nature, two data sets deal with quantitative structure activity relationships (QSARs) and one data set is concerned with modeling. All five data sets confirm that selection of a subset without consideration to order secures the best results with PCR. One data set is also compared using partial least squares 1.

Return to Component selection

T


J. Tellinghuisen
Inverse vs. classical calibration for small data sets
Fresenius Journal of Analytical Chemistry, 368 (2000) 585-588

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.

Return to Reliability of univariate calibration

J. Tellinghuisen
A simple, all-purpose nonlinear algorithm for univariate calibration
Analyst, 125 (2000) 1045-1048

The goal of univariate calibration is the estimation of some unknown concentration or amount and its uncertainty, from the sample's response to a probe measurement, by reference to a set of known samples whose responses are similarly measured. For this purpose, a simple nonlinear algorithm has been devised, in which the unknown sample is included as the (n + 1)-th data point in the data set and is fitted directly to the unknown concentration while the n calibration points are fitted to the calibration function. For example, in straight-line calibration, y = a + bx, the response y0 of the unknown is fitted to y0 = a + bx0. The standard error in the estimated concentration x0 is obtained directly from the variance-covariance matrix for the fit. The method handles homo- and heteroscedastic data equally well, and is easily extended to more complex linear calibration functions (e.g., polynomials in x) and to nonlinear functions (exponentials, logs). Its implementation is illustrated for a typical microcomputer data analysis program.

Return to Reliability of univariate calibration

J. Tellinghuisen
Simple algorithms for nonlinear calibration by the classical and standard additions methods
Analyst, 130 (2005) 370-378

In univariate calibration by both the classical method and the standard additions method, calibration data are fitted to a response function y = f(x), from which the amount of an unknown x0 is estimated by solving an equation of form y0 = f(x0). Most such calibrations are limited to linear response functions f, for which the uncertainty in x0 can be estimated from well-known expressions. The present work describes and illustrates one-step algorithms, in which x0 is treated as an adjustable parameter in a nonlinear fit of the calibration data, with its standard error thus obtained numerically as a direct outcome of the fit. The computations are easily implemented with a number of data analysis programs and are illustrated here for a representative one, KaleidaGraph™. The methods handle heteroscedastic data as easily as homoscedastic and nonlinear functions as easily as linear, permitting the analyst to experiment with different response functions in the quest for an optimum calibration. The estimates of sx0 are obtained from the variance-covariance matrix V for the fit, so for weighted fitting with commercial programs, it is important to know which Va priori or a posteriori—is being used.

Return to Reliability of univariate calibration

E.V. Thomas
Non-parametric statistical methods for multivariate calibration model selection and comparison
Journal of Chemometrics, 17 (2003) 653-659

Model selection is an important issue when constructing multivariate calibration models using methods based on latent variables (e.g. partial least squares regression and principal component regression). It is important to select an appropriate number of latent variables to build an accurate and precise calibration model. Inclusion of too few latent variables can result in a model that is inaccurate over the complete space of interest. Inclusion of too many latent variables can result in a model that produces noisy predictions through incorporation of low-order latent variables that have little or no predictive value. Commonly used metrics for selecting the number of latent variables are based on the predicted error sum of squares (PRESS) obtained via cross-validation. In this paper a new approach for selecting the number of latent variables is proposed. In this new approach the prediction errors of individual observations (obtained from cross-validation) are compared across models incorporating varying numbers of latent variables. Based on these comparisons, nonparametric statistical methods are used to select the simplest model (least number of latent variables) that provides prediction quality that is indistinguishable from that provided by more complex models. Unlike methods based on PRESS, this new approach is robust to the effects of anomalous observations. More generally, the same approach can be used to compare the performance of any models that are applied to the same data set where reference values are available. The proposed methodology is illustrated with an industrial example involving the prediction of gasoline octane numbers from near-infrared spectra.

Return to Component selection

M.E. Tipping and C.M. Bishop
Probabilistic principal component analysis
Journal of the Royal Statistical Society B, 61 (1999) 611-622

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss, with illustrative examples, the advantages conveyed by this probabilistic approach to PCA.

Return to Reliability of principal component analysis

U


V


H. van der Voet
Comparing the predictive accuracy of models using a simple randomization test
Chemometrics and Intelligent Laboratory Systems, 25 (1994) 313-323

A simple randomization t-test is proposed for testing the equality of performance of two prediction methods. The application of the test is shown to prevent unjustified conclusions about method superiority. Previous approaches to the problem of comparing predictive methods are discussed, and the proposed test is compared to other tests for paired data in a small simulation study. It is shown that the test can also be applied for classification problems where the predicted entity is qualitative rather than quantitative.

Return to Validation or return to Component selection

H. van der Voet, W.J. de Boer, W.G. de Ruig and J.A. Van Rijn
Detection of residues using multivariate modelling of low-level GC-MS measurements
Journal of Chemometrics, 12 (1998) 279-294

The chemometric analysis of low-level analytical data is hampered by the common presence of interfering compounds, by the frequent absence of measurement signals and by a non-constant measurement variability which is related to concentration level in a non-linear way. A model is presented to handle this type of data in the context of the practical problem of multivariate detection from gas chromatographic/mass spectrometry (GC-MS) data. The model, based on log ratio modelling, is compared with previous approaches to parts of the problem. The basic idea behind the model is to define for the multivariate detection problem a null hypothesis for the values of log ratio measurements and to estimate variability as a function of total measured intensity. In practice it is often impossible to anticipate all kinds of interference which may occur. Therefore we propose to use expert assessments of the probability that certain expected peak ratios are generated by the analyte rather than by interferences. These expert assessments can then be used to define a proper null hypothesis for the multivariate detection test. The application of the model is illustrated for the detection of the illegal growth promoter clenbuterol in urine by selected ion-monitoring GC-MS.

Return to Limit of detection

H. van der Voet
Pseudo-degrees of freedom for complex predictive models: the example of partial least squares
Journal of Chemometrics, 13 (1999) 195-208

This paper considers models that are relatively complex considering the extent of the calibration data. Such data frequently arise in chemometric applications, near-infrared spectroscopy (NIRS) being a well-known example. Commonly used models are multiple linear regression (MLR) with variable selection or partial least squares (PLS) regression. The concept of degrees of freedom is undefined for such models; this paper proposes a definition for pseudo-degrees of freedom (PDF) based on predictive performance and an analogy with the standard linear model. The generalization is intended for all models which assume independent and identically distributed errors. Pseudo-degrees of freedom are very easily calculated from ordinary and cross-validation residuals. An example from a real-life NIRS application is given to illustrate the new concept.

Return to Reliability of multivariate calibration

L. Vega-Montoto and P.D. Wentzell
Maximum likelihood parallel factor analysis (MLPARAFAC)
Journal of Chemometrics, 17 (2003) 237-253

Algorithms for carrying out maximum likelihood parallel factor analysis (MLPARAFAC) for three-way data are described. These algorithms are based on the principle of alternating least squares, but differ from conventional PARAFAC algorithms in that they incorporate measurement error information into the trilinear decomposition. This information is represented in the form of an error covariance matrix. Four algorithms are discussed for dealing with different error structures in the three-way array. The simplest of these treats measurements with non-uniform measurement noise which is uncorrelated. The most general algorithm can analyze data with any type of noise correlation structure. The other two algorithms are simplifications of the general algorithm which can be applied with greater efficiency to cases where the noise is correlated only along one mode of the three-way array. Simulation studies carried out under a variety of measurement error conditions were used for statistical validation of the maximum likelihood properties of the algorithms. The MLPARAFAC methods are also shown to produce more accurate results than PARAFAC under a variety of conditions.

Return to Maximum likelihood calibration

L. Vega-Montoto, H. Gu and P.D. Wentzell
Mathematical improvements to maximum likelihood parallel factor analysis: theory and simulations
Journal of Chemometrics, 19 (2005) 216-235

A number of simplified algorithms for carrying out maximum likelihood parallel factor analysis (MLPARAFAC) for three-way data affected by different error structures are described. The MLPARAFAC method was introduced to establish the theoretical basis to treat heteroscedastic and/or correlated noise affecting trilinear data. Unfortunately, the large size of the error covariance matrix employed in the general formulation of this algorithm prevents its application to solve standard three-way problems. The algorithms developed here are based on the principle of alternating least squares, but differ from the generalized MLPARAFAC algorithm in that they do not use equivalent alternatives of the objective function to estimate the loadings for the different modes. Instead, these simplified algorithms tackle the loss of symmetry of the PARAFAC model by using only one representation of the objective function to estimate the loadings of all of the modes. In addition, a compression step is introduced to allow the use of the generalized algorithm. Simulation studies carried out under a variety of measurement error conditions were used for statistical validation of the maximum likelihood properties of the algorithms and to assess the quality of the results and computation time. The simplified MLPARAFAC methods are also shown to produce more accurate results than PARAFAC under a variety of conditions.

Return to Maximum likelihood calibration

L. Vega-Montoto and P.D. Wentzell
Mathematical improvements to maximum likelihood parallel factor analysis: experimental studies
Journal of Chemometrics, 19 (2005) 236-252

In this paper, the application of a number of simplified algorithms for maximum likelihood parallel factor analysis (MLPARAFAC) to experimental data is explored. The algorithms, described in a companion paper, allow the incorporation of a variety of correlated error structures into the threeway analysis. In this work, three experimental data sets involving fluorescence excitation-emission spectra of synthetic three-component mixtures of aromatic compounds are used to test these algorithms. Different experimental designs were employed for the acquisition of these data sets, resulting in measurement errors that were correlated in either two or three modes. A number of dataanalysis methods were applied to characterize the error structures of these data sets. In all cases, the introduction of statistically meaningful information translated to estimates of better quality than the conventional PARAFAC estimates of concentrations and spectra. The use of the algorithms that employ the error structure suggested by the analysis of the error covariance matrix yielded the best results for each data set.

Return to Maximum likelihood calibration

L. Vega-Montoto and P.D. Wentzell
Approaching the direct exponential curve resolution algorithm from a maximum likelihood perspective
Analytica Chimica Acta, 556 (2006) 383-399

The implementation of maximum likelihood parallel factor analysis (MLPARAFAC) in conjunction with the direct exponential curve resolution algorithm (DECRA) is described. DECRA takes advantage of the intrinsic exponential structure of some bilinear data sets to produce trilinear data by a simple shifting scheme, but this manipulation generates an error structure that is not optimally handled by traditional three-way chemometrics methods such as TLD and PARAFAC. In this work, the effects of these violations are studied using simulated and experimental data used in conjunction with the well-established TLD and PARAFAC. The results obtained by both methods are compared with the results obtained by MLPARAFAC, which is a method designed to optimally accomodate a variety of measurement error structures. The impact on the estimates of different parameters linked to the data sets and the DECRA method is investigated using simulated data. The results indicate that PARAFAC produces estimates of much poorer quality than TLD and MLPARAFAC. Also, it was found that the quality TLD estimates was comparable or only marginally poorer than the MLPARAFAC estimates. A number of commonly used algorithms were also compared to MLPARAFAC using two sets of published experimental data from kinetic studies. The MLPARAFAC estimates of rate constants were more precise than the other methods examined.

Return to Maximum likelihood calibration

J. Vessman, R.I. Stefan, J.F. van Staden, K. Danzer, W. Lindner, D.T. Burns, A. Fajgelj and H. Müller
Selectivity in analytical chemistry
Pure & Applied Chemistry, 73 (2001) 1381-1386

The correct use of the term "selectivity" and its clear distinction from the term "specificity" are discussed. A definition of selectivity is given, and it is recommended that the use of this term be promoted and that the use of the term "specificity" be discouraged.

Return to Analytical figures of merit

E. Voigtman
Comparison of signal-to-noise ratios
Analytical Chemistry, 69 (1997) 226-234

The probability distribution (density) function for the experimental signal-to-noise ratio (SNR) defined as x/s, where x is the sample mean and s is the customary sample standard deviation, has been derived and found to be in excellent agreement with accurate Monte Carlo simulation results. The SNR probability distribution function is a hypergeometric function which has no closed-form expression in elementary functions. The same applies to the probability distribution function for the relative standard deviation. In contrast, the probability distribution function for the approximate SNR defined by m/s', where m is the population mean parameter and s'= s[(N - 1)/ N]½, has a closed-form expression but is inaccurate for small numbers of measurements. The experimental SNR is a biased estimator of the true SNR, but the bias is easily correctable. Monte Carlo simulation methods were used to derive critical value tables for comparison of experimental SNRs and relative standard deviations. The critical value tables presented herein are accurate to about 1% for confidence levels of 75%, 90%, and 95%, and to about 5% for 99% confidence level.

Return to Analytical figures of merit

E. Voigtman
Comparison of signal-to-noise ratios, Part 2
Communications in Mathematical and in Computer Chemistry, 60 (2008) 333-348

It has been shown that previously published [1] probability density functions (PDFs) for several common signal-to-noise ratio (SNR) definitions are simply minor algebraic variants of the noncentral t distribution- based PDF results recently published by Nadarajah and Kotz [2]. The previously published, but unevaluated, integral expression for the PDF of quotients of SNRs has been shown to be in excellent quantitative agreement with Monte Carlo results. Furthermore, it has been shown that the same integral expression also yields the PDF for quotients of relative standard deviations (RSDs) and the PDF for quotients of simple detection limits. The latter was validated by comparison with detailed Monte Carlo simulations, with the result that accurate expectation values and detection limit 95% confidence intervals were obtained. As a consequence, a major step has been taken toward the goal of being able to compare simple detection limits on a fair basis and being able to perform a statistical test, precisely analogous to a standard F test, to determine whether a given pair of experimental detection limits are plausibly from the same chemical measurement system with the same measurement system parameters and measurement protocol.

Return to Analytical figures of merit

E. Voigtman
Limits of detection and decision. Part 1
Spectrochimica Acta B, 63 (2008) 115-128

Extensive Monte Carlo studies of instrumental limits of detection were performed on a simple univariate chemical measurement system having homoscedastic, Gaussian measurement noise and using ordinary least squares (OLS) processing of tens of millions of independent calibration curve data sets. It was found that prediction interval-based experimental detection limits were significantly negatively biased, in both the net response domain and the chemical content domain, resulting in substantially higher rates of false negatives than specified via customary critical t values. The diagnostic fix for the bias problem provided clear proof that hypothesis-based detection limits need not be unique, even as distributions of random variates, if the alternate hypothesis is non-unique. It was also demonstrated that hypothesis-based decision and detection limits have finite support that does not include the region near zero analyte content, so that both have finite moments and finite confidence intervals.

Return to Limit of detection

E. Voigtman
Limits of detection and decision. Part 2
Spectrochimica Acta B, 63 (2008) 129-141

Extensive Monte Carlo studies of instrumental limits of detection (LODs) were performed on a simple univariate chemical measurement system having homoscedastic, Gaussian measurement noise and using ordinary least squares (OLS) processing of tens of millions of independent calibration curve data sets. It was found that experimental decision and detection limits in the content domain were distributed as scaled reciprocals of noncentral t variates. In the response domain, the decision and detection limits were distributed as scaled c variates. Rates of false negatives were found to be as expected statistically and no bias was found. However, use of detection limit expressions based on critical values of the noncentrality parameter of the noncentral t distribution were found to be significantly biased, resulting in substantial bias in rates of false negatives.

Return to Limit of detection

E. Voigtman
Limits of detection and decision. Part 3
Spectrochimica Acta B, 63 (2008) 142-153

It has been shown that the MARLAP (Multi-Agency Radiological Laboratory Analytical Protocols) for estimating the Currie detection limit, which is based on 'critical values of the non-centrality parameter of the non-central t distribution', is intrinsically biased, even if no calibration curve or regression is used. This completed the refutation of the method, begun in Part 2. With the field cleared of obstructions, the true theory underlying Currie's limits of decision, detection and quantification, as they apply in a simple linear chemical measurement system (CMS) having heteroscedastic, Gaussian measurement noise and using weighted least squares (WLS) processing, was then derived. Extensive Monte Carlo simulations were performed, on 900 million independent calibration curves, for linear, "hockey stick" and quadratic noise precision models (NPMs). With errorless NPM parameters, all the simulation results were found to be in excellent agreement with the derived theoretical expressions. Even with as much as 30% noise on all of the relevant NPM parameters, the worst absolute errors in rates of false positives and false negatives, was only 0.3%.

Return to Limit of detection

E. Voigtman
Limits of detection and decision. Part 4
Spectrochimica Acta B, 63 (2008) 154-165

Probability density functions (PDFs) have been derived for a number of commonly used limit of detection definitions, including several variants of the Relative Standard Deviation of the Background-Background Equivalent Concentration (RSDB-BEC) method, for a simple linear chemical measurement system (CMS) having homoscedastic, Gaussian measurement noise and using ordinary least squares (OLS) processing. All of these detection limit definitions serve as both decision and detection limits, thereby implicitly resulting in 50% rates of Type 2 errors. It has been demonstrated that these are closely related to Currie decision limits, if the coverage factor, k, is properly defined, and that all of the PDFs are scaled reciprocals of noncentral t variates. All of the detection limits have well-defined upper and lower limits, thereby resulting in finite moments and confidence limits, and the problem of estimating the noncentrality parameter has been addressed. As in Parts 1-3, extensive Monte Carlo simulations were performed and all the simulation results were found to be in excellent agreement with the derived theoretical expressions. Specific recommendations for harmonization of detection limit methodology have also been made.

Return to Limit of detection

E. Voigtman and K.T. Abraham
Statistical behavior of ten million experimental detection limits
Spectrochimica Acta B, 66 (2011) 105-113

Using a lab-constructed laser-excited fluorimeter, together with bootstrapping methodology, the authors have generated many millions of experimental linear calibration curves for the detection of rhodamine 6G tetrafluoroborate in ethanol solutions. The detection limits computed from them are in excellent agreement with both previously published theory and with comprehensive Monte Carlo computer simulations. Currie decision levels and Currie detection limits, each in the theoretical, chemical content domain, were found to be simply scaled reciprocals of the non-centrality parameter of the non-central t distribution that characterizes univariate linear calibration curves that have homoscedastic, additive Gaussian white noise. Accurate and precise estimates of the theoretical, content domain Currie detection limit for the experimental system, with 5% (each) probabilities of false positives and false negatives, are presented.

Return to Limit of detection

W


I.N. Wakeling and J.J. Morris
A test of significance for partial least squares regression
Journal of Chemometrics, 7 (1993) 291-304

Partial least squares (PLS) regression is a commonly used statistical technique for performing multivariate calibration, especially in situations where there are more variables than samples. Choosing the number of factors to include in a model is a decision that all users of PLS must make, but is complicated by the large number of empirical tests available. In most instances predictive ability is the most desired property of a PLS model and so interest has centred on making this choice based on an internal validation process. A popular approach is the calculation of a cross-validated r2 to gauge how much variance in the dependent variable can be explained from leave-one-out predictions. Using Monte Carlo simulations for different sizes of data set, the influence of chance effects on the cross-validation process is investigated. The results are presented as tables of critical values which are compared against the values of cross-validated r2 obtained from the user's own data set. This gives a formal test for predictive ability of a PLS model with a given number of dimensions.

Return to Component selection

J.-H. Wang and P.K. Hopke
Estimation of the heteroscedastic noise in large data arrays
Analytica Chimica Acta, 412 (2000) 177-184

An approach has been developed to estimate the uncertainties in experimental data that follows a heteroscedastic model. The method presented is based on a hypothesis that the size of data is sufficiently large such that the data values over a limited domain have approximately homoscedastic variance. The implementation of the procedure has two steps. The first step is to estimate an approximate error for each data element. The second step is to estimate the standard deviation for the data points using the errors estimated for the neighbouring data. The related mathematical theory is presented, and several simulated data examples are used to illustrate this approach.

Return to General on uncertainty estimation

R. Wehrens and W.E. Van der Linden
Bootstrapping principal component regression models
Journal of Chemometrics, 11 (1997) 157-171

Bootstrap methods can be used as an alternative for cross-validation in regression procedures such as principal component regression (PCR). Several bootstrap methods for the estimation of prediction errors and confidence intervals are presented. It is shown that bootstrap error estimates are consistent with cross-validation estimates but exhibit less variability. This makes it easier to select the correct number of latent variables in the model. Using bootstrap confidence intervals for the regression vectors, it is possible to select a subset of the original variables to include in the regression, yielding a more parsimonious model with smaller prediction errors. The methods are illustrated using PCR, but can be applied to all regression models yielding a vector or matrix of regression coefficients.

Return to Reliability of multivariate calibration

R. Wehrens, E. Pretsch and L.M.C. Buydens
The quality of optimisation by genetic algorithms
Analytica Chimica Acta, 388 (1999) 265-271

The recently introduced quality criteria for optimisation, describing the coverage of the search and solution spaces as well as the reproducibility of both, are applied in combination with experimental design to fine-tune parameter settings and fitness function of a genetic algorithm for the structure optimisation of a heptapeptide. A series of influences of the investigated parameters are revealed by these criteria, while none of them seem significant from the fitness values of the last population alone. It is therefore suggested to apply these criteria, which are not based on the fitness values of the final population, when developing genetic algorithms. It is shown that they are easily adaptable to specific problems.

Return to Numerical optimization

R. Wehrens, H. Putter and L.M.C. Buydens
The bootstrap: a tutorial
Chemometrics and Intelligent Laboratory Systems, 54 (2000) 35-52

Bootstrap methods have gained wide acceptance and huge popularity in the field of applied statistics. The bootstrap is able to provide accurate answers in cases where other methods are simply not available, or where the usual approximations are invalid. The number of applications in chemistry, however, has been rather limited. One possible cause for this is the overwhelming number of techniques available. This tutorial aims to introduce the basic concepts of bootstrap methods, provide some guidance as to what bootstrap methods are appropriate in different situations, and illustrate several potential application areas in chemometrics by worked examples.

Return to General on uncertainty estimation

P.D. Wentzell, D.T. Andrews, D.C. Hamilton, N.M. Faber and B.R. Kowalski
Maximum likelihood principal component analysis
Journal of Chemometrics, 11 (1997) 339-366

The theoretical principles and practical implementation of a new method for multivariate data analysis, maximum likelihood principal component analysis (MLPCA), are described. MLCPA is an analog to principal component analysis (PCA) that incorporates information about measurement errors to develop PCA models that are optimal in a maximum likelihood sense. The theoretical foundations of MLPCA are initially established using a regression model and extended to the framework of PCA and singular value decomposition (SVD). An efficient and reliable algorithm based on an alternating regression method is described. Generalization of the algorithm allows its adaptation to cases of correlated errors provided that the error covariance matrix is known. Models with intercept terms can also be accommodated. Simulated data and near-infrared spectra, with a variety of error structures, are used to evaluate the performance of the new algorithm. Convergence times depend on the error structure but are typically around a few minutes. In all cases, models determined by MLPCA are found to be superior to those obtained by PCA when non-uniform error distributions are present, although the level of improvement depends on the error structure of the particular data set.

Return to Maximum likelihood calibration

P.D. Wentzell, D.T. Andrews and B.R. Kowalski
Maximum likelihood multivariate calibration
Analytical Chemistry, 69 (1997) 2299-2311

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.

Return to Method comparison studies or return to Maximum likelihood calibration

P.D. Wentzell and M.T. Lohnes
Maximum likelihood principal component analysis with correlated measurement errors: theoretical and practical considerations
Chemometrics and Intelligent Laboratory Systems, 45 (1999) 65-85

Procedures to compensate for correlated measurement errors in multivariate data analysis are described. These procedures are based on the method of maximum likelihood principal component analysis MLPCA , previously described in the literature. MLPCA is a decomposition method similar to conventional PCA, but it takes into account measurement uncertainty in the decomposition process, placing less emphasis on measurements with large variance. Although the original MLPCA algorithm can accommodate correlated measurement errors, two drawbacks have limited its practical utility in these cases: 1 an inability to handle rank deficient error covariance matrices, and 2 demanding memory and computational requirements. This paper describes two simplifications to the original algorithm that apply when errors are correlated only within the rows of a data matrix and when all of these row covariance matrices are equal. Simulated and experimental data for three-component mixtures are used to test the new methods. It was found that inclusion of error covariance information via MLPCA always gave results which were at least as good and normally better than PCA when the true error covariance matrix was available. However, when the error covariance matrix is estimated from replicates, the relative performance depends on the quality of the estimate and the degree of correlation. For experimental data consisting of mixtures of cobalt, chromium and nickel ions, maximum likelihood principal components regression showed an improvement of up to 50% in the cross-validation error when error covariance information was included.

Return to Maximum likelihood calibration

P.D. Wentzell and L. Vega Montoto
Comparison of principal components regression and partial least squares regression through generic simulations of complex mixtures
Chemometrics and Intelligent Laboratory Systems, 65 (2003) 257-279

Two of the most widely employed multivariate calibration methods, principal components regression (PCR) and partial least squares regression (PLS), are compared using simulation studies of complex chemical mixtures which contain a large number of components. Details of the complex mixture model, including concentration distributions and spectral characteristics, are presented. Results from the application of PCR and PLS are presented, showing how the prediction errors and number of latent variables (NLV) used vary with the relative abundance of mixture components. Simulation parameters varied include the distribution of mean concentrations, spectral correlation, noise level, number of mixture components, number of calibration samples, and the maximum number of latent variables available. In all cases, except when artificial constraints where placed on the number of latent variables retained, no significant differences were reported in the prediction errors reported by PCR and PLS. PLS almost always required fewer latent variables than PCR, but this did not appear to influence predictive ability.

Return to Method comparison studies

F. Westad and H. Martens
Variable selection in near infrared spectroscopy based on significance testing in partial least squares regression
Journal of Near Infrared Spectroscopy, 8 (2000) 117-124

A jack-knife based method for variable selection in partial least squares regression is presented. The method is based on significance tests of model parameters, in this paper applied to regression coefficients. The method is tested on a near infrared (NIR) spectral data set recorded on beer samples, correlated to extract concentration and compared to other methods with known merit. The results show that the jack-knife based variable selection performs as well or better than other variable selection methods do. Furthermore, results show that the method is robust towards various cross-validation schemes (the number of segments and how they are chosen).

Return to Reliability of multivariate calibration

R. Wolthuis, G.C.H. Tjiang, G.J. Puppels, T.C. Bakker Schut
Estimating the influence of experimental parameters on the prediction error of PLS calibration models based on Raman spectra
Journal of Raman Spectroscopy, 37 (2006) 447-466

Partial least squares (PLS) calibration is often the method of choice for making multivariate calibration models to predict analyte concentrations from Raman spectral measurements. In the development of such models, it is often difficult to assess beforehand what the prediction error will be, and whether instrumental or model factors limit the lower limit of the prediction error. Here, we present a method to assess the influence of experimental errors such as power fluctuations and spectral shifts, on the PLS prediction errors using simulated datasets. Assumptions that are implicit to PLS calibration and their implications with respect to the choice of experimental parameters for collecting a proper set of Raman spectra are discussed. The influence of various experimental parameters and signal pre-processing steps on PLS prediction error is demonstrated by means of simulations. The results of simulations are compared with the outcome of PLS calibrations of an experimental dataset.

Return to Reliability of multivariate calibration

X


Q.-S. Xu and Y.-Z. Liang
Monte Carlo cross validation
Chemometrics and Intelligent Laboratory Systems, 56 (2001) 1-11

In order to choose correctly the dimension of calibration model in chemistry, a new simple and effective method named Monte Carlo cross validation (MCCV) is introduced in the present work. Unlike leave-one-out procedure commonly used in chemometrics for cross validation (CV), the Monte Carlo cross validation developed in this paper is an asymptotically consistent method in determining the number of components in calibration model. It can avoid an unnecessary large model and therefore decreases the risk of over-fitting for the calibration model. The results obtained from simulation study showed that MCCV has an obviously larger probability than leave-one-out CV in choosing the correct number of components that the model should contain. The results from real data sets demonstrated that MCCV could successfully choose the appropriate model, but leave-one-out CV could not.

Return to Component selection

Q.-S. Xu, Y.-Z. Liang and Y.-P. Du
Monte Carlo cross-validation for selecting a model and estimating the prediction error in multivariate calibration
Journal of Chemometrics, 18 (2004) 112-120

A new simple and effective method named Monte Carlo cross validation (MCCV) has been introduced and evaluated for selecting a model and estimating the prediction ability of the model selected. Unlike the leave-one-out procedure widely used in chemometrics for cross-validation (CV), the Monte Carlo cross-validation developed in this paper is an asymptotically consistent method of model selection. It can avoid an unnecessarily large model and therefore decreases the risk of overfitting of the model. The results obtained from a simulation study showed that MCCV has an obviously larger probability than leave-one-out CV (LOO-CV) of selecting the model with best prediction ability and that a corrected MCCV (CMCCV) could give a more accurate estimation of prediction ability than LOO-CV or MCCV. The results obtained with real data sets demonstrated that MCCV could successfully select an appropriate model and that CMCCV could assess the prediction ability of the selected model with satisfactory accuracy.

Return to Component selection

Y


Z



[A][B][C][D][E][F][G][H][I][J][K][L][M][N][O][P][Q][R][S][T][U][V][W][X][Y][Z]