Multiway calibration is a powerful generalization of multivariate calibration in the sense that more complex analytical problems can be solved. Think of correct predictions in the presence of unexpected interferences  the solution to the universal background problem. Moreover, multiway calibration often provides additional information such as the purecomponent signals.
There exist two entirely different approaches to the calibration of multiway data. The first approach separates analytes and interferences by resolving the data into purecomponent signals, and then applies a univariate calibration strategy for the property of interest. The second approach can be interpreted in terms of solving systems of equations that are coupled for the various 'ways' of the data. The first approach is mainly developed within chemometrics, whereas the second approach appears to have more input from conventional regression statistics.
Unlike the second approach, deconvolutionbased multiway calibration lacks a direct analogue for multivariate (i.e. oneway) data. To keep a clear distinction, these approaches will therefore be referred to as deconvolutionbased and regressionbased multiway calibration, respectively.
It is not surprising that interest in these techniques rapidly increases. For example, deconvolution (also known as curve resolution) has been announced as the 'Sleeping Giant of Chemometrics' at the FACSS meeting of 2003.
This page is organized as follows:
Official literature
To the best of our knowledge, there are no guidelines issued for multiway calibration.
However, potential multiway extensions of generally accepted univariate methodology are listed in:
 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) 633661
Download (=645 kB: © IUPAC 2006)

Hyphenated techniques

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A broad class of instruments that deliver multiway data is formed by the hyphenated techniques. The term hyphenation applies when one instrument modulates another one. In particular, combinations of chromatographic separation and multichannel spectroscopic detection abound, i.e. chromatographyspectroscopy[spectroscopy[&c.]] or chromatographychromatography[spectroscopy[&c.]]. Prime examples are HPLCUV and GCMS, which generate secondorder or twoway data.
Consider the simulated HPLCUV data set presented as a 'landscape' in Figure TAC 3. Because the data are simulated, all underlying elution profiles and spectra are known:


Figure MWC 1: Elution profiles (left panel) and spectra (right panel) corresponding to the simulated HPLCUV landscape of Figure TAC 3.

Traditional methods for analyzing 'spectrochromatographic' data start with determining the number of components under a chromatographic peak using, for example, the max plot or its second derivative:


Figure MWC 2: Max plot (left panel) and its second derivative (right panel).

In this case, one would conclude that there are at least three overlapping components. The following step would be to improve the resolution by changing the chromatography. Eventually, one might discover that there are four components under the peak.
However, since around 1980 various methods have been developed for 'mathematically' resolving the underlying purecomponent profiles. A nice example of a method that even works without operator intervention is presented in:
 H. Shen, B. Grung, O.M. Kvalheim and I. Eide
Automated curve resolution applied to data from multidetection instruments
Analytica Chimica Acta, 446 (2001) 313328
The resulting deconvolution information  in the form of purecomponent profiles  brings several benefits:
 With little additional effort, the scope of instruments is markedly increased. For example, a method originally developed for screening purposes only can often be adapted to yield acceptable quantitative results.
 If (near) baseline resolution is nevertheless required (owing to regulation), the optimization of the chromatographic parameters can be guided by detailed knowledge about the purecomponent profiles, e.g. relative peak areas, separation and degree of tailing or fronting.
Suitable calibration scenarios have already been described in:
 B.G.M. Vandeginste, F. Leyten, M. Gerritsen, J.W. Noor and G. Kateman
Evaluation of curve resolution and iterative target transformation factor analysis in quantitative analysis by liquid chromatography
Journal of Chemometrics, 1 (1987) 5771
Since the purecomponent profiles are free from interferences, deconvolution enables a reduction of multiway data to concentrationrelated scalars. This transformation to univariate data then allows one to calculate figures of merit such as limit of detection in the traditional way, see:
 J. Saurina, C. Leal, R. Compañó, M. Granados, M. Dolors Prat and R. Tauler
Estimation of figures of merit using univariate statistics for quantitative secondorder multivariate curve resolution
Analytica Chimica Acta, 432 (2001) 241251
We note that deconvolution is applied here to excitationemission fluorescence data (see below). However, the resolved analyte emission spectra consist of single peaks, making them mathematically equivalent to elution profiles. Moreover, Saurina et al. fully acknowledge that they build on the pioneering work of Vandeginste et al. in the spectrochromatographic area.
A representative result, adapted from the study of Saurina et al. is:

Clearly, this plot demonstrates the potential of mathematical deconvolution for calibrating spectrochromatographic instruments.

Excitationemission fluorescence

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The practical aspects of modeling this type of data are reviewed in:
 C.M. Andersen and R. Bro
Practical aspects of PARAFAC modeling of fluorescence excitationemission data
Journal of Chemometrics, 17 (2003) 200215
Excitationemission fluorescence combines high sensitivity with good linearity over a large range of signal. The potential utility of this technique can be markedly increased if nonfluorescent analytes of interest are converted into fluorescent species by a chemical reaction that is followed over time. The additional temporal dimension in the resulting data set enhances selectivity, see:
 A.C. Olivieri, J.A. Arancibia, A. Muñoz de la Peña, I. DuránMerás and A. Espinosa Mansilla
Secondorder advantage achieved with fourway fluorescence excitationemissionkinetic data processed by parallel factor analysis and trilinear leastsquares. Determination of methotrexate and leucovorin in human urine
Analytical Chemistry, 76 (2004) 56575666
The following plots show the signal for analytes of interest and interfering background:


Figure MWC 4: Contour plot of the data for an aqueous solution (pH 9.4) containing methotrexate 0.39 mg L^{1} and leucovorin 0.38 mg L^{1} (left panel) and a typical human urine sample diluted 1:250 (right panel). The analyte maxima are indicated (MTX and LV), as well as the presence of diffraction grating harmonics (H) and Rayleigh scatter (R). The region selected for calibration with PARAFAC and TLLS is highlighted by the dashed rectangle. Fluorescence intensity has been coded in colors.

A visualization of the timedependent data for a representative sample is:

These plots illustrate that the kinetics introduce a considerable amount of variation in the data, which is conveniently exploited using multiway calibration methods.
Prediction intervals for excitationemission fluorescence data can be obtained from:
 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) 6976
 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) 95109
 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) 133149
 M. Linder and R. Sundberg
Precision of prediction in secondorder calibration, with focus on bilinear regression methods
Journal of Chemometrics, 16 (2002) 1227
 A.C. Olivieri and N.M. Faber
Standard error for prediction in parallel factor (PARAFAC) analysis of threeway data
Chemometrics and Intelligent Laboratory Systems, 70 (2004) 7582
Click here for more information about the multivariate analogue.
It is worth mentioning that the methods under consideration differ greatly with respect to underlying assumptions hence scope, viz:
 The generalized rank annihilation method (GRAM) applies when only a single calibration sample is available. Unlike common regression methods such as partial least squares (PLS) regression and principal component regression (PCR), GRAM jointly models the calibration and prediction sample instrumental responses. From this joint model one obtains concentration ratios that can be combined with the corresponding calibration values to obtain the desired predictions. Constructing a new model for each prediction sample ensures that the secondorder advantage is obtained, i.e., the analyte of interest can be determined correctly without explicitly accounting for all interferences in the prediction sample.
 Bilinear least squares (BLLS), the method studied by Linder and Sundberg in their prizewinning paper, is a genuine leastsquares method and, in principle, the method of choice for 'clean' data. However, in its basis form it does not ensure the secondorder advantage. Consequently, its superior statistical efficiency may suffer when being modified to obtain the secondorder advantage. Furthermore, it can only be used if the calibration is done with at least as many samples as the number of independently varying spectrally active consitituents.
 Parallel factor analysis (PARAFAC) achieves the same objectives as GRAM, but is much more generally applicable. The performance is similar when only a single calibration sample is available. However, it is preferable to GRAM when having multiple calibration samples. Moreover, it can be generalized to any order of the instrumental reponses and it can easily handle missing values. A drawback is the computational burden because, unlike GRAM, it is an iterative procedure.
 Unlike the other methods, multiway PLS does not assume the rather restrictive secondorder bilinear model. Furthermore, it can be generalized to any order of the instrumental reponses.

References & further information

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