Three-Mode Abstracts, Part H

INDEX

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• Haack, M. B., Eliasson, A. & Olsson, L. (2004).
  On-line cell mass monitoring of Saccharomyces cerevisiae cultivations by multi-wavelength fluorescence. Journal of Biotechnology, 114, 199-208.
  The catalyst in bioprocesses, i.e. the cell mass, is one of the most challenging and important variables to monitor in bioprocesses. In the present study, cell mass in cultivations with Saccharomyces cerevisiae was monitored on-line with a non-invasive in situ placed sensor measuring multi-wavelength culture fluorescence. The excitation wavelength ranged from 270 to 550 nm with 20 nm steps and the emission wavelength range was from 310 to 590 nm also with 20 nm steps. The obtained spectra were analysed chemometrically with the multi-way technique, parallel factor analysis (PARAFAC), resulting in a decomposition of the multivariate fluorescent landscape, whereby underlying spectra of the individual intrinsic fluorophors present in the cell mass were estimated. Furthermore, gravimetrically determined cell mass concentration was used together with the fluorescence spectra for calibration and validation of multivariate partial least squares (PLS) regression models. Both two- and three-way models were calculated, the models behaved similarly giving root mean square error of prediction (RMSEPs) of 0.20 and 0.19 g l(-1), respectively, when test set validation was used. Based on this work, it is evident that on-line monitoring of culture fluorescence can be used for estimation of the cell mass concentration during cultivations.

• Halstensen, M., & Esbensen, K. (2004).
  New developments in acoustic chemometric prediction of particle sice distribution - 'the problem is the solution'. Journal of Chemometrics, 14, 463-481.
  We present a new prototype acoustic chemometric approach for prediction of powder particle size distributions, intended for in-line implementation. The standard basic solutions demand that calibration be carried out on representative, 'non-segregated' reference powder samples. However, as practical powder flow with no segregation is extremely difficult to achieve with the precision needed for calibration, there will always be a significant uncertainty in the reference values relative to what is actually measured. The problem is flow segregation. In order to solve this problem, we have designed a completely new acoustic chemometric approach, which by way of contrast forces the flowing powder mass to segregate as much as possible by various mechanical means. The new approach measures the acoustic signals from an integrated series of segregated, part-sample characteristics. The calibration X-data matrix now becomes a three-way matrix, which demands a three-way calibration solution to 'unscramble' the latent information in the maximally segregated powder sample. Thus the problem is now, the solution. Our earlier forays into this matter, which were based on two-way calibrations, have all been limited by a severe 'particle size ratio' bracket outside which destructive self-damping has effectively negated practical, useful accuracy and precision. The new approach allows a much greater range of contrasting particle sizes. Our first-generation results achieved by using three-way PLS-R as well as the standard two-way calibrations show that it is more precise than all earlier attempts and can be used for many-component mixtures without extensive further modifications. We also look at the feasibility of quantifying for prediction of in-line particle size distributions in an industrial environment.

• Hammers, W.E., Janssen, R.H.A.M., Baars, A.G., & De Ligny, C.L. (1978).
  Standardization and determination of the selectivity of octadecylsilylsilica in high-performance liquid chromatography. Journal of Chromatography, 167, 273-289.
  Net retention volumes per gram of octadecylsilylsilica have been measured for a large number of mono- and disubstituted benzene derivatives, some monosubstituted hexanes and cyclohexanes and a number of polycyclic aromatic compounds, using n-hexane, methylene chloride and a mixture of both solvents as the eluent at 25° and 43.5°. The retention data are interpreted in terms of the semi-empirical adsorption model, developed by Snyder for bare adsorbents. The effects of adsorbent deactivation temperature, solute and eluent localization, change of the charge distribution in the solute molecule and adsorption mode of the solute on retention and selectivity are discussed quantitatively.

• Hammers, W.E., Spanjer, M.C., & De Ligny, C.L. (1979).
  Selectivity of nucleosil 10 NH₂ as an adsorbent in high-performance liquid chromatography. Journal of Chromatography, 174, 291-305.
  Net retention volumes per gram of Nucleosil 10 NH₂ have been measured for a large number of mono- and disubstituted benzene derivatives and of polycyclic aromatic hydrocarbons, using n-hexane, dichloromethane and a mixture of both as eluents at 25°. The retention data are interpreted in terms of the semi- empirical adsorption model, developed by Snyder for bare adsorbents, using octadecylsilylsilica as a reference adsorbent. The effects of the bound monomers on adsorbent deactivation, solute and eluent localization, change of the charge distribution in the solute molecule and adsorption mode of the solute are evaluated and discussed in terms of donor-acceptor interaction (including hydrogen bonding).

• Hanafi, M., & Lafosse, R. (2001).
  Généralisations de la régression simple pour analyser la dépendance de K ensembles de variables avec un K + 1ème. Revue Statistique Appliquée, 49, 5-30.
  By introducing a projector, the simultaneous dependency analysis considered between a matrix and K matrices by Lafosse & Hanafi (1997) boils down to an analysis between two matrices. Some simultaneous simple regressions then lead to a calculus of the contributions to this dependency for every row of each matrix. The simultaneous calculus of the regression coefficient now comes from the solution of one optimization problem.

• Hanke, B., Lohmöller, J.-B., & Mandl, H. (1980).
  Schülerbeurteilung in der Grundschule: Ergebnisse der Augsburger Längsschnittuntersuchung [Evaluation of students at primary school: Results of the Augsburger longitudinal study]. München, FRG.: Oldenbourg Verlag.
  In the Augsburg longitudinal study about 2500 primary school- children were tested for four years. Chapter 6 reports four three-mode studies based on correlation matrices (8 school subjects; 6 occasions; 8 rating scales with teacher judgements, 4 occasions; 8 primary mental abilities, 3 occasions; 8 sociometric tests, 3 occasions). The variable modes are varimax rotated. Target rotations to orthogonal polynomials, symmetric rotation of core matrices. Results are interpreted in terms of stability, transformation, and continuity (see also Lohmöller, 1982).

• Hansen, P. W., Van Brakel, A. S., Garman, J., & Norgaard, L. (1999).
  Detection of specific sugars in dairy process samples using multivariate curve resolution. Journal of Dairy Science, 82, 1351-1360.
  Dairy process monitoring by application of multivariate curve resolution using alternating least squares is presented. Alternating least squares was used for resolving Fourier transform infrared spectral data from a dairy batch process in which lactose is enzymatically hydrolyzed to glucose and galactose. It was possible to extract four compounds (fat, lactose, and two other sugar components) from the spectral data obtained from nine process runs. Subsequently, the pure spectra obtained in this way were used to monitor the content of these compounds in two new process runs. In this way, alternating least squares made it possible to follow the hydrolysis process by Fourier transform infrared spectroscopy without the need for reference analyses. When the results were correlated to reference results for lactose, the accuracy was similar to that obtained when a partial least squares regression was performed on the same data; lactose correlation was 0.980 when alternating least squares was used and was 0.987 when partial least squares was used.

• Harris, M.L., & Romberg, T.A. (1974).
  An analysis of content and task dimensions of mathematics items designed to measure level of concept attainment. Journal for Research in Mathematics Education, 5, 72-86.
  Used factor analytic techniques to study content and task dimensions of mathematics items, developed to measure concept attainment, using a crossed design involving 30 concepts and 12 tasks. Conventional factor analyses were performed separately for 196 female 5th graders and 195 male 6th graders for concept and task scores. Also, three-mode factor analyses were performed. From results of the conventional factor analyses, it is concluded that all 30 of the concepts are measures of a single functional relationship existing among the concepts and that all 12 tasks are measures of a single underlying ability or latent trait. The three-mode results indicate that there are no important concept-task interactions for idealized persons.

• Harshman, R.A. (1970).
  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis. UCLA Working Papers in Phonetics, 16, 1-84.
  Proposes the same model, CANDECOMP, as Carroll & Chang (1970).

  

• Harshman, R.A. (1972a).
  Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics,22, 111-117.
  A relatively simple proof that gives some "minimal" data characteristics required for the PARAFAC (PARAFAC1) solution to be unique. These requirements are weaker than in Jennrich (in Harshman, 1970), in that only two levels of Mode C (or B or A) are necessary for uniqueness, provided that each factor changes by a distinct percentage. Although less general than the proof by Kruskal (1976) and Kruskal (1977), this proof shows how uniqueness can break down "partially" (i.e., for some factors but not others), which Kruskal's proof does not cover. (see, also, Carroll & Wish, 1974, and Carroll & Arabie, 1980).

• Harshman, R.A. (1972b).
  PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30-44.
  A model for the analysis of scalar products which allows for common oblique axis projections of the stimuli and individual differences in weights or saliences (and thus is related to INDSCAL and IDIOSCAL). The model is a special case of T2, but with PARAFAC-like uniqueness properties. [These properties are only conjectured here (and are subsequently questioned in Carroll & Wish, 1974, and Carroll & Arabie, 1980) but are later proven as a special case of PARATUCK2 (see Harshman & Lundy, 1996).]

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• Harshman, R.A. (1976).
  PARAFAC: Methods of three-way factor analysis and multidimensional scaling according to the principle of proportional profiles. Unpublished doctoral thesis, University of California. (Dissertation Abstracts International, 37, 2478-2479).
  
  
• Harshman, R.A. (1994a).
  Substituting statistical for physical decomposition: Are there applications for parallel factor analysis (PARAFAC) in non- destructive evaluation? In P.V. Malague (Ed.), Advances in Signal Processing for Nondestructive Evaluation of Materials (pp. 469-483).
  Dordrecht: Kluwer.
  We describe the three-way factor analysis method PARAFAC (PARAllel FACtor analysis) and its possible application to Non-destructive Evaluation (NDE). Because standard analysis methods usually seperate a signal into abstract, mathematically convenient parts such as frequency bands, orthogonal variance components, etc., irrelevant signals often remain mixed with important ones. In contrast, Parallel Factor Analysis separates mixtures into functionally distinct parts, i.e., parts showing distinct patterns of variation in magnitude across varying measurement conditions. It is used, for example, to decompose the mixtures of curves generated by fluorescence spectroscopy of complex samples into the individual spectra of the constituent chemical compounds in the mixture; this is feasible because each compound shows distinct patterns of variation in fluorescence intensity across varying stimulus frequencies. In NDE, PARAFAC could similarly separate the mixture of signals (or image patches) from an object under test into those arising from each causally/physically distinct component of interest in the object, provided that there are several parallel test conditions, or multiple time slices, where the signals from different components of interest show distinct patterns of variation in magnitude accross the conditions. Various normal and anomalous signals can then be isolated onto distinct factors, allowing anomalous ones to be identified and linked to their physical sources.

• Harshman, R.A. (1999).
  Use of angle bracket notation to simplify expression of multilinear models. (Research Bulletin, no. 757). London, Canada: University of Western Ontario, Department of Psychology.
  To adapt matrix notation to three- or higher-way structures, the array structure is often represented by considering only a two-way subpart, an arbitrary slice. The description of a two-way structure for that slice is then understood to generalize (with appropriate change of weights) to all slices and hence to the array as a whole. In this report, a new approach to handling three-way structure is proposed, called 'angle-bracket notation'.

• Harshman, R.A. (2000). Generalization of matrix notation and algebra to n-way arrays (Research Bulletin, no. 758). London, Canada: University of Western Ontario, Department of Psychology.
  Matrix notation and the rules of matrix algebra are generalized to n-way arrays. The resulting language seems easy to use; all the capabilities of matrix notation are retained and most appear to carry over naturally to the n-way context. For example, one can multiply a three-way array times a four-way array to obtain a three-way product. Many of the language's key characteristics are based on the rules of tensor notation and algebra. The most important example of this is probably the incorporation of subscript/index related information into both the names of array objects and the rules used to operate on them. Some topics that emerge are relatively unexplored, such as inverses of n-way arrays; these might prove interesting for future theoretical study. It is hoped that this language will facilitate expression of multilinear and quasi-multilinear models and exploration of new directions in n-way research, after the critical scrutiny of other researchers who will no doubt suggest useful improvements or modifications. Thus, the language described here is offered as a working draft.

• Harshman, R.A. (2001).
  An index to formalism that generalizes the capabilities of matrix notation and algebra to n-way arrays. Journal of Chemometrics, 15, 689-714.
  The capabilities of matrix notation and algebra are generalized to n-way arrays. The resulting language seems easy to use; all the capabilities of matrix notation are retained and most carry over naturally to the n-way context. For example, one can multiply a three-way array times a four-way array to obtain a three-way product. Many of the language's key characteristics are based on the rules of tensor notation and algebra. The most important example of this is probably the incorporation of subscript/index-related information into both the names of array objects and the rules used to operate on them. Some topics that emerge are relatively unexplored, such as inverses of n-way arrays; these might prove interesting for future theoretical study.

• Harshman, R.A., & Berenbaum, S.A. (1981).
  Basic concepts underlying the PARAFAC-CANDECOMP three-way factor analysis model and its application to longitudinal data. In D.H. Eichorn, J.A. Clausen, N. Haan, M.P. Honzik & P.H. Mussen (Eds.), Present and past in middle life (pp. 435-459). New York: Academic Press.
  This paper presents the basic concepts and assumptions underlying the PARAFAC-CANDECOMP factor analysis model and consider possible benefits and disadvantages of this approach to analyzing longitudinal data.

  

• Harshman, R. A., & Hong, S. (2002).
  'Stretch' vs. 'slice' methods for representing three-way structure via matrix notation. Journal of Chemometrics, 16, 198-205.
  A three-way array must be represented in two-way form if its structure is to be described and manipulated by means of matrix notation. Historically, two methods, here called 'array stretching' and 'array slicing', have been used. More recently, however, array slicing has often been overlooked, resulting in a loss of mathematical flexibility. 'Stretching' involves matricizing (unfolding) the threeway array and applying one's mathematical operations to the resulting two-way matrix; this results in expressions that are often quite useful for parameter estimation but which are relatively long and require practice to interpret properly. 'Slicing' involves taking a representative two-way subarray and applying operations to it; this often gives compact and easily understood expressions but requires the introduction of extra matrix names and becomes awkward if the array is not 'slicewise regular'. In this paper the advantages of each approach are demonstrated and compared by applying them to a set of models from the Tucker and Parafac families. In addition, we show how slicewise representation can be improved by using (i) angle brackets to eliminate the need for extra diagonal matrices, and (ii) 'encapsulated summation' notation to allow representation of array structure that is orderly but not slicewise regular.

• Harshman, R. A., Hong, S. J. & Lundy M. E.(2003).
  Shifted factor analysis - Part I: Models and properties. Journal of Chemometrics, 17, 363-378.
  The factor model is modified to deal with the problem of factor shifts. This problem arises with sequential data (e.g. time series, spectra, digitized images) if the profiles of the latent factors shift position up or down the sequence of measurements: such shifts disturb multilinearity and so standard factor/component models no longer apply. To deal with this, we modify the model(s) to include explicit mathematical representation of any factor shifts present in a data set; in this way the model can both adjust for the shifts and describe/recover their patterns. Shifted factor versions of both two- and three (or higher)-way factor models are developed. The results of applying them to synthetic data support the theoretical argument that these models have stronger uniqueness properties; they can provide unique solutions in both two-way and three-way cases where equivalent non-shifted versions are under-identified. For uniqueness to hold, however, the factors must shift independently; two or more factors that show the same pattern of shifts will not be uniquely resolved if not already uniquely determined. Another important restriction is that the models, in their current form, do not work well when the shifts are accompanied by substantial changes in factor profile shape. Three-way factor models such as Parafac, and shifted factor models such as described here, may be just two of many ways that factor analysis can incorporate additional information to make the parameters identifiable. Copyright (C) 2003 John Wiley Sons, Ltd.

• Harshman, R.A., Ladefoged, P., & Goldstein, L. (1977).
  Factor analysis of tongue shapes. Journal of the Acoustical Society of America, 62, 693-707.
  A new analytic procedure PARAFAC has been applied to the description of the shape of the tongue in English vowels. The procedure models the data in terms of a unique set of possibly explanatory factors. It solves in parallel for factors in several data sets, simultaneously describing the differences among data sets in terms of different relative involvement of these common factors. Tracings were made of x-rays taken during the pronunciation of 10 English vowels by five speakers. The position of the tongue in these 50 vowels were quantized in terms of 13 superimposed grid lines. PARAFAC analysis shows that the data can be described in terms of two factors. One factor generates a forward movement of the root of the tongue accompanied by an upward movement of the front of the tongue. The second factor generates an upward and backward movement of the tongue. Movements from front to back vowels involve decreasing amounts of factor one. Movements from high to low vowels involve decreasing amounts of factor two. Different speakers use the two factors to different degrees which may be associated with their individual anatomy. The correlation between the observed data and that predicted by the model is greater than 0.96.

• Harshman, R. A., & Lundy, M. E. (1984a).
  The PARAFAC model for three-way factor analysis and multidimensional scaling. In H. G. Law, C. W. Snyder Jr, J. A. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis (pp. 122-215). New York: Praeger.
  This is a comprehensive discussion of the different forms of the PARAFAC(-CANDECOMP) three-way factor analysis model, its "uniqueness" properties, and its relationship with other two- and three-way factor analysis models. First, the PARAFAC1 model for raw score or profile data ("direct fit") is derived as a generalization of the two-way factor model, issues of scaling and interpretation of the factor loading matrices are discussed (pp. 128-139; 192-203), and "system" vs. "object" variation as they relate to the model are examined (pp. 125-133). Next, the PARAFAC1 model for covariances ("indirect fit") is derived, first from the raw score model and then from more general assumptions. The first derivation shows PARAFAC1 to be a special case of PARAFAC2 (pp. 136-7). Indirect and direct fitting are compared and the advantages and disadvantages of the orthogonality assumption required by the indirect fit model are discussed (pp. 137-8), along with why correlations are inappropriate (p. 141), and whether the principal components model differs appreciably from the common factors model in the three-way indirect fit context (pp. 141-3). Finally, the relationship between factor analysis (PARAFAC) and metric MDS (INDSCAL, IDIOSCAL) is examined (pp. 144-7). The next major section in the chapter deals with the uniqueness properties of the PARAFAC model (pp. 147-169). "Uniqueness" or "intrinsic axes" means that given certain assumptions, the solution determined from the data by PARAFAC has no alternative, equal-fitting form (i.e., any other rotation would reduce its fit to the data). Why this is important (pp. 147-150; 163-9) and the minimum conditions for uniqueness (pp. 161-2) are explained. The value of empirically confirming a PARAFAC solution by split-half, bootstrapping, and/or jackknifing procedures is also discussed. The final section compares PARAFAC with other models, most notably Tucker's T3 (pp. 169-182), Corballis' three-way model (pp. 184-5), and Sands and Young's ALSCOMP (pp. 188-190). PARAFAC1 is described as a special case of the T3 model and vice versa, and a diagram on p. 175 shows how Carroll transforms a T3 representation into the corresponding PARAFAC one (two other methods of embedding T3 in PARAFAC are also discussed on pp. 176-178 and pp. 203-7). A family of related models for three-way profile data, arranged from most general to most restricted, is presented on p. 184. PARAFAC3 is listed between T3 and PARAFAC1 and is discussed on pp. 185-6. PARAFAC2 and DEDICOM, not in the table, are also discussed in relation to PARAFAC3 and T3 (p. 187). Full text

• Harshman, R. A., & Lundy, M. E. (1984b).
  Data preprocessing and the extended PARAFAC model. In H. G. Law, C. W. Snyder Jr, J. A. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis (pp. 216-284). New York: Praeger.
  The data preprocessing discussed here is restricted to additive adjustments (centering) and multiplicative adjustments (rescaling or normalization) that may be applied to three-way profile data, for example, before direct fitting of the PARAFAC model. It does not include conversion of profile data to covariances or cross-products, nor does it include conversion of proximity data to scalar products. Eight reasons for preprocessing are given, the most important of which is "to make the data appropriate for the PARAFAC model" (p. 218), which is accomplished by centering. It is shown algebraically that "fiber"-centering and "slab"-reweighting are the only appropriate ways to preprocess data for the trilinear PARAFAC model (see p. 231 for a diagram of fibers and slabs). Also shown is the effect the proper preprocessing has on the original factor loadings (e.g., centering Mode A of the data centers the Mode A factors; rescaling Mode B similarly reweights the rows of the Mode B factor matrix). Practical guidelines for deciding how to preprocess data are given on pp. 257-259. Preprocessing is also discussed from the perspective of extending the PARAFAC model to more general data (instead of making the data appropriate for a restricted model, as above). In this context, "degenerate" solutions and their possible causes are described. Degenerate solutions are characterized by two or more factors whose loadings are highly correlated across all three modes, with a negative triple-product. Appropriate centering can sometimes correct this problem, but not always. Other times an orthogonally constrained PARAFAC procedure can block the high correlations and yield an interpretable solution (alternatively, zero-correlation constraints can be used, e.g., if the solution requires positive loadings). In these instances, it seems that while more general Tucker-type structure is present in the data, the constraints allow a subset of Tucker variations to be meaningfully expressed via orthogonal/uncorrelated PARAFAC factors. Full text

• Harshman, R.A., & Lundy, M.E. (1994).
  PARAFAC: Parallel factor analysis. Computational Statistics & Data Analysis, 18, 39-72.
  We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or 'slices' of a three-way array in terms of a common set of factors with differing relative weights in each 'slice'. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (x_ij = Sigma_r^R = ₁a_irb_jr) to a trilinear model (x_ijk = Sigma_r^R = Sigma_ra_irb_jrc_kr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need of a separate 'rotation' phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor 'interactions'. Of key importance is the need to increase the method's robustness against nonstationary factor structures and qualitative (nonproportional) factor change.

• Harshman, R.A., & Lundy, M.E. (1996).
  Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC-CANDECOMP. Psychometrika, 61, 133-154.
  Some existing three-way factor analysis and MDS models incorporate Cattell's "Principle of Parallel Proportional Profiles". These models can - with appropriate data - empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called "interactions" among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, X_k = A ^AD_k H ^BD_k B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where ^AD_k = ^BD_k). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.

• Hart, S. J., & Jiji, R. D. (2002).
  Light emitting diode excitation emission matrix fluorescence spectroscopy. Analyst, 127, 1693-1699.
  An excitation emission matrix (EEM) fluorescence instrument has been developed using a linear array of light emitting diodes ( LED). The wavelengths covered extend from the upper UV through the visible spectrum: 370-640 nm. Using an LED array to excite fluorescence emission at multiple excitation wavelengths is a low-cost alternative to an expensive high power lamp and imaging spectrograph. The LED-EEM system is a departure from other EEM spectroscopy systems in that LEDs often have broad excitation ranges which may overlap with neighboring channels. The LED array can be considered a hybrid between a spectroscopic and sensor system, as the broad LED excitation range produces a partially selective optical measurement. The instrument has been tested and characterized using fluorescent dyes: limits of detection (LOD) for 9,10-bis(phenylethynyl)- anthracene and rhodamine B were in the mid parts-per-trillion range; detection limits for the other compounds were in the low parts-per-billion range (< 5 ppb). The LED-EEMs were analyzed using parallel factor analysis ( PARAFAC), which allowed the mathematical resolution of the individual contributions of the mono- and dianion fluorescein tautomers a priori. Correct identification and quantitation of six fluorescent dyes in two to six component mixtures ( concentrations between 12.5 and 500 ppb) has been achieved with root mean squared errors of prediction (RMSEP) of less than 4.0 ppb for all components.

• Hartley, R. I. (1997).
  Lines and points in three views and the trifocal tensor. International Journal of Computer Vision, 22, 125-140.
  This paper discusses the basic role of the trifocal tensor in scene reconstruction from three views. This 3 x 3 x 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two view case. In particular, the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views. It is further shown in this paper, that the trifocal tensor is essentially identical to a set of coefficients introduced by Shashua to effect point transfer in the three view case. This observation means that the 13 line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences. From the trifocal tensor the camera matrices of the images may be computed, and the scene may be reconstructed. For unrelated uncalibrated cameras, this reconstruction will be unique up to projectivity. Thus, projective reconstruction of a set of lines and points may be carried out linearly from three views.

• Hartmann, W. (1986).
  Canonical decomposition with linear equality and inequality constraints on parameters. In W. Gaul & M. Schader (Eds.), Classification as a tool of research (pp. 191-199). Amsterdam: Elsevier.
  A numerical method is developed for the restricted canonical decomposition with which the single components of the parameter matrices can be subjected not only to linear equality but also linear inequality constraints. By this method especially boundary and/or order constraints can be handled very effectively. With simple modifications the technique can be used also for the minimizing of related models.

• Harvey, D., & Greenway, P. (1982).
  How parent attitudes and emotional reactions affect their handicapped child's self-concept. Psychological Medicine, 12, 357-370.
  The parents of 24 physically handicapped children were grouped according to their responses to a Primary Mood Factors grid and comparisons were made with their children's responses to a self-concept scale. It was found that parents who were close together in their primary mood reactions had handicapped children who were more positive in self-esteem than those dyads who were divided in their primary mood reaction.

• Hasegawa, K., Arakawa, M., & Funatsu, K. (1999).
  3D-QSAR study of insecticidal neonicotinoid compounds based on 3-way partial least squares model. Chemometrics and Intelligent Laboratory Systems, 47, 33-40.
  A choice of an active conformer and the corresponding alignment rule is an important problem for determining the success of 3D-QSAR study. For flexible molecules, this problem is the most difficult one and construction of the method with appropriate chemometric tools has been required. Recently, Bro has proposed a trilinear PLS algorithm as the trilinear extension of standard bilinear PLS in the field of analytical chemistry. This method seems to be suitable to the 3D-QSAR problem but only few attempts have so far been made at the subject. The object of this study is to investigate the ability of Bro's method for solving the conformer/alignment problem in 3D-QSAR study. The structure-activity data of insecticidal neonicotinoid compounds were used as a test example. The 3-way arrays were constructed from eight sample vectors and eight electrostatic similarity matrices derived from eight combinations of conformers and alignment rules. The correlation between the 3-way arrays and the insecticidal activity vector was investigated by the 3-way PLS method. A model with three significant components was obtained, and from its PLS loading the best combination of conformer and alignment rule could be selected.

• Hasegawa, K., Arakawa, M., & Funatsu, K. (2000).
  Rational choice of bioactive conformations through use of conformation analysis and 3-way partial least squares modeling Chemometrics and Intelligent Laboratory Systems, 50, 253-261.
  Comparative molecular field analysis (CoMFA) has become widely used in three-dimensional (3D) QSAR studies. Although CoMFA has been of general use, there are some critical problems in the proper application. A major problem of CoMFA, including most other 3D QSAR methodologies, is that the results are dependent on the chosen bioactive conformations and the corresponding alignment rules of molecules. Recently, we have proposed a novel method with a 3-way PLS formulation for solving the conformation/alignment problem in 3D QSAR studies (Hasegawa, Arakawa, & Funatsu, 1999). The purpose of the present study is to demonstrate the general utility of our approach by applying to a real CoMFA data set. The data set of Protein-Tyrosine Kinase (PTK) inhibitors was used as a test sample. The possible 3D conformations of all molecules were generated by conformational analysis and they were characterized by field variables of CoMFA. To each unique conformation of the most active compound, one sample-variable sheet comprising of the most similar conformations was defined. The 3-way arrays for 3-way PLS analysis were created by collecting all sample-variable sheets. From the regression coefficient values of the 3-way PLS model, conformations largely contributing to inhibitory activity were selected and the resulting final CoMFA model could give the reasonable 3D coefficient contour maps.

• Hasegawa, K., Arakawa, M., & Funatsu, K. (2003a).
  Simultaneous determination of bioactive conformations and alignment rules by multi-way PLS modeling. Computational Biology and Chemistry, 27, 211-216.
  In this study, we propose a new three-dimensional quantitative structure-activity relationship (3D-QSAR) method for selecting bioactive conformations and alignment rule simultaneously. The possible conformations of all molecules are generated by conformational analysis and they are superimposed on template conformer with possible alignment rules. The field variables are calculated as 3D descriptor of structures. Four-way partial least-squares (PLS) analysis is applied, and the conformations and alignment rule largely contributing to biological activity are selected. In order to demonstrate this method, the data set of benzodiazepine derivatives, antagonists of (CCK-B), was used as a test sample. As a result, appropriate conformers and alignment rule were selected and significant PLS model was obtained. The resulting final model could give the reasonable 3D coefficient contour maps. Moreover, external prediction was carried out by use of external data sample and its prediction was proved to be high enough.

• Hasegawa, K., Matsuoka, S., Arakawa, M., & Funatsu, K. (2003b).
  Multi-way PLS modeling of structure-activity data by incorporating electrostatic and lipophilic potentials on molecular surface. Computational Biology and Chemistry, 27, 381-386.
  We devised and elaborated a surface-based three-dimensional-quantitative structure-activity relationship (3D-QSAR) method, which had been proposed in the previous study. This approach can be applied to more general case where both the electrostatic and lipophilic potentials on molecular surface simultaneously change. The 3D coordinates of all sampling points on molecular surface are projected into a 2D map by Kohonen neural network (KNN). Each node in the map is coded by the associated molecular electrostatic potential (MEP) or molecular lipophilic potential (MLP) values. The electrostatic and lipophilic KNN maps are generated for each compound and the four-way array is constructed by collecting two KNN maps of all samples. The correlation between four-way array and biological activity is examined by four-way partial least-squares (PLS). For validation, the structure-activity data of estrogen receptor antagonists was investigated. The four-way PLS model gave the high statistics at calibration and validation stages. The coefficients of the four-way PLS model back-projected on molecular surface had a reasonable 3D distribution and it was nicely consistent with active site of the estrogen receptor which was recently made clear by X-ray crystallography.

• Hayashi, C. (1981).
  Distance or generalized variance. Classification Automatique et Perception par Ordinateur, 129-141.
  The various criteria or measures are discussed and used for clustering, pattern recognition and representation of goodness of fit or discrimination in data analysis [2,6,8,10]. In the present paper, some discussions of the topic of the title will be given.

  

• Hayashi, C. (1982).
  An algorithm for the solution of PARAFAC model and analysis of the data by the international rice adaptation experiments. Bulletin of the Biometric Society of Japan, 3, 77-91.
  PARAFAC is a three mode factor analytic method developed by R.A. Harshman and is useful for data analysis. However, the computer program of PARAFAC developed by Harshman seems to require great skill to understand how to handle it and to select the optimal solution through many steps. Here a new fool-proof algorithm is presented that does not require any skill, with an application to the data of the international rice adaptation experiments.

• Hayashi, C. (1985).
  Recent theoretical and methodological developments in multidimensional scaling and its related methods in Japan. Behaviormetrika, 18, 67-79.
  This is a review paper. The term multidimensional scaling (MDS) is used in two senses. One is a narrow sense, a technical term which is divided into metric MDS and non-metric MDS. The other is a wider sense and has many variations in multidimensional data analysis. In this review, MDS in a narrow sense and its closely related methods, which are a part of MDS in a wider sense, are taken into consideration. And Japanese works are introduced.

• Hayashi, C. (1989).
  Multiway data matrices and method of quantification of qualitative data as a strategy of data analysis. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 131-142). Amsterdam: Elsevier.
  In the present paper, the analysis of multiway data matrices is discussed in the light of a method of quantification of qualitative data. The following three topics will be discussed. 1. Multiway frequency data matrix; 2. Three-way data matrix of dissimilarity indices among three elements; 3. Generalization of the method of quantification of response pattern (correspondence analysis) to a three-way matrix.

• Hayashi, C. & Hayashi, F. (1982).
  A new algorithm to solve PARAFAC-model. Behaviormetrika, 11, 49-60.
  PARAFAC is a three-mode factor analytic method developed by R.A. Harshman and is useful for data analysis. The fundamental idea says: x(ijk) ~ Sum{a(is)b(js)c(ks) where x(ijk) is given by measurement for i=1,...,I, j=1,...,J, k=1,...,K; a(is), b(js), and c(ks) are unknown.

  

• Hayashi, F., Yamaoka, K., & Terao, H. (1982).
  A comparative study between two algorithms of PARAFAC model (in Japanese). Japanese Journal of Behaviormetrics, 1, 47-62.
  In utilizing the PARAFAC model, when Dr. Harshman's program was used, there seems to be too many steps involved in leading to the optimum solution. In order to avoid such complexities and difficulties, Dr. Hayashi has suggested that a different, more fool proof method be used. A comparison was made between these two methods using data about the physical constitution of children. The results showed that these two methods yielded two different solutions. Yet the solution from Dr. Hayashi's mathod was found to be much more understandable than from Dr. Harshman's method.

  
         

• Heimdal, H., Bro, R., Larsen, L.M., & Poll, L. (1997).
  Prediction of polyphenol oxidase activity in model solutions containing various combinations of chlorogenic acid,(-)-epicatechnin, O₂, CO₂, temperature, and pH by multiway data analysis. Journal of Agricultural and Food Chemistry,45, 2399-2406
  Controlled and modified packing of lettuce is simulated in model solutions containing various chemicals under various conditions and enzymatic browning is measured. Results were analysed by multiway data analysis. A very descriptive five-component multiplicative Parafac model was obtained. Moreover it was shown that it is possible to predict polyphenol oxidase activity even though the samples are of very different constitutions.

  

• Heiser, W.J. (1989).
  The city-block model for three-way multidimensional scaling. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 395-404). Amsterdam: Elsevier.
  The city-block representation of three-way proximity data, with diagonal dimension weights, is studied for four reasons: (1) since it is the model of choice in certain domains, (2) because it naturally enjoys a unique axes property, (3) a large class of city-block (dis)similarity coefficients exists, and (4) there are some algorithmic problems to solve. A convergent algorithm based on iterative majorization and alternating least squares is presented, and a number of special applications with Gower's (1971) general coefficient of similarity are discussed.

• Heiser, W. J., & Bennani, M. (1997).
  Triadic distance models: Axiomatization and least squares representation. Journal of Mathematical Psychology,41, 189-206.
  Distance models for three-way proximity data, which consist of numerical values assigned to triples of objects that indicate their joint (lack of) homogeneity or resemblance. require a generalization of the usual distance concept defined on pairs of objects. An axiomatic framework is given for characterizing triadic dissimilarity, triadic similarity, and triadic distance, where the term triadic implies that each element of the triple is treated on an equal footing. Two kinds of distance models are studied in detail: the Minkowski-p or M-p model, which is based upon dyadic components and includes the perimeter model as an important special case, and several models based on presence-absence variables. They are shown to satisfy the tetrahedral inequality, a condition that is characteristic for the present axiomatization. Two monotonically convergent algorithms are described that find weighted least squares representations of three-way proximity data under the Euclidean M-1 model and the Euclidean M-2 model. To enable a scalefree evaluation of the quality of the fit, an additive decomposition of the sum of squares of the dissimilarities is derived. As illustrated in one of the examples, distance analysis of three-way, three-mode tables is possible by a suitable manipulation of the least squares weights.

• Henrion, G., Henrion R., Bacsó J., & Uzonyi I. (1990).
  Ausgewählte Methoden der Korrelationsanalyse am Beispiel der Spurenelementrelationen im menschlichen Körper [Selected methods of correlation-analysis for the example of trace-element relations in human-body]. Zeitschrift für Chemie, 30, 204-211.**
  Selected methods from uni- and multivariate analysis of correlation are discussed and applied to a 3-dimensional data array of the object-variable occasion type. The data resulted from trace element determinations (by x-ray fluorescence analysis in human tissues. Special attention is devoted to 3-way principal component analysis, partial least squares, linear characteristics and the nonparametric rank correlation coefficient by Kendall. It is shown that the correlations between interior human tissues (brain, liver, kidney) are strongly governed by a specific set of toxic trace elements (As, Se, Hg, Pb, Ni). However the relations between tissues and hair, which are of special interest in medicine, turn out to be rather weak. This results in an unsatisfactory prediction of toxic elements in tissues by those in hair.

• Henrion, G., Nass, D., Michael, G., & Henrion, R. (1995).
  Multivariate 3-way data analysis of amino acid patterns of lakes. Fresenius' Journal of Analytical Chemistry, 352, 431-436.
  Time dependent patterns of amino acid concentrations have been studied by HPLC for different lakes of the Berlin area. Data analysis has been performed by conventional principal component analysis as well as by its more recent N-way extension. It turns out that lakes mainly differ by their general amino acid production as a function of time and season. Apart from this, in a single case there occurs a specific pattern which might be related to an exterior influence. This pattern, although clearly detected, has not been stable over time. Measurements are reproducible with respect to time (comparison of two succeeding years) and to position (comparison of isolated parts of a lake).

• Henrion, R. (1993).
  Body diagonalization of core matrices in three-way principal components analysis: Theoretical bounds and simulation. Journal of Chemometrics, 7, 477-494.
  In contrast with conventional PCA, a direct superposition and joint interpretation of loading plots is not possible in three-way PCA, since there may be data variance which is described by unequal components of different modes. The contributions to variance of all possible combinations of components are described in the core matrix. Body diagonalization, which is achieved by appropriate rotation of component matrices, is an essential tool for simplifying the core matrix structure. The maximum degree of body diagonality which may be obtained from such transformations is analysed from both the mathematical and simulation viewpoints. It is shown that, at least in the average case, high degrees can be expected, which makes the procedure reasonable for many practical applications. Furthermore, simulation as well as theoretical derivation show that the success of body diagonality depends on the so-called polarity of the core array. The methodology is illustrated by a three-way data example from environmental chemistry.

• Henrion, R. (1994).
  N-way principal component analysis: Theory, algorithms and applications. Chemometrics and Intelligent Laboratory Systems, 25, 1-23.
  Due to sophisticated experimental designs and to modern instrumental constellations the investigation of N-dimensional (or N-way or N-mode) data arrays is attracting more and more attention. Three-dimensional arrays may be generated by collecting data tables with a fixed set of objects and variables under different experimental conditions, at different sampling times, etc. Stacking all the tables along varying conditions provides a cubic arrangement of data. Accordingly the three index sets or modes spanning a three-way array are called objects, variables and conditions. In many situations of practical relevance even higher-dimensional arrays have to be considered. Among numerous extensions of multivariate methods to the three-way case the generalization of principal component analysis (PCA) has central importance. There are several simplified approaches of three-way PCA by reduction to conventional PCA. One of them is unfolding of the data array by combining two modes to a single one. Such procedure seems reasonable in some specific situations like multivariate image analysis, but in general combined modes do not meet the aim of data reduction. A more advanced way of unfolding which yields separate component matrices for each mode is the Tucker 1 method. Some theoretically based models of reduction to two-way PCA impose some specific structure on the array. A proper model of three-way PCA was first formulated by Tucker (so-called Tucker3 model among other proposals). Unfortunately the Tucker 1 method is not optimal in the least squares sense of this model. Kroonenberg and De Leeuw demonstrated that the optimal solution of Tucker's model obeys an interdependent system of eigenvector problems and they proposed an iterative scheme (alternating least squares algorithm) for solving it. With appropriate notation Tucker's model as well as the solution algorithm are easily generalized to the N-way case (N > 3). There are some specific aspects of three-way PCA, suc as complicated ways of data scaling or interpretation and simple-structure-transformation of a so-called core matrix, which make it more difficult to understand than classical PCA. An example from water chemistry serves as an illustration. Additionally, there is an application section demonstrating several rules of interpretation of loading plots with examples taken from environmental chemistry, analysis of complex round robin tests and contamination analysis in tungsten wire production.

• Henrion, R. (1994).
  Simultaneous simplification of loading and core matrices in N-way PCA: application to chemometric data arrays Fresenius Journal of Analytical Chemistry, 361, 15-22.
  In the Tucker3 model of N-way principal components analysis (NPCA), a so-called core matrix describes the possible interactions between components from different modes. For an easy interpretation of solutions, it is necessary to have as few interactions as possible (in conventional PCA of data tables, such interactions can always be avoided). This goal may be realized by various approaches of core matrix transformations. At the same time, it is desirable to have simple component (or loading) matrices. Usually, the simplicity of the core conflicts to a certain degree with the simplicity of the components. The paper demonstrates how the conditional optimization of both goals can be used to find a compromise. For the purpose of illustration, the procedure is first applied to a small three-way data array from heavy metal analysis of tissues in different samples of game. Later, a data array of bigger size from a three-way interlaboratory study is considered.

• Henrion, R. (2000).
  On global, local and stationary solutions in three-way data analysis. Journal of Chemometrics, 14, 261-274.
  The issue of global and local solutions to optimization problems is of much interest in the context of three-way analysis, in particular when dealing with the PARAFAC and Tucker3 models or core transformations within the latter. For clarity of statements, it is useful to consider the most simple yet reasonable situation, namely one-component PARAFAC decomposition or, closely related, maximization of the leading squared core entry in Tucker3. In the paper, necessary and sufficient conditions for global solutions are derived. Furthermore, it is shown that, in general, the usual cyclic co-ordinate optimization scheme of three-way methods does not converge towards a local minimum (or maximum) even if the iterates yield global solutions in each co- ordinate direction. Finally, an example for a proper local minimum in one- component PARAFAC is given.

• Henrion, R. & Andersson, C.A. (1999).
  A new criterion for simple-structure transformations of core arrays in N-way principal components analysis. Chemometrics and Intelligent Laboratory Systems, 47, 189- 204.
  Among the possible (orthogonal) transformations of core arrays in N-way principal components analysis (PCA), the conventional approach of body diagonalization turns out not to provide the simplest structure (in the sense of minimizing the number of significant entries). As an alternative, the maximization of the variance-of-squared core entries is proposed. Both criteria are equivalent in a two-way constellation but may differ markedly for N greater than or equal to 3. Actually, using the variance criterion may provide more insight into the rank structure of the given data, and it is also easily applied to general rectangular core arrays. In order to clarify the relation between body diagonality and variance-of-squares, we prove the following main result of the paper: If some cubic N-way core array can be transformed to exact body diagonality, then the same transformation yields maximum variance-of-squared entries. This result implies the equivalence in the two-way case mentioned above. A solution algorithm is formulated and illustrated with a small numerical example. The application to data examples from environmental chemistry and chromatographic analysis is briefly discussed.

• Henrion, R., Henrion, G., Böhme, M. & Behrendt, H. (1997).
  Three-way principal components analysis for fluorescence spectroscopic classification of algae species. Fresenius' Journal of Analytical Chemistry, 357, 522-526.
  In this paper it is illustrated how three-way prinicpal components analysis as the appropriate generalisation of conventional prinicpal components analysis may serve as a powerful method for clasisifcation of algae species using the excitation-emission matrices from fluorescence spectroscopy from different species.

• Henrion, R., Henrion, G., Heininger, P., & Steppuhn, G. (1991).
  Statistical analysis of complex round robin tests. Acta Hydrochimica et Hydrobiologica, 19, 603-614.
  Different possibilities for the evaluation of complex round robin tests are presented. The advantages of Three-way over classical principal components analysis when applied to Three-way data arrays are discussed. The method allows extreme data reduction without essential loss of information. This is useful first of all for a graphically oriented evaluation of laboratories. The results of 4 round robin tests (8 laboratories, 5 parallel estimations, 4 trace element concentrations) for the analysis of heavy metals in waters (synthetic solutions with known concentrations) serve as an example. Special attention is paid to the recognition of different kinds of errors (random and systematic errors or unstable working).

• Henrion, R., Henrion, G., & Onuoha, G.C. (1992).
  Multiway principal components-analysis of a complex data array resulting from physicochemical characterization of natural-waters. Chemometrics and Intelligent Laboratory Systems, 16, 87- 94.**
  Multi-way principal components analysis (MPCA) is an efficient tool for reducing higher dimensional data arrays. Using the Kroonenberg algorithm: which originally was developed for three-dimensional data arrays but may be generalized to arbitrary dimensions in a straightforward manner: MPCA is applied to a complex example from the chemistry of waters. The data originated from the measurements of fifteen physicochemical parameters (variables) at ten different locations (objects) within some specific area of the Niger delta. These measurements were consistently recorded 22 times (occasions) in the course of a year. MPCA allows the detection of spatial and temporal factors of influence and the classification of the parameters considered according to these factors.

• Henshaw, J.M., Burgess, L.W., Booksh, K.S., & Kowalski, B.R. (1994).
  Multicomponent determination of chlorinated hydrocarbons using a reaction-based chemical sensor .1. Multivariate calibration of fujiwara reaction-products. Analytical Chemistry, 66, 3328-3336.**
  Multicomponent analysis for 1,1,1-trichloroethane, trichloroethylene, and chloroform is demonstrated by partial least squares modeling of absorbance data collected from the Fujiwara reaction of these analytes over time. Optimal calibration times are determined for each analyte dependent upon its reaction rate. This is the simplest example of trying to use the selectivity gained from a second data dimension or order. Problems associated with first-order calibration are demonstrated when interferences unaccounted for in the calibration model are present. Determination of 1,1,1-trichloroethane in the presence of the other two species is also demonstrated using the generalized rank annihilation method and trilinear decomposition for second-order calibration. a subsequent paper describes an approach using multivariate curve resolution to extract the analytical information from the spectral and temporal profiles of analytes in the Fujiwara reaction.

• Hentschel, U. & Klintman, H. (1974).
  A 28-variable semantic differential. I. On the factorial identification of content. Psychological Research Bulletin, 14.
  T3 was used to assess the structure of semantic differential data of 4 concepts (primarily self-description) scored on 28 bi-polar scales by 209 subjects who did, however, not all score all concepts. Therefore, only very partial results could be obtained.

  

• Hernandez-Arteseros, J. A., Beltran, J. L., Compano, R., & Prat, M. D. (2002).
  Fast-scanning fluorescence spectroscopy as a detection system in liquid chromatography or confirmatory analysis of flumequine and oxolinic acid. Journal of Chromatography A, 942, 275-281.
  Oxolinic acid and flumequine were analysed by reversed-phase liquid chromatography after extraction from the sample matrix with dichloromethane and partitioning with NaOH. The detection system consisted of a fast-scanning fluorescence detector, which provides the full spectra of the eluting peaks and can thus be used to confirm the identity of analytes. Determination was performed by partial least squares (PLS) and three-way PLS over the three-dimensional data, i.e. fluorescence intensity versus retention time and excitation wavelength. In both cases, similar results, with prediction errors around 4%, were obtained. The method was successfully applied to the analysis of salmon, pork and chicken muscle spiked up to 300 ng g(-1).

• Herrero, A., Zamponi, S., Marassi, R., Conti, P., Ortiz, M. C. & Sarabia, L. A. (2002).
  Determination of the capability of detection of a hyphenated method: application to spectroelectrochemistry.Chemometrics en Intelligent Laboratory Systems, 61, 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 alpha and beta 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 the presence of this interferent, avoiding the need to previously identify and determine the quantity of the interferent. (C) 2002 Elsevier Science B.V. All rights reserved.

• Herrmann, W.M., Röhmel, J., Streitberg, B., & Willmann, J. (1983).
  Example for applying the COMSTAT multimodal factor analysis algorithm to EEG data to describe variance sources. Neuropsychobiology, 10, 164-172.
  An example is given for the application of the COMSTAT algorithm for multimodal factor analysis to EEG power spectra data. The COMSTAT algorithm enlarges Tucker's three-mode factor analysis to a multimodal one, and improves his algorithm by a least squares solution. The EEG power spectral data from 65 healthy subjects with an occipital rythm between 8 and 12 Hz were taken. For demonstration purposes we selected three modes, which have been used by other authors: mode 1: 29 frequency classes, ln of relative power, in delta f = 1.0-Hz steps between 1 and 30 Hz; mode 2: 16 segments, 40 s each, during the two situational vigilance conditions reaction time (RT) and resting (RS), and mode 3: 65 persons. The frequency mode could be described sufficiently by five factors. The factor-loading profiles were similar to those described earlier in independent data. Thus, in the three-mode model we obtained results comparable to those of two models. On the basis of these data, we formulate the hypothesis that in EEG data from subjects with occipital alpha-EEG and for the occipital lead, the most important P-variance sources are the alpha power, the dominant alpha frequency, and variance due to dynamic changes which can be caused by a shift in vigilance or a dissociative shift in vigilance. The aim of applying our model in EEG data is to describe variance sources in a multidimensional space. The ultimate goal, however, is to arrive at procedures which allow us to keep more natural sources of variance in our models, and not to exclude these variance sources by rigorous selection of subjects.

  
         

• Hiden, H. G., Willis, M. J., Tham, M. T., & Montague, G. A. (1999).
  Non-linear principal components analysis using genetic programming. Computers & Chemical Engineering, 23, 413-425.
  Principal components analysis (PCA) is a standard statistical technique, which is frequently employed in the analysis of large highly correlated data sets. As it stands, PCA is a linear technique which can limit its relevance to the non-linear systems frequently encountered in the chemical process industries. Several attempts to extend linear PCA to cover non-linear data sets have been made, and will be briefly reviewed in this paper. We propose a symbolically oriented technique for non-linear PCA, which is based on the genetic programming (GP) paradigm. Its applicability will be demonstrated using two simple non-linear systems and data collected from an industrial distillation column.

• Hildebrandt, L., & Klapper, D. (1994).
  The analysis of three-way three-mode data: A program based on GAUSS. In F. Faulbaum (Ed.), Advances in Statistical Software, 4. (pp. 527-534). Stuttgart - Jena - New York: Gustav Fischer.
  Four different methods are commonly proposed for the exploratory analysis of three-way three-mode data (TUCKALS-3, TUCKALS-2, CANDECOMP/PARAFAC, PFCORE). Available computer programs are in general stand alone versions and do not permit an interactive stepwise analysis of the data. The program TREMODA (ThRee MOde Data Analysis), written in GAUSS, allows to combine the four methods and can handle large data sets on a PC. The output is available with a high performance graphics. After a short presentation of the methodological and mathematical background of these methods the application of TREMODA to a problem of consumer research is shown.

• Hildebrandt, L., & Klapper, D. (2001).
  The analysis of price competition between corporate brands. International Journal of Research in Marketing, 18, 139-159.
  The methodology developed in this paper provides a means to analyze price competition between corporate brands. Corporate brands are considered to be produced and marketed by the same company. We establish price competition from an array of cross-price elasticities across time, which provides the necessary information to uncover the competitive interaction effects between corporate brands. The cross-price elasticities across time form a three-mode three-way array. The Constrained TUCKALS3-approach is developed and introduced for the analysis of the complex array of cross-elasticities. The approach takes explicitly a priori information about the competitive reaction or pattern of the marketing activities in a competitive market into account. This new methodology will enable brand management to gain further and deeper insight into the competitive interaction effects in the market. The Constrained TUCKALS3-model parameters provide the basis to investigate cannibalistic effects between corporate brands and thus helps to improve the marketing-mix, especially the price management of these brands. Furthermore, the results of the Constrained TUCKALS3-approach can serve to determine idealized market share estimates for certain a priori defined competitive conditions. The applicability and the methodological advantages of this approach will be shown by an empirical study on the price competition between two corporate brands. The reported results provide managerial useful information for the development and improvement of marketing-mix-strategies of these corporate brands.

• Hindmarch, P., Kavianpour, K., & Brereton, R.G. (1997).
  Evaluation of parallel factor analysis for the resolution of kinetic data by diode-array high-performance liquid chromatography. Analyst, 122, 871-877.
  The Parafac algorithm for factor analysis of three or higher way datasets is summarised. A series of simulations of kinetic profiles of two-way diode-array HPLC data is described. A three-phase reaction system of reactant, intermediate and product is used to illustrate the method, each closely eluting and with similar spectra based on experimental HPLC with diode-array detection of chlorophyll degradation products. A kinetic parameter is varied to change the relative concentration of the intermediate in each series of simulations. Several indices of quality of reconstruction are introduced. It is concluded that the number of factors used to model the data is crucial to the quality of reconstruction. A good approach is first to use fewer factors than are expected, then increasing the number until each elution profile shows a single maximum.

• Hirschberg, N. (1980).
  Individual differences in social judgment: A multivariate approach. In M. Fishbein (Ed.), Progress in social psychology. Hillsdale, NJ: Erlbaum.
  Contains an overview of multivariate methods to assess individual differences, like MDS, point-of-view analysis, INDSCAL, T3, and preference analyses. One of the examples is a detailed summary of Wiggins & Blackburn (1976).

  
             

• Hoffman, E.L. & Tucker, L.R. (1964).
  Three-way factor analysis of a multitrait-multimethod matrix. (Technical Report), Urbana: University of Illinois, Department of Psychology, and the Office of Naval Research.
  H & T reanalyse a multitrait-multimethod (MM) matrix (with communalities in the diagonal) of Fiske by using the T3 model. Some special formulas were derived to obtain the method and the trait correlation matrices and the core matrix as the individual scores were not available. The factors of the trait-, method-, and MM correlation matrices and the core matrix were interpreted, and compared with Fiske's results.

• Hogden, J., Lofqvist, A., Gracco, V., Zlokarnik, I., & Rubin, P., & Saltzman, E. (1996).
  Accurate recovery of articulator positions from acoustics: New conclusions based on human data. Journal of the Acoustical Society of America, 4, 4.**
  Vocal tract models are often used to study the problem of mapping from the acoustic transfer function to the vocal tract area function (inverse mapping). Unfortunately, results based on vocal tract models are strongly affected by the assumptions underlying the models. In this study, the mapping from acoustics (digitized speech samples) to articulation (measurements of the positions of receiver coils placed on the tongue, jaw, and lips) is examined using human data from a single speaker: Simultaneous acoustic and articulator measurements made for vowel-to-vowel transitions, /g/ closures, and transitions into and out of /g/ closures. Articulator positions were measured using an emma system to track coils placed on the lips, jaw, and tongue. Using these data, look-up tables were created that allow articulator positions to be estimated from acoustic signals. On a data set not used for making look-up tables, correlations between estimated and actual coil positions of around 94% and root-mean-squared errors around 2 mm are common for coils on the tongue. An error source evaluation shows that estimating articulator positions from quantized acoustics gives root-mean-squared errors that are typically less than 1 mm greater than the errors that would be obtained from quantizing the articulator positions themselves. This study agrees with and extends previous studies of human data by showing that for the data studied, speech acoustics can be used to accurately recover articulator positions. (c) 1996 acoustical society of america.

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• Hohn, M.E. (1979).
  Principal components analysis of three-way tables. Journal of the International Association of Mathematical Geology, 11, 611-626.
  A concise, straight forward description of T3 based on Tucker (1966). Illustrated with geological data: 4 localities of sample collection from the Early Jurassic scales of the Paris Basin, 3 fractions of the organic extracts and 6 organic elements or elemental ratio's. Data standardized by elements. Varimax rotation for all components. Neat solutions. Uses a 'classical' score matrices based on combination-modes for interpretation.

  

• Hohn, M.E. & Friberg, L.M. (1979).
  A generalized principal components model in petrology. Lithos, 12, 317-324.
  An exposition of T3 is presented and its usefulness in petrology is explained. The method is illustrated with 4 samples of Machas charnockites of which 9 cations and 1 ratio were determined from 3 minerals. The data were standardized per sample; factor scores on combination-mode components were used. A second example comprises a set of data from the Spuhler Peale Formation in Montana (9 cations, 4 minerals and 3 samples).

• Holmes, S. (1989).
  Using the bootstrap and the RV coefficient in the multivariate context. In E. Diday (Ed.), Data analysis, learning symbolic and numeric knowledge (pp. 119-131). New York: Nova Science.
  When bootstrapping in the multivariate context several difficulties arise, one of these is the need to find an appropriate vector space where the different bootstraps can be studied. One solution to this problem is provided through the analysis of a cube of data as in Conjoint Analysis. This method uses the concepts of characteristic operators and their inner product which will be defined in the first part of this paper. Another possibility is to bootstrap a uni-dimensional statistic, namely the RV coefficient, this provides a way of choosing the number of components to be retained in PCA, without the indecision of graphical procedures and the distributional assumptions in the usual parametric procedures.

  

• Hong, S., & Harshman, R.A. (2000).
  Shifted factor analysis, Part II: Algorithms and applications. Unpublished manuscript, University of Western Ontario Psychology Department, London, Canada.
  The authors previously proposed a family of models that overcome the problem of factor position shift in sequential data and used the information provided by the shifts to make all the model parameters identifiable. Now they consider methods of parameter estimation, and study some properties of the algorithms and their underlying models using both error-free and fallible position shifted data. The Alternating Least Squares (ALS) approach is not fully suitable for estimation because factor position shifts destroy multilinearity of the latent structure. Therefore, and alternative "quasi-ALS" approach is developed, and some of its practical and theoretical problems are dealt with. Quasi-ALS algorithms for the two-way, three-way, and n-way cases are described in detail. A four-way chromatographic dataset previously analyzed by Bro et al. is reanalyzed, and (two or) three out of four factors are recovered. The reason for the incomplete success may be factor shape changes combined with the lack of distinct shift patterns for two of the factors. SFA is compared with Parafac2 from both theoretical and practical points of view.

• Hong, S., & Harshman, R. A. (2003a).
  Shifted factor analysis, Part II: Algorithms. Journal of Chemometrics, 17, 379-388.
  We previously proposed a family of models that deal with the problem of factor position shift in sequential data. We conjectured that the added information provided by fitting the shifts would make the model parameters identifiable, even for two-way data. We now derive methods of parameter estimation and give the results of experiments with synthetic data. The alternating least squares (ALS) approach is not fully suitable for estimation, because factor position shifts destroy the multilinearity of the latent structure. Therefore an alternative ‘quasi-ALS’ approach is developed, some of its practical and theoretical properties are dealt with and several versions of the quasi-ALS algorithm are described in detail. These procedures are quite computation-intensive, but analysis of synthetic data demonstrates that the algorithms can recover shifting latent factor structure and, in the situations tested, are robust against high error levels. The results of these experiments also provide strong empirical support for our conjecture that the two-way shifted factor model has unique solutions in at least some circumstances.

• Hong, S., & Harshman, R. A. (2003b).
  Shifted factor analysis - Part III: N-way generalization and application Journal of Chemometrics, 17, 389-399.
  The 'quasi-ALS' algorithm for shifted factor estimation is generalized to three-way and n-way models. We consider the case in which mode A is the only shifted sequential mode, mode B determines shifts, and modes above B simply reweight the factors. The algorithm is studied using error-free and fallible synthetic data. In addition, a four-way chromatographic data set previously analyzed by Bro et al. (J. Chemometrics 1999; 13: 295-309) is reanalyzed and (two or) three out of four factors are recovered. The reason for the incomplete success may be factor shape changes combined with the lack of distinct shift patterns for two of the factors. The shifted factor model is compared with Parafac2 from both theoretical and practical points of view.

• Hong-Ping, X., Jian-Hui, J. Guo-Li, S., & Ru-Qin, Y. (2002).
  Estimation of the chemical rank for the three-way data: a principal norm vector orthogonal projection approach. Computers & Chemesty, 26, 183-190.
  A new approach for estimating the chemical rank of the three-way array called the principal norm vector orthogonal projection method has been proposed. The method is based on the fact that the chemical rank of the three-way data array is equal to one of the column space of the unfolded matrix along the spectral or chromatographic mode. A vector with maximum Frobenius norm is selected among all the column vectors of the unfolded matrix as the principal norm vector (PNV). A transformation is conducted for the column vectors with an orthogonal projection matrix formulated by PNV. The mathematical rank of the column space of the residual matrix thus obtained should decrease by one. Such orthogonal projection is carried out repeatedly till the contribution of chemical species to the signal data is all deleted. At this time the decrease of the mathematical rank would equal that of the chemical rank, and the remaining residual subspace would entirely be due to the noise contribution. The chemical rank can be estimated easily by using an F-test. The method has been used successfully to the simulated HPLC-DAD type three-way data array and two real excitation-emission fluorescence data sets of amino acid mixtures and dye mixtures. The simulation with added relatively high level noise shows that the method is robust in resisting the heteroscedastic noise. The proposed algorithm is simple and easy to program with quite light computational burden.

• Hoole, P. (1999).
  On the lingual organization of the German vowel system. Journal of the Acoustical Society of America, 106, 1020-1032.
  A hybrid PARAFAC and principal-component model of tongue configuration in vowel production is presented, using a corpus of German vowels in multiple consonant contexts (fleshpoint data for seven speakers at two speech rates from electromagnetic articulography). The PARAFAC approach is attractive for explicitly separating speaker-independent and speaker-dependent effects within a parsimonious linear model. However, it proved impossible to derive a PARAFAC solution of the complete dataset (estimated to require three factors) due to complexities introduced by the consonant contexts. Accordingly, the final model was derived in two stages. First, a two-factor PARAFAC model was extracted. This succeeded; the result was treated as the basic vowel model. Second, the PARAFAC model error was subjected to a separate principal-component analysis for each subject. This revealed a further articulatory component mainly involving tongue- blade activity associated with the flanking consonants. However, the subject- specific details of the mapping from raw fleshpoint coordinates to this component were too complex to be consistent with the PARAFAC framework. The final model explained over 90% of the variance and gave a succinct and physiologically plausible articulatory representation of the German vowel space.

• Hopke, P.K., Paatero, P., Jia, H., Ross, R.T., & Harshman, R.A. (1998).
  Three-way (PARAFAC) factor analysis: examination and comparison of alternative computational methods as applied to ill-conditioned data. Chemometrics and Intelligent Laboratory Systems, 43, 25- 42.
  Four different approaches to solving the trilinear three-way factor analysis problem are compared, and their performance with 'difficult' (i.e., ill-conditioned) data is tested. These approaches ape represented by four different computer programs: one using a simple alternating least squares (ALS) algorithm with only minimal extrapolation (HL-PARAFAC), one in which the ALS is supplemented by a sophisticated extrapolation to speed convergence (TPALS), one using a non-linear curve fitting method (PMF3), and one using a non-iterative closed-form approximation (DTDMR). The options provided by these programs (e.g., with regard to missing values, weighted least squares, non-negativity and other constraints) are compared, criteria for choosing synthesized test data and a method for synthesizing exponential test data are described. A numerical index is introduced to characterize the ill-conditioning of n-way arrays (n > 2). Two well characterized synthetic data sets serve as 'difficult' (ill-conditioned) test data. Intercomparisons among HL-PARAFAC, TPALS, DTDMR and PMF3 were implemented with these test data. Consequently, their limitations and strengths are determined, in addition, these trilinear analysis approaches are applied to a difficult set of ill-conditioned real data: a set of fluorescence spectroscopy measurements that characterize the steady-state fluorescence of an amino acid in aqueous solution. When converged, the results produced by the three least-squares techniques (but not DTDMR) agree. However, there are large differences in convergence speed when these difficult problems are solved: TPALS is faster than PARAFAC by a factor of ten, and PMF3 is faster than TPALS, again by a factor of ten. The program DTDMR is the fastest, bur it only solves half of the problems.

• Hopke, P. K. (2003).
  The evolution of chemometrics. Analytica Chimica Acta, 500, 365-377.
  mathematical methods to chemical problems to permit maximal collection and extraction of useful information. The development of advanced chemical instruments and processes has led to a need for advanced methods to design experiments, calibrate instruments, and analyze the resulting data. For many years, there was the prevailing view that if one needed fancy data analyses, then the experiment was not planned correctly, but now it is recognized that most systems are multivariate in nature and univariate approaches are unlikely to result in optimum solutions. At the same time, instruments have evolved in complexity, computational capability has similarly advanced so that it has been possible to develop and employ increasing complex and computationally intensive methods. In this paper, the development of chemometrics as a subfield of chemistry and particularly analytical chemistry will be presented with a view of the current state-of-the-art and the prospects for the future will be presented.

  

• Houle, D., Mezey, J., & Galpern, P. (2002).
  Interpretation of the results of common principal components analyses. Evolution, 56, 433-440.
  Common principal components (CPC) analysis is a new tool for the comparison of phenotypic and genetic variance-covariance matrices. CPC was developed as a method of data summarization, but frequently biologists would like to use the method to detect analogous patterns of trait correlation in multiple populations or species. To investigate the properties of CPC, we simulated data that reflect a set of causal factors. The CPC method performs as expected from a statistical point of view, but often gives results that are contrary to biological intuition. In general, CPC tends to underestimate the degree of structure that matrices share. Differences of trait variances and covariances due to a difference in a single causal factor in two otherwise identically structured datasets often cause CPC to declare the two datasets unrelated. Conversely, CPC could identify datasets as having the same structure when causal factors are different. Reordering of vectors before analysis can aid in the detection of patterns. We urge caution in the biological interpretation of CPC analysis results.

             

• Huang, J., Wium, H., Qvist, K. B., & Esbensen, K. H. (2003a).
  Multi-way methods in image analsis-relationships and applications. Chemometrics and Intelligent Laboratory Systems, 66, 141-158.
  This paper gives an overview of multi-way methods in image analysis, termed N-way image analysis. Both weak and strong multi-way methods are applied in order to decompose and characterize image data, and obtain insight into their abilities to capture and model the interpretable data structure. Multivariate Image Analysis (MIA) is a typical example based on weak multi-way methods like unfold-PCA/PLS. Strong multi-way methods such as PARAFAC, Tucker3, N-PLS are also introduced and applied to image analysis in this work. Which method to use is problem-dependent. Through macroscopic satellite images, virtual fluorescence images and microscopic functional property image examples, the performance of each alternative method is presented, as well as comparisons between weak and strong multi-way models. It is demonstrated that efficient handling of multiple images requires a clear a priori overview of the relationship between problem formulation and data array configuration. Appropriate preprocessing techniques, such as 2-D FFT and Wavelet transform, may also be needed in order to transform and configure some special types of image data to forms specifically suited for multi-way modeling. Application I shows the possibility for application of strong multi-way methods on multispectral images, otherwise conventionally analyzed by MIA. By contrast, application II attempts to investigate the feasibility of applying MIA models on typical three-way data, normally handled by the strong multi-way methods and provides a new perspective of dealing with fluorescence spectra as images. In application III, attempts have been made to predict rheological parameters from microscopic cheese images by multi-way methods. The present didactic exposition allows to draw some tentative first conclusions as to the dominant relationships between strong and weak multi-way data decompositions, their pros and cons and their relative merits.

• Huang, J. R., Li, T. H., Chen, K., & Qi, Y. P. (2003b).
  A preprocessing method applied in multi-dimension data - NOSC. Chemical Journal of Chinese Universities-Chinese, 24, 1009-1012.
  Orthogonal signal correction (OSC) is a novel spectral preprocessing method, which is based on the orthogonal projection. This pre-processing way removes from the spectral vector or matrix(X) only the part that definitely is unrelated to Y-vector or Y-matrix. Due to the merits of OSC, much interesting is: focus on it, and several modified algorithms have been presented to improve it. In this paper, we present a modified multi-dimension method, named NOSC, which is applied in preprocessing three-dimension data. The three-dimension data is the measurement of a drug mixture by HPLC-DAD, in which there are three components, that are enoxacin, norfloxacin, and ciprofloxacin. Compared with, those only processed by NPLS, the multi-dimension data preprocessed by NOSC have an optimal analytic result, and the covariance sauqre root is 0.33, 0.21 and 0.16 for enoxacin, norfloxacin, and ciprofloxacin respectively. NOSC was recommended for processing multi-dimension data with NPLS, PRFA and GRAFA, etc..

• Hubert, L., & Arabie, P. (1995).
  Iteratieve projectiion strategies for the least-squares fitting of tree structures to proximity data. British Journal of Mathematical & Statistical Psychology, 48, 281-317.
  A least squares optimization strategy is first reviewed and applied to the task of fitting a given collection of symmetric proximity values defined between the objects from one set by a collection of reconstructed proximity values, satisfying a fixed set of constraints, generated from some specified graph- theoretic structure, such as an ultrametric or an additive tree, selected for representing the objects. Our method uses iterative projection on to closed convex sets defined by the collection of given constraints characterizing the structural representation specified, and in contrast to least squares optimization methods that impose such constraints through the use of penalty functions, avoids the use of the latter, as well as the implementation of any gradient-based optimization technique. Secondly, just as various penalty-function/gradient- based optimization techniques have been turned into heuristic search strategies for such particular structures of interest as ultrametrics or additive trees, the use of iterative projection is suggested as a general heuristic search strategy for locating the best structural representations to impose in the first place, where the collection of constraints used may vary over the course of the optimization process. Our evaluation of the expected results uses several data sets previously analysed in the literature. Finally, several other applications of iterative projection as a heuristic optimization technique are discussed, including the consideration of data beyond that of a single symmetric proximity matrix (for example, extensions to two-mode proximity matrices, i.e. between two distinct object sets, and to three-way proximity matrices either symmetric or not), and to representations based on sums of matrices where each is constrained separately to conform to some desired representational structure.

• Hunt, L. A., & Basford, K. E. (1999).
  Fitting a mixture model to three-mode three-way data with categorical and continuous variables. Journal of Classification, 16, 283-296.
  The mixture likelihood approach to clustering is most often used with two-mode two-way data to cluster one of the modes (e.g., the entities) into homogeneous groups on the basis of the other mode (e.g., the attributes). In this case, the attributes can either be continuous or categorical. When the data set consists of a three-mode three-way array (e.g., attributes measured on entities in different situations), an analogous procedure is needed to enable the clustering of the entities (i.e., one of the modes) on the basis of both of the other modes simultaneously (i.e., the attributes measured in different situations). In this paper, it is shown that the finite mixture approach to clustering can be extended to analyze three-mode three-way data where some of the attributes are continuous and some are categorical. The methodology is illustrated by clustering the genotypes in a three-way soybean data set where various attributes were measured on genotypes grown in several environments.

• Hunt, L.A., & Basford, K.E. (2001).
  Fitting a mixture model to three-mode three-way data with missing information. Journal of Classification, 18, 209-226.
  When the data consist of certain attributes measured on the same set of items in different situations, they would be described as a three-mode three-way array. A mixture likelihood approach can be implemented to cluster the items (i.e., one of the modes) on the basis of both of the other modes simultaneously (i.e, the attributes measured in different situations). In this paper, it is shown that this approach can be extended to handle three-mode three-way arrays where some of the data values are missing at random in the sense of Little and Rubin (1987). The methodology is illustrated by clustering the genotypes in a three-way soybean data set where various attributes were measured on genotypes grown in several environments.

• Huo, R., Wehrens, R., Van Duynhoven, J., & Buydens, L. M. C. (2003).
  Assessment of techniques for DOSY NMR data processing. Analytica Chimica Acta, 490, 231-251.
  Diffusion-ordered spectroscopy (DOSY) NMR is based on a pulse-field gradient spin-echo NMR experiment, in which components experience diffusion. Consequently, the signal of each component decays with different diffusion rates as the gradient strength increases, constructing a bilinear NMR data set of a mixture. By calculating the diffusion coefficient for each component, it is possible to obtain a two-dimensional NMR spectrum: one dimension is for the conventional chemical shift and the other for the diffusion coefficient. The most interesting point is that this two-dimensional NMR allows non-invasive "chromatography" to obtain the pure spectrum for each component, providing a possible alternative for LC-NMR that is more expensive and time-consuming. Potential applications of DOSY NMR include identification of the components and impurities in complex mixtures, such as body fluids, or reaction mixtures, and technical or commercial products, e.g. comprising polymers or surfactants. Data processing is the most important step to interpret DOSY NMR. Single channel methods and multivariate methods have been proposed for the data processing but all of them have difficulties when applied to real-world cases. The big challenge appears when dealing with more complex samples, e.g. components with small differences in diffusion coefficients, or severely overlapping in the chemical shift dimension. Two single channel methods, including SPLMOD and continuous diffusion coefficient (CONTIN), and two multivariate methods, called direct exponential curve resolution algorithm (DECRA) and multivariate curve resolution (MCR), are critically evaluated by simulated and real DOSY data sets. The assessments in this paper indicate the possible improvement of the DOSY data processing by applying iterative principal component analysis (IPCA) followed by MCR-alternating least square (MCR-ALS).

           

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