Three-Mode Abstracts, Part W

INDEX

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• Wainer, H., Gruvaeus, G., & Blair, M. (1974).
  TREBIG: A 360/75 FORTRAN program for three-mode factor analysis designed for big data sets. Behavioral Research Methods and Instrumentation, 6, 53- 54.
  A computational outline of Tucker's (1966) Method II is des- cribed.

  

• Wainer, H., Gruvaeus, G., & Zill, N., II. (1973a).
  Senatorial decision making: I. Determination of structure. Behavioral Science, 18, 7-19.
  A T3 analysis (Tucker's (1966), Method II) of US Senate roll call voting was performed (8 issues, 6 years, 132 senators). The entries in the data matrix were the point-biserial correlations of senator's votes on a particular issue in a particular year with the popularity of the roll calls in that set. Varimax rotation. Introduction of the mixed-modal matrices (MMM), which give the loadings of the variables of one mode on the factors of an other. These MMMs turned out be useful in interpreting the Senator factors. These factors were validated using indices that purport to measure various facets on senatorial voting behavior.

• Wainer, H., Zill, N., II, & Gruvaeus, G. (1973b).
  Senatorial decision making: II. Prediciton. Behavioral Science, 18, 20-26.
  Not a three-mode paper, but uses the results of Wainer, Gruvaeus & Zill (1973a).

  

• Walsh, J.A. (1964).
  An IBM 709 program for factor analyzing three-mode matrices. Educational and Psychogical Measurement, 24, 669-773.
  Describes a computer program for Tucker's (1966) Method I which needs relative large amounts of storage.

• Walsh, J.A., & Walsh, R. (1976).
  A revised Fortran program for three-mode factor analysis. Educational & Psychological Measurement, 36, 169-170.
  A revision of the Walsh (1964) program. Main improvements are organizational with respect to the program itself.

  

• Walter, J. (1976).
  Komplexe taaksituaties en hartsnelheidsvariabiliteit in de psychiatrie. Technical Report, Stichting Centrum St.-Bavo, Noordwijkerhout, The Netherlands.
  During 15 conditions of varying mental stress hart frequences were measured in three ways from 67 members of the staff and patients of a psychiatric institution. T2 (as implemented by Kroonenberg & De Leeuw, 1980) was applied to the data of all subjects together and to the data of the staff and those of the patients separately.

• Wang, J.H., & Hopke, P.K. (2001).
  Equation-oriented system: an efficient programming approach to solve multilinear and polynomial equations by the conjugate gradient algorithm. Chemometrics and Intelligent Laboratory Systems, 55, 13-22.
  The factor analysis problem can be conceptualized as an expansion of polynomial equations that are solvable using least-squares methods. The equation-oriented system (EOS) is introduced as a method for solving polynomial equations using a preconditioned conjugate gradient (CG) algorithm for the normal equations. EOS is a fast, easy to program, low computer memory requirement method for accomplishing this task. EOS can be used to solve multilinear and PARAFAC problems. The practical aspects of implementing EOS in MATLAB are discussed.

• Wang, J. H., Hopke, P. K., Hancewicz, T. M., & Zhang, S. L. L. (2003).
  Application of modified alternating least squares regression to spectroscopic image analysis. Analytica Chimica Acta, 476, 93-109.
  Analysis of synthetically generated and real Raman imaging data sets were used to show the significance of modified alternating least squares (MALS) regression as a superior method of analysis compared with several other well established mathematical algorithms. The performance of MALS was compared with that of ordinary alternating least squares regression (ALS) and fast non-negative least squares (FNNLS) regression in applications of spectroscopic image analysis and self-modeling curve resolution (SMCR). The MALS algorithm is shown to be superior in terms of computational speed, stability, and component resolution ability in the analysis of both real and synthetic data sets. Results of the analysis show that MALS is significantl y faster than FNNLS and generally produces equivalent or superior results. This work also shows that MALS is superior to ordinary ALS in all performance aspects. A detailed description of the regression equations is given along with a discussion of the application of MALS to the general spectroscopic image analysis problem.

• Wang, K., Begleiter, H., & Porjesz, B. (2000).
  Trilinear Modeling of Event-Related Potentials. Brain Topography, 12, 263-271.
  This paper describes a method for estimating a set of spatial components (brain maps) and temporal components (waveforms) of brain potentials. These components play the role of bases of a coordinate system, in the sense that the brain potentials of any subject can be represented as superpositions of these components. The representation is unique given the spatial and temporal components, and this decomposition is particularly appealing for comparing the brain potentials of different subjects (say alcoholics and controls). It can also be used for single trial modeling, clinical classification of patients, and data filtering. The method is based on the topographic component model (TCM, Mocks 1988) which models brain potentials in a trilinear form. We extend the TCM in two aspects. First, the diagonal amplitude matrix is replaced by a general loading matrix based on some neurophysiological considerations. Secondly, the number of spatial components and the number of temporal components can be different. The spatial components and temporal components are obtained respectively by performing singular value decomposition (SVD). This method is illustrated with visual P3 data.

• Wang, Y.D., Borgen, O.S., & Kowalski, B.R. (1993).
  Comments on the residual bilinearization method. Journal of Chemometrics, 7, 439-445.
  Through theoretical analysis and computer simulation, this short communication comments on the residual bilinearization (RBL) method and compares it with non-bilinear rank annihilation (NBRA) for the treatment of second-order calibration with non-bilinear data. It is found that these two methods are mathematically equivalent but have different noise propagation properties. The second-order advantage, namely quantitation in the presence of unknown interferences, can be carried over to non-bilinear data only if there exists a net analyte rank (NAR) for the analyte of interest.

• Wang, Y.D., Borgen, O.S., Kowalski, B.R., Gu, M., & Turecek, F. (1993).
  Advances in second-order calibration. Journal of Chemometrics, 7, 117-130.
  Several multivariate methods are now available for the calibration of second-order or hyphenated instruments (e.g. GC/MS). When applied to bilinear data, it has been shown that calibration can be performed in the presence of unknown interferences - a significant advantage over first-order calibration. In this paper, non-bilinear rank annihilation (NBRA), a method which has the potential of handling, second-order non-bilinear data, is studied through theoretical analysis and computer simulation. It is found that the second-order advantage can be carried over to non-bilinear data if a property defined as net analyte rank (NAR) holds for the analyte of interest. The net analyte signal (NAS) is defined accordingly for second-order calibration and the analogy to and difference from lower-order calibration are discussed. With NAS, some analytical figures of merit such as signal-to noise ratio, selectivity, sensitivity and limit of determination can be calculated for second order calibration. An application to MS/MS data is also given.

• Wang, Y.D., & Kowalski, B.R. (1993).
  Standardization of second-order instruments. Analytical Chemistry, 65, 1174-1180.
  A method is presented to standardize two-dimensional responses (e.g., GC/MS, LC/UV) measured on multiple instruments or on a single instrument under different operational conditions. This second-order standardization method proceeds by calculating two banded diagonal transformation matrices, using the responses of a common standard sample, to simultaneously correct for the response channel shift and intensity variations in both dimensions or orders. Different from first-order standardization, these two transformation matrices must be estimated from a set of simultaneous nonlinear equations via the Gauss-Newton method. The effects of noise and transformation matrix bandwidth on the standardization performance are studied through computer simulation. When tested with experimental LC/UV data, the proposed standardization method can reduce the variation between two runs from 0.15-0.20 to 0.025-0.03 AU. From both the computer simulation and experimental data study, it is found that the design of the standard sample is critical for the parameter estimation and response standardization.

• Wansbeek, T., & Buyze, J. (1981).
  Multi-dimensional matrices and multiplicative interdependence in regression analysis. (Research Report, BPA no. 13593-81-M1), Voorburg, The Netherlands: Department for Statistical Methods.
  A model for multiplicative interdependence in regression is introduced, and interpreted as a way to describe the relation between two multi-dimensional matrices. For several versions of this model, least-squared estimation and identification is considered.

• Wansbeek, T., & Verhees, J. (1989).
  Models for multidimensional matrices in econometrics and psychometrics. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 543-552). Amsterdam: Elsevier.
  The potential of notation and calculus introduced by Kapteyn, Neudecker and Wansbeek (1986) is used to formulate a number of models for multidimensional matrices, and to analyse estimation of their parameters.

• Wansbeek, T., & Verhees, J. (1990).
  The algebra of multimode factor analysis. Linear Algebra and its Applications, 127, 631-639.
  Extending the work of Bentler and Lee, the multimode factor-analysis model is introduced. First-order conditions are given for maximum-likelihood, weighted least-squares, and unweighted least-squares estimation, using the calculus for handling multidimensional matrices due to Kapteyn, Neudecker, and Wansbeek. Also, the asymptotic distribution of the various estimators is discussed.

• Warmerdam, M. (1990).
  Affectieve reakties op reclame [Affective reactions to advertising] . Master's Thesis. Department of Psychology, Leiden University.
  Introduction
  Reactionprofiles
  Aims
  Research methods
  Respondents; procedure; stimulusmaterial; questionnaire.
  Analyses
  Three-mode principal components analysis; PROFIT-analyses; reliability analyses; regression-analyses.
  Results
  Scores; relationships between affetive reactions, advertising and respondents: Three-way analysis; relationships between affective reactions, advertising and communication effects: PROFIT-analyses; scale-constructions; the predictive validity of the affective reaction profile.
  Conclusions and discussion

• Wasserman, S., & Iacobucci, D. (1991).
  Statistical modeling of one-mode and 2-mode networks - Simultaneous analysis of graphs and bipartite graphs. Britisch Journal of Mathematical & Statistical Psychology, 44, 13-43.
  A bipartite graph, in which the nodes (or actors in a social network) are partitioned into two sets, can be studied using recent statistical models for dyadic interactions. These models, which are longlinear for the probabilities of dyadic choices or interactions, allow not only arcs or relationships to exist between the sets but also within the sets. Thus, the methods described here are applicable not only to bipartite graphs, consisting of arcs existing between nodes in different sets, but also to directed graphs that are defined within the two sets of nodes. Data on both types of graphs can be analysed simultaneously. A bipartite graph has an adjacency matrix (or sociomatrix) with two 'modes'. The set of nodes in the row mode differs from the set of nodes in the column mode. For example, in marketing, one could study the dyadic relations in a 'buyers-by-sellers' network. Generally, the relations observed in a one-mode network, which has a square sociomatrix (row mode = column mode) are bidirectional-the actors indexing the columns may also 'relate to' the actors indexing the rows. The relations observed in a two-mode network are generally unidirectional-the row actors relate to or choose the column actors, but the column actors do not relate to the row actors. Referring to our example, a buyer might pay a seller for some item, but a seller would not pay a buyer. Statistical models for the separate analysis of these one-mode and two-mode matrices are extended in this paper to the simultaneous analysis of both types of networks. A superordinate one-mode sociomatrix is created in which the rows and columns consist of all actors (that is, all buyers and sellers). This larger matrix contains both the one-mode matrices and the two-mode matrices. Multivariate analysis of unidirectional and bidirectional relations in social networks and complex directed graphs becomes possible with this simultaneous consideration of both types of matrices.

• Watkins, D., & Hattie, J. (1985).
  A longitudinal study of the approaches to learning of Australian tertiary students. Human Learning, 4, 127-141.
  A longitudinal study of 540 college students using the Approaches to Studying Inventory provided little evidence that students' approaches to learning became deeper during the course of their tertiary studies. This is despite many of the most disillusioned of the original sample having withdrawn from their studies or not responding to the follow-up survey. Contrary to predictions, the changes that did occur were independent of the faculty and the age of the students. An improved method of conducting a three mode factor analysis (McDonald, 1984) was used to compare the factor analytic models over time. In this study covariances were used with the means unconstrained over time and the residual covariance matrices patterned with every submatrix diagonal, to allow for stable specific components (See McDonald, 1984, for details).

  
         

• Weesie, J., & Van Houwelingen, H. (1983).
  GEPCAM users' manual: Generalized principal components analysis with missing values. (Technical report), Utrecht: University of Utrecht, Institute of Mathematical Statistics.
  This very important report is the only full exposition of GEPCAM (Generalised PCA with Missing values). An alternating least squares algorithm is described in which not only the component matrices but also the slices of the core array are estimated via regression. The standard approach in regression, using a binary weight matrix is used to handle missing data. It contains the first explicit statement about the three-way orthogonality of the core array. A possible extension to robust regression is mentioned. The program described is no longer available in that form, however. For an adapted version please contact the three-mode company. Published applications include Spanjer, et al.; De Ligny, e.a.
  
  
• Wei, W.Z., Zhu, W.H., & Yao, S.Z. (1992).
  Application of ridge trace analysis in direct spectrophotometric calibration. Talanta, 39, 1629-1636.**
  The use of ridge trace analysis for multivariate calibration problems is described. Application is made to the simultaneous quantitative determination of components for several pharmaceutical systems with unknown sample UV spectra by the direct calibration technique. The results indicate that with the help of ridge trace analysis and ridge regression, the parameter k can be selected to obtain estimation of the colinearity of a system based on the stability of relative concentration as a function of k. Much information about the system to be analyzed can be obtained, such as whether there is colinearity among the variables, which measurement wavelength range is suitable for performing quantitative analysis, which components cannot be measured and which components can only be measured as a sum, and whether the quantitative model is correct.

• Weinberg, S. L., & Menil, V. C. (1993).
  The recovery of structure in linear and ordinal data - indscal versus alscal. Multivariate Behavioral Research, 28, 215-233.
  The ability of three-way INDSCAL and ALSCAL to recover true structure in proximity data is examined in a Monte Carlo study using data based on two-dimensional configurations. Extending earlier work in this area, data were simulated to vary on four factors: number of subjects (15 and 30), number of stimuli (12 and 20), amount of error (0.10 and 0.70), and type of monotonic transformation (linear, square, logarithmic, general cubic, and rank order). Recovery was measured using two indicators, corresponding to the two sets of parameters estimated by these techniques: recovery of true stimulus dimensions, and recovery of true subject weights. In this study, INDSCAL outperformed metric and nonmetric versions of ALSCAL under all conditions. Implications for practice and future research are discussed.

• Wentzell, P. D., & Lohnes, M. T. (1999).
  Maximum likelihood principal component analysis with correlated measurement errors: theoretical and practical considerations. Chemometrics and Intelligent Laboratory Systems, 45, 65-85.
  Procedures to compensate for correlated measurement errors in multivariate data analysis are described. These procedures are based on the method of maximum likelihood principal component analysis (MLPCA), previously described in the literature. MLPCA is a decomposition method similar to conventional PCA, but it takes into account measurement uncertainty in the decomposition process, placing less emphasis on measurements with large variance. Although the original MLPCA algorithm can accommodate correlated measurement errors, two drawbacks have limited its practical utility in these cases: (1) an inability to handle rank deficient error covariance matrices, and (2) demanding memory and computational requirements. This paper describes two simplifications to the original algorithm that apply when errors are correlated only within the rows of a data matrix and when all of these row covariance matrices are equal. Simulated and experimental data for three-component mixtures are used to test the new methods. It was found that inclusion of error covariance information via MLPCA always gave results which were at least as good and normally better than PCA when the true error covariance matrix was available. However, when the error covariance matrix is estimated from replicates, the relative performance depends on the quality of the estimate and the degree of correlation. For experimental data consisting of mixtures of cobalt, chromium and nickel ions, maximum likelihood principal components regression showed an improvement of up to 50% in the cross-validation error when error covariance information was included.

• Wentzell, P. D. Nair, S. S., & Guy, R. D. (2001).
  Three-way analysis of fluorescence spectra of polycyclic aromatic hydrocarbons with quenching by nitromethane. Analytical Chemistry, 73, 1408-1415.
  The application of trilinear decomposition (TLD) to the analysis of fluorescence excitation-emission matrices of mixtures of polycyclic aromatic hydrocarbons (PAHs) is described. The variables constituting the third-order tenser are excitation wavelength, emission wavelength, and concentration of a fluorescence quencher (nitromethane), The addition of a quencher to PAH mixtures selectively reduces the fluorescence intensity of mixture components according to the Stern-Volmer equation. TLD allows the three-way matrix to be decomposed to give unique solutions for the excitation spectrum, emission spectrum, and quenching pro files for each component. The availability of spectra and calculated Stern-Volmer constants can aid in the identification of unknown components. Preprocessing of the data to correct for Rayleigh/Raman scatter and primary absorption by the quencher is necessary. Both three-component (anthracene, pyrene, l-methylpyrene) and four-component (fluoranthene, anthracene, pyrene, 2,3-benzofluorene) synthetic mixtures are successfully resolved by TLD using quencher concentrations up to 100 mM, Results are compared using both alternating least-squares and direct trilinear decomposition algorithms. The reproducibility of extracted Stern-Volmer constants is determined from replicate experiments. To illustrate the application of TLD to a real sample, a chromatographic cut from the analysis of a light gas oil sample was used. Analysis of the TLD extracted spectra and quenching constants suggests the presence of three classes of polycyclic aromatic hydrocarbons consistent with data from a second dimension of chromatography and mass spectrometry.

• Westerhuis, J.A., Kourti, T. & MacGregor, J.F. (1999).
  Comparing alternative approaches for multivariate statistical analysis of batch process data. Journal of Chemometrics, 13, 397-413.
  Batch process data can be arranged in a three-way matrix (batch × variable × time). This paper provides a critical discussion of various aspects of the treatment of these multiway data. First, several methods that have been proposed for decomposing three-way data matrices are discussed in the context of batch process data analysis and monitoring. These methods are multiway principal component analysis (MPCA)-also called Tucker1-parallel factor analysis (PARAFAC) and Tucker3. Secondly, different ways of unfolding, mean centering and scaling the three-way matrix are compared and discussed with respect to their effects on the analysis of batch data. Finally, the role of the time variable in batch process data is considered and methods suggested to predict the per cent completion of batch runs with unequal duration are discussed.

• Westerhuis, J. A., Gurden, S. P., & Smilde, A. K. (2000a).
  Standardized Q-statistic for improved sensitivity in the monitoring of residuals in MSPC. Journal of Chemometrics, 14, 335-349.
  This paper presents the standardized Q-statistic for monitoring residuals of latent variable models in multivariate statistical process control (MSPC). Before the summation of the squared residuals, they are scaled according to their expected variation obtained from normal operating conditions (NOC) data. Data from a simulated batch process and from an industrial batch process are used to show that this scaling improves the sensitivity of the Q-statistic considerably. The standardized Q-statistic is introduced for the off-line monitoring of batch processes, but it can also be used for the monitoring of continuous processes as well as for the on-line monitoring of batch processes.

• Westerhuis, J. A., Gurden, S. P., & Smilde, A. K. (2000b).
  Generalized contribution plots in multivariate statistical process monitoring. Chemometrics and Intelligent Laboratory Systems, 51, 95-114.
  This paper discusses contribution plots for both the D-statistic and the Q-statistic in multivariate statistical process control of batch processes. Contributions of process variables to the D-statistic are generalized to any type of latent variable model with or without orthogonality constraints. The calculation of contributions to the Q-statistic is discussed. Control limits for both types of contributions are introduced to show the relative importance of a contribution compared to the contributions of the corresponding process variables in the batches obtained under normal operating conditions. The contributions are introduced for off-line monitoring of batch processes, but can easily be extended to on-line monitoring and to continuous processes, as is shown in this paper.

       

• Whitman, D. A., Seasholtz, M. B., Christian, G. D., Ruzicka, J., & Kowalski, B. R. (1991).
  Double-injection flow-injection analysis using multivariate calibration for multicomponent analysis. Analytical Chemistry, 63, 775-781.
  A flow injection analysis (FIA) system is presented in which the reagent and sample are simultaneously injected for multicomponent analysis. A simple eight-port valve is described to perform the double injection. The method is illustrated by the determination of nickel and iron in a model plating bath solution. First-order calibration methods including classical least squares (CLS), principal components regression (PCR), and partial least squares (PLS) are used to analyze the complex time profiles that this method provides. This study shows that very moderate sampling rates and small calibration data sets can be used with multivariate calibration techniques.

• Whitman, B.Y. (1980).
  The relationship of client personality variables and worker skill variables in the production of client behavioral self-disclosure in the therapeutic interview. Dissertation Abstracts International, 41, 1779.
  The Tucker model was used to analyze self-disclosure measures of trained workers and comparable students. Only the measure space was deemed of interest. No further details in abstract.

   

• Wiberg, K., Andersson, M., Hagman, A., & Jacobsson, S. P. (2003).
  Use of control sample for estimation of prediction error in multivariate determination of lidocaine solutions with non-column chromatographic diode array UV spectroscopy. Journal of Pharmaceutical and Biomedical Analysis, 33, 859-869.
  The aim of this study was to investigate the ability of a control sample, of known content and identity, to diagnose and correct errors in the predictions when the same multivariate calibration model was used for analysis of new samples over time. A calibration set consisting of 16 samples with a known content of lidocaine was analysed and two external test sets, A and B, were used for the validation. Test set A contained 15 samples with different concentrations of lidocaine and test set B contained three samples with different lidocaine content, which were analysed six times in order to obtain a measure of repeatability. The multivariate calibration was done with PLS regression on UV spectra collected between 245 and 290 nm. A representative UV spectrum was exported from the collected DAD files by two methods, average spectrum over the whole file and average spectrum over the sample plug. Test set A was analysed further on another three occasions together with a control sample. The results showed that the control sample could be used to give a diagnosis and estimate of the prediction error. Moreover, the measured prediction error of the control sample could also be used to correct the predictions, thereby reducing the prediction error. Finally, some practical considerations regarding use of the proposed DAD method with a control sample are presented. The procedure suggested could lead to an efficient analytical approach where the same calibration model could be used over time without recalibration, which may be attractive in industrial quality control or screening analysis in pharmaceutical research.

• Wiberg, K., Andersson, M., Hagman, A., & Jacobsson, S. P. (2004a).
  Peak purity determination with principal component analysis of high-performance liquid chromatography-diode array detection data. Journal of Chromatography A, 1029, 13-20.
  A method is proposed for the determination of chromatographic peak purity by means of principal component analysis (PCA) of high-performance liquid chromatography with diode array detection (HPLC-DAD) data. The method is exemplified with analysis of binary mixtures of lidocaine and prilocaine with different levels of separation. Lidocaine and prilocaine have very similar spectra and the chromatograms used had substantial peak overlap. The samples analysed contained a constant amount of lidocaine and a minor amount of prilocaine (0.02-2conc.%) and hence the focus was on determining the purity of the lidocaine peak in the presence of much smaller levels of prilocaine. The peak purity determination was made by examination of relative observation residuals, scores and loadings from the PCA decomposition of DAD data over a chromatographic peak. As a reference method, the functions for peak purity analysis in the chromatographic data system used (Chromeleon) were applied. The PCA method showed good results at the same level as the detection limit of baseline- separated prilocaine, outperforming the methods in Chromeleon by a factor of ten. There is a discussion of the interpretation of the result, with some comparisons with evolving factor analysis (EFA). The main advantage of the PCA method for determination of peak purity over methods like EFA lies in its simplicity, short time of calculation and ease of use.

• Wiberg, K., & Jacobsson, S. P. (2004b).
  Parallel factor analysis of HPLC-DAD data for binary mixtures of lidocaine and prilocaine with different levels of chromatographic seperation. Analytica Chimica Acta, 514, 203-209.
  A set of 17 samples containing a constant amount of lidocaine (667 muM) and a decreasing amount of prilocaine (667-0.3 muM) was analysed by LC-DAD at three different levels of separation, followed by parallel factor analysis (PARAFAC) of the data obtained. In Case 1 no column was connected, the chromatographic resolution (R-s) therefore being zero, while Cases 2 and 3 had partly separated peaks (R-s = 0.7 and 1.0). The results showed that in Case 1, analysed without any separation, the PARAFAC decomposition with a model consisting of two components gave a good estimate of the spectral and concentration profiles of the two compounds. In Cases 2 and 3, the use of PARAFAC models with two components resolved the underlying chromatographic, spectral and concentration profiles. The loadings related to the concentration profile of prilocaine were used for regression and prediction of the prilocaine content. The results showed that prediction of prilocaine content was possible with satisfactory prediction (RMSEP & 0.01). This study shows that PARAFAC is a powerful technique for resolving partly separated peaks into their pure chromatographic, spectral and concentration profiles, even with completely overlapping spectra and the absence or very low levels of separation.

• Wicker, F.W. (1966).
  A scaling study of synesthetic thinking. ETS Res. Bull. RB-66-25); doctoral thesis, Princeton, New Jersey, 1966. (Dissertation Abstracts International, 27 (6-B), 2173).
  Two types of multivariate procedure - based, respectively, on similarity and semantic judgments - were used in the attempt to map intersensory associations between colors and tones. With both procedures it was possible to investigate individual differences as well as group trends. For both procedures stable associations were found for the group as a whole. The two methods agreed considerably in the types of intersensory analogy indicated. Individual differences were great, but the attempt to relate these differences to other attributes of individuals met with little success. In particular, people who reported synesthetic imagery did not differ systematically in their judgments from those who did not. Two types of alignment were concluded to underlie the intersensory relationships obtained. Bright colors were aligned with high pitched tones, and loud tones were aligned with colors which contrasted sharply with a gray background.

• Wickremasinghe, W. N., & Johnson, D. E. (2002)
  Testing subhypotheses and estimating sigma(2) in the nonreplicated three-way multiplicative interaction model Communication in Statistics-simulation and Computation, 31, 605-618.
  Standard statistical techniques do not provide methods for analyzing data from nonreplicated factorial experiments. Such experiments occur for several reasons. Many experimenters may prefer conducting experiments having a large number of factor levels with no replications than conducting experiments with a few factor levels with replications particularly in pilot studies. Such experiments may allow one to identify factor combinations to be used in follow-up experiments. Another possibility is when the experimenter thinks that an experiment is replicated when in fact it is not. This occurs when a naive researcher believes that sub-samples are replicates when in reality they are not. Nonreplicated two-way experiments have been extensively studied. This paper discusses the analysis of nonreplicated three-way experiments. In particular, estimation of sigma(2) is discussed and a test is derived for testing whether three-factor interaction is absent in sub-areas of three-way data using a nonreplicated three-way multiplicative interaction model with a single multiplicative term. Approximate null distribution of the derived test statistic is studied using Monte Carlo studies and results are illustrated through an example.

• Widen, B., Andersson, S., Rao, G. Y., & Widen, M. (2002)
  Population divergence of genetic (co)variance matrices in a subdivided plant species, Brassica cretica. Journal of Evolutionary Biology, 15, 961-970.
  The present study of Brassica cretica had two objectives. First, we compared estimates of population structure (Q(st)) for seven phenotypic characters with the corresponding measures for allozyme markers (F-st) to evaluate the supposition that genetic drift is a major determinant of the evolutionary history of this species. Secondly, we compared the genetic (co) variance (G) matrices of five populations to examine whether a long history of population isolation is associated with large, consistent differences in the genetic (co) variance structure. Differences between estimates of Fst and Qst were too small to be declared significant, indicating that stochastic processes have played a major role in the structuring of quantitative variation in this species. Comparison of populations using the common principal component (CPC) method rejected the hypothesis that the G matrices differed by a simple constant of proportionality: most of the variation involved principal component structure rather than the eigenvalues. However, there was strong evidence for proportionality in comparisons using the method of percentage reduction in mean-square error (MSE), at least when characters with unusually high (co) variance estimates were included in the analyses. Although the CPC and MSE methods provide different, but complementary, views of G matrix variation, we urge caution in the use of proportionality as an indicator of whether genetic drift is responsible for divergence in the G matrix.

  

• Wicker, F.W. (1968).
  Mapping the intersensory regions of perceptual space. American Journal of Psychology, 81, 178-188.
  Presents a summary of a T3 analysis (25 semantic scales, 26 tones and colours, 59 subjects) which attempted to map inter- sensory associations between colour vision and the perception of tones on the basis of similarity and semantic judgements. Details reported in Wicker (1966).

• Wicker, F.W., Thorelli, I.M., Barron, III, W.L., & Ponder, M.R. (1981).
  Relationships among affective and cognitive factors in humor. Journal of Research in Personality, 15, 359-370.
  In a study to assess the relationships between various factors in humor, 2 lists of 37 jokes were scored on 14 scales by 125 students. Tucker's methods were employed, and just the scale spaces were compared.

• Wienke, D., Van den Broek, & Buydens, L. (1995).
  Identification of plastics among nonplastics in mixes wase by remote-sensing near-infrared spectroscopy. 2. Multivariate image rank analysis for rapid classification. Analytical Chemistry, 67, 3760-3766..
  Macroscopic samples of household waste were experimentally characterized by a sequence of images, taken in four distinct wavelength regions by NIRIS. The obtained three-dimensional stack of images serves as individual fingerprint for each sample, A rapid data compression, followed by an abstract factor rotation of this stack into a spectroscopically meaningful intermediate four-element vector by a method called multivariate image rank analysis (MIRA), finally provided a single number. This number serves as decision limit for detection of plastics among nonplastic waste. The MIRA results are independent of sample size and sample position within the camera image. They are sensitive to only the type of sample material, MIRA was also found to be robust against image errors such as shadow or slight sample replacements between measurements.

• Wienke, D., Van den Broek, W., Melssen, W., Buydens, L., Feldhoff, R., HuthFehre, T., Kantim, T., Winter, F., & Cammann, K. (1996).
  Near-infrared imaging spectroscopy (NIRIS) and image rank analysis for remote identification of plastics in mixed waste. Fresenius Journal of Analytical Chemistry, 354, 823-828.
  An infrared camera with focal plane InSb array detector has been applied to the characterization of macroscopic samples of household waste over distances up to two meters. Per waste sample (singelized), a sequence of images was taken at six optical wavelength ranges in the near infrared region (1100 nm - 2500 nm). The obtained three-dimensional data stack served as individual fingerprint per sample. An abstract factor rotation of this stack of six images into a spectroscopical meaningful intermediate six-element vector by Multivariate Image Rank Analysis (MIRA) finally provided a decision limit for the discrimination of plastics and nonplastics. A correct classification of better than 80% has been reached. The experimental NIRIS set-up has been automated so far to allow an on-line identification of a real world waste sample within a few seconds.

  

• Wiggins, N., & Blackburn, M.C. (1976).
  Implicit theories of personality: an individual differences approach. Multivariate Behavioral Research, 11, 267-285.
  Twenty ratees were described on 20 bipolar personality trait adjectives by 51 raters in an own-control design. Both close friends and complete strangers were judged to assess possible individual differences in personality perception. T3 was applied and the rater dimensions were rotated using binorma oblique rotation, the ratee and trait dimensions by varimax. The resulting dimensions were correlated with a host of other variables. Two subject factors were interpreted using the core matrix.

• Wiggins, N., & Fishbein, M. (1969).
  Dimensions of semantic space: A problem of individual differences. In J.R. Snider & C.E. Osgood (Eds.), The semantic differential technique: A book of readings (pp. 183-193). Chicago: Aldine Press.
  Despite the general findings of intra- and inter-cultural comparability of semantic dimensions and their extension into the personality realm, a few exceptions to the usual lack of individual differences in meaning structure have been noted. The major purpose of this paper is to point out that individual differences in semantic structure do indeed exist. The authors are arguing the question of whether there is a single universal set of indicants for this underlying framework within or across given population groupings. The dimensionality of the subject space was investigated with the Tucker & Messick (1963) procedure.

• Wilkinson, C., Schipper, M., & Leguijt, T. (2000).
  Weighted analysis for missing values in generalized procrustes analysis. Food Quality and Preverence, 11, 85-90.
  Generalized Procrustes Analysis (GPA), a popular tool in sensory science, is generally carried out on panelist data matrices averaged over replicates. This paper addresses the problem of missing values arising when panelists miss sessions. Because this does not necessarily result in missing values in the final averaged data matrices, a weighted analysis is proposed with weights set proportional to the number of replicates for each panelist product combination. In a simulation study the weighted analysis gives a better match of the rotated panelist matrices (a lower loss) than the unweighted analysis although the resulting average configuration is not significantly closer to the true configuration. The weighted analysis is a straightforward extension to GPA for dealing with missing sessions and offers an improved basis on which to evaluate panelist performance.

• Williams, R. P., Swinkels, D. A. J., & Maeder, M. (1992).
  Identification and application of a prognostic vector for use in multivarate calibration. Chemometrics and Intelligent Laboratory Systems, 15, 185-193.
  Examination of the mathematical structures of principal components regression and partial least-squares regression has led to the identification of a prognostic vector for both techniques that contains information about the contribution of each feature in a sample spectrum to the quality of the sample. The properties and use of the prognostic vector are demonstrated by a practical application involving the estimation of electrolytic manganese dioxide battery activity from X-ray diffraction patterns. Results from the study suggest that the prognostic vectors derived from both PCR and PLS are similar, and provide information that can ultimately be used to improve sample quality.

• Williams, W.T., & Stephenson, W. (1973).
  The analysis of three-dimensional data (sites × species × times) in marine ecology. Journal of Experimental Marine Biological Ecology, 11, 207-227.
  A new numerical model is defined for the partition of three-dimensional data; it is based on the analysis of variance, and is shown to be equivalent to an unstandardized Euclidean distance preceded by a specified form of standardization. The resulting quantity can be used as a dissimilarity measure for classification; it will serve to define a two-way table and provides a simple and elegant method for data-reduction requiring only a single subjective decision. The concept of 'mean variance per comparison' is introduced, and it is shown that such mean variances can be used to assess the relative importance of the three dimensions; two important symmetry properties of mean variance are established. The model is applied to a small real-life problem of 15 sites × 56 species × 8 seasons, and is shown to produce ecologically satisfactory results.

• Wilson, B.E., Sanchez, E., & Kowalski, B.R. (1989).
  An improved algorithm for the generalized rank annihilation method. Journal of Chemometrics, 3, 493-498.
  An improved algorithm for the generalized rank annihilation method (GRAM) is presented. GRAM is a method for multicomponent calibration using two- dimensional instruments, such as GC-MS. In this paper an orthonormal base is first computed and used to project the calibration and unkown sample response matrices into a lower-dimensional subspace. The resulting generalized eigenproblem is then solved using the QZ algorithm. The result of these improvements is that GRAM is computationally more stable, particularly in the case where the calibration sample contains chemical constituents not present in the unknown sample and the unknown contains constituents not present in the calibration (the most general case).

  

• Windig, W., Antalek, B., Robbins, M.J., Zumbulyadis, N., & Heckler, C.E. (2000).
  Applications of the direct exponential curve resolution algorithm (DECRA) to solid state nuclear magnetic resonance and mid-infrared spectra. Journal of Chemometrics, 14, 213-227.
  DECRA (direct exponential curve resolution algorithm) is a fast multivariate method used to resolve spectral data with concentration profiles that are linear combinations of exponential functions. DECRA has been previously applied to a wide variety of spectroscopies. Results are presented in this paper for two new application areas: solid state nuclear magnetic resonance spectra of polymorphic crystal mixtures and mid-infrared spectroscopy of chemical reactions. Furthermore, the paper will show the effect of the way the data set is split, which is a part of the algorithm, on the results.

• Windig, W., Antalek, B., Sorriero, L.J., Bijlsma, S., Louwerse, D.J., & Smilde, A.K. (1999a).
  Applications and new developments of the direct exponential curve resolution algorithm (DECRA). Examples of spectra and magnetic resonance images. Journal of Chemometrics, 13, 95-110.
  Recently, a new multivariate analysis tool was developed to resolve mixture data sets, where the contributions ('concentrations') have an exponential profile. The new approach is called DECRA (direct exponential curve resolution algorithm). DECRA is based on the generalized rank annihilation method (GRAM). Examples will be given of resolving nuclear magnetic resonance spectra resulting from a diffusion experiment, spectra in the ultraviolet/visible region of a reaction and magnetic resonance images of the human brain.

• Windig, W., & Antalek, B. (1999b).
  Resolving nuclear magnetic resonance data of complex mixtures by three-way methods: Examples of chemical solutions and the human brain. Chemometrics and Intelligent Laboratory Systems, 46, 207-219.
  Despite the use of hyphenated and/or high-resolution instruments in analytical spectroscopy, the resulting spectral data often represent mixtures of several components. When no reference data in the form of reference spectra or concentration profiles are available, self-modeling mixture analysis techniques can be utilized to obtain the spectra of the pure components and their concentration profiles, There are many different algorithms to resolve mixture spectra, and the mathematical procedures involved are not always simple. This paper will discuss some of the aspects and problems of self-modeling mixture analysis, with the focus on the three-way method and without going into the mathematical details. Practical examples will be shown of methods applied to nuclear magnetic resonance data. The techniques discussed can also be applied to magnetic resonance images and an example will be shown of the human brain.

  
  

• Windig, W., Hornak, J. P., & Antalek, B. (1998).
  Multivariate image analysis of magnetic resonance images with the direct exponential curve resolution algorithm (DECRA) - Part 1: Algorithm and model study. Journal of Magnetic Resonance, 132, 298-306.
  Antalek and Windig recently presented a fast method to resolve a series of NMR mixture spectra, where the contribution of the components varies with a decaying exponential [B. Antalek and W. Windig, J. Am. Chem. Sec. 118, 10,331-10,332 (1996); W. Windig and B. Antalek, Chemom. Intell. Lab. Syst. 37, 241-254(1997)]. The method was called DECRA (direct exponential curve resolution algorithm). In this paper DECRA will be applied to two series of magnetic resonance images. The signal of one series is based upon T-2 relaxation, and the other is based upon T-1 relaxation. In order to evaluate the technique, the magnetic resonance images of a phantom where used. A transformation is introduced to enable the application of DECRA to a T-1 series of magnetic resonance images. A separate paper in this issue will describe the application of the techniques to magnetic resonance images of the human brain.

• Winsberg, S., & Carroll, J.D. (1989).
  A quasi-nonmetric method for multidimensional scaling of multiway data via a restricted case of an extended INDSCAL model. In R. Coppi & S. Bolasco (Eds.) Multiway data analysis, (pp. 405-414). Amsterdam: North Holland.
  An extended three-way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the "standard" (three-way) INDSCAL MDS model. In this extended INDSCAL model the N stimuli (or objects) are assumed to be characterized not only by coordinates on P common dimensions, but in addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. A numerical procedure alternating a modified Newton-Raphson algorithm with one for fitting an optimal spline (or linear function) is used to obtain maximum likelihood estimates of the parameters. AIC and/or BIC statistics can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition to the P common) dimensions, and/or the necessity for individual weights for the dimensions, and/or the necessity for nonlinear transformations. Thus a wide variety of models can be compared statistically. This approach is illustrated with applications to both artificial data and data on judged similarity of adjectives relating to emotions as well as other real data sets.

• Winsberg, S.L., & De Soete, G. (1993b).
  A latent class approach to fitting the weighted Euclidean model, CLASCAL. Psychometrika, 58, 315-330.
  A weighted Euclidean distance model for analyzing three-way proximity data is proposed that incorporates a latent class approach. In this latent class weighted Euclidean model, the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class. This model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions, and drasticlly reduces the number of parameters in the traditional INDSCAL model. The probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities. The maximum likelihood function is optimized by means of an EM algorithm; a modified Fisher scoring method is used to update the parameters in the M-step. A model selection strategy is proposed and illustrated on both real and artificial data.

• Wise, B.M., & Gallagher, N.B. (1996).
  The process chemometrics approach to process monitoring and fault detection. Journal of Process Control, 6, 329-348.
  Chemometrics, the application of mathematical and statistical methods to the analysis of chemical data, is finding ever widening applications in the chemical process environment. This article reviews the chemometrics approach to chemical process monitoring and fault detection. These approaches rely on the formation of a mathematical/statistical model that is based on historical process data. New process data can then be compared with models of normal operation in order to detect a change in the system. Typical modelling approaches rely on principal components analysis, partial least squares and a variety of other chemometric methods. Applications where the ordered nature of the data is taken into account explicitly are also beginning to see use. This article reviews the state-of-the-art of process chemometrics and current trends in research and applications.

• Wise, B. M., Gallagher, N. B., Bulter, S. W., White, D. D., & Barna, G. G. (1999).
  A comparison of principal component analysis, multiway principal component analysis, trilinear decomposition and parallel factor analysis for fault detection in a semiconductor ETCH process. Journal of Chemometrics, 13, 379-396.
  Multivariate statistical process control (MSPC) tools have been developed for monitoring a Lam 9600 TCP metal etcher at Texas Instruments. These tools are used to determine if the etch process is operating normally or if a system fault has occurred. Application of these methods is complicated because the etch process data exhibit a large amount of normal systematic variation. Variations due to faults of process concern can be relatively minor in comparison. The Lam 9600 used in this study is equipped with several sensor systems including engineering variables (e.g. pressure, gas flow rates and power), spatially resolved optical emission spectroscopy (OES) of the plasma and a radio-frequency monitoring (RFM) system to monitor the power and phase relationships of the plasma generator. A variety of analysis methods and data preprocessing techniques have been tested for their sensitivity to specific system faults. These methods have been applied to data from each of the sensor systems separately and in combination. The performance of the methods on a set of benchmark fault detection problems is presented and the strengths and weaknesses of the methods are discussed, along with the relative advantages of each of the sensor systems.

  Wise, B. M., Gallagher, N. B., & Martin, E. B. (2001).
  Application of PARAFAC2 to fault detection and diagnosis in semiconductor etch. Journal of Chemometrics, 15, 285-298.
  Monitoring and fault detection of batch chemical processes are complicated by stretching of the time axis, resulting in batches of different length. This paper offers an approach to the unequal time axis problem using the parallel factor analysis 2 (PARAFAC2) model. Unlike PARAFAC, the PARAFAC2 model does not assume parallel proportional profiles, but only that the matrix of profiles preserves its ‘inner product structure’ from sample to sample. PARAFAC2 also allows each matrix in the multiway array to have a different number of rows. It has previously been demonstrated how the PARAFAC2 model can be used to model chromatographic data with retention time shifts. Fault detection and, to a lesser extent, diagnosis in a semiconductor etch process are considered in this paper. It is demonstrated that PARAFAC2 can effectively model batch process data from semiconductor manufacture with unequal dimension in one of the orders, such as the unequal batch length problem. It is shown that the PARAFAC2 model has approximately the same sensitivity to faults as other competing methods, including principal component analysis (PCA), unfold PCA (often referred to as multiway PCA), trilinear decomposition (TLD) and conventional PARAFAC. The advantage of PARAFAC2 is that it is easier to apply than MPCA, TLD and PARAFAC, because unequal batch lengths can be handled directly rather than through preprocessing methods. It also provides additional diagnostic information: the recovered batch profiles. It is likely, however, that it is less sensitive to faults than conventional PARAFAC.

• Wish, M., & Carroll, J.D. (1972).
  Multi-dimensional scaling with differential weighting of dimensions. In F.R. Hodson, D.G. Kendall & P. Tautu, Mathematics in the Archeological and Historical Sciences (pp. 150-167). Edinburgh: Edinburgh University Press.
  Recently a new method was developed by Carroll and Chang (1970) based on a model that related structures from different sources in a strong way, yet also permits large differences among them. This method has been implemented in a computer program called INDSCAL (for INdividual Differences SCALing). In this chapter, the authors illustrate INDSCAL by applying it to data from three studies of human perception. In the first and third illustrations they re-analyze data from earlier experiments dealing respectively with 'psychological distances' among colours and with confusions among acoustically degraded English consonants. The data analyzed in the second example are from a recent study of individual differences in perception of rhythm and accent in English speech (Wish 1969). The authors conclude the paper with a discussion of potential applications of INDSCAL in archeology.

• Wish, M., & Carroll, J.D. (1982).
  Multidimensional scaling and its applications. Handbook of Statistics, 2, 317-345.
  This review chapter gives an overview of both two-way (such as MDScal, KYST) and three-way MDS (such as INDSCAL, IDIOSCAL, PARAFAC and CANDELINC) up to 1982. The techniques are illustrated with the Rothkopf Morse-data, the societal-problems data and Wish' perception-of-nations data.

• Witzke, D.B. (1975).
  Determining developmental changes in Holtzman inkblot technique factors using three-mode factor analysis. Unpublished doctoral thesis, University of Texas, Austin. ( Disserta tion Abstracts International, 1975, 36 (5-A), 2727.)

             

• Wold, S., Geladi, P., Esbensen, K., & Öhman, J. (1987).
  Multi-way principal components and PLS-analysis. Journal of Chemometrics, 1, 41-56.
  The Lohmöller-Wold decomposition of multi-way (three-way, four-way, etc.)data arrays is combined with the non-linear partial least squares (NIPALS) algorithms to provide multi-way solutions of principal components analysis (PCA) and partial least squares modelling in latent variables (PLS). The decomposition of a multi-way array is developed as the product of a score vector and a loading array, where the score vectors have the same properties as those of ordinary two-way PCA and PLS. In image analysis, the array would instead be decomposed as the product of a loading vector and an image score matrix. The resulting methods are equivalent to the method of unfolding a multi-way array to a two-way matrix followed by ordinary PCA or PLS analysis. This automatically proves the eigenvector and least squares properties of the multi-way PCA and PLS methods. The methodology is presented; the algorithms are outlined and illustrated with a small chemical example.

• Wold, J. P., & Kvaal, K. (2000).
  Mapping lipid oxidation in chicken meat by multispectral imaging of autofluorescence. Applied Spectroscopy, 54, 900-909.
  Multispectral imaging of autofluorescence was carried out to investigate the feasibility of mapping the degree of lipid oxidation in ground chicken meat. Meat samples from both breast and thighs were collected from 32 chickens, ground and freeze-stored for different time intervals. Sixteen samples were imaged at the time, making up two sets of multispectral images, A and B. Lipid oxidation was measured by a method using 2-thiobarbituric acid reactive substances (TBARS), and samples were in the range 0.15-3.23. Principal component analysis was performed on image set A, and variation in score images of the two first components corresponded well with lipid oxidation reference values. The multivariate image regression model based on image set A was tested on set B. Pixel-wise prediction gave large individual errors, but averaging predicted values within samples improved accuracy and resulted in a correlation of 0.98. Increasing the amount of spatial variation (number of pixel vectors) in the regression models led to more robust models with lower prediction errors. The technique has potential for nondestructive investigation of distribution and kinetics of lipid oxidation in food.

• Wold, J. P., Westad, F., & Heia, K. (2001).
  Detection of parasites in cod fillets by using SIMCA classification in multispectral images in the visible and NIR region. Applied Spectroscopy, 55, 1025-1034.
  The presence of parasitic nematodes in fillets of commercially important fish species has been a serious quality problem for the fishing industry for several decades. Various approaches have been tried to develop an efficient method to detect the parasites, but so far the only reasonable solution is manual inspection and trimming of each fish fillet on a candling table. In this study we have investigated how multispectral imaging in combination with SIMCA classification can be used for automatic detection of parasites. The results indicate that the spectral characteristics of nematodes differ sufficiently from those of fish flesh to allow one to obtain fairly good classifications. The method is able to detect parasites at depths down to about 6 mm into the fish muscle. The method shows promising results, but further studies are required to verify feasibility for the fish industry.

• Wong, R. S. K. (2001).
  Multidimensional association models - A multilinear approach. Sociological Methods & Research, 30, 197-240.
  This article develops several multidimensional multilinear association models for sociologists and other social science researchers to analyze the relationship between categorical variables in multiway cross-classification tables. The proposed multilinear approach not only provides satisfactory fit by conventional standards in the illustrative examples but also offers better understanding of the complex relationship between variables. This study highlights the relationship between two alternative decompositions in the multilinear framework-the PARAFAC/CANDECOMP and the Tucker 3-mode methods to decompose log-linear parameters-as well as the relationship between the multilinear approach and the log-multiplicative association models developed by Goodman and others. In addition, the author discusses empirical strategies to determine whether some or all cross-dimensional and other identifying restrictions can be relaxed in certain restricted models and to account for the proper degrees of freedom for these models.

• Workman, J. (2002).
  The state of mulivariate thinking for scientists in industry: 1980-2000. Chemometrics and Intelligent Laboratory Systems, 60, 13-32.
  Chemometrics has enjoyed tremendous success in the areas related to calibration of spectrometers and spectroscopy-based measurements. These chemometric-based spectrometers have been widely applied for process monitoring and quality assurance. However, chemometrics has the potential to revolutionize the, very intellectual roots of problem solving. Are there barriers to a more rapid proliferation of chemometric-based thinking, particularly in industry? What are the potential effects of chemometrics technology and the New Network Economy (NNE) working in concert? Who will be the winners in the race for faster, better, cheaper systems and products? These questions are reviewed in terms of the principles of the NNE and in the promise of chemometrics for industry. What then is the state of multivariate thinking in industry? Several powerful principles are derived from an evaluation of the NNE and chemometrics which could allow chemometrics to proliferate much more rapidly as a key general problem-solving tool.

             

• Wu, H. L., Shibukawa, M., & Oguma, K. (1997a).
  Second-order calibration based on alternating trilinear decomposition: A comparison with the traditional PARAFAC algorithm. Analytical Sciences, 13, 53-58.
  Quantitative analysis of p-chlorotoluene in the presence of o- chlorotoluene as an interferent was attempted by using HPLC with diode array detection and second-order calibration methods. An emphasis was placed on a further comparison between two methods, i.e., the traditional parallel factor analysis (PARAFAC) and the alternating trilinear decomposition (ATLD) methods. The following conclusions have been confirmed for the calibration examples tested. PARAFAC does not always converge to chemical meaningful solutions and its convergence rate is slow for the. This seems to be owing to the deficiency of a real trilinear sense in PARAFAC. The convergence rate of the ATLD algorithm is extremely fast. The ATLD-based calibration not only retains the advantage of second-order calibration but can give satisfactory concentration predictions.

• Wu, H.L., Feng, Y.Q., Shibukawa, M., & Oguma, K. (1997b).
  Alternative algorithm for simultaneous determinations of components poorly resolved by liquid chromatography with multiwavelength detection. Analytical Sciences, 13, 99-108.
  This paper describes an alternative algorithm for a simultaneous multicomponent analysis with three-mode three-dimensional array data generated by liquid chromatography with diode array detection (LC-DAD). Mainly based on a combination of two-step Moore-Penrose pseudoinverse computations based on a truncated singular value decomposition (TSVD) and one-step direct bilinear decomposition. It is modeled with a series of pure or mixed reference samples and then used to analyze several unknown samples. The algorithm was evaluated by Monte-Carlo simulations of two synthetic LC-DAD data sets with different noise levels, and applied to the simultaneous determination of chlorobenzene and toluene by means of HPLC with a photodiode array detector. The obtained results show that the proposed algorithm is suitable for simultaneous determinations of several components, of which the chromatographic and absorption heavily overlap each other, provided that their three-dimensional data have trilinear properties.

• Wu, H. L., Shibukawa, M., & Oguma, K. (1998).
  An alternating trilinear decomposition algorithm with application to calibration of HPLC-DAD for simultaneous determination of overlapped chlorinated aromatic hydrocarbons. Journal of Chemometrics, 12, 1-26.
  In this paper an alternating trilinear decomposition (ATLD) algorithm that is an improvement of the traditional PARAFAC algorithm without any constraints is described as an alternative algorithm for decomposition of three-way data arrays. It is based on an alternating least squares principle and an improved iterative procedure that, in a real trilinear sense, uses the Moore-Penrose generalized inverse with singular value decomposition. Its performance is compared with that of the traditional PARAFAC algorithm by a series of Monte Carlo simulations with different noise levels. It was found that the ATLD algorithm has a capability to converge faster than the traditional PARAFAC algorithm. The ATLD-based second-order calibration retains the second-order advantage that calibration in the presence of unknown interferents can be performed to provide satisfactory concentration estimates. Both algorithms have been used for simultaneous determination of overlapped chlorinated aromatic hydrocarbons measured by means of a high-performance liquid chromatograph with a diode array detector.

• Wu, H. L., Yu, R. Q., & Oguma, K. (2001).
  Trilinear component analysis in modern analytical chemistry. Analytical Sciences, 17, i483-i486.
  The purpose of this paper is to present an overview of the recent developments in three-mode three-way trilinear component analysis with its applications in modern analytical chemistry. Emphasis is placed on the relatively new triadic algorithms that provide new ways to decompose the three-way data arrays from HPLC/DAD or two-dimensional excitation-emission fluorescence spectra, the rank estimation of three-way data as well as the uniqueness of the presentations. It has been shown that the combination of three-way data analysis methods with fluorescence spectroscopy or HPLC/DAD is a practical way to uniquely estimate the spectra and concentration of the analyte(s) of interest in the presence of the unknown interferents.

• Wu, H. L., Yu, R. Q., Shibukawa, M., & Oguma, K. (2000).
  Second-order standard addition method based on alternating trilinear decomposition. Analytical Sciences, 16, 217-220.
  In this study, a second-order standard addition method based on alternating trilinear decomposition (ATLD-SOSAM) was developed. It was applied to second-order HPLC-DAD data and compared to methods employing direct trilinear decomposition (DTLD) and PARAFAC. The results show that ATLD-SOSAM is slightly superior to both DTLD-SOSAM and PARAFAC-SOSAM.

• Wu, W, Guo, Q., Jouan-Rimbaud, D., & Massart, D. L. (1999).
  Using contrasts as data pretreatment method in pattern recognition of multivariate data. Chemometrics and Intelligent Laboratory Systems, 45, 39-53.
  A contrast method originally proposed by Spiegelman [C.H. Spiegelman, Calibration: a look at the mix of theory, methods and experimental data, presented at Compana ’95, Wuerzburg, Germany.] is modified to pretreat multivariate data for classification. Three NIR data sets and one pollution data set are used as examples. Our results show that the contrast method greatly improves the ratios of between- to within-class variance. It is more powerful than offset correction, SNV, first- and second-derivative methods in the cases studied. This conclusion does not depend on the type of classifier used. Regularised discriminant analysis (RDA) and partial least squares (PLS2) with univariate feature selection based on Fisher’s ratio were applied here. There is a risk that chance correlations occur after the contrast pretreatment. The chance correlation decreases after first eliminating un-informative variables using the modified Uninformative Variable Elimination (UVE)-PLS method.

• Wu, W., Guo, Q., Massart, D.L., Boucon, C. & De Jong, S. (2003).
  Structure preserving feature selection in PARAFAC using a genetic algorithm and Procrustes analysis. Chemometrics and Intelligent Laboratory Systems, 65, 83-95.
  In this paper, a method is proposed to select subsets of variables in parallel factor analysis (PARAFAC), such that information in the complete multi-way data set is preserved as much as possible. The information retained is measured by means of the percentage of consensus in Procrustes analysis. The best N-way subset is obtained by applying a genetic algorithm (GA) to optimize the consensus between the subset and the complete N-way data set in order to prevent exhaustive searching. The method was applied to two industrial data sets: a three-way sensory data set and a four-way gas chromatography (GC) data set. The results showed that the proposed method successfully identified structure-bearing variables in both data sets and that it led to better subsets of variables than feature selection based on loadings.

           

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