Three-Mode Abstracts, Part B

INDEX

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• Bader, B. W., & Kolda, T. G. (2006).
  Efficient MATLAB computations with sparse and factored tensors. Sandia Report, Sandia National Laboratories, Albuquerque, New Mexico and Livermore, California.
  In this paper, the term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: a Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.

• Bader, B. W. , & Kolda, T. G. (2007).
  Efficient MATLAB computations with sparse and factored tensors. SIAM Journal on Scientific Computing, 30, 205-231.
  In this paper, the term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation first, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: A Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.

• Baerwald, T.J. (1976).
  The emergence of a new "downtown".The Geographical Review, 68, 308-318.
  A geographical application of T3 to changes in land use over time with land use classes, time periods and districts as modes. No numerical results presented.

  

• Bagozzi, R. P., & Yi, Y. J. (1990).
  Assessing method variance in multitrait multimethod matrices - the case of self reported affect and perceptions at work. Journal of Applied Psychology, 75, 547-560.
  Spector (1987) concluded that there was little evidence of method variance in multitrait-multimethod data from 10 studies of self-reported affect and perceptions at work, but Williams, Cote, and Buckley (1989) concluded that method variance was prevalent. We extended these studies by examining several important but often neglected issues in assessing method variance. We describe a direct-product model that can represent multiplicative method effects and propose that model assumptions, individual parameters, and diagnostic indicators, as well as overall model fits, be carefully examined. Our reanalyses indicate that method variance in these studies is more prevalent than Spector concluded but less prevalent than Williams et al. asserted. We also found that methods can have multiplicative effects, supporting the claim made by Campbell and O'Connell (1967, 1982).

• Bailey, R.A., & Rowley, C.A. (1993).
  Maximal rank of an element of a tensor product. Linear Algebra and Its Applications, 182, 1-7.
  Upper bounds are given for the maximal rank of an element of the tensor product of three vector space.

• Baird, I. S., Sudharsan, D., & Thomas, H.(1988).
  Adressing temporal change in strategic groups analysis: a three-mode factor analysis approach. Journal of Management, 14, 425-139.
  he advantages of using 3-mode factor (TMF) analysis to capture temporal effects when formulating strategic groups are examined. TMF analysis is proposed as a procedure to permit joint consideration of temporal changes in strategy and temporal variations in the importance of strategic criterion variables when analyzing the strategic group structure within industries. The TMF demonstration for strategic grouping across time is based upon the 46 firms in the computer industry for whom financial data are available on COMPUSTAT tapes for 1977-1981. The empirical data set has the dimensionality of a 46 X 6 X 5 matrix (46 firms arrayed on 6 financial risk variables over a time period of 5 years). The financial mode analysis enables the financial risk dimensions to be synthesized into 3 factors: 1. investor treatment orientation, 2. liquidity management sophistication, and 3. debt aversion. The time-based analysis identified 3 periods of strategic change in the industry. Financial risk traits of the 6 strategic groups identified in the firm-mode analysis may be examined with reference to the core matrix.

• Baltink.G.J.H. (1969).
  Driemodale faktoranalyse in een differentieelpsychologisch onderzoek naar de beoordeling van abstracte schilderijen.Nederlands Tijdschrift voor de Psychologie, 24, 529-540.
  T3 and its possible rotations are discussed at a conceptual level. The model is illustrated with an analysis of 15 abstract (non-figurative) paintings, scored on 10 bipolar (semantic) scales by 34 subjects. The relation between neuroticism and extraversion (measured independently), and the resulting factors were analysed using the core matrix.

  

• Barbieri, P., Andersson, C.A., Massart, D.L. & Predonzani, S., Adami, G., & Reisenhofer, E. (1999a).
  Modeling bio-geochemical interactions in the surface waters of the gulf of Trieste by three-way principal component analysis (PCA). Analytica Chimica Acta, 398, 227-235.
  Data of temperature, salinity, dissolved oxygen, nutrients and chlorophyll measured on samples of surface seawater and collected monthly during 2 years in different sites of the gulf of Trieste are modeled by means of three-way principal component analysis (PCA). Missing values are handled using an expectation maximization algorithm, regression or substitution with random numbers, depending on their origin. Physicochemical parameters are described by three different components that explain the effect of the river input on the seawater pattern, the effect of temperature, and metabolic-catabolic activity of the phytoplankton, respectively. One spatial component accounts for the gradient of influence of the estuarine waters in the gulf, and three temporal components characterize three main seasonal conditions. Anomalous situations, generated by meteoclimatic events, are highlighted.

• Barbieri, P., Adami, G., & Reisenhofer, E. (1999b).
  Searching for a 3-way model of spatial and seasonal variations in the chemical composition of karstic freshwaters. Annali Di Chimica, 89, 639-648.
  A procedure is described for the search of a three,way;principal components model, characterizing a data set concerning the spatial and temporal distribution of the physico-chemical parameters which govern the composition of waters collected from springs, ponds and;rivers of the Karst of Trieste. Ten physico-chemical parameters were determined-for eleven sampling sites and eleven sampling times. A:graphic method was applied in order to find the number of components in each of the three ways of the model, explaining a relatively high quantity of variation of the. data, with a limited number of components, i.e. with descriptive parsimony, and generating interpretable factors. The examination of 125 possible Tucker3 models, having from 1 to 5 components in each of the ways, allowed us to identify the model having two components in each of the three ways as the one satisfying the desired criteria. The chance of reducing to a simpler PARAFAC model has been successfully explored, and two trilinear components were then computed. The first one is mainly related to a spatial factor conditioning the considered waters, while the second is related to a seasonal factor.

• Barbieri, P., Adami, G., Predonzani, S., Reisenhofer, E., & Massart, D. L. (1999c).
  Survey of environmental complex systems: pattern recognition of physicochemical data describing coastal water quality in the Gulf of Trieste. Journal of Environmental Monitoring, 1, 39-74.
  A data set reporting temperature, salinity,dissolved oxygen, nitrogen as ammonia, nitrite and nitrate, silicate, chlorophyll a and phaeopigment values, determined in seawaters sampled during two years with a monthly frequency in 16 stations in the Gulf of Trieste, and at different depths of the water column, has been studied. In order to find synthetic descriptors useful for following the spatial and temporal variations of biogeochemical phenomena occurring in the considered ecosystem, the data set has been factorized using principal component analysis. A graphical display of scores, by means of boxplots and biplots, helped in the interpretation of the data set. The first factor conditioning the system is related to the input of freshwater from the estuary of the Isonzo River and to the stratification of the seawater(thermohaline discontinuity), while the second and third components describe interactions between biological activity, nutrients and physicochemical parameters; typical spring and autumn phytoplankton blooms were identified, in addition to an exceptional winter bloom conditioned by anomalous meteorological/climatic conditions. The fourth principal component explains the reducing activity of seawaters, which often increases when the decomposition of organic matter is relevant. The simple linear model proposed, and the related graphs, are shown to be useful tools for monitoring the main features of such a complex dynamic environmental system. The outlined approach to the considered complex data structure presents in a cognitive easy way (graphical outputs) the significant variations of the data, and allows for a detailed interpretation of the results of the monitoring campaign. Temporal and spatial effects are outlined, as well as those related to the depth in the water column.

• Barbieri, P., Adami, G., Piselli, S., Gemiti, F., & Reisenhofer, E. (2002a).
  A three-way principal factor analysis for assessing the time variability of freshwaters related to a municipal water supply. Chemometrics and Intelligent Laboratory Systems, 62, 89-100.
  Chemical analyses (total hardness, HARD; dissolved oxygen, DO; chlorides; sulfates; nitrates; nitrites; ammonia; orthophosphates; and UV-absorbing organic constituents, UV-ORG), physical data (turbidity, TURB; temperature, TEMP; conductivity, COND), and biological monitors (total and faecal coliforms, FAEC; faecal streptococci, STREPTO) constitute the 15 parameters, monitored with monthly frequency in the space of 4 years on freshwaters sampled at seven sites in a karstic area of northeastern Italy. The data set was used for a three-way principal factor analysis aimed at exploring the pattern of information about the environmental quality of the monitored freshwaters, since four wells are feeding the municipal water supply of the Province of Trieste, and the other water courses can influence them. The selected three-way (3,3,2) model uses three components for describing the analytical parameters, three for temporal variations and two for spatial variations. The method optimising the ‘variance of squares’ of the core elements has permitted a simple and meaningful interpretation of the Tucker-3 solution. The procedure succeeded in decomposing the overall temporal variation in three parts, thus highlighting nonperiodic critical events, a periodic seasonal component and a constant term. The seasonality has been confirmed by the examination of the autocorrelation function of the second temporal component. An environmental interpretation and an estimate of the relative relevance of phenomena conditioning the considered water body, detected by the multiway analysis, have been proposed.

• Barbieri, P., Adami, G., Piselli, S., Gemiti, F., & Reisenhofer, E. (2002b). A three-way principal factor analysis for assessing the time variability of freshwaters related to a municipal water supply. Chemometrics and Intelligent Laboratory Systems, 60, 89-100.
  Chemical analyses and biological monitors constitute the 15 parameters, monitored with monthly frequency in the space of 4 years on freshwaters sampled at seven sites in a karstic area of northeastern Italy. The data set was used for a three-way principal factor analysis aimed at exploring the pattern of information about the environmental quality of the monitored freshwaters, since four wells are feeding the municipal water supply of the Province of Trieste, and the other water courses can influence them. The selected three-way (3,3,2) model uses three components for describing the analytical parameters, three for temporal variations and two for spatial variations. The method optimising the 'variance of squares' of the core elements has permitted a simple and meaningful interpretation of the Tucker-3 solution. The procedure succeeded in decomposing the overall temporal variation in three parts, thus highlighting nonperiodic critical events, a periodic seasonal component and a constant term. The seasonality has been confirmed by the examination of the autocorrelation function of the second temporal component. An environmental interpretation and an estimate of the relative relevance of phenomena conditioning the considered water body, detected by the multiway analysis, have been proposed.

• Baronti, S., Casini, A., Lotti, F., & Porcinai, S. (1997).
  Principal component analysis of visible and near-infrared multispectral images of works of art. Chemometrics and Intelligent Laboratory Systems, 39, 103-114.
  Principal component analysis (PCA) was applied to a very simple case of a tempera panel painted with four known pigments (cinnabar, malachite, yellow ochre and chromium oxide). The four pigments were spread pure as well as dilute with carbon black (5% w/w, 10% w/w) thus creating 12 homogeneous areas of the same size. The panel was imaged by a Vidicon camera in the visible and near-infrared regions (420-1550 nm) resulting in a set of 29 images. PCA was applied by taking various subsets of the input data. From the analysis of this simple and predictable case study some guidelines are synthesized and proposed for the application to actual work of art. Results are presented for the painted panel. Preliminary results are also reported for the Luca Signorelli's ''Predella della Trinita''. The multivariate image analysis results in the visible and near-infrared regions show that it is possible to use the multispectral image data in order to get a segmentation and a classification of painted zones by pigments with different chemical composition or physical properties.
  
  
• Barré, A., & Fichet, B. (1985).
  Analyse des correspondances et rotations procustéennes représentation hiérarchique et ordres compatibles. Statistiques et Analyse de Données, 10, 16-26.
  Contingency tables depending on time define the evolution of items for two qualitative variables. We study this evolution by means of a correspondence factor analysis and a hierarchical classification at each time. Moreover, Procrustes analysis and the search of a common order on units for the hierarchies help in comparing the graphical representations.
  
  
• Bartussek, D. (1973).
  Zur Interpretation der Kernmatrix in der dreimodalen Faktorenanalyse von R.L. Tucker [On the interpretation of the core-matrix in the three-mode factor analysis of R.L. Tucker]. Psychologische Beiträge, 15, 169-184.
  After a rather clear exposition of T3, generalizing from PCA on two-mode matrices, B. proposes to scale the component matrices such that the components have lengths equal to the corresponding eigenvalues. These eigenvalues are themselves adjusted by division through the total number of elements in the other two modes. The reciprocal scaling is performed for the core matrix elements. These elements become in this way independent of the size of the sum of squares of the components and may therefore be interpreted as 'classical' factor scores. In the same sense the elements of the components correspond to 'classical' factor loadings rather than being just elements of orthogonal eigenvectors. Standardization of the raw data and interpretation of T3 results by comparing them with external variables are discus- sed as well.

• Bartussek, D. (1980).
  Die dreimodale Faktorenanalyse als Methode zur Bestimmung von EEG-Frequenzbändern [Trimodal factor analysis as a method of determining EEG frequency bands]. In St. Kubicki, W.M. Herrmann & G. Laudahn (Eds.), Faktorenanalyse und Variablenbildung aus dem Elektroenzephalogramm (pp. 15-26). Stuttgart, FRG.: Gustav Fischer Verlag.
  T3 is outlined, its relation to Cattell's (1966) data box is indicated, and the interpretation of the core matrix for EEG data is discussed. Also included is a discussion of the subject and situation selection, the norming of the EEG frequency spectra to be calculated, the standardization of the spectrum values and the choice of a time basis for the frequency analysis.

  

• Bartussek, D. & Gräser, H. (1980).
  Ergebnisse dreimodaler Faktorenanalysen von EEG-Frequenzspektren. In S. Kubicki, W.M. Herrmann & G. Laudahn (Eds.), Faktorenanalyse und Variablenbildung aus dem Electroenzephalogramm (pp. 79-87). Stuttgart, FRG: Gustav Fischer Verlag.
  The results of two unpublished studies are reported. Of 40 students 30 values of the frequency spectrum for six activity situations were measured in 2 ways. T3 was performed on data (40x30x12) standardized per spectral value over all student/situation combinations. Frequencies and situations were varimaxed. Special attention was paid to the interpretation of the core matrix and the effects of standardization. In the other study 3 spectral values were collected from 20 subjects in 24 situations. Data standardized as above. Frequencies were varimaxed; the situations and subjects were obliquely rotated. Again detailed attention to the core matrix.

• Basford, K.E., Federer, W.T., & Miles-McDermott, N.J. (1987).
  Illustrative examples of clustering using the mixture method and two comparable methods from SAS. Computational Statistics Quarterly, 4, 219-233.
  The technique of clustering uses the measurements on a set of elements to identify clusters or groups of elements such that there is relative homgeneity within the groups and heterogeneity between the groups. Under the mixture model approach to clustering, the elements are assumed to be a sample from a mixture of several populations in various proportions. In addition to the formal definition, the practical application to two real data sets is considered, with the density function in each underlying population assumed to be Normal. To provide a base for comparison, two SAS clustering methods with similar assumptions are also considered. The data are analyzed using: KMM, SAS (CLUSTER) - Ward's method, and SAS (CLUSTER) - EML method; the results are discussed.

  

• Basford, K.E., & Kroonenberg, P.M. (1989).
  An investigation of multi-attribute genotype response across environments using three-mode principal component analysis. Euphytica,44, 109-123.
  The usefullness of three-mode principal component analysis to explore multi- attribute genotype-environment interaction is investigated. The technique provides a general description of the underlying patterns present in the data in terms of interactions of the three quantities involved. As an example, data from an Australian experiment on the breeding of soybean lines are treated in depth.

• Basford, K.E., Kroonenberg, P.M., & DeLacy, I.H. (1991).
  Three-way methods for multiattribute genotype by environment data: An illustrated partial survey. Field Crops Research, 27, 131-157.
  Several ordination and clustering techniques are discussed with respect to their usefulness in analysing multiattribute genotype×environment data. The methods are briefly described and illustrated by application to data from the Australian Cotton Cultivar Trials (ACCT), a series of regional variety trials designed to investigate various cotton (Gossypium hirsutum (L.)) lines in several locations each year. Multivariate techniques applicable to three-way data are necessary to assess these lines using yield and lint-quality data. By the choice of complementary methods, it is possible to make both global and detailed statements about the relative performance of the cotton lines. These techniques can enhance the researcher's ability to make informed decisions about the genotype×environment data collected from these trials using simultaneous analysis of the attributes of interest.

• Basford, K.E., Kroonenberg, P.M., DeLacy, I.H., & Lawrence, P.K. (1990).
  Multiattribute evaluation of regional cotton variety trials. Theoretical and Applied Genetics, 79, 225-234.
  Two multivariate techniques applicable to three-way data are described and used to analyse the data: the mixture maximum likelihood method of clustering and three-mode principal component analysis. Applied together, the methods enhance each other's usefullness in interpreting the information on the line response patterns across the locations. The methods provide a good integration of the responses across environments of the entries for the different attributes in the trials, and a less subjective, relativly easy to apply and interpret analytical method of describing the patterns of performance and associations in complex multiattribute and multilocation trials. This should lead to more efficient selection among lines in such trials.

  

• Basford, K.E., & McLachlan, G.J. (1985a).
  Estimation of allocation rates in a cluster analysis context. Journal of the American Statistical Association, 80, 286-293.
  A sample of multivariate observations is assumed to be drawn from a mixture of a given number of underlying populations. The mixture likelihood approach to clustering is used to allocate each individual in the sampe to its population of origin on the basis of the estimated posterior probabilities of population membership. Estimation of the correct allocation rate is considered for individual populations as well as for the overall mixture by averaging functions of the maximum of these posterior probabilities. The estimates of the correct allocation rates provide a means of assessing the performance of the mixture approach to clustering. The bootstrap technique is investigated for its effectiveness in reducing the bias of the estimates so obtained. Results are reported for three real data sets and a simulation study. It is demonstrated that the proposed estimates generally provide useful information on the unobservable allocation rates of the mixture approach. Encouraging results are obtained for the bootstrap method of bias correction applied to the estimates of the individual and overall allocation rates.
  
  
• Basford, K.E., & McLachlan, G.J. (1985b).
  Likelihood estimation with normal mixture models. The Journal of the Royal Statistical Society, 34, 282-289.
  Considered are some of the problems associated with likelihood estimation in the context of a mixture of multivariate normal distributions. In mixture models, the likelihood equation usually has multiple roots and so there is the question of which root to choose. In the case of equal covariance matrices the choice of root is straightforward in the sense that the maximum likelihood estimator exists and is consistent. However, an example is presented to demonstrate that the adoption of a homoscedastic normal model in the presence of some heteroscedasticity can considerably influence the likelihood estimates, in particular of the mixing proportions, and hence the consequent clustering of the sample at hand.
  
  
• Basford, K.E., & McLachlan, G.J. (1985c).
  The mixture method of clustering applied to three-way data. Journal of Classification, 2, 109-125.
  This article shows that by appropriate specification of the underlying model, the mixture maximum likelihood approach to clustering can be applied in the context of a three-way table. It is illustrated using a soybean data set which consists of multiattribute measurements on a number of genotypes each grown in several environments. Although the problem is set in the framework of clustering genotypes, the technique is applicable to other types of three-way data sets.
  
• Batchelder, W. H., Kumbasar, E., & Boyd, J. P. (1997).
  Consensus analysis of three way social network data. Journal of Mathematical Sociology, 22, 29-58.
  Three way social network data occurs when every actor in a social network generates a digraph of the entire network. This paper presents a statistical model based on cultural consensus analysis for aggregating these separate digraphs into a single consensus digraph. In addition, the model allows estimation of separate hit and false alarm rates for each actor that can vary within each actor in different regions of the digraph. Several standard signal detection models are used to interpret the hit and false alarm parameters in terms of knowledge and response bias. A published three way data set by Kumbasar, Romney, and Batchelder (American Journal of Sociology, 1994) is analyzed, and the model reveals that both response bias and knowledge decrease with distance from ego.
  
  
• Baunsgaard , D., Andersson, C. A., Arndal, A., & Munck, L., (2000a).
  Multi-way chemometrics for mathematical separation of fluorescent colorants and colour precursors from spectrofluorimetry of beet sugar and beet sugar thick juice as validated by HPLC analysis. Food Chemistry, 70, 113-121.
  In previous analyses of colour impurities in processed sugar, a multi-way chemometric model, CANDECOMP-PARAFAC (CP), has been used to model fluorescence excitation-emission landscapes of sugar samples. Four fluorescent components were found, two of them tyrosine and tryptophan, correlating to important quality and process parameters. In this paper HPLC analyses are used to chemically verify and extend the CP models of sugar. Thick juice, an intermediate in the sugar production, was analysed by size exclusion HPLC. Tyrosine and tryptophan were confirmed as constituents in thick juice. Colorants were found to be high molecular weight compounds. Fluorescence landscapes on collected column fractions were modelled by the CP model and seven fluorophores were resolved. Apart from tyrosine and tryptophan, four of the fluorophores were identified as high molecular weight compounds, three of them possible Maillard reaction polymers, whereas the seventh component resembled a polyphenolic compound. It is concluded that the relevance of CP for mathematical separation of fluorescence landscapes has been justified on two levels by HPLC; firstly as a screening method of fluorophores in complex samples and secondly as a confirmation of peak purity in chromatographic separation.
  
  
• Baunsgaard, D., Munck, L., & Norgaard, L. (2000b).
  Analysis of the effect of crystal size and color distribution on fluorescence measurements of solid sugar using chemometrics. Applied Spectroscopy, 54, 1684-1689.
  Fluorescence from sugar crystal samples has previously been used to obtain information about factory imprint and sugar quality. Solid-phase fluorescence has potential as a fast screening method, but the spectra are highly influenced by the measurement geometry and sugar crystal sample. The aim of the present study was to examine how the fluorescence measurements are related to the sugar crystals for a better understanding of both. Initially, five sugar samples of varied composition were sieved into five crystal size fractions. Fluorescence excitation-emission landscapes of the fractions were measured with solid transmission and reflection techniques and in solution. The transmission fluorescence was quenched at ultraviolet wavelengths, and light scatter highly influenced the reflection fluorescence. Principal component analysis (PCA) showed that large crystals favored the transmission fluorescence, whereas smaller crystals improved the reflection fluorescence measurements. The multi-way method PARAFAC (parallel factor analysis) was used to resolve spectra of individual components from the fluorescence landscapes. Transmission and solution components had similar spectral profiles at higher wavelengths, characterizing a colorant and a colorant intermediate. The resolved components of the reflection data were very influenced by scatter. Color predictions based on a few significant wavelength variables equaled the model results of full-spectrum models using partial least-squares regression (PLS). The variables corresponded to wavelength maxima of the resolved colorants and ultraviolet wavelengths characterizing colorant precursors.
  
  
• Baunsgaard, D., Norgaard, L., & Godshall, H. A. (2000c).
  Fluorescence of raw cane sugars evaluated by chemometrics. Journal of Agricultural and Food Chemistry, 48, 4955-4962.
  In a fluorescence study of raw cane sugar samples, two-way and three-way chemometric methods have been used to extract information about the individual fluorophores in the sugar from fluorescence excitation-emission landscapes. A sample set of 47 raw sugar samples representing a varied selection was analyzed, and three individual fluorophores with (275, 350) nm, (340, 420) nm, and (390, 460) nm as their approximate excitation and emission maxima were found. The spectral profiles of the fluorophores were estimated with the three-way decomposition model PARAFAC. Two-way principal component analysis (PCA) of unfolded fluorescence landscapes confirmed the PARAFAC results and showed patterns of samples related to time of storage; Partial least squares (PLS) calibration models of color at 420 nm had a high model error due to the very high color range of the raw sugars, but variable selection; performed on the fluorescence data revealed that all three fluorophores were correlated to color. The (275, 350) nm fluorophore is considered as a color precursor to the color developed on storage and the (340, 420) nm and (390, 460) nm fluorophores show colorant polymer characteristics.
  
  
• Baunsgaard, D., Norgaard, L., & Godshall, H. A. (2001).
  Specific screening for color precursors and colorants in beet and cane sugar liquors in relation to model colorants using spectrofluorometry evaluated by HPLC and multiway data analysis. Journal of Agricultural and Food Chemistry, 49, 1687-1694.
  A comparison was made of the fluorophores in beet thick juice and cane final evaporator syrup, which are comparable in the production of cane and beet sugar; that is, both represent the final stage of syrup concentration prior to crystallization of sugar. To further elucidate the nature of the color components in cane and beet syrup, a series of model colorants was also prepared, consisting of mildly alkaline-degraded fructose and glucose and two Maillard type colorants, glucose-glycine and glucose-lysine. Fluorescence excitation-emission landscapes resolved into individual fluorescent components with PARAFAC modeling were used as a screening method for colorants, and the method was validated with size exclusion chromatography using a diode array UV-vis detector. Fluorophores from the model colorants were mainly located at visible wavelengths. An overall similarity in chromatograms and absorption spectra of the four model colorant samples indicated that the formation of darker color was the distinguishing characteristic, rather than different reaction products. The fluorophores obtained from the beet and cane syrups consisted of color precursor amino acids in the W wavelength region. Tryptophan was found in both beet and cane syrups. Tyrosine as a fluorophore was resolved in only beet syrup, reflecting the higher levels of amino acids in beet processing. In the visible wavelength region, cane syrup colorant fluorophores were situated at higher wavelengths than those of beet syrup, indicating formation of darker colorants. A higher level of invert sugar in cane processing compared to beet processing was suggested as a possible explanation for the darker colorants.
  
• Baxter, D. C., & Ohman, J. (1990).
  Multicomponent standard additions and partial least-squares modeling: A multivariate calibration approach to the resolution of spectral interferences in graphite-furnace atomic-absorption spectrometry. Spectrochimica Acta Part B - Atomic Spectroscopy, 45, 481-491.
  Spectral interferences in graphite furnace atomic absorption spectrometry (GFAAS) represent a considerable problem, often making the direct determination of certain elements in specific matrices impossible, particularly when continuum source background correction is employed. The possibilities to resolve such spectral interferences mathematically by applying multivariate calibration have been investigated. Resolution is achieved using multi-component standard additions (the so-called generalised standard addition method or GSAM) combined with partial least squares (PLS) modelling. This multivariate calibration method, PLS-GSAM, is described and its use illustrated by application to the GFAAS determination of gold in the presence of cobalt at the 242.8 nm wavelength, where severe spectral interference problems are observed using continuum source background correction. Two requirements for the successful application of PLS-GSAM are that the sample constituent causing the spectral interference is known and that its concentration can be increased by standard additions. It is shown that more accurate results are obtained by PLS-GSAM than by conventional (single-component) standard addition methods.
  
         
• Bechmann, I.E. (1997).
  Second-order data by flow injection analysis with spectrophotometric diode-array detection and incorporated gel-filtration chromatographic column. Talanta, 44, 585-591.
  A flow injection analysis (fia) system furnished with a gel-filtration chromatographic column and with photodiode-array detection was used for the generation of second-order data. The system presented is a model system in which the analytes are blue dextran, potassium hexacyanoferrate(iii) and heparin. It is shown that the rank of the involved sample data matrices corresponds to the number of chemical components present in the sample. The Parafac (parallel factor analysis) algorithm combined with multiple linear regression and the tri-PLS (tri-linear partial least-squares regression), which allows unknown substances to be present in the sample, are implemented for fia systems and it is illustrated how these three-way algorithms can handle spectral interferents. The prediction ability of the two methods for pure two-component samples and also the predictions ability in the presence of unknown interferents are satisfactory. However, the predictions obtained by tri-PLS are slightly better than those obtained using Parafac regression algorithm.

• Beckmann, C. F., & Smith, S. M. (2005).
  Tensorial extensions of independent component analysis for mulisubject FMRI analysis. NeuroImage, 25, 294-311.
  We discuss model-free analysis of multisubject or multisession FMRI data by extending the single-session probabilistic independent component analysis model (PICA; Beckmann and Smith, 2004. IEEE Trans. on Medical Imaging, 23 (2) 137-152) to higher dimensions. This results in a three-way decomposition that represents the different signals and artefacts present in the data in terms of their temporal, spatial, and subject-dependent variations. The technique is derived from and compared with parallel factor analysis (PARAFAC; Harshman and Lundy, 1984. In Research methods for multimode data analysis, chapter 5, pages 122-215. Praeger, New York). Using simulated data as well as data from multisession and multisubject FMRI studies we demonstrate that the tensor PICA approach is able to efficiently and accurately extract signals of interest in the spatial, temporal, and subject/session domain. The final decompositions improve upon PARAFAC results in terms of greater accuracy, reduced interference between the different estimated sources (reduced cross-talk), robustness (against deviations of the data from modeling assumptions and against overfitting), and computational speed. On real FMRI dactivationT data, the tensor PICA approach is able to extract plausible activation maps, time courses, and session/subject modes as well as provide a rich description of additional processes of interest such as image artefacts or secondary activation patterns. The resulting data decomposition gives simple and useful representations of multisubject/ multisession FMRI data that can aid the interpretation and optimization of group FMRI studies beyond what can be achieved using modelbased analysis techniques.

• Bécue-Bertaut, M. (1992).
  In A. Rizzi, M. Vichi, & H.-H. Bock (Eds.) Advances in data science and classification (pp. 457-464). Berlin: Springer.
  Methods are presented for analysing open-ended answers in surveys having the form of three-way arrays, such as comparing answers given to the same questions in different surveys, and examining individuals as described by their answers to several open questions. In essence the methods come down to replicate use of standard correspondence analysis of different varieties of tables constructed from the original three-way array. Specific software routines have been developed.

• Bécue-Bertaut, M., & Pagès, J. (2004).
  A principal axes method for comparin contingency tables: MFACT. Computational Statistics & Data Analysis, 45, 481-503.
  A new methodology is introduced for comparing the structures of several contingency tables. The latter,built up from di erent samples or populations,present the same rows and di erent columns (or vice versa).This methodology combines some aspects of principal axes methods (global maximum dispersion axes),canonical correlation techniques (canonical dispersion axes) and Procrustes analysis (superimposed representations)but takes into account the particularities of contingency tables in order to extend correspondence analysis to multiple contingency tables. Two main problems arise:the di erences between the margins of the common dimension and the need for balancing the in uence of the di erent tables in global processing.A study of the four structures induced on Spanish regions by mortality causes (by gender)and by age distribution (by gender),in conjunction,will illustrate the methodology.

• Beffy, J.L. (1992).
  Application de l'analyse en composantes principales à trois modes pour l'étude physico-chimique d'un écosystème lacustre d'altitude: Perspectives en écologie.[Application of three-mode principal component analysis for a chemo-physical study of the ecosystem of a high altitude lake: Perspectives in ecology]. Revue Statistique Appliquée, 40 (1), 37-56.
  Physical and chemical parameters were measured by CHACORNAC (1986) in an oligotrophic high-mountain lake. The presence of a thick ice-cover during eight months involves a complex spatio-temporal evolution of the parameters. The application of three-mode principal component analysis brings to the fore the interaction between spacial and temporal patterns of these parameters. The results obtained on this particular case allow to discuss a more intensive use in ecological research.

• Begin, M., Roff, D. A., & Debat, V. (2004).
  The effect of temperature and wing morphology on quantitative genetic variation in the cricket Gryllus firmus, with an appendix examining the statistical properties of the Jackknife-manova method of matrix comparison. Journal of Evolutionary Biology, 17, 1255-1267.
  We investigated the effect of temperature and wing morphology on the quantitative genetic variances and covariances of five size-related traits in the sand cricket, Gryllus firmus. Micropterous and macropterous crickets were reared in the laboratory at 24, 28 and 32 degreesC. Quantitative genetic parameters were estimated using a nested full-sib family design, and (co)variance matrices were compared using the T method, Flury hierarchy and Jackknife-MANOVA method. The results revealed that the mean phenotypic value of each trait varied significantly among temperatures and wing morphs, but temperature reaction norms were not similar across all traits. Micropterous individuals were always smaller than macropterous individuals while expressing more phenotypic variation, a finding discussed in terms of canalization and life-history trade-offs. We observed little variation between the matrices of among-family (co)variation corresponding to each combination of temperature and wing morphology, with only one matrix of six differing in structure from the others. The implications of this result are discussed with respect to the prediction of evolutionary trajectories.

• Beh, E. J., & Davy, P. J. (1998).
  Partitioning Pearson's chi-squared statistic for a completely ordered three-way contingency table. Journal of Marketing Research, 11, 156-163.
  The paper presents a partition of the Pearson chi-squared statistic for triply ordered three-way contingency tables. The partition invokes orthogonal polynomials and identifies three-way association terms as well as each combination of two-way associations. This partition provides information about the structure of each variable by identifying important bivariate and trivariate associations in terms of location (linear), dispersion (quadratic) and higher order components. The significance of each term in the partition, and each association within each term can also be determined. The paper compares the chi-squared partition with the log-linear models of Agresti (1994) for multi-way contingency tables with ordinal categories, by generalizing the model proposed by Haberman (1974).

• Belk, R.W. (1974).
  An exploratory assessment of situational effects in buyer behavior. Journal of Marketing Research, 11, 156-163.
  The variance in selected purchase decisions was explored as function of consumption and purchase context. T3 was used for data from 100 subjects in 10 situations with 10 snack products, and in 9 situations with 11 meat products. Solutions were obtained via Tucker's Method III. Both situation and product spaces were varimax rotated. The same data were also analysed with a three-way mixed effects analysis of variance model.

  

• Belk, R.W. (1979).
  Gift-giving behavior. In J.N. Sheth (Ed.), Research in marketing, Vol. 2 (pp. 95-126). Greenwich, CT: JAI Press Inc.
  As part of a larger study 12 characteristics in each of 15 gift-giving situations were rated by 219 respondents. The components were varimax rotated, and the two person components were analysed using the core matrix. One component matrix and the core matrix are presented in detail.

• Bell, T. S., Dirks, D. D., & Carterette, E. C. (1989). Interactive factors in consonant confusion patterns Journal of the Acoustical Society of America, 85, 339-346.
  Confusion patterns among English consonant were examined using log-linear modeling techniques to assess the influence of low-pass filtering, shaped noise, presentation level, and consonant position. Ten normal-hearing listeners were presented consonant-vowel (C-V) and vowel-consonant (V-C) syllables containing the vowel/a/. Stimuli were presented in quiet and in noise, and were either filtered or broadband. The noise was shaped such that the effective signal level in each « octave band was equivalent in quiet and noise listening conditions. Three presentation levels were analyzed corresponding to the overall rms level of the combined speech stimuli. Error patterns were affected significantly by presentation level, filtering, and consonant position as a complex interaction. The effect of filtering was dependent on presentation level and consonant position. The effects stemming from the noise were less pronounced Specific confusions responsible for these effects were isolated, and an acoustical interaction is suggested stressing the spectral characteristics of the signals and their modification by presentation level and filtering.

• Beltrán, J. L., & Ferrer, R., & Guiteras, J. (1998a).
  Parallel factor analysis of partially resolved chromatographic data Determination of polycyclic aromatic hydrocarbons in water samples. Journal of Chromatography A, 802, 263-275.
  A procedure, based on parallel factor analysis (PARAFAC), has been used for the analysis of polycyclic aromatic hydrocarbons in water samples. The chromatographic system has been set to obtain short-time chromatograms containing several unresolved peaks. The detection system consisted of a fast-scanning fluorescence spectra detector, which allowed three-dimensional data – where retention time, emission wavelengths and fluorescence intensity were represented – to be obtained. The procedure has been applied to spiked tap water samples with good results.

• Beltrán, J. L., & Ferrer, R., & Guiteras, J. (1998b).
  Multivariate calibration of polycyclic aromatic hydrocarbon mixtures from excitation-emission fluorescence spectra. Analytica Chimica Acta, 373, 311-319.
  The excitation–emission fluorescence spectra (EEM) of mixtures of 10 polycyclic aromatic hydrocarbons (PAHs) have been analyzed using different multivariate calibration procedures (partial least squares regression, PLSR; and parallel factor analysis, PARAFAC). The compounds studied were anthracene, benz[a]anthracene, benzo[a]pyrene, chrysene, phenanthrene, fluoranthene, fluorene, naphthalene, perylene and pyrene.

• Beltrán, J. L., Guiteras, J., & Ferrer, R., (1998c).
  Three-way multivariate calibration procedures applied to high-performance liquid chromatography coupled with fast-scanning fluorescence spectrometry detection. Determination of polycyclic aromatic hydrocarbons in water samples. Analytical Chemistry, 70, 1949-1955.
  Three-way partial least-squares and n factor parallel factor analysis have been compared for the analysis of polycyclic aromatic hydrocarbons in water samples. Data were obtained with a chromatographic system set to record short-time chromatograms containing several unresolved peaks. The detection system consisted of a fast-scanning fluorescence spectra detector, which allows one to obtain three-dimensional data, where retention time, emission wavelengths, and fluorescence intensity are represented. The combined use of a multivariate calibration method and the three-dimensional data obtained from the HPLC-FSFS system allows resolution of closely eluting compounds, thus making a complete separation unnecessary. The procedure has been applied to tap water samples (spiked at 0.10 and 0.20 mu g L-1 levels) with good results, similar to those obtained with a HPLC system with a conventional fluorescence detector.

• Bendtsen, A. B., Glarborg, P., & Dam-Johansen, K. (2001).
  Visualization methods in analysis of detailed chemical kinetics modelling. Computers & Chemistry, 25, 161-170.
  Sensitivity analysis, principal component analysis of the sensitivity matrix, and rate-of-production analysis are useful tools in interpreting detailed chemical kinetics calculations. This paper deals with the practical use and communication of the sensitivity analysis and the related methods are discussed. Some limitations of sensitivity analysis, originating from the mathematical concept (e.g. first-order or brute force methods) or from the software-specific implementation of the method, are discussed. As supplementary tools to the current methods, three novel visual tools for analysis of detailed chemical kinetics mechanisms are introduced: (a) scaled sensitivity analysis which is especially suited for studying initiation reactions where the span of reaction rates is high; (b) automated generation of reaction pathway plots which provides an immediate graphical illustration of the chemical processes occurring; (c) explorative (or chemometric) analysis of accumulated rate of progress matrices which assist in the identification of reaction subsets. The application of these tools are demonstrated by analysing NO, enhanced oxidation of methane at 700-1200 K.

• Benito, M., & Peña, D. (2005).
  A fast approach for dimensionality reduction with image data. Pattern Recognition, 38, 2400-2408.
  An important objective in image analysis is dimensionality reduction. The most often used data-exploratory technique with this objective is principal component analysis, which performs a singular value decomposition on a data matrix of vectorized images. When considering an array data or tensor instead of a matrix, the high-order generalization of PCA for computing principal components offers multiple ways to decompose tensors orthogonally. As an alternative, we propose a new method basedon the projection of the images as matrices andsho w that it leads to a better reconstruction of images than previous approaches.

• Bennani Dosse, M. (1995).
  Positionnement multidimensionnel d'un tableau à 3 voies. Revue Statistique Appliquée, 43(4), 63-75.
  We propose a new Multidimensional scaling method which represents 3-way tables where all of the three ways are of equal consideration from the point of exploratory data analysis and whose elements are interpreted as measures of proximity. The treatment of real data is performed to illustrate the proposed methods.

• Benninghoff, L., Vonczarnowski, D., Denkhauw, E., & Lemke, K. (1997).
  Analysis of human tissues by total reflection x ray fluorescence: application of chemometrics for diagnostic cancer recognition. Spectrochimica Acta Part B Atomic Spectroscopy, 52, 1039-1046.
  For the determination of trace element distributions of more than 20 elements in malignant and normal tissues of the human colon, tissue samples (approx. 400 mg wet weight) were digested with 3 ml of nitric acid (sub boiled quality) by use of an autoclave system. The accuracy of measurements has been investigated by using certified materials. The analytical results were evaluated by using a spreadsheet program to give an overview of the element distribution in cancerous samples and in normal colon tissues. A further application, cluster analysis of the analytical results, was introduced to demonstrate the possibility of classification for cancer diagnosis. To confirm the results of cluster analysis, multivariate three way principal component analysis was performed. Additionally, microtome frozen sections (10 mu m) were prepared from the same tissue samples to compare the analytical results, i.e. the mass fractions of elements, according to the preparation method and to exclude systematic errors depending on the inhomogeneity of the tissues. (C) 1997 Elsevier Science B.V.

• Bentler, P.M. & Lee, S.-Y. (1978).
  Statistical aspects of a three-mode factor analysis model. Psychometrika, 43, 343-352.
  A special case of Bloxom's version (1968) of T3 is developed statistically. A distinction is made between fixed and random modes. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. Estimation by a weighted least squares method based upon Gauss-Newton. Example based upon self-report and peer-report measures (see also B. & L., 1979).

  

• Bentler, P.M. & Lee, S.-Y. (1979).
  A statistical development of three-mode factor analysis. British Journal of Mathematical & Statistical Psychology, 32, 87-104.
  B & L consider a factor analytic random vector version of T3. The parameters of the model are associated with two fixed modes and the covariance matrix of the random vectors. Their approach brings three-mode FA in the realm of structural equation models. Their model does not treat all three modes symmetrically as Tucker (1966) and Kroonenberg & De Leeuw (1980) do. With B & L's model a confirmatory approach to T3 is possible, and standard errors and a goodness-of-fit statistic become available. B & L's model has some similarity to the treatment of T3 by Tucker (1966) through his method III. The model is illustrated by a multi-trait multi-method matrix example.

• Bentler, P.M., Poon, W.-Y., & Lee, S.-Y. (1988).
  Generalized multimode latent variable models: Implementation by standard programs. Computational Statistics & Data Analysis 6, 107-118.
  Three-mode models in factor analysis have not been used very frequently due in part to their mathematical, statistical, and computational complexity. It is shown that standardly-available computer programs such as LISREL and EQS can be used to estimate and test such models. The models are generalized to permit more complex measurement structures, as well as to allow linear structural regressions among the latent variables. These generalized multimode models can be similarly easily computationally implemented. An example is used to illustrate the ideas.

• Bentler, P.M., & Weeks, D.G. (1979).
  Interrelations among models for the analysis of moment structures Multivariate Behavioral Research, 14, 169-186.
  Factor analysis in several populations, covariance structure models, three-mode factor analysis, structural equation systems with measurement model, and analysis of covariance with measurement model are all shown to be specializations of a general moment structure model by Bentler. Some new structured linear models are also described that may be considered either generalizations or special cases of existing models.

  

• Bernstein, A.L. & Wicker, F.W. (1969).
  A three-mode factor analysis of the concept of novelty. Psychonomic Science, 14, 291- 292.
  A rather simplistic inquiry into the concept of 'novelty' using the unscaled scores of 30 students on an 18 item semantic differential type scale with 10 realistic and unrealistic animals. T3 on cross-products. No serious interpretation.

• Berrueta, L.A., Fernandez, L.A., & Vicente, F. (1991).
  Fluorim: A computer program for the automated data collection and treatment using commercial spectrofluorimeters. Computers & chemistry, 15, 307-312.
  A general purpose program that uses a pc microcomputer has been designed for the control of experiments in fluorimetry. This program allows the digital collection of the main types of scans that a commercial spectrofluorimeter can perform: Emission, excitation and synchronic; as well as the measurement of the fluorescent intensity as a function of time. The program also allows a series of operations with previously stored spectra. These operations include screen and paper plots of the spectra, calculation of linear combinations of them and the nine first derivatives, as well as their relative maxima and minima.

• Bertero, H. D., de la Vega, A. J., Correa, G., Jacobsen, S. E., & Mujica, A. (2004).
  Genotype and genotype-by-environment interaction effects for grain yield and grain size of quinoa(Chenopodium quinoa Willd.) as revealed by pattern analysis of international multi-environment trials. Field Crops Research, 89, 299-318.
  The size and nature of the genotype (G) and genotype environment (GxE) interaction effects for grain yield, its physiological determinants, and grain size exhibited by the Andean grain crop quinoa at low latitudes were examined in a multi-environment trial involving a diverse set of 24 cultivars tested in 14 sites under irrigation across three continents. These environments included a wide latitudinal (from 21 30'N to 16 21'S), altitudinal (from 5 to 3841 m a.s.l.) and temperature (average daily temperatures during crop cycle varied from 9 to 22.1 C) range; while average daily photoperiods exhibited a smaller variation, from 11.2 to 12.8 h. The GxE interaction to G component of variance ratio was 4:1 and 1:1 for grain yield and grain size, respectively. Two-mode pattern analysis of the environment-standardised matrix of grain yield revealed four genotypic groups of different response pattern across environments. This clustering, which separates cultivars from mid-altitude valleys of the northern Andes, northern altiplano, southern altiplano and sea level, showed a close correspondence with adaptation groups previously proposed. The results of the genotype clustering can be used to choose genotypes of contrasting relative performance across environments for further studies aimed at assessing the opportunity to select for broad or specific adaptation. Classification of sites for grain yield grossly discriminated between cold highland sites, tropical valleys of moderate altitude, and warmer, low altitude sites. As expected from the size of the G E interaction component, no single genotype group showed consistently superior grain yield across all environment groups. The G and GxE interaction effects observed for the duration of the crop cycle had a major influence on the average cultivar performance and on the form of GxE interactions observed for total above-ground biomass and grain yield. Although different environment types showed contrasting effects on the physiological attributes underlying grain yield variation among cultivars, it was observed that good average performance and broad adaptation could come from the combination of medium–late maturity and high harvest index. Correlation analysis revealed no association between the average cultivar responses for grain yield and grain size. Three-mode pattern analysis have also shown no association between the GxE interaction effects for both traits. Both observations indicate that simultaneous progress for grain yield and grain size can be expected from selection.

• Beylkin, G., & Mohlenkamp, M. J. (2005).
  Algorithms for numerical analysis in high dimensions. Society for Industrial and Applied Mathemetics, 26, 2133-2159.
  Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to be surprisingly efficient; (ii) addressing the issue of conditioning; (iii) presenting algorithms for solving linear systems within this framework; and (iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schr¨odinger equation in quantum mechanics. Numerical examples are given.

• Bezemer, E., & Rutan, S. C. (2001a).
  Multivariate curve resolution with non-linear fitting of kinetic profiles. Chemometrics and Intelligent Laboratory Systems, 59, 19-31.
  This paper describes the incorporation of a hard modeling step based on a kinetic model, into a soft modeling multi-variate curve resolution technique. The soft modeling technique allows for the determination of the retention and spectral profiles from overlapped components while the hard modeling step allows for the simultaneous prediction of the rate constants of the various steps in the reaction pathway. The program uses standard MATLABw functions for determining the solutions of the differential equations as well as for finding the optimal rate constants to describe the kinetic profiles. The kinetic model is entered by a set of command line parameters and can describe any order chemical reaction with multiple reaction pathways. This paper uses simulated first- and second-order reaction data as well as real data to characterize the performance of the program. The algorithm is able to resolve overlapped retention and spectral profiles and predict the rate constants for the reaction.

• Bezemer, E., & Rutan, S. C. (2001b).
  Study of the Hydrolysis of a Sulfonylurea Herbicide Using Liquid Chromatography with Diode Array Detection and Mass Spectrometry by Three-Way Multivariate Curve Resolution-Alternating Least Squares. Analytical Chemistry, 73, 4403-4409.
  This research is focused on the development of a novel, automated chemometric method for obtaining relevant chemical information from time-course measurements of an evolving chemical system. This paper describes an investigation of the hydrolysis of Ally, which is a sulfonylurea herbicide. The hydrolysis of this compound is observed at different pHs and temperatures by reversedphase liquid chromatography using a diode array detector. The data are analyzed using a three-way, multivariate curve resolution technique. Of special interest was the application of a closure constraint in the kinetic dimension followed by the determination of the rate constants for each step of the pathway by using a differential equation solver and nonlinear fitting of the data.

• Bezemer, E., & Rutan, S.C. (2002).
  Three-way alternating least squares using three-dimensional tensors in MATLAB. Chemometrics and Intelligent Laboratory Systems, 60, 239-251.
  This paper describes an improved three-way alternating least-squares multivariate curve resolution algorithm that makes use of the recently introduced multi-dimensional arrays of MATLAB. Multi-dimensional arrays allow for a convenient way to apply chemically sound constraints, such as closure, in the third dimension. The program is designed for kinetic studies on liquid chromatography with diode array detection but can be used for other three-way data analysis. The program is tested with a large number of synthetic data sets and its flexibility is demonstrated, especially when non-trilinear data sets are fit. In this case, the algorithm finds a solution with a better fit than direct trilinear decomposition (DTD). When trilinear data are used, the optimal fit is not as good as when a direct decomposition method is used. Most real data sets, however, have some degree of non-trilinearity. This makes this method a better choice to analyze non-trilinear, three-way data than direct trilinear decomposition.

• Bezemer, E., & Rutan, S. C. (2003).
  Evaluation of synthetic liquid chromatography-diode array detection-mass spectrometry data for the determination of enzyme kinetics. Analytica Chimica Acta, 490, 17-29.
  In this paper, we investigate the accuracy and precision of the results from diode array detector (DAD) data and mass spectrometry (MS) data as obtained subsequent to chromatographic separations using computer simulations. Special attention was given to simulations of multiple injections from a developing enzymatic reaction. These simulations result in three-way LC-DAD-MS kinetic data; LC-DAD and LC-MS data were also evaluated independently in this investigation. The noise characteristics of the MS detector prevent accurate determination of the individual reaction rate constants by the analysis method. Using the data from the DAD in combination with the MS detector results in improved estimation of the rate constants. The results also indicate that the higher resolving power of the MS information compensates for the lower signal-to-noise ratio in these data, compared to DAD data. (C) 2003 Elsevier Science B.V. All rights reserved.

• Bezemer, E., & Rutan, S. C. (2006).
  Analysis of three- and four-way data using multivariate curve resolution-alternating least squares with global multi-way kinetic fitting. Chemometrics and Intelligent Laboratory Systems, 81, 82-93.
  This paper demonstrates a novel implementation of an alternating least squares (ALS) algorithm for resolving three- and four-way data. Computer-simulated multi-way data are studied as well as the multi-way data obtained from typical kinetic experiments observed using liquid chromatography with diode array detection (LC-DAD) and UV–visible spectroscopy. Each data set is analyzed using this new multi-way ALS algorithm, not only providing estimates of the spectral profiles (and retention profiles in the case of LC-DAD measurements) for each of the components involved, but also simultaneously estimating the rate constants for the reaction steps at different experimental conditions using a global kinetic analysis. However, when the reaction conditions do not require that all the rate constants are identical for each experiment, as is the case when the reactions are observed at different temperatures, the data analysis still benefits from the common information present in the data, such as spectral and retention profiles, as well as a common reaction mechanism.
       
• Bharati, M. H., & MacGregor, J. F. (1998).
  Multivariate image analysis for real-time process monitoring and control. Industrial & Engineering Chemistry Research, 37, 4715-4724.
  Information from on-line imaging sensors has great potential for the monitoring and control of spatially distributed systems. The major difficulty lies in the efficient extraction of information from the images in real-time, information such as the frequencies of occurrence of specific features and their locations in the process or product space. This paper uses multivariate image analysis (MIA) methods based on multiway principal component analysis to decompose the highly correlated data present in multispectral images. The frequencies of occurrence of certain features in the image, regardless of their spatial locations, can be, easily monitored in the space of the principal components (PC). The spatial locations of these features in the original image space can then be obtained by transposing highlighted pixels from the PC space' into the original image space. In this manner it is possible to easily detect and locate (even very subtle) features from real-time imaging sensors for the purpose of performing statistical process control or feedback control of spatial processes. Due to; the current lack of availability of such multispectral sensors in industrial processes, the concepts and potential of this approach are illustrated using a sequence of multispectral images obtained from a LANDSAT satellite, as it passes over a certain geographical region of the earth's surface.

• Bhattacharya, P., & Mukherjee, N.P. (1994).
  On the representation of uncertain information by multidimensional arrays. IEEE Transactions on Systems, Man, and Cybernetics, 24, 107-111.
  A multidimensional approach is introduced to the representation of uncertain information in conjunction with the Dempster-Schafer theory. A multidimensional array, called a transition array, is defined, which stores the joint probabilities of the occurences of a set of variables taking values in different sets. Using this array, it is shown how to compute the information regarding the probability of occurences of the variables as certain matrix products.

  

• Bhonske, J. B., Wang, Z., Tamamura, H., Fujii, N., Peiper, S. C., & Trent, J. O. (2005).
  A simple, automated quasi-4D-QSAR, quasi-multi way PLS approach to develop highly predictive QSAR models for highly predictive QSAR models for highly flexible CXCR4 inhibitor cyclic pentapeptide ligands using scripted common molecular modeling tools. QSAR & Combinatorial Science, 24, 620-630.
  A methodology for developing highly predictive (r2>0.9) 3D-QSAR models (q2>0.7) based on sixteen flexible CXCR4 cyclic pentapeptide inhibitors is reported. The effective automated use of common molecular modeling tools such as Macromodel and Sybyl is demonstrated. The recently developed multi-way Partial Least Square (PLS) approach for discovering the bioactive conformers and alignment was used in a quasi-multi-way PLS approach. Twenty-five conformers for each compound were generated by Monte Carlo conformational searches and alignments (seventy five in total) were based on the templates from the three most active compound conformers. These were aligned in Sybyl Molecular Databases and Sybyl Molecular Spreadsheets. All repetitive tasks were automated by use of simple Unix shell, python and Sybyl Programming Language (SPL) scripts. This efficient protocol furnished three 3D-QSAR models with q2 values of 0.714, 0.734 and 0.657 and predictive r2 values of 0.951, 0.990, and 0.956 respectively. The best 3D-QSAR model predicted the biological activities of nine test compounds from all activity ranges within 0.5 log units.

  
   

• Bieber, S. L. (1986).
  A hierarchical approach to multigroup factorial invariance. Journal of Classification, 3, 113-134.
  A procedure is presented which permits the analysis of factor analytic problems in which several groups exist. The analysis incorporates a hierarchical scheme of searching for factorial invariance and is an extension of Meredith's (1964) Method One procedure. By overlaying a contextual frame of reference on a traditional factor analysis solution, it is possible to use this technique to examine structural similarity and dissimilarity between groups. The procedure is exhibited in an example and in addition a comparison is made to discriminant analysis.

• Bijlsma, S. (2000).
  Estimating Rate Constants of Chemical Reactions using Spectroscopy. Unpublished doctoral thesis, University of Amsterdam, The Netherlands. General introduction; Theory of two-way methods; theory of three-way methods; quality assessment of reaction rate constant estimates; description of datasets and experimental set-up; applications of two-way methods; applications of three- way methods; comparison between two-way and three-way methods; the use of constraints in classical curve resolution; general conclusions and future work.

• Bijlsma, S., Boelens, H. F. M., & Smilde, A. K. (2001).
  Determination of rate constants in second-order kinetics using UV-visible spectroscopy. Applied Spectroscopy, 55, 77-83.
  A general method for estimating reaction rate constants of chemical reactions using ultraviolet-visible (UV-vis) spectroscopy is presented. The only requirement is that some of the chemical components involved be spectroscopically active. The method uses the combination of spectroscopic measurements and techniques from numerical mathematics and chemometrics. Therefore, the method can be used in cases where a large spectral overlap of the individual reacting absorbing species is present. No knowledge about molar absorbances of individual reacting absorbing species is required for quantification. The reaction rate constants and the individual spectra of the reacting absorbing species of the two-step consecutive reaction of 3-chlorophenylhydrazonopropane dinitrile,vith 2-mercaptoethanol were estimated simultaneously from UV-vis recorded spectra in time. The results obtained were excellent.

• Bijlsma, S., Boelens, H. F. M., Hoefsloot, H. C. J., & Smilde, A. K. (2002).
  Constrained least squares methods for estimating reaction rate constants from spectrscopic data. Journal of Chemometrics, 16, 28-40.
  Model errors, experimental errors and instrumental noise influence the accuracy of reaction rate constant estimates obtained from spectral data recorded in time during a chemical reaction. In order to improve the accuracy, which can be divided into the precision and bias of reaction rate constant estimates, constraints can be used within the estimation procedure. The impact of different constraints on the accuracy of reaction rate constant estimates has been investigated using classical curve resolution (CCR). Different types of constraints can be used in CCR. For example, if pure spectra of reacting absorbing species are known in advance, this knowledge can be used explicitly. Also, the fact that pure spectra of reacting absorbing species are non-negative is a constraint that can be used in CCR. Experimental data have been obtained from UV-vis spectra taken in time of a biochemical reaction. From the experimental data, reaction rate constants and pure spectra were estimated with and without implementation of constraints in CCR. Because only the precision of reaction rate constant estimates could be investigated using the experimental data, simulations were set up that were similar to the experimental data in order to additionally investigate the bias of reaction rate constant estimates. From the results of the simulated data it is concluded that the use of constraints does not result self-evidently in an improvement in the accuracy of rate constant estimates. Guidelines for using constraints are given.

• Bijlsma, S., Louwerse, D.J., & Smilde, A.K. (1999).
  Estimating rate constants and pure UV-vis spectra of a two-step reaction using trilinear models. Journal of Chemometrics, 13, 311-329.
  This paper describes the estimation of reaction rate constants and pure species UV-vis spectra of the consecutive reaction of 3-chlorophenylhydrazonopropane dinitrile with 2-mercaptoethanol. The reaction rate constants were estimated from the UV-vis measurements of the reacting system using the generalized rank annihilation method (GRAM) and the Levenberg-Marquardt/PARAFAC (LM-PAR) algorithm. Both algorithms can be applied in cases where the contribution of different species in the mixture spectra is of exponentially decaying character. From a single two-way array, two two-way data sets are formed by means of splitting such that there is a constant time lag between the two two-way data sets. By stacking these two two-way data sets, the reaction rate constants can be estimated very easily from the third dimension. GRAM, which is fast and non- iterative, decomposes the trilinear structure using a generalized eigenvalue problem (GEP). The iterative algorithm LM-PAR consists of a combination of the Levenberg-Marquardt algorithm and alternating least squares steps of the PARAFAC model using GRAM results as a set of initial starting values. Pure spectra of the absorbing species were estimated and compared with their measured pure spectra. LM-PAR performed the best, giving the lowest relative fit error. However, the relative fit error obtained with GRAM was acceptable. Since a lot of measurements are based on exponentially decaying functions, GRAM and LM-PAR can have many applications in chemistry.

• Bijlsma, S., Louwerse, D.J., Windig, W., & Smilde, A.K. (1998).
  Rapid estimation of rate constants using on-line SW-NIR and trilinear models. Analytica Chimica Acta, 376, 339-355.
  In this paper, two algorithms are presented to estimate reaction rate constants from on-line short-wavelength near-infrared (SW-NIR) measurements. These can be applied in cases where the contribution of the different species in the mixture spectra is of exponentially decaying character. From a single two-dimensional data set two two-way data sets are formed by splitting the original data set such that there is a constant time lag between the two two-way data sets. Next, a trilinear structure is formed by stacking these two two-way data sets into a three-way array. In the first algorithm, based on the generalized rank annihilation method (GRAM), the trilinear structure is decomposed by solving a generalized eigenvalue problem (GEP). Because GRAM is sensitive to noise it leads to rough estimations of reaction rate constants. The second algorithm (LM- PAR) is an iterative algorithm, which consists of a combination of the Levenberg-Marquardt algorithm and alternating least squares steps of the parallel factor analysis (parafac) model using the GRAM results as initial values. Simulations and an application to a real data set showed that both algorithms can be applied to estimate reaction rate constants in case of extreme spectral overlap of different species involved in the reacting system.

• Bijlsma, S., & Smilde, A. K. (2000).
  Estimating reaction rate constants from a two-step reaction: a comparison between two-way and three-way methods. Journal of Chemometrics, 14, 541-560. In this paper, two different spectral datasets are used in order to estimate reaction rate constants using different algorithms. Dataset 1 consists of short-wavelength near-infrared (SW-NIR) spectra taken in time of the two-step epoxidation of 2,5-di-tert-butyl-1,4-benzoquinone using tert-butyl hydroperoxide and Triton B catalyst. This dataset showed moderate reproducibility. Dataset 2 consists of UV-VIS recorded spectra of the consecutive reaction of 3-chlorophenylhydrazonopropane dinitrile with 2-mercaptoethanol. This dataset showed good reproducibility. Two-way and three-way methods were used in order to estimate the reaction rate constants for both datasets. For the SW-NIR dataset the lowest standard deviations for the reaction rate constants were obtained with a two-way method. The lowest standard deviations for the reaction rate constant estimates for the UV-VIS dataset were obtained with a two-way method which uses spectral information that is known in advance. In this case the pure spectrum of two reacting absorbing species is known in advance and this information was used by the two-way method. For one two-way method and a few three-way methods which do not use spectral information that is known in advance, pure spectra of the reacting absorbing species of the UV-VIS dataset were estimated which showed excellent agreement with the recorded pure spectra. The pure spectra of the reacting absorbing species for the SW-NIR dataset were not estimated, because it was not possible to record the real pure spectra of these species. For both spectral datasets, quality assessment has been performed using a jackknife method.

       

• Bloxom, B. (1968).
  A note on invariance in three-mode factor analysis. Psychometrika, 33, 347-350.
  B. proposes a 'true' factor analysis variant of T3, where the derived factor scores, the scores of the subjects on the combination variables and the errors are random variables rather than matrices of parameters for a finite number of individuals (see also Bentler & Lee, 1978, 1979). Conditions for the invariance across subpopulations for the factor pattern matrices, the core matrix and the residual covariance matrix are discussed.

• Bloxom, B. (1984).
  Tucker's three-mode factor analysis model. In H.G. Law, C.W. Snyder Jr, J.A. Hattie & R.P. McDonald (Eds.), Research methods for multimode data analysis (pp. 104-121). New York: Praeger.
  

  
       

• Bocci, L., Vicari, D., & Vichi, M. (2006).
  A mixture model for the classification of three-way proximity data. Computational Statistics & Data Analysis, 50, 1625-1654.
  Large data sets organized into a three-way proximity array are generally difficult to comprehend and specific techniques are necessary to extract relevant information. The existing classification methodologies for dissimilarities between objects collected in different occasions assume a unique common underlying classification structure. However, since the objects’ clustering structure often changes along the occasions, the use of a single classification to reconstruct the taxonomic information frequently appears quite unrealistic. The methodology proposed here models the dissimilarities in a likelihood framework. The goal is to identify a (secondary) partition of the occasions in homogeneous classes and, simultaneously, a (primary) consensus partition of the objects within each of such classes. Furthermore, a class-specific dimensionality reduction operator is also included which allows to identify classes of occasions such that the within-class variability is minimized. The model is formalized as a finite mixture of multivariate normal distributions and solved by a numerical method based on ECM strategy.

• Boente, G.(2002).
  Influence functions and outlier detection under the common principal components model: A robust approach. Biometrika, 89, 861-875.
  The common principal components model for several groups of multivariate obser-vations assumes equal principal axes but dierent variances along these axes among the groups. Influence functions for plug-in and projection-pursuit estimates under a common principal component model are obtained. Asymptotic variances are derived from them. Outlier detection is possible using partial influence functions.

• Boik, R.J. (1990).
  A likelihood ratio test for three-mode singular values: Upper percentiles and an application to three-way ANOVA. Computational Statistics & Data Analysis, 10, 1-9.
  This paper considers the rank-1 three-mode model for an n₂×n₃×n₁ matrix, Y. In vector form, the model is y = (v₁¤v₂¤v₃)z + e, where y = vec(Y), v_j is an n_j×1 vector of parameters, v_j'v_j = 1 for j = 1, 2, 3, and e~ N(0, s²I). The likelihood ratio test of H0: ? = 0 is given and, employing a Jacobi polynomial expansion, upper percentiles of the null distribution of the test statistic are computed. As an illustration, the results are applied to the problem of testing additivity in unreplicated three-way classifications.

• Boik, R.J., & Marasinghe, M.G. (1989).
  Analysis of nonadditive multiway classifications. Journal of the American Statistical Assocation, 84, 1059-1064.
  This article considers the problems of testing additivity and estimating s² in unreplicated multiway classifications. To model nonadditivity and jointly estimate s², the interaction parameter space must be restricted; otherwise the model is saturated. The parameterization used is a multiway extension of the two- way multiplicative interaction model of Mandel (1971) and Johnson and Graybill (1972a). An exact test of ? = 0 is constructed and an estimator of s² is proposed that can be used when interaction has been detected. The test is an approximation to the likelihood ratio test (LRT) of H₀: ? = 0. Selected percentiles of the null distribution are given for three-way classifications. For large ??, a transformation of the test statistic is shown to be approximately distributed as a noncentral F and can be used to compute the power of the test. The test and estimator are illustrated on a data set.

  

• Bolton, B. (1988).
  Multivariate approaches to human learning. In J.R. Nesselroade & R.B. Cattell (Eds.) Handbook of multivariate experimental psychology. Perspectives on individual differences., pp. 789-819. Plenum Press, New York, NY.
  A comprehensive, mathematically precise learning formulation, called structured learning theory (SLT), is outlined. The relation between factor analysis and learning and the analysis of generalized learning curves by factor analysis are discussed, as well as the three-mode factor analysis of learning performance.

• Bolck, A., Smilde, A. K., & Bruins, C. H. P. (1999).
  Monitoring aged reversed-phase high performance liquid chromatography columns. Chemometrics and Intelligent Laboratory Systems, 46, 1-12.
  In this paper, a new approach for the quality assessment of routinely used reversed-phase high performance liquid chromatography columns is presented. A used column is not directly considered deteriorated when changes in retention occur. If attention is paid to the type and magnitude of the changes, columns often can still be used. Therefore, columns have to be monitored at regular time points. This means that, in the first place, a few well chosen measurements have to be done on the used column. With statistical techniques, Hotelling's T-2 statistic in combination with three-way analysis, the type and magnitude of changes in retention then can be detected. The type of changes can be divided in hydrophobicity changes, selectivity changes and both hydrophobicity and selectivity changes. This paper describes the approach in theory, completed with examples. At the end, a strategy for monitoring during routine use is proposed, which is visualized in a monitoring scheme.

• Booksh, K. S. (around 1997).
  Three-way calibration with hyphenated data. Department of Chemistry and Biochemistry, Arizona State University.
  Three-way calibrations methods, such as the generalized rank annihilation method (GRAM) and parallel factor analysis (PARAFAC), are becoming increasing prevalent tools to solve analytical challenges. The main advantage of three-way calibration is estimation of analyte concentrations in the presence of unknown, uncalibrated spectral interferents. These methods also permit the extraction of analyte, and often interferent, spectral profiles from complex and uncharacterized mixtures. In this tutorial a theoretical and practical overview throughout the progression of three-way calibration methods from the simplest rank annihilation factor analysis (RAFA) to the more flexible PARAFAC is presented. Extensions of many three-way methods are covered to highlight the paradigms flexibility to solve particular analytical calibration problems.

• Booksh, K., Henshaw, J.M., Burgess, L.W., & Kowalski, B.R. (1995).
  A second-order standard addition method with application to calibration of a kinetics-spectroscopic sensor for quantitation of trichloroethylene. Journal of Chemometrics, 9, 263-282.
  Presented here is an algorithm for analysis of second order data by the method of standard additions. The method of standard additions is applicable when matrix effects make traditional calibration unreliable. The algorithm employs a generalized eigenproblem to mathematically separate the instrument response of the analyte from the instrument response of any interfering species. A scheme for determining the eigenvectors (and hence the concentration estimate) that uniquely correspond to the analyte of interest is given. These eigenvectors can readily be distinguished from any eigenvector that corresponds to the spectrum of the interferents or both the interferents and analyte. The stability of the estimated analyte concentration is verified by monte carlo simulations. The algorithm is applied to the analysis of trichloroethylene in samples that have matrix effects caused by an interaction with chloroform.

  ---

• Booksh, K.S., & Kowalski, B.R. (1994a).
  Comments on the DATa ANalysis (DATAN) algorithm and rank annihilation factor analysis for the analysis of correlated spectral data. Journal of Chemometrics, 8, 287-292.
  It is shown that the data analysis (DATAN) algorithm can be expressed in terms of rank annihilation factor analysis (RAFA). Subsequent advances in RAFA are applied to DATAN to eliminate the problems and restrictions associated with DATAN. The extension of DATAN in terms of the trilinear decomposition algorithm is discussed.

• Booksh, K.S., & Kowalski, B.R. (1997).
  Calibration method choice by comparison of model basis functions to the theoretical instrumental response function. Analytica Chimica Acta, 348, 1-9.
  Sorting through the large array of calibration methods available for first and second order calibration is often a daunting task for initiates into the field of chemometrics. Justifying the selected method as the most appropriate one is even more difficult. Presented here is a justification for calibration method selection based on matching the model employed in the calibration method with the instrumental response function. This is applied to the disparate types of nonlinearities found in both first and second order calibration. Matching the calibration method to the instrumental response function is employed to parse the decision making process for choosing between branches in the first order parsimony tree. The different types of nonlinearities present in second order data and their implications on calibration model selection are discussed.

• Booksh, K.S., Lin, Z.H., Wang, Z.Y., & Kowalski, B.R. (1994).
  Extension of trilinear decomposition method with an application to the flow probe sensor. Analytical Chemistry, 66, 2561-2569.
  The trilinear decomposition algorithm (TLD) is a method for calibration of second-order instrumentation (e.g., lc-uv). This method, like the generalized rank annihilation method (GRAM), estimates the intrinsic profiles (e.g., spectra and chromatograms) of each component present in each sample by solving an eigenvector/eigenvalue problem. The relative concentration of each component between the samples is found by the least squares fitting of the intrinsic profiles to the instrument response of the samples. The advantage the TLD algorithm has over GRAM is the ability to analyze data from multiple samples simultaneously. The previously published algorithm provided unreliable calibration estimates when imaginary eigenvectors were included in the solution of the eigenproblem. An improved TLD algorithm is presented to correct this problem. The TLD algorithm is also extended to provide reliable calibration in the case where the instrument response to analyte concentration is nonlinear. This extension assumes the intrinsic profiles of the analyte are identical at all analyte concentrations. The improved and extended TLD algorithm is demonstrated on two simulated data sets as well as the flow optrode analysis of Pb(II) and Cd(II).

• Booksh, K.S., Muroski, A.R., & Myrick, M.L. (1996).
  Single-measurement excitation/emission matrix spectrofluorometer for determination of hydrocarbons in ocean water. 2. Calibration and quantitation of naphthalene and styrene. Analytical Chemistry, 68, 3539-3544.
  An excitation/emission matrix imaging spectrofluorometer was employed for quantitation of two fluorescent compounds, naphthalene and styrene, contained in ocean water exposed to gasoline. Multidimensional parallel factor (Parafac) analysis models were used to resolve the naphthalene and styrene fluorescence spectra from a complex background signal and overlapping spectral interferents not included in the calibration set. Linearity was demonstrated over two orders of magnitude for determination of naphthalene with a detection limit of eight parts per billion. Similarly, nearly two orders of magnitude of linearity was demonstrated in the determination of styrene with an 11 ppb limit of detection. Furthermore, the synthesis of the EEM spectrofluorometer and the Parafac analysis for unbiased prediction of naphthalene and styrene concentration in mixture samples containing uncalibrated spectral interferents was demonstrated.

• Bonnet, N., & Zahm, J. M. (1998).
  Analysis of image sequences in fluorescence videomicroscopy of stationary objects. Cytometry, 31, 217-228.
  Fluorescence videomicroscopy allows the temporal behavior of biological specimens to be studied at the cellular level. We describe two types of methods that can be used for extracting quantitative information from image sequences: the modelling approach, which is mainly local, and multivariate statistical analysis, which provides a global approach. The potentials for use of these two methods are illustrated through a simulation example and actual examples dealing with the study of chloride secretion by airway epithelial cells. We define some guidelines for making a choice between these two approaches, bearing in mind that a blend of these two methodologies is also possible.

• Boqué, R., Larrechi, M. S., & Ruis, F., X. (1999a).
  Multivariate detection limits with fixed probabilities of error. Chemometrics and Intelligent Laboratory Systems, 45, 397-408.
  In this paper, a new approach to calculate multivariate detection limits (MDL) for the commonly used inverse calibration model is discussed. The derived estimator follows the latest recommendations of the International Union of Pure and Applied Chemistry (IUPAC) concerning the detection capabilities of analytical methods. Consequently, the new approach: (a) is based on the theory of hypothesis testing and takes into account the probabilities of false positive and false negative decisions, and (b) takes into account all the different sources of error, both in calibration and prediction steps, which affect the final result. The MDL is affected by the presence of other analytes in the sample to be analysed; therefore, it has a different value for each sample to be tested and so the proposed approach attempts to find whether the concentration derived from a given response can be detected or not at the fixed probabilities of error. The estimator has been validated with and applied to real samples analysed by NIR spectroscopy.

• Boqué, R., Ferré, J., Faber, N. M.& Rius, F. X. (2002).
  Limit of detection estimator for second-order bilinear colibration. Analytic Chimica Acta, 451, 313-321.
  A new approach is developed for estimating the limit of detection in second-order bilinear calibration with the generalized rank annihilation method (GRAM). The proposed estimator is based on recently derived expressions for prediction variance and bias. It follows the latest IUPAC recommendations in the sense that it concisely accounts for the probabilities of committing both types I and II errors, i.e. false positive and false negative declarations, respectively. The estimator has been extensively validated with simulated data, yielding promising results. (C) 2002 Elsevier Science B.V. All rights reserved.

• Boqué, R., & Smilde, A.K. (1999b).
  Monitoring and diagnosing batch processes with multiway covariates regression models. AIChE Journal, 45, 1504-1520.
  Multivariate statistical procedures for monitoring the behavior of batch processes are presented. A new type of regression, called multiway covariates regression, is used to find the relationship between the process variables and the quality variables of the final product. The three-way structure of the batch process data is modeled by means of a Tucker3 or a PARAFAC model. The only information needed is a historical data set of past successful batches. Subsequent new batches can be monitored using multivariate statistical process control charts. In this way the progress of the new batch can be tracked and possible faults can be easily detected. Further detailed information from the process can be obtained by interrogating the underlying model. Diagnostic tools, such as contribution plots of each of the variables to the observed deviation, are also developed. Finally, on-line predictions of the final quality variables can be monitored; providing an additional tool to see whether a particular batch will produce an out-of-spec product. These ideas are illustrated using simulated and real data of a batch polymerization reaction.

• Borg, I., & Lingoes, J.C. (1978).
  What weight should weights have in individual differences scaling? Quality and Quantity, 12, 223-237.
  This paper reanalyzes some data collected by Green and Rao (1972) via PINDIS. The results are compared with those produced by INDSCAL which is a) persently the most popular procedure, and b) also the method of analysis chosen originally by Green and Rao (1972).

  

• Borg, I., & Lingoes, J. (1987).
  Multidimensional Similarity Structure Analysis. New York: Springer- Verlag. (Review by P.M. Kroonenberg)
  Chapter 20 (Individual differences models) contains a discussion of three-way data analysis. In particular various aspects of Procrustes analysis for several configurations are presented and the PINDIS and INDSCAL procedures are explained in some detail. (see also Lingoes & Borg, 1978)

• Borgatti, S. P., & Everett, M. G. (1992).
  Regular blackmodels of multiway, multimode matrices. Social Networks, 14, 91-120.
  Blockmodels are used to collapse redundant elements in a system in order to clarify the patterns of relationships among the elements. Traditional blockmodels define redundancy in terms of structural equivalence. This choice serves many analytic purposes very well, but is inadequate for others. In particular, role systems would be better modeled by blockmodels based on regular equivalence. The first goal of this paper is to generalize blockmodels to incorporate both structural and regular equivalence. Another limitation of traditional blockmodels is that they are defined only for (collections of) 2-way 1-mode adjacency matrices. This excludes common datasets such as actor-by-event, actor-by-organization, item-by-use and case-by-variable matrices. It also excludes 3-way data such as actor-by-actor-by-time or subject-by-verb-by-object matrices. The second goal of this paper is to define blockmodels for multiway, multimode matrices in general. In so doing, we also shift the focus of attention away from the blocking of actors (or other entities) and toward the blocking of ties (or multiway cells).

• Bouroche, J.-M. & Dussaix, A.-M. (1975).
  Several alternatives for three-way data analysis. Metra, 14, 299-319.
  A method called 'Double PCA' is proposed for the analysis of three-way data, say subjects x variables x time points. First PCA is performed on the variables x time points matrix averaged over subjects to assess general trends. Then per time point PCA is performed over the subjects x variables matrix centred per variable. Finally four different procedures are discussed to obtain a 'best' common subject-space for all time points. Plots showing the 'trajectory' of each subject in the common space are given. Illustrated with a study of the French car market.

• Bove, G., & Di Ciaccio, A. (1989).
  Comparisons among three factorial methods for analysing three-mode data. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 103-113). Amsterdam: Elsevier.
  Methods for analusing three-mode data are compared by focusing on their different approaches and models. Some interesting results concerning the different adopted distances in the geometrical representations of the variables are also obtained. An application to French census data is provided to emphasize the differences.

• Bove, G., & Di Ciaccio, A. (1994).
  A user-oriented overview of multiway methods and software. Computational Statistics & Data Analysis, 18, 15-37.
  This paper provides a brief overview of the most widely known methods and software dealing with multiway data. The main features are described focusing on applicative capabilities, in order to make the choices of users easier.

  
       

• Bramston, P., Snyder Jr, C.W., Leah, J.A. & Law, H.G. (1983).
  Assessment of assertiveness in the intellectually handicapped. Multivariate Experimental Clinical Research, 6, 143-159.
  Earlier work in the structural analysis of self-reported difficulty in assertiveness had indicated that individuals differed in terms of a two-facet model - response type (positive vs. negative assertiveness) by referents (close vs. distant interpersonal encounters). This study replicated the individual differences structure for an intellectually handicapped sample, thus extending the generalizability of that model. However, although the dimensions were found in three different methods of assessment, self-report, behavioral rating, and role play, little agreement was found between the methods in accounting for individual profiles. Additionally, there were hints that the four interaction dimensions of assertiveness might actually reflect different difficulty positions on a non-linear unidimensional scale of assertiveness. Using a Rasch model to derive the single scale, role play and self-report were significantly correlated in their assessments, but the correlation was not very great. It was hypothesized that method differences might reflect legitimately different perspectives of close-distant referent raters.

• Bridgman, R.P., Snyder Jr, C.W., & Law, H.G. (1981).
  Individual differences in conceptual behaviour following manipulated controllability. Personality and Individual Differences, 2, 197- 205.
  The present study examined the influence of manipulated controllability on the intrinsic individual differences among 30 female undergraduates in a disjunctive conceptual behavior recovery task. Three-mode factor analysis was used to explore the process variability in a multivariate time-series design. Results indicate that intrinsic task processes were altered by the controllability pretreatment, but the nature of the impact reflected substantial individual differences in reaction.

• Bro, R. (1995).
  Algorithm for finding an interpretable simple neural network solution using PLS. Journal of Chemometrics, 9, 423-430.
  This communication describes the combination of a feedforward neural network (NN) with one hidden neuron and partial least squares (PLS) regression. Through training of the neural network with an algorithm that is a combination of a modified simplex, PLS and certain numerical restrictions, one gains an NN solution that has several feasible properties: (i) as in PLS the solution is qualitatively interpretable; (ii) it works faster than or comparably with ordinary training algorithms for neural networks; (iii) it contains the linear solution as a limiting case. Another very important aspect of this training algorithm is the fact that outlier detection as in ordinary PLS is possible through loadings, scores and residuals. The algorithm is used on a simple non-linear problem concerning fluorescence spectra of white sugar solutions.

• Bro, R. (1996).
  Multiway calibration. Multilinear PLS. Journal of Chemometrics, 10, 47-61.
  A new multiway regression method called N-way partial least squares (N-PLS) is presented. The emphasis is on the three-way PLS version (tri-PLS), but it is shown how to extend the algorithm to higher orders. The developed algorithm is superior to unfolding methods, primarily owing to a stabilization of the decomposition. This stabilization potentially gives increased interpretability and better predictions. The algorithm is fast compared with e.g. PARAFAC, because it consists of solving eigenvalue problems. An example of the developed algorithm taken from the sugar industry is shown and compared with unfold-PLS. Fluorescence excitation-emission matrices (EEMs) are measured on white sugar solutions and used to predict the ash content of the sugar. The predictions are comparable by the two methods, but there is a clear difference in the interpretability of the two solutions. Also shown is a simulated example of EEMs with very noisy measurements and a low relative signal from the analyte of interest. The predictions from unfold-PLS are almost twice as bad as from tri-PLS despite the large number of samples (125) used in the calibration.

• Bro, R. (1997).
  PARAFAC. Tutorial & applications. Chemometrics and Intelligent Laboratory Systems, 38, 149-171.
  This paper explains the multi-way decomposition method PARAFAC and its use in chemometrics. PARAFAC is a generalization of PCA to higher order arrays, but some of the characteristics of the method are quite different from the ordinary two-way case. An important advantage of using multi-way methods instead of unfolding methods is that the estimated models are very simple in a mathematical sense, and therefore more robust and easier to interpret. The applications presented include subjects as: Analysis of variance by PARAFAC, a five-way application of PARAFAC, PARAFAC with half the elements missing, PARAFAC constrained to positive solutions and PARAFAC for regression as in principal component regression.

  

• Bro, R. (1998).
  Multi-way Analysis in the Food Industry. Unpublished doctoral thesis, University of Amsterdam, Amsterdam The Netherlands.
  1. Background
  Introduction; Multi-way analysis; How to read this thesis.
  2. Multi-way data
  Introduction; Unfolding; Rank of multi-way arrays.
  3. Multi-way models
  Introduction; The Kathri-Rao product; Parafac; Parafac2; Paratuck2; Tucker models; Multilinear partial least squares regression; Summary.
  4. Algorithms
  Introduction; Alternating least squares; Parafac; Parafac2; Paratuck2; Tucker models; Multilinear partial least squares regression; Improving alternating least squares algorithm; Summary.
  5. Validation
  What is validation; Preprocessing; Which model to use; Number of components; Checking convergence; Degeneracy; Assessing uniqueness; Influence and residual analysis; Assessing robustness; Frequent problems and questions; Summary.
  6. Constraints
  Introduction; Constraints; Alternating least squares revisited; Algorithms; Summary.
  7. Applications
  Introduction; Sensory analysis of bread; Comparing regression models; Rank- deficient spectral FIA data; Exploratory study of sugar production; Enzymatic activity; Modeling chromatographic retention time shifts.
  8. Conclusion

• Bro, R. (1999).
  Exploratory study of sugar production using fluorescence spectroscopy and multi-way analysis. Chemometrics and Intelligent Laboratory Systems, 46, 133-147.
  This paper is concerned with the possibility of obtaining chemically meaningful models of complicated processes by the use of fluorescence spectroscopy screening and the unique parallel factor analysis (Parafac) model. The second-order nature of fluorescence excitation emission data and the fact that the Parafac model has no rotational indeterminacy mean that in certain cases, it is possible to decompose complex mixture signals into contributions from individual chemical components. Relating the thus obtained information to, e.g., important quality parameters, it is possible to analyze, understand, predict and monitor the quality based on a chemical foundation. The proposed approach thus gives a direct link between process analytical chemistry and multivariate statistical process control.

• Bro, R. (2003).
  Multivariate calibration - What is in chemometrics for the analytical chemist? Analytica Chimica Acta, 500, 185-194.
  Chemometrics has been used for some 30 years but there is still need for disseminating the potential benefits to a wider audience. In this paper, we claim that proper analytical chemistry (1) must in fact incorporate a chemometric approach and (2) that there are several significant advantages of doing so. In order to explain this, an indirect route will be taken, where the most important benefits of chemometric methods are discussed using small illustrative examples. Emphasis will be on multivariate data analysis (for example calibration), whereas other parts of chemometrics such as experimental design will not be treated here. Four distinct aspects are treated in detail: noise reduction; handling of interferents; the exploratory aspect and the possible outlier control. Additionally, some new developments in chemometrics are described.

• Bro, R., & Andersson, C.A. (1998).
  Improving the speed of multiway algorithms. Part II: Compression. Chemometrics and Intelligent Laboratory Systems, 42, 105-113.
  In this paper an approach is developed for compressing a multiway array prior to estimating a multilinear model with the purpose of speeding up the estimation. A method is developed which seems very well-suited for a rich variety of models with optional constraints on the factors. It is based on three key aspects: (1) a fast implementation of a Tucker3 algorithm, which serves as the compression method, (2) the optimality theorem of the CANDELINC model, which ensures that the compressed array preserves the original variation maximally, and (3) a set of guidelines for how to incorporate optional constraints. The compression approach is tested on two large data sets and shown to speed up the estimation of the model up to 40 times. The developed algorithms can be downloaded from http:\\www.models.kvl.dk\source.

• Bro, R., Andersson, C.A., & Kiers, H.A.L. (1999).
  PARAFAC2 - Part II. Modeling chromatic data with retention time shifts. Journal of Chemometrics, 13, 295-309.
  This paper offers an approach for handling retention time shifts in resolving chromatographic data using the PARAFAC2 model. In Part I of this series an algorithm for PARAFAC2 was developed and extended to N-way arrays. It was discussed that the PARAFAC2 model has a number of attractive features. It is unique under mild conditions though it puts fewer restrictions on the data than the well-known PARAFAC1 model. This has important implications for the modeling of chromatographic data in which retention time shifts can be regarded as a violation of the assumption of parallel proportional profiles underlying the PARAFAC1 model. The PARAFAC2 model does not assume parallel proportional elution profiles, but only that the matrix of elution profiles preserve its 'inner- product structure' from sample to sample. This means that the cross-products of the matrix holding the elution profiles in its columns remain constant. Here an application using chromatographic separation based on the molecular size of thick juice samples from the beet sugar industry illustrates the benefit of using the PARAFAC2 model.

• Bro, R., & De Jong, S. (1997).
  A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics, 11, 392-401.
  In this paper a modification of the standard algorithm for nonnegativity constrained linear least squares regression method is proposed. The algorithm is specifically designed for use in multiway decomposition methods like PARAFAC and N-mode principal component analysis. In those methods the typical situation is that there is a high ratio between the number of objects and variables in the regression problems solved. Furthermore, very similar regression problems are solved many times during iterative procedures used. The algorithm proposed is based on the de facto standard algorithm NNLS by Lawson and Hanson, but modified to take advantage of the special characteristics of iterative algorithms involving repeated use of nonnegativity constraints. The principle behind the NNLS algorithm is described in detail and a comparison is made between this standard algorithm and the new algorithm called FNNLS (fast NNLS).

• Bro, R., & Heimdal, H. (1996).
  Enzymatic browning of vegetables. Calibration and analysis of variance by multiway methods. Chemometrics and Intelligent Laboratory Systems, 34, 85-102.
  This paper describes the chemometrical aspects of an investigation of the enzymatic browning of vegetables. Enzymatic browning is caused by polyphenol oxidase, PPO. Kinetic UV/VIS spectra and experimental design variables of PPO incubated samples are used for predicting enzymatic activity and substrate consumption. The mathematical models used are multiway PLS (N-PLS) and fiveway PARAFAC. Both methods are available from Internet in MATLAB code. Throughout the results of the multiway methods are compared to competing methods (PLS, PCR, Tucker, feedforward neural networks, locally weighted regression, ANOVA and others). The result of the investigation is, that the multiway methods have clear advantages with respect to predictions and interpretability, both mathematically and technologically.

• Bro, R., & Jakobsen, M. (2002).
   Exploring complex interactions in designed data using GEMANOVA. Color changes in fresh beef during storage. Journal of Chemometrics, 16, 294-304.
  Data from a severely reduced experimental design are investigated in order to obtain detailed information on important factors affecting the changes in quality of meat during storage under different conditions. It is possible to model the response, meat color, using traditional ANOVA (analysis of variance) techniques, but the exploratory and explanatory value of this model is somewhat restricted owing to the number of factors and the fact that several interactions exist. For those reasons, it is not possible to visualize the model in a simple way and therefore not possible to have a clear overview of the total variation in the data. Using a recently suggested alternative to traditional analysis of variance, GEMANOVA (generalized multiplicative ANOVA), it is possible to analyze the data effectively and obtain a more interpretable solution that enables a simple overview of the whole sampling domain. Whereas traditional analysis of variance typically seeks a model with main effects and as few and simple interactions and cross-products as possible, the GEMANOVA model seeks to describe the data primarily by means of higher-order interactions, albeit in a straightforward way. The two approaches are thus complementary. It is shown that the GEMANOVA model is simple to interpret, primarily because the GEMANOVA structure is in agreement with the nature of the data. It is shown that the GEMANOVA model used is mathematically unique, which leads to attractive simplified ways of interpreting the model. The results presented are the first published results, where the GEMANOVA model is not simply equivalent to an ordinary PARAFAC model, thus taking full advantage of the additional structural power of GEMANOVA. A new algorithm for fitting the GEMANOVA model is developed and is available from the authors.

• Bro, R., & Kiers, H. A. L. (2003a).
  A new efficient method for determining the number of components in PARAFAC models. Journal of Chemometrics, 17, 274-286.
  A new diagnostic called the core consistency diagnostic (CORCONDIA) is suggested for determining the proper number of components for multiway models. It applies especially to the parallel factor analysis (PARAFAC) model, but also to other models that can be considered as restricted Tucker3 models. It is based on scrutinizing the 'appropriateness' of the structural model based on the data and the estimated parameters of gradually augmented models. A PARAFAC model (employing dimension-wise combinations of components for all modes) is called appropriate if adding other combinations of the same components does not improve the fit considerably. It is proposed to choose the largest model that is still sufficiently appropriate. Using examples from a range of different types of data, it is shown that the core consistency diagnostic is an effective tool for determining the appropriate number of components in e.g. PARAFAC models. However, it is also shown, using simulated data, that the theoretical understanding of CORCONDIA is not yet complete. Copyright (C) 2003 John Wiley Sons, Ltd.

• Bro, R., Nielsen, H. H., Stefánsson, G. & Skara, T. (2002).
  A phenomenological study of ripening of salted herring. Assessing homogeneity of data from different countries and laboratories. Journal of Chemometrics, 16, 81-88.
  Data from ripening experiments of herring carried out at three Nordic fishery research institutions in the period 1992-1995 were collected and analyzed by multivariate analysis. The experiments were carried out at different times, with different stocks as raw material, using different types of treatments and analyzed in different laboratories. The question considered here is whether these data can be assumed to be one homogeneous set of data pertaining to ripening of salted herring or whether data from different labs, stocks, etc. must be considered independently. This is of importance for further research into ripening processes with these and similar data. It is shown in this paper that all data can be considered as one homogeneous data set. This is verified using resampling where latent structures are compared between different sample sets. This is done indirectly by testing regression models, that have been developed on one sample set, on other sample sets. It is also done directly by monitoring the deviation in latent structure observed between different sample sets. No formal statistical test is developed for whether samples can be assumed to stem from the same population. Although this can easily be envisioned, it was exactly the need for a more intuitive and visual test that prompted this work, developing different exploration tools that visually make it clear how well the data can be assumed to derive from the same population. Subsequently analyzing the data as one homogeneous group provides new information about factors that govern the ripening of salted herring and can be used in new strategic research as well as in industrial practice.

• Bro, R., Rinnan, A., & Faber, N. K. M. (2005).
  Standard error of prediction for multilinear PLS 2. Practical implementation in fluorescence spectroscopy. Chemometrics and Intelligent Laboratory Systems, 75, 69-76.
  In Part 1 of this series, a new simplified expression was derived for estimating sample-specific standard error of prediction in inverse multivariate regression. The focus was on the application of this expression in multilinear partial least squares (N-PLS) regression, but its scope is more general. In this paper, the expression is applied to a fluorescence spectroscopic calibration problem where N-PLS regression is appropriate. Guidelines are given for how to cope in practice with the main assumptions underlying the proposed methodology. The sample-specific uncertainty estimates yield coverage probabilities close to the stated nominal value. Similar results were obtained for standard (i.e., linear) PLS regression and principal component regression on data rearranged to ordinary two-way matrices. The two-way results highlight the generality of the proposed expression.

• Bro, R., & Smilde, A. K. (2003b).
  Centering and scaling in component analysis. Journal of Chemometrics, 17, 16-33.
  

• Bro, R. & Sidiropoulos, N.D. (1998).
  Least squares algorithms under unimodality and non-negativity constraints. Journal of Chemometrics, 12, 223-247.
  In this paper a least squares method is developed for minimizing //Y-XB'//2 over the matrix B subject to the constraint that the columns of B are unimodal. This method is directly applicable in curve resolution and in improving stability when unimodality is known to be a valid assumption. Unimodality least squares regression turns out to be no more difficult than two simple Kruskal monotone regressions. The method is useful in and exemplified with two- and multiway methods (such as PARAFAC and PARATUCK2) based upon least squares regression solving problems in chromotography and flow injection analysis.

• Bro, R., Sidiropoulos, N. D., & Smilde, A. K. (2002).
  Maximum likelihood fitting using ordinary least squares algorithms. Journal of Chemometrics, 16, 387-400.
  In this paper a general algorithm is provided for maximum likelihood fitting of deterministic models subject to Gaussian-distributed residual variation (including any type of non-singular covariance). By deterministic models is meant models in which no distributional assumptions are valid (or applied) on the parameters. The algorithm may also more generally be used for weighted least squares (WLS) fitting in situations where either distributional assumptions are not available or other than statistical assumptions guide the choice of loss function. The algorithm to solve the associated problem is called MILES (Maximum likelihood via Iterative Least squares EStimation). It is shown that the sought parameters can be estimated using simple least squares (LS) algorithms in an iterative fashion. The algorithm is based on iterative majorization and extends earlier work for WLS fitting of models with heteroscedastic uncorrelated residual variation. The algorithm is shown to include several current algorithms as special cases. For example, maximum likelihood principal component analysis models with and without offsets can be easily fitted with MILES. The MILES algorithm is simple and can be implemented as an outer loop in any least squares algorithm, e.g. for analysis of variance, regression, response surface modeling, etc. Several examples are provided on the use of MILES.

• Bro, R., Smilde, A.K., & De Jong, S. (2001).
  On the difference between low-rank and subspace approximation: improved model for multi-linear PLS Regression. Chemometrics and Intelligent Laboratory Systems, 58, 3-13.
  While both Tucker3 and PARAFAC models can be viewed as latent variable models extending principal component analysis (PCA) to multi-way data, most fundamental properties of PCA do not extend to both models. This has practical importance, which will be explained in this paper. The fundamental difference between the PARAFAC and the Tucker3 model can be viewed as the difference between so-called low-rank and subspace approximation of the data. This insight is used to pose a modification of the multi-linear partial least squares regression (N-PLS) model. The modification is found by exploiting the basic properties of PLS and of multi-way models. Compared to the current prevalent implementation of N-PLS, the new model provides a more reasonable fit to the independent data and exactly the same predictions of the dependent variables. Thus, the reason for introducing this improved model is not to obtain better predictions, but rather the aim is to improve the secondary aspect of PLS: the modeling of the independent variables. The original version of N-PLS has some built-in problems that are easily circumvented with the modification suggested here. This is of importance, for example, in process monitoring, outlier detection and also, implicitly, for jackknifing of model parameters. Some examples are provided to illustrate some of these points.

• Bro R., Workman Jr, J.J., Mobley, P.R. & Kowalski, B. (1997).
  Review of chemometrics applied to spectroscopy: 1985-95, Part 3 - Multi-way analysis. Applied Spectroscopy Reviews, 32, 237-261.
  I. INTRODUCTION. A. Multi-way data; B. Important technology; C. Software; D. Books, reviews, and tutorials.
  II. MULTI-WAY MODELS AND ALGORITHMS. A. PARAFAC/GRAM; B. N-mode PCA; C. Other models; D. Preprocessing.
  III. APPLICATIONS OF MULTI-WAY METHODS. A. Mass spectrometry; B. UV/Visible spectroscopy; C. Fluorescence; D. Other.

  

• Bro, R., Berg, van den, F., Thybo, A., Andersen, C. M., Jorgensen, B. M., & Andersen, H. (2002).
  Multivariate data analysis as a tool in advanced quality monitoring in the food production chain. Trends in Food Science & technology, 13, 235-244.
  This paper summarizes some recent advances in mathematical modeling of relevance in advanced quality monitoring in the food production chain. Using chemometrics - multivariate data analysis - it is illustrated how to tackle problems in food science more efficiently and, moreover, solve problems that could not otherwise be handled before. The different mathematical models are all exemplified by food related subjects to underline the generic use of the models within the food chain. Applications will be given from meat storage, vegetable characterization, fish quality monitoring and industrial food processing, and will cover areas such as analysis of variance, monitoring and handling of sampling variation, calibration, exploration/data mining and hard modeling.

• Brouwer, P., & Kroonenberg, P.M. (1991).
  Some notes on the diagonalization of extended three-mode core matrices. Journal of Classification, 8, 93-98.
  We extend previous results of Kroonenberg and de Leeuw (1980) and Kroonenberg (1983, Ch. 5) on transformations of the extended core matrix of the Tucker2 model (Kroonenberg and de Leeuw 1980). In particular, it is shown that non-singular transformations to diagonalize the core matrix will leac to PARAFAC solutions (Harshman 1970; Harshman and Lundy 1984), if such solutions exist.

• Brown, S.D. (1998).
  Information and data handling in chemistry and chemical engineering: the state of the field from the perspective of chemometrics. Computers and Chemical Engineering, 23, 203-216.
  The basic trends in current researc of chemometrics are reviewed from the perspective of soft modelling. Included is a discussion of the role of second-order and higher calibration methods which make use of three-mode and multimode data. Advantages and limitations of these methods are indicated. The logical connections between efforts made to improve or extend chemometric methods and defects inherent in soft modelling are identified and briefly explored.

• Browne, M.W. (1989).
  Relationships between an additive model and a multiplicative model for multitrait-multimethod matrices. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 507-520). Amsterdam: Elsevier.
  An additive model and a multiplicative model for multitrait-multimethod correlation matrices are described and are related to the Campbell-Fiske conditions. Approximations are provided for the parameters of the multiplicative model in terms of the parameters of the additive model and situations in which the two models coincide are considered. An example where both models are fitted to the same correlation matrix is provided.

       

• Burdick, D.S. (1995).
  Tutorial. An introduction to tensor products with applications to multiway data analysis. Chemometrics and Intelligent Laboratory Systems, 28, 229-237.
  The concepts of tensor algebra and vector space geometry provide a unifying framework for multilinear data analysis which simplifies notation and leads to economy of thought. Avoiding too much abstraction too soon in defining tensor products makes these concepts accessible. Examples are given of the use of tensor algebra in the analysis of bilinear and trilinear models arising in fluorescence spectroscopy.

• Burdick, D.S., Tu, X.M., McGown, L.B., & Millican, D.W. (1990).
  Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission-frequency array. Journal of Chemometrics, 4, 15-28.
  Fluorescence lifetime provides a third independent dimension of information for the resolution of total luminescence spectra of multicomponent mixtures. The incorporation of this parameter into the excitation-emission matrix (EEM) by the phase modulation technique results in a three-dimensional excitation-emission-frequency array (EEFA). Multicomponent analysis based on the three-dimensional EEFA brings a qualitative change for the resolved spectra, i.e. individual spectra can be uniquely resolved, which is impossible with any two-dimensional analysis. In this paper we present a method for analyzing the EEFA. We show mathematically that with the three-dimensional analysis of the EEFA individual spectra and lifetimes can be obtained. Our algorithm is developed in mathematical detail and is demonstrated by its application to a two-component mixture.

• Burgess, L.W. (1995).
  Absorption-based sensors. Sensors and actuators B - Chemical, 29, 10-15.
  Many chemical sensors based on fiber optics and absorption spectroscopy have been reported in applications ranging from biomedical and environmental monitoring to industrial process control. In these diverse applications, the analyte can be probed directly, by measuring its intrinsic absorption, or by incorporating some transduction mechanism such as a reagent chemistry to enhance sensitivity and selectivity. Physical and performance requirements are placed on a device depending on its intended use. In applications such as chemical process monitoring, survivability and the assurance of the long-term quality of the analytical data are paramount. The above needs have resulted in devices that now employ multivariate data analysis, complex sampling interfaces, and reagent renewal mechanisms. The response from such systems can provide information not only about target analyte(s), but can also signal the presence of interferences, and may potentially be used to follow sensor degradation. Examples are given of devices currently being investigated along with a discussion of some of the remaining material, chemical, and optical challenges.

• Burnham, A.J., Macgregor, J.F. & Viveros, R. (1999).
  A statistical framework for multivariate latent variable regression methods based on maximum likelihood. Journal of Chemometrics, 13, 49-65.
  A statistical framework is developed to contrast methods used for parameter estimation for a latent variable multivariate regression (LVMR) model. This model involves two sets of variables, X and Y, both with multiple variables and sharing a common latent structure with additive random errors. The methods contrasted are partial least squares (PLS) regression, principal component regression (PCR), reduced rank regression (RRR) and canonical co-ordinate regression (CCR). The framework is based on a constrained maximum likelihood analysis of the model under assumptions of multivariate normality. The constraint is that the estimates of the latent variables are restricted to be linear functions of the X variables, which is the form of the estimates for the methods being contrasted. The resulting framework is a continuum regression that goes from RRR to PCR depending on the ratio of error variances in the X and Y spaces. PLS does not arise as a member of the continuum; however, the method does offer some insight into why PLS would work well in practice. The constrained maximum likelihood result is also compared with the unconstrained maximum likelihood analysis to investigate the impact of the constraint. The results are illustrated on a simulated example.

  
  

• Burnham, A. J., Viveros, R., & Macgregor, J. F. (1996).
  Frameworks for latent multivariate regressin. Journal of Chemometrics, 10, 31-45.
  A set of frameworks for latent variable multivariate regression method is developed. The first two of these frameworks describe the objective functions satisfied by the latent variables chosen in canonical coordinates regression (CCR), reduced rank regression (RRR) and SIMPLS. These frameworks show the methods as a natural progression from CCR (maximizing correlation) to SIMPLS (maximizing covariance) via RRR (which is an intermediate method). These frameworks are unique in that they look at these methods in terms of latent variables in both the X- and Y-spaces. This adds insight to the nature of the latent variables being chosen. These frameworks are then extended to include PLS for latent variables beyond the first component. This new framework provides a detailed description ofthe objective function satisfied by PLS latent variables for the multivariate case. It also includes CCR, RRR and SIMPLS, allowing comparisons between the methods. A further framework suggests a new method, undeflated PLS (UDPLS), which adds insight to the effect of the deflation process on PLS. The impact of the objective functions on each of the methods is illustrated on real data from a mineral sorting plant.

  
         

• Bylund, D., Danielsson, R., Malmquist, G., & Markides, K. E. (2002).
  Chromatographic alignment by warping and dynamic programming as a pre-processing tool for PARAFAC modelling of liquid chromatography-mass spectrometry data. Journal of Chromatography A, 961, 237-244.
  Solutes analysed with LC-MS are characterised. by their retention times and mass spectra, and quantified by the intensities measured. This highly selective information can be extracted by multiway modelling. However, for full use and interpretability it is necessary that the assumptions made for the model are valid. For PARAFAC modelling, the assumption is a trilinear data structure. With LC-MS, several factors, e.g. non-linear detector response and ionisation suppression may introduce deviations from trilinearity. The single largest problem, however, is the retention time shifts not related to the true sample variations. In this paper, a time warping algorithm for alignment of LC-MS data in the chromatographic direction has been examined. Several refinements have been implemented and the features are demonstrated for both simulated and real data. With moderate time shifts present in the data, pre-processing with this algorithm yields approximately trilinear data for which reasonable models can be made.

   

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