Three-Mode Abstracts, Part L

INDEX

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• Ladefoged, P., Harshman, R., Goldstein, L., & Rice, L. (1978).
  Generating vocal tract shapes from formant frequencies. Journal of the Acoustical Society of America, 64, 1027-1035.
  An algorithm that uses only the first three formant frequencies has been devised for generating vocal tract shapes as seen on midsagittal x-ray diagrams of most English vowels. The shape of the tongue is characterized in terms of the sum of two factors derived from PARAFAC analysis: a front raising component and a back raising component. Stepwise multiple regression techniques were used to show that the proportions of these two components, and of a third parameter corresponding to the distance between the lips, are highly correlated with the formant frequencies in 50 vowels. The recovery algorithm developed from these correlations was tested on a number of published sets of tracings from x-ray diagrams, and appears to be generalizable to other speakers.

• LaFaye, J.Y. (1985).
  Application du modèle INDSCAL. Statistique et Analyse de Données, 10, 83-102.
  According to the frame of GRECO CNRS ANADI work, that aims at comparing several statistical approaches in the study of evolutive data, the authors present an application of the INDSCAL model to the analysis of a series of contingency tables. The data refer to the distribution of the S.P.C. among 42 Languedoc-Roussilon rural counties. These data have been collected for the 1954, 1962, 1968 and 1975 population census. A prior analysis, published in the "Bulletin de la société languedocienne de géographie" dealing with the first three years is given in the bibliography and will be used as a reference.

• LaFosse, R. (1985).
  Generalized redundancy analyses and maximal agreement. (Research Report), Toulouse: University Paul-Sabatier.
  Two analyses of occurrences of a set of quantitative variables are developed in order to explain an occurrence from the others. Depending on whether pairing of the variables is taken into account or not, these provide generalizations of reduced rank regression or explanatory analysis built on agreement and redundancy, or redundancy alone. This work is justified by a simplification of the crossed ties between an occurrence and the other occurrences. The underlying linear models can be used, among other things, to describe sensorial data or to reduce the multivariate prediction problem to several univariate ones.

• LaFosse, R. (1997). Analyse de concordance de deux tableaux: Monogames, simultanéités et découpages. Revue de Statistique Appliquée, 45, 45- 72.
  The components of factor analyses of one or two matrices are defined using a zero-correlation property rather than through an optimization problem. For two matrices this yields new arguments for simultaneous simple regression. The variation in a table analysed with respect to its concordance with another is partitioned into three parts indicating concordance, discordance and noise. Such a partition relative to the prediction potential of a matrix gives the concordance for the explicative part. If applied to the comparison of two multivariate distributions, the discordance can be viewed as a sort of stereo effect. For each part, measures of the importance of the contributions of the individuals and partial participations of the variables are developed and a biplotting technique described.

• LaFosse, R., & Hanafi, M. (1997).
  Concordance d'un tableau avec K tableaux: Définition de K + 1 uples synthétiques. Revue de Statistique Appliquée, 45, 111-126.
  The interbatter Tucker's analysis (1958) is extended over two matrices with an aymmetric viewpoint, one of the matrices being a reference matrix. A particular choice of the reference matrix leads to a symmetric analysis close to known analyses.

• Lalonde, R. N., Stroink, M. L., & Aleem, M. R. (2002).
  Representations and preferences of responses to housing and employment discrimination. Group Processes & Intergroup Relations, 5, 83-102.
  In two studies, the behavioral preferences of majority (White) and visible minority (non-White) individuals in response to a hypothetical situation of discrimination were examined. In addition, the characteristics and dimensions perceived to relate to these behaviors were also examined. In the first study, 120 primarily White undergraduate students first rated the likelihood of engaging in each of 14 behaviors in response to a situation of discrimination, and then rated each behavior on a number of attributes representing key dimensions of behavior identified in intergroup theories (individual–collective; active–passive; non-normative–normative) and phenomenological studies on the experience of discrimination (e.g. risk). A multimode factor analysis of the behaviors and attributes provided a three-component solution. While the dimensions underlying these components reflected dimensions of behavior identified by intergroup theorists, they were also qualitatively different from them. Further analysis revealed that behaviors associated with higher preference ratings were perceived as more normative, preparatory, and low in cost and risk. The behavioral preferences, and the dimensions underlying these preferences were replicated in a second study, which comprised 70 Black and South Asian participants. The patterns of results were similar for the White and non-White participants, although these two groups did differ in their endorsement and ratings of some of the behaviors.

• Lammers, C.J. (1974).
  Groei en ontwikkeling van de ziekenhuisorganisaties in Nederland. Interimrapport, Institute of Sociology, University of Leiden, Leiden, The Netherlands.
  To assess the growth of the organizational structure of 188 Dutch hospitals, data on 27 variables were collected over 11 years. The T3 analysis is discussed and presented in great detail. The results are related to the average values of the variables over time.

  

• Landis, D., & O'Shea, W.A. (2000).
  Cross-cultural aspects of passionate love: An individual differences analysis. Journal of Cross-cultural Psychology, 31, 752-777.
  Recent work has suggested that passionate love may be conceived of within cultures as an emic that may consist of several dimensions. This study explored the factor dimensionality and cultural relativity of passionate love using the Passionate Love Scale (PLS) with data from 9 samples from North America, Europe, the Middle East, and a Pacific Island and analyzed with 3 Mode Factor Analysis with Point-of-View solutions (3M-POV). A 6-factor group-common structure was found to best explain variance in PLS responses collapsed across cultures. 3M-POV hierarchical clustering procedures yielded 6 idealized cultures separated by gender. Principal component analysis of PLS responses by idealized culture groups revealed unique factor structures for each of these groups. These results suggest passionate love to be a multifactorial construct uniquely defined within cultures.

• Langeheine, R. (1982).
  Statistical evaluation of measures of fit in the Lingoes-Borg Procrustean individual differences scaling. Psychometrika, 47, 427-442.
  PINDIS, as recently presented by Lingoes and Borg (1978) not only marks the latest development within the scope of individual differences scaling, but, may be of benefit in some closely related topics, such as target analysis. Decisions on whether the various models available from PINDIS fit fallible data are relatively arbitrary, however, since a statistical theory of the fit measures is lacking. Using Monte Carlo simulation, expected fit measures as well as some related statistics were therefore obtained by scaling sets of 4(4)24 random configurations of 5(5)30 objects in 2, 3, and 4 dimensions (individual differences case) and by fitting one random configuration to a fixed random target for 5(5)30 objects in 2, 3, and 4 dimensions (target analysis case). Applications are presented.

• Langlois, A. (1983).
  Les transformations de l'espace social de la ville: Une application de l'analyse factorielle à trois entrées. Le Géographe Canadien, 27, 67-73.
  Taylor and Parkes (1975) aimed to understand to what extent the social geography of a city is a dynamic phenomenon which can be analysed both in terms of location in the city and time of day. In this paper, Taylor and Parkes' problem is re-analysed with three-mode component analysis. The results correspond to a large extent with those of Taylor and Parkes, but it also allowed a good synthesis of the spatial and temporal descriptions and their interconnectedness.

• Langsrud, O., & Naes, T. (1995).
  On the structure of PLS in orthogonal designs. Journal of Chemometrics, 9, 483-487.
  This paper presents an interpretation of PLS applied to orthogonal explanatory variables. In particular, it is shown that in fractional factorial multiresponse experiments PLS2 gives identical results to ordinary least squares applied to principal components of the response variables. The general relationship is that the reduced-rank regression algorithm which first projects Y onto the X-space and then truncates this matrix by principal component analysis before performing ordinary least squares estimation gives the same predictor as PLS2 and SIMPLS if all the non-zero eigenvalues of XTX are identical.

• Lappe, H. (1991).
  Perceptions of drug risk: Individual and public responses to drugs. Unpublished doctoral thesis. Technische Universität, Berlin.
  1. Introduction:
  The relevance of drug risk perception; A frame: The psychometric paradigm for the analysis of drug risk perception; Previous research; New approaches - The perspective; Objectives.
  2. Current state of research:
  The definition of risk; General findings of research on risk perception; Risk perception of pharmaceuticals; On multivariate approaches in risk perception research.
  3. Conceptual frame and hypotheses.
  4. Analysis of data:
  Description of database; Preparation of data; Results of TUCKALS2 and 3 analyses; subgroup analyses of optimal TUCKALS2 solution and TUCKALS2 analyses of specified subgroups; Analysis of behavioral intentions, informational needs, and trust in the pharmaceutical industry; Analyses of TUCKALS subgroup data; Analysis of concept group data.
  5. Summary and discussion.

• Lastovicka, J.L. (1981).
  The extension of component analysis to four-mode matrices. Psychometrika, 46, 47-57.
  L. presents the direct generalization of Tucker's (1966) model to four-mode data. The computational procedure is the direct analogue of Tucker's Method I. The illustration is taken from advertising (27 subjects, 5 exposure occasions, 6 advertise- ments, 16 items). All component matrices were varimax rotated.

• Latkoczy, C., Hutter, H., Grasserbauer, M., & Wilhartitz, P. (1995).
  Classification of secondary-ion mass-spectrometry (SIMS) micrographs to characterize chemical-phases. Mikrochimica Acta, 119, 1-12.
  This work demonstrates the potential of multivariate image analysis methods in the extraction of useful, problem dependent information from SIMS images. Specific algorithms have been developed to classify SIMS micrographs manually as well as automatically. A feature selection has been achieved by means of principal component analysis with a subsequent image classification. As an application example for these improved digital image processing tools chemical phases within a soldered industrial metal sample have been identified. This is of highly practical value as it was assumed that during the soldering process inhomogeneities occur along the joint site which cause a cracking of the brazed material under mechanical strain conditions.

• Lavit, C. (1985).
  Application de la methode STATIS. Statistique et Analyse de Données, 10, 103-116.
  STATIS is a data-analytic method which constructs graphs for the information contained in a set of individuals by variables matrices. In particular, the following plots are made: a plot where each matrix is represented as a point, a plot of the individuals based on all matrices, and a plot which shows the differences for each individual in his position in all matrices.

• Lavit, C. (1986).
  Analyse conjointe de plusieurs tableaux: Exemple d'utilisation du programme STATIS de la bibliothèque MODULAD [Simultaneous analysis of several data matrices. An example of the use of the STATIS program as incorporated in the MODULAD software library]. Paper presented at the Statistical Meeting of the association for Methodology and Research in Psychiatry, Lille.
  After a short introduction into the STATIS method, an extensive discussion is presented of the input and output of the STATIS program. The working of the program is illustrated with an example of patients with depression symptoms which are measured several times and which have been given different treatments.

• Lavit, C., & Pernin, M.-O. (1987).
  Multivariate and longitudinal data on growing children: Solution using STATIS. In J. Janssen, F. Marcotorchino & J. M. Proth (Eds.), Data analysis. The ins and outs of solving real problems, (pp. 13-29). New York: Plenum.
  These data can be taken as a succession of twelve similar data matrices. The first one describes in a multivariate way the morphology of the population at the fourth birthday, the fifth birthday, and so on until the fifteenth birthday. STATIS is a good tool to extract information from this data and gather it by means of graphic output. The result is an overall description on a plot where each point represents a year, a plot of compromise individuals in two or three axes which can be explained by the variables and, in addition, each individual's evolution around its compromise point can be followed on this plot.

• Lavit, C. (1988).
  Analyse conjointe de tableaux quantitatifs [Simultaneous analysis of several quantitative matrices]. Paris: Masson.
  1. General remarks on the STATIS method.
  2. Linear functions in euclidean vector-spaces.
  3. Non-centred principal component analysis of a set of weighted points.
  4. Theoretical description of the STATIS method.
  5. Programming the STATIS method.
  6. Detailed explanation of the results generated by the STATIS method, applied to growth data.
  7. Economic development of Spanish provinces between 1960 and 1979.
  8. Typing the rural cantons of Herault, described by the changes in the working population between 1954 and 1982.

• Lavit, C., Escoufier, Y., Sabatier, R., & Traissac, P. (1994).
  The ACT (STATIS method). Computational Statistics & Data Analysis, 18, 97-119.
  ACT (STATIS method) is a data analysis technique which computes Euclidean distances between configurations of the same observations obtained in K different circumstances, and thus handles three-way data as a set of K matrices. In this article, the recent developments of the ACT technique are fully described - concepts and theorems related to Euclidean scaling being discussed in the Appendix - and the software manipulation is illustrated on real data.

  

• Lavit, C., & Roux, C. (1984).
  Analyse conjointe de plusieurs tableaux de données par la méthode STATIS [Simultaneous analysis of several datamatrices using the STATIS method]. (Technical report No. 8402), Montpellier: Unité de Biométrie.
  After a short introduction into the STATIS method, an extensive discussion is presented of the input and output of the STATIS program. The working of the program is illustrated with data from four censuses of districts in the Languedoc-Rouisillon.

• Law, H.G. & Snyder Jr C.W. (1979).
  Three-mode models for the analysis of psychological data. Australian Psychologist, 14, 214 (conference abstract).
  The paper gives a non-technical overview of recently developed three-mode techniques for the analysis of psychological data: T3, Harshman's PARAFAC, INDSCAL, ALSCAL, PINDIS, a version of Analysis of Covariance Structures. The techniques are discussed with reference to simple examples.

• Law, H.G. & Snyder Jr, C.W. (1981).
  An introduction to the analysis of covariance structures: A general model for data analysis. In J.M. Morris (Ed.), Proceedings of a seminar on measuring social behaviour in road research. Vermont South, Vic., Australia: Australian Road Research Board (Pp. 49-60).
  The general approach of McDonald (1978) for the analysis of covariance structures is shown to include Tucker's (1966) common factor model as well.

  

• Law, H.G., Snyder Jr, C.W., Hattie, J.A., & McDonald, R.P. (Eds.) (1984).
  Research methods for multimode data analysis. New York: Praeger.
  Part I: Introduction
  C.W. Snyder Jr, H.G. Law & J.A. Hattie; J.B. Kruskal.
  Part II: Factor-Analytic Tradition
  P.M. Kroonenberg; B. Bloxom; R.A. Harshman & M.E. Lundy; idem; R.P. McDonald; H. Swaminathan; R.B. Cattell, D.D. Blaine & J.M. Brennan.
  Part III: Multidimensional Scaling Tradition.
  J.D. Carroll & S. Pruzansky; J.C. Lingoes & I. Borg; F.W. Young; Y. Takane; J.D. Carroll & P. Arabie.
  Part IV: Reflections.
  " Destructive retrieval in the realm of three-mode thinking", by P. Gould J.A. Hattie, R.P. McDonald, C.W. Snyder Jr, & H.G. Law.
  Appendices.
  R.A. Harshman; " Multidimensional Scaling Displays", by J.C. Gower; R.A. Harshman & W. DeSarbo.

• Lawes, R. A., Wegener, M. K., Basford, K. E., & Lawn, R. J. (2004).
  The evaluation of the spatial and temporal stability of sugarcane farm performance based on yield and commercial cane sugar. Australian Journal of Agricultural Research. 55, 335-344.
  In broader catchment scale investigations, there is a need to understand and ultimately exploit the spatial variation of agricultural crops for an improved economic return. In many instances, this spatial variation is temporally unstable and may be different for various crop attributes and crop species. In the Australian sugar industry, the opportunity arose to evaluate the performance of 231 farms in the Tully Mill area in far north Queensland using production information on cane yield (t/ha) and CCS ( a fresh weight measure of sucrose content in the cane) accumulated over a 12-year period. Such an arrangement of data can be expressed as a 3-way array where a farm x attribute x year matrix can be evaluated and interactions considered. Two multivariate techniques, the 3-way mixture method of clustering and the 3-mode principal component analysis, were employed to identify meaningful relationships between farms that performed similarly for both cane yield and CCS. In this context, farm has a spatial component and the aim of this analysis was to determine if systematic patterns in farm performance expressed by cane yield and CCS persisted over time. There was no spatial relationship between cane yield and CCS. However, the analysis revealed that the relationship between farms was remarkably stable from one year to the next for both attributes and there was some spatial aggregation of farm performance in parts of the mill area. This finding is important, since temporally consistent spatial variation may be exploited to improve regional production. Alternatively, the putative causes of the spatial variation may be explored to enhance the understanding of sugarcane production in the wet tropics of Australia.

  
         

• Leah, J.A., Law, H.G. & Snyder Jr, C.W. (1979).
  The structure of self- reported difficulty in assertiveness: an application of three-mode common factor analysis. Multivariate Behavioral Research, 14, 443-462.
  T3 used in the development of assertiveness inventory of 56 items (considered as a completely crossed two-facet design of 7 referents and 8 response classes). Used on two samples of 140 and 130 subjects respectively. Varimax transformation of first sample used on second sample to test congruence. High overall and detailed agreement between the structure of the two samples. Uses Tucker's (1966) Method III.

• Leardi, R., Armanino, C., Lanteri, S., & Alberotanza, L. (2000).
  Three-mode principal component analysis of monitoring data from Venice lagoon. Journal of Chemometrics, 14, 187-195.
  A data set obtained by 44 monthly determinations of 11 variables from 13 sampling sites in the Venice lagoon has been treated by three-mode principal component analysis. The results show that the sampling sites are grouped according to their geographical location, following an inner-outer lagoon direction. In terms of sampling periods, a very strong seasonal effect has been detected, together with an almost linear decrease in nutrients (P and NO^- ₃) and increase in eutrophication.

• Le Bihan, N., & Ginolhac, G. (2004).
  Three-mode data set analysis using higher order subspace method: application to sonar and seismo-acoustic signal processing. Signal Processing, 84, 919-942.
  In this paper, a three-mode subspace technique based on higher order singular value decomposition (HOSVD) is presented. This technique is then used in the context of wave separation. It can be regarded as the extension to three-mode arrays of the well-known subspace technique proposed by Eckart and Young (Psychometrica 1 (1936) 211) for matrices. Three-mode data sets are increasingly encountered in signal processing and are classically processed using matrix algebra techniques. The proposed approach aims to process naturally three-mode data with multilinear algebra tools. So in the proposed algorithms, the structure of the data set is preserved and no reorganization is performed on it. The choice of HOSVD for subspace method is explained, studying the rank definition for three-mode arrays and orthogonality between subspaces. A projector formulation for three-mode signal and noise subspaces is also given and the improvement of separation with the three-mode approach over a componentwise approach is shown. We study two applications for the proposed Higher Order Subspace approach: the reverberation problem in sonar, and the polarized seismo-acoustic wave separation problem. For the first application, we propose a three-mode version of the Principal Component Inverse algorithm (IEEE Trans. Aerospace Electron. Systems 30(l) (1994) 55). We apply the proposed technique on simulated data as well as on real sonar data where the three modes are angle, delay and distance. For the second application, we consider the polarization of the seismic wave as the third mode (in addition to time and distance modes) and show the resulting improvement of wave separation using the proposed Higher Order approach.

• Lechevalier, F. (1987).
  L'Analyse de l'évolution dans STATIS: Une solution et des généralisations. (Technical report), Lille: University of Science and Technology, Faculty of Economic and Social Sciences.
  In the simultaneous analysis of multiple data matrices, it is essential to be able to make the changing patterns of the individuals or of the variables from one matrix to another, explicit. A method is proposed which permits the description of all the individuals in all the matrices (or all the variables) in the same space. On the basis of this it is possible to determine a projection in the sub-space of dimension s. For two dimensions the changing patterns may be described by trajectories in a plane. Even though it is inspired by STATIS, the method described is an independent development. It may therefore be used instead of or complementary to STATIS.

• Lechevalier, F. (1990a).
  L'Analyse en composantes conjointes d'une famille de triplets indexés. Statistique et Analyse des Données, 15, 35-75.
  Let {X_k, k element of K} be a set of data matrices. Each X_k is an n rows and p_k columns (variables) matrix. We are looking for a simultaneous representation of the objects in a common space. A method is presented which, applied to a set of data matrices, allows a generalization of PCA results. For each set of objects an exact mapping is defined, and the simultaneous representation takes into account and respects the properties of duality.

• Lee, C.-H. (1988).
  Multilinear analysis of fluorescence spectra of photosynthetic systems. Unpublished doctoral thesis, The Ohio State University, Columbus, OH.
  I. Introduction.
  Photosynthesis; Fluorescence and photosynthesis; Absorption and fluorescence spectra of CP complexes; Effect of Cations on Chlorophyll fluorescence; Photosynthetic systems in cyanobacteria; Multilinear analysis; Multilinear analysis in spectroscopy; Statement of the problem.
  II. Theory.
  Two-way data: singular value decomposition (SVD) and iterative SVD, combined vector (COV) algorithm; Three-way data; Four-way data; Successive Overrelaxation; Data arrays with missing data or unequal data; Constraints; Convergence criteria.
  III. Experimental methods.
  Spectra acquisition and data analysis; Sources of error in data collection: background noise, backlash, cutwidth, stray light and temperature effect; Standard dyes and reagents; Deazaflavin; Growth of plants and isolation of chloroplasts; Growth of cyanobactera and Calium ion depletion; Proteins.
  IV. Performance of two-way codes.
  Successive overrelaxation; Comparison with other algorithms.
  V. Artificial spectra.
  Model dicrimination.
  VI. Fluorescent dyes.
  Single dye solution: two -way and three-way analysis; three-daye mixture; discussion.
  VII. Deazaflavins.
  VIII. Photosynthetic systems of peas.
  Steady state: Effects of Mg2+; Steady state: Effect of temperature, Four- way analysis; More components?: Four-way analysis; Kinetics; Temperature effect on the kinetics; Discussion.
  IX. Photosynthetic systems of cyanobacteria.
  X. Aromatic residues in proteins.
  Human serum albumin; Bovine serum albumin; Spinach plastocyanin.
  XI. Plastocyanin absorption data.
  Two-way analysis; Three-way analysis; Four-way analysis; A new data set.
  XII. Data collection strategies.
  A 'cross-hatch' pattern data collection; Importance of peaks; Importance of antistokes.
  XIII. Conclusion.

• Lee, C.-H., Kim, K., & Ross, R.T. (1991).
  Trilinear analysis for the resolution of overlapping fluorescence spectra. Korean Biochemical Journal, 24, 374-379.
  The information contained in vivo fluorescence spectra of biological specimens is often obscured by severe overlapping of component spectra. Fluorescence intensity is seperately linear in functions of excitation wavelength, emission wavelength, and any chemical treatment which alters overall fluorescence yield. When data are collected for several values of each of these three independent variables, strongly overlapping spectra can be resolved without the use of any a priori information about their shapes. Using concentration of a fluorescence quencher, potassium iodide, as a third variable, a mixture of rhodamine 6G, rhodamine B, and sulforhodamine 101 was used as a test system; the spectra resolved agreed closely with the spectra of the pure dyes. Previously published applications of trilinear analysis to luminescence spectrosopy have required either an instrument capable of time resolution or multiple specimens with varying relative concentrations of fluorophores; use of quencher concentration as a variable permits use of the method with a steady-state fluorometer and withour changing fluorophore concentrations. Requiring that the spectra deduced be non-negative can increase the accuracy of the results.

• Lee, J.K., & Ross, R.T. (1998)
  Absorption and Fluorescence of Tyrosine Hydrogen-Bonded to Amide-like Ligands Journal of Physical Chemistry, 102, 4612-4618.
  The absorption and steady-state fluorescence of N-acetyl-L-tyrosinamide (NAYA), chosen to model the tyrosine in proteins, was measured in four solvents in the presence of N,N-dimethylacetamide, N-methulacetamide, and urea, chosen as ligands that model the amide of a protein backbone. The data were fit to a multilinear mathematical model to resolve the overlapping spectra of NAYA with and without ligand. The ligand binding constant was between 0.18 and 4 M-1, increasing with solvent polarity; lack of a strong dependence of binding constant on N-methylation suggests that the carbonyl oxygen is responsible for the hydrogen bonding. The emission spectrum of NAYA was essentially identical in all solvents and with and without added ligand. In contrast, the excitation spectrum shifted by up to 10 nm, depending on both solvent and ligand; this shift is described as a sum of three terms: a blue- shift due to hydrogen bonding to the phenolic oxygen which is proporional to ligand donor acidity, a red-shift due to hydrogen bonding to the phenolic hydrogen which is proportional to ligand acceptor basicity, and a blue-shift proportional to solvent dielectric effect. The extinction coefficient varies by up to 40%, depending on solvent and complex formation. The fluorescence quantum yield of the hydrogen-bonded complex varies between 0.102 and 0.044, increasing slightly with N-methylation, but more dependent on solvent. In methanol, acetone, and dioxane, the amide-model complex has a 3-10 times lower fluorescence quantum yield than that of NAYA in pure solvent; in water the complex has a higher quantum yield. Complex formation did not explain all of the fluorescence quenching by ligand in water and in dioxane, suggesting that the ligand also causes dynamic quenching of the excited state in these solvents. Many of the experimental findings are in good agreement with semiempirical molecular orbital calculations.

• Lee, J.K., Ross, R.T., Thampi, S., & Leurgans, S. (1992).
  Resolution of the properties of hydrogen-bonded tyrosine using a trilinear model of fluorescence. Journal of Physical Chemistry, 96, 9158-9162.
  The fluorescence of any dilute specimen is separately linear in functions of each of the independent variables excitation wavelength, emission wavelength, and any treatment which alters concentration or fluorescence quantum yield. The resulting trilinear models have a structure that permits the mathematical dissection of spectra from complex specimens without the use of any other information about the properties of the specimen. Using this technique, the steady-state fluorescence at aqueous n-acetyl-l-tyrosinamide in the presence of five proton acceptors was resolved into three components, corresponding to normally solvated side chain, side chain hydrogen-bonded to the added proton acceptor, and impurity. The excitation spectrum of the hydrogen-bonded complex is red-shifted about 2 nm. The emission maximum of the complex is 310 nm for phosphate monoanion, 320 nm for acetate, 330 run for phosphate bianion, and 335 nm for imidazole and tris(hydroxymethyl)aminomethane. All binding constants for complex formation and reciprocal quenching constants for fluorescence are between 0.2 and 0.3 m, except for phosphate monoanion, for which both constants are 3.3 m. The temperature dependence of the binding constants is small, giving a deltah for complex dissociation between -0.8 and 0.0 kcal/mol.

• Lee, S. Y. (1979).
  Constrained estimation in covariance structure-analysis. Biometrika, 66, 539-545.
  In covariance structure analysis, the method of weighted least squares for estimating parameters that are subject to functional constraints is developed. Statistical properties of the estimator are studied and a goodness-of-fit statistic is presented. Based on the penalty function technique, an algorithm is developed. This algorithm is able to produce not only the constrained weighted least squares estimator but also the constrained maximum likelihood estimator, if we choose an appropriate weight matrix iteratively. The feasibility of the proposed algorithm is demonstrated by an example in factor analysis.

• Lee, S.-Y., & Fong, W.-K. (1983).
  A scale invariant model for three-mode factor analysis. British Journal of Mathematical and Statistical Psychology, 36, 217-223.
  The statistical model for three-mode factor analysis developed by Bentler & Lee (1979) represented a model for the covariance structure of random variables. In this model, arbitrary rescalings of the variables will destroy the covariance structure. A slightly modified model is proposed in this paper to handle the problem. A statistical development of the model is discussed based on the generalized least squares approach. Estimates, standard error estimates and a goodness-of-fit statistic are obtained via the Gauss-Newton algorithm. It is shown that either the sample correlation matrix or the sample covariance matrix can be used for estimation. An example is presented to illustrate the theory.

  

• Leenen, I., Van Mechelen, I., & De Boeck, P. (1999).
  INDCLAS: A three- way hierarchical classes model. Psychometrika, 64, 9-24.
  A three-way three-mode extension of De Boeck and Rosenberg's (1988) two-way two- mode hierarchical classes model is presented for the analysis of individual differences in binary object × attribute arrays. In line with the two-way hierarchical classes model, the three-way extension represents both the association relation among the three modes and the set-theoretical relations among the elements of each mode. An algorithm for fitting the model is presented and evaluated in a simulation study. The model is illustrated with data on psychiatric diagnosis. Finally, the relation between the model and extant models for three-way data is discussed.

• Lefkovitch, L.P. (1993).
  Consensus principal components. Biometrical Journal, 35, 567-580.
  From the polar forms of principal components correspondingn with each of a set of covariance (or correlation) matrices, a linear combination based on their inner products is defined as the polar form of the consensus. The corresponding eigenvectors form an orthogonal matrix which rotates each of the covariance matrices to approximate diagonal form. From the norms of the polar forms, these eigenvectors can be used to estimate a common covariance matrix. These procedures are illustrated by a numerical example.

• Leibovici, D., & El Maache, H. (1997).
  Une décomposition en valeurs singulières d'un élément d'un produit tensoriel de kappa espaces de hilbert séparables [A singular value decomposition of an element belonging to a tensor product of kappa separable hilbert spaces.] Comptes Rendus de l'Académie des Sciences Série I - Mathématique, 325, 779-782.**
  Using a tensorial approach, Leibovici has already explained a very simple theoretic and practical way to obtain the singular value decomposition of multi-way arrays. Based on this presentation, we derive the same analysis in separable hilbert spaces. This permits us to find again the pca of several curves and pca of a process, but we can also derive a pca of a multivariate process via the pta-3modes method, and so on via the pta-kmodes method. Proofs of convergence properties of pca are redescribed within this approach and extended for the pta-kmodes method.

• Leibovici, D., & Sabatier, R. (1998).
  A singular value decomposition of a k-way array for a principal component analysis of multiway data, pta-k. Linear Algebra and its Applications, 269, 307-329.**
  Employing a tensorial approach to describe a k-way array, the singular value decomposition of this type of multiarray is established. The algorithm given to attain a singular value, based on a generalization of the transition formulae, has a gauss-seidel form. A recursive algorithm leads to the decomposition termed svd-k. A generalization of the eckart-young theorem is introduced by consideration of new rank concepts: The orthogonal rank and the free orthogonal rank. The application of this generalization in data analysis is illustrated by a principal component analysis (PCA) over k modes, termed pta-k, which conserves most of the properties of a PCA.

• Leichner, R. (1975).
  Zur Verarbeitung psychiatrischer Information I. [Processing psychiatric information: I.]Diagnostica, 21, 147-166.
  Five psychiatrists judged 15 patients (schizophrenics, cyclothymics and neurotics) on a semantic differential scale of 21 bipolar items at 3 occasions in a carefully designed study. T3 was applied for each occasion and the (only partially presented) results were compared with interesting results. Some discussion on standardization of the data. Cursory examination of the core matrix.

  

• Lescourret, F. (1994).
  Temporal data modeling and role of graphism. Veterinary Research, 25, 140-146.
  The importance of graphism for temporal data modelling, which is often used in ecopathology, is illustrated through 2 examples. The first concerns a univariate time series (tank milk germ measurements). It uses correlograms (choice of the model), plots of residuals of the model (diagnostics) and principal component analysis graphical outputs (comparison of time series). The second concerns climate multivariate monthly time series. A 3-way analysis and graphical outputs are used to give a mark to each station, describing its climate for a set of months.

• Let, M. B., Jacobsen, C., & Meyer, A. S. (2004).
  Effects of fish oil type, lipid antioxidants and presence of rapeseed oil on oxidative flavour stability of fish oil enriched milk. European Journal of Lipid Science and Technology, 106, 170-182.
  As a part of our ongoing experiments on optimization of the oxidative stability of fish oils in genuine food systems, this study investigated the oxidative deterioration of fish oil enriched milk emulsions during cold storage. The experimental data showed that addition of rapeseed oil to fish oil (1:1) prior to emulsification into milk significantly protected the emulsions against oxidative deterioration. Addition of propyl gallate and a citric acid ester to the fish oil prior to emulsification also protected the fish oil enriched milk during storage. Emulsions containing a rapeseed:fish oil mixture were oxidatively stable during 11 d at 2 degreesC. Thus, no additional inhibitory effect of the added antioxidants was observed. The peroxide value and concentrations of five selected volatiles derived from n-3 PUFA degradation in rapeseed:fish oil mixture emulsions were not significantly different from the corresponding levels in neither the emulsion containing only rapeseed oil nor the milk. It is proposed that the tocopherols in rapeseed oil may be the protective factor. Three-way chemometric exploratory data analysis was implemented in form of a parallel factor analysis (PARAFAC). The PARAFAC model provided an overview of the obtained data with significantly enhanced interpretability, and revealed information about groupings and correlations in our data.

• Leurgans, S.E. (1991).
  Multilinear models: Array formulas. (Technical report 473), Columbus, OH: Ohio State University, Department of Statistics.
  Multilinear models for arrays can be represented in terms of a product of an array that is the outer product of matrices, one matrix for each way, with a core array. The bilinear case reduces to a matrix product, and the trilinear case to a product defined by Kruskal (1977). A distributivity property is established for the product. Formulas for vectorized arrays and for matrices derived from arrays reduce to known formulas. The notation is applied to show that the canonical representation of Kapteyn and coauthors (1986) is a subarray of the maximal invariant of arrays under a group of orthogonal multiliner tranformations.

• Leurgans, S., & Ross, R.T. (1992).
  Multilinear models: Applications in spectroscopy (with discussion). Statistical Science, 7, 289-319.
  Multilinear models are models in which the expectation of a multiway array is the sum of products of parameters, where each parameter is associated with only one of the ways. In spectroscopy, multilinear models permit mathematical decompositions of data sets when chemical decomposition of specimens is difficult or impossible. This paper presents a unified description of the model in an array notation. The spectroscopic context shows how to interpret one initialization of the nonlinear least-squares fits of these models. Several examples show that these models can be applied successfully.

• Leurgans, S.E., Ross, R.T., & Abel, R.B. (1993).
  A decomposition for three-way arrays. SIAM Journal on Matrix Analysis and Applications, 14, 1064- 1083.
  An i-by-j-by-k array has rank 1 if the array is the outer product of an i-, a j-, and a k-vector. The authors prove that a three-way array can be uniquely decomposed as the sum of f rank-1 arrays if the f vectors corresponding to two of the ways are linearly independent and the f vectors corresponding to the third way have the property that no two are collinear. Several algorithms that implement the decomposition are described. The algorithms are applied to obtain initial values for nonlinear least-squares calculations. The performances of the decompositions and of the nonlinear least-squares solutions on real and on simulated data are compared. An extension to higher-way arrays is introduced, and the method is compared with those of other authors.

• Levi, M. A. B., Scarminio, L. S., Poppi, R. J. & Trevisan, M. G. (2004).
  Three-way chemometric method study and UV-Vis absorbance for the study of simultaneous degradation of anthocyanins in flowers of the Hibiscus rosa-sinensys species. Talanta, 62, 299-305.
  Ultraviolet-visible spectra of flower extracts of the Hibiscus rosa-sinensys L. var. regius maximus species have been measured between 240.02 and 747.97 nm at pH values ranging from 1.1 to 13.0. Deconvolution of these spectra using the Parallel Factor Analysis (PARAFAC) model permitted the study of anthocyanin systems without isolation and purification of the individual species. Seven species were identified: flavilium cation, carbinol, quinoidal base, and E and Z-chacone and their ionized forms. The concentration changes of flavilium cation, quinoidal basen, and E and Z ionized chalcones were determined as function of pH at the different wavelengths. The flavilium cation, quinoidal base, and ionized E-clacone are involved in tho stage kinetic processes, a fast one followed by a slower one. Ionized Z-chalcone obeys a simple first-order processes. The spectral profiles recovered by PARAFAC model are in excellent agreement with bands of experimental spectra reported in the literature for the individual species measured at specific pH values. These results complement those obtained using chemical and simple mathematical techniques and demonstrate how chemometric methods can resolve problems for complex systems. (C) 2003 Elsevier B.V. All rights reserved.

• Levin, J. (1963).
  Three-mode factor analysis. Unpublished doctoral thesis, University of Illinois, Urbana, Ill. (Dissertation Abstracts International, 1964, 24 (12), 5530-5531)
  (See Levin, 1965).

• Levin, J. (1965).
  Three-mode factor analysis. Psychological Bulletin, 64, 442-452.
  A medium-level explanation of T3 via generalization of the singular value decomposition of two-mode matrices. T3 illustrated with 2 data sets. Semantic differential data (31 widely differing concepts rated on 20 scales by 60 subjects) analysed with Tucker's (1966) Method I or II. Stimulus- Response Inventory of Anxiousness data (14 responses, 11 situations, 169 subjects) were analysed using Method III (Tucker, 1966). Detailed numerical results.

  

• Levin, J. (1966).
  Simultaneous factor analysis of several Gramian matrices. Psychometrika, 31, 413-419.
  Given several Gramian matrices, a least-square fit to all the matrices by one factor matrix, with a predetermined number of factors, is shown to be the principal axes solution of the average of the matrices.

  
       

• L'Hermier des Plantes, H. (1976).
  Structuration des tableaux à trois indices de la statistique: Théorie et application d'une méthode d'analyse conjointe. Doctoral thesis. University of Science and Technology of Languedoc.
  1. Preface.
  2. Propositions.
  Duality diagram; Structures; Comparison of structures; Analysis of differences; Analysis of change; Extensions.
  3. Comparisons.
  Method of L.R. Tucker (1972); Method of J.D. Carroll & J.J. Chang (1972); J.M. Bouroche's double PCA (1975).
  4. Applications.
  Comparison of classifications; Analysis of order relations; Analysis of longitudinal data.

   

• Li, J. (1999).
  Quantitative analysis of cosmetics waxes by using supercritical fluid extraction (SFE)/supercritical fluid chromatography (SFC) and multivariate data analysis. Chemometrics and Intelligent Laboratory Systems, 45, 385-395.
  Waxes play an indispensable role in the formulation of modern cosmetics. Quantitative measurement of the level of different waxes in cosmetics is essential to the understanding of the performance characteristics of the cosmetics. Conventionally, waxes in cosmetics are extracted by Soxhlet extraction using an organic solvent. This extraction is not quantitative. By using SFE, quantitative extraction of the waxes was easily obtained. Due to the complexity of the natural waxes and the coexistence of several types of waxes in a cosmetics product, the level of different waxes in a mixture is best determined by using SFC and multivariate data analysis. Application of this approach in analysis of waxes in mascaras is reported.

• Li, S.S., & Gemperline, P.J. (1993).
  Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method. Journal of Chemometrics, 7, 77-88.
  The direct trilinear decomposition method (dtdm) is an algorithm for performing quantitative curve resolution of three-dimensional data that follow the so-called trilinear model, e.G. Chromatography-spectroscopy or emission-excitation fluorescence. Under certain conditions complex generalized eigenproblem is solved in dtdm. Previous publications never treated those cases. In this paper we show how similarity transformations can be used to eliminate the imaginary part of the complex eigenvalues and eigenvectors, thereby increasing the usefulness of dtdm in practical applications. The similarity transformation technique was first used by our laboratory to solve the similar problem in the generalized rank annihilation method (gram). Because unique elution profiles and spectra can be derived by using data matrices from three or more samples simultaneously, dtdm with similarity transformations is more efficient than gram in the case where there are many samples to be investigated.

  ---

• Li, S.S., Gemperline, P.J., Briley, K., & Kazmierczak, S. (1994).
  Identification and quantitation of drugs of abuse in urine using the generalized rank annihilation method of curve resolution. Journal of Chromatography B: Biomedical Applications, 655, 213-223.
  A rapid analytical method which is of practical use for the identification and quantitation of drugs of abuse in urine using hplc with a diode-array detection is described. Because the method utilizes mathematical resolution of partially resolved peaks, greatly simplified sample preparation procedures and very short run times can be used. The generalized rank annihilation method (gram) is used to eliminate response due to unknown background peaks and separate partially resolved peaks. An optimized gradient elution program was found for which morphine, phenylpropanolamine, ephedrine, benzoylecgonine, lidocaine, cocaine, diphenhydramine, nortriptyline, norpropoxyphene, nordiazepam, codeine, d-amphetamine, meperidine, and amitriptyline elute from the hplc column in less than 8.5 min. A commercially available system for hplc analysis of drugs of abuse is currently available, however, the commercially available system takes 21 min to complete its analysis. Two modified sample pre-treatment methods were also developed to simplify sample treatment procedures substantially. In this paper, the gram technique is shown to be extremely powerful in identifying drugs of abuse from large overlapping peaks.

• Li, Y., Jiang, J. H., Wu, H. L., Chen, Z. P., & Yu, R. Q. (2000).
  Alternating coupled matrices resolution method for three-way arrays analysis. Chemometrics and Intelligent Laboratory Systems, 52, 33-43.
  An alternating coupled matrices resolution (ACOMAR) method is developed for decomposition of three-way data arrays. By utilizing alternating least squares algorithm to minimize the proposed coupled matrices resolution error, the intrinsic profiles are found. Moreover, it yields simultaneously a numerically exact solution for all analytes present in the samples. This method retains the second-order advantage of quantization for analyte(s) of interest in the presence of potentially unknown interferents. The performance of a simulated experiment and a real analytical example shows that the proposed method works well when the number of components is chosen to be equal to or greater than the actual model dimensionality. The insensitivity of the ACOMAR method to the estimated component number escapes the difficulty of determining a proper component number for the model, which is hard to handle for the PARAFAC algorithm. Furthermore, this method circumvents the two-factor degeneracy, which is intrinsic in the PARAFAC algorithm.

• Liang, Y. Z., Kvalheim, O. M., & Manne, R. (1993).
  White, gray and black multicomponent systems - a classification of mixture problems and methods for their quantitative analysis. Chemometrics and Intelligent Laboratory Systems, 18, 235-250.
  Multivariate calibration and resolution methods for handling samples of chemical mixtures are examined from the point of view of the analytical chemist. The methods are classified concordant to three different kinds of analytical mixture systems; i.e. 'white', 'grey', and 'black' analytical systems. Advantages and limitations of available multivariate calibration and resolution methods are discussed with respect to the proposed classification of the analytical mixture problem.

• Librando, V., Drava, G., & Forina, M. (1998).
  3-way principal component analysis applied to the evaluation of water quality of underground waters in the area of Siracusa. Annali Di Chimica, 88, 867-878.
  The present work reports the results of evaluation of the water quality of underground waters in the area of Siracusa (Sicily). Due to the high density of industries, this area has been considered a high hazard site for theenvironment. The area under study is located in the Southeast of the Ionic coast of Sicily, and extends for about 20 Km inland. In this site, 487 samples of groundwater were collected during 1994. The study has been focused on the chemical-physical characteristics of the waters and on the space-time relationships pointed out in the data by the application of 3-way Principal Component Analysis. This method has been applied in addition to the classical multivariate analysis techniques, in order to study the three-dimensional data structure which is typical of environmental studies (samples x variables x sampling times). The further information obtained on the similarity of underground waters can be very useful for the improvement of criteria of water resource management in Sicily.

• Lickteig, T. (1985).
  Typical tensorial rank. Linear Algebra and its Applications, 69, 95-120.
  Upper bounds on the typical rank R(n, m, l) of tensors (=maximal border rank = rank of almost all tensors) of a given shape (n, m, l) are presented. These improve previous results by Atkinson and Lloyd. For cubic shape tensors the typical rank is determined exactly: R(n, n, n) = [n3/(3n-2)] (n>3).

  

• Lied, T. T., & Esbensen, K. H. (2001).
  Principles of MIR, multivariate image regression I: Regression typology and representative application studies. Chemomectrics and Intelligent Laboratory Systems, 58, 213-226.
  We present an introduction to Multivariate Image Regression (MIR) with a selection of illustrative application studies. Generalisation from two-way multivariate calibration to the three-way regimen leads to-at least-three alternative image regression cases depending on the nature of the available Y-data: IPLS-Y-discrim; IPLS-Y-grid; IPLS-Y-total. A systematic image regression typology is briefly introduced. We here present the core of the principles of applied MIR. Two major MIR application studies are worked through, a food mass product industrial inspection study (IPLS-Y-discrim) and a food product (fruit) storage stability image analytical monitoring (IPLS-Y-grid). These exemplifications are presented as archetypes, representing a much wider range of potential industrial/technological application areas. Based on simple three-channel imagery (in order to simulate many industrial systems), they nevertheless represent all higher-dimensional multivariate image cases as well, since the pertinent MIR principles and software are invariant w.r.t. any number of channels/variables employed. The present paper represents one major element of our work towards establishing a complete, stand-alone facility for Multivariate Image Regression (MIR); the second paper in this series deals with the development, implementation and extensive exemplifications of a complementary cross-validation facility.

• Lied, T. T., Geladi, P., & Esbensen, K. H. (200).
  Multivariate image regression (MIR): implementation of image PLSR-first forays. Journal of Chemomectrics, 14, 585-598.
  In the effort of analysing multivariate images, image PLS has been considered interesting. In this paper, image PLS (MIR) is compared with image PCA (MW) by studying a comparison data set. While MIA has been commercially available for some time, image PLS has not. The kernel PLS algorithm of Lindgren has been implemented in a development environment which is a combination of G (LabVIEW) and MATLAB. In this presentation the power of this environment, as well as an early example in image regression, will be demonstrated. With kernel PLS, all PLS vectors (eigenvectors and eigenvalues) can be calculated from the joint variance-covariance (X'Y and Y'X) and association (Y'Y and X'X) matrices. The dimensions of the kernel matrices X'YY'X and Y'XX'Y are K x K (K is the number of X-variables) and M x M (M is the number of Y-variables) respectively. Hence their size is dependent only on the number of X and Y-variables and not on the number of observations (pixels), which is crucial in image analysis. The choice of LabVIEW as development platform has been based on our experience of a very short implementation time combined with user-friendly interface possibilities. Integrating LabVIEW with MATLAB has speeded up the decomposition calculations, which otherwise are slow, Also, algorithms for matrix calculations are easier to formulate in MATLAB than in LabVIEW. Applying this algorithm on a representative test image which shows many of the typical features found in technical imagery, we have shown that image PLS (MIR) decomposes the data differently than image PCA (MIA), in accordance with chemometric experience from ordinary two-way matrices. In the present example the Y-reference texture-related image used turned out to be able to force a rather significant 'tilting' compared with an 'ordinary MIA' of the primary structures in the original, spectral R/G image.

• Lilly, R.S. (1965).
  A developmental study of the semantic differential. ETS Research Bulletin 65-28; Unpublished doctoral dissertation, Princeton University, Princeton, N.J. (Dissertation Ab stracts International, 1966, 26,(7), 4063-4064).
  The same 20 concepts (of various nature) were rated on the same 28 11-point adjective scales by 96, 110, 107 and 100 parochial school children in grades 3, 4, 6 and in high school. Both the adjectives and concepts were high frequency words. Separate T3 analyses were performed for each of the grades. Apparently only the component matrices were interpreted and not the core matrix.

• Lim, L.-H., & Comon, P. (2014).
  Blind multilinear identification IEEE Transactions on Information Theory, 60, 1260-1280. PDF
  We discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible. These instances include: 1) closely located or highly correlated sources in antenna array processing; 2) highly correlated spreading codes in code division multiple access (CDMA) radio communication; and 3) nearly dependent spectra in fluorescence spectroscopy. These have important implications. In the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescence spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that can be specialized to the usual ones for matrices and operators and also be applied to hypermatrices and tensors.

• Lin, Z.H., Booksh, K.S., Burgess, L.W., & Kowalski, B.R. (1994).
  Second-order fiber optic heavy metal sensor employing second-order tensorial calibration. Analytical Chemistry, 66, 2552-2560.
  A fully selective and versatile fiber optic heavy metal sensor based on the concept of second-order instrumentation has been fabricated and evaluated. This sensor uses chemically facilitated donnan dialysis as the means of temporal species discrimination and reagent-assisted spectroscopy for spectral species discrimination. The signals in both orders (time and wavelength) are combined and analyzed with second-order tensorial analysis algorithms-generalized rank annihilation method (gram) and trilinear decomposition (tld)-in order to extract the information of analytes form the sensor responses that are interfered by unknown interferents. Principal component analysis (pca) is used to evaluate sensor characteristics. With second-order calibration, the sensor can measure pb(ii) or cd(ii) in the presence of other interfering transition metal ions. The prediction accuracy is affected by the response linearity of the sensor and the competition effect of co-existing cations with the analyte ions in the ion-exchange process. The specificity of the sensor can be easily switched from pb(ii) measurement into cd(ii) measurement by just changing the calibration standard. The sensor was also tested with ''real-world'' samples. With the stopped-flow preconcentration measuring mode, the sensor gains in sensitivity so that low concentration pb(ii) in tap water and lake water samples can be measured. For tap water samples, the sensor agrees with the graphite furnace atomic absorption (gfaa) verification very well. Due to complexation of pb(ii) in the lake water samples, some complexed pb(ii) ions are rejected by the cation-exchanging membrane, causing the sensor results to be lower than the gfaa data.

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• Linder, M., & Sundberg, R. (1998).
  Second-order calibration: bilinear least squares regression and a simple alternative. Chemometrics and Intelligent Laboratory Systems, 42, 159-178.
  We consider calibration of second-order, or hyphenated instruments generating bilinear two-way data for each specimen. The bilinear regression model is to be estimated from a number of specimens of known composition. We propose a simple estimator and study how it works on real and simulated data. The estimator, which we call the SVD (singular value decomposition) estimator is usually not much less efficient than bilinear least squares. The advantages of our method over bilinear least squares are that it is faster and more easily computed, its standard errors are explicit (and derived in the paper), and it has a simpler correlation structure.

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

• Lindgren, F., & Geladi, P. (1992).
  Multivariate spectrometric image analysis: An illustrative study with two constructed examples of metal ions in solution. Chemometrics and Intelligent Laboratory Systems, 14, 397-412.
  A general introduction to multivariate image analysis (mia) as a useful tool in chemistry is given, by presenting the analysis of two artificial examples. A comparison of a spectrophotometer and a mia system is described. Similarities between the two systems are discussed and the advantages of both systems are highlighted. In earlier publications, a classification of pure solid substances with mia using their reflectances was introduced. This paper further develops the method by considering two examples where aqueous solutions of the metal ions, co2+, ni2+ and cu2+, have been investigated. A chemical introduction to the problem and an explanation of the method is given.

• Lindgren, F., & Geladi, P. (1992b).
  Multivariate spectrometric image-analysis - an illustrative study with 2 constructed examles of metal-ions in solution. Chemomectrics and Intelligent Laboratory Systems, 14, 397-412.
  A general introduction to multivariate image analysis (MIA) as a useful tool in chemistry is given, by presenting the analysis of two artificial examples. A comparison of a spectrophotometer and a MIA system is described. Similarities between the two systems are discussed and the advantages of both systems are highlighted. In earlier publications, a classification of pure solid substances with MIA using their reflectances was introduced. This paper further develops the method by considering two examples where aqueous solutions of the metal ions, Co2+, Ni2+ and Cu2+, have been investigated. A chemical introduction to the problem and an explanation of the method is given.

• Lindgren, F., Geladi, P., & Wold, S. (1994).
  The Kernel Algorithm for PLS. Journal of Chemomectrics, 7, 45-59.
  A fast and memory-saving PLS regression algorithm for matrices with large numbers of objects is presented. It is called the kernel algorithm for PLS. Long (meaning having many objects, N) matrices X (N x K) and Y (N x M) are condensed into a small (K x K) square 'kernel' matrix X(T)YY(T)X of size equal to the number of X-variables. Using this kernel matrix X(T)YY(T)X together with the small covariance matrices X(T)X (K x K), X(T)Y (K x M) and Y(T)Y (M x M), it is possible to estimate all necessary parameters for a complete PLS regression solution with some statistical diagnostics. The new developments are presented in equation form. A comparison of consumed floating point operations is given for the kernel and the classical PLS algorithm. As appendices, a condensed matrix algebra version of the kernel algorithm is given together with the MATLAB code.

• Lindgren, F., Geladi, P., & Wold, S. (1994).
  Kernel-based PLS regression cross-validation and applications to spectral data. Journal of Chemomectrics, 8, 377-389.
  Multivariate images are very large data structures and any type of regression for their analysis is very computer-intensive. Kernel-based partial least squares (PLS) regression, presented in an earlier paper, makes the calculation phase more rapid and less demanding in computer memory. The present paper is a direct continuation of the first paper. In this study the kernel PLS algorithm is extended to include cross-validation for determination of the optimal model dimensionality. To show the applicability of the kernel algorithm, two examples from multivariate image analysis are used. The first example is an image from an airborne scanner of size 9 x 512 x 512. It consists of nine images which are regressed against a constructed dependent image to test the accuracy of the kernel algorithm when used on large data structures. The second example is a satellite image of size 7 x 512 x 512. Several different regression models are presented together with a comparison of their predictive capabilities. The regression models are also used as examples for showing the use of cross-validation.
• Lingoes, J.C. (1982).
  Progressively complex linear transformations for finding geometric similarities among data structures. In H.C. Hudson (Ed.), Classifying Social Data (pp. 108-126). San Francisco: Jossey-Bass.
  This paper contains a survey of the PINDIS method and mathematical expressions are given to intuitive obvious geometric changes in configurations. Similarity is quantified in terms of predictable variance. Examples: 1. Thompson's (1948) graphical analysis of the common porcupine fish and the sun fish; 2. Tobacco leaves; 3. Skull loci of early and modern man.

• Lingoes, J.C. & Borg, I. (1978).
  A direct approach to individual differences scaling using increasingly complex transformations. Psychometri ka, 43, 491-520.
  A family of models for the representation and assessment of individual differences for two-mode three-way data is embodied in a hierarchically organized and sequentially applied procedure (PINDIS), which uses increasingly complex transformations of some common or hypothesized space. The method is a mixture of simultaneous analysis of the matrices of all 'individuals' and separate analyses on each individual. It allows assessing the appropriateness of models like INDSCAL and three-mode scaling.

  

• Linker, W.J. (1982).
  Articulatory and acoustic correlates of labial activity in vowels: A cross-linguistic study. Dissertation Abstracts International, 43, 2336. PhD Thesis, University of California, Los Angeles.
  This dissertation addresses the question of whether or not there exist differences among languages as to the lip gestures they use in the production of vowels. In Chapter 3, PARAFAC is used to solve the the problems found in the previous chapter. Solutions of differing dimensionalities for each language that is used are found, and criteria for choosing among them are discussed.

• Lipovetskii, S.S. (1984).
  Component analysis of tables with many inputs. Industrial Laboratory, 50, 474-481.
  In this paper, the Parafac model is re-invented. The parameters of the model are estimated by subtracting earlier components, a method which is however suboptimal.

• Lipovetsky, S., & Tishler, A. (1994).
  Linear methods in multimode data-analysis for decision-making. Computers & Operations Research, 21, 169-183.
  This paper presents and analyses several methods for the evaluation of information given in the form of many-way matrices. These methods are based on the least squares approximation of a matrix by a many-vector product which can be represented as a nonlinear eigenvector problem. Using real data about university choice by high school graduates in Israel, we develop and compare the following three families of methods: parallel proportional profiles, various types of methods based on the use of cyclic matrices (canonical correlations, principal components, and planes' approximation), and minimization of relative deviations.

• Liu, S. (1999).
  A study on the applicability on multicomponent calibration methods in chemometrics. Chemometrics and Intelligent Laboratory Systems, 45, 131-145.
  Twelve multivariate calibration method alternatives are compared to establish the effect of spectral nonlinearity and collinearity on accuracy and precision of determined results. Simulated and real spectral data are used in this research. This study can help us to select an optimum method for determination.

• Liu, Y., & Dengler, N.G. (1994).
  Bundle sheath and mesophyll cell differentiation in the C4 dicotyledon Atriplex rosea: Quantitative ultrastructure. Canadian Journal of Botany, 72, 644-657.
  In leaves of most C₄ species, both bundle sheath and mesophyll cells are derived from ground meristem, yet at maturity differ in photosynthetic enzyme complement and in cell size, shape, and subcellular ultrastructure. Multigroup principal components analysis (M-PCA) of the data emphasizes that the greatest source of variation is overall size change as both cell types expand. M-PCA also identifies patterns of allometry within the data; for instance, mesophyll cell vacuoles and chloroplast peripheral reticulum undergo greater relative growth than do bundle sheath microbody area and number.

• Liu, X., & Sidiropoulos, N. D. (2001).
  Cramér–Rao Lower Bounds for Low-Rank Decomposition of Multidimensional Arrays. IEEE Transacions on Signal Processing, 49, 2074-2086.
  

  
             

• Lohmöller, J.-B. (1978a).
  Stabilität und Kontinuität in Längsschnittdaten, analysiert durch T- und trimodale Faktorenanalyse. [Stability and continuity in longitudinal data, analysed by T-factor analysis and three-mode factor analysis].Internal report Hochschule des Bundeswehr München, Fachbereich Pädagogik, München, Germany (revision 1981)

• Lohmöller, J.B. (1978b).
  How longitudinal factor stability, continuity, differentiation, and integration are portrayed into the core matrix of three-mode factor analysis. Paper presented at the European Meeting on psychometrics and mathematical psychology, Uppsala, Sweden, June 16.
  Concepts for the study of change, like stability, continuity, integration, differentiation, and transformation are defined within the framework of a multivariate autoregressive model. How these concepts show up in the parameters of a three-mode factor analysis is investigated by an artificial example, and data from the Augsburg longitudinal study (5 attributes, 4 time periods, and roughly 1900 school children).

  

• Lohmöller, J.B. (1979).
  Die trimodale Faktorenanalyse von Tucker: Skalierungen, Rotationen, andere Modelle. Archiv für Psychologie, 131, 137-166(a).
  L. provides a very comprehensive survey of many aspects of T3, such as the model itself, the relationships between T3 and three-way ANOVA, rotations of factor matrices and core matrix, factor scores, relationships with two-mode FA, scaling of input data, additive versus multiplicative models and types of programs. Most of the concepts are illustrated with data of roughly 1900 school children from the Augsburg longitudinal study (3 time periods) of 8 subtests of the primal mental abilities test.

• Lohmöller, J.-B. (1979b).
  Programmbeschreibung von FA-3- Trimodale Faktorenanalyse. In J.-B. Lohmöller, Das COR-Programm-system zur Korrelationsanalyse. Fachbereich Pädagogik, Hochschule der Bundeswehr München, Neubiberg, FRG. *
  As part of a larger system for the analysis of correlation matrices, three programs for T3 are described, which use the three methods of Tucker (1966) respectively.

  

• Lohmöller, J.-B. (1978).
  Stabilität und Kontinuität in Längsschnittdaten, analysiert durch T- und trimodale Faktorenanalyse. Internal report Hochschule de Bundeswehr München, Fachbereich Pädagogik, München, Germany (revision 1981))

• Lohmöller, J.-B. (1989).
  Latent variable path modeling with partial least squares (Chapter 6: Latent variables three-mode path (LVP3) analysis). Heidelberg: Physica.
  Contents:
  Three-way data models; the Kronecker principal component (KPC) model; the three- mode LV path (LVP3) model; special cases and properties; the PLS estimation of LVP3 models; application: longitudinal data; concluding remarks.

• Lohmöller, J.-B. & Oerter, R. (Eds.) (1979).
  Medien in der Erzieherausbildung:Erprobung des Medienverbundes "Vorschulische Erziehung im Ausland". München, FRG.: Oldenbourg Verlag (pp.114-130). *
  800 subjects judged 8 media packages on 27 rating scales. The scale components were varimax rotated, and the core matrix was symmetrically rotated. Factor scores for the subjects were computed. The means were analysed by three-way ANOVA. The six variable components were introduced in a path analysis model, together with various other variables, such as social and per- sonal background, job-related motivations and attitudes towards learning.

  

• Lohmöller, J.-B. & Wold, H. (1980; revised 1982).
  Three-mode path models with latent variables and partial least squares (PLS) parameter estimation. Forschungsbericht 80.03, Fachbereich Paedagogik, Hochschule der Bundeswehr München, Neubiberg, FRG. (presented at the European Meeting of the Psychometric Society, Groningen, The Netherlands, June 18-21, 1980).
  An exposition of various three-mode models is given, including two new models. One for indicators observed both over time and cases, one for three-mode path analysis with latent variables. Includes algorithm for three-mode factor and path analysis within PLS (or ALS) context. Example: 3-mode path analysis for 8 school subjects measured at 6 occassions. Correlation matrices from 2500 children.

• Lombardo, R. (1994).
  Modelli di decomposizione per l'analisi della dipendenza nelle tabelle di contingenza a tre vie [Decomposition models for dependency analysis in three-way contingency tables]. Unpublished doctoral thesis, Department of Mathematics and Statistics, University of Napels.
  1. Models for three-mode principal component analysis.
  2. A measure for nonsymmetrical association.
  3. Three-mode nonsymmetrical correspondence analysis (NSCA) and decomposition models.
  4. Applications of three-mode NSCA.
  Appendix I: Tucker methods.
  Appendix II: Multiple NSCA and the prediction index delta.
  Appendix III: The three-mode NSCA program.

• Lombardo, R., Carlier, A., & D'Ambra, L. (1996).
  Nonsymmetric correspondence analysis for three-way contingency tables. Methodologica, 4, 59-80
  Non-symmetric correspondence analysis is extended to three-way tables in a parallel fashion to Carlier and Kroonenberg's (1996) extension of regular correspondence analysis. The extension is based on a decomposition of the tau-index which allows the variables from two modes to predict the third one. The decomposition of the Tucker3 model is used as well as Lancaster's (1951) additive decomposition of interactions in three-way tables. The example concerns the prediction of (7) fields of study by French students by (34) parents occupations and (2) sex of the student using a 3 by 4 by 2 Tucker3 model. Extensive discussions of graphical representations for this situation are presented as well.

  

• London, M., Crandall, R. & Fitzgibbons, D. (1977).
  The psychological structure of leisure: acti vities, needs, people. Journal of Leisure Re- search, 9, 252-263.
  For 83 students the presence of 15 need-satisfying attributes was measured with a seven-point rating scale for 30 leisure activities. The stability of the T3 solution was checked by split-half analysis. Discussion of the presented results of the factor loading and core matrices.

• Lopes, J. A., & Menezes, J. C. (2003).
  Industrial fermentation end-product modelling with multilinear PLS. Chemometrics and Intelligent Laboratory Systems, 68, 75-81.
  In this paper, a trilinear version of the partial least squares (PLS) algorithm was used to model the performance of an industrial fed-batch fermentation process. Trilinear data obtained from process operation were used to derive a model for the end-process active product ingredient (API) concentration prediction. Obtained mutilinear PLS models were compared with the correspondent bilinear models. A genetic algorithm was used to select appropriate calibration sets (to reduce the influence of nominal batches). A validation coefficient of determination (QY2) of 91.4% was obtained for the multilinear PLS model after batch selection (prediction intervals were estimated using bootstrapping). Examination of the multilinear PLS model weights led to the delimitation of a small time region (from 50 to 75 processing hours) almost exclusively responsible for the fermentation performance.

• Lopes, J. A., Menezes, J. C., Westerhuis, J. A., & Smilde, A. K. (2002).
  Multiblock PLS analysis of an industrial pharmaceutical process. Biotechnology and Bioengineering, 80, 419-427.
  The performance of an industrial pharmaceutical process (production of an active pharmaceutical ingredient by fermentation, API) was modeled by multiblock partial least squares (MBPLS). The most important process stages are inoculum production and API production fermentation. Thirty batches (runs) were produced according to an experimental planning. Rather than merging all these data into a single block of independent variables (as in ordinary PLS), four data blocks were used separately (manipulated and quality variables for each process stage). With the multiblock approach it was possible to calculate weights and scores for each independent block. It was found that the inoculum quality variables were highly correlated with API production for nominal fermentations. For the nonnominal fermentations, the manipulations of the fermentation stage explained the amount of API obtained (especially the pH and biomass concentration). Based on the above process analysis it was possible to select a smaller set of variables with which a new model was built. The amount of variance predicted of the final API concentration (cross-validation) for this model was 82.4%. The advantage of the multiblock model over the standard PLS model is that the contributions of the two main process stages to the API volumetric productivity were determined.

• Lorber, A., Faber, K., & Kowalski, B.R. (1997).
  Net analyte signal calculation in multivariate calibration. Analytical Chemistry, 69, 1620-1626.**
  Net analyte signal plays an important role in the calculation of figures of merit for characterizing a calibration model. Until now, its computation has only been feasible for the direct calibration model, which requires knowledge of pure spectra or concentrations of all contributing species in the calibration samples. An increasingly important calibration model is the inverse calibration model, which also allows for quantitation if the knowledge about the interferents is incomplete. This paper shows that net analyte signal computation is possible for the inverse calibration case. Application to the determination of protein content in wheat samples by near-infrared spectrometry is presented. Net analyte signal calculation was used to estimate selectivities (ratio of signal available for quantitation to the total measured signal). The selectivities were found to range between 0 and 2% of the measured reflectance signal. A new measure for outlier diagnosis based on the correlation of the net analyte signal to the regression coefficients vector is introduced and tested on the same data.

• Losada, M.A., Medina, R., Vidal, C., & Losada, I.J. (1992).
  Temporal and spatial cross-shore distributions of sediment at "El Puntal" spit, Santander, Spain. In Coastal Engineering 1992, Proceedings of the Twenty-Third International Conference Held October 4-9, Venice, Italy (pp. 2251-2264).
  Sediment samples and beach profile evolution data, collected along one profile line at "El Puntal" Spit, Santaner, Spain, are studied by means of Principal Component Analysis (PCA). This analysis technique is used to separate the temporal, spatial and grain size distribution variability of the data. The results show that there is a seasonality in the grain size distribution affecting the fine sand as well as the coarse sand. Further, a "master" grain size distribution, which is constructed by adding all the grain samples, taken from all over the profile, is shown to be constant in time.

• Louwerse, D.J., & Smilde, A.K. (2000).
  Multivariate statistical process control of batch processes based on three-way models. Chemical Engineering Science, 55, 1225-1235.
  The theory of batch MSPC control charts is extended and improved control charts are developed. Unfold-PCA, PARAFAC and Tucker3 models are discussed and used as a basis for these charts. The results of the different models are compared and the performance of the control charts based on these models is investigated. It is found that this performance depends on the type of fault occurring in the batch process. A strategy is provided to partition reference data describing the normal operating conditions, in order to be able to monitor a new incomplete batch on-line.

• Louwerse, D.J., Smilde, A.K., & Kiers, H.A.L. (1999a).
  Cross-validation of multiway component models. Journal of Chemometrics, 13, 491-510.
  Two cross-validation methods are presented for multiway component models. They are used for choosing the numbers of components to use in Tucker3 models describing three-way data. The approach is general and can easily be adapted to other three-way and multiway models. A model is estimated after leaving out a small part of the multiway data array. The predictive residual error sum of squares (PRESS) is calculated for the eliminated part of the data by comparing the model values with the actual data. PRESS of the entire data set can be calculated like this sequentially. The methods are the leave-bar-out cross-validation method, which leaves out data slices in all modes, and the EM cross-validation method, which handles eliminated data as missing values. A method to calculate the statistical significance of the PRESS reduction for an additional component, the so called W-statistic, is provided for Tucker3 models. A strategy is proposed to search along an efficient path, to reduce computation time, since the number of feasible models as a function of the total number of components summed over the modes increases rapidly.

• Louwerse, D. J., Tates, A. A., Smilde, A. K., Koot, G. L. M., & Berndt, H. (1999b).
  PLS discriminant analysis with contribution plots to determine differences between parallel batch reactors in the process industry. Chemometrics and Intelligent Laboratory Systems, 46, 197-206.
  PLS discriminant analysis, applied to a PVC polymerisation batch process, is used to determine performance differences of parallel batch reactors. Weight contribution plots of time points and of variables are used to physically interpret the modelled differences; the main time points in which deviations occur and variables that cause the observed differences are assigned, A simple step-wise procedure is suggested to implement this method in the process industry. It was found that a systematic difference between the polymerisation time of the PVC batch reactors was caused by sensor failure or due to drifting thermocouples.

• Love, W.D. & Tucker, L.R. (1970).
  A three-mode factor analysis of serial learning. Report of the Office of Naval Research.
  Variations in serial position learning curves over 10 stages of learning and 33 individuals using 19 trials were studied by means of T3 analysis for a list of 20 CVC trigrams. Moderately detailed analysis and interpretation.

             

• Lu, J.Z., Wu, H.L., Jiang, J.H. Long, N., Mo, C.Y. & Ru, R.Q. (2003).
  An improved trilinear decomposition algorithm based on a Lagrange operator Analytical Sciences, 19, 1037-1043
  An improved trilinear decomposition algorithm based on a Lagrange operator (LO) is developed in this paper, which introduces a Lagrange operator and penalty terms in the loss function to improve the performance of the algorithm. Compared to the traditional parallel factor (PARAFAC) algorithm, the algorithm not only may converge much faster, but also overcome the sensibility to estimate the number of components. A set of simulated and measured excitation/emission fluorescence data were treated by both the proposed and traditional PARAFAC algorithm to compare their efficiencies. The analytical results obtained with real chemical system containing aspirin and its metabolic products show that the trilinear decomposition methodology is a promising tool to obtain spectral and composition information from mixtures without chemical separation.

• Lu, J. Z., Wu, H. L., Sun, X. Y., Cui, H., Sun, J. Q. & Yu, R. Q.(2004).
  Simultaneous decomposition and determination of the complex dimethylphenol isomers by alternating trilinear decomposition algorithm combined with region selection. Chinese Journal of Analytical Chemistry 32 1278-1282.
  Resolution and determination of complex isomer system: 2,3-dimethylphenol, 2,4-dimethylphenol, 2,5-dimethylphenol, 2.,6-dimethylphenol, 3,4-dimethylphenol of serious overlapped chromatograms and spectra were studied. Before the system was resolved by alternating trilinear decomposition (ATLD) algorithm, the better region of the chromatograms and spectra was selected purposely. The complex system was simultaneously determined among the five dimethylphenol isomers, and satisfactory results were obtained.

• Luan, S., Pang, H.M., & Houk, R.S. (1995).
  Application of generalized standard additions method to inductively-coupled plasma-atomic emission-spectroscopy with an echelle spectrometer and segmented-array charge-coupled detectors. Spectrochimica Acta Part B: Atomic Spectroscopy, 50, 791- 801.**
  Simultaneous correction for both spectral interferences and matrix effects in inductively coupled plasma atomic emission spectrometry can be accomplished by using the generalized standard additions method (gsam). Results obtained with the application of the gsam to the perkin-elmer optima 3000 icp atomic emission spectrometer are presented. The echelle-based polychromator with segmented-array charge-coupled device detectors enables the direct, visual examination of the overlapping lines cd(i) 228.802 nm and as(i)228.812 nm. The slit translation capability allows a large number of data points to be sampled; therefore, the advantage of noise averaging is gained. Pure spectra of each of the spectrally active components in the sample can be extracted through the gsam.

• Lundy, M.E., & Harshman, R.A. (1985).
  Reference manual for the PARAFAC analysis package. Scientific Software Associates. London, Ontario, Canada.
  This manual attempts to present, using nontechnical language, the basic information necessary to use the programs in the PARAFAC Analysis Package and to understand their output. It is assumed throughout that the user is reasonably familiar with his or her own computer system.

• Lundy, M.E., Harshman, R.A., & Kruskal, J.B. (1989).
  A two-stage procedure incorporating good features of both trilinear and quadrilinear models. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 123-130). Amsterdam: Elsevier. Published version of paper presented at the annual Meeting of the Classification Scoiety, St. John, Newfoundland, Canada, 1985.
  Applications of trilinear (PARAFAC-CANDECOMP) factor/component analysis to data requiring the more complex quadrilinear (Tucker T3 or T2) model sometimes produces uninterpretable "degenerate" solutions, in which two or more factors are highly negatively correlated. The more general model does not have this problem, but it is subject to an axis indeterminacy that leaves some interesting questions unanswered. Described here is a two-stage procedure that combines the strengths of the trilinear (PARAFAC) and quadrilinear (Tucker T3) models to better deal with such problems. An application to real data illustrates how it provides unique meaningful axes along with a core matrix that can give substantive insights into the data complexities that caused the degeneracies. More general models are also discussed.

           

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