ThreeMode Abstracts, Part L
With one can go to the index of
this part of the Abstracts, with
one can go to other
parts (letters) of the Abstracts.
INDEX
La  Lb 
Lc  Ld 
Le  Lf 
Lg  Lh 
Li  Lj 
Lk  Ll 
Lm  Ln 
Lo  Lp 
Lq  Lr 
Ls  Lt 
Lu  Lv 
Lw  Lx 
Ly  Lz 


Ladefoged, P., Harshman, R., Goldstein, L., & Rice, L. (1978).
Generating vocal tract shapes from formant frequencies.
Journal of the Acoustical Society of America, 64,
10271035.
An algorithm that uses only the first three formant frequencies has been
devised for generating
vocal tract shapes as seen on midsagittal xray 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 xray diagrams, and appears
to be generalizable
to other speakers.

LaFaye, J.Y. (1985).
Application du modèle INDSCAL. Statistique et Analyse de
Données, 10, 83102.
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 LanguedocRoussilon 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 PaulSabatier.
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
zerocorrelation 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, 111126.
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, 83102.
In two studies, the behavioral preferences of majority (White) and visible minority (nonWhite)
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; nonnormative–normative) and
phenomenological studies on the experience of discrimination (e.g. risk). A multimode factor
analysis of the behaviors and attributes provided a threecomponent 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 nonWhite
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).
Crosscultural aspects of passionate love: An individual differences analysis.
Journal of Crosscultural Psychology, 31, 752777.
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 PointofView solutions (3MPOV). A 6factor groupcommon
structure was found to best explain variance in PLS responses collapsed across
cultures. 3MPOV 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
LingoesBorg Procrustean individual differences scaling. Psychometrika,
47, 427442.
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, 6773.
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 reanalysed with threemode 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, 483487.
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
reducedrank regression algorithm which first projects Y onto the Xspace 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 nonzero
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
fourmode
matrices. Psychometrika, 46, 4757.
L. presents the direct generalization of Tucker's (1966) model
to fourmode 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 secondaryion massspectrometry (SIMS) micrographs to characterize chemicalphases.
Mikrochimica Acta, 119, 112.
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, 103116.
STATIS is a dataanalytic 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. 1329). 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 vectorspaces.
3. Noncentred 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, 97119.
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 threeway 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
LanguedocRouisillon.

Law, H.G. & Snyder Jr C.W. (1979).
Threemode models for the
analysis of
psychological data. Australian Psychologist, 14, 214
(conference abstract).
The paper gives a nontechnical overview of recently developed
threemode 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. 4960).
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: FactorAnalytic 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 threemode 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, 335344.
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 12year
period. Such an arrangement of data can be expressed as a 3way array where a farm x
attribute x year matrix can be evaluated and interactions considered. Two multivariate
techniques, the 3way mixture method of clustering and the 3mode 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 threemode
common factor analysis. Multivariate Behavioral Research,
14, 443462.
T3 used in the development of assertiveness inventory of 56
items (considered as a completely crossed twofacet 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).
Threemode principal component analysis of monitoring data from Venice lagoon.
Journal of Chemometrics, 14, 187195.
A data set obtained by 44 monthly determinations of 11 variables from 13
sampling sites in the Venice lagoon has been treated by threemode principal
component analysis. The results show that the sampling sites are grouped
according to their geographical location, following an innerouter 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).
Threemode data set analysis using higher order subspace method: application to
sonar and seismoacoustic signal processing.
Signal Processing, 84, 919942.
In this paper, a threemode 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 threemode arrays of
the wellknown subspace technique proposed by Eckart and Young (Psychometrica 1 (1936)
211) for matrices. Threemode data sets are increasingly encountered in signal processing
and are classically processed using matrix algebra techniques. The proposed approach aims
to process naturally threemode 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 threemode arrays and orthogonality between subspaces. A projector formulation for
threemode signal and noise subspaces is also given and the improvement of separation
with the threemode 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 seismoacoustic wave separation problem. For the first
application, we propose a threemode 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 subspace 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, 3575.
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.
Twoway data: singular value decomposition (SVD) and iterative SVD,
combined vector (COV)
algorithm; Threeway data; Fourway 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 twoway codes.
Successive overrelaxation; Comparison with other algorithms.
V. Artificial spectra.
Model dicrimination.
VI. Fluorescent dyes.
Single dye solution: two way and threeway analysis; threedaye mixture;
discussion.
VII. Deazaflavins.
VIII. Photosynthetic systems of peas.
Steady state: Effects of Mg2+; Steady state: Effect of temperature, Four
way analysis; More
components?: Fourway 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.
Twoway analysis; Threeway analysis; Fourway analysis; A new data
set.
XII. Data collection strategies.
A 'crosshatch' 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, 374379.
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 steadystate fluorometer and
withour changing fluorophore concentrations. Requiring that the spectra
deduced be nonnegative can increase the accuracy of the results.

Lee, J.K., & Ross, R.T. (1998)
Absorption and Fluorescence of Tyrosine HydrogenBonded to Amidelike Ligands
Journal of Physical Chemistry, 102, 46124618.
The absorption and steadystate fluorescence of NacetylLtyrosinamide
(NAYA), chosen to model the tyrosine in proteins, was measured in four
solvents in the presence of N,Ndimethylacetamide, Nmethulacetamide,
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 M1, increasing with solvent polarity; lack of a strong dependence
of binding constant on Nmethylation 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 redshift due to hydrogen bonding to the phenolic
hydrogen which is proportional to ligand acceptor basicity, and a blueshift
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 hydrogenbonded complex varies between 0.102 and 0.044,
increasing slightly with Nmethylation, but more dependent on solvent.
In methanol, acetone, and dioxane, the amidemodel complex has a 310 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 hydrogenbonded
tyrosine using a trilinear model of fluorescence.
Journal of Physical Chemistry, 96, 91589162.
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
steadystate fluorescence at aqueous
nacetylltyrosinamide in the presence of five
proton acceptors was resolved into three
components, corresponding to normally solvated
side chain, side chain hydrogenbonded to the
added proton acceptor, and impurity. The
excitation spectrum of the hydrogenbonded complex
is redshifted 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 structureanalysis.
Biometrika, 66, 539545.
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 goodnessoffit 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 threemode factor analysis. British
Journal of
Mathematical and Statistical Psychology, 36, 217223.
The statistical model for threemode 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
goodnessoffit statistic are obtained via the GaussNewton 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, 924.
A threeway threemode extension of De Boeck and Rosenberg's (1988) twoway two
mode hierarchical classes model is presented for the analysis of individual
differences in binary object × attribute arrays. In line with the twoway
hierarchical classes model, the threeway extension represents both the
association relation among the three modes and the settheoretical 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 threeway data is discussed.

Lefkovitch, L.P. (1993).
Consensus principal components. Biometrical Journal, 35,
567580.
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, 779782.**
Using a tensorial approach, Leibovici
has already explained a very simple theoretic and
practical way to obtain the singular value
decomposition of multiway 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 pta3modes method,
and so on via the ptakmodes method. Proofs of
convergence properties of pca are redescribed
within this approach and extended for the
ptakmodes method.

Leibovici, D., & Sabatier, R. (1998).
A singular value decomposition of a kway array
for a principal component analysis of multiway
data, ptak.
Linear Algebra and its Applications, 269, 307329.**
Employing a tensorial approach to describe a kway
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
gaussseidel form. A recursive algorithm leads to
the decomposition termed svdk. A generalization
of the eckartyoung 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 ptak, which conserves
most of the properties of a PCA.

Leichner, R. (1975).
Zur Verarbeitung psychiatrischer Information
I.
[Processing psychiatric information: I.]Diagnostica,
21, 147166.
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, 140146.
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 3way 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, 170182.
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 n3 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. Threeway 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, 289319.
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 leastsquares 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 threeway arrays.
SIAM Journal on Matrix Analysis and Applications, 14, 1064
1083.
An ibyjbyk array has rank 1 if the array is
the outer product of an i, a j, and a kvector.
The authors prove that a threeway array can be
uniquely decomposed as the sum of f rank1 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 leastsquares calculations. The
performances of the decompositions and of the
nonlinear leastsquares solutions on real and on
simulated data are compared. An extension to
higherway 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).
Threeway chemometric method study and UVVis absorbance for the study of simultaneous
degradation of anthocyanins in flowers of the Hibiscus rosasinensys species.
Talanta, 62, 299305.
Ultravioletvisible spectra of flower extracts of the Hibiscus rosasinensys 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 Zchacone 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 Eclacone are involved in tho
stage kinetic processes, a fast one followed by a slower one. Ionized Zchalcone
obeys a simple firstorder 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).
Threemode factor analysis. Unpublished doctoral
thesis, University of Illinois, Urbana, Ill. (Dissertation Abstracts
International, 1964, 24 (12), 55305531)
(See Levin, 1965).

Levin, J. (1965).
Threemode factor analysis. Psychological
Bulletin,
64, 442452.
A mediumlevel explanation of T3 via generalization of the
singular value decomposition of twomode 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, 413419.
Given several Gramian matrices, a leastsquare 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, 385395.
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, 7788.
The direct trilinear decomposition method (dtdm)
is an algorithm for performing quantitative curve
resolution of threedimensional data that follow
the socalled trilinear model, e.G.
Chromatographyspectroscopy or emissionexcitation
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, 213223.
A rapid analytical method which is of practical
use for the identification and quantitation of
drugs of abuse in urine using hplc with a
diodearray 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, damphetamine, 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 pretreatment 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 threeway arrays analysis.
Chemometrics and Intelligent Laboratory Systems, 52, 3343.
An alternating coupled matrices resolution (ACOMAR) method is
developed for decomposition of threeway 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
secondorder 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 twofactor 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, 235250.
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).
3way principal component analysis applied to the evaluation of water quality of
underground waters in the area of Siracusa.
Annali Di Chimica, 88, 867878.
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 chemicalphysical characteristics of the waters
and on the spacetime relationships pointed out in the data by the application of 3way
Principal Component Analysis. This method has been applied in addition to the classical
multivariate analysis techniques, in order to study the threedimensional 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, 95120.
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/(3n2)] (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, 213226.
We present an introduction to Multivariate Image Regression (MIR) with a
selection of illustrative application studies. Generalisation from twoway multivariate
calibration to the threeway regimen leads toat leastthree alternative image regression cases
depending on the nature of the available Ydata: IPLSYdiscrim; IPLSYgrid; IPLSYtotal. 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 (IPLSYdiscrim) and a food
product (fruit) storage stability image analytical monitoring (IPLSYgrid). These exemplifications
are presented as archetypes, representing a much wider range of potential industrial/technological
application areas. Based on simple threechannel imagery (in order to simulate many industrial
systems), they nevertheless represent all higherdimensional 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,
standalone facility for Multivariate Image Regression (MIR); the second paper in this series
deals with the development, implementation and extensive exemplifications of a complementary
crossvalidation facility.

Lied, T. T., Geladi, P., & Esbensen, K. H. (200).
Multivariate image regression (MIR): implementation of image PLSRfirst forays.
Journal of Chemomectrics, 14, 585598.
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 variancecovariance (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 Xvariables) and M x M (M is the number of Yvariables)
respectively. Hence their size is dependent only on the number of X and Yvariables 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 userfriendly 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 twoway matrices. In the present example
the Yreference texturerelated 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 6528; Unpublished doctoral dissertation,
Princeton University, Princeton, N.J. (Dissertation Ab
stracts International, 1966, 26,(7), 40634064).
The same 20 concepts (of various nature) were rated on the
same 28 11point 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, 12601280.
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 lowrank
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).
Secondorder fiber optic heavy metal sensor employing
secondorder tensorial calibration.
Analytical Chemistry, 66, 25522560.
A fully selective and versatile fiber optic heavy metal sensor based on the concept of
secondorder instrumentation has been fabricated and evaluated. This sensor uses chemically
facilitated donnan dialysis as the means of temporal species discrimination and
reagentassisted spectroscopy for spectral species discrimination. The signals in both orders
(time and wavelength) are combined and analyzed with secondorder tensorial analysis
algorithmsgeneralized 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 secondorder 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 coexisting cations with the
analyte ions in the ionexchange 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 ''realworld'' samples. With the stoppedflow 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 cationexchanging
membrane, causing the sensor results to be lower than the gfaa data.

Linder, M., & Sundberg, R. (1998).
Secondorder calibration: bilinear least squares regression and a simple alternative.
Chemometrics and Intelligent Laboratory Systems, 42, 159178.
We consider calibration of secondorder, or hyphenated instruments
generating bilinear twoway 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 secondorder calibration, with focus on bilinear regression methods.
Journal of Chemometrics, 16, 1227.

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, 397412.
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 imageanalysis  an illustrative study with 2 constructed examles
of metalions in solution.
Chemomectrics and Intelligent Laboratory Systems, 14, 397412.
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, 4559.
A fast and memorysaving 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 Xvariables. 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).
Kernelbased PLS regression crossvalidation and applications to spectral data.
Journal of Chemomectrics, 8, 377389.
Multivariate images are very large data structures and any type of regression
for their analysis is very computerintensive. Kernelbased 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 crossvalidation 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 crossvalidation.

Lingoes, J.C. (1982).
Progressively complex linear transformations for finding
geometric similarities among data structures. In H.C. Hudson (Ed.),
Classifying Social Data (pp. 108126). San Francisco:
JosseyBass.
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, 491520.
A family of models for the representation and assessment of
individual differences for twomode threeway 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 threemode scaling.

Linker, W.J. (1982).
Articulatory and acoustic correlates of labial activity in
vowels: A crosslinguistic 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, 474481.
In this paper, the Parafac model is reinvented. 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 dataanalysis for decisionmaking.
Computers & Operations Research, 21, 169183.
This paper presents and analyses several methods for the
evaluation of information given in the form of manyway
matrices. These methods are based on the least squares
approximation of a matrix by a manyvector 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, 131145.
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, 644657.
In leaves of most C_{4} 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 (MPCA)
of the data emphasizes that the greatest source of variation is overall
size change as both
cell types expand. MPCA 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 LowRank Decomposition of Multidimensional Arrays.
IEEE Transacions on Signal Processing, 49, 20742086.

Lohmöller, J.B. (1978a).
Stabilität und
Kontinuität in Längsschnittdaten, analysiert durch T
und trimodale Faktorenanalyse. [Stability and continuity in
longitudinal data, analysed by Tfactor analysis and threemode
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 threemode 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 threemode
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, 137166(a).
L. provides a very comprehensive survey of many aspects of T3,
such as the model itself, the relationships between T3 and
threeway ANOVA, rotations of factor matrices and core matrix,
factor scores, relationships with twomode 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 FA3
Trimodale
Faktorenanalyse. In J.B. Lohmöller, Das
CORProgrammsystem 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 threemode path (LVP3) analysis).
Heidelberg: Physica.
Contents:
Threeway 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.114130). *
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 threeway ANOVA. The six
variable components were introduced in a path analysis model,
together with various other variables, such as social and per
sonal background, jobrelated motivations and attitudes
towards learning.

Lohmöller, J.B. & Wold, H. (1980; revised 1982).
Threemode 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 1821, 1980).
An exposition of various threemode models is given, including
two new models. One for indicators observed both over time and
cases, one for threemode path analysis with latent variables.
Includes algorithm for threemode factor and path analysis
within PLS (or ALS) context. Example: 3mode 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 threeway contingency tables]. Unpublished doctoral thesis,
Department of Mathematics and Statistics, University of Napels.
1. Models for threemode principal component analysis.
2. A measure for nonsymmetrical association.
3. Threemode nonsymmetrical correspondence analysis (NSCA) and decomposition
models.
4. Applications of threemode NSCA.
Appendix I: Tucker methods.
Appendix II: Multiple NSCA and the prediction index delta.
Appendix III: The threemode NSCA program.

Lombardo, R., Carlier, A., & D'Ambra, L. (1996).
Nonsymmetric correspondence analysis for threeway contingency
tables. Methodologica, 4, 5980
Nonsymmetric correspondence analysis is extended to threeway
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 tauindex 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 threeway 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, 252263.
For 83 students the presence of 15 needsatisfying attributes
was measured with a sevenpoint rating scale for 30 leisure
activities. The stability of the T3 solution was checked by
splithalf analysis. Discussion of the presented results of
the factor loading and core matrices.

Lopes, J. A., & Menezes, J. C. (2003).
Industrial fermentation endproduct modelling with multilinear PLS.
Chemometrics and Intelligent Laboratory Systems, 68, 7581.
In this paper, a trilinear version of the partial least squares (PLS) algorithm was used to model the performance of an
industrial fedbatch fermentation process. Trilinear data obtained from process operation were used to derive a model for the
endprocess 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, 419427.
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 (crossvalidation) 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, 16201626.**
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
nearinfrared 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 crossshore distributions of sediment at "El
Puntal" spit, Santander, Spain. In Coastal Engineering 1992,
Proceedings of the TwentyThird International Conference Held
October 49, Venice, Italy (pp. 22512264).
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 threeway
models. Chemical Engineering Science, 55, 12251235.
The theory of batch MSPC control charts is
extended and improved control charts are
developed. UnfoldPCA, 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 online.

Louwerse, D.J., Smilde, A.K., & Kiers, H.A.L. (1999a).
Crossvalidation of multiway component models. Journal of Chemometrics,
13, 491510.
Two crossvalidation methods are presented for
multiway component models. They are used for
choosing the numbers of components to use in
Tucker3 models describing threeway data. The
approach is general and can easily be adapted to
other threeway 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 leavebarout
crossvalidation method, which leaves out data
slices in all modes, and the EM crossvalidation
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 Wstatistic,
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, 197206.
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 stepwise 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 threemode 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, 10371043
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 12781282.
Resolution and determination of complex isomer system: 2,3dimethylphenol,
2,4dimethylphenol, 2,5dimethylphenol, 2.,6dimethylphenol, 3,4dimethylphenol 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 inductivelycoupled plasmaatomic
emissionspectroscopy with an echelle spectrometer
and segmentedarray chargecoupled 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 perkinelmer optima
3000 icp atomic emission spectrometer are
presented. The echellebased polychromator with
segmentedarray chargecoupled 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 twostage procedure incorporating good features of both
trilinear and quadrilinear models. In R. Coppi & S. Bolasco
(Eds.), Multiway data analysis (pp. 123130). Amsterdam:
Elsevier. Published version of paper presented at the annual
Meeting of the Classification Scoiety, St. John, Newfoundland,
Canada, 1985.
Applications of trilinear (PARAFACCANDECOMP) 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 twostage
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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Algemene en Gezinspedagogiek  Datatheorie

Centre for Child and
Family Studies 
Department of Educational
Sciences 
The ThreeMode Company 
ThreeMode bibliography

P.M. Kroonenberg
Leiden Institute of Education and Child Studies, Leiden University
Wassenaarseweg 52, 2333 AK Leiden, The Netherlands
Tel. *31715273446/5273434 (secr.); fax *31715273945
Email:
kroonenb@fsw.leidenuniv.nl
First version : 12/02/1997;