Three-Mode Abstracts, Part T
With one can go to the index of
this part of the Abstracts, with
one can go to other
parts (letters) of the Abstracts.
|Ta | Tb |
Tc | Td |
Te | Tf |
Tg | Th |
Ti | Tj |
Tk | Tl |
Tm | Tn |
To | Tp |
Tq | Tr |
Ts | Tt |
Tu | Tv |
Tw | Tx |
Ty | Tz |
Takane, Y., Young, F.W., & De Leeuw, J. (1977).
Nonmetric individual differences multidimensional scaling: An
alternating least squares method with optimal scaling features.
Psychometrika, 42, 7-67.
A new procedure is discussed which fits either the weighted or simple
Euclidean model to data that may (a) be defined at either the nominal, ordinal, interval or ratio
levels of measurement; (b) have missing observations; (c) be symmetric or
asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f)
be continuous or discrete. Various special cases of the procedure include the most
commonly used indiviual differences multidimensional scaling models, the familiar nonmetric
multidimensional scaling model, and several other previously undiscussed variants. The procedure
optimizes the fit of the model directly to the data (not to scalar products determined from
the data) by an alternating least squares procedure which is convergent, very quick, and
relatively free from local minimum problems. The procedure is evaluated via both Monte Carlo
and empirical data. It is found to be robust in the face of measurement error, capable of
recovering the true underlying configuration in the Monte Carlo situation, and capable of
obtaining structures equivalent to those obtained by other less general procedures in the
Takane, Y., Kiers, H. A. L., & De Leeuw, J. (1995).
Component analysis with different sets of constrainst on different dimensions.
Psychometrika, 60, 259-280.
Many of the ''classical'' multivariate data analysis and
multidimensional scaling techniques call far approximations by lower dimensional
configurations. A model is proposed, in which different sets of linear
constraints are imposed on different dimensions in component analysis and
''classical'' multidimensional scaling frameworks. A simple, efficient, and
monotonically convergent algorithm is presented for fitting the model to the
data by least squares. The basic algorithm is extended to cover across-dimension
constraints imposed in addition to the dimensionwise constraints, and to the
case of a symmetric data matrix. Examples are given to demonstrate the use of
Matrices with special reference to applications in psychometrics.
Linear Algebra and its Applications, 388, 341-361.
Multidimensional scaling, item response theory, and factor analysis may
be considered three major contributions of psychometricians to statistics. Matrix theory
played an important role in early developments of these techniques. Unfortunately,
nonlinear models are currently very prevalent in these areas. Still, one can identify
several areas of psychometrics where matrix algebra plays a prominent role. They include
analysis of asymmetric square tables, multiway data analysis, reduced-rank regression
analysis, and multiple-set (T-set) canonical correlation analysis among others. In this
article we review some of the important matrix results in these areas and suggest future
Takeuchi, H., Kroonenberg, P.M., Taya, H., & Miyano, H.
An analysis of Japanese language on thermal sensation.
Mathematical Linguistics, 15, 201-209.
An analysis was conducted by the semantic differential method to clarify the
relation of the words which express thermal sensation. Fifty subjects were
required to rate twenty-two stimulus words by twenty-six semantic differential
scales. The data were analyzed by three mode principal component analysis. And
the meaning structure on thermal sensation was obtained.
Tan, Y.X., Jiang, J.H., Wu, H.L., Cui, H. & Yu, R.Q. (2000).
Resolution of kinetic system of simultaneous
degradations of chlorophyll a and b by PARAFAC.
Analytica Chimica Acta, 412, 195-202.
A three-way resolution method based on PARAFAC
analysis of fluorescence excitation/emission
matrices (EEMs) is presented to study the black
kinetic system of simultaneous degradations of
chlorophyll a and b extracted with other
interferents from fresh spinach. The excitation
and emission spectral profiles as well as the
kinetic concentration profiles of chlorophyll a,
b and their degradation products, pheophytin a, b
were resolved for this multi-component kinetic
system. The degradation of chlorophyll to
pheophytin under the optimized experimental
conditions is confirmed to follow the first-order
reaction model. The most obvious advantage of
this method is that the complex black
multi-component kinetic system can be resolved
and studied in the presence of unknown
interferents by applying PARAFAC, which is
capable of resolving the three-way data array
provided by EEMs and giving a unique solution to
the analytical problem.
Tanaka, K., & Shinotake, T. (1993).
Analysis of clinical cases adopting three-mode factor analysis.
Japanese Journal of Psychology, 64, 284-288.
Studied sequentially the evaluations of the self
and others in psychotherapy using the rating grid method (a
repertory grid technique). Ss were a 25-yr-old female with
somatization disorder and a 30-yr-old male with conversion
disorder. The evaluations of each S were submitted to a three-mode
Tates, A. A., Louwerse, D. J., Smilde, A. K., Koot, G. L. M., &
Berndt, H. (1999).
Monitoring a PVC batch process with multivariate statistical process control
Industrial & Engineering Chemistry Research, 38, 4769-4776.
Multivariate statistical process control charts
(MSPC charts) are developed for the industrial
batch production process of poly(vinyl chloride)
(PVC). With these MSPC charts different types of
abnormal batch behavior were detected on-line.
With batch contribution plots, the probable
causes of these abnormalities were located.
Examples are given for two different types of
abnormalities, i.e., a bias in the batch loading
and a control valve problem during
polymerization. These examples show that MSPC
charts in combination with batch contribution
plots are a simple and powerful method for fault
detection and identification of their cause in
the operation and control of batch processes.
Tassinari, G., & Vichi, M. (1994).
La dinamica economica dei Paesi avanzati negli anni Ottanta: Riflessioni
sulle traiettorie risultanti dall'analisi di matrici a tre vie. (Economic
Dynamics of Advanced Countries in the Eighties: A Three-Way Matrices
Analysis. With English summary.)
Giornale-degli-Economisti-e-Annali-di-Economia, 53, 101-133.
The paper deals with the comparison of the levels of economic
performances among G7 countries in the 1980-86 period and with the analysis of
their evolution. A three-way matrices analysis method (known as factorial
matrices analysis, FAMA) is used, where the three dimensions are,
respectively, countries, times and economic variables. The results show
that growth rates of the same order might be matched by lacks of balance on
the financial side, so that a synthesis of many economic variables is to be
used to represent the economic performance of a country. The use of FAMA
allows to identify, in every year, the changes in the variables that have
caused the changes in the countries positions.
Tatsunami, K., Kuwabara, R., Yago, N., Mimaya, J., Yamada, K.,
Sato, M. & Sato, Y. (1998).
Application of three-way data clustering to analysis of lymphocyte
subset numbers in Japanese hemophiliacs infected with HIV-1. In A.
Rizzi, M. Vichi & H.-H. Bock (Eds.),
Advances in data science and classification (pp.
627-632). Berlin: Springer
Three-way clustering developed by Sato and Sato (1994) was used to
analyse a number of CD4+ and CD8+ cells from 131 Japanese hemophiliacs
infected with HIV-1 resulting in four clusters. In particular,
times-series data of the CD4+ and CD8+ cell counts were analysed.
Tauler, R. (1995).
Multivariate curve resolution applied to second
Chemometrics and Intelligent Laboratory Systems, 30, 133-146.
Application of multivariate curve resolution to
second order data from hyphenated liquid
chromatography with spectrometric diode array
detection is shown. Chromatographic analysis of
samples giving unresolved mixtures produces
different data structures depending on the
reproducibility of the elution process: (a) second
order data where elution peaks of the same
component in the different chromatographic runs
have the same shape and appear at exactly the same
elution times (synchronized); (b) second order
data where elution peaks of the same component in
the different chromatographic runs appear at
different elution times (non-synchronized)
although they are still of the same shape; and (c)
second order data where elution peaks of the same
component in the different chromatographic runs
have different shapes and appear at different
elution times. Multivariate curve resolution is
easily adapted to analyze all these situations
taking advantage in every case of the particular
data structure. Multivariate curve resolution is
also easily adapted to those situations where
second order data has not a complete trilinear
Tauler, R. (2001).
Calculation of maximum and minimum band boundaries of feasible solutions for
species profiles obtained by multivariate curve resolution.
Journal of Chemometrics, 15, 627-646.
A method for the calculation of maximum and minimum band boundaries of feasible
solutions corresponding to the species profiles estimated by multivariate curve
resolution is presented. The method is based on a non-linear constrained
optimization of an objective function defined by the ratio of the signal
contribution of a particular species to the whole measured signal. Implementation
of constraints such as normalization, closure, nonnegativity, unimodality and
local rank/selectivity during the non-linear optimization and their effect
on the calculation of the feasible solutions are studied in detail. Calculation
of the band boundaries is shown for different simulated and real two-way data
examples of increasing complexity.
Tauler, R., & Barceló, D. (1993).
Multivariate curve resolution applied to
liquid chromatography-diode array detection.
TrAC: Trends in Analytical Chemistry, 12,, 319-327.
Multivariate curve resolution methods can be used
for the quantitation of the overlapping components
in a chromatographic peak. Initial qualitative
solutions obtained by self-modelling curve
resolution methods, such as evolving factor
analysis, can be further optimized by simultaneous
analysis of multiple chromatographic runs with
alternating least-squares regression. Quantitation
is achieved by constraining the pure unit spectra
and elution profiles of the common analytes in the
different chromatographic runs to be equal. The
relationship between the proposed method and other
higher-order calibration and resolution methods is
Tauler, R., & Casassas, E. (1992a).
Application of factor-analysis to speciation in
Analusis, 20, 255-268.**
Multiequilibria systems in solution have
traditionally been investigated by least-squares
curve-fitting methods. These approaches are
difficult to apply to complicated systems because
a chemical model must be defined based on the
constituent species and their constants of
formation. Model-free factor analysis is mainly
seen as an independent method for investigating
the number of species present in the system.
Further developments such as target factor
analysis, rank annihilation, evolving factor
analysis and tensorial resolution, all of which
can elucidate the nature and concentrations of the
species involved, are still, however, very little
used, in spite of the their clear advantages for
understanding multiequilibria systems. A subject
on which many controversies exist in the
literature. Here we summarize results of the
application of our new SPFAC procedure (based on
recent advances in factor analysis techniques) to
the study of multiequilibria systems in solution.
SPFAC is a new and powerful method for the study
of ionic multiequilibria systems; when
macromolecules are present it offers an
independent approach which avoids the rather
cumbersome difficulties intrinsic to the
multifactor contributions in such systems.
Tauler, R., & Casassas, E. (1992b).
Spectroscopic resolution of macromolecular complexes using factor-analysis: CU (II)-
Chemometrics and Intelligent Laboratory Systems, 14, 305-317.
The complexes of Cu(II) ion with polyethyleneimine
formed during a VIS spectroscopic titration of
solutions containing different amounts of Cu(II)
and polyethyleneimine at various pH values have
been studied using a new self-resolving approach
based on different factor analysis techniques,
including evolving factor analysis, error in
factor analysis, cross-validation, target factor
analysis, and rank annihilation. Three different
macromolecular complex species between Cu(II) and
polyethyleneimine were detected in the system, and
their concentration profiles and individual
spectra were estimated without any previous
knowledge of the underlying chemical model (set of
stoichiometric coefficients and set of stability
constants). The method developed is proposed for
use in the deduction of the metal complexing
properties of macromolecular systems.
Tauler, R., Kowalski, B., & Fleming, S. (1993).
Multivariate curve resolution applied to spectral
data from multiple runs of an industrial process.
Analytical Chemistry, 65, 2040-2047.
A method for extracting information from
spectroscopic data gathered during process
monitoring is described and applied to an
industrial problem. The method allows the
estimation of the changes in the concentrations of
the components in the process as well as their
pure spectroscopic responses. Three key aspects of
the new method are as follows: (1) the use of
evolving factor analysis to have an initial
estimation of how the concentrations of the
constituents change during the process; (2) the
implementation of an alternating and constrained
least-squares method to optimize both the spectra
and the concentrations of the components in the
process; (3) the development of a new approach for
the simultaneous analysis of various runs of the
same process to estimate the ratio of
concentrations between the common components in
the different runs.
Tauler, R., Lacorte, S., Guillamon, M., Cespedes, R., Viana, P., &
Chemometric modeling of main contamination sources in surface waters of Portugal.
Environmental Toxicology and Chemistry, 23, 565-575.
Various chemometric data analysis methods, such as principal components analysis,
multivariate curve resolution-alternating least squares, parallel factor analysis,
and Tucker3, are proposed and compared for the resolution and modeling of main
contamination sources in a large environmental data array obtained in an exhaustive
environmental monitoring program that examined the quality of surface waters of
Portugal. The study covered the analysis of 19 priority semivolatile organic compounds
(SVOCs) frequently found in a total number of 644 surface water samples, including
46 different sites from throughout Portugal and corresponding to a period of 14 months,
from April 1999 to May 2000. Main contamination sources of the analyzed SVOCs were
identified and interpreted according to their chemical composition and according to
their resolved geographical and temporal distribution profiles.
Tauler R., Marqués, I. & Casassas, E. (1998).
Multivariate curve resolution applied to three-way trilinear data:
Study of a spectrofluorimetric acid-base titration of salicylic acid at three
excitation wavelengths. Journal of Chemometrics, 12, 55-76.
Application of multivariate curve resolution (MCR) is shown for three-way data
obtained in acid-base titrations of salicylic acid monitored using emission
spectrofluorimetry at three different excitation wavelengths. Rank analysis of
the augmented column- and row-wise matrices showed that the experimental fluorescent
data have a trilinear structure. MCR allows unambiguous recovery of the species
profiles in both orders. Results are compared with those obtained using the trilinear
decomposition and generalized rank annihilation methods.
Tauler, R., Smilde, A.K., Henshaw, J.M., Burgess, L.W., &
Kowalski, B.R. (1994).
Multicomponent determination of chlorinated
hydrocarbons using a reaction-based chemical
sensor. 2. Chemical speciation using multivariate
Analytical Chemistry, 66, 3337-3344.
A new multivariate curve resolution method that
can extract analytical information from UV/visible
spectroscopic data collected from a reaction-based
chemical sensor is proposed. The method is
demonstrated with the determination of mixtures of
chlorinated hydrocarbons by estimating the kinetic
and spectral profiles of the chemical species
formed in the Fujiwara reaction. The three key
aspects of the proposed method are (1) the initial
estimation of the kinetic concentration profiles
from evolving factor analysis; (2) the
implementation of an alternating and constrained
least squares method to optimize the determination
of both the spectral and concentration profiles of
the species present in the reaction, and (3) the
development of a quantitative approach based on
the simultaneous analysis of standards and
unknowns for the determination of the initial
concentration of the analytes in the mixtures.
Tauler, R., Smilde, A., & Kowalski, B. (1995).
Selectivity, local rank, three-way data analysis and
ambiguity in multivariate curve resolution.
Journal of Chemometrics, 9, 31-58.
A new multivariate curve resolution method is
presented and tested with data of various levels
of complexity. Rotational and intensity
ambiguities and the effect of selectivity on
resolution are the focus. Analysis of simulated
data provides the general guidelines concerning
the conditions for uniqueness of a solution for a
given problem. Multivariate curve resolution is
extended to the analysis of three-way data
matrices. The particular case of three-way data
where only one of the orders is common between
slices is studied in some detail.
Tauler, R., Gargallo, R., Vives., M., & Izquierdo-Ridorsa, A. (1999).
Resolution of temperature dependent conformational multiequilibria processes.
Chemometrics and Intelligent Lagoratory Systems, 46, 275-+.
Multivariate Curve Resolution (MCR) is applied to
the study of temperature dependent conformational
multiequilibria evolving processes. Experimental
data sets are obtained by UV spectral monitoring
of the melting behavior of the
acid-poly(uridylic) acid (poly(A)-poly(U)) and of
the hompolynucleotides poly(adenylic) (poly(A))
acid and poly(urydilic) acid (poly(U)), i.e.,
recording UV spectra at different temperatures
during the melting process of these
polynucleotides, Separate study of every
individual melting experiment by MCR did not give
a satisfactory resolution of the
heteropolynucleotide poly(A)-poly(U) melting
process because of unresolved rotational
ambiguities and rank deficiency problems.
Conversely, the simultaneous MCR analysis of the
melting process of poly(A)-poly(U)
heteropolynucleotide together with the separate
melting processes of the poly(A) and poly(U)
homopolynucleotides, allowed the resolution of
the species profiles and the elimination of the
rank deficiency problems present in the
individual analysis of the melting behavior of
Ten Berge, J.M.F. (1972).
Difficulty factors, distribution effects, and the least squares
simplex data matrix solution. Educational and Psychological
Measurement, 32, 911-920.
Horst (1965) raises Ferguson's dilemma again. He introduces a method for
distribution factors. The method fails to deal adequately with these
factors. A distinction is
made between factors resulting from the effect of distribution shape
product-moment correlation, and a factor referring to an attribute like
"capacity for solving
difficult problems." Some well-known alternative solutions for the problem
factors are mentioned.
Ten Berge, J.M.F. (1977).
Orthogonal Procrustes rotation for two or more matrices.
Psychometrika, 42, 267-276.
Necessary and sufficient conditions for rotating matrices to maximal
agreement in the
least-squares sense are discussed. A theorem by Fischer and Roppert,
which solves the case of
two matrices, is given a more straightforward proof. A sufficient
condition for a best
least-squares fit for more than two matrices is formulated and shown to
be not necessary. In
addition, necessary conditions suggested by Kristof and Wingersky are
shown to be not
sufficient. A rotation procedure that is an alternative to the one by
Kristof and Wingersky
is presented. Upper bounds are derived for determining the extent to
which the procedure
falls short of attaining the best least-squares fit. The problem of
scaling matrices to
maximal agreement is discussed. Modifications of Gower's method of
analysis are suggested.
Ten Berge, J.M.F. (1983).
A generalization of Kristof's Theorem on the trace of certain matrix
products. Psychometrika, 48, 519-523.
Kristof has derived a theorem on the maximum and minimum of the trace
of matrix products of the form
where the matrices Li are diagonal and fixed and the
Xi vary unrestri
Kristof has derived a theorem on the maximum and minimum of the trace
of matrix products of
where the matrices Li are diagonal and fixed and the
unrestrictedly and independently over the set of orthogonality
constraints. The present
paper contains a generalization of Kristof's theorem to the case where
are merely required to be submatrices of orthonormal
Ten Berge, J.M.F. (1986a).
Rotatie naar perfecte congruentie en de multipele groep methode.
Nederlands Tijdschrift voor de Psychologie en haar
grensgebieden, 41, 218-225.
This article contains an analysis of component comparisons across
congruence. For any given component in a first population, a parallel
component in any second
population can be defined which has, up to a constant of proportionality,
variable-component correlations or the same component scores coefficients,
provided that the
same variables have been used in the two populations. For this reason,
should not be concerned with amounts of congruence obtained, but should
focus on the behavior
of perfectly congruent components in terms of amounts of variance
explained. From this
perspective, congruence rotation applied to component weights appears to
be closely related to
Multiple Group Analysis. The present article is a non- technical summary
of results published
elsewhere in greater detail, and additionally, contains a description of a
Ten Berge, J.M.F. (1986b).
Rotation to perfect congruence and the cross-validation of
component weights across populations. Multivariate Behavioral
Research, 21, 41-64.
This paper deals with srearegies of congruence studies, aimed at
evaluating recoverableness of
a given set of components from a first population in a second population
where the same
variables have been used. Five decisions inherent to congruence studies
are analysed in
detail. Confirmatory evidence with respect to recoverableness can be
obtained from an
independent component analysis for the second population, parallel to that
of the first
population. Disconfirmatory evidence requires oblique rotation to perfect
can always be attained. Rotation to perfect congruence is advocated as a
new strategy, in
which amounts of variance explained are of major concern. The perfect
congruence strategy can
be applied to variable-component correlations and to weights. The latter
approach is to be
preferred for two reasons. First, rotating weights to perfect congruence
can be easily
understood as a cross-validation method, closely related to the well-known
method. Second, this approach appears to give more satisfactory results in
applications than are obtained from rotating variable-component
correlations to perfect
Ten Berge, J.M.F. (1986c)
Some relationships between descriptive comparisons of components
from different studies. Multivariate Behavioral Research,
This paper considers descriptive methods of comparing components
from different studies, based on correlations between columns of
component scores matrices, on congruence coefficients between
columns of pattern, structure or component scores coefficient
matrices, or on coefficient invariance. Contrary to common
belief, it is shown that coefficients of invariance are unrelated
to correlations between component scores. On the other hand,
having a perfect coefficient of invariance is shown to be
equivalent to having a perfect congruence between corresponding
columns of the component scores coefficient matrices. A similar
but weaker relationship between the latter congruence and
congruence between columns of pattern matrices is demonstrated.
Ten Berge, J.M.F. (1989).
Convergence of PARAFAC preprocessing procedures and the
Deming-Stephan method of iterative proportional fitting. In R.
Coppi & S. Bolasco (Eds.), Multiway data analysis (pp.
53-63). Amsterdam: Elsevier.
Before three-way arrays can be analyzed by PARAFAC, the arrays typically
have to be
preprocessed by removing means and/or standardizing mean squares in
certain directions. This
paper contains a taxonomy of the main preprocessing problems for PARAFAC.
Deming-Stephan method of iterative proportional fitting is shown to
converge, except for a
specific class of matrices. The PARAFAC preprocessing method of iterative
within two modes is expressed in a form equivalent to the Deming-Stephan
the convergence properties obtained for the Deming-Stephan method are
shown to hold for
iterative rescaling within two modes as well.
Ten Berge, J.M.F. (1991a).
A general solution for a class of weakly constrained linear
regression problems. Psychometrika, 56, 601-609.
This paper contains a globally optimal solution for a class of functions
composed of a linear
regression function and a penalty function for the sum of squared
regression weights. Global
optimality is obtained from inequalities rather than from partial
derivatives of a Lagrangian
function. Applications arise in multidimensional scaling of symmetric or
of squared distances, in Procrustes analysis, and in ridge regression
analysis. The similarity
of existing solutions for these applications is explained by considering
them as special cases
of the general class of functions addressed.
Ten Berge, J.M.F. (1991b).
Kruskal's polynomial for 2×2×2 arrays and a generalization to
2×n×n arrays. Psychometrika,
A remarkable difference between the concept of rank for matrices and that
for three-way arrays has to do with the occurrence of non-maximal rank. Kruskal pointed
out that a 2 × 2 × 2 array has rank three or less, and that the subsets of
those 2 × 2 × 2 arrays for which the rank is two or three both have positive
volume. These subsets can be distinguished by the roots of a certain polynomial. The present
paper generalizes Kruskal's results to 2 × n × n arrays.
Incidentally, it is shown that two n × n matrices can be diagonalized
simultaneously with positive probability.
Ten Berge, J. M. F. (2000).
The typical rank of tall three-way arrays. Psychometrika, 65, 525-532.
The rank of a three-way array refers to the smallest number of rank-one arrays (outer
products of three vectors) that generate the array as their sum. It is also the number of components
required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way
array refers to the rank a three-way array has almost surely. The present paper deals with typical rank,
and generalizes existing results on the typical rank of I × J × K arrays with K = 2 to a particular
class of arrays with K ? 2. It is shown that the typical rank is I when the array is tall in the sense that
JK - J < I < JK. In addition, typical rank results are given for the case where I equals JK - J .
Ten Berge, J. M. F. (2004a).
Partial uniqueness in CANDECOMP/PARAFAC.
Journal of Chemometrics, 18, 12-16.
A key property of CANDECOMP/PARAFAC is the essential uniqueness it displays
under certain conditions. It has been known for a long time that, when these conditions are
not met, partial uniqueness may remain. Whereas considerable progress has been made in the
study of conditions for uniqueness, the study of partial uniqueness has lagged behind. The
only well known cases are those of overfactoring, when more components are extracted than
are required for perfect fit, and those cases where the data do not have enough system
variation, resulting in proportional components for one or more modes. The present paper
deals with partial uniqueness in cases where the smallest number of components is
extracted that yield perfect fit. For the case of K x K x 2 arrays of rank K, randomly
sampled from a continuous distribution, it is shown that partial uniqueness, with some
components unique and others differing between solutions, arises with probability zero.
Also a closed-form CANDECOMP/PARAFAC solution is derived for 5 x 3 x 3 arrays when these
happen to have rank 5. In such cases, any two different solutions share four of the five
components. This phenomenon will be traced back to a sixth degree polynomial having six
real roots, any five of which can be picked to construct a solution.
Ten Berge, J. M. F. (2004b).
Simplicity and typical rank of three-way arrays, with applications to Tucker-3 analysis
with simple cores.
Journal of Chemometrics, 18, 17-21.
In chemometric applications of Tucker three-way principal component analysis, core arrays
are often constrained to have a large majority of zero elements. This gives rise to
questions of non-triviality (are the constraints active, or can any core of a given format
be transformed to satisfy the constraints?) and uniqueness (can we transform the components
in one or more directions without losing the given pattern of zero elements in the core?).
Rather than deciding such questions on an ad hoc basis, general principles are to be
preferred. This paper gives an overview of simplicity transformations on the one hand,
and typical rank results on the other, which are suitable to determine whether or not
certain constrained cores are trivial.
Ten Berge, J.M.F., Bekker, P.A., & Kiers, H.A.L. (1994).
Some clarifications of the TUCKALS2 algorithm applied to the IDIOSCAL
problem. Psychometrika, 59, 193-201.
Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model
by the TUCKALS2 algorithm for three-way components analysis. In theory,
this is problematic because TUCKALS2 produces two possibly different
coordinate matrices, that are useless for IDIOSCAL unless they are
equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by
TUCKALS2, the resulting coordinate matrices will be identical. In the
present paper, this claim is proven valid when the data matrices are
semidefinite. However, counterexamples for indefinite matrices are also
constructed, by examining the global minimum in the case where the data
matrices have the same eigenvectors. Similar counterexamples have been
considered by ten Berge and Kiers in the related context of CANDECOMP/
PARAFAC to fit the INDSCAL model.
Ten Berge, J.M.F., De Leeuw, J., & Kroonenberg, P.M. (1987).
Some additional results on principal components analysis of
three-mode data by means of alternating least squares algorithms.
Psychometrika, 52, 183-191.
Kroonenberg and De Leeuw (1980) have developed an alternating least-
squares method TUCKALS-3
as a solution for Tucker's three-way principal components model. This
paper offers some
additional features of their method. Starting from a reanalysis of
Tucker's problem in terms
of a rank-constrained regression problem, it is shown that the fitted sum
of squares in
TUCKALS-3 can be partitioned according to elements of each mode of the
three-way data matrix.
An upper bound to the total fitted sum of squares is derived. Finally, a
special case of
TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC
Ten Berge, J.M.F., & Kiers, H.A.L. (1989a).
Convergence properties of an iterative procedure of ipsatizing and
standardizing a data matrix, with applications to Parafac/Candecomp
preprocessing. Psychometrika, 54, 231-236.
Centering a matrix row-wise and rescaling it column-wise to a
unit sum of squares requires an iterative procedure. It is shown
that this procedure converges to a stable solution. This solution
need not be centered row-wise if the limiting point of the
iterations is a matrix of rank one. The results of the present
paper bear directly on several types of preprocessing methods in
Ten Berge, J.M.F., & Kiers, H.A.L. (1989b).
Fitting the off-diagonal DEDICOM model in the least-squares sense by a
generalization of the Harman and Jones minres procedure of factor
analysis. Psychometrika, 54, 333-337.
Harshman's DEDICOM model provides a framework for analyzing
square but asymmetric matrices of directional relationships among
n objects or persons in terms of a small number of components.
One version of DEDICOM ignores the diagonal entries of the
matrices. A straightforward computational solution for this model
is offered in the present paper. The solution can be interpreted
as a generalized Minres procedure suitable for handling
Ten Berge, J.M.F., & Kiers, H.A.L. (1990).
Simultane componenten-analyse voor twee of meer groepen personen.
Nederlands Tijdschrift voor de Psychologie, 45,
Principal Components Analysis (PCA) is a familiar technique for
summarizing a matrix of standard scores from persons on a set of
variables. Occasionally, researchers have to deal with scores
from several groups of persons on the same variables. In this
paper a generalization of PCA, that provides common components
for groups of persons, is described and illustrated. This
generalization was first suggested by Millsap and Meredith, and
was treated in an alternating least squares framework.
Ten Berge, J.M.F., & Kiers, H.A.L. (1991a).
A numerical approach to the approximate and the exact minimum rank
of a covariance matrix. Psychometrika, 56, 309-315.
A concept of approximate minimum rank for a covariance matrix
is defined, which contains the (exact) minimum rank as a special
case. A computational procedure to evaluate the approximate
minimum rank is offered. The procedure yields those proper
communalities for which the unexplained common variance, ignored
in low-rank factor analysis, is minimized. The procedure also
permits a numerical determination of the exact minimum rank of a
covariance matrix, within limits of computational accuracy. A set
of 180 covariance matrices with known or bounded minimum rank was
analyzed. The procedure was successful throughout in recovering
the desired rank.
Ten Berge, J.M.F., & Kiers, H.A.L. (1991b).
Some clarification of the CANDECOMP algorithm applied to INDSCAL.
Psychometrika, 56, 317-326.
Carroll and Chang have claimed that CANDECOMP applied to
symmetric matrices yields equivalent coordinate matrices, as
needed for INDSCAL. Although this claim had appeared to be valid
for all practical purposes, it has gone without a rigorous
mathematical footing. The purpose of the present paper is to
clarify CANDECOMP in this respect. It is shown that equivalent
coordinate matrices are not granted at global minima when the
symmetric matrices are not Gramian, or when these matrices are
Gramian but the solution not globally optimal.
Ten Berge, J.M.F., & Kiers, H.A.L. (1996a).
Optimality criteria for principal component analysis and
generalizations. British Journal of Mathematical and Statistical
Psychology, 49, 335-345.
Principal components analysis can be derived from various criteria.
Because these give essentially the same results, the question of
which criterion should be used has not received much attention.
Nevertheless, it can be argued that the approach of Pearson and
Eckart & Young, based on variance explained by components, is
more elegant and flexible than the (more popular) approach of
Hotelling, which is concerned with variance that components have, rather
than explain. When two or more correlation or covariance matrices,
based on the same variables, are to be analyzed in generalized component
analysis, the question of which criterion is used becomes of utmost
importance. A taxonomy of generalized principal component methods is
given. It appears than generalized component analysis based on the
Hotelling criterion coincides with one particular generalization based
on the criterion of Pearson and Eckart & Young.
Ten Berge, J.M.F., & Kiers, H.A.L. (1996b).
Some uniqueness results for PARAFAC2. Psychometrika,
Whereas the unique axes properties of PARAFAC1 have been examined
extensively, little is known about uniqueness properties for the
PARAFAC2 model for covariance matrices. This paper is concerned with
uniqueness in the rank two case of PARAFAC2. For this case, Harshman
and Lundy have recently shown, subject to mild assumptions, that
PARAFAC2 is unique when five (covariance) matrices are analyzed. In
the present paper, this result is sharpened. PARAFAC2 is shown to be
usually unique with four matrices. With three matrices it is not unique
unless a certain additional assumption is introduced. If, for instance,
the diagonal matrices of weights are constrained to be non-negative,
three matrices are enough to have uniqueness in the rank two case of
Ten Berge, J.M.F., & Kiers, H.A.L. (1999).
Simplicity of core arrays in three-way principal component analysis and the
typical rank of p × q × 2 arrays. Linear Algebra
and its Applications, 294, 169-179.
Interpreting the solution of a Principal Component Analysis of a three-way array
is greatly simplified when the core array has a large number of zero elements.
The possibility of achieving this has recently been explored by rotations to
simplicity or to simple targets on the one hand, and by mathematical analysis on
the other. In the present paper, it is shown that a p × q
× 2 array, with p greater than q greater than or equal to 2,
can almost surely be transformed to have all but 2q elements zero. It is
also shown that arrays of that form have three-way rank p at most. This
has direct implications for the typical rank of p × q
× 2 arrays, also when p = q. When p is greater than
or equal to 2q, the typical rank is 2q; when q is less than
p is less than 2q it is p, and when p=q, the
rank is typically (almost surely) p or p+1. These typical rank
results pertain to the decomposition of real valued three-way arrays in terms of
real valued rank one arrays, and do not apply in the complex setting, where the
typical rank of p × q × 2 arrays is also
min[p,2q] when p is less than q, but it is p
Ten Berge, J.M.F., Kiers, H.A.L., & Commandeur, J.J.F.
Orthogonal procrustes rotation for matrices with missing values.
British Journal of Mathematical and Statistical Psychology,
A method is offered for orthogonal Procrustes rotation of two
or more matrices with missing values, and an extension to
Generalized Procrustes Analysis is given. The method based on the
idea that missing values, in arbitrary places, can be replaced by
optimal values according nto the least squares criterion.
Commandeur has offered a method of orthogonal Procrustes rotation
and Generalized Procrustes Anlysis for the case where entire rows
of the data matrices are missing etc.
Ten Berge, J.M.F., Kiers, H.A.L., & De Leeuw, J. (1988).
Explicit candecomp/parafac solutions for a contrived 2x2x2 array of
rank three. Psychometrika, 53, 579-584.
Kruskal, Harshman and Lundy have contrived a special 2 x 2 x 2
array to examine formal properties of degenerate
CANDECOMP/PARAFAC solutions. It is shown that for this array the
CANDECOMP/PARAFAC loss has an infimum of 1. In addition, the
array will be used to challenge the tradition of fitting INDSCAL
and related models by means of the CANDECOMP/PARAFAC process.
Ten Berge, J.M.F., Kiers, H.A.L., & Krijnen, W.P. (1993).
Computational solutions for the problem of negative saliences and
nonsymmetry in INDSCAL. Journal of Classification, 10,
Carroll and Chang have derived the symmetric CANDECOMP model from
the INDSCAL model, to fit symmetric matrices of approximate
scalar products in the least squares sense. Typically, the
CANDECOMP algorithm is used to estimate the parameters. In the
present paper it is shown that negative weights may occur with
CANDECOMP. This phenomenon can be surpressed by updating the
weights by the Nonnegative Least Squares Algorithm. A potential
drawback of the resulting procedure is that it may produce two
different versions of the stimulus space matrix. To obviate this
possibility, a symmetry preserving algorithm is offered, which
can be monitored to produce non-negative weights as well.
Ten Berge, J.M.F., Kiers, H.A.L., Murakami, T., & Van der Heijden, R.
Transforming three-way arrays to multiple orthonormality. Journal of
Chemometrics, 14, 275-284.
This paper is concerned with the question to what extent the concept of rowwise
or columnwise orthonormality can he generalized to three-way arrays. Whereas
transforming a three-way array to multiple orthogonality is immediate,
transforming it to multiple orthonormality is far from straightforward. The
present paper offers an iterative algorithm for such transformations, and gives
a proof of monotonical convergence when only two modes are orthonormalized.
Also, it is shown that a variety of three-way arrays do not permit double
orthonormalization. This is due to the order of the arrays, and holds regardless
of the particular elements of the array. Studying three-way orthonormality has
proven useful in exploring the possibilities for simplifying the core, to guide
the search for equivalent direct transformations to simplicity; see Murakami et al. (1998) as an example. Also, it
appears in various contexts of the mathematical study of three-way
Ten Berge, J.M.F., Kiers, H.A.L., & Van der Stel, V. (1992).
Simultaneous components analysis.
Statistica Applicata, 4, 377-392.
Simultaneous Components Analysis is a generalization of Principal
Components Analysis for the situation where the same variables have been measured in two or more
populations. A common matrix of component weights is used to ensure that the components are the
same linear combinations of the same variables in each population. This common matrix
of weights is determined such that the explained variance, summed over populations, is
a maximum. The present paper deals with certain unknown properties of simultaneous
components. The main result is that one always finds simultaneous components that explain a
considerable amount of variance. Furthermore, it will be shown that neither using small samples,
nor replacing optimal weights by simple weights, need to detract seriously from the
usefulness of simultaneous components.
Ten Berge, J.M.F., Knol, D.L., & Kiers, H.A.L. (1988).
A treatment of the orthomax rotation family in terms of diagonalization, and a
re-examination of a singular value approach to varimax rotation. Computational
Statistics Quarterly, 3, 207-217.
The two most familiar members of the Orthomax rotation family are Varimax and
Quartimax rotation. It is well-known that Varimax rotation can be interpreted as
diagonalizing a set of symmetric matrices simultaneously in the least-squares sense.
This paper contains a generalization of this result to the entire family of Orthomax
rotations. It follows that Varimax and Quartimax rotations can be carried out by
applying the same rotation angle formulas to differently derived symmetric matrices.
The Horst-Sherin method of Varimax rotation is based on iterative singular value
decompositions. This method is re-examined as a diagonalization method. It is shown
that the method converges monotonely if the symmetric matrices to be diagonalized
are positive or negative semi-definite, and that this condition can always be implemented
by properly adjusting the diagonal entries of the symmetric matrices. The SVD-approach
can also be generalized to the case of column-wise orthonormal 'rotation' matrices,
thus providing a solution for the orthogonally constrained case of INDSCAL.
Ten Berge, J.M.F., & Sidiropoulos, N. D.(2002a).
On uniqueness in CANDECOMP/PARAFAC.
Psychometrika, 67, 399-409.
One of the basic issues in the analysis of three-way arrays by CANDECOMP/ PARAFAC (CP) has been the question of
uniqueness of the decompositon. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of the
k-ranks of the three component matrices involved is at least twice the rank of the solution plus 2.
Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove
that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity
have not been detected. The present paper gives a method for generating the class of all solutions (or at least a
subset of that class), given a CP ssolution that satisfies certain conditions. This offers the possibility to
examine uniqueness for a great variety of specific CP solutions. It wil be shown that Kruskal's condition is
necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition
is not satisfied, when the rank is four or higher.
Ten Berge, J. M. F. & Smilde, A. K. (2002b)
Non-triviality and identification of a constrained Tucker3 analysis Journal of Chemometrics,
Properties of estimated parameters of models of chemical systems are important.
This paper focuses on two properties of such estimated parameters: triviality and
uniqueness. If a chemical system is analyzed using a Tucker3 model, then the resulting
core can often be rotated to a simple structure containing zeros. This means that
it is possible that a prespecified pattern of zero and non-zero elements of the
core, as used in e.g. constrained Tucker3 models, is not an active constraint, that
is, the zeros can be obtained trivially for free. Once a non-trivial pattern of
zeros in the core is specified, the question arises whether this pattern is sufficient
for obtaining unique loadings. Both issues are discussed in this paper and it is
shown that the model used by Gurden et al. (J. Chemometrics 2001; 15: 101-121) does
essentially involve a non-trivial core and implies rotationally unique parameter
estimates. Copyright (C) 2002 John Wiley Sons, Ltd.
Ten Berge, J. M. F., & Sidiropoulos, N. D. (2002).
On uniqueness in CANDECOMP/PARAFAC.
Psychometrika, 67, 399-409.
One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP)
has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness
is guaranteed when the sum of the k-ranks of the three component matrices involved is at least twice
the rank of the solution plus 2. Since then, little has been achieved that might further qualify
Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except
for rank I or 2) have failed, but counterexamples to necessity have not been detected. The present
paper gives a method for generating the class of all solutions (or at least a subset of that class),
given a CP solution that satisfies certain conditions. This offers the possibility to examine
uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition
is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold
even if the condition is not satisfied, when the rank is four or higher.
Ten Berge, J. M. F., Sidiropoulos, N. D., & Rocci, R. (2004).
Typical rank and indscal dimensionality for symmetric three-way arrays of order Ix2x2 or Ix3x3.
Linear Algebra and its Applications, 388, 363-377.
A peculiar property of three-way arrays is that the rank they typically
have does not necessarily coincide with the maximum possible rank, given their order.
Typical tensorial rank has much been studied over algebraically closed fields. However,
very few results have been found pertaining to the typical rank of three-way arrays over
the real field. These results refer to arrays sampled randomly from continuous
distributions. Arrays that consist of symmetric slices do not fit into this sampling
scheme. The present paper offers typical rank results (over the real field) for arrays,
containing symmetric slices of order 2 x 2 and 3 x 3. Symmetric arrays often appear to
have lower typical ranks than their asymmetric counterparts. This paper also examines
whether or not the rank of a symmetric array coincides with the smallest number of
dimensions that allow a perfect fit of INDSCAL. For all cases considered, this is indeed
true. Thus, a full INDSCAL solution may require fewer dimensions for a symmetric array
than a full CP decomposition applied to an asymmetric array of the same size. The reverse
situation does not seem to arise. Next, we examine in which cases CP solutions inevitably
are INDSCAL solutions. Finally, the rank-reducing impact of double standardizing the
slices is discussed.
Teppola, P., Mujunen, S. P., & Minkkinen, P. (1999).
Addaptive Fuzzy C.Means clustering in process monitoring.
Chemometrics and Intelligent Laboratory Systems, 45, 23-38.
Quite often, quality control models fail because, e.g., the mean values are changing continuously.
These kinds of changes, e.g., process drifts due to seasonal fluctuations, are common in an activated sludge
waste-water treatment plant in Finland. Different Fuzzy C-Means (FCM) clustering algorithms were tested in order to
cope with these kinds of seasonal effects. Firstly, a Principal Component Analysis (PCA) model was constructed in
order to visualize the data set and reduce the dimensionality of the problem. Then, score values of the PCA were
used in the FCM. The cluster centers represented the different process conditions (winter and summer seasons).
Different algorithms were used to update the cluster centers or to give them some flexibility. The testing of
different FCM algorithms was carried out by using a separate test set. The adaptive and the flexible FCM
algorithms were compared to the basic non-adaptive FCM. For both cases, modifications are proposed and a simple
strategy for updating the cluster centers is given.
Teppola, P., & Minkkinen, P. (2000).
Wavelet-PLS regression models for both exploratory data analysis and process monitoring.
Journal of Chemometrics, 14, 383-399.
Two novel approaches an presented which take into account the collinearity among variables and
the different phenomena occurring at different scales. This is achieved by combining partial least squares (PLS)
and multiresolution analysis (MRA). In this work the two novel approaches are interconnected. First, a standard
exploratory PLS model is scrutinized with MRA. In this way, different events at different scales and latent
variables are recognized. In this case, especially periodic seasonal fluctuations and long-term drifting
introduce problems. These low-frequency variations mask and interfere with the detection of small and
moderate-level transient phenomena. As a result, the confidence limits become too wide. This relatively common
problem caused by autocorrelated measurements can be avoided by detrending. In practice, this is realized by
using fixed-size moving windows and by detrending these windows Based on the MRA of the standard model, the
second PLS model for process monitoring is constructed based on the filtered measurements. This filtering is
done by removing the low-frequency scales representing low-frequency components, such as seasonal fluctuations
and other long-term variations, prior to standard PLS modeling. For these particular data the results are shown
to be superior compared to a conventional PLS model based on the non-filtered measurements. Often, model updating
is necessary owing to non-stationary characteristics of the process and variables. As a big advantage, this
new approach seems to remove any further need for model updating, at least in this particular case. This is
because the presented approach removes low-frequency fluctuations and results in a more stationary filtered
data set that is more suitable for monitoring.
Terbeek, D., & Harshman, R. (1971).
Cross-language differences in the perception of natural vowel
sounds. UCLA Working Papers in Phonetics, 19, 26-38.
One of the first applications of Harshman's
Parafac algorithm and program. Twelve by twelve stimulus similarity matrices
were presented to 5 German, 6 Thai and 6 English speakers. Separate Parafac
analyses were used for each language group. The presence or absence of
uniqueness was used to determine the dimensionality.
Teufel, S. (1969).
TUCK, Tuckers Modell einer drei-dimensionalen Faktorenanalyse. Ein
FORTRAN IV-Programm. In F. Gebhardt, Statistische Programme des
DRZ. Teil B: Einzelbeschreibungen. Programm Information PI-33 des
Deutschen Rechenzentrum, Darmstadt.
Thake, A. J., McLellan, P.J. & Forbes, J.F. (1999).
Controller approximation using semidefinite programming.
Industrial & Engineering Chemistry Research, 38, 2699-2708.
Controller design, and the resulting control system performance, is frequently limited by the
capabilities of the available control system hardware. When the control system hardware provides
only a small set of fixed controller structures (e.g., proportional-integral-derivative, PID) or
limited programming capabilities, it may be desirable to use the available functional elements
in a manner such that the implemented fixed structure controller matches the performance of
some reference advanced controller as closely as possible. This paper presents a systematic
method for approximating the performance of an advanced controller with simpler fixed structure
controllers using semidefinite programming. Tuning of the simple, fixed structure controller is
obtained by matching a specific desired behavior of the advanced controller. The proposed method
is illustrated using two case studies: the approximation of a linear quadratic regulator for a
distillation column by a multiloop PID controller and cross-direction profile control for a paper
Thomas, K., & Otway, H.J. (1980).
Public perceptions of energy system risks: Some policy
implications. In T. O'Riordan & K. Turner, Progress in
resource management and environmental planning, Vol. 2 (pp.
109-131). New York: Wiley.
A questionnaire of 39 beliefs with respect to 5 energy sources was administered
to 234 Austrian subjects via a Tucker 3 analysis. A solution of three source
components, three subject components (the first two of which were considered
related to response style), and five belief components was chosen for
interpretation. For details reference is made to an unpublished report Thomas et
al. (1980) [not in bibliography].
Thomas, P.R., & Bain, J.D. (1982).
Consistency in learning strategies. Higher Education,
This article examines the reported use of surface and deep level learning
first-year student teachers at an Australian College of Advanced
Education. Students responded
to a brief questionnaire measuring the learning strategies they adopted in
assessment situations. The article describes the development of this
factorial structure, and the predictive validity of its factors. High
livels of achievement,
on both objective tests and essay assignments, were found to be associated
with the reported
use of deep strategies. Three-mode factor analyses revealed high levels of
consistency in the
strategies reported for various learning contexts, implying that these
behaviours rather than strategic approaches to learning which were
situation specific. The
notion of consistency in learning strategies was considered in light of
suggesting a greater extent of cognitive flexibility.
Thomas, P.R., & Bain, J.D. (1984).
Contextual dependence of learning approaches: The effects of
assessments. Human Learning, 3, 227-240.
A conceptual framework is presented in which learning activities are
described in terms of
cognitive operations applied to various features of study material. This
framework was used to
generate a learning activities questionnaire which college students
completed after each of
four assessments in their course. Three-mode factor analyses of the
learning activities identified transformational, reproductive, and
skimming approaches to
learning. Closed and open-ended assessment contexts were also identified
among course assessments. Finally, three-mode procedures were used to
correlational interactions of the approach and context factors. In
general, the two main
approaches (transformational and reproductive) varied directly in each of
contexts, contrary to the view (e.g. Marton) that they are mutually
Thygesen, L. G., Rinnan, R., Barsberg, S., & Moller, J. K. S. (2004).
Stabilizing the PARAFAC decomposition of fluorescence spectra by insertion of zeros
outside the data area.
Chemometrics and Intelligent Laboratory Systems, 71, 97-106.
The use of fluorescence spectroscopy for recording multiple excitation
and corresponding emission wavelengths and the subsequent technique of analyzing the
resulting fluorescence landscapes is a rather new method as opposed to the use of just a
single excitation wavelength. In a fluorescence landscape, several light-scatter effects
are usually present, and often the part of the landscape containing information on the
chemical and/or physical characteristics of the sample is surrounded by two Rayleigh
scatter lines. When such landscapes are decomposed using parallel factor analysis (PARAFAC) ,
the scatter effects may have detrimental effects on the resolved spectra, especially if
the peaks from the analytes lie close to or on the Rayleigh scatter lines. Normally, all
values close to and outside the Rayleigh scatter lines are set to missing values before
decomposing the fluorescence landscapes by PARAFAC. In this paper, we introduce a novel
pretreatment method applicable for two-dimensional fluorescence landscapes, where instead
of inserting only missing values a mixture of zeros and missing values are inserted close
to and outside the Rayleigh scatter lines. It is shown that, by the use of this technique,
a physically and chemically meaningful decomposition is obtained, and furthermore the
modeling converges faster. Constraining the PARAFAC solution to positive values in all
modes gave results similar to those obtained for the unconstrained model, except that the
loadings where less smooth and the number of iterations before convergence was smaller.
Timmerman, M. E. (2001).
Component analysis of multisubject multivariate longitudinal data.
PhD. Dissertation, Department of Psychology, University of Groningen,
Groningen, The Netherlands.
Longitudinal research; Sampling from the longitudinal axis; Two types of multisubject longitudinal
data: longitudinal two-way data and multiple time series; Two types of multivariate multisubject
2. Component models for longitudinal three-way data
Introduction; Notational issues and some matrix algebraic properties; Component models for three-way
data;Fitting the component models for three-way data; Transformational freedom within the component
models for three-way data; Issues in the application of the component models to three-way data;
Preprocessing three-way data; Choice of a specific model and the numbers of components;
Interpretation of three-way component models applied to longitudinal three-way data.
3. Occasion components as evaluations of latent curves: possibilities for constraints to the
4. The CP and Tucker3 models with smoothness constraints
Introduction; The choice of a smoother; How to smooth in the Tucker3 model and CP model?; Comparing
constrained with unconstrained CP and Tucker3 models; Construction of the data for the simulation
study; Analyses of simulation data; Criteria of interest; Results of the simulation studies;
Empirical example: Learning to read study (I); Discussion and conclusion.
5. Structured latent curve component models for longitudinal three-way
Introduction; Structured latent curve two-way component models for growth data; The SLC two-way
component model for data measured at equal time points; Fitting the SLC two-way component model to
data with equal measurements; The SLC two-way component model for growth data measured at unequal
time points; The SLC Tucker3 model for longitudinal three-way data; Fitting the SLC Tucker3 model
to data; Transformational freedom and interpretation in the SLC Tucker3 model; Empirical example:
Learning to read study (II); Discussion and conclusion.
6. Simultaneous Component Models of Multisubject Multivariate Time
Introduction; Four models for simultaneous component analysis; Preprocessing of raw data before
fitting the SCA model to data; SCA with invariant Pattern (SCA-P); Constrained versions of SCA-P;
SCA with PARAFAC2 constraints (SCA-PF2); SCA with INDSCAL constraints (SCA-IND); SCA with Equal
average Cross-Products constraints (SCA-ECP); Transformational freedom in the SCA-P, SCA-PF2,
SCA-IND and SCA-ECP models; Model selection; Fitting the four SCA models to the data; Fitting the
SCA-PF2 model; Fitting the SCA-IND model to data; Fitting the SCA-ECP model to data; Starting
values of the parameters; Empirical examples of simultaneous component analyses; Empirical example
1: Mood in individuals with Parkinson's disease; Empirical example 2: The Big Five as states;
Discussion and conclusion.
7. Lagged Simultaneous Component Models of Multisubject Multivariate Time Series
Introduction; The dynamic factor model; Lagged SCA models; LSCA-P; LSCA-PF2, LSCA-IND and LSCA-ECP;
Fitting the LSCA models to data; Transformational freedom in the LSCA models; Testing the LSCA-P
and LSCA-IND algorithms; Empirical example: Mood in individuals with Parkinson's disease;
Discussion and conclusion.
Summary; Discussion and future work
Timmerman, M.E., & Kiers, H.A.L. (2000).
Three-mode principal components analysis: Choosing the numbers of components and
sensitivity to local optima. British Journal of Mathematical and Statistical
Psychology, 53, 1-16.
A method that indicates the numbers of components to use in fitting the three-
mode principal components analysis (3MPCA) model is proposed. This method,
called DIFFIT, aims to find an optimal balance between the fit of solutions for
the 3MPCA model and the numbers of components. The achievement of DIFFIT is
compared with that of two other methods, both based on two-way PCAs, by means of
a simulation study. It was found that DIFFIT performed considerably better than
the other methods in indicating the numbers of components. The 3MPCA model can
be estimated by the TUCKALS3 algorithm, which is an alternating least squared
algorithm. In a study of how sensitive TUCKALS3 is at hitting local optima, it
was found that, if the numbers of components are specified correctly, TUCKALS3
never hits a local optimum. The occurrence of local optima increased as the
difference between the numbers of underlying components and the numbers of
components as estimated by TUCKALS3 increased. Rationally initiated TUCKALS3
runs hit local optima less often than randomly initiated runs.
Timmerman, M.E., & Kiers, H.A.L. (2002).
Three-way component analysis with smoothness constraints. Computational Statistics
& Data Analysis, 40, 447-470.
Tucker3 Analysis and CANDECOMP/PARAFAC (CP) are closely related methods for
three-way component analysis. Imposing constraints on the Tucker3 or CP solutions
can be useful to improve estimation of the model parameters. In the present paper,
a method is proposed for applying smoothness constraints on Tucker3 or CP solutions,
which is particularly useful in analysing functional three-way data. The usefulness
of smoothness constraints on Tucker3 and CP solutions is examined by means of a
simulation experiment. Generally, the results of the experiments indicate better
estimations of the model parameters. An empirical example illustrates the use of
smoothness constraints. The constrained model is more stable and easier to interpret
than the unconstrained model. (C) 2002 Elsevier Science B.V. All rights reserved.
Timmerman, M. E., & Kiers, H. A. L. (2003).
Four simultaneous component models for the analysis of multivariate time series
from more than one subject to model intraindividual and interindividual differences.
Psychometrika, 68, 105-121.
A class of four simultaneous component models for the exploratory analysis of
multivariate time series collected from more than one subject simultaneously is discussed. In each
of the models, the multivariate time series of each subject is decomposed into a few series of
component scores and a loading matrix. The component scores series reveal the latent data structure
in the course of time. The interpretation of the components is based on the loading matrix. The
simultaneous component models model not only intraindividual variability, but interindividual
variability as well. The four models can be ordered hierarchically from weakly to severely
constrained, thus allowing for big to small interindividual differences in the model. The use of the
models is illustrated by an empirical example.
Togkalidou, T., Tung, H. H., Sun, Y. K., Andrews, A., & Braatz, RD(2005).
Solution concentration prediction for pharmaceutical crystallization
processes using robust chemometrics and ATR FTIR spectroscopy.
Organic Process Research & Development, 6, 317-322.
In the pharmaceutical industry, a vast number of compounds are
produced by solution crystallization, making the design and development of such
processes of critical importance. The kinetics of crystal growth and nucleation,
the fundamental mechanisms of a solution crystallization process, are strongly
dependent on supersaturation (the difference between solution concentration and
the saturation concentration). The present study uses attenuated total reflection
(ATR) Fourier transform infrared (FTIR) spectroscopy, coupled with robust
chemometric techniques, for the on-line measurement of solution concentration of
two pharmaceutical compounds in multicomponent systems in the presence of
impurities and over a wide range of temperature. To our best knowledge, this is
the first time that ATR FTIR spectroscopy has been applied to a multicomponent
pharmaceutical system. The resulting models show high accuracy, in predicting
the solution concentration and are applied successfully in measuring the solubility
for the cases of cooling and antisolvent crystallization.
Tomasi, G., & Bro, R. (2005).
PARAFAC and missing values.
Chemometrics and Intelligent Laboratory Systems, 75, 163-180.
Missing values are a common occurrence in chemometrics data, and different approaches have been proposed to deal with them. In this
work, two different concepts based on two algorithms are compared in their efficiency in dealing with incomplete data when fitting the
PARAFAC model: single imputation (SI) combined with a standard PARAFAC-alternating least squares (ALS) algorithm, and fitting the
model only to the existing elements using a computationally more expensive method (Levenberg–Marquadt) appropriately modified and
The performance of these two algorithms and the effect of the incompleteness of the data on the final model have been evaluated on the
basis of a Monte Carlo study and real data sets with different amounts and patterns of missing values (randomly missing values, randomly
missing spectra/vectors, and systematically missing spectra/vectors).
The evaluation is based on the quality of the solution as well as on computational aspects (time requirement and number of iterations).
The results show that a PARAFAC model can be correctly determined even when a large fraction of the data is missing (up to 70%), and that
the pattern matters more than the fraction of missing values. Computationally, the Levenberg–Marquadt-based approach appeared superior
for the pattern of missing values typical of fluorescence measurements when the fraction of missing elements exceeded 30%.
Treat, T., McRall, R. M., Viken, R. J., Nosofsky, R. M., MacKay, D. B.,
& Kruschke, J. K. (2002).
Assessing clinically relevant perceptual organization with multidimensional scaling techniques.
Psychological Assessment, 14, 239-252.
Multidimensional scaling (MDS) techniques provide a promising measurement strategy for characteriz-ing
individual differences in cognitive processing, which many clinical theories associate with the
development, maintenance, and treatment of psychopathology. The authors describe the use of deter-ministic
and probabilistic MDS techniques for investigating numerous aspects of perceptual organization,
such as dimensional attention, perceptual correlation, within-attribute organization, and perceptual
variability. Additionally, they discuss how formal quantitative models can be used, in conjunction with
MDS-derived representations of individual differences in perceptual organization, to test theories about
the role of cognitive processing in clinically relevant phenomena. They include applied examples from
their work in the areas of eating disorders and sexual coercion.
Trendafilov, N.T. (2002).
GIPSCAL revisited. A projected gradient approach.
Statistics and Computing, 12, 135-145.
A model for analysis and visualization of asymmetric data-GIPSCAL-is reconsidered
by means of the projected gradient approach. GIPSCAL problem is formulated as
initial value problem for certain first order matrix ordinary differential
equations. This results in a globally convergent algorithm for solving GIPSCAL.
Additionally, first and second order optimality conditions for the solutions
are established. A generalization of the GIPSCAL model for analyzing three-way
arrays is also considered. Finally, results from simulation experiments are
Trendafilov, N.T. & Lippert, R.A.(2002).
The multimode Procrustes problem.
Linear Algebra and its Applications, 349, 245-264.
In this paper, we consider a generalization of the well-known Procrustes problem
relevant to principal component analysis of multidimensional data arrays.
This multimode Procrustes problem is a complex constrained minimization problem
which involves the simultaneous least-squares fitting of several matrices.
We propose two solutions of the problem: the projected gradient approach which
leads to solving ordinary differential equations on matrix manifolds, and
differential-geometric approach for optimization on products of matrix manifolds.
A numerical example concerning the three-mode Procrustes illustrates the
developed algorithms. (C) 2002 Elsevier Science Inc. All rights reserved.
Trevisan, M. G., & Poppi, R. J. (2003).
Determination of doxorubicin in human plasma by excitation-emission matrix fluorescence
and multi-way analysis. Analytica Chimica Acta, 493,69-81.
Direct determination of doxorubicin (DXR), a cytotoxic anthracycline antibiotic,
in human plasma was accomplished based on excitation-emission matrix (EEM) fluorescence
measurements and multi-way chemometric methods based on parallel factor analysis
(PARAFAC) and N-PLS. Several different procedures, such as residual analysis, core
consistency diagnostic (CONCORDIA) and split-half analysis were employed to determine
the correct number of factors in PARAFAC. These procedures converged to a choice of
two factors, attributed to DXR and to the sum of two fluorescence species present
in the plasma. Sample PARAFAC loadings were employed to build a regression model
against concentration, resulting in a RMSECV of 0.060 mug ml(-1). N-PLS using two
factors produced a RMSECV of 0.045 mug ml(-1). Figures of merit (FOM), such as sensitivity
(SEN), selectivity (SEL) and limit of detection (LD) were determined for both PARAFAC
Triandis, H.C. (Ed.) (1972).
The analysis of subjective culture. (pp. 47-51). New York, NY: Wiley-Interscience.
Contains a summary of an unpublished study by Triandis et al.
(1967) on interpersonal attitudes (p.49,50), and a completely
non-technical and non-numerical detailed description of
results of obtained with T3 (H.C. Triandis & V. Vassiliou, A
comparative analysis of subjective culture, p.299-335).
Triandis, H.C. (1977).
Subjective culture and interpersonal relations across cultures.
Annals of the New York Academy of Sciences,
Triandis, H.C., Feldman, J.M., Weldon, D.E. & Harvey, W.M.
Ecosystem distrust and the hard-to-employ. Journal of Applied
Psychology, 60, 44-56.
T3 was used as one of the analyses in a large scale project.
It was employed to examine a concepts x judgments x subjects
matrix. The precise dimensions are not given. The results are
only partially presented, and the effectiveness of the method
for the data is difficult to assess or check.
Triandis, H.C., Malpass, R.S., & Feldman, J. (1976).
Method and a sample of results. In H.C. Triandis (Ed.),
Variations in black and white perceptions of the social
environment. Urbana, IL: University of Illinois Press.
In chapter 5 an overview is given of the methods of data
collection and analysis which are the basis of the other
chapters. As an example, a three-mode analysis of 20
behavioral item scales, 104 role pairs, 89 individuals based
on cross-products matrices is included. Oblique factor
rotations with counterrotation of core matrix.
Triandis, H.C., Tucker, L.R., Koo, P. & Stewart, T. (1967).
Three-mode factor analysis of the behavioral component of
interpersonal attitudes. Technical Report No.50, Department of
Psychology, University of Illinois, Urbana, Ill.
250 subjects from Japan, India and USA responded to semantic
and behavioral differential scales. The interpersonal
attitudes of the subjects were assessed with respect to
stimulus persons varying in sex, age, occupation, and
religion. In part reported in H.C. Triandis (Ed.), 1972, p.49-
Trick, L., & Katz, A.N. (1986).
The domain interaction approach to metaphor processing: Relating
individual differences and metaphor characteristics. Metaphor
and Symbolic Activity, 1, 185-213.
Tourangeau and Sternberg (1981) proposed a sophisticated model to explain
comprehend and appreciate metaphors of the form "Robert Redford is the
peacock of actors".
This study conceptually extended this work by incorporating methodological
permit the examination of individual differences. The paper contains an
PARAFAC analyses and a complete four-factor solution.
Tsuychiya, T. (1996).
Scaling methods for qualitative three-mode data by partitioning items into G groups.
Japanese Journal of Educational Psychology, 44, 425-434.
This paper proposes three scaling methods for qualitative
three-mode three-may data. A real data set collected to
investigate the impression of pictures by the semantic
differential (SD) method is analyzed as numerical examples. The
data consists of three modes; raters, SD items and pictures.
All the models classify I items into G groups to construct uni-
dimensional scale in each group by introducing fuzzy c-means
criterion into homogeneity analysis. In the first model, each
scale score is expressed as a function of a design matrix. The
data is analyzed assuming that scale scores of all raters to
the same picture are equal. Appropriate scales are, however,
not constructed indicating that there are differences among the
raters. In the second model, rank of scale scores is restricted
to R(g) in order to explore the differences of raters. In the
third model, raters are clustered into D groups to find which
raters are different. The selection of R(g) or D is perfomed by
means of increasing the parameter value until appropriate
classification is obtained.
Tu, X.M. (1995).
Nonparametric-estimation of survival distributions with censored initiating
time, and censored
and truncated terminating time: application to transfusion data for
Applied statistics, 44, 3-16.
The analysis of survival data with censored
initiating time, and censored and truncated
terminating time arises in same recent
epidemiological studies. The transfusion-related
acquired immune deficiency syndrome (AIDS) data of
the centers for disease control (CDC) are a
typical example. The initiating time in this case
is the time of infection by the human
immunodeficiency virus and is not observed for
every patient either because of unrecorded
transfusion times or multiple transfusions. The
terminating time here is the onset of AIDS and is
truncated, the result at being able to report
within an observational period only a proportion
of the infected cases which came down with AIDS in
the time period. We consider nonparametric
estimation of the survival as well as the
initiating time distributions assuming that they
are independent and non-informatively censored. We
propose a simple algorithm to obtain the maximum
likelihood estimates for the discrete formulation
of the problem and apply it to estimating the aids
latency and infection distributions for four age
groups of transfusion-related aids from the CDC
Tu, X.M., & Burdick, D.S. (1992).
Resolution of trilinear mixtures: Application in
Statistica Sinica, 2, 577-593.
Recent methodological development in three-way
array analysis allows unique resolution of
multicomponent linear mixtures in the absence of
noise. When noise is present, an iterative
linearization procedure based on a least squares
formulation was proposed by appellof and davidson,
but it requires good initial guesses which may be
difficult to find in practical situations. In the
paper, we describe an alternative procedure based
on eigenanalysis, and we discuss the relationship
between the two procedures. Even though the
alternative procedure does not aim to minimize the
squared residuals, it is however, non-iterative an
may be used to find initial values for the least
squares procedure. An example using phase-resolved
fluorescence spectroscopy data seems to indicate
that only minor improvement may be expected from
using the least squares procedure, and, for
practical purposes, the estimates from the
eigenanalysis procedure may be close enough to
dispense with the least squares procedure.
Tu, X.M., Meng, X.L., & Pagano, M. (1994).
On the use of conditional maximization in
Journal of Chemometrics, 8, 365-370.
The purpose of this short communication is to
illustrate the use of conditional maximization
(CM) in chemometric applications. The CM algorithm
is useful in reducing the computational complexity
when a high-dimensional and complicated
maximization problem arises from fitting
chemometric models. It can also be efficiently
combined with the expectation-maximization (EM)
algorithm for handling incomplete data, a problem
that sometimes arises when only a part of the
intended data can be collected. Three models from
fluorescence spectroscopy are used for
Determination of parameters of a functional relation by factor analysis.Psychometrika, 23,
Consideration is given to determination of parameters of a
functional elation between two variables by the means of factor analysis techniques.
If the function can be separated into a sum of products of functions of the
individual parameters and corresponding functions of the independent
variable, particular values of the functions of the parameters and of the
functions of the independent variables might be found by factor analysis.
Otherwise approximate solutions may be determined. These solutions may
represent important results from experimental investigations.
Tucker, L.R. (1963a).
Implications of factor analysis of three-way matrices for
measurement of change. In C.W. Harris (Ed.), Problems in
measuring change. Madison: University of Wisconsin Press,
The first introduction in the literature of T3. The main
principles are presented, and some computational problems are
discussed. The missing details appear in later papers (Tucker,
1964, 1966; Levin, 1965), illustrated with an articial
Tucker, L. R. (1963b).
An individual differences model for multidimensional scaling.Psychometrika, 28,
A quantitative system is presented to permit the determination of
separate multidimensional perceptual spaces for individuals having different
viewpoints about stimulus interrelationships. The structure of individual
differences in the perception of stimulus relationships is also determined to
provide a framework for ascertaining the varieties of consistent individual
viewpoints and their relutionships with other variables.
Tucker, L.R. (1964).
The extension of factor analysis to three- dimensional matrices.
In H. Gullikson & N. Frederiksen (Eds.),
Contributions to mathematical psychology. New
York: Holt, Rinehart and Winston (pp. 110-119).
A more detailed discussion of T3 than Tucker's (1963) paper,
but the mathematics are still somewhat awkward. The
transformational freedom in the model is also treated in more
detail. The same artificial example is used as in 1963.
Tucker, L.R. (1966a).
Experiments in multimode factor analysis. In
A. Anastasi (ed.), Testing Problems in Perspective (pp. 369-
Washington, DC: American Council on Education. Reprinted from:
Proceedings of the 1964 Invitational Conference on
Problems. Princeton, N.J.: Educational Testing Service, 1965
Presents a very global description of T3. Illustrated by
analyses presented in detail in Hoffman & Tucker (1964),
Tucker (1967), and Levin (1963).
Tucker, L.R. (1966b).
Learning theory and multivariate experiment. Illustration by
determination of generalized learning curves. In R.B. Cattell
(Ed.), Handbook of multivariate experimental psychology
(pp. 476-501). Chicago, IL: Rand McNally.
This paper is a precursor of Tucker's work in three-mode component analysis.
Several groups were analysed separately and the results were informally compared
after rotation for learning curves.
Tucker, L.R. (1966c).
Some mathematical notes on three-mode factor analysis.
Psychometrika, 31, 279-311.
The basic paper on T3 which presents a detailed and mathema-
tically coherent description of the model. Also a full
discussion on the freedom of rotation is given. It contains an
outline of the notation which is used by many later authors,
and introduces some new terminology. Three computational
methods are presented, all in the spirit of principal
component analysis rather than factor analysis. Method I is a
straightforward application of the basic formulas. Method II and III are
applicable to the
analysis of data with one (very)
large mode. Method III is particular appropriate for
'multitrait-multimethod' type matrices. An extension of the
general method is described in which allowance is made for
unique variances for each combination variable. This model
uses a version of Method III for the analysis. A fictitious
body of data is used to illustrate several points.
Tucker, L.R. (1967).
Three-mode factor analysis of Parker-Fleishman complex tracking
behavior data. Multivariate Behavioral Research,
T3 was used to analyse data from a simulation of tracking
during airborne radar intercept missions (4 accuracy measures,
10 stages of practices and 203 individuals). More in
particular the available multimeasures-multistage method
matrix of correlations with estimated communalities was used
with Tucker's method III (1966). Extensive information on, and
interpretation of the solution.
Tucker, L.R. (1972a).
Relations between multidimensional scaling and three-mode factor
analysis. Psychometrika, 37, 3-27(a).
The presented multidimensional scaling variant of T3, which
later came to be called 'three-mode scaling', assumes that the
first two modes are equal, and that the input consists of a
scalar product matrix for each of subjects. This results in an
object space, a person space and a core matrix, of which the
frontal planes contain the subject components' weights for the
components of the common object space as well as their angles
between those dimensions. Also the extended core matrix is in-
troduced as an aid for interpretation. Some special transfor-
mations of the core matrix are suggested. The technique is
illustrated with data from an adjective similarity study (all
66 pairs from 12 adjectives were judged as to their similarity
by 87 students). The adjectives were designed to form a
circle, which indeed was found, but not properly recognized.
The interpretation of the core matrix and person space was
made by using 'conceptual' individuals.
Tucker, L.R. (1972b).
Use of three-mode factor analysis in MDS. Paper presented at the
Workshop on Multidimensional Scaling, University of Illinois, 7-10
Summary presentation of theory in Tucker (1972a).
Illustrations from Helm's color data and Wish' relations-
Tucker, L.R. (1975).
Three-mode factor analysis applied to multidimensional scaling.
Paper presented to the U.S.-Japan Seminar on Theory, Methods, and
Applications of Multidimensional Scaling and Re- lated Techniques,
La Jolla, Calif., August 20-24.
A concise overview of three-mode factor analysis, the T2
model, three-mode scaling. Illustrated with artificial
Tucker, L.R. (1992).
Remarks on the study of the variety of individuals. Multivariate
Behavioral Research, 27, 635-647.
Techniques for the study of the variety of individuals are
discussed. A contrast is pointed out between the study of mean
performances and the study of covariations. These are two
components of the performance of individuals. For the study of
the variety of individuals, instead of keeping these two
components seperate, these two components should be combined into
a single analysis. Examples of techniques which combine these two
components are discussed.
Tucker, L.R. & Messick, S. (1963).
An individual differences model for multidimensional scaling.
Psychometrika, 28, 333-367.
The point-of-view approach to the analysis of individual
differences for dissimilarity data (a fore-runner of T2/T3) is
developed. The subjects are factored to construct 'idealized
individuals' representing different points-of-view. The
coordinates of the 'idealized individuals' in the subject
space are used to construct estimated distances of all
stimulus-pairs for each of the idealized individuals. Separate
MDS analyses are performed to obtain separate stimulus spaces
for each 'point-of-view'. Illustrated with political judgment
Turecek, F., & Gu, M. (1995).
Differentiation of olefin isomers by survivor-ion
Journal of mass spectrometry, 30, 144-152.**
Survivor-ion mass spectrometry is a method that relies on the selective
non-dissociating ions that underwent collisional neutralization acid
charge-permutation processes are found to modulate the relative intensities
of precursor ions,
both molecular ions and fragments, formed by electron impact from
C6H12, C6H12O, and xylene
isomers. The relative
intensities of highly-unsaturated ions, such as
C4H2+., C3H3+, and
and enol ions are enhanced in the survivor-ion spectra in dependence on the
Isomer differentiation through survivor-ion spectra is achieved for
1,4-hexadiene and cyclohexene, but not for hexenols and xylenes.
Tzeng, O.C.S. (1972).
Differentiation of affective and denotative meaning systems in
personality ratings via three-mode factor analysis. Unpublished
doctoral thesis, University of Illinois, Urbana - Champaign.
( Dissertation Abstracts International, 1973, 34
(See Tzeng, 1975, 1977a).
Tzeng, O.C.S. (1975).
Differentiation of affective and denotative meaning systems and
their influence in personality ratings. Journal of
Personality and Social Psychology, 32,
Fifty Belgian males scored 40 person concepts on 40 semantic
scales to investigate person perception. Component analysis
loadings from T3 were rotated by a complex scheme on so-called
marker scales of the E.P.A. dimensions and on non-marker
scales with the object to separate the affective from the
denotative semantic space. In conjunction with the core matrix
three subject components were interpreted.
Tzeng, O.C.S. (1977a).
Differentiation of affective and denotative semantic subspaces.
Annals of the New York Academy of Sciences,
In each of four countries 80-160 subjects scored 24-42
concepts on 40-60 semantic differential scales. The scales
were separated in marker (affective) and non-marker
(denotative scales); see Tzeng, 1975. T3 was applied for each
country. Comparisons between the component matrices and the
core matrices of the four countries are given.
Tzeng, O.C.S. (1977).
Individual differences in self-conception: multivariate approach.
Perceptual and Motor Skills, 45,
Investigation of determinants of the affective and denotative
semantic structures in the process of self-conception. T3 ap-
plied to semantic differential ratings of 11 ego-related con-
cepts on a representative set of 29 (?) scales from a homoge-
neous group of 29 male college sophomores. The retained
factors (after varimax) were correlated with other variables.
Virtually no numerical details.
Tzeng, O. C. S. (1994).
Insufficiency of product-moment correlation for factor-analysis of bipolar measurements.
Perceptual and Motor Skills, 79, 1647-1665.
This study examined the problems in using product-moment
correlations for the 2 general functions of factor analysis,
identification (or verification) and use of psychological
constructs embedded in bipolar ratings. For illustration, a
set of fictitious semantic differential ratings, with known
structural similarities and differences between concept
profiles, were factored. The results were assessed against 12
criteria for evaluation derived from simultaneous
considerations of issues in statistics, measurement, and
psychosemantics. It was concluded that solutions from factor
analysis of r coefficients do not identify or verify underlying
attributional relationships among concepts implicit in
subjects' cognitive processes and manifest in ratings. An
alternative index, Beta Coefficient, was then used to
illustrate its advantage in analysis of the hypothetical
bipolar ratings. Implications of the present study are
Tzeng, O. C., & Everett, A.V. (1985).
A cross-cultural perspective of self-related conceptions in
adolescence. International Journal of Psychology, 20,
Compared intercultural similarities and differences in
conceptions of 5 social determinants involving adolescents from
30 language/culture communities. The ratings of 30 concepts from the
Atlas that represent the 5 major problem areas of adolescence
(work, social and sexual identities, economy, and a philosophy of
life) were analyzed with the 3-mode multidimensional scaling with
points-of-view solution technique developed by the 1st author and
Tzeng, O.C.S. & Landis, D. (1978).
Three-mode multidimensional scaling with points of view solutions.
Multivariate Behavioral Research, 13,
A rather hybrid stringing together of multivariate procedures
called "three-mode point of view analysis" (3M-POV). Including
in sequence Tucker's (1972) analysis, hierarchical clustering,
averaging over individuals in a cluster to obtain coefficients
for an 'idealized individual', computing frontal core planes
for these idealized individuals, subjecting these planes to an
eigen-decomposition and some additional rotation(s).
Illustrated with data from Osgood's cross-cultural
Tzeng, O.C.S., & Osgood, C.E. (1976).
Validity tests for componential analyses of conceptual domains: A
cross-cultural study in methodology. Behavioral Science,
An analytic prediction model for evaluation of the validities of intuited
in componential analyses is outlined and illustrated for a set of concepts
referring to time.
The intuited components function as independent variables and the measured
of the time-related concepts function as dependent variables, both
represented in Euclidean
spaces. Predictions of attributions of affect from the denotative
components proved to have
very high predictive, construct and content validities for both the cross-
cultural (mean) and
the majority of the culture-indigenous interconcept squared distance
matrices. Although this
model for validity checking is limited to those components which are
predictive of affect
attribution, the external criterion, it represents a type of social system
analysis in which
objective culture as reflected in the denotative components for a
conceptual domain is related
to subjective culture, i.e., evaluative, potency and activity feelings,
via the mediation of
the cognitive systems of the individual human beings who make the semantic
judgments. Similarities and differences in both objective (weights given
to components) and
subjective (variations in affect attribution) cultures can be
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Institute of Education and Child Studies |
The Three-Mode Company |
Three-Mode bibliography |
Faculty of Social and Behavioural Sciences, Leiden University
The Three-Mode Company, Leiden, The Netherlands
E-mail: kroonenb at fsw.leidenuniv.nl
First version : 12/02/1997;