Three-Mode Abstracts, Part K
With one can go to the index of
this part of the Abstracts, with
one can go to other
parts (letters) of the Abstracts.
|Ka | Kb |
Kc | Kd |
Ke | Kf |
Kg | Kh |
Ki | Kj |
Kk | Kl |
Km | Kn |
Ko | Kp |
Kq | Kr |
Ks | Kt |
Ku | Kv |
Kw | Kx |
Ky | Kz |
Kapteyn, A., Neudecker, H., & Wansbeek, T. (1982).
An approach to n-mode components analysis. Psychometrika,
As an extension of Lastovicka's four-mode components analysis an n-mode
components analysis is developed. Using a convenient notation, both a
canonical and a least squares solution are derived. The relation between
both solutions and their computational aspects are discussed. An n-mode
extension of the three-mode model and algorithm developed by Kroonenberg
& De Leeuw (1980) is proposed. By using some new notation an elegant
and concise presentation of the theory is given. Their model is compared
with the standard Tucker (1966) approach and its extension to four modes
by Lastovicka (1981).
Karnas, G. (1975).
Note sur une procédure d'analyse de
données relatives à une correspondance ternaire ou pseudo-ternaire par la
méthode d'analyse binaire de Faverge. Le Travail Humain, 38,
Proposal to string out a three-mode matrix in one of three
ways with the object to perform on the resultant (two-mode)
matrix a variant of the singular value decomposition (called
the binary method of Faverge). One of the stringing-out
proposals is identical to forming a combination-mode (Tucker,
1966). The other two are variants of the same idea.
Illustrated with beer appreciation data, and data of tram
conductors judging aspects of their profession.
Karukstis, K.K., Suljak, S.W., Waller, P.J., Whiles, J.A.,
& Thompson, E.H.Z. (1996).
Fluorescence analysis of single and mixed micelle systems of SDS
and DTAB. Journal of Physical Chemistry, 100,
Structural features of the single and mixed micellar systems of sodium
dodecyl sulfate (SDS) and dodecyltrimethylammonium bromide (DTAB) were
characterized using the fluorescence probe 6-propionyl-2-(dimethylamino)naphthalene
(Prodan). In particular, our investigations capitalized on the spectral
sensitivity of Prodan to its environment as well as the extensive solubility
of Prodan in solvents of varying polarity and/or hydrophobicity to effectively
use a three-mode factor analysis technique to resolve the coincident emission
from Prodan in multiple microenvironments of single and mixed micelle
Kelly, E.F., Lenz, J.E., Franaszczuk, P.J., & Truong, Y.K.
A general statistical framework for
frequency-domain analysis of EEG topographic
Computers and Biomedical Research, 30, 129-164.**
A wide variety of rhythmic electrophysiological phenomena-including driven,
induced, and endogenous activities of cortical neuronal masses-lend themselves
naturally to analysis using frequency-domain techniques applied to multichannel
recordings that discretely sample the overall spatial pattern of the rhythmic
activity. For such cases, a large but so far poorly utilized body of statistical
theory supports a third major approach to topographic analysis, complementing
the more familiar mapping and source-recovery techniques. These methods, many
of which have only recently become computationally feasible, collectively
provide general solutions to the problem of detecting and characterizing
systematic differences that arise-not only in the spatial distribution of
the activity, but also in its frequency-dependent between-channel covariance
structure-as a function of multiple experimental conditions presented in
conformity with any of the conventional experimental designs. This application-
oriented tutorial review provides a comprehensive outline of these resources,
including: (1) real multivariate analysis of single-channel spectral measures
(and measures of between-channel relationships such as coherence and phase),
(2) complex multivariate analysis based on multichannel fourier transforms,
and (3) complex multivariate analysis based on multichannel parametric models.
Special emphasis is placed on the potential of the multichannel autoregressive
model to support eeg (and meg) studies of perceptual and cognitive processes.
(c) 1997 academic press.
Kettenring, J. R. (1971).
Canonical analysis of several sets of variables.
Biometrika, 58, 433-460.
Five extensions of the classical two-set theory of canonical correlation
analysis to three or more sets are considered. For each one, a model of the general principal
component type is constructed to aid in motivating, comparing and understanding the methods.
Procedures are developed for finding the canonical varialbles associated with the different
approaches. Some practical considerations and an example are also included.
Keramidas, E.M., Devlin, S.J., & Gnanadesikan, R. (1987).
A graphical procedure for comparing the principal components of
several covariance matrices. Communications in Statistics, Part B -
Simulation and Computation, 16, 161-191.
Principal components analysis is an extensively used tool for reduction
of dimensionality in multivariate analyses. In many applications, however, little attempt is
made to compare principal components solutions (i.e., eigenvectors) across many samples.
Methods are needed for assessing the degree of similarity of corresponding eigenvectors, a
problem that is meaningful in the presence of clearly separated eigenvalues. This paper
proposes a gamma probability plotting procedure for a measure of the angle between a pair
of eigenvectors, or equivalently, the distance between points on the unit sphere defined by
such vectors. One of the vectors in the pair is the principal component of a sample, and the
other can be either a prespecified vector or a "typical" vector obtained from the corresponding
eigenvectors in all samples. Simulations, as well as real-data examples, are
Kettenring, J.R. (1983a).
A case study in data analysis. Proceedings of Symposia in
Applied Mathematics, 28, 105-139.
A set of data consisting of annual revenues of Bell operating companies
broken down into several components is analyzed using two different models.
The intent is not only to describe the results but also to convey as many
facets of the data analytic process as possible.
Kettenring, J.R. (1983b).
Components of interaction in analysis of variance models with no
replications. In P.K. Sen (Ed.), Contributions to Statistics:
Essays in Honour of Norman L. Johnson (pp. 283-297). Amsterdam:
An old problem in statistics is how to properly analyze data in a multi-
way table with only
one observation per cell. Standard analysis of variance models for such
data, which include
all possible "main effect" and "interaction" terms, are unsatisfactory
because the number of
independent parameters involved equals the number of data points.
two-way tables, a
popular antidote is to impose a particular structural form on the
interactions. One approach,
suggested by Gollob (1968) and Mandel (1971), is based on the singular
value decomposition of
the matrix of two-way interaction terms estimated from the data.
Probability plots of the
singular values from this decomposition vs. computer-generated null
values provide an
effective way of informally implementing this approach.
these ideas to deal
with three-way interactions are also feasible. In some contexts, it makes
sense to treat
these interactions as if they were layers of two-way interactions and to
apply methods which
exploit this viewpoint. More generally, an extended form of the singular
based on the work of Harshman (1970) and Carroll and Chang (1970), can be
applied to these
interactions although this method is not as simple as in the two-way
case. Each of these
approaches could be broadened to treat higher-way interactions as
develops all of the methods mentioned and illustrates them using a three-
way array of revenue
data for Bell System operating telephone companies.
Kiers, H.A.L. (1988).
Comparison of "Anglo-Saxon" and "French" three-mode methods.
Statistique et Analyse des Données, 13, 14-
Seven methods for the analysis of three-mode data are described in terms
of minimizing loss
functions. On the basis of this description global and specific
comparisons are made between
a number of the methods presented here. Special attention is paid to the
french methods and anglo-saxon methods.
Kiers, H.A.L. (1989a).
INDSCAL for the analysis of categorical data. In R. Coppi & S.
Bolasco (Eds.), Multiway data analysis (pp. 155-167).
A method for PCA of categorical variables is proposed which does not only
yield loadings for
the categorical variables, but also coordinates for the observation
units. The method is
based on INDSCAL applied to a set of matrices whose entries are
similarities between the
observation units with respect to the categorical variables. The
resulting methods is shown
to be a compromise between Multiple Correspondence Analysis and a method
proposed by Cazes,
Bonnefous, Baumerder and Pagès. Some properties of the method
proposed here are
Kiers, H.A.L. (1989c).
An alternating least squares algorithm for fitting the two- and three-
way DEDICOM model and the idioscal model. Psychometrika,
The DEDICOM nodel is a model for representing asymmetric relations
a set of objects by means of a set of coordinates for the objects on a
number of dimensions. The present paper offers an alternating least
algorithm for fitting the DEDICOM model. The model can be generalized to
represent any number of sets of relations among the same set of objects.
algorithm for fitting this three-way DEDICOM model an algorithm is
for fitting the IDIOSCAL model in the least squares sense.
Kiers, H.A.L. (1989d).
A computational short-cut for INDSCAL with orthonormality
constraints on positive semi-definite matrices of low rank.
Computational Statistics Quarterly, 5, 119-135.
When INDSCAL with orthonormality constraints (INDORT) is to be applied
to a set of very
large similarity matrices, one is confronted with huge computational
problems. When those
very large matrices are positive semi-definite (p.s.d.) matrices of low
can be facilitated to a large extent. The present paper describes a
simplified algorithm for
the INDORT analysis of such matrices. One of the applications of INDORT
on low-rank p.s.d.
matrices is that of INDORT on a set of quantification matrices for
quantitative variables. The implications of using the simplified
algorithm for this INDORT
analysis are worked out. It is concluded that, in case INDORT is applied
matrices for the qualitative variables exclusively, the computation of
the solution uses
only the categories frequencies and the bivariate frequencies of pairs
of categories from
two different variables. In this way INDORT for qualitative data can be
applied when one
only has the total bivariate contingency table for all
Kiers, H.A.L. (1990a).
Procrustes-pc v2.0: Een programma voor gegeneraliseerde
Procrustes-analyse. [Procrustes-pc v2.0: A program for generalized
Procrustes-analysis.] Kwantitatieve Methoden, 11,
In this paper a description and evaluation are given of a pc-program for
Procrustes-analysis. After a short discussion of the aim of generalized
the possibilities and restrictions of the program are discussed. The
user-friendliness of the program are looked into using several simple
Kiers, H.A.L. (1991a).
Hierarchical relations among three-way methods.
Psychometrika, 56, 449-470.
Several methods have been developed for the analysis of a mixture of
quantitative variables, and one, called PCAMIX, includes ordinary
principal component analysis
(PCA) and multiple correspondence analysis (MCA) as special cases. The
present paper proposes
several techniques for simple structure rotation of a PCAMIX solution
based on the rotation of
component scores and indicates how these can be viewed as generalizations
of the simple
structure methods for PCA. In addition, a recently developed technique for
the analysis of
mixtures of qualitative and quantitative variables, called INDOMIX, is
shown to construct
possible sets of component scores. A numerical example is used to
illustrate the implication
that when used for qualitative variables, INDOMIX provides axes that
discriminate between the
observation units better than do those generated from MCA.
Kiers, H.A.L. (1991b).
Simple structure in component analysis techniques for mixtures of
qualitative and quantitative variables. Psychometrika, 56,
The present paper proposes several techniques for simple structure
a PCAMIX solution based on the rotation of component scores and indicates
these can be viewed as generalizations of the simple structure methods
In addition, a recently developed technique for the analysis of mixtures
qualitative and quantitaive variables, called INDOMIX, is shown to
component scores (without rotational freedom) maximizing the quartimax
over all possible sets of component scores. A numerical example is used
illustrate the implication that when used for qualitative variables,
provides axes that discriminate between the observation units better than
generated from MCA.
Kiers, H.A.L. (1992).
TUCKALS core rotations and constrained TUCKALS modelling.
Statistica Applicata, 4, 659-667.
TUCKALS3 is a method for analyzing a three-way data set. The method
decomposes the data into component matrices for all three sets of
The solution for the component matrices and the core array are determined
a (possibly oblique) rotation only. In the present paper some procedures
proposed for rotating the solution such that the core becomes optimally
a particular way. An alternative procedure for obtaining a simple core is
to fit the
TUCKALS3 model subject to the constraint that the core has zeros at
places. An algorithm for this procedure is sketched and the procedure is
by means of an example.
Kiers, H.A.L. (1993a).
An alternating least squares algorithm for PARAFAC2 and three-way
DEDICOM. Computational Statistics and Data Analysis, 16,
PARAFAC2 is a method for analyzing three-way data consisting of
frontal slices. Three-way DEDICOM can be considered a generalization of
PARAFAC2 in that it fits essentially the same model to three-way data
of square frontal slices that may be asymmetric. In the present paper, an
alternating least squares algorithm is developed for three-way DEDICOM,
algorithm for PARAFAC2 is derived from it. The performance of the
is studied for some empirical and synthetical data.
Kiers, H.A.L. (1993b).
A comparison of techniques for finding components with simple structure.
In C.M. Cuadras & C.R. Rao (Eds). Multivariate Analysis: Future
(pp. 67-86). Amsterdam: Elsevier.
Principal component analysis (PCA) is usually followed by rotation to
simple structure to
facilitate the interpretation of the components. Recently, some
alternatives to PCA have been
developed, in which simple structure is part of the criterion optimized,
and is no longer seen
as a secondary objective. These methods, Principal cluster components
INDOMIX, 'Varimax Optimization' and INDSCAL applied to quantification
quantitative variables, are designed for situations where, apart from fit,
parsimony of the
solution is deemed valuable as well. In the present study, these
techniques are described in
some detail, and compared on theoretical grounds. Next, in a simulation
study, the methods are
compared to PCA followed by three different simple structure rotations
(Varimax, Promax, and
Orthoblique rotation). All techniques have been applied to artificial data
sets with a known
simple structure, to which noise had been added. It is studied to what
extent the original
structure is recovered, how 'simple' the different solutions are, and how
a possible gain in
simplicity is offset by a loss of fit. The main conclusion is that, if it
is desired to gain
simplicity and one is prepared to incur a small loss of fit, then INDOMAX
Optimization should not be used, but both PCCA and INDSCAL can be used,
with a slight
preference for INDSCAL because it gives a better ratio of gain of
simplicity versus loss of
fit. The paper is concluded by the INDSCAL analysis of an empirical data
Kiers, H.A.L. (1993c).
Handling ordinal variables in three-way analysis of
quantification matrices for variables of mixed measurement levels.
British Journal of Mathematical and Statistical Psychology,
For the analysis of variables of mixed measurement levels a class of
methods can be used that
is based on three-way analysis of quantification matrices for nominal or
variables. This class of methods incorporated some well-known techniques
but also offers a
series of interesting new alternatives for the analysis of nominal or
Ordinal variables have received hardly any attention in this class of
methods, and are
usually treated as if they are quantitative variables. In the present
paper this gap is
filled by constructing quantification matrices for ordinal variables via
optimal scaling of
the ordinal variables, thus yielding optimal quantification matrices for
Algorithms for this optimal scaling procedure are developed, and the
procedures are compared to optimal scaling of ordinal variables in the
context of principal
components analysis and multidimensional scaling.
Kiers, H.A.L. (1993d).
The exploratory analysis of qualitative variables by means of
three-way analysis of two types of quantification matrices.
Applied Stochastic Models and Data Analysis, 9, 301-317.
A comparison is made between a number of techniques for the exploratory
analysis of qualitative variables. The paper mainly focuses on a comparison between
multiple correspondence analysis (MCA) and Gower's principal co-ordinates analysis
(PCO), applied to qualitative variables. The main difference between these methods is in
how they deal with infrequent categories. It is demonstrated that MCA solutions can be
dominated by infrequent categories, and that, especially in such cases, PCO is a useful
alternative to MCA, because it tends to downweight the influence of infrequent categories. Apart from
studying the difference between MCA and PCO, other alternatives for the analysis of
qualitative variables are discussed, and compared to MCA and PCO.
Kiers, H. A. L. (1995).
Maximization of sums of quotients of quadratic-forms and some generalizations.
Psychometrika, 60, 221-245.
Monotonically convergent algorithms are described for maximizing
six (constrained) functions of vectors x, or matrices X with columns x(1),..., x(r). T
hese functions are h(1)(x) = Sigma(k) (x'A(k)x)(x'C(k)x)(-1), H-1(X) = Sigma(k) tr
(X'A(k)X)(X'C(k)X)(-1), (h) over tilde(1)(X) = Sigma(k) Sigma(l)(x'(l)A(k)x(l))(x'(l)C(k)x(l))(-1)
with X constrained to be columnwise orthonormal, h(2)(x) = Sigma(k) (x'A(k)x)(2)(x'C(k)X)(-1)
subject to x'x = 1, H-2(X) = Sigma(k) tr (X'A(k)X)(X'A(k)X)'(X'C(k)X)(-1) subject to X'X = I, and
(h) over tilde(2)(X) = Sigma(k) Sigma(l) (x'(l)A(k)x(l))(2)(x'(l)C(k)x(l))(-1) subject to X'X = I.
In these functions the matrices C-k are assumed to be positive definite. The matrices A(k) can be
arbitrary square matrices. The general formulation of the functions and the algorithms allows for
application of the algorithms in various problems that arise in multivariate analysis. Several
applications of the general algorithms are given. Specifically, algorithms are given for reciprocal
principal components analysis, binormamin rotation, generalized discriminant analysis, variants of
generalized principal components analysis, simple structure rotation for one of the latter variants,
and set component analysis. For most of these methods the algorithms appear to be new, for the others
the existing algorithms turn out to be special cases of the newly derived general algorithms.
Kiers, H.A.L. (1997a).
A modification of the SINDCLUS algorithm for fitting the ADCLUS
and INDCLUS models Journal of Classification, 14,
The SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models
deals with a parameter matrix that occurs twice in the model by
considering the two occurrences as independent parameter
matrices. This procedure has been justified empirically by the
observation that upon convergence of the algorithm to the global
optimum, the two independently treated parameter matrices turn
out to be equal. In the present paper, results are presented that
contradict this finding, and a modification of SINDCLUS is
presented which obviates the need for independently treating two
occurrences of the same parameter matrix.
Kiers, H.A.L. (1997b).
Three-mode orthomax rotation.
Psychometrika, 62, 579-598.
Factor analysis and principal components analysis
(pca) are often followed by an orthomax rotation
to rotate a loading matrix to simple structure.
The simple structure is usually defined in terms
of the simplicity of the columns of the loading
matrix. In three-mode pca, rotational freedom of
the so called core (a three-way array relating
components for the three different modes) can be
used similarly to find a simple structure of the
core. Simple structure of the core can be defined
with respect to ail three modes simultaneously,
possibly with different emphases on the different
modes. The present paper provides a fully flexible
approach for orthomax rotation of the core to
simple structure with respect to three modes
simultaneously. Computationally, this approach
relies on repeated (two-way) orthomax applied to
supermatrices containing the frontal, lateral or
horizontal slabs, respectively. The procedure is
illustrated by means of a number of exemplary
analyses. As a by-product, application of the
three-mode orthomax procedures to two-way arrays
is shown to reveal interesting relations with and
interpretations of existing two-way simple
structure rotation techniques.
Kiers, H.A.L. (1997c).
Weighted least squares fitting using ordinary
least squares algorithms.
Psychometrika, 62, 251-266.
A general approach for fitting a model to a data
matrix by weighted least squares (wls) is studied.
This approach consists of iteratively performing
(steps of) existing algorithms for ordinary least
squares (ols) fitting of the same model. The
approach is based on minimizing a function that
majorizes the wls loss function. The generality of
the approach implies that, for every model for
which an ols fitting algorithm is available, the
present approach yields a wls fitting algorithm.
In the special case where the wls weight matrix is
binary, the approach reduces to missing data
Kiers, H. A. L. (1997d).
Discrimination by means of components that are orthogonal in the data space.
Jounal of Chemometrics, 11, 533-545.
Krzanowski (J. Chemometrics, 9, 509 (1995)) proposed a method for
obtaining so-called orthogonal canonical variates (henceforth called components) for
discrimination purposes. In contrast with ordinary discriminant analysis, this method
employs components that are orthogonal in the original data space. These components are
derived in a successive way, thus optimizing discrimination of a component given the
previously extracted components. Two alternative procedures are proposed to extract the
desired number of components simultaneously, yielding a better overall discrimination.
The simultaneous approaches are applied to the same two data sets as analysed by
Krzanowski, as well as to Anderson's Iris data, and a comparison of discriminatory
quality of the solutions is presented.
Kiers, H.A.L. (1998a).
Recent developments in three-mode factor analysis: Constrained
three-mode factor analysis and core rotations. In C. Hayashi, N. Ohsumi,
K. Yajima, Tanaka, Y., H.-H. Bock, & Y. Baba, Data Science,
Classification, and Related Methods (pp. 563-574). Tokyo: Springer-
A review is presented of some recent developments in three-mode factor
analysis, that are all
aimed at reducing the difficulties in interpreting three-mode factor
First, variants of three-mode factor analysis with zero constraints on the
core are described,
and attention is paid to algorithms for fitting these models, as well as
to uniqueness of the
representations. Next, various methods for rotation of the core to simple
discussed and related to two-way simple structure rotation techniques. In
section, new perspectives for simplification of the interpretation of
analysis solutions are discussed.
Kiers, H.A.L. (1998b)
An overview of three-way analysis and some recent developments.
In A. Rizzi, M. Vichi
& H.-H. Bock (Eds.), Advances in data science and
classification (pp. 593-602). Berlin: Springer.
A brief overview is presented of techniques for the analysis of
three-way data. Techniques will be distinguished in terms of the
type of data they are meant to analyze, on the basis of the type
of analysis pursued, and the three-way models are distinguished
with respect to their uniqueness. Next, recent approaches are
discussed for eliminating nonuniqueness by imposing constraints
and rotation to simple structure.
A three-step algorithm for CANDECOMP/PARAFAC analysis of large
data sets with multicollinearity.
Journal of Chemometrics, 12, 155-171.
Fitting the CANDECOMP/PARAFAC model by the standard alternating
least squares algorithm often requires very many iterations. One
case in point is that of analysing data with mild to severe
multicollinearity. If, in addition, the size of the data is
large, the computation of one CANDECOMP/PARAFAC solution is very
time-consuming. The present paper describes a three-step
procedure which is much more efficient than the ordinary
CANDECOMP/PARAFAC algorithm, by combining the idea of compression
with a form of regularization of the compressed data array.
Joint orthomax rotation of the core and
resulting from three-mode principal components analysis. Journal of
The analysis of a three-way data set using three-mode principal
components analysis yields
component matrices for all three modes of the data, and a three-way array
called the core,
which relates the components for the different modes to each other. To
freedom in the model, one may rotate the core array (over all three
modes) to an optimally
simple form, for instance by three-mode orthomax rotation. However, such
a rotation of the
core may inadvertently detract from the simplicity of the component
matrices. One remedy is
to rotate the core only over those modes in which no simple solution for
matrices is desired or available, but this approach may in turn reduce
the simplicity of the
core to an unacceptable extent. In the present paper, a general approach
is developed, in
which a criterion is optimized that not only takes into account the
simplicity of the core,
but also, to any desired degree, the simplicity of the component
matrices. This method (in
contrast to methods for either core or component matrix rotation) can be
used to find
solutions in which the core and the component matrices are all
Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core
to simple structure.
Computational Statistics & Data Analysis, 28, 307-324.
SIMPLIMAX is proposed as a procedure for oblique rotation of a factor
loadings matrix to
simple structure. The distinguishing feature of this method is that it
rotates the loading
matrix so that after rotation the m smallest elements have a
minimal sum of squares
(where m is specified in advance). In the present paper, the
SIMPLIMAX method is
generalized to handle three-way arrays: Three-way SIMPLIMAX finds oblique
rotations of the core matrix that results from a three-mode factor
three-way SIMPLIMAX minimizes the sum of the m smallest elements
of the rotated core
array. An algorithm for three-way SIMPLIMAX is presented, the performance
of the algorithm is
discussed, some applications are shown, and it is indicated how the
method can be used for
rotation of solutions of N-mode factor analysis, and for rotation
over a subset of the
modes of an N-mode core array.
Kiers, H.A.L. (2000a).
Some procedures for displaying results from three-way methods. Journal of
Chemometrics, 14, 151-170.
Three-way Tucker analysis and CANDECOMP/PARAFAC are popular methods for the
analysis of three-way data (data pertaining to three sets of entities). To
interpret the results from these methods, one can, in addition to inspecting the
component matrices and the core array, inspect visual representations of the
outcomes. In this paper, first an overview is given of plotting procedures
currently in use with three-way methods. Not all of these optimally correspond
to the actual approximation of the data furnished by the three-way method at
hand. Next it is described how plotting procedures can be designed that do
correspond exactly to the low-dimensional description of the data by means of
the three-way method at hand, and it is indicated to what extent these
correspond to the ones currently in use. Specifically, procedures are described
for displaying either one set of entities (e.g. a set of chemical samples) in
two- or three-dimensional plots, or a set of combinations of entities (e.g.
pertaining to each object at each time point, thus providing "trajectories" for
each object). Furthermore, it is shown how, in these plots, the other entities
can be plotted simultaneously (e.g. superimposing the variables on a plot with
trajectories for objects). Both procedures are summarized in an
Kiers, H.A.L. (2000b).
Towards a standardized notation and terminology in multiway analysis.
Journal of Chemometrics, 14, 105-122.
This paper presents a standardized notation and terminology to be used for
three- and multiway analyses, especially when these involve (variants of) the
CANDECOMP/PARAFAC model and the Tucker model. The notation also deals with basic
aspects such as symbols for different kinds of products, and terminology for
three- and higher-way data. The choices for terminology and symbols to be used
have to some extent been based on earlier (informal) conventions. Simplicity and
reduction of the possibility of confusion have also played a role in the choices
Kiers, H. A. L. (2004).
Bootstrap confidence intervals for three-way methods.
Journal of Chemometrics, 18, 22-36.
Results from exploratory three-way analysis techniques such as
CANDECOMP/PARAFAC and Tucker3 analysis are usually presented without giving insight
into uncertainties due to sampling. Here a bootstrap procedure is proposed that produces
percentile intervals for all output parameters. Special adjustments are offered for
handling the non-uniqueness of the solutions. The percentile intervals indicate the
instability of the sample solutions. By means of a simulation study it is demonstrated
that the percentile intervals can fairly well be interpreted as confidence intervals for
the output parameters.
Kiers, H.A.L., Cléroux, R., & Tenberge, J.M.F. (1994).
Generalized canonical-analysis based on optimizing
matrix correlations and a relation with idioscal.
Computational Statistics & Data Analysis, 0, 0.**
Carroll's method for generalized canonical
analysis of two or more sets of variables is shown
to optimize the sum of squared inner-product
matrix correlations between a consensus matrix and
matrices with canonical variates for each set of
variables. In addition, the method that
analogously optimizes the sum of squared rv matrix
correlations (proposed by escoufier, 1973) between
a consensus matrix and matrices with canonical
variates, can be interpreted as an application of
carroll and chang's idioscal. A simple algorithm
is developed for this and other applications of
idioscal where the similarity matrices are
Kiers, H.A.L., & Der Kinderen, A. (2003).
A fast method for choosing the numbers of components in Tucker3 analysis
British Journal of Mathematical and Statistical Psychology, 56,
Recently, Timmerman and Kiers proposed an effective procedure for choosing
the numbers of components in Tucker3 analysis, a kind of component analysis
of three-way data. The procedure, however, is rather time-consuming, relying
on very many complete Tucker3 analyses. Here, an alternative procedure is
proposed, which basically relies on a single, quick analysis of the three-way
data set. In a simulation study it was found that the new procedure is
comparable in its effect to the original procedure.
Kiers, H.A.L., & Harshman, R.A. (1997).
Relating two proposed methods for speedup of algorithms for fitting
two- and three-way principal component and related multilinear
models. Chemometrics and Intelligent Laboratory Systems, 36,
Multilinear analysis methods such as component (and three-way component)
analysis of very large data sets can become very computationally
demanding and even infeasible unless some method is used to compress the
data and/or speed up the algorithms. We discuss two previously proposed
speedup methods. (a) Alsberg and Kvalheim have proposed use of data
simplification along with some new analysis algorithms. We show that
their procedures solve the same problem as (b) the more general approach
proposed (in a different context) by Carroll, Pruzansky, and Kruskal. In
the latter approach, a speed improvement is attained by applying any
(three-mode) PCA algorithm to a small (three-way) array derived from
the original data. Hence, it can employ the new algorithms by Alsberg
and Kvalheim, but, as is shown in the present paper, it is easier and
often more efficient to apply standard (three-mode) PCA algorithms to
the small array. Finally, it is shown how the latter approach for speed
improvement can also be used for other three-way models and analysis
methods (e.g., PARAFAC/CANDECOMP and constrained three-mode PCA).
Kiers, H.A.L., & Krijnen, W.P. (1991).
An efficient algorithm for PARAFAC of three-way data with large
numbers of observation units. Psychometrika, 56,
The CANDECOMP algorithm for the PARAFAC analysis of n x m x p three-
way arrays is adapted to handle arrays in which n > mp more
efficiently. For such
arrays, the adapted algorithm needs less memory space to store the data
iterations, and uses less computation time than the original CANDECOMP
algorithm. The size of the arrays that can be handled by the new
algorithm is in
no way limited by the number of observation units (n) in the data.
Kiers, H.A.L., Kroonenberg, P.M., & Ten Berge, J.M.F. (1992).
An efficient algorithm for TUCKALS3 on data with large numbers of
observation units. Psychometrika, 57, 415-422
A modification of the TUCKALS3 algorithm is proposed that handles
arrays of order I x J x K for any I. When I is much larger than JK, the
algorithm needs less work space to store the data during the iterative
part of the
algorithm than does the original algorithm. Because of this and the
feature that execution speed is higher, the modified algorithm is highly
for use on personal computers.
Kiers, H.A.L., & Marchetti, G.M. (1994).
Handling large numbers of observation units in
three-way methods for the analysis of qualitative and
quantitative two-way data.
Computational statistics, 9, 41-64.
Recently, a number of methods have been proposed for the exploratory
analysis of mixtures of
qualitative and quantitative variables. In these methods for each variable
an object by object
similarity matrix is constructed, and these are consequently analyzed by
means of three-way
methods like indscal, idioscal and tuckals-3. When the number of
observation units (objects) is
large, algorithms for indscal, idioscal and tuckals-3 become. Inefficient
or even infeasible.
The present paper offers variants of these algorithms that can handle large
numbers of objects
in case the similarity matrices are of rank much smaller than the number of
objects, which is
usually the case. In addition, it is shown that results of the three-way
methods at hand are
essentially based only on certain aggregate measures for the variables,
like variances and
covariances for numerical variables, and bivariate and marginal frequencies
Kiers, H.A.L., & Smilde, A.K. (1995).
Some theoretical results on second-order calibration
methods for data with and without rank overlap.
Journal of Chemometrics, 9, 179-195.
Gram, a method for second-order calibration, has
been introduced by Sánchez and kowalski and later
modified by wilson, Sánchez and kowalski. The
methods are based on the claim that, in cases
without measurement error they yield correct
estimates for the concentration ratios and
profiles of (rank-one) analytes present in sample
and mixture. This claim has not been proven
rigorously. In the present paper, rigorous proofs
are given for situations where the claims are
valid indeed. In addition, it is shown that
parafac, an alternative method for second-order
calibration, can be used to obtain the same
results. Next it is shown that the claims do not
hold in cases with 'rank overlap' (partly
overlapping profiles) and it is proven that a
procedure by wang et al. Can still be used to
assess some of the concentration ratios. A general
framework is provided for a variety of
second-order calibration problems and the extent
to which quantitative and qualitative information
can be expected is given.
Kiers, H.A.L., & Smilde, A.K. (1998).
Constrained three-mode factor analysis as a tool for parameter
estimation with second-order instrumental data Journal of
Chemometrics, 12, 125-147.
Three-mode factor analysis models are often used in exploratory
analysis of three-way data. However, in some situation it is a
priori known that a particular constraint three-mode factor
analysis (C3MFA) model describes an underlying process exactly.
In such situations, fitting C3MFA model to a data set can be used
for both quantitative analysis (e.g. estimating concentrations of
a chemical substance in a mixture) and qualitative analysis (e.g.
on the basis of certain subsets of parameters one can identify
the substance in a mixture). In this paper a general algorithm
for fitting a range of such C3MFA models is proposed. Whether
C3MFA is used for qualitative or quantitative analyses, in both
cases it is crucial that the relevant parameter estimates are
uniquely determinable. In the present paper it is discussed how
and to what extent uniqueness of certain model parameters can be
Kiers, H.A.L., & Ten Berge, J.M.F. (1989).
Alternating least squares algorithms for simultaneous components
analysis with equal component weight matrices in two or more
populations. Psychometrika, 54, 467-473.
Millsap and Meredith (1988) have developed a generalization of
components analysis for the simultaneous analysis of a number of
observed in several populations or on several occasions. The algorithm
provide has some disadvantages. The present paper offers two alternating
squares algorithms for their method, suitable for small and large data
respectively. Lower and upper bounds are given for the loss function to
minimized in the Millsap and Merdith method. These can serve to indicate
whether or not a global optimum for the simultaneous components analysis
problem has been attained.
Kiers, H.A.L., Ten Berge, J.M.F., & Bro, R. (1999).
PARAFAC2 - Part I. A
direct fitting algorithm for the PARAFAC2 model. Journal of Chemometrics,
PARAFAC is a generalization of principal component analysis (PCA) to the
situation where a set of data matrices is to be analysed. If each data matrix
has the same row and column units, the resulting data are three-way data and can
be modelled by the PARAFAC1 model. If each data matrix has the same column units
but different (numbers of) row units, the PARAFAC2 model can be used. Like the
PARAFAC1 model, the PARAFAC2 model gives unique solutions under certain mild
assumptions, whereas it is less severely constrained than PARAFAC1. It may
therefore also be used for regular three-way data in situations where the
PARAFAC1 model is too restricted. Usually the PARAFAC2 model is fitted to a set
of matrices with cross-products between the column units. However, this model-
fitting procedure is computationally complex and inefficient. In the present
paper a procedure for fitting the PARAFAC2 model directly to the set of data
matrices is proposed. It is shown that this algorithm is more efficient than the
indirect fitting algorithm. Moreover, it is more easily adjusted so as to allow
for constraints on the parameter matrices, to handle missing data, as well as to
handle generalizations to sets of three- and higher-way data. Furthermore, with
the direct fitting approach we also gain information on the row units, in the
form of 'factor scores'. As will be shown, this elaboration of the model in no
way limits the feasibility of the method. Even though full information on the
row units becomes available, the algorithm is based on the usually much smaller
cross-product matrices only.
Kiers H.A.L., Ten Berge, J.M.F., & Rocci, R. (1997).
Uniqueness of three-mode factor analysis models with sparse cores:
The 3×3×3 case.
Psychometrika, 62, 349-374.
Three-mode PCA and PARAFAC are methods to describe three-way data.
The three-mode PCA model also uses a three-way core array for
linking all components to each other, which makes it far more
general than the PARAFAC model, but also more complicated. In
contrast with PARAFAC, the three-mode PCA model has non-unique
components and it seems hard to choose among all possible
solutions. The present paper introduces a class of three-mode PCA
models in between three-mode PCA and PARAFAC that share good
properties of both models. They are relatively simple and they fit
(almost) as well as the former model and they have the same
uniqueness properties as the latter model.
Kiers, H.A.L., & Van Mechelen, I. (2001).
Three-way component analysis: Principles and illustrative application.
Psychological Methods, 6, 84-110.
Three-way component analysis techniques are designed for descriptive
analysis of 3-way data, for example, when data are collected on individuals,
in different settings, and on different measures. Such techniques summarize
all information in a 3-way data set by summarizing, for each way of the 3-way
data set, the associated entities through a few components and describing
the relations between these components. In this article, 3-mode principal
component analysis is described at an elementary level. Guidance is given
concerning the choices top be made in each step of the process of analyzing
3-way data by this technique. The complete process is illustrated with a
detailed description of the analysis of an empirical 3-way data set.
Kjerulff, K. & Wiggins, N.H. (1976).
Graduate student styles
with stressful situations. Journal of Educational Psychology,
34 graduate students were asked to rate 26 stressful
situations encountered since entering graduate school on 11
characteristics. Data centred by subtracting the grand mean per
rating scale. T3 was applied with varimax rotation for
situations, scales and the two subject dimensions. Reasonable
amount of detail presented. Validation with outside
Klauer, K.C., & Carroll, J.D. (1995).
Network models for scaling proximity data. In R.D. Luce, M.
D'Zmura, D. D. Hoffman, G.J. Iverson, & A.K. Romney (Eds.),
Geometric representations of perceptual phenomena: Papers in
honor of Tarow Indow on his 70th birthday (pp. 319-342).
Mahwah, NJ: Erlbaum.
Network models aim at representing proximity data by means of the minimum-
of connected and weighted graphs. Fundamental representation and
uniqueness results underlying
network models as psychological representations of stimuli, given both
ordinal-scale as well
as interval-scale proximity measures, are discussed. In addition,
computational methods for
network analyses are reviewed and compared. Methods now exist to scale
metric as well as
nonmetric data, symmetric and nonsymmetric proximity measures, and two-way
and three-way data.
They are compared with respect to the factors of (a) computational cost,
(b) accuracy of
recovery of an underlying network, and (c) goodness of fit to the observed
Knobloch, E.M. (1972).
Einschätzung von leistungsrelevanten
Unpublished master thesis, University of Hamburg, Hamburg,
Kohler, A. (1980).
Das Trimod-Programm-System (TRIPSY) zur
der dreimodalen Faktorenanalyse nach Orlik (manuscript
Description of a computer program implementing Orlik's (1981)
Kofides, E., & Regalia, P. A. (2002).
On the best rank-1 approximation of higher-order supersymmetric tensors.
Siam Journal of Matrix Analysis and Applications,23, 863-884.
Recently the problem of determining the best,in the least-squares sense,rank-1
approximation to a higher-order tensor was studied and an iterative method that extends the well-
known power method for matriceswasproposed for itssolution.Thishigher-order power method
is also proposed for the special but important class of supersymmetric tensors,with no change.
simpli .ed version,adapted to the special structure of the supersymmetric problem,is deemed unreli-
able,asitsconvergence isnot guaranteed.The aim of thispaper isto show that a symmetric version
of the above method converges under assumptions of convexity (or concavity)for the functional in-
duced by the tensor in question,assumptions that are very often satis .ed in practical applications.
The use of this version entails signi .cant savings in computational complexity as compared to the
unconstrained higher-order power method.Furthermore,a novel method for initializing the iterative
processisdeveloped which hasbeen observed to yield an estimate that liescloser to the global op-
timum than the initialization suggested before.Moreover,its proximity to the global optimum is a
priori quanti .able.In the course of the analysis,some important properties that the supersymmetry
of a tensor implies for its square matrix unfolding are also studied.
Kolda, T.G. (2001).
Orthogonal tensor decompositions.Siam Journal of Matrix Analysis and Applications,
We explore the orthogonal decomposition of tensors (also known as multidimensional
arrays or n-way arrays) using two di erent de nitions of orthogonality. We present
numerous examples to illustrate the diculties in understanding such decompositions.
We conclude with a counterexample to a tensor extension of the Eckart-Young SVD
approximation theorem by Leibovici and Sabatier.
Kolda, T. G. (2003).
A couterexample to the possibility of an extension of the eckart-young low-rank
approximation theorem for the orthogonal rank tensor decomposition.
Siam Journal of Matrix Analysis and Applications,24, 762-767.
Earlier work has shown that no extension of the Eckart –Young SVD approximation
theorem can be made to the strong orthogonal rank tensor decomposition.Here,we present a
counterexample to the extension of the Eckart –Young SVD approximation theorem to the orthogonal
rank tensor decomposition, answering an open question previously posed by Kolda [SIAM J.Matrix
Anal.Appl.,23 (2001),pp.243–355 ].
Korzhnev, D. M., Ibraghimov, I. V., Billeter, M., & Orekhov, V. Y. (2001).
MUNIN: Application of three-way decomposition to the analysis of heteronuclear NMR
relaxation data. Journal of Biomolecular NMR,21, 263-268.
MUNIN (Multidimensional NMR Spectra Interpretation), a recently introduced
approach exploiting the mathematical concept of three-way decomposition, is proposed for
separation and quantitative relaxation measurements of strongly overlapped resonances in
sets of heteronuclear two-dimensional spectra that result from typical relaxation experiments.
The approach is general and may also be applied to sets of two-dimensional spectra with arbitrary
modulation along the third dimension (e.g., J-coupling, diffusion). Here, the method is applied
for the analysis of 15N rotating frame relaxation data.
Kosanovich, K.A., Dahl, K.S., & Piovoso, M.J. (1996).
Improved process understanding using multiway
principal component analysis.
Industrial & Engineering Chemistry Research, 35, 138-146.
Producing a uniform polymer by batch processing is
important for the following reasons: To improve
the downstream processing performance, to enable
material produced at one site to be used by
another, and to remain competitive. Eliminating
the sources of batch-to-batch variability and
tightening the control of key variables are but
two ways to accomplish these objectives. In this
work, it is shown that multiway principal
component analysis (mpca) can be used to identify
major sources of variability in the processing
steps. The results show that the major source of
batch-to-batch variability is due to reactor
temperature variations arising from disturbances
in the heating system and other heat-transfer
limitations. Correlations between the variations
in the processing steps and the final product
properties are found, and recommendations to
reduce the sources of variations are discussed.
Kourti, T., & MacGregor, J.F. (1995).
Process analysis, monitoring and diagnosis, using
multivariate projection methods.
Chemometrics and Intelligent Laboratory Systems, 28, 3-21.
Multivariate statistical methods for the analysis,
monitoring and diagnosis of process operating
performance are becoming more important because of
the availability of on-line process computers
which routinely collect measurements on large
numbers of process variables. Traditional
univariate control charts have been extended to
multivariate quality control situations using the
hotelling t-2 statistic. Recent approaches to
multivariate statistical process control which
utilize not only product quality data (y), but
also all of the available process variable data
(x) are based on multivariate statistical
projection methods (principal component analysis,
(pca), partial least squares, (pls), multi-block
pls and multi-way pca). An overview of these
methods and their use in the statistical process
control of multivariate continuous and batch
processes is presented. Applications are provided
on the analysis of historical data from the
catalytic cracking section of a large petroleum
refinery, on the monitoring and diagnosis of a
continuous polymerization process and on the
monitoring of an industrial batch process.
Kourti, T., Nomikos, P., & MacGregor, J.F. (1995).
Analysis, monitoring and fault diagnosis of batch
processes using multiblock and multiway PLS.
Journal of Process Control, 5, 277-284.
Multivariate statistical procedures for the
analysis and monitoring of batch processes have
recently been proposed. These methods are based on
multiway principal component analysis (pca) and
partial least squares (pls), and the only
information needed to exploit them is a historical
database of past batches. In this paper, these
procedures are extended to allow one to use not
only the measured trajectory data on all the
process variables and information on measured
final quality variables but also information on
initial conditions for the batch such as raw
material properties, initial ingredient charges
and discrete operating conditions. Multiblock
multiway projection methods are used to extract
the information in the batch set-up data and in
the multivariate trajectory data, by projecting
them onto low dimensional spaces defined by the
latent variables or principal components. This
leads to simple monitoring charts, consistent with
the philosophy of spc, which are capable of
tracking the progress of new batch runs and
detecting the occurrence of observable upsets.
Powerful procedures for diagnosing assignable
causes for the occurrence of a fault by
interrogating the underlying latent variable model
for the contributions of the variables to the
observed deviation are also presented. The
approach is illustrated with databases from two
industrial batch polymerization processes.
Kouwer, B.J. (1967).
(GRON. PSYCH. 07+07BIS). Orthogonale rotaties (GRON.PSYCH.12).
Reports, Institute of Psychology, University of Groningen,
Groningen, The Netherlands.
Kreiman, J., & Gerratt, B. R. (1996).
The perceptual structure of pathologic voice quality.
Journal of the Acoustical Society of America, 100, 1787-1795.
Although perceptual assessment is included in most protocols for evaluating
pathologic voices, a standard set of valid scales for measuring voice quality has never been
established. Standardization is important for theory and for clinical acceptance, and also
because validation of objective measures of voice depends on valid perceptual measures. The
present study used large sets (n=80) of male and female voices, representing a broad range of
diagnoses and vocal severities. Eight experts judged the dissimilarity of each pair of voices,
and responses were analyzed using nonmetric individual differences multidimensional scaling.
Results indicate that differences between listeners in perceptual strategy are so great that
the fundamental assumption of a common perceptual space must be questioned. Because
standardization depends on the assumption that listeners are similar, it is concluded that
efforts to standardize perceptual labels for voice quality are unlikely to succeed. However,
analysis by synthesis may provide an alternate means of modeling quality as a function of both
voices and listeners, thus avoiding this problem. (C) 1996 Acoustical Society of America.
Krijnen, W.P. (1993).
The analysis of three-way arrays by constrained PARAFAC
methods. Leiden: DSWO Press.
In this book - a companion volume to Kroonenberg's Three-mode Principal
Analysis - much attention is directed to the most salient property of
the PARAFAC model,
the uniqueness of its components. It turns out that the theoretical
property of uniqueness
does not exclude the existence of alternative representations that fit the
data almost as well
as the standard PARAFAC solution. In such cases, the uniqueness is called
weak. A constrained
PARAFAC variant is developed specifically to determine the degree of
uniqueness of PARAFAC
components. For data having weak uniqueness, three new constrained PARAFAC
introduced that allow for easier interpretations. The first one employs
importances of the components across occasions, the second one determines
components, and the third one determines components that correspond to
clusters of variables. The usefulness of the constrained PARAFAC methods
is illustrated by
various analyses of empirical three-way data, and by simulations.
Krijnen, W.P. (1993).
Non-contrast components according to the PARAFAC model. In R. Steyer, K.F. Wender,
& K.F. Widaman (Eds.), Psychometric Methodology, Proceedings of the 7th
European Meeting of the Psychometric Society in Trier (pp. 237-241). Stuttgart
and New York: Gustav Fischer Verlag.
If a three-way array with scores of for instance a number of persons on a number
of variables that measure intelligence on a number of occasions is analyzed with
the PARAFAC method, contrast components may be found. That is, one may find
components that correlate positively with some variables and negatively with other
variables and/or have positive and negative component regression weights for
the variables (pattern elements). It is illustrated with results from a PARAFAC
analysis of empirical data that these contrast components can be less nicely
interpreted than non-contrast components. In case one finds contrast components
with the PARAFAC method, the uniqueness property prevents using a rotation to find
non-contrast components. A restricted PARAFAC method is presented that identifies
non-contrast components that optimally represents the variables. In case the fit
of the non-contrast components is close to the fit of he ordinary PARAFAC components
the non-contrast solution may be preferred because its components are more simple
to interpret than the PARAFAC components.
Krijnen, W.P., & Kiers, H.A.L. (1993).
Clustered variables in PARAFAC. In J.H.L. Oud & R.A.W. van Blokland-
Vogelesang (Eds.), Advances in longitudinal and multivariate
analysis in the behavioral sciences: Proceedings of the SMABS 1992
conference. (pp. 165-177). Nijmegen: ITS.
In this paper a constrained PARAFAC method is proposed with which it
can be verified whether or not the variables can be partitioned into
non-overlapping clusters. In addition a second variant of PARAFAC is
proposed that identifies an optimal PARAFAC representation which has
components corresponding to non-overlapping clusters of variables. Both
methods are illustrated by an analysis of empirical data, and some
relationships with other methods are outlined.
Krijnen, W.P., & Kiers, H.A.L. (1995).
An efficient algorithm for weighted PCA. Computational
Statistics, 10, 299-306.
The method for analyzing three-way data where one of the three
components matrices in TUCKALS3 is chosen to have one column is called
Replicated PCA. The corresponding algorithm is relatively inefficient.
This is shown by offering an alternative algorithm called Weighted PCA.
Specifically, it is proven that the algorithms Replicated PCA and
Weighted PCA produce identical convergent sequences of loss-function
values, and that Weighted PCA attains these values using fewer floating
Krijnen, W.P., & Ten Berge, J.M.F. (1991).
Contrastvrije oplossingen van het CANDECOMP/PARAFAC-model
[Contrast-free solutions of the CANDECOMP/PARAFAC-model].
Kwantitatieve Methoden, 12, 87-96.
The CANDECOMP/PARAFAC decomposition of three-way arrays produces
components and loadings that
are unique in the sense that rotation results in a worse fit and is
therefore not allowed.
This paper investigates whether the absence of rotational freedom causes
problems in contrast
with ordinary principal components for data which can be beautifully
analysed using rotational
freedom. This is especially true for variables which correlate positively
with each other such
that they can be rotated to components with contrast-free loadings.
indeed able to produce contrast components; the lack of rotational freedom
is a problem in
that case. The solution can be found by imposing non-negativity
constraints. It turns out that
with a minimal loss, constrast-free components can be found. This result
calls into question
the uniqueness of the CANDECOMP/PARAFAC solution.
Krijnen, W.P., & Ten Berge, J.M.F. (1992).
A constrained PARAFAC method for positive manifold data.
Applied Psychological Measurement, 16, 295-305.
A set of non-negativity correlated variables, referred to as positive
display a peculiar pattern of loadings in principal components analysis
(PCA). If a small set
of principal components is rotated to a simple structure, the variables
with all components, thus displaying positive manifold. However,
this phenomenon is
critically dependent on the freedom of rotation, as is evident from the
That is, although the first principal component is without contrast (which
means that all
variables correlate either positively or negatively with the first
components have mixtures of positive and negative loadings - which means
manifold is absent. PARAFAC is a generalization of PCA that has unique
components, which means
that rotations are not allowed. This paper examines how PARAFAC behaves
when applied to
positive manifold data. It is shown that PARAFAC does not always produce
solutions. For cases in which PARAFAC does not produce a positive manifold
constrained PARAFAC method is offered that restores positive manifold by
non-negativity constraints. Thus, noncontrast PARAFAC components can be
found that explain
only a negligible amount of variance less than the PARAFAC components.
components cannot be degenerate and cannot be partially unique in the
Krolak-Schwerdt, S. (1991a).
Modelle der dreimodalen Faktorenanalyse. Frankfurt am
Main: Peter Lang.
2. From the co-variation chart to three-mode factor analysis: A historic
3. Three-mode factor analytical models: Basic equations, algorithms and
4. Formal characteristics and specifics of the three-mode factor analytical
model: A classification.
5. Coherence and theoretical relations between three-mode factor analytical
6. Application of the three-mode factor analytical models: Preliminary
7. Re-analysis of a model-experiment by Orlik concerning the psychophysics of
8. Dimensions of colour perception: A comparison between MDS-models and
individual differences structure-analysis.
9. Different types of sorting techniques: Rosenberg and Kim's study of
10. Osgood and Luria: A case of multiple personalities - an empirical comparison
of three-mode data models.
Krolak-Schwerdt, S. (1991b).
Modelle der dreimodalen Faktorenanalyse: Formale Eigenschaften,
theoretische Zusammenhänge und ihre Implikationen für das
Konzept individueller Differenzen. Psychologische
Beiträge, 33, 314-346.
The present paper is concerned with methods of three-mode factor analysis to
obtain a dimensional representation of three-way data. Classifying the methods
by the number of derived spaces and their interrelations yields two distinct
classes of models: CANDECOMP (Carroll &
Chang, 1970), PARAFAC (Harshman, 1976)
and SUMMAX (Orlik, 1980) rest on a basic
trilinear decomposition for the data defining a separate space for each mode,
whereas Tucker's (1964a) three-mode factor
analysis and SUMMAX in its extended form use a quadrilinear model specifying an
additional core matrix. Associated with the current classification are different
properties of the two types of models which refer to the number of substantial
dimensions, their interpretation and the orientation of dimensions which is
subject to rotations within the quadrilinear class and uniquely determined by
trilinear methods. Considering the different characteristics of the methods,
formal relations between the models have been found under very restrictive
conditions only. However, there exist some general connections between trilinear
and quadrilinear models. CANDECOMP and PARAFAC derive from the trilinear SUMMAX
model by rescaling and permutation of axes, and the methodological link between
the Tucker model and SUMMAX is given by orthogonal rotations of the SUMMAX
configuration. These relationships are shown in an empirical example and their
implications for the distinct concepts of individual differences within the two
classes of methods are discussed.
Krolak-Schwerdt, S., Orlik, P., & Ganter, B. (1994).
TRIPAT: A model for analyzing three-mode binary data. In H.H.
Bock, W. Lenski, & M.M. Richter (Eds.), Information Systems
and Data Analysis: Prospects-Foundations-Applications (pp.
298-307). Berlin: Springer.
A discrete, categorical model is presented for three-mode (conditions by
attributes) data arrays with binary entries xijk element
Basically, the model attempts a simultaneous classification of the
elements of the three modes
in a number of common clusters. Clusters are defined by three-mode
submatrices of maximum size
with entries xijk = 1. In performing a discrete
representation of the data
structure, the model may be classified as a non-hierarchical clustering
procedure. It involves
a reorganization of the data array such that the final clustering solution
directly on the data, and it allows for overlapping as well as
nonoverlapping clusters. The
method is similar to three-mode component models such as CANDECOMP and
SUMMAX in the model
function to predict the data. An application concerning recall data in a
study of social
perception is provided.
Kroonenberg, P.M. (1981).
User's Guide to TUCKALS3. A program for
mode principal component analysis. WEP-reeks, WR 81-6-RP, Vakgroep
W.E.P., University of Leiden, Leiden, The Netherlands,
A description of the implementation of the algorithm described
in Kroonenberg & De Leeuw (1980). Includes a detailed example
from Dutch politics.
Kroonenberg, P.M. (1981b).
Scaling of input data for three-mode
component analysis. WEP-reeks, WR 81-21-EX, Vakgroep W.E.P.,
University of Leiden, Leiden, The Netherlands.
A number of proposals for scaling of input data are collected
within one framework. Examples of some of the more common
scaling procedures are given, and some effects on three-mode
component analysis are considered.
Kroonenberg, P.M. (1981c).
User's guide to TUCKALS2. A
program for three- mode principal component analysis with
extended core matrix. WEP- reeks, WR-81-35-RP, Vakgroep W.E.P.,
University of Leiden, Leiden, The Netherlands.
Description of the implementation of the algorithm described
in Kroonenberg & De Leeuw (1977, 1978). Includes an example
from the Dutch political scene.
Kroonenberg, P.M. (1982).
TUCKALS3: A program for three-mode principal
component analysis. Kwantitatieve Methoden, 3, 65-94.
After a relatively non-technical account of three-mode component analysis
of three-way data,
several features of a computer program to perform such an analysis,
TUCKALS3, are described. A
detailed analysis of data on the similarities between Dutch political
parties is presented to
illustrate how three-mode principal component analysis may be used to
Kroonenberg, P.M. (1983a).
Annotated bibliography of three-mode factor analysis. British
Journal of Mathematical and Statistical Psychology, 36,
Published and unpublished theoretical and applied papers on three-mode
analysis and factor analysis have been annotated. In addition, the
applications have been
classified according to subject matter, data type and language.
Theoretical papers have been
classified according to problem, model, method or computer program
Kroonenberg, P.M. (1983b).
Correlational structure of the subtests of the Snijders-Oomen
non-verbal intelligence scale. Kwantitatieve Methoden,
Using three-mode principal component analysis on correlation matrices for
three age groups of
both hearing and deaf children, it is shown that the structure of the
subtests is virtually
the same in all six groups. This structure might be described by a
component shared by all
tests and two other components of almost equal importance.
Kroonenberg, P.M. (1983c).
Three-Mode Principal Component Analysis: Theory and
Applications. Leiden: DSWO Press.
Part I: THEORY
4. Methods and algorithms
5. Transformations of core matrices
Part II: THEORY FOR APPLICATIONS
6. Scaling and interpretations
Part III: APPLICATIONS
8. Standard three-mode data: Attachment study
9. Semantic differential data: Triple personality study
10. Asymmetric similarity data: ITP study
11. Similarities and adjective ratings: Cola study
12. Correlation matrices: Four ability-factors study
13. Multivariate longitudinal data: Hospital study
14. Growth curves: Learning-to-read study
15. Three-mode correspondence analysis: Leiden electorate study
(Errata, 1989; available from author).
Kroonenberg, P.M. (1984a).
Centring three-mode data: Views, problems, and queries. A
discussion with Harshman and Lundy. (WEP Reeks, WR 84-54-
IN), Leiden: University of Leiden, Department of Education.
This contains a number of comments on the question of "appropriate" centring and
normalization. The discussion is continued in Harshman & Lundy (1985b).
Kroonenberg, P.M. (1984b).
Three-mode principal component analysis: Illustrated with an
example from attachment theory. In H.G. Law, C.W. Snyder Jr, J.A.
Hattie & R.P. McDonald (Eds.), Research methods for
multimode data analysis (pp. 64-103). New York: Praeger.
In this chapter, the three-mode principal component model is presented on
a conceptual level
by providing various informal ways of looking at it. Secondly, an outline
is provided of some
technical aspects connected with analyzing this type of model. Finally, an
data from attachment theory is used to illustrate some of the major
aspects and possibilities
of analyzing three-mode data with the three-mode principal component
Kroonenberg, P.M. (1985).
Three-mode principle components analysis of semantic differential data:
The case of a
triple personality. Applied Psychological Measurement, 9,
This paper shows how three-mode principal components analysis can be
useful for the analysis
of semantic differential ratings, in particular because no summation is
necessary over any
one mode. The use of "joint plots" (a variant of the biplot) and sums-of-
interpretations is explained and illustrated.
Kroonenberg, P.M. (1986).
The three-mode world. (WEP Reeks, WR 86-01-LE), Leiden:
of Leiden, Department of Education.
This report is a rough attempt to show Who is Who in three-mode land.
The emphasis is on
three-mode factor and component analysis (i.e. three-mode three-way
data), rather than on
individual differences multidimensional scaling (i.e. two-mode three-way
Kroonenberg, P.M. (1988b).
TUCKALS2. Three-mode principal component analysis with extended
core matrix. In A. di Ciaccio & G. Bove (Eds.), Multiway
'88. Software Guide (pp. 93-103). Roma:
Università di Roma "La Sapienza".
The main principles of three-mode PCA with extended core array (or Tucker2-
model) are described, as well as the capabilities and features of the computer
program TUCKALS2 based on this model. Relations with PARAFAC and INDSCAL are
Kroonenberg, P.M. (1988c).
TUCKALS3. Three-mode principal component analysis. In A. di
Ciaccio & G. Bove (Eds.), Multiway '88. Software guide
(pp. 105-114). Roma: Università di Roma "La
The main principles of three-mode PCA with full core array (or Tucker2-model)
are described, as well as the capabilities and features of the computer program
TUCKALS3 based on this model.
Kroonenberg, P.M. (1989b).
Singular value decompositions of interactions in three-way
contingency tables. In R. Coppi & S. Bolasco (Eds.),
Multiway data analysis (pp. 169-184). Amsterdam: Elsevier.
In this paper generalizations of the singular value decomposition are used
interactions from three-way contingency tables. These decompositions are
primarily applied to
standardized residuals from various loglinear models to produce three-way
Kroonenberg, P.M. (1989c).
The analysis of multiple tables in factorial ecology. III.
Three-mode principal component analysis: "Analyse triadique
complète". Acta Oecologica. Oecologica Generalis,
Thioulouse and Chessel's (1987) "partial" triadic analysis to handle
multiple tables in
ecology can be extended to a complete triadic analysis. This method was
already developed by
Tucker (1966) under the name of three-mode factor (or principle
components) analysis. This
technique is applied to Thioulouse and Chessel's data on the water quality
Méaudret. The compact and efficient data condensation of the method
is emphasized and
Kroonenberg, P.M. (1990b).
Three-mode analysis by example. In Metodoloxía da
Científica (pp. 105-126). Santiago de Compostela, Spain:
de Publicacíons e Intercambio Científico.
In this paper three-mode analysis, in particular, three-mode principal
and, to a lesser extent, parallel factor analysis are presented. The
level of explanation is
exclusively on a conceptual level, and formulas are entirely avoided. The
example is based on
the data from a psychophysiological experiment. Twin pairs were given an
acute dose of
alcohol and several measures were taken before and three times after the
drinking. Many other
domains of enquiry also yield data which have been fruitfully handled by
techniques. For instance, the plant breeders' problem of evaluating
genotypes of soy beans in
different locations on various attributes for further selection has been
three-mode techniques, as well as, intelligence scores from normal and
retarded children. In
the latter case, only the correlation matrices were available, but not
the original scores.
Thus both cross-sectional data bases and repeated measures data can be
with three-way methods.
Kroonenberg, P.M. (1992a).
PARAFAC in three-way land. [Comment on the article "Multilinear models:
applications in spectroscopy"]. Statistical Science, 7,
The purpose of this comment is to provide a somewhat wider background to
the PARAFAC model
discussed in Leurgans and Ross' paper on three-way methods in
spectroscopy. The literature on
the PARAFAC model is briefly sketched.
Kroonenberg, P.M. (1992b).
Three-mode component models: A survey of the literature.
Statistica Applicata, 4, 619-633.
This paper is a part of a larger review paper on three-way techniques. In
component models are reviewed, with special emphasis on PARAFAC and the
Kroonenberg, P.M. (1993).
Three-way methods for multivariable-multioccasion
matrices. Paper presented at the 8th European Meeting of the
Psychometric Society, Barcelona, Spain.
A preliminary investigation is made into the usefulness of three-mode PCA for
the analysis of multivariable-multioccasion (or multitrait-multimethod)
matrices. A brief comparison is made with stochastic three-mode models (see
Bentler & Lee, 1978, 1979; Browne, 1984;
Kroonenberg, P.M. (1994).
The Tuckals line: A suite of programs for three-way data analysis.
& Data Analysis, 18, 73-96.
This paper describes two programs (tuckals2 en
tuckals3) with which three-way data can be
analysed. Both are based on generalisations of
standard (two-way) principal component analysis.
The working of the programs, and the basic theory
behind them, is explained, and is illustrated with
data on the influence of alcohol on the behaviour
of Australian twins.
Kroonenberg, P.M. (1995).
Introduction to biplots for G×E tables. (Research
report, no. 51). Brisbane: University of Queensland, Centre for Statistics.
This report contains an introduction to biplots, a technique to display large
tables in a graph. The construction and interpretation is explained at a fairly
basic level and is directed at plant breeders. The technique is illustrated with
several artificial data sets as well as a real one from maize breeding in
Kroonenberg, P.M. (1996a).
3WAYPACK Menu Structure. (Technical report+software),
Leiden: University of Leiden, Education and Child Studies.
This document is primarily an annotated overview of all menu entries in
INTERFACE3, and as such it may serve as a manual for the program package. In
essence the document is a formatted version of the Help File with the
Kroonenberg, P.M. (1996b).
3WAYPACK User's manual: A package of three-way programs.
(Technical report+software), Leiden: University of Leiden, Department of
The collection of programs which constitute 3WAYPACK have been designed for the
analysis of three-way data. The package consists of three analysis programs,
i.e., TUCKALS3, TUCKALS2, and TRILIN and four additional programs: PREPROC3,
RESIDUAL, ROTATE, and JOINTPLT. All of these programs can be accessed through of
a user-friendly interface, INTERFACE3.
Kroonenberg, P.M. (1997a).
Recent developments in three-way data analysis: A showcase of methods and
In R. Klar & O. Opitz (Eds.), Classification and Knowledge
Organization (pp. 44-62). Berlin: Springer.
In this paper a compact idiosyncratic overview will be provided of the areas
into which three-way data analysis has expanded. The historical introduction is
followed by a scheme presenting an indication of the techniques involved. Then
four condensed examples give a feel of the scope of applications, while the
final section is devoted to publicly available programs to perform the
Kroonenberg, P.M. (1998)
Studying the diffusion of three-mode
analysis in chemistry: Design considerations. In A. Rizzi, M. Vichi
& H.-H. Bock (Eds.), Advances in data science and
classification (pp. 603-611). Berlin: Springer.
The paper presents an example of studying the introduction of
methodological innovation in a science, in particular that of
three-mode (component) analysis into chemistry, especially in
such areas as chromatography, fluorescence, spectrometry, and
analytical chemistry in general.
Kroonenberg, P.M. (2001).
Three-mode correspondence analysis: An illustrated exposé.
In: Actes des XXXIIIèmes Journée de Statistique
(pp. 101-108). Nantes, France: 14-18 May.
In this presentation, André Carlier's contributions to three-mode
correspondence analysis will be presented. The special merit of his work
is that such tables are treated as genuine three-way arrays, rather than
as a matriced (or flattened) two-way matrix. The procedure will be
illustrated with an example describing the changes in play quality of
Kroonenberg, P.M., & Basford, K.E. (1989).
An investigation of multi-attribute genotype response across
environments using three-mode principal component analysis.
Euphytica, 44, 109-123.
The usefulness of three-mode principal component analysis to explore multi-
attribute genotype-environment interaction is investigated. The technique
provides a general description of the underlying patterns present in the data in
terms of interactions of the three quantities (attributes, genotypes, and
environments) involved. As an example, data from an Australian experiment on the
breeding of soybean lines are treated in depth.
Kroonenberg, P.M., & Basford, K.E. (2002).
Applied Three-Mode Data Analysis. Chapter: Three-Mode Clustering.
Research Report No. 104, Centre of Statistics University of
Queensland, Brisbane Australia.
In this chapter the question is addressed of how to group individuals
given that we have measured three-mode profiles, i.e. individuals
having scores on variables under different conditions. We require the same
groups to exist in all conditions under consideration. After the
theoretical introduction, the search for groups is illustrated with
a small example taken from a study of the effect of pollution on blue
crabs (Gemperline et al., 1992). The subsequent sections will provide
guidance for applying the clustering technique in practical problems and
a fully fledged illustration will deal with the attachment relations
between mother and infant and how different infants have different types
of relationships with their mothers.
Kroonenberg, P.M., Basford, K.E., & Ebskamp, A.G.M. (1995).
Three-way cluster and component analysis of maize
Euphytica, 84, 31-42.
Data from the Dutch variety list trials for maize
were analysed with three-way mixture method
clustering and three-mode component analysis. The
main objective of the paper is to demonstrate the
usefulness of such multivariate analysis
techniques for plant breeding data. In particular,
attention is paid how one may gain insight into
the complex patterns that are embodied in this
type of data sets.
Kroonenberg, P.M., Basford, K.E., & Van Dam, M. (1995).
Classifying infants in the Strange Situation with three-way
mixture method clustering. British Journal of Psychology,
The quality of the attachment relationship between mother and infant is
typically determined in the Strange Situation. The assignments of infants to the
A (avoidant), B (secure), and C (resistant) attachment classes are largely but
not exclusively based on measurements during the reunion episodes. In this
paper, the measurements in the reunion episodes are used to derive a clustering
of the infants via three-way mixture method of clustering, a technique
especially designed for clustering three-way data (here: infants, variables and
episodes). The results are compared with the A-B-C classification, and the
relevance of the outcomes for attachment research are discussed. At the same
time, the paper aims to demonstrate the use and usefulness of the three-way
clustering procedure for data from the social and behavioural
Kroonenberg, P.M., Basford, K.E., & Van Dam, M. (1992).
Three-way mixture method clustering. Annual Meeting
Classification Society of North America, 12-13 June, 1992, East
This paper briefly explains the data - from 326 Dutch infants observed
Strange Situation - and three-way mixture method clustering. It also
upon the ordination technique to portray the cluster results. The major
is to show three-way mixture method of clustering at work.
Kroonenberg, P.M. & De Leeuw, J. (1977).
TUCKALS2: A principal component
analysis of three mode data. Res. Bull. RB. 001-77, Department
of Data Theory, University of Leiden, Leiden, The Netherlands.
An ALS method to estimate the T2 is presented, in which the
principle components are computed for two of the three modes,
resulting in an extended core matrix. Two examples from the
1968 Dutch political scene, i.e. 11 psychologists indicating
which of 12 parties had which of 17 aspects, and 100
psychology students rating the similarity on a rating scale of
the nine major Dutch parties. A method for producing joint
plots of two modes is introduced, as well as an algorithm for
orthonormally rotating an extended core matrix.
Kroonenberg, P.M. & De Leeuw, J. (1978).
hoofdassenanalyse voor drieweggegevens. Methoden en Data
Nieuwsbrief (vd SWS vd VVS), 3 (3), 30-53.
A condensed (Dutch) version of Kroonenberg & De Leeuw
Kroonenberg, P.M. & De Leeuw, J. (1980).
component analysis of three-mode data by means of alternating
least squares algorithms. Psychometrika, 45,
A new method to estimate T3 is discussed, and the convergence
properties of the ALS algorithm are considered. A special case
of T3, using an extended core matrix, i.e. T2 (which was
treated extensively in Kroonenberg & De Leeuw, 1977), is
outlined as well. The Miller & Nicely data on the confusion of
English consonants (16 consonant spoken, 16 consonants heard
and 17 degrading conditions of the spoken sound) are used as
illustration. Very clear interpretable solutions and core
matrices. Contains illustrations of rotation of T2 core matrix
to diagonality simultaneous for all frontal planes, and of
joint plots of the components of two modes. The joint plots
are related to the mixed mode matrices of Wainer et al.
Kroonenberg, P. M., Dunn III, W. J., & Commandeur, J. J. F. (2003).
Consensus molecular alignment based on generalized Procrustes analysis.
Journal of Chemical Information and Computer Science,45, 69-97.
One of the most serious problems in three-dimensional quantitative structure-activity relationship (3D-QSAR)
studies is selection of an alignment rule for molecular super position of the compounds in the data
set. In 3D-QSAR analyses of structure-activity data, a reference compound in a defined conformation is
chosen, and all structures in the data set are aligned with the reference in a pairwise manner. In subsequent
steps, conformation/alignment-dependent descriptors are computed for the compounds and compared to
those of the reference. This approach gives much weight to the arbitrarily chosen reference molecule and
can introduce a bias in the results. Here an alternative, and more general, approach to molecular alignment
is presented that is based on Generalized Procrustes Analysis (GPA). The result is a consensus alignment
that uses all molecules in the data set and avoids the bias introduced in the pairwise alignment strategy.
Kroonenberg P.M., & Heiser, W.J. (1997).
Parallel factor analysis with constraints on the configurations:
An overview. In C.Hayashi, N. Ohsumi, K. Yajima, Y. Tanaka, H.-H.
Bock, & Y. Baba (Eds.), Data science, classification, and
related methods (pp. 587-597). Tokyo: Springer.
The paper presents an overview of recent developments with respect
to the use of constraints with the parallel factor analysis
model. Constraints and the way they can be incorporated
in the estimation process of the model are reviewed. Emphasis is
placed on the relatively new triadic algorithm which provides a
large number of new ways to use the Parafac model.
Kroonenberg, P.M., & Kashima, Y. (1997).
Rules in context. A three-mode principal component analysis of
Mann et al.'s data on cross-cultural differences in respect for
Journal of Cross-Cultural Psychology, 28, 463-480.
The paper reports a secondary analysis of data from a study in
which Australian and Japanese children's perceptions of
interpersonal rules were compared. The per country/format matrices
of Acts by Targets were double-centred and both separately and
jointly analysed with three-mode principal component
analysis using a 3x3x2 solution. Especially joint plot
representations were found to be useful.
Kroonenberg, P.M., Lammers, C.J., & Stoop, I. (1985).
Three-mode principal component analysis of multivariate
longitudinal organizational data. Sociological Methods &
Research, 14, 99-136.
The exploratory role three-mode principal component analysis can play in
analyzing multivariate longitudinal organizational data is outlined by an
exposition of the technique itself, and by its application to organizational
data from Dutch hospitals. Relationships with some other techniques for such
data are indicated.
Kroonenberg, P.M., & Miyano, H. (1986).
Tucker2 moderuniyoru tahenryô jikeiretu dêta no kaiseki
- takaku seityô dêta [Analysis of multivariate
longitudinal data by the Tucker2 model - growth curves data].
Bulletin of Industrial Products Research Institute,
Data on the physical growth of Japanese girls between 6 and 14 years old were
analysed to illustrate the application of three-mode principal component
analysis, in particular the Tucker2 model, to growth curves. The individual
differences in growth and growth speed were investigated using the deviation
scores with respect to the average growth curves. The results were compared with
a similar study of French girls, and with previous analyses of the data by C. Hayashi & F. Hayashi
(1982). In addition, the problem of preprocessing data before a three-mode
analysis is discussed.
Kroonenberg, P.M. & Miyano, H. (1996c).
Three-way data analysis and its recent developments.
Japanese Psychological Review, 39, 386-407.
Three-way data analysis methods are reviewed and explained from
the viewpoint of Tucker models. A simple description of a
three-way data set is followed by an introduction of typical
tucker models, including the Tucker2, Tucker3, and Parafac models.
A brief explanation of algorithms and preprocessing is given. The
example of Kroonenberg & Miyano (1985) is discussed in some
detail and several examples of recent,less common applications are
presented (see also Kroonenberg, 1997a)
Kroonenberg, P.M., Murakami, T., & Coebergh, J.W.W. (2002).
Added value of three-way methods for the analysis of mortality trends
illustrated with worldwide female cancer mortality (1968-1985).
Statistical Methods in Medical Research, 11, 275-292.
Trends in mortality rates are usually presented per tumour site or per
country without an overall analysis of the complete data encompassing all
three aspects (tumour sites, countries, trends). This paper presents a
methodology for such an overall analysis using three-way methods applied
to a data set on female mortality rates for 17 tumour sites of 43
countries for the years 1968-1985. Multivariate techniques like biplots
and three-mode principal component analysis within an overall three-way
analysis-of-variance framework were used. We confirmed the known patterns
of comparatively high mortality for women due to cancer of the bladder,
intestines, pancreas, rectum, breast, ovary, skin and leukaemia and the
relatively low mortality rates for liver cancer in Western and Northern
Europe, the USA, Australia and New Zealand. Also, the reverse pattern was
observed for Middle and Southern Europe, Hong Kong, Singapore, and in
Japan, and in some but not all Latin American countries. The relatively
mortality due to cancer was high in the lungs, mouth, larynx and oesophagus
in the British Isles, but was much less in other European countries.
Mortality due to cancer of the thyroid, uterus, gall bladder and stomach
was high in Middle European countries, as was the case in Japan, Chile
and Costa Rica. Rates were low for Southern European countries, North
America, Australia and New Zealand. Specific deviating patterns in the
data were the more rapidly decreasing mortality rates for stomach cancer
in Chile and Japan and the more rapidly increasing mortality rates for
lung cancer in the USA, Scotland and Denmark. In conclusion, using
three-way methods, it was feasible to analyse the cancer mortality data
in their entirety. This enabled the simultaneous comparison of trends in
relative mortality rates between all countries due to all tumour sites,
as well as the identification of specific deviating trends for specific
tumour sites in specific countries.
Kroonenberg, P. M. & Oort, F. J. (2003).
Three-mode analysis of multimode covariance matrices.
British Journal of Mathematical & Statistical Psychology, 56, 305-335.
Multimode covariance matrices, such as multitrait-multimethod matrices, contain
the covariances of subject scores on variables for different occasions or conditions.
This paper presents a comparison of three-mode component analysis and three-mode
factor analysis applied to such covariance matrices. The differences and similarities
between the non-stochastic and stochastic approaches are demonstrated by two
examples, one of which has a longitudinal design. The empirical comparison is
facilitated by deriving, as a heuristic device, a statistic based on the maximum
likelihood function for three-mode factor analysis and its associated degrees of
freedom for the three-mode component models. Furthermore, within the present
context a case is made for interpreting the core array as second-order components.
Kroonenberg, P.M., & Snyder Jr, C.W. (1989).
Individual differences in assimilation resistance and affective
responses in problem solving. Multivariate Behavioral
Research, 24, 257-284.
Data, comprising 6 judgment scales by 8 problem solving tasks by 32 thirteen-
year-old boys, collected within the framework of Eckblad's (1981b) cognitive
theory of affect are analyzed with three-mode principal component analysis. In
general, this study illustrates the effectiveness of the three-mode principal
component analysis (TUCKALS) method for the assessment of the nomothetic
validity of a theoretical framework as it pertains to within and across person
Kroonenberg, P.M., & Ten Berge, J.M.F. (1987).
Cross-validation of the WISC-R factorial structure using
three-mode principal components analysis and perfect congruence
analysis. Applied Psychological Measurement, 11, 195-210.
By using three-mode principal components analysis and perfect congruence
analysis in conjunction, the factorial structure of the 11 correlation matrices
of the Wechsler Intelligence Scale for Children-Revised was analyzed within a
single framework. This allows a unified description showing both the strong
similarities between the standardization samples and some small differences
related to age. Furthermore, claims about comparability between the WISC-R
factorial structure, the structures of other independently conducted studies,
and those of several translations of the WISC-R were evaluated. Again the
overall similarity was striking, albeit most studies showed lower explained
variances. Some age effects seemed to be present here as well. The contribution
of three-mode principal components analysis was found to lie primarily in the
simultaneous analysis of the standardization samples, while perfect congruence
analysis allowed the evaluation of the strengths and the correlations of the
common WISC-R components in all studies without encountering rotation
Kroonenberg, P.M., & Ten Berge, J.M.F. (1989).
Three-mode principal component analysis and perfect congruence
analysis for sets of covariance matrices. British Journal of
Mathematical and Statistical Psychology, 42, 63-80.
In this paper three-mode principal component analysis and perfect congruence
analysis for weights applied to sets of covariance matrices are explained and
detailed, and the relationships between the two techniques are explored. It is
shown that given several assumptions are made for three-mode principal component
analysis close links between the two techniques exist. The methods are
illustrated with data pertaining to a theory of self-concept.
Kroonenberg, P.M., Ten Berge, J.M.F., Brouwer, P.,
& Kiers, H.A.L. (1989).
Gram-Schmidt versus Bauer-Rutishauser in alternating
least-squares algorithms for three-mode principal component
analysis. Computational Statistics Quarterly, 5,
The effect of replacing a Bauer-Rutishauser step using an eigendecomposition by
a Gram-Schmidt orthogonalization step in an algorithm for three-mode principal
component analysis was explored both theoretically and empirically. The results
showed that the latter procedure has a slight to moderate advantage over the
Kroonenberg, P.M., & Van der Voort, T.H.A. (1987).
Multiplicatieve decompositie van interacties bij oordelen over de
werkelijkheidswaarde van televisiefilms [Multiplicative
decomposition of interactions for judgements of realism of
television films]. Kwantitatieve Methoden, 8, 117-144.
Interactions of three-way factorial designs can be clarified by using
multiplicative decompositions, such as the singular value decomposition and
three-mode principal component analysis. This procedure is illustrated for a
three-way analysis-of-variance design with a data set on the reality perception
of television films by children.
Kroonenberg, P.M., & Van IJzendoorn, M.H. (1987).
Exploring children's behavior in the Strange Situation. In L.W.C.
Tavecchio, & M.H. van IJzendoorn (Eds.), Attachment in
social networks: Contributions to the Bowlby-Ainsworth attachment
theory (pp. 379 -425). Amsterdam: North Holland.
Using data from six different countries but disregarding nationality, an
analysis was made of the behaviour of children and of subgroups of children in
the Strange Situation. Employing three-mode principal component analysis, trends
in behaviour were studied both for the Mother episodes, and for the Strange
episodes separately, and for most episodes jointly. With continuous components,
compact descriptions could be given of the behaviour, both in terms of idealised
individuals and as members of the subgroups of Ainsworth's classification
system. The rather complex patterns of avoidance towards the mother were studied
and commented upon. Details are presented on the development of these components
over the episodes. It was also shown that the components succeed to a reasonable
degree to separate the subgroups.
Kruskal, J.B. (1976).
More factors than subjects, tests and treatments:
An indeterminacy theorem for canonical decomposition and individual
scaling. Psychometrika, 41, 281-293.
Some methods that analyze three-way arrays of data (including
INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not
subject to arbitrary rotation. This property is studied in this
paper by means of the "triple product" [A, B, C] of three matrices.
The question is how well the triple product determines the three
factors. The answer: up to permutation of columns and multiplication
of columns by scalars - under certain conditions. In this paper
we greatly expand the conditions under which the result is known to
hold. A surprising fact is that the nonrotability characteristic can
hold even when the number of factors extracted is greater than every
dimension of the three-way array, namely, the number of subjects,
the number of tests, and the number of treatments.
Kruskal, J.B. (1977).
Three-way arrays: Rank and uniqueness of trilinear
decompositions, with application to arithmetic complexity and
statistics. Linear Algebra and Its Applications, 18,
95-138. (Corrections, 17-1-1984; available from author or The Three-mode
A triad is a multiplicative array. Analogous to the rank and the row rank of a
matrix, we define rank (X) to be the minimum number of triads,
xijk = aIbjck, whose
sum is X, and dim1(X) to be the dimensionality of the
space of matrices generated by the 1-slabs of X. We prove several lower
bounds on rank, generalising a matrix theorem of Frobenius. We prove several
sufficient conditions for the factors of a triple product,
Sigmarairbjrckr, to be
essentially unique. The results have applications to arithmetic complexity
theory and to statistical models used in three-way multidimensional
Kruskal, J.B. (1981).
Multilinear models for data analysis. Behaviormetrika,
This paper is about structural models which are bilinear, trilinear, or
multilinear of higher order, such as the PARAFAC model, used to analyze two-way
arrays, three-way arrays or many-way arrays of higher order. We will also
discuss the geometrical meaning which many of these models have, using primarily
the geometrical concepts of inner product and distance. Finally, we will present
as an application the analysis of Harshman, Ladefoged & Goldstein (1977)
tongue shape data, to illustrate these ideas.
Kruskal, J.B. (1983a).
Multilinear methods. Proceedings of Symposia in Applied
Mathematics, 28, 75-104.
This paper is about structural models which are bilinear, trilinear, or
of higher order, used to analyze 2-way arrays, 3-way arrays or many-way
of higher order. Many of these models have geometrical content, which is
explained briefly. The bilinear models touched on include factor
component analysis, and multidimensional scaling. Trilinear models
include INDSCAL and PARARAFAC. An application of the latter is presented.
Also, the new methods of preprocessing are described in an appendix,
other description is available yet in the published literature.
Kruskal, J.B. (1983b).
Statement of some current results about three-way arrays.
The concept of rank extends naturally to three-dimensional matrices, and the
extension has applications to complexity theory and statistics. Calculating rank
is difficult even for tiny matrices. New results are presented up to 3 × 3
× 3. Max 2 × 2 × 2 rank is 3; surprisingly, both rank 2 and
rank 3 occur with positive measure. Their geometrical arrangement is
Kruskal, J.B. (1985).
Rank of N-way arrays and the geometry of 2×2×2
(Technical Memorandum), Murray Hill, NJ: AT&T Bell Laboratories.
The concept of rank, which is fundamental to the theory of matrices, can be
extended to many-way arrays of all orders. This extension is so useful and
natural, particularly for 3-way arrays, that it has been introduced on several
separate occasions for different purposes. The smallest 3-way arrays which are
not effectively the same as matrices are 2 × 2 × 2. The maximum rank
of such arrays is three. The rank 2 × 2 × 2 arrays is explored in
considerable detail. The maximum rank of a 2 × J × J
array is [3J/2]. More generally, if J is smaller than or equal to
K, the maximum rank of a 2 × J × K array is
J + min(J, [K/2]). This bound is sharp.
Kruskal, J.B. (1989).
Rank, decomposition, and uniqueness for 3-way and N-way arrays.
In R. Coppi & S. Bolasco (Eds.), Multiway data analysis
(pp. 7-18). Amsterdam: Elsevier.
Decomposition of a matrix underlies both the bilinear methods (factor analysis,
principal components analysis, and correspondence analysis) and the fundamental
concept of matrix rank. The decomposition of a 3-way array as a sum or linear
combination of outer product matrices underlies PARAFAC, and can be used to
define rank of 3-way arrays. The many differences between 3-way arrays and 2-way
arrays with respect to decomposition and rank are discussed. For 2-way arrays,
rotational uniqueness of decompositions holds only in trivial cases, but for 3-
way arrays, it holds for many decompositions of interest, including most PARAFAC
solutions. Many people consider the rotational uniqueness of PARAFAC solutions
to be a major advantage of this model. This paper also introduces the
dimensionality vector of N-way arrays, which is closely connected to the
number of factors used in 3-mode factor analysis.
Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1985).
Several mathematical relationships between PARAFAC-CANDECOMP and
3-mode factor analysis. Paper presented at the Annual
Meeting of the Classification Society, St. John, Newfoundland,
Canada. [Published as Kruskal, Harshman &
Cf. Kruskal, Harshman, & Lundy, 1989.
Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1989).
How 3-MFA data can cause degenerate PARAFAC solutions, among
other relationships. In R. Coppi & S. Bolasco (Eds.),
Multiway data analysis (pp. 115-122). Amsterdam: Elsevier.
This paper discusses relationships among three models: (1) 3-mode factor
analysis, (2) PARAFAC-CANDECOMP, and (3) CANDELINC. The most interesting
relationship is that data satisfying model (1) can cause degenerate solutions
when analyzed with model (2), as described by Theorem 1 and its corollary.
Another interesting relationship connecting all three models at once is
described by Theorem 2 and its corollaries.
Krzanowski, W.J. (1979).
Between-groups comparison of principal components. Journal of
the American Statistical Association, 74,
A method is given for comparing principal
component analyses conducted on the same variables in two
different groups of individuals, and an extension to the case of
more than two groups outlined. The technique leads to a latent
root and vector problem, which has also arisen in the comparison
of factor patterns in seperate factor analyses. Emphasis in the
present article is on the underlying geometry and interpretation
of the results. An illustrative example is provided.
Krzanowski, W.J. (1984).
Principal Component Analysis in the Presence of Group Structure Applied Statistics
, 33, 164-168.
A nested series of hypotheses on dispersion structure is identified when observations are grouped in a
multivariate sample. A simple method of estimation is suggested for one of these hypotheses, and
results using this method are compared with those previously obtained by maximum likelihood methods.
Using these hypotheses, an analogy may be drawn between comparison of principal components between
groups and comparison of regressions between groups.
Krzanowski, W.J. (1990).
Between-group analysis with heterogeneous covariance matrices: The
common principal component model. Journal of Classification,
This paper proposes two methods for between-group analysis when
the common principal component model replaces the equal
dispersion matrix assumption. One method is by extension of the
two-stage approach to canonical variate analysis using the
sequential principal component analysis as described by Campbell
and Atchey (1981). The second method is by definition of a
distance function between populations satislying the common
principal component model, followed by metric scaling of the
resulting between-populations distance matrix. The two methods
are compared with each other and with ordinary canonical variate
analysis on the previously introduced data.
Kubista, M., Ismail, I. H., Forootan, A., & Sjogren, B. (2004).
Determination of protolytic constants by trilinear fluorescence spectroscopy.
Journal of Fluorescene,14, 139-144.
Protolytic equilibria often have profound effects on chemical activity,
since protolytic species usually behave quite differently. It is therefore important to
characterize the protolytic properties of important chemicals. Here we present a new
approach to study protolytic equilibria of fluorescent species that is extremely accurate
and relies on minimum assumptions. We show that by measuring 2-dimensional excitation/
emission scans of samples at different pH, the 3-dimensional experimental data set, I
(lambda(ex), lambda(em), C(pH)), can be unambiguously decomposed into the spectral responses
of the protolytic species present as well as their concentration. The approach is
demonstrated on the protolytic equilibrium of fluorescein. Although the fluorescein
monoanion cannot be obtained in pure form, the spectra and concentrations of both
fluorescein species, as well as the protolytic constant, are determined with excellent
accuracy. The proposed method is general and can be applied not only for studies of
protolytic equilibria, but on any chemical equilibria and chemical reactions involving
Kumar, A. & Dillon, W. R. (1992).
An integrative look at the use of additive and multiplicative covariance
structure models in the analysis of MTMM data.
Journal of Marketing Research,29, 51-64.
First-order confirmatory factor analytic models have had
widespread use in the analysis of multitrait-multimethod (MTMM) data. In
contrast to the usual first-order confirmatory factor analytic model for
the analysis of MTMM data, other covariance structure models have recently
been proposed and advocated. Two such models are Wothke's covariance
component analysis model andBrowne's direct product model. The authors
provide a conceptual and analytic discussion of those alternative procedures
and compare them with the conventional first-order confirmatory factor
analytic model. They consider the relationship between method factors and
trait factors assumed under each model specification. General remarks about
the nature of method factors and the likely reasons for lack of fit and
ill-defined solutions frequently encountered with use of first-order factor
models are presented. The authors also attempt to integrate the various
approaches to modeling MTMM data and in so doing provide some perspective
on selection of a particular covariance structure model for use in applied
Kunert, J., & Qannari, E. M. (1999).
A simple alternative to generalized procrustes analysis: Application to sensory
Journal of Sensory Studies,14, 197-208.
A statistical method for analyzing sensory profiling data obtained by
means of fixed vocabulary or free choice profiling is discussed the most interesting
feature of this method is that it involves only simple statistical treatment and can
therefore be performed using standard software packages. The outcomes of this method are
compared to those of generalized procrustes analysis on the basis of two data sets
obtained, respectively, by means of fixed vocabulary and free choice profiling. a
significance test is also discussed in order to assess whether the overall configuration
of the products is meaningful. This significance test is based upon a simulation study
involving the permutation procedure.
Kuze, T., Goto, M., Ninomiya, K., Asano, K., Miyazawa, S., Munekata,
H., Ohno, H., & Uchiyama, I. (1985).
A longitudinal study on development of adolescents' social
attitudes. Japanese Psychological Research, 27,
The purpose of this study is to identify the factor structure of social
attitudes among contemporary Japanese adolescents, and to examine the process of
change in social attitudes through adolescence. Longitudinal data covering six
years through junior high and high school years were obtained from 70 boys and
70 girls. Subjects were asked to respond to the 39-item social attitude
questionnaire once a year. In order to explore the factor structure, the Quasi
Three-Mode Principal Component Analysis
(Murakami, 1983b) was employed on the items
by time by sex data array. All factors were found to have basic stability.
Kvaal, K., Wold, J. P., Indahl, U. G., Baardseth, P., & Naes, T. (1998).
Multivariate feature extraction from textural images of bread.
Chemomectrics and Intelligent Laboratory Systems, 42, 141-158.
In order to compute the classical texture measures there is often a need to
perform extensive calculations on the images and do a preprocessing in a specialised manner.
Some of these texture measures are constructed to estimate specific information. Other texture
measures seem to be more global in nature. The techniques presented in this paper define
algorithms applied on the raw image without extensive preprocessing. We want to show that
mathematical transformations of images on a vectorised form will easily enable the use of
multivariate techniques and possibly model several features hidden in the images at the same
time. In this paper we will compare five different methods of extracting features from textural
images in food by multivariate modelling of the sensory porosity of wheat baguettes. The sample
images are recorded from factorial designed baking experiments on wheat baguettes. The
multivariate feature extraction methods to be treated are the angle measure technique (AMT), the
singular value decomposition (SVD), the autocorrelation and autocovariance functions (ACF) and
the so-called size and distance distribution (SDD) method. The methods will be tested on equal
basis and the modelling of sensory porosity from extracted features is done using principal
component regression (PCR) and partial least square regression (PLS). The difference between the
behaviour of the methods will be discussed. The results show that all the methods are suited to
extract sensory porosity but the AMT method prove to be the best in this case.
Go to other sections of the Abstracts
| A | B |
C | D |
E | F |
G | H |
I | J |
K | L |
M | N |
O | P |
Q | R |
S | T |
U | V |
W | X |
Y | Z |
Algemene en Gezinspedagogiek - Datatheorie
Centre for Child and
Family Studies |
Education and Child Studies
The Three-Mode Company |
Education and Child Studies, Leiden University
Wassenaarseweg 52, 2333 AK Leiden, The Netherlands
Tel. *-31-71-5273446/5273434 (secr.); fax *-31-71-5273945
First version : 12/02/1997;